Datasets:

zelalt

/

scientific-papers

like 1

0704.0313

Possibility of Gapless Spin Liquid State by One-dimensionalization

arXiv:0704.0313v1 [cond-mat.str-el] 3 Apr 2007 Typeset with jpsj2.cls <ver.1.2> Letter Possibility of Gapless Spin Liquid State by One-dimensionalization Yuta Hayashi∗ and Masao Ogata Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113-0033 Motivated by the observation of a gapless spin liquid state in κ-(BEDT-TTF)2Cu2(CN)3, we analyze the anisotropic triangular lattice S = 1/2 Heisenberg model with the resonating valence bond mean-field approximation. Paying attention to the small quasi-one-dimensional anisotropy of the material, we take an approach from one-dimensional (1D) chains coupled with frustrating zig-zag bonds. By calculating one-particle excitation spectra changing anisotropy parameter J ′/J from the decoupled 1D chains to the isotropic triangular lattice, we find almost gapless excitations in the wide range from the 1D limit. This one-dimensionalization by frustration is considered to be a candidate for the mechanism of the gapless spin liquid state. KEYWORDS: gapless spin liquid, κ-(BEDT-TTF)2Cu2(CN)3, anisotropic triangular lattice, frustration, one-dimensionalization Organic conductors are one of the fascinating materi- als which have low-dimensionality and relatively strong electron correlations. So far, various physical states have been observed and investigated intensively.1 Among them, magnetism in the Mott insulating phase next to the unconventional superconductivity has been attract- ing considerable attention. This phase is observed in the family of κ-(BEDT-TTF)2X, where BEDT-TTF (ET) denotes bis(ethylenedithio)-tetrathiafulvalene and X rep- resents a monovalent anion. Similarities to that of high- Tc cuprates are worthy of note. Another stimulating problem concerning magnetism is ground state proper- ties of geometrically frustrated spin systems such as a tri- angular lattice and a Kagomé lattice. These two intrigu- ing issues meet in a material κ-(ET)2Cu2(CN)3, which is a Mott insulator having a nearly isotropic triangular lattice, and it has been in the spotlight of late. According to 1H NMR measurements at ambient pres- sure,2 κ-(ET)2Cu2(CN)3 shows no indication of long- range magnetic order (LRMO) down to 32mK. This is 4 orders of magnitude below the exchange constant J ∼ 250K estimated from the temperature dependence of susceptibility. Recently, a similar result has been ob- tained by zero-field muon spin relaxation measurements, which have observed no LRMO down to 20mK.3 These results suggest that a quantum spin liquid state is real- ized in the ground state. On the other hand, the static susceptibility remains finite down to 1.9K, and spin- lattice relaxation rate 1/T1 shows power-law temperature dependence below 1K. These imply that almost gapless spin excitation exists. This fact is a significant feature of the spin liquid phase observed in this material. Since Anderson’s proposal of a resonating valence bond (RVB) state,5 enormous number of studies have been made on the triangular lattice spin system. It is now a general view that the ground state of the isotropic triangular lattice Heisenberg model has LRMO, such as the 120◦ structure.6–9 On the other hand, if one ne- glects the LRMO and assumes a disordered ground state, the mean-field theory of RVB state gives a spin-gap ∗E-mail address: [email protected] Table I. Anisotropy of effective transfer integrals in κ-(ET)2X. The definition of t and t′ are not as usual (see the text). Anion X t′/t Cu2(CN)3 0.94 Cu(NCS)2 1.19 Cu[N(CN)2]Br 1.33 Cu[N(CN)2]Cl 1.47 Cu(CN)[N(CN)2] 1.47 Ag(CN)2·H2O 1.67 I3 1.72 state with dx2−y2+idxy-wave symmetry, which is called “d+id state”.10–12 This RVB state, describing an insu- lating spin system, corresponds to a projected BCS state at half-filling in which doubly occupied states are ex- cluded. Thus, the existing theories show that the ground state has LRMO in general, and if the magnetic order is destroyed in some reason, the d+id fullgap state will appear. If we regard the Mott insulating phase of κ- (ET)2Cu2(CN)3 in low temperatures as an isotropic tri- angular lattice spin system, the results of NMR and sus- ceptibility measurements, which suggest neither LRMO nor spin gap, cannot be explained. In this letter, we pay attention to small anisotropy of κ-(ET)2Cu2(CN)3 and propose a new possibility for un- derstanding its gapless spin liquid state. As shown in Ta- ble I, only κ-(ET)2Cu2(CN)3 has an opposite anisotropy among the family of κ-(ET)2X studied in the past. Here, the effective transfer integrals t and t′ are defined in- versely to the conventional way; t = 0 corresponds to the square lattice, and t′ = 0 the decoupled chains. Therefore, κ-(ET)2Cu2(CN)3 has quasi-one-dimensional (Q1D) anisotropy rather than an isotropic triangular lat- tice. Considering that the pure 1D spin system has no LRMO and gapless spin excitation, it is likely that this Q1D anisotropy is concerned with the formation of the gapless spin liquid state in κ-(ET)2Cu2(CN)3. Based on the above consideration, we study the Heisenberg model on an anisotropic triangular lattice, which is equivalent to 1D chains coupled with zig-zag http://arxiv.org/abs/0704.0313v1 2 J. Phys. Soc. Jpn. Letter Author Name bonds as shown in Fig. 1. The Hamiltonian is given by <i,i′> JSi · Si′ + <i,j> J ′Si · Sj , (1) where <i, i′> and <i, j> represent the summation over intrachain and interchain nearest-neighbor pairs with an- tiferromagnetic coupling constant J and J ′, respectively (see Fig. 1). We investigate the anisotropy parameter range J ′/J = 0.0-1.0, in which the model interpolates between the decoupled chains (J ′ = 0) and the isotropic triangular lattice (J ′ = J). In the following, we consider a projected BCS state defined as ∣p-BCS , (2) where PG is the Gutzwiller projection operator which ex- cludes double occupancy and is a BCS mean-field wave function. Since it is difficult to treat the Gutzwiller projection analytically, we apply an RVB mean-field ap- proximation to the Hamiltonian (1) and calculate the one-particle excitation spectra. To put it more con- cretely, we introduce mean fields ∆ij ≡ ci↑cj↓ , ξij ≡ and obtain its excitation spectrum by diagonalizing the mean-field Hamiltonian. This approxi- mation is equivalent to the “Gutzwiller approximation” which replaces the effect of the Gutzwiller projection op- erator with the statistical weight gs as p-BCS ∣Si ·Sj ∣p-BCS ∣Si ·Sj . (3) In the simplest Gutzwiller approximation, the statisti- cal weight is given as gs = 4/(1 + δ) 2 where δ is the density of holes,15 and in the case of half-filling (δ = 0), gs = 4. Although double occupancy is no longer excluded from wave functions in this approximation, it is known in the research of high-Tc superconductivity that the RVB mean-field (Gutzwiller) approximation gives quali- tatively good results. The spin operators Si ·Sj in the Hamiltonian (1) can be rewritten by the fermion operators as Si · Sj = ci↑ − c†i↓ci↓ cj↑ − c†j↓cj↓ cj↑ + c . (4) Fig. 1. The anisotropic triangular lattice Heisenberg model with intrachain coupling J and interchain zig-zag coupling J ′. τ1, τ2, τ3 are lattice vectors. By introducing the mean fields, we can rewrite the Hamiltonian as HMF = ck↑+ c + h.c. except for constant terms. Here, ξk and ∆k are given by ξk ≡ −3Jξτ1cos(k · τ 1) − 3J ′ ξτ 2cos(k · τ 2) + ξτ 3cos(k · τ 3) , (6) ∆k ≡ 3J∆τ1cos(k · τ 1) + 3J ′ ∆τ 2cos(k · τ 2) + ∆τ3cos(k · τ 3) , (7) where τ 1 = (1, 0), τ 2 = (1/2, 3/2), τ 3 = (1/2,− as shown in Fig. 1, and ci+τ↑ ci+τ↓ , ∆τ ≡ ci↑ci+τ↓ . (8) On the analogy of BCS theory, we obtain self-consistent equations at zero temperature ξτ i = − eik·τ i ∆τ i = e−ik·τ i with a quasiparticle excitation spectrum + |∆k|2. (10) We determine the order parameters ∆τ i , ξτ i (i = 1, 2, 3) by solving self-consistent equations (9) numerically, and obtain the one-particle excitation spectrum Ek. Firstly, we verify our method in 1D limit (J ′/J = 0). According to the exact solution, the ground state is a spin disordered state and the excitation spectrum is “des Cloizeaux-Pearson mode” with S = 1.16 In the present RVB mean-field theory, the one-particle excitation spec- trum becomes Ek = 3J + |∆τ 1 | 2 |cos kx| (11) in the 1D limit. This clearly realizes gapless excitations at kx = ±π/2. Note that this one-particle excitation describes a spin singlet breaking, i.e. S = 1/2 spinon excitation, whereas the des Cloizeaux-Pearson mode de- scribes S = 1 spin-wave (magnon) excitation. Thus, two- spinon excitations with kx = π/2 and kx = −π/2 form an S = 1 magnon with kx = 0. This means that the present gapless excitation spectrum obtained in the RVB mean- field theory is consistent with the exact des Cloizeaux- Pearson mode. Nextly, we show the results of 0 ≤ J ′/J ≤ 1 case, focusing on the following parameters + |∆τ 1 | D23 ≡ + |∆τ2 | + |∆τ3 | Because of the SU(2) degeneracy at half-filling,10, 15 these parameters are determined uniquely regardless of the de- generate ground states. Actually, the excitation spectrum J. Phys. Soc. Jpn. Letter Author Name 3 can be written as = 9J2D21 cos + 9J ′2D223 + cos2 Therefore, D1, D23 determine the dispersion relations along the chains (τ 1) and between the chains (τ 2,τ 3), respectively. Their J ′/J dependence calculated in the system size L = 1200 (N = L2) are plotted in Fig. 2. A notable feature is that D23 remains very small com- pared to D1, in spite of the comparatively large J to J ′/J ∼ 0.25. When D23 = 0 the system is a pure 1D chain. Indeed, when J ′/J = 0, the right-hand side of the self-consistent equations of ξτ 2 , ξτ3 , ∆τ2 , ∆τ 3 become all equal to zero. As we show later, D23 is very small for J ′/J . 0.25 and vanishes when J ′/J → 0. This in- dicates that there are scarcely any correlations between spins of different chains, and practically 1D state is real- ized. As J ′/J approaches unity, D23 gradually increases and becomes equal to D1. Finally, we show in Fig. 3 the J ′/J dependence of the one-particle excitation spectra Ek in (12). We find that the structure of excitation spectra in 0 ≤ J ′/J . 0.25 has little difference from that of the decoupled chains (J ′/J = 0.0). As a result, almost gapless excitations are realized in this wide parameter range. This means that practically 1D state is realized, which is also expected from the behavior of D23 in Fig. 2. When J ′/J exceeds 0.25, the excitation gap gradually increases globally in the first Brillouin zone (1BZ). However, the shape of the whole spectrum is almost unchanged until the J ′/J be- comes as large as about 0.6. Moreover, focusing on the lowest energy excitations (dark areas in the contour plot shown in Fig. 3), their locations in the 1BZ do not deviate from those in the 1D limit (kx = ±π/2) for J ′/J . 0.8. Additionally, when kx = ±π/2, the excitation spectrum Ek is independent of ky, i.e., Ek = 3J ′D23. This is be- cause the frustration of two interchain couplings (corre- sponding to the lattice vector τ 2 and τ 3) cancel the ky dependence. This fact is rather important, since it indi- cates that the excited quasiparticles along the kx = ±π/2 lines feel free to move along the ky direction. This is the same condition as in the 1D limit, except for the exis- Fig. 2. Anisotropy dependence of D1 and D23 for L = 1200. Note that D23 is very small compared to D1 in a wide range 0 ≤ J ′/J .0.25. tence of a finite energy gap. Figure 4 shows the minimum gap energy in the 1BZ as a function of anisotropy J ′/J , changing the system size L. We can see the almost gapless excitations in the wide parameter range 0 ≤ J ′/J . 0.25, as is already expected. It is quite natural that this behavior is simi- Fig. 3. Anisotropy dependence of the one-particle excitation spectra. Contour plots of the spectra are on the left, and sections along ky = 0 line are on the right. The hexagons with broken lines represent 1BZ of the triangular lattice. Up to J ′/J ∼ 0.25, the spectra for each anisotropy are hardly distinguishable, and the one-dimensionality strongly remains for large J ′/J . 4 J. Phys. Soc. Jpn. Letter Author Name lar to that of D23, considering that the minimum energy excitations are located along kx = ±π/2 for J ′/J . 0.6. By plotting the same data for various system size, L, in a semi-log scale (Fig. 4), we can see a discontinuous jump for every size. We find that this critical value J ′c/J vanishes very slowly as (lnL)−1. Thus, the discontinu- ity is an artifact of finite-size calculation. We also find that the minimum gap energy is finite when infinitesimal J ′ is introduced. Actually, we can fit the J ′ dependence as aJ ′ exp(−bJ/J ′)17 for J ′/J . 0.6 as shown in Fig. 4. Considering that the minimum gap energy is already about 3 orders of magnitude below J at J ′/J ∼ 0.25, it can be said that almost gapless excitation is realized in 0 ≤ J ′/J . 0.25. This result is fairly suggestive com- pared with the previous series expansion18 and linear spin wave19, 20 studies, all of which suggest a spin dis- ordered state in the parameter range J ′/J . 0.25. From the above results, we conclude that there is a strong tendency to form a 1D-like excitation spectrum for the triangular lattice spin system with anisotropy 0 ≤ J ′/J . 0.6. Furthermore, even if the anisotropy is as large as 0.6 . J ′/J . 0.8, we can still expect 1D- like behavior for quasiparticles except for the existence of the excitation gap. Let us here discuss the relation to κ- (ET)2Cu2(CN)3. The anisotropy of spin exchange inter- actions in this material can be estimated from J = 4t2/U (U being the onsite Coulomb repulsion) as J ′/J ∼ 0.89. At this anisotropy, a rather large excitation gap exists as shown in Fig. 4. We consider two possibilities to under- stand the gaplessness. One is that the small gap region in Fig. 4 expands to large values of J ′/J by some factors not considered in the present model. For example, If long- distance exchange interactions, quantum fluctuation or multiple spin exchange effect14 (higher order terms of the Heisenberg model) suppress not only LRMO but also the spin gap, we can reproduce the gapless spin liquid state at large J ′/J . These possibilities remain as future problems. Another possibility is that the anisotropy J ′/J of κ-(ET)2Cu2(CN)3 deviates from the above estimation Fig. 4. (Color Online) Anisotropy dependence of the minimum gap energy in the 1BZ (right axis) for L=60(diamond), 120(plus), 300(square), 600(cross) and 1200(triangle). The semi-log plots of the same quantity are also shown (left axis). The solid line is a fitted exponential function aJ ′ exp(−bJ/J ′), where a = 3.50 and b = 1.61. We find that the observed critical behavior is an artifact of finite size calculation (see the text). due to, for example, a finite U effect.21 If it is in the range J ′/J < 0.25, the excitation gap is sufficiently small and the susceptibility behavior (finite at 1.9K whereas J ∼ 250K) can be explained. In summary, we analyzed an anisotropic triangular lat- tice Heisenberg model using RVB mean-field approxima- tion in order to investigate the physical origin of the gap- less spin liquid state observed in κ-(ET)2Cu2(CN)3. We payed attention to the Q1D anisotropy of this material, and took an approach from the 1D limit. As a result of calculations, we found that a practically 1D state with almost gapless excitations is realized in the wide range of the anisotropy parameter 0 ≤ J ′/J . 0.25. Furthermore, one-dimensionality remained strongly even in J ′/J > 0.25 due to the geometrical frustration of interchain cou- plings. We consider this “one-dimensionalization by frus- tration” as a candidate for the mechanism of the gapless spin liquid state, although the full understanding has not yet been achieved. This work was partly supported by a Grant-in-Aid for Scientific Research on Priority Areas of Molecular Conductors (No. 15073210) from the Ministry of Edu- cation, Culture, Sports, Science and Technology, Japan, and also by a Next Generation Supercomputing Project, Nanoscience Program, MEXT, Japan. 1) For a review, see T.Ishiguro, K.Yamaji and G.Saito: Organic Superconductors (Springer-Verlag, Berlin, 1998), 2nd ed. 2) Y.Shimizu, K.Miyagawa, K.Kanoda, M.Maesato and G.Saito: Phys. Rev. Lett. 91 (2003) 107001. 3) S.Ohira, Y.Shimizu, K.Kanoda and G.Saito: J. Low Temp. Phys. 142 (2006) 153. 4) T.Komatsu, N.Matsukawa, T.Inoue and G.Saito: J. Phys. Soc. Jpn 65 (1996) 1340. 5) P.W.Anderson: Mater. Res. Bull. 8 (1973) 153. 6) B.Bernu, P.Lecheminant, C.Lhuillier and L.Pierre: Phys. Rev. B 50 (1994) 10048. 7) N.Elstner, R.R.P.Singh and A.P.Young: Phys. Rev. Lett. 71 (1993) 1629. 8) P.Lecheminant, B.Bernu, C.Lhuillier and L.Pierre: Phys. Rev. B 52 (1995) 9162. 9) L.Capriotti, A.E.Trumper and S.Sorella: Phys. Rev. Lett. 82 (1999) 3899. 10) M.Ogata: J. Phys. Soc. Jpn. 72 (2003) 1839. 11) G.Baskaran: Phys. Rev. Lett. 91 (2003) 097003. 12) T.Watanabe, H.Yokoyama, Y.Tanaka, J.Inoue and M.Ogata: J. Phys. Soc. Jpn 73 (2004) 3404. 13) Y.Shimizu, K.Miyagawa, K.Kanoda, M.Maesato and G.Saito: Prog. Theor. Phys. Suppl. 159 (2005) 52. 14) G.Misguich, C.Lhuillier, B.Bernu and C.Waldtmann: Phys. Rev. B 60 (1999) 1064. 15) F.C.Zhang, C.Gros, T.M.Rice and H.Shiba: Supercond. Sci. Technol. 1 (1988) 36. 16) J.des Cloizeaux and J.J.Pearson: Phys. Rev. 128 (1962) 2131. 17) We would like to thank T.Misawa for pointing out this possi- bility. 18) W.Zheng, R.H.McKenzie and R.R.P.Singh: Phys. Rev. B 59 (1999) 14367. 19) J.Merino, R.H.McKenzie, J.B.Marston and C.H.Chung: J. Phys. Condens. Matter 11 (1999) 2965. 20) A.E.Trumper: Phys. Rev. B 60 (1999) 2987. 21) H.Otsuka: Phys. Rev. B 57 (1998) 14658.

0704.0314

Extra dimensions and Lorentz invariance violation

Extra dimensions and Lorentz invariance violation Viktor Baukh∗ and Alexander Zhuk† Department of Theoretical Physics and Astronomical Observatory, Odessa National University, 2 Dvoryanskaya St., Odessa 65026, Ukraine Tina Kahniashvili‡ CCPP, New York University, 4 Washington Place, New York, NY 10003, USA National Abastumani Astrophysical Observatory, 2A Kazbegi Ave, Tbilisi, GE-0160 Georgia We consider effective model where photons interact with scalar field corresponding to conformal excitations of the internal space (geometrical moduli/gravexcitons). We demonstrate that this interaction results in a modified dispersion relation for photons, and consequently, the photon group velocity depends on the energy implying the propagation time delay effect. We suggest to use the experimental bounds of the time delay of gamma ray bursts (GRBs) photons propagation as an additional constrain for the gravexciton parameters. PACS numbers: 04.50.+h, 11.25.Mj, 98.80.-k Lorentz invariance (LI) of physical laws is one of the corner stone of modern physics. There is a number of ex- periments confirming this symmetry at energies we can approach now. For example, on a classical level, the ro- tation invariance has been tested in Michelson-Morley experiments, and the boost invariance has been tested in Kennedy-Torhndike experiments [1]. Although, up to now, LI is well established experimentally, we can- not say surely that at higher energies it is still valid. Moreover, modern astrophysical and cosmological data (e.g. UHECR, dark matter, dark energy, etc) indicate for a possible LI violation (LV). To resolve these chal- lenges, there are number of attempts to create new phys- ical models, such as M/string theory, Kaluza-Klein mod- els, brane-world models, etc. [1]. In this paper we investigate LV test related to photon dispersion measure (PhDM). This test is based on the LV effect of a phenomenological energy-dependent speed of photon [2, 3, 4, 5, 6, 7, 8], for recent studies see Ref. [9] and references therein. The formalism that we use is based on the analogy with electromagnetic waves propagation in a magnetized medium, and extends previous works [8, 10, 11]. In our model, instead of propagation in a magnetized medium, the electromagnetic waves are propagating in vacuum filled with a scalar field ψ. LV occurs because of an in- teraction term f(ψ)F 2 where F is an amplitude of the electromagnetic field. Such an interaction might have different origins. In the string theory ψ could be a dila- ton field [12, 13]. The field ψ could be associated with geometrical moduli. In brane-world models the similar term describes an interaction between the bulk dilaton and the Standard Model fields on the brane [14]. In Ref. [15], such an interaction was obtained in N = 4 ∗Electronic address: bauch˙[email protected] †Electronic address: [email protected] ‡Electronic address: [email protected] super-gravity in four dimensions. In Kaluza-Klein mod- els the term f(ψ)F 2 has the pure geometrical origin, and it appears in the effective, dimensionally reduced, four dimensional action (see e.g. [16, 17]). In particular, in reduced Einstein-Yang-Mills theories, the function f(ψ) coincides (up to a numerical prefactor) with the volume of the internal space. Phenomenological (exactly solv- able) models with spherical symmetries were considered in Refs. [18]. To be more specific, we consider the model which is based on the reduced Einstein-Yang-Mills the- ory [17], where the term ∝ ψF 2 describes the interaction between the conformal excitations of the internal space (gravexcitons) and photons. It is clear that the similar LV effect exists for all types of interactions of the form f(ψ)F 2 mentioned above. Obviously, the interaction term f(ψ)F 2 modifies the Maxwell equations, and, consequently, results in a mod- ified dispersion relation for photons. We show that this modification has rather specific form. For example, we demonstrate that refractive indices for the left and right circularly polarized waves coincide with each other. Thus, rotational invariance is preserved. However, the speed of the electromagnetic wave’s propagation in vac- uum differs from the speed of light c. This difference implies the time delay effect which can be measured via high-energy GRB photons propagation over cosmological distances (see e.g. Ref. [9]). It is clear that gravexcitons should not overclose the Universe and should not result in variations of the fine structure constant. These de- mands lead to a certain constrains for gravexcitons (see Refs. [17, 19]). We use the time delay effect, caused by the interaction between photons and gravexcitons, to get additional bounds on the parameters of gravexcitons. The starting point of our investigation is the Abelian part of D-dimensional action of the Einstein-Yang-Mills theory: SEM = − |g|FMNFMN , (1) http://arxiv.org/abs/0704.0314v4 mailto:[email protected] mailto:[email protected] mailto:[email protected] where the D-dimensional metric, g = gMN (X)dX dXN = g(0)(x)µνdx µ ⊗ dxν + a21(x)g(1), is defined on the product manifold M = M0 × M1. Here, M0 is the (D0 = d0 + 1)-dimensional external space. The d1- dimensional internal space M1 has a constant curvature with the scale factor a1(x) ≡ LPl expβ1(x). Dimensional reduction of the action (1) results in the following effec- tive D0-dimensional action [17] S̄EM = − |g̃(0)| [(1−Dκ0ψ)FµνFµν ] , (2) which is written in the Einstein frame with the D0- dimensional metric, g̃ µν = (exp d1β̄ 1)−2/(D0−2)g Here, κ0ψ ≡ −β̄1 (D0 − 2)/d1(D − 2) ≪ 1 and β̄1 ≡ β1 − β10 are small fluctuations of the internal space scale factor over the stable background β10 (0 subscript de- notes the present day value). These internal space scale- factor small fluctuations/oscillations have the form of a scalar field (so called gravexciton [20]) with a mass mψ defined by the curvature of the effective potential (see for detail [20]). Action (2) is defined under the approximation κ0ψ < 1 that obviously holds for the condition1 ψ < MPl. κ 0 = 8π/M Pl is four dimen- sional gravitational constant, MPl is the Plank mass, D = 2 d1/[(D0 − 1)(D − 1)] is a model dependent con- stant. The Lagrangian density for the scalar field ψ reads: |g̃(0)|(−g̃µνψ,µψ,ν−m2ψψψ)/2. For simplicity we assume that g̃0 is the flat Friedman-Lemaitre-Robertson- Walker (FLRW) metric with the scale factor a(t). Let’s consider Eq. (2). It is worth of noting that the D0-dimensional field strength tensor, Fµν , is gauge in- variant.2 Secondly, action (2) is conformally invariant in the case when D0 = 4. The transform to the Einstein frame does not break gauge invariance of the action (2), and the electromagnetic field is antisymmetric as usual, Fµν = ∂µAν − ∂νAµ. Varying (2) with respect to the electromagnetic vector potential, −g (1−Dκ0ψ)Fµν = 0. (3) The second term in the round brackets Dκ0ψFµν reflects the interaction between photons and the scalar field ψ, and as we show below, it is responsible for LV. In par- ticular, coupling between photons and the scalar field ψ makes the speed of photons different from the standard speed of light. Eq. (3) together with Bianchi identity (which is preserved in the considered model due to gauge- invariance of the tensor, Fµν [17]) defines a complete set 1 In the brane-world model the prefactor κ0 in the expression for κ0ψ is replaced by the parameter proportional to M [14]. Thus, the smallness condition holds for ψ < MEW . 2 Eq. (2) can be rewritten in the more familiar form S̄EM = −(1/2) |g̃(0)|F̄µν F̄ µν [17]. The field strength tensor F̄µν is not gauge invariant here. of the generalized Maxwell equations. As we noted, ac- tion (2) is conformally invariant in the 4D dimensional space-time. So, it is convenient to present the flat FLRW metric g̃0 in the conformally flat form: g̃0µν = a 2ηµν , where ηµν is the Minkowski metric. Using the standard definition of the electromagnetic field tensor, Fµν , we obtain the complete set of the Maxwell equations in vacuum, ∇ ·B = 0 , (4) ∇ ·E = Dκ0 1−Dκ0ψ (∇ψ ·E) , (5) ∇×B = ∂E − Dκ0ψ̇ 1−Dκ0ψ 1−Dκ0ψ [∇ψ ×B] , (6) ∇×E = −∂B , (7) where all operations are performed in the Minkowski space-time, η denotes conformal time related to physi- cal time t as dt = a(η)dη, and an overdot represents a derivative with respect to conformal time η. Eqs. (4) and (7) correspond to Bianchi identity, and since it is preserved, Eqs. (4) and (7) keep their usual forms. Eqs. (5) and (6) are modified due to interactions between photons and gravexcitons (∝ κ0ψ). These mod- ifications have simple physical meaning: the interaction between photons and the scalar field ψ acts as an effective electric charge eeff . This effective charge is proportional to the scalar product of the ψ field gradient and the E field, and it vanishes for an homogeneous ψ field. The modification of Eq. (6) corresponds to an effective cur- rent Jeff , which depends on both electric and magnetic fields. This effective current is determined by variations of the ψ field over the time (ψ̇) and space (∇ψ). For the case of a homogeneous ψ field the effective current is still present and LV takes place. The modified Maxwell equations are conformally invariant. To account for the expansion of the Universe we rescale the field components asB,E → B,E a2 [21]. To obtain a dispersion relation for photons, we use the Fourier transform between position and wavenumber spaces as, F(k, ω) = dη d3x e−i(ωη−k·x)F(x, η) , F(x, η) = (2π)4 dω d3kei(ωη−k·x)F(k, ω) . (8) Here, F is a vector function describing either the elec- tric or the magnetic field, ω is the angular frequency of the electro-magnetic wave measured today, and k is the wave-vector. We assume that the field ψ is an oscilla- tory field with the frequency ωψ and the momentum q, so ψ(x, η) = Cei(ωψη−q·x) , C = const . Eq. (4) implies B ⊥ k. Without loosing of generality, and for simplic- ity of description we assume that the wave-vector k is oriented along the z axis. Using Eq. (7) we get E ⊥ B. A linearly polarized wave can be expressed as a super- position of left (L, −) and right (R, +) circularly polar- ized (LCP and RCP) waves. Using the polarization basis of Sec. 1.1.3 of Ref. [22], we derive E± = (Ex± iEy)/ Rewriting Eqs. (4) - (7) in the components,3 for LCP and RCP waves we get, (1 − n2+)E+ = 0, (1− n2(−))E − = 0 , (9) where n+ and n− are refractive indices for RCP and LCP electromagnetic waves n2+ = k2 [1−Dκ0ψ(1 + qz/k)] ω2 [1−Dκ0ψ(1 + ωψ/ω)] = n2− . (10) In the case when LI is preserved the electromagnetic waves propagating in vacuum have n+ = n− = n = k/ω ≡ 1. For the electromagnetic waves propagating in the magnetized plasma, k/ω 6= 1, and the difference be- tween the LCP and RCP refractive indices describes the Faraday rotation effect, α ∝ ω(n+ − n−) [23]. In the considered model, since n+ = n− the rotation effect is absent, but the speed of electromagnetic waves propaga- tion in vacuum differs from the speed of light c (see also Ref. [24] for LV induced by electromagnetic field cou- pling to other generic field). This difference implies the propagation time delay effect, ∆t = ∆l(1−∂k/∂ω) (∆l is a propagation distance), ∆t is the difference between the photon travel time and that for a ”photon” which travels at the speed of light c. Here, t is physical synchronous time. This formula does not take into account the evo- lution of the Universe. However, it is easy to show that the effect of the Universe expansion is negligibly small. Solving the dispersion relation as a square equation, we obtain ω2ψ − q2z (Dκ0ψ)2 , (11) where ± signs correspond to photons forward and back- ward directions respectively. The modified inverse group velocity (11) shows that the LV effect can be measured if we know the gravexciton frequency ωψ, z-component of the momentum qz and its amplitude ψ. For our estimates, we assume that ψ is the oscillatory field, satisfying (in local Lorentz frame) the dispersion relation, ω2ψ = m ψ + q 2, where mψ is the mass of gravexcitons4. Unfortunately, we do not have 3 We have defined the system of 6 equations with respect to 6 components of the vectors E and B. This system has non-trivial solutions only if its determinant is nonzero. From this condition we get the dispersion relation. The Faraday rotation effect is absent if the matrix has a diagonal form. 4 To get physical values of the corresponding parameters we should rescale them by the scale factor a. any information concerning parameters of gravexcitons (some estimates can be found in [17, 19]). Thus, we intend to use possible LV effects (supposing it is caused by interaction between photons and gravexcitons) to set limits on gravexciton parameters. For example, we can easily get the following estimate for the upper limit of the amplitude of gravexciton oscillations: |ψ| ≈ 1√ MPl , (12) where for ω and mψ we can use their physical values. In the case of GRB with ω ∼ 1021 ÷ 1022Hz ∼ 10−4 ÷ 10−3GeV and ∆l ∼ 3 ÷ 5 × 109y ∼ 1017sec the typical upper limit for the time delay is ∆t ∼ 10−4sec [9]. For these values the upper limit on gravexciton amplitude of oscillations is5 |κ0ψ| ≈ 10−13GeV . (13) This estimate shows that our approximation κ0ψ < 1 works for gravexciton masses mψ > 10 −13GeV. Future measurements of the time-delay effect for GRBs at fre- quencies ω ∼ 1 − 10GeV would increase significantly the limit up to mψ > 10 −9GeV. On the other hand, Cavendish-type experiments [26, 27]) exclude fifth force particles with masses mψ . 1/(10 −2cm) ∼ 10−12GeV which is rather close to our lower bound for ψ field masses. Respectively we slightly shift the considered mass lower limit to be mψ ≥ 10−12GeV. These masses considerably higher than the mass corresponding to the equality between the energy densities of the matter and radiation (matter/radiation equality), meq ∼ Heq ∼ 10−37GeV, where Heq is the Hubble ”constant” at mat- ter/radiation equality. It means that such ψ-particles start to oscillate during the radiation dominated epoch (see appendix). Another bound on the ψ-particles masses comes from the condition of their stability. With re- spect to decay ψ → γγ the life-time of ψ-particles is τ ∼ (MPl/mψ)3tPl [17], and the stability conditions re- quires that the decay time should be greater than the age of the Universe. According this we consider light gravex- citons with masses mψ ≤ 10−21MPl ∼ 10−2GeV ∼ 20me (where me is the electron mass). As an additional restriction arises from the condi- tion that such cosmological gravexcitons should not overclose the observable Universe. This reads mψ . meq(MPl/ψin) 4 which implies the following restriction for the amplitude of the initial oscillations: ψin . (meq/mψ) MPL << MPl [19]. Thus, for the range of masses 10−12GeV ≤ mψ ≤ 10−2GeV, we obtain respec- tively ψin . 10 −6MPl and ψin . 10 −9MPl. According to 5 We thank R. Lehnert to point that in addition of the time de- lay effect the Cherenkov effect could be used to constrain the electromagnetic field and ψ field coupling strength [25]. Eq. (A.3), we can also get the estimate for the amplitude of oscillations of the considered gravexciton at the present time. Together with the non-overcloseness condition, we obtain from this expression that |κ0ψ| ∼ 10−43 for mψ ∼ 10−12GeV and ψin ∼ 10−6MPl and |κ0ψ| ∼ 10−53 for mψ ∼ 10−2GeV and ψin ∼ 10−9MPl. Obviously, it is much less than the upper limit (13). Note, as we men- tioned above, gravexcitons with masses mψ & 10 −2GeV can start to decay at the present epoch. However, taking into account the estimate |κ0ψ| ∼ 10−53, we can easily get that their energy density ρψ ∼ (|κ0ψ|2/8π)M2Plm2ψ ∼ 10−55g/cm3 is much less than the present energy density of the radiation ργ ∼ 10−34g/cm3. Thus, ρψ contributes negligibly in ργ . Otherwise, the gravexcitons with masses mψ & 10 −2GeV should be observed at the present time, which, obviously, is not the case. Additionally, it follows from Eq. (42) in Ref. [17] that to avoid the problem of the fine structure constant variation, the amplitude of the initial oscillations should satisfy the condition: ψin . 10 −5MPl which, obviously, completely agrees with our upper bound ψin . 10 −6GeV. Summarizing we shown that LV effects can give addi- tional restrictions on parameters of gravexcitons. First, we found that gravexcitons should not be lighter than 10−13GeV. It is very close to the limit following from the fifth-force experiment. Moreover, experiments for GRB at frequencies ω > 1GeV can result in significant shift of this lower limit making it much stronger than the fifth- force estimates. Together with the non-overcloseness con- dition, this estimate leads to the upper limit on the am- plitude of the gravexciton initial oscillations. It should not exceed ψin . 10 −6GeV. Thus, the bound on the ini- tial amplitude obtained from the fine structure constant variation is one magnitude weaker than our one even for the limiting case of the gravexciton masses. Increasing the mass of gravexcitons makes our limit stronger. Our estimates for the present day amplitude of the gravexci- ton oscillations, following from the obtained above lim- itations, show that we cannot use the LV effect for the direct detections of the gravexcitons. Nevertheless, the obtained bounds can be useful for astrophysical and cos- mological applications. For example, let us suppose that gravexcitons with masses mψ > 10 −2GeV are produced during late stages of the Universe expansion in some re- gions and GRB photons travel to us through these re- gions. Then, Eq. (A.3) is not valid for such gravexcitons having astrophysical origin and the only upper limit on the amplitude of their oscillations (in these regions) fol- lows from Eq. (13). In the case of TeV masses we get |κ0ψ| ∼ 10−16. If GRB photons have frequencies up to 1 TeV, ω ∼ 1TeV, then this estimate is increased by 6 orders of magnitude. Acknowledgments We thank G. Dvali, G. Gabadadze, A. Gruzinov, G. Melikidze, B. Ratra, and A. Starobinsky for stimulating discussions. T. K. and A. Zh. acknowledge hospital- ity of Abdus Salam International Center for Theoreti- cal Physics (ICTP) where this work has been started. A.Zh. would like to thank the Theory Division of CERN for their kind hospitality during the final stage of this work. T.K. acknowledges partial support from INTAS 061000017-9258 and Georgian NSF ST06/4-096 grants. A. Appendix: Dynamics of Light Gravexcitons In this appendix we briefly summarize the main prop- erties of the light gravexcitons necessary for our inves- tigations. The more detail description can be found in Refs. [17, 19]. The effective equation of motion for massive cosmolog- ical gravexciton6 is ψ + (3H + Γ) ψ +m2ψψ = 0 , (A.1) where H ∼ 1/t and Γ ∼ m3ψ/M2Pl are the Hubble pa- rameter and decay rate (ψ → γγ) correspondingly. This equation shows that at times when the Hubble parame- ter is less than the gravexciton mass: H . mψ the scalar field begins to oscillate (i.e. time tin ∼ H−1in ∼ 1/mψ roughly indicates the beginning of the oscillations): ψ ≈ CB(t) cos(mψt+ δ) . (A.2) We consider cosmological gravexcitons with masses 10−12GeV ≤ mψ ≤ 10−2GeV. The lower bound fol- lows both from the fifth-force experiments and Eq. (13). The upper bound follows from the demand that the life- time of these particles (with respect to decay ψ → γγ) is larger than the age of the Universe: τ = 1/Γ ∼ (MPl/mψ) tPl ≥ 1019sec > tuniv ∼ 4 × 1017 sec. Thus, we can neglect the decay processes for these gravexci- tons. Additionally, it can be easily seen that these par- ticles start to oscillate before teq ∼ H−1eq when the en- ergy densities of the matter and radiation become equal to each other (matter/radiation equality). According to the present WMAP data for the ΛCDM model it holds Heq ≡ meq ∼ 10−56MPl ∼ 10−28eV. Thus, considered particles have masses mψ >> meq and start to oscil- late during the radiation dominated stage. They will not overclose the observable Universe if the following condi- tion is satisfied: mψ . meq(MPl/ψin) 4, where ψin is the amplitude of the initial oscillations at the moment tin (see Eq. (18) in Ref. [19]). Prefactors C and B(t) in Eq. (A.2) for con- sidered light gravexcitons respectively read: C ∼ (ψin/MPl) (MPl/mψ) and B(t) ∼ MPl (MPlt)−3s/2. Here, s = 1/2, 2/3 for oscillations during the radiation 6 We have seen that the interaction between gravexcitons and or- dinary matter (in our case it is 4D-photons) is suppressed by the Planck scale. Thus, gravexcitons are weakly interacting massive particles (WIMPs). dominated and matter dominated stages, correspond- ingly. We are interested in the gravexciton oscillations at the present time t = tuniv. In this case s = 2/3 and for B(tuniv) we obtain: B(tuniv) ∼ t−1univ ≈ 10−61MPl. Thus, the amplitude of the light gravexciton oscillations at the present time reads: |κ0ψ| ∼ 10−60 . (A.3) [1] V. A. Kostelecký, Third meeting on CPT and Lorentz Symmetry (World Scientific, Singapore, 2005); G. M. Shore, Nucl. Phys. B 717, 86 (2005). D. Mattingly, Liv- ing Rev. Rel. 8, 5 (2005); T. Jacobson, S. Liberati, and D. Mattingly, Ann. Phys. 321, 150 (2006). [2] G. Amelino-Camelia, et al., Nature, 393, 763, (1998). [3] J. R. Ellis, K. Farakos, N. E. Mavromatos, V. A. Mit- sou and D. V. Nanopoulos, Astrophys. J. 535, 139 (2000); J. R. Ellis, N. E. Mavromatos, D. V. Nanopoulos, A. S. Sakharov and E. K. G. Sarkisyan, Astropart. Phys. 25, 402 (2006). [4] V. A. Kostelecký and M. Mewes, Phys. Rev. Lett. 87, 251304 (2001); G. Amelino-Camelia and T. Piran, Phys. Rev. D 64, 036005 (2001). [5] S. Sarkar, Mod. Phys. Lett. A 17, 1025 (2002) [6] R. C. Myers and M. Pospelov, Phys. Rev. Lett. 90, 211001 (2003). [7] T. Piran, in Planck Scales Effects in Astrophysics and Cosmology, eds. J. Kowalski-Glikman and G. Amelino- Camelia (Springer, Berlin, 2005), p. 351. [8] T. Jacobson, S. Liberati, and D. Mattingly, Nature 424, 1019 (2003). [9] M. Rodŕıguez Mart́ınez and T. Piran, J. Cosmo. As- tropart. Phys. 4, 006 (2006). [10] S. M. Carroll, G. B. Field, and R. Jackiw, Phys. Rev. D 41, 141601 (1990). [11] T. Kahniashvili, G. Gogoberidze, and B. Ratra, Phys. Lett. B 643, 81 (2006). [12] M.B. Green, J.H. Schwarz and E. Witten, 1987 Super- string Theory, (Cambridge: Cambridge Univ. Press). [13] T. Damour and A.M. Polyakov, Nucl. Phys. B 423, 532 (1994). [14] A. Zhuk, Int. J. Mod.Phys. D 11, 1399 (2002). [15] V.A. Kostelsky, R. Lehnert and M. Perry, Phys.Rev. D 68, 123511 (2003). [16] P. Loren-Aguilar, E. Garcia-Berro, J. Isern and Yu.A. Kubyshin, Class.Quant.Grav. 20, 3885, (2003). [17] U. Günther, A. Starobinsky and A. Zhuk, Phys. Rev. D69, 044003 (2004). [18] K.P. Stanyukovich and V.N. Melnikov, 1983 Hydrody- namics, fields and constants in theory of gravity, (in Rus- sian); U. Bleyer, K.A.Bronnikov, S.B.Fadeev and V.N. Melnikov, gr-qc/9405021. [19] U. Günther and A. Zhuk, Int. J. Mod. Phys. D 13, 1167 (2004). [20] U. Günther and A. Zhuk, Phys. Rev. D 56, 6391 (1997). [21] B. Ratra, Astrophys. J. Lett. 391, L1 (1992); D. Grasso and H. R. Rubinstein, Phys. Rept. 348, 163 (2001); M. Giovannini, Class. Quant. Grav. 22, 363 (2005). [22] D. A. Varshalovich, A. N. Moskalev, and V. K. Kher- sonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988). [23] N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics (McGraw-Hill, New York, 1973). [24] M. B. Cantcheff, Eur. Phys. J. C 46, 247 (2006). [25] R. Lehnert and R. Potting, Phys. Rev. Lett. 93, 110402 (2004), Phys. Rev. D 70, 125010 (2004). [26] G. R. Dvali and M. Zaldarriaga, Phys. Rev. Lett. 88, 091303 (2002). [27] E.G. Adelberger, B.R. Heckel, A.E. Nelson, Ann.Rev.Nucl.Part.Sci. 53, 77 (2003). http://arxiv.org/abs/gr-qc/9405021

0704.0315

The small deviations of many-dimensional diffusion processes and rarefaction by boundaries

arXiv:0704.0315v1 [math.PR] 3 Apr 2007 THE SMALL DEVIATIONS OF MANY-DIMENSIONAL DIFFUSION PROCESSES AND RAREFACTION BY BOUNDARIES Vitalii A. Gasanenko Abstract. We lead the algorithm of expansion of sojourn probability of many-dimensional diffusion processes in small domain. The principal member of this expansion defines nor- malizing coefficient for special limit theorems. Introduction. Let ξ(t) be a random process with measurable phase space (X,Σ(X)). Consider the measurable connected domain D ∈ Σ(X) and small parameter ǫ. The investigations of asymptotics of sojourn probability (small deviations) P (ξ(t) ∈ ǫD, t ∈ [0, T ]) (1) is jointed with many practice and theoretical problems [1-4]. In the literature, it was researched both rough asymptotics of principal member of (1)(log from it)[5] and exact asymptotics of diffusion processes of (1)[6-8]. In the works [9,10] was proved of algo- rithms of expansions of exact asymptotics of small deviation for diffusion and piecewise deterministic random processes for one-dimensional case. The purpose this article is to present the algorithm of expansion of small deviation for many-dimensional diffusion processes and to define all constants of principal member. In Section 1 our main result is stated and proved. In section 2 we consider the limits theorems about numbers of unabsorbed diffusion particles by boundaries of small domain. I. The expansion. We shall investigate of asymptote of following probability P (ǫ, x) = P (ξ(t) ∈ ǫD, 0 ≤ t ≤ T ) , ǫ → 0, where ξ(t) ∈ Rd is solution of the following stochastic differential equation dξ(t) = a(t, ξ(t))dt+ bi(ξ(t))dwi(t), ξ(0) = x ∈ ǫD. (2) where functions 1991 Mathematics Subject Classification. 60 J 65. Key words and phrases. parabolic problem,small domain, algorithm of expansion, number of unab- sorbed processes. Typeset by AMS-TEX http://arxiv.org/abs/0704.0315v1 2 VITALII A. GASANENKO bi(x), a(t, x) : R d → Rd and R+ ×R d → Rd. are differentiable. Set σij(x) = bik(x)b k(x). It is known that P (ǫ, x) = uǫ0(T, x). Here u 0(t, x) is solution of the following parabolic boundary problem at 0 ≤ t ≤ T ∂uǫ0(t, x) i,j=1 σij(x) ∂2uǫ0(t, x) ∂xi∂xj ai(T − t, x) ∂uǫ0(t, x) , x ∈ Dǫ; u(t, x)|t=0 = 1; x ∈ Dǫ; u(t, x) = 0 x ∈ ∂Dǫ, 0 ≤ t ≤ T. (3) where Dǫ = ǫD. It is assumed that D is a connected bounded domain from R m; the boundary ∂Q is the Lyapunov surface C(1,λ) and 0 ∈ D. We interest of the asymptotic expansion ǫ → 0 of solution this problem uǫ0(t, x) at ǫ → 0. We define the differential operator A : 1 1≤i,j≤d σij(0) ∂xi∂xj . Let σ be a matrix with the following property 1≤i,j≤d σij(0)zizj ≥ µ|~z| Here µ, there is a fixed positive number, and ~z = (z1, · · · , zd) is an arbitrary real vector. This operator acts in the following space HA = {u : u ∈ L2(D) ∩ Au ∈ L2(D) ∩ u(∂D) = 0} with inner product (u, v)A = (Au, v). Here (, ) is inner product in L2(Q). The opera- tor A is a positive operator[11]. It is known that the following eigenvalue problem Au = −λu, u(∂D) = 0 has infinite set of real eigenvalues λi → ∞ and 0 < λ1 < λ2 < · · · < λs < · · · . The corresponding eigenfunctions f11, . . . , f1n1 , · · · , fs1, . . . , fsns , · · · form the complete system of functions both in HA and L 2(Q) := {u : u ∈ L2(Q) ∩ u(∂Q) = 0}. Here the number nk is equal to multiplicity of eigenvalue λk. It is often convenient to present the system of eigenfunctions by one index: {fn(z)}. The corresponding system of eigenvalues {λn} will be with recurrences. We shall use it We introduce the spectral function e(x, y, λ) = fj(x)fj(y). We shall need in the following theorem from the monograph [12]. THE SMALL DEVIATION OF MANY-DIMENSIONAL DIFFUSION PROCESSES 3 Theorem 1 ([12].Th.17.5.3). . There exists such constant Cα that x,y∈D |Dαx,ye(x, y, λ)| ≤ Cαλ (n+|α|)/2 Here α is multi-index. Theorem 2. . If the surface ∂D is Lyapunov surface and (t,z)∈[0,T ]×D,1≤i,j≤d ∂ai(t, z) ∂bi(z) ∂ai(T − t, z) then the following relation takes place at ǫ → 0 P (ǫ, zǫ) = exp µ(t)dt c1mf1m(z) (1 +O(ǫ)) , at z ∈ D, where µ(t) = σij(0)ai(t, 0)aj(t, 0)− δijai(t, 0)aj(t, 0) and c1m = f1m(z)dz. Proof. Make the change of variables and function xi = ziǫ, u 1 = u 0 exp ak(T − t, 0)zk Now we obtain the following parabolic problem for function uǫ1 ∂uǫ1(t, z) i,j=1 σij(ǫz) ∂2uǫ1(t, z) ∂zi∂zj ai(T − t, ǫz)− σij(ǫz)aj(T − t, 0) ∂uǫ1(t, z) σij(ǫz)ai(T − t, 0)aj(T − t, 0)− δijai(T − t, 0)aj(T − t, ǫz)− ǫ ∂ai(T − t, 0) uǫ1, z ∈ D; 1(t, z)|t=0 = exp ak(T, 0)zk ; z ∈ D; uǫ1(t, z) = 0 z ∈ ∂D, 0 ≤ t ≤ T. We will construct the asymptotic expansion of solution for this initial - boundary problem in the following form uǫ1(t, z) = vk(t, z)ǫ k. (5) Note that the famous expansion 4 VITALII A. GASANENKO ak(T, 0)zk = 1 + ǫ ak(T, 0)zk + ak(T, 0)zk + · · · , defines the initial conditions for vk, k ≥ 0: v0(0, z) = 1, v1(0, z) = ak(T, 0)zk, v2(0, z) = ak(T, 0)zk · · · . Using the first fragment of Taylor series in zero point under conditions of theorem we can obtain the following representations σij(ǫz) = σij(0) + ǫσ ij(z), ai(T − t, ǫz) = ai(T − t, 0) + ǫa i(T − t, z), 1 ≤ i, j ≤ d (6) where z∈D,ǫ∈[0,1],1≤i,j≤d |σǫij(z)| < ∞, sup z∈D,t∈[0,T ],ǫ∈[0,1],1≤i≤d |aǫi(T − t, z)| < ∞ Now, after substitution of (5),(6) to (4) we conclude that the v0 satisfies the problem i,j=1 σij(0) ∂zi∂zj  v0 + µ(t)v0 (7) v0|∂D = 0; v0(0, z) = 1, z ∈ D. µ(t) = σij(0)ai(T − t, 0)aj(T − t, 0)− δijai(T − t, 0)aj(T − t, 0) Further, let us denote by Bǫ(t, z) the operator C 2(D) → C(D), for f ∈ C2(D) it’s defined as follows: ǫ(t, z)f = i,j=1 σǫij(z) ∂zi∂zj ai(T − t, ǫz)− σij(ǫz)aj(T − t, 0) i,j=1 σǫij(z)ai(T − t, 0)aj(T − t, 0)− δijai(T − t, 0)a j(T − t, z)− ∂ai(T − t, 0) i,j=1 σǫij(z) ∂zi∂zj Aǫ1(t, z)f + ǫA 2(t, z). THE SMALL DEVIATION OF MANY-DIMENSIONAL DIFFUSION PROCESSES 5 Now, formally the functions vk, k ≥ 1 are defined by the following recurrence system problems i,j=1 σij(0) ∂zi∂zj  vk +Bǫ(t, z)vk−1 (8) v0|∂D = 0; vk(0, z) = ak(T − t, 0)zk , z ∈ D. We shall solve the problems of (7),(8) by method of separation of variables. According to this method the solutions are defined in the form vk(t, z) = qk,n(t)fn(z). (9) For definition of principal number it suffices to construct of the v0. If we substitute (9) at k = 0 to (7) then we obtain −q̇0,n(t)− q0,n(t) + µ(t)q0,n(t) fn(z) = 0. Set c0,n = fn(z)dz (coefficients of expansion of indicator of set D). The initial condition of v0 has the following stating v0(0, z) = q0,n(0)fn(z) = c0,nfn(z) = c0,lmflm(z), z ∈ D. By definition of system of functions {fn(z)}, now we have the system of ordinary differential equations q̇0,n(t) + − µ(t) q0,n(t) = 0, q0,n(0) = c0,n. From the latter one we have q0,n(t) = c0,n exp µ(s)ds A0 = sup ǫ≤1,z∈D;i,j |σǫij(z)|, L0 = l≥1,1≤m≤nl (c0,ml) A1 = sup 0≤ǫ≤1,z∈D,t∈[0,T ];i,j ai(T − t, ǫz)− σij(ǫz)aj(T − t, 0) = sup 0≤ǫ≤1,z∈D,t∈[0,T ];i,j ij(z)ai(T − t, 0)aj(T − t, 0)− δijai(T − t, 0)a j(T − t, z)− ∂ai(T − t, 0) We have the following relations for eigenvalues λl 6 VITALII A. GASANENKO 2/d ≤ λl ≤ k2l 2/d, max(k1, k2) < ∞ Applying Cauchy-Bunyakovskii inequality, Theorem 1 and the latter one, we get aǫi,j(z) ∂zi∂zj −λltǫ µ(s)ds c0,ml aǫi,j(z) ∂2fml(z) ∂zi∂zj ≤ A0d −λltǫ µ(s)ds (c0,ml) ∂2fml(z) ∂zi∂zj ≤ A0dC2,2L0 µ(s)ds l ≤ exp K0. (10) Here K0 < ∞. Reasoning similarly we convince ourselves that for other parts of Bǫ(t, z)v0 the fol- lowing estimations take place |Aǫ1(t, z)v0| ≤ A1dC1,1L0 µ(s)ds l ≤ exp K0,1; (11) |Aǫ2(t, z)v0| ≤ A2dC0,0L0 µ(s)ds l ≤ exp K0,2, (12) where max{K0,1,K0,2} < ∞. Now let us estimate the coefficients βǫn(t) of expansion of B ǫ(t, z)v0 by system {fn}n≥1. Applying (10)-(12) and Cauchy-Bunyakovskii inequality, we get |βǫn(t)| = | Bǫ(t, z)v0(t, z)fn(z)dz| ≤ (Bǫ(t, z)v0) n(z)dz ≤ exp(−λ1tǫ K0 +K0,1 + ǫK0,2 The latter one now gives βǫn(s)ds| ≤ ǫγǫ(t), (13) THE SMALL DEVIATION OF MANY-DIMENSIONAL DIFFUSION PROCESSES 7 where 0≤ǫ≤1,t∈[0,T ] γǫ(t) < ∞. Finally, let us estimate the difference rǫ(t, z) = uǫ1(t, z)−v0(t, z). By definition, r ǫ(t, z) is solution of the following problem i,j=1 σij(0) ∂zi∂zj  rǫ +Bǫ(T − t, z)v0 z ∈ D; (14) rǫ(t, z)|t=0 = exp ak(T, 0)zk − 1; z ∈ D; rǫ(t, z) = 0 z ∈ ∂D, 0 ≤ t ≤ T. It is clear that rǫ(0, z) we can present as ǫrǫ1(0, z), where r 1(0, z) is uniform bounded function of variables ǫ ∈ [0, 1] and z ∈ D. So, the coefficients of expansion this function by system {fn(z)} have the following forms rǫ(0, z)fn(z)dz = ǫµ n, where sup 0≤ǫ≤1 (µǫn) = M < ∞. (13) Now we have the solution of (14) in the following form rǫ(t, z) = ǫ µǫn exp{−λntǫ βǫn(s)ds}fn(z) Applying latter one ,(13),(15), Theorem 1 and Cauchy-Bunyakovskii inequality we get at t > 0 |rǫ(t, z)ǫ−1| ≤ (µǫn) exp{−λntǫ βǫn(s)ds}λ n } ≤ ≤ MC0,0 exp{−λ1tǫ −2}K0,3, where K0,3 < ∞. The proof of theorem is completed. Remark 1. According to the above system of problems for definition of the functions vk, k ≥ 1, we outline the construction of coefficients qk,n)(t) for the series (8): q̇k,n(t) = + + µǫk−1,n(t) qk,n(t), qk,n(0) = vk(0, z)fn(z)dz = am(T, 0)zm fn(z)dz Here µǫk−1,n(t) = fn(z)B ǫ(t, z)vk−1(t, z)dz. 8 VITALII A. GASANENKO Remark 2. Theorem 2 is coordinated with results of works [6-8] where the principal member of small deviations in ball are investigated for more simple SDE. II. The rarefaction of set of diffusion processes by boundaries of small domains. The following problem was investigated in works[13,14]. Let a set identical diffusion random processes start at the initial time from the different points of domain D. These processes are diffusion processes with absorbtion on the boundary ∂D. We are interested in distribution of the number yet absorbed at the moment T . The initial number and initial position of diffusion processes are defined either a random Poisson measure[14] or deterministic measure [13]. The proved limits theorems described the situation when T → ∞ and initial number of diffusion processes depended on T and it increased at the rise of T . The role of normalizing function played principal member of asymptote of solution of according parabolic problem at T → ∞. Henceforth we shall assume that considered diffusion processes satisfy of the SDE (2) with different initial points. Now we consider the situation when initial number of absorbing diffusion processes in small domain ǫD depends on ǫ → 0 and it increase under the condition of decrease of ǫ. It is not hard to show, that now normalizing function is the principal member of parabolic problem (3) at ǫ → 0. The proofs of stated below theorems repeat the proofs of according theorems from [13,14] almost word for word. We will denote by η(ǫ, T ) the number of remaining processes in the region ǫD at the moment T . We will also assume that σ-additive measure ν is given on the Σν- algebra sets from D, ν(D) < ∞. All eigenfunctions fij : D → R 1 are (Σν ,ΣY ) measurable. Here ΣY is system of Borel sets from R1. Let ⇒ denote the weak convergence of random values or measures. At the beginning we assume that initial number and position of diffusion processes are defined by deterministic measure N(ǫB, ǫ), B ∈ D. Thus, N(ǫB, ǫ) is equal to number of starting points in the set ǫB. Let us denote by νǫ(·) the measure νǫ(ǫB) = exp N(ǫB, ǫ). where B ∈ Σν . By definition of measure νǫ(·), we have dνǫ(x) = , if x = xk, k = 1, · · · , N(ǫD, ǫ) 0, otherwise. Theorem 3. Under the assumptions of the Theorem 2 let the N(ǫ·, ǫ) satisfies the con- dition νǫ(ǫ ·) ⇒ ν(·). THE SMALL DEVIATION OF MANY-DIMENSIONAL DIFFUSION PROCESSES 9 Then η(ǫ, T ) ⇒ η(T ) if ǫ → 0 where η(T ) has Poisson distribution function with parameter a(T ) = exp µ(s)ds F (z)dν(z), where F (z) = f1i(z)c1i, c1i = f1i(z)dz and µ(t) is the function from Theorem 2. Now we consider the case when the initial number and positions of processes are defined by the random Poisson measure µ(·, ǫ) in ǫD: P (µ(ǫA, ǫ) = k) = mk(ǫA, ǫ) −m(ǫA,ǫ) where m(ǫ ·, ǫ) is finitely additive positive measure on ǫD for fixed ǫ. We assign g(ǫ) = exp Theorem 4. Under the assumptions of the Theorem 2 we suppose that m(ǫ·, ǫ) holds the condition m(ǫB, ǫ)g(ǫ) = ν(B), B ∈ Σν . Then η(ǫ, T ) ⇒ η(T ) if ǫ → 0 where η(T ) has the Poisson distribution function with the parameter a(T ) from Theorem 3. References 1. GrahamR.,Path integral formulation of general diffusion processes, Z.Phys.(1979),B 26,pp.281-290. 2. Onsager L. andMachlup S. Fluctuation and irreversible processes, I,II, Phys.Rev.(1953) 91,pp.1505-1512,1512-1515. 3. Li W. V.,Shao Q.-M., Gaussian processes:inequalities, small ball probabilities and applications, in : Stochastic Processes:Theory and Methods, in : Handbook of Statistics, vol.19, 2001, pp. 533-597. 4. Lifshits M.A., Asymptotic behavior of small ball probabilities, in Probab. Theory and Math.Statist., Proc. VII International Vilnius Conference (1998), pp. 453-468. 5. Lifshits M., Simon T., Small deviations for fractional stable processes, Ann. I. H. Poincare - PR 41 (2005) pp. 725-752. 6. Mogulskii A.A, The method of Fourier for determination of asymptotics of small deviations of Wiener process, Siberian Math. Journ. (1982),v.22,no.3,pp.161-174. 7. Fujita T. and Kotani S., The Onsager - Machlup Function for diffusion processes, J.Math.Kyoto Uneversity.- 1982.-vol.22,no.22.pp.131-153. 8. Zeitoni O., On the Onsager-Machlup functional of diffusion processes around non C2 curves, Ann. Probab.(1989),vol.17, no.3, pp.1037-1054. 10 VITALII A. GASANENKO 9. Gasanenko V.A., The total asymptotic expansion of sojourn probability of diffusion process in thin domain with moving boundaries, Ukraine Math. Journ. (1999),v.51, no. 9, pp.1155-1164. 10. Gasanenko V.A., The jump like processes in thin domain, Analytic questions of stochastic system, Kyiv:Institute of Mathematics (1992), pp. 4-9. 11. Mihlin S.G. Partial differential linear equations (1977), Vyshaij shkola, Moskow, 12.L.Hörmander, The analysis of Linear Partial differential Operators III (1985), Spinger-Verlag. 13.Fedullo A., Gasanenko V.A., Limit theorems for rarefaction of set of diffusion processes by boundaries, Theory of Stochastic Processes vol. 11(27), no.1-2,2005, pp.23- 14.Fedullo A., Gasanenko V.A.,Limit theorems for number of diffusion processes, which did not absorb by boundaries, Central European Journal of Mathematics 4(4), 2006, pp.624-634. Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivska 3, 252601, Kiev, Ukraine E-mail address: [email protected] or [email protected]

0704.0319

Spin-orbit coupling effect on the persistent currents in mesoscopic ring with an Anderson impurity

Spin-orbit coupling effect on the persistent currents in mesoscopic ring with an Anderson impurity Guo-Hui Ding and Bing Dong Department of Physics, Shanghai Jiao Tong University, Shanghai, 200240, China (Dated: November 4, 2018) Abstract Based on the finite U slave boson method, we have investigated the effect of Rashba spin- orbit(SO) coupling on the persistent charge and spin currents in mesoscopic ring with an Anderson impurity. It is shown that the Kondo effect will decrease the magnitude of the persistent charge and spin currents in this side-coupled Anderson impurity case. In the presence of SO coupling, the persistent currents change drastically and oscillate with the strength of SO coupling. The SO coupling will suppress the Kondo effect and restore the abrupt jumps of the persistent currents. It is also found that a persistent spin current circulating the ring can exist even without the charge current in this system. PACS numbers: 73.23.Ra, 71.70.Ej, 72.25.-b http://arxiv.org/abs/0704.0319v1 I. INTRODUCTION Recently the spin-orbit(SO) interaction in semiconductor mesoscopic system has attracted a lot of interest[1]. Due to the coupling of electron orbital motion with the spin degree of freedom, it is possible to manipulate and control the electron spin in SO coupling system by applying an external electrical field or a gate voltage, and it is believed that the SO effect will play an important role in the future spintronic application. Actually, various interesting effects resulting from SO coupling have already been predicted, such as the Datta-Das spin field-effect transistor based on Rashba SO interaction[2] and the intrinsic spin Hall effect[3]. In this paper we shall focus our attention on the persistent charge current and spin cur- rent in mesoscopic semiconductor ring with SO interaction. The existence of a persistent charge current in a mesoscopic ring threaded by a magnetic flux has been predicted decades ago[4], and has been extensively studied in theory[5, 6, 7, 8, 9] and also observed in various experiments[10, 11, 12]. The reason that a persistent charge current exists may be inter- preted as that the magnetic flux enclosed by the ring introduces an asymmetry between electrons with clockwise and anticlockwise momentum, thus leads to a thermodynamic state with a charge current without dissipation. For a mesoscopic ring with a texture like inho- mogeneous magnetic field, D. Loss et al.[13] predicted that besides the charge current there are also a persistent spin current. The origin of the persistent spin current can be related to the Berry phase acquired when the electron spin precesses during its orbital motion. The persistent spin current has also been studied in semiconductor system with Rashba SO cou- pling term[14, 15, 16]. Recently it is shown that a semiconductor ring with SO coupling can sustain a persistent spin current even in the absence of external magnetic flux[17]. For the system of a mesoscopic ring with a magnetic impurity, the persistent charge current has been investigated in the context of a mesoscopic ring coupled with a quantum dot[18, 19, 20, 21, 22, 23, 24], where the quantum dot acts as an impurity level and will introduce charge or spin fluctuations to the electrons in the ring. The Kondo effect arising from a localized electron spin interacting with a band of electrons will be essential in the charge transport in the ring. But to our knowledge in these systems the SO effect hasn’t been considered. It might be expected that the interplay between the Kondo effect and the SO coupling in the ring can give new features in the persistent currents. In this paper we shall address this problem and investigate the SO effect on persistent charge and spin currents in the ring system with an Anderson impurity. The Anderson impurity can act as a magnetic impurity when the impurity level is in single electron occupied state and as well as a barrier potential in empty occupied regime. The outline of this paper is as follows. In section II we introduce the model Hamiltonian of the system and also the method of calculation by finite-U slave boson approach[25, 26, 27, 28]. In section III the results of persistent charge current and spin current are presented and discussed. In Section IV we give the summary. II. MESOSCOPIC RING WITH AN ANDERSON IMPURITY The electrons in a closed ring with SO coupling of Rashba term can be described by following Hamiltonian in the polar coordinates[14, 29] Hring = ∆(−i [(σx cosϕ+ σy sinϕ)(−i ) + h.c.] , (1) where ∆ = h̄2/(2mea 2), a is the radius of the ring. αR will characterize the strength of Rashba SO interaction. Φ is the external magnetic flux enclosed by the ring, and Φ0 = 2πh̄c/e is the flux quantum. We can write the above Hamiltonian in terms of creation and annihilation operators of electrons in the momentum space, Hring = mσcmσ + 1/2 [tm(c m+1↓cm↑ + c m−1↑cm↓) + h.c.] , (2) where ǫm = ∆(m+φ) 2, tm = αR(m+φ),(m = 0,±1, · · · ,±M) with φ = Φ/Φ0. One can see that the SO interaction causes the m mode electrons coupled with m + 1 and m − 1 mode electrons and spin-flip process. We consider the system with a side-coupled impurity which can be described by the Anderson impurity model, σdσ + Und↑nd↓ . (3) The tunneling between the impurity level and the ring are given by Hd−ring = tD (d†σcmσ + h.c) . (4) Then the total Hamiltonian for the system should be H = Hring +Hd +Hd−ring . (5) In order to treat the strong on-site Coulomb interaction in the impurity level. we adopt the finite-U slave boson approach[25, 26]. A set of auxiliary bosons e, pσ, d is introduced for the impurity level, which act as projection operators onto the empty, singly occupied(with spin up and spin down), and doubly occupied electron states on the impurity, respectively. Then the fermion operators dσ are replaced by dσ → fσzσ, with zσ = e †pσ + p σ̄d. In order to eliminate un-physical states, the following constraint conditions are imposed : σpσ + e†e+ d†d = 1, and f †σfσ = p σpσ + d †d(σ =↑, ↓). Therefore, the Hamiltonian can be rewritten as the following effective Hamiltonian in terms of the auxiliary boson e, pσ, d and the pesudo- fermion operators fσ: Heff = mσcmσ + 1/2 [tm(c m+1↓cm↑ + c m−1↑cm↓) + h.c.] σfσ + Ud σcmσ + h.c.) + λ p†σpσ + e †e+ d†d− 1) λ(2)σ (f σfσ − p σpσ − d †d) , (6) where the constraints are incorporated by the Lagrange multipliers λ(1) and λ(2)σ . The first constraint can be interpreted as a completeness relation of the Hilbert space on the impurity level, and the second one equates the two ways of counting the fermion occupancy for a given spin. In the framework of the finite-U slave boson mean field theory[25, 26], the slave boson operators e, pσ, d and the parameter zσ are replaced by real c numbers. Thus the effective Hamiltonian is given as HMFeff = mσcmσ + 1/2 [tm(c m+1↓cm↑ + c m−1↑cm↓) + h.c.] ǫ̃dσf σfσ + (t̃Dσf σcmσ + h.c.) + Eg , (7) where t̃Dσ = tDzσ represents the renormalized tunnel coupling between the impurity and the mesoscopic ring. zσ can be regarded as the wave function renormalization factor. ǫ̃dσ = σ is the renormalized impurity level and Eg = λ 2+d2−1)− d2) + Ud2 is an energy constant. In this mean field approximation the Hamiltonian is essentially that of a non-interacting system, hence the single particle energy levels can be calculated by numerical diagonalization of the Hamiltonian matrix. Then the ground state of this system |ψ0 > can be constructed by adding electrons to the lowest unoccupied energy levels consecutively . By minimizing the ground state energy with respect to the variational parameters a set of self-consistent equations can be obtained as in Ref.[27,28], and they can be applied to determine the variational parameters in the effective Hamiltonian. III. THE PERSISTENT CHARGE CURRENT AND SPIN CURRENT In this section we will present the results of our calculation of the persistent charge current and spin current circulating the mesoscopic ring. Since there is still some controversial in the literature for the definition of the spin current operator in the ring system with SO coupling term[30]. We give both the formula of charge and spin currents used in this paper explicitly. It is easy to obtain that the ϕ component of electron velocity operator in this SO coupled ring is [2∆(−i + φ) + αR(σx cosϕ+ σy sinϕ)] . (8) Thereby the charge current operator is define as Î = −evϕ, and in terms of creation and annihilation operator it can be written as Î = − c†mσcmσ(m+ φ) + αR m+1↓cm↑ + c m−1↑cm↓)] . (9) At zero temperature, the persistent charge current is given by the expectation value of the above charge current operator in the ground state, I = 1 < ψ0|Î|ψ0 >, and it can also be calculated from the expression I = −c < ψ0| |ψ0 > , (10) where Egs is the ground state energy. In Fig.1 the persistent charge current vs. the enclosed magnetic flux is plotted for a set of values for the SO coupling strength. Here we have taken the model parameters ∆ = 0.01, tD = 0.3, U = 2.0 and the total number of electrons N is around 100. In this case one can obtain the Fermi energy of the system EF = 6.25 and the level spacing δ = 0.5 around the Fermi surface. We consider the energy level of the Anderson impurity is well below the Fermi energy( with ǫd − EF = −1.0), therefore the Anderson impurity is in the Kondo regime. One can see in Fig.1 that the characteristic features of persistent charge current depends on the parity of the total number of electrons(N), and can be distinguished by two cases with N odd and N even. This is attributed to the different occupation patterns of the highest occupied single particle energy level in the mean field effective Hamiltonian. The persistent charge current for the system with N +2 electrons is different from that with N electrons by a π phase shift IN+2(φ) = IN(φ+ π). In case (I) where the electron number is odd(N = 4n− 1 and N = 4n + 1), one electron is almost localized on the impurity level and forming a singlet with electron cloud in the conducting ring. This phenomena leads to the well known Kondo effect. Fig.1 shows that the Kondo effect decreases the magnitude of the persistent charge current, and also makes its curve shape resemble sinusoidal. In the presence of finite SO coupling(αR < ∆), the spin-up and spin-down electrons are coupled and it causes the splitting of the twofold degenerated energy levels in the effective Hamiltonian. It turns out that the Kondo effect is suppressed and the abrupt jumps of the persistent charge current with similarity to that of ideal ring case appears. It is explained in Ref.[14] that the jumps of the persistent charge current in the case of odd number of electrons are due to a crossing of levels with opposite spin. In case (II) where N is even (N = 4n and N = 4n+2), The Kondo effect is manifested that the magnitude of persistent charge current is significantly suppressed compared with ideal ring case and the rounding of the jumps of persistent charge current due to the level crossing. In the presence of finite SO coupling, the persistent charge current decreases with increasing the SO coupling strength when αR < ∆. Fig.2 displays the persistent charge current as a function of the SO coupling strength αR at different enclosed magnetic flux. The persistent charge current exhibits oscillations with increasing the value of αR for both the systems with even or odd number of electrons. Therefore by tuning the SO coupling strength, the magnetic response of this system can change from paramagnetic to diamagnetic and vice versa. It indicates that SO coupling can play a important role in electron transport in this mesoscopic ring. The curve of the persistent charge current for odd number of electrons shows discontinuity in its derivation, this can be attributed the level crossing in the energy spectrum by changing αR. It is also noted that the position of this discontinuity for odd N also corresponds to the peak or valley in even N case. Since the electron has the spin degree of freedom as well as the charge, the electron motion in the ring may give rise to a spin current besides the charge current. Now we turn to study the persistent spin current in the ground state. The spin current operator is defined by Ĵv = (v ϕσv + σvv ϕ)/2, which can be written explicitly as Ĵv = {2∆(−i + φ)σv + [(σx cosϕ+ σy sinϕ)σv + h.c.]} , (11) Therefore the three component of spin current operator in terms of creation and annihi- lation operators are given by Ĵz = m↑cm↑ − c m↓cm↓)(m+ φ)] , (12) Ĵx = m↑cm↓ + c m↓cm↑)(m+ φ) + m+1σ + c m−1σ)cmσ] , (13) Ĵy = [−2i∆ m↑cm↓ − c m↓cm↑)(m+ φ)− i m+1σ − c m−1σ)cmσ] , (14) The expectation value of the spin current Jv = < ψ0|Ĵv|ψ0 >. In our calculation we find that only the z component of the spin current is nonzero in the ground state. Fig.3 shows the persistent spin current Jz vs. magnetic flux at different SO coupling strength. The persistent spin current is a periodic function of the magnetic flux φ, which has the even parity symmetry Jz(−φ) = Jz(φ) and also an additional symmetry Jz(φ) = Jz(π−φ). It is noted that the persistent spin current has quite different dependence behaviors on magnetic flux compared with the persistent charge current in Fig.1. In the presence of finite SO coupling, the persistent spin current is nonzero both for the systems with odd N and even N at zero magnetic flux, it indicates that a persistent spin current can be induced solely by SO interaction without accompany a charge current. This phenomena is also shown in Ref.[17] where a SO coupling/normal hybrid ring was considered. In Fig.4 the persistent spin current Jz as a function of SO coupling strength is plotted. In the absence of SO coupling αR = 0, the persistent spin current is exactly zero for both even and odd number electron system. In the presence of SO coupling, The persistent spin current becomes nonzero and shows oscillations with increasing αR. It can change from positive to negative values or vice versa by tuning the SO coupling strength. The sign of the persistent spin current also shows dependence on the enclosed magnetic flux. For the system with odd N , there is abrupt jumps in the curve of persistent spin current at certain value of αR, the reason for the jump is the same as that in the charge current, and is due to the level crossing in the energy spectrum. It is noted that the position of the jump coincides with that of the persistent charge current. This kind of characteristic feature of the persistent currents might provide a useful way to detect the SO coupling effects in semiconductor ring system. IV. CONCLUSIONS In summary, we have investigated the Rashba SO coupling effect on the persistent charge current and spin current in a mesoscopic ring with an Anderson impurity. The Anderson impurity leads to the Kondo effect and decreases the amplitude of the persistent charge and spin current in the ring. In the semiconducting ring with SO interaction, the persistent charge current changes significantly by tuning the SO coupling strength, e.g. from the paramagnetic to diamagnetic current. Besides the persistent charge current, there also exists a persistent spin current, which also oscillates with the SO coupling strength. It is shown that at zero magnetic flux a persistent spin current can exist even without the charge current. Since the persistent spin current can generate an electric field[31], one might expect that experiments on semiconductor ring with Rashba SO coupling can detect the persistent spin current. Acknowledgments This project is supported by the National Natural Science Foundation of China, the Shanghai Pujiang Program, and Program for New Century Excellent Talents in University (NCET). [1] I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). [2] S. Datta and B. Das, Appl. Phys. Lett. 56, 665(1990). [3] S. Murakami, N. Nagaosa, and S. C. Zhang, Science 301, 1348 (2003); J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T.Jungwirth, and A. H. MacDonald, Phys. Rev. Lett., 92, 126603(2004). [4] M. Büttiker, Y. Imry, and R. Landauer, Phys. Lett.96A, 365 (1983). [5] H. F. Cheung, Y. Gefen, E. K. Riedel, and W. H. Shih, Phys. Rev. B 37, 6050 (1988). [6] D. Loss and P. Goldbart, Phys. Rev. B 43, 13762 (1991). [7] G. Montambaux, H. Bouchiat, D. Sigeti, and R. Friesner, Phys. Rev. B 42, 7647 (1990). [8] Y. Meir, Y. Gefen, and O. Entin-Wohlman, Phys. Rev. Lett. 63, 798 (1989). [9] B. L. Altshuler, Y. Gefen, and Y. Imry, Phys. Rev. Lett. 66, 88(1991). [10] L. P. Lévy, G. Dolan, J. Dunsmuir, and H. Bouchiat, Phys. Rev. Lett. 64, 2074 (1990). [11] V. Chandrasekhar, R. A. Webb, M. J. Brady, M. B. Ketchen, W. J. Gallagher, and A. Klein- sasser, Phys. Rev. Lett. 67, 3578 (1991). [12] D. Mailly, C. Chapelier, and A. Benoit, Phys. Rev. Lett. 70, 2020 (1993). [13] D. Loss, P. Goldbart, and A. V. Balatsky, Phys. Rev. Lett., 65, 1655 (1990); D. Loss and P. M. Goldbart, Phys. Rev. B 45, 13544 (1992). [14] J. Splettstoesser, M. Governale, and U. Zülicke, Phys. Rev. B 68, 165341 (2003). [15] J. S. Shen and K. Chang, Phys. Rev. B 74,235315(2006). [16] R. Citro and F. Romeo, Phys. Rev. B75,073306(2007). [17] Q. F. Sun, X. C. Xie, and J. Wang, cond-mat/0605748. [18] M. Büttiker and C. A. Stafford, Phys. Rev. Lett. 76, 495 (1996). [19] V. Ferrari, G. Chiappe, E. V. Anda, and M. A. Davidovich, Phys. Rev. Lett. 82, 5088 (1999). [20] I. Affleck and P. Simon, Phys. Rev. Lett.86, 2854 (2001); Phys. Rev. B64, 085308 (2001). [21] K. Kang and S. C. Shin, Phys. Rev. Lett. 85, 5619 (2000). [22] S. Y. Cho, K. Kang, C. K. Kim, and C. M. Ryu, Phys. Rev. B 64, 033314 (2001) [23] H. P. Eckle, H. Johannesson, and C. A. Stafford, Phys. Rev. Lett. 87, 016602 (2001). [24] H. Hu, G. M. Zhang, and L. Yu, Phys. Rev. Lett. 86, 5558 (2001). [25] G. Kotliar and A. E. Ruckenstein, Phys. Rev. Lett. 57, 1362 (1986). [26] V. Dorin and P. Schlottmann, Phys. Rev. B 47, 5095 (1993). [27] B. Dong and X. L. Lei, Phys. Rev. B 63, 235306 (2001); Phys. Rev. B 65, R241304 (2002). [28] G. H. Ding and B. Dong, Phys. Rev. B 67, 195327 (2003). [29] F. E. Meijer, A. F. Morpurgo, and T. M. Klapwijk, Phys. Rev. B 66, 033107 (2002). [30] Q. F. Sun and X. C. Xie, Phys. Rev. B 72, 245305(2005). [31] F. Meir and D. Loss, Phys. Rev. Lett. 90, 167204 (2003). http://arxiv.org/abs/cond-mat/0605748 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (d)(b) 0.0 0.2 0.4 0.6 0.8 1.0 FIG. 1: The persistent charge current vs. magnetic flux for a set of values for the spin-orbit coupling strength(αR/∆ = 0.0(solid line),0.5(dashed line), 0.7(dotted line),1.0(dash-dotted line)). The total number of electrons N = 99 (a), 100(b), 101(c), 102(d). We take the other parameters ∆ = 0.01, td = 0.3, ǫd − EF = −1.0, U = 2.0 in the calculation. The persistent charge current is measured in units of I0 = eN∆. 0 1 2 3 4 -0.15 -0.10 -0.05 0 1 2 3 4 -0.10 -0.05 (c)(a) 0 1 2 3 4 -0.10 -0.05 0 1 2 3 4 -0.10 -0.05 FIG. 2: The persistent charge current as a function of the spin-orbit coupling strength. The magnetic flux (Φ/Φ0 = 0.125(solid line),0.25(dashed line), 0.375(dotted line)). 0.0 0.2 0.4 0.6 0.8 1.0 -0.15 -0.10 -0.05 0.0 0.2 0.4 0.6 0.8 1.0 -0.10 -0.05 0.0 0.2 0.4 0.6 0.8 1.0 -0.05 0.0 0.2 0.4 0.6 0.8 1.0 -0.05 FIG. 3: FIG.3: The persistent spin current Jz vs. magnetic flux for a set of values for the spin- orbit coupling strength( with αR/∆ = 0.5(solid line),0.7(dashed line), 1.0(dotted line)). The panel (a), (b), (c) and (d) corresponds the system with total number of electrons N = 99, 100, 101, 102, respectively. The persistent spin current is measured in units of J0 = N∆, and we have taken the other parameter values the same as that in Fig.1. 0 1 2 3 4 -0.10 -0.05 0 1 2 3 4 -0.10 -0.05 0 1 2 3 4 -0.10 -0.05 0 1 2 3 4 -0.10 -0.05 FIG. 4: FIG.4: The persistent spin current Jz as a function of the spin-orbit coupling strength. The magnetic flux takes the value (Φ/Φ0 = 0.0(solid line),0.125(dashed line), 0.25(dotted line), 0.5(dash-dotted line)). introduction Mesoscopic ring with an Anderson impurity the persistent charge current and spin current conclusions Acknowledgments References

0704.0320

Probability distributions generated by fractional diffusion equations

FRACALMO PRE-PRINT www.fracalmo.org Probability distributions generated by fractional diffusion equations1 Francesco MAINARDI(1), Paolo PARADISI(2) and Rudolf GORENFLO(3) (1) Department of Physics, University of Bologna, and INFN, Via Irnerio 46, I-40126 Bologna, Italy. [email protected] [email protected] (2) ISAC: Istituto per le Scienze dell’Atmosfera e del Clima del CNR, Strada Provinciale Lecce-Monteroni Km 1.200, I-73100 Lecce, Italy. [email protected] (3) Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 3, D-14195 Berlin, Germany. [email protected] Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . p. 2 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . p. 2 2. The Standard Diffusion Equation . . . . . . . . . . . . . . p. 4 3. The Time-Fractional Diffusion Equation . . . . . . . . . . . p. 8 4. The Cauchy Problem for the Time-Fractional Diffusion Equation p.10 5. The Signalling Problem for the Time-Fractional Diffusion Equation p.13 6. The Cauchy Problem for the Symmetric Space-Fractional Diffusion Equation . . . . . . . . . . . . . . . . . . . . p.15 7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . p.21 A. The Riemann-Liouville Fractional Calculus . . . . . . . . . p.22 B. The Stable Probability Distributions . . . . . . . . . . . . p.31 References . . . . . . . . . . . . . . . . . . . . . . . p.41 1This paper is based on an invited talk given by Francesco Mainardi at the International Workshop on Econophysics held at Bolyai College, Eötvös University, Budapest, on July 21-27, 1997. The paper was originally edited as a contribution for the book J. Kertesz and I. Kondor (Editors), Econophysics: an Emerging Science, Kluwer Academic Publishers, Dordrecht (NL) that should contain selected papers presented at that Workshop and should have appeared in 1998 or 1999. Unfortunately the book was not published. The present e-print is a revised version (with up-date annotations and references) of that unpublished contribution, but essentially represents our knowledge of that early time. http://arXiv.org/abs/0704.0320v1 Abstract Fractional calculus allows one to generalize the linear, one-dimensional, diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of these generalized diffusion equations are shown to provide probability density functions, evolving on time or variable in space, which are related to the peculiar class of stable distributions. This property is a noteworthy generalization of what happens for the standard diffusion equation and can be relevant in treating financial and economical problems where the stable probability distributions are known to play a key role. 1 Introduction Non-Gaussian probability distributions are becoming more common as data models, especially in economics where large fluctuations are expected. In fact, probability distributions with heavy tails are often met in economics and finance, which suggests to enlarge the arsenal of possible stochastic models by non-Gaussian processes. This conviction started in the early sixties after the appearance of a series of papers by Mandelbrot and his associates, who point out the importance of non-Gaussian probability distributions, formerly introduced by Pareto and Lévy, and related scaling properties, to analyse economical and financial variables, as reported in the recent book by Mandelbrot (1997). Some examples of such variables are common stock prices changes, changes in other speculative prices, and interest rate changes. In this respect many works by different authors have recently appeared, see e.g. the recent books by Bouchaud & Potter (1997), Mantegna & Stanley (1998) and the references therein quoted. It is well known that the fundamental solution (or Green function) of the Cauchy problem for the standard linear diffusion equation provides at any time the probability density function (pdf) in space of the Gauss (or normal) law. This law exhibits all moments finite thanks to its exponential decay at infinity. In particular, the space variance of the Green function is proportional to the first power of time, a noteworthy property that can be understood by means of an unbiased random walk model for the Brownian motion, see e.g. Feller (1957). Less known is the property for which the fundamental solution of the Signalling problem for the same diffusion equation, provides at any position a unilateral pdf in time, known as Lévy law, using the terminology of Feller (1966-1973). Because of its algebraic decay at infinity as t−3/2 , this law has all moments of integer order divergent, and consequently its expectation value and variance are infinite. Both the Gauss and Lévy laws belong to the general class of stable probability distributions, which are characterized by an index α (0 < α ≤ 2), called index of stability or characteristic exponent. In particular, the index of the Gauss law is 2 , whereas that of the Lévy law is 1/2 . In this paper we consider two different generalizations of the diffusion equation by means of fractional calculus, which allows us to replace either the first time derivative or the second space derivative by a suitable fractional derivative. Correspondingly, the generalized equation will be referred to as the time-fractional diffusion equation or the symmetric, space-fractional diffusion equation. Here we show how the fundamental solutions of this equation for the Cauchy and Signalling problems provide probability density functions related to certain stable distributions, so providing a natural generalization of what occurs for the standard diffusion equation. The plan of the paper is as follows. First of all, for the sake of convenience and completeness, we provide the essential notions of Riemann-Liouville Fractional Calculus and Lévy Stable Probability Distributions in Appendix A and B, respectively. In Section 2, we recall the basic results for the standard diffusion equation concerning the fundamental solutions of the Cauchy and Signalling problems. In particular we provide the derivation of these solutions by the Fourier and Laplace transforms and the interpretation in terms of Gauss and Lévy stable pdf , respectively. In Section 3, we consider the time-fractional diffusion equation and we formulate for it the basic Cauchy and Signalling problems to be treated in the subsequent two sections. Here we adopt the Riemann-Liouville approach to Fractional Calculus, and the related definition for the Caputo time-fractional derivative of a causal function of time. In Section 4, we solve the Cauchy problem for the time-fractional diffusion equation by using the technique of Fourier transform and we derive the corresponding fundamental solution in terms of a special function of Wright type in the similarity variable. In this case the solution can be interpreted as a noteworthy symmetric pdf in space with all moments finite, evolving in time. In particular, its space variance turns out to be proportional to a power of time equal to the order of the time-fractional derivative. In Section 5, we derive the fundamental solution for the Signalling problem of the time-fractional diffusion equation by using the technique of Laplace transform. In this case the solution, still expressed in terms of a special function of Wright type, can be interpreted as a unilateral stable pdf in time, depending on position, with index of stability given by half of the order of the time-fractional derivative. In Section 6, we consider the symmetric, space-fractional diffusion equation. Here we adopt the Riesz approach to Fractional Calculus, and the related definition for the symmetric space-fractional derivative of a function of a single space variable. Here we treat the Cauchy problem by technique of Fourier transform and we derive the series representation of the corresponding Green function. In this case the fundamental solution is interpreted in terms of a symmetric stable pdf in space, evolving in time, with index of stability given by the order of the space-fractional derivative. To approximate such evolution we propose a random walk model, discrete in space and time, which is based on the Grünwald-Letnikov approximation of the fractional derivative. Finally, Section 7 is devoted to conclusions and remarks on related work. 2 The standard diffusion equation For the standard diffusion equation we mean the linear partial differential equation u(x, t) = D u(x, t) , u = u(x, t) , (2.1) where D denotes a positive constant with the dimensions L2 T−1 , x and t are the space-time variables, and u = u(x, t) is the field variable, which is assumed to be a causal function of time, i.e. vanishing for t < 0 . The typical physical phenomenon related to such an equation is the heat conduction in a thin solid rod extended along x , so the field variable u is the temperature. In order to guarantee the existence and the uniqueness of the solution, we must equip (1.1) with suitable data on the boundary of the space-time domain. The basic boundary-value problems for diffusion are the so-called Cauchy and Signalling problems. In the Cauchy problem, which concerns the space-time domain −∞ < x < +∞ , t ≥ 0 , the data are assigned at t = 0+ on the whole space axis (initial data). In the Signalling problem, which concerns the space-time domain x ≥ 0 , t ≥ 0 , the data are assigned both at t = 0+ on the semi-infinite space axis x > 0 (initial data) and at x = 0+ on the semi-infinite time axis t > 0 (boundary data); here, as mostly usual, the initial data are assumed to be vanishing. Denoting by g(x) and h(t) two given, sufficiently well-behaved functions, the basic problems are thus formulated as following: a) Cauchy problem u(x, 0+) = g(x) , −∞ < x < +∞ ; u(∓∞, t) = 0 , t > 0 ; (2.2a) b) Signalling problem u(x, 0+) = 0 , x > 0 ; u(0+, t) = h(t) , u(+∞, t) = 0 , t > 0 . (2.2b) Hereafter, for both the problems, we derive the classical results which will be properly generalized for the fractional diffusion equation in the subsequent sections. Let us begin with the Cauchy problem. It is well known that this initial value problem can be easily solved making use of the Fourier transform and its fundamental solution can be interpreted as a Gaussian pdf in x. Adopting the notation g(x) ÷ ĝ(κ) with κ ∈ R and ĝ(κ) = F [g(x)] = e+iκx g(x) dx , g(x) = F−1 [ĝ(κ)] = 1 e−iκx ĝ(κ) dκ , the transformed solution satisfies the ordinary differential equation of the first order ( + κ2 D û(κ, t) = 0 , û(κ, 0+) = ĝ(κ) , (2.3) and consequently it turns out to be û(κ, t) = ĝ(κ) e−κ 2 D t . (2.4) Then, introducing Gdc (x, t) ÷ Ĝdc (κ, t) = e−κ 2 D t , (2.5) where the upper index d refers to (standard) diffusion, the required solution, obtained by inversion of (2.4), can be expressed in terms of the space convolution u(x, t) = −∞ Gdc (ξ, t) g(x − ξ) dξ , where Gdc (x, t) = t−1/2 e−x 2/(4D t) . (2.6) Here Gdc (x, t) represents the fundamental solution (or Green function) of the Cauchy problem, since it corresponds to g(x) = δ(x) . It turns out to be a function in x , even and normalized, i.e. Gdc (x, t) = Gdc (|x|, t) and∫ +∞ −∞ Gdc (x, t) dx = 1 . We also note the identity |x| Gdc (|x|, t) = Md(ζ) , (2.7) where ζ = |x|/( D t1/2) is the well-known similarity variable and Md(ζ) = 2/4 . (2.8) We note that Md(ζ) satisfies the normalization condition d(ζ) dζ = 1 . The interpretation of the Green function (2.6) in probability theory is straightforward since we easily recognize Gdc (x, t) = pG(x;σ) := 2/(2σ2) , σ2 = 2D t , (2.9) where pG(x;σ) denotes the well-known Gauss or normal pdf spread out over all real x (the space variable), whose moment of the second order, the variance, is σ2 . The associated cumulative distribution function (cdf) is known to be PG(x;σ) := ′;σ) dx′ = 1 + erf , (2.10) where erf (z) := (2/ 0 exp (−u2) du denotes the error function. Furthermore, the moments of even order of the Gauss pdf turn out to be∫ +∞ 2n pG(x;σ) dx = (2n − 1)!!σ2n , so x2n Gdc (x, t) dx = (2n − 1)!! (2D t)n , n = 1, 2, . . . . (2.11) Let us now consider the Signalling problem. This initial-boundary value problem can be easily solved by making use of the Laplace transform. Adopting the notation h(t) ÷ h̃(s) with s ∈ C and h̃(s) = L [h(t)] = e−st h(t) dt , h(t) = L−1 h̃(t) est h̃(s) ds , where Br denotes the Bromwich path, the transformed solution of the diffusion equation satisfies the ordinary differential equation of the second order ũ(x, s) = 0 , ũ(0+, s) = h̃(s) , ũ(+∞, s) = 0 . (2.12) and consequently it turns out to be ũ(x, s) = h̃(s) e−(x/ D) s1/2 . (2.13) Then introducing Gds (x, t) ÷ G̃ds (x, s) = e−(x/ D) s1/2 , (2.14) the required solution, obtained by inversion of (2.13), can be expressed in terms of the time convolution, u(x, t) = 0 Gds (x, τ)h(t − τ) dτ , where Gds (x, t) = t−3/2 e−x 2/(4D t) . (2.15) Here Gds (x, t) represents the fundamental solution (or Green function) of the Signalling problem, since it corresponds to h(t) = δ(t) . We note that Gds (x, t) = pLS(t;µ) := 2π t3/2 e−µ/(2t) , t ≥ 0 , µ = x , (2.16) where pLS(t;µ) denotes the one-sided Lévy-Smirnov pdf spread out over all non negative t (the time variable). The associated cdf is, see e.g. Feller (1966-1971) and Prüss (1993), PL(t;µ) := ′;µ) dt′ = erfc = erfc , (2.17) where erfc (z) := 1 − erf (z) denotes the complenatary error function. The Lévy-Smirnov pdf has all moments of integer order infinite, since it decays at infinity as t−3/2 . However, we note that the absolute moments of real order ν are finite only if 0 ≤ ν < 1/2 . In particular, for this pdf the mean is infinite, for which we can take the median as expectation value. From PLs(tmed;µ) = 1/2 , it turns out that tmed ≈ 2µ , since the complementary error function gets the value 1/2 as its argument is approximatively 1/2. We note that in the common domain x > 0 , t > 0 the Green functions of the two basic problems satisfy the identity xGdc (x, t) = tGds (x, t) , (2.18) that we refer to as the reciprocity relation between the two fundamental solutions of the diffusion equation. Furthermore, in view of (2.7) and (2.18) we recognize the role of the function of the similarity variable, Md(ζ) , in providing the two fundamental solutions; we shall refer to it as to the normalized auxiliary function of the diffusion equation for both the Cauchy and Signalling problems. 3 The time-fractional diffusion equation By the time-fractional diffusion equation we mean the linear evolution equation obtained from the classical diffusion equation by replacing the first- order time derivative by a fractional derivative (in the Caputo sense) of order α with 0 < α ≤ 2. In our notation it reads , u = u(x, t) , 0 < α ≤ 2 , (3.1) where D denotes a positive constant with the dimensions L2 T−α . From Appendix A we recall the definition of the Caputo fractional derivative of order α > 0 for a (sufficiently well-behaved) causal function f(t) , see (A.9), Dα∗ f(t) := Γ(m − α) (t − τ)m−α f (m)(τ) dτ , (3.2) where m = 1, 2, . . . , and 0 ≤ m − 1 < α ≤ m . According to (3.2) we thus need to distinguish the cases 0 < α ≤ 1 and 1 < α ≤ 2 . In the the latter case (3.1) may be seen as a sort of interpolation between the standard diffusion equation and the standard wave equation. Introducing Φλ(t) := tλ−1+ , λ > 0 , (3.3) where the suffix + is just denoting that the function is vanishing for t < 0 , we easily recognize that the equation (3.1) assumes the explicit forms : if 0 < α ≤ 1 , Φ1−α(t) ∗ Γ(1 − α) (t − τ)−α dτ = D ; (3.4) if 1 < α ≤ 2 , Φ2−α(t) ∗ Γ(2 − α) (t − τ)1−α dτ = D ∂ . (3.5) Extending the classical analysis for the standard diffusion equation (2.1) to the above integro-differential equations (3.4-5), the Cauchy and Signalling problems are thus formulated as in equations (2.2), i.e. a) Cauchy problem u(x, 0+) = g(x) , −∞ < x < +∞ ; u(∓∞, t) = 0 , t > 0 ; (3.6a) b) Signalling problem u(x, 0+) = 0 , x > 0 ; u(0+, t) = h(t) , u(+∞, t) = 0 , t > 0 . (3.6b) However, if 1 < α ≤ 2 , the presence in (3.5) of the second order time derivative of the field variable requires to specify the initial value of the first order time derivative ut(x, 0 +) , since in this case two linearly independent solutions are to be determined. To ensure the continuous dependence of our solution on the parameter α also in the transition from α = 1− to α = 1+ , we agree to assume ut(x, 0 +) = 0 . We recognize that our fractional diffusion equation (3.1), when subject to the conditions (3.6), is equivalent to the integro-differential equation u(x, t) = g(x) + (t − τ)α−1 dτ , (3.7) where 0 < α ≤ 2 . Such integro-differential equation has been investigated by several authors, including Schneider & Wyss (1989), Fujita (1990), Prüss (1993) and Engler (1997). In view of our subsequent analysis we find it convenient to put , 0 < ν < 1 . (3.8) In fact the analysis of the time-fractional diffusion equation turns out to be easier if we adopt as a key parameter the half of the order of the time-fractional derivative. In future we shall provide the symbol α with other relevant meanings, as the index of stability of a stable probability distribution or the order of the space derivative in the space-fractional diffusion equation. Henceforth, we agree to insert the parameter ν in the field variable, i.e. u = u(x, t; ν) . By denoting the Green functions of the Cauchy and Signalling problems by Gc(x, t; ν) and Gs(x, t; ν) , respectively, the solutions of the two basic problems are obtained by a space or time convolution, u(x, t; ν) =∫ +∞ −∞ Gc(ξ, t; ν) g(x−ξ) dξ , u(x, t; ν) = 0 Gs(x, τ ; ν)h(t−τ) dτ , respectively. It should be noted that Gc(x, t; ν) = Gc(|x|, t; ν) , since the Green function turns out to be an even function of x . In the following two sections we shall compute the two fundamental solutions with the same techniques (based on Fourier and Laplace transforms) used for the standard diffusion equation and we shall provide their interpretation in terms of probability distributions. Most of the presented results are based on the papers by Mainardi (1994), (1995), (1996), (1997) and by Mainardi & Tomirotti (1995), (1997). 4 The Cauchy problem for the time-fractional diffusion equation For the fractional diffusion equation (3.1) subject to (3.6a) the application of the Fourier transform leads to the ordinary differential equation of order α = 2ν , + κ2 D û(κ, t; ν) = 0 , û(κ, 0+; ν) = ĝ(κ) , (4.1) Using the results of Appendix A, see (A.22-30), the transformed solution is û(κ, t; ν) = ĝ(κ)E2ν −κ2 D t2ν , (4.2) where E2ν(·) denotes the Mittag-Leffler function of order 2ν , and conse- quently for the Green function we have Gc(x, t; ν) = Gc(|x|, t; ν) ÷ Ĝc(k, t; ν) = E2ν −κ2D t2ν . (4.3) Since the Green function is a real and even function of x, its (exponential) Fourier transform can be expressed in terms of the cosine Fourier transform and thus is related to its spatial Laplace transform as follows Ĝc(k, t; ν) = 2 Gc(x, t; ν) cos κx dx = G̃c(s, t; ν) s=+ik + G̃c(s, t; ν) s=−ik (4.4) Indeed, a split occurs also in (4.3) according to the duplication formula for the Mittag-Leffler function, see (A.26), Ĝc(k, t; ν) = E2ν(−κ2 D t2ν) = [Eν(+iκ D tν) + Eν(−iκ D tν)]/2 . (4.5) When ν 6= 1/2 the inversion of the Fourier transform in (4.5) cannot be obtained by using a standard table of Fourier transform pairs; however, for any ν ∈ (0, 1) such inversion can be achieved by appealing to the Laplace transform pair (A.37) with r = |x| , and s = ±iκ . In fact, taking into account the scaling property of the Laplace transform, we obtain from (4.5) and (A.37) Gc(|x|, t; ν) = ( |x|√ , (4.6) where M(ζ; ν) is the special function of Wright type, defined by (A.31-33), , (4.7) the similarity variable. We note the identity |x| Gc(|x|, t; ν) = M(ζ; ν) , (4.8) which generalizes to the time-fractional diffusion equation the identity (2.7) of the standard diffusion equation. Since 0 M(ζ; ν) dζ = 1 , see (A.40), the function M(ζ; ν) is the normalized auxiliary function of the fractional diffusion equation. We note that for the time-fractional diffusion equation the fundamental solution of the Cauchy problem is still a bilateral symmetric pdf in x (with two branches, for x > 0 and x < 0 , obtained one from the other by reflection), but is no longer of Gaussian type if ν 6= 1/2 . In fact, for large |x| each branch exhibits an exponential decay in the ”stretched” variable |x|1/(1−ν) as can be derived from the asymptotic representation (A.36) of the auxiliary function M(·; ν) . In fact, by using (4.7-8) and (A.36), we obtain Gc(x, t; ν) ∼ a∗(t) |x|(ν−1/2)/(1−ν) exp −b∗(t)|x|1/(1−ν) , (4.9) as |x| → ∞ , where a∗(t) and b∗(t) are certain positive functions of time. Furthermore, the exponential decay in x provided by (4.9) ensures that all the absolute moments of positive order of Gc(x, t; ν) are finite. In particular, using (4.8) and (A.39) it turns out that the moments (of even order) are x2n Gc(x, t; ν) dx = Γ(2n + 1) Γ(2νn + 1) (Dt2ν)n , n = 0 , 1 , 2 , . . . (4.10) The formula (4.10) provides a generalization of the corresponding formula (2.11) valid for the standard diffusion equation, ν = 1/2 . Furthermore, we recognize that the variance associated to the pdf is now proportional to Dt2ν , which for ν 6= 1/2 implies a phenomenon of anomalous diffusion. According to a usual terminology in statistical mechanics, the anomalous diffusion is said to be slow if 0 < ν < 1/2 and fast if 1/2 < ν < 1 . In Figure 1, as an example, we compare versus |x| , at fixed t , the fundamental solutions of the Cauchy problem with different ν (ν = 1/4 , 1/2 , 3/4 ). We consider the range 0 ≤ |x| ≤ 4 and assume D = t = 1 . 0 1 2 3 4 Figure 1: The Cauchy problem for the time-fractional diffusion equation. The fundamental solutions versus |x| with a) ν = 1/4 , b) ν = 1/2 , c) ν = 3/4 . We note the different behaviour of the pdf in the cases of slow diffusion (ν = 1/4 ) and fast diffusion (ν = 3/4 ) with respect to the Gaussian behaviour of the standard diffusion (ν = 1/2). In the limiting cases ν = 0 and ν = 1 we have Gc(x, t; 0) = e−|x| , Gc(x, t; 1) = δ(x − D t) + δ(x + . (4.11) We also recognize from the appendix B that for 1/2 ≤ ν < 1 any branch of the fundamental solution is proportional to the corresponding positive branch of an extremal stable pdf with index of stability α = 1/ν , which exhibits an exponential decay at infinity. In fact, applying (B.29) with α = 1/ν and y = ζ = |x|/( Dtν) , from (4.7-8) we obtain Gc(|x|, t; ν) = |x|/( D tν) ; − (2 − 1/ν) · p1/ν (|x|; +1, 1, 0) , 1 < 1/ν ≤ 2 . (4.12) We also note that the stable distribution in (4.12) satisfies the condition p1/ν (x; +1, 1, 0) dx = ν , 1 < 1/ν ≤ 2 . (4.13) 5 The Signalling problem for the time-fractional diffusion equation For the fractional diffusion equation (3.1) subject to (3.6b) the application of the Laplace transform leads to the ordinary differential equation of order ũ(x, s; ν) , ũ(0+, s; ν) = h̃(s) , ũ(+∞, s; ν) = 0 . (5.1) Thus the transformed solution reads ũ(x, s; ν) = h̃(s) e−(x/ D) sν , (5.2) so for the Green function we have Gs(x, t; ν) ÷ G̃s(x, s; ν) = e−(x/ D) sν . (5.3) When ν 6= 1/2 the inversion of this Laplace transform cannot be obtained by looking in a standard table of Laplace transform pairs. Also here we appeal to a Laplace transform pair related to the Wright-type function M(ζ; ν). In fact, using (A.40) with r = t , and taking into account the scaling property of the Laplace transform, we obtain Gs(x, t; ν) = ν D t1+ν . (5.4) Introducing the similarity variable ζ = x/( Dtν) , we recognize the identity tGs(x, t; ν) = ν ζ M(ζ; ν) , (5.5) which is the counterpart for the Signalling problem of the identity (4.8) valid for the Cauchy problem. Comparing (5.5) with (4.8) we obtain the reciprocity relation between the two fundamental solutions of the time-fractional diffusion equation, in the common domain x > 0 , t > 0 , 2ν xGc(x, t; ν) = tGs(x, t; ν) . (5.6) The interpretation of Gs(x, t; ν) as a one-sided stable pdf in time is straightforward: in this respect we need to apply (B.28), with index of stability α = ν and variable y = ζ−1/ν = t ( D/x)1/ν , in (5.5). We obtain Gs(x, t; ν) = ; − ν  = pν (t; −1, 1, 0) . (5.7) In Figure 2, as an example, we compare versus t , at fixed x , the fundamental solutions of the Signalling problem with different ν (ν = 1/4 , 1/2 , 3/4 ). We consider the range 0 ≤ t ≤ 3 and assume D = x = 1 . We note the different behaviour of the pdf in the cases of slow diffusion (ν = 1/4 ) and fast diffusion (ν = 3/4 ) with respect to the Lévy pdf for the standard diffusion (ν = 1/2). In the limiting cases ν = 0 , 1 , we have Gs(x, t; 0) = δ(t) , Gs(x, t; 1) = δ(t − x/ D) . (5.8) 0 1 2 3 Figure 2: The Signalling problem for the time-fractional diffusion equation. The fundamental solutions versus t with a) ν = 1/4 , b) ν = 1/2 , c) ν = 3/4 . 6 The Cauchy problem for the symmetric space- fractional diffusion equation The symmetric space-fractional diffusion equation is obtained from the classical diffusion equation by replacing the second-order space derivative by a symmetric space-fractional derivative (explained below) of order α with 0 < α ≤ 2 . In our notation we write this equation as ∂|x|α , u = u(x, t;α) , x ∈ R , t ∈ R+0 , 0 < α ≤ 2 , (6.1) where D is a positive coefficient with the dimensions Lα T−1 . The fundamental solution for the Cauchy problem, Gc(x, t;α) is the solution of (6.1), subject to the initial condition u(x, 0+;α) = δ(x) . The symmetric space-fractional derivative of any order α > 0 of a sufficiently well-behaved function φ(x) , x ∈ R , may be defined as the pseudo- differential operator characterized in its Fourier representation by d|x|α φ(x) ÷ −|κ|α φ̂(κ) , x , k ∈ R , α > 0 . (6.2) According to a usual terminology, −|κ|α is referred to as the symbol of our pseudo-differential operator, the symmetric space-fractional derivative, of order α . Here, we have adopted the notation introduced by Zaslavski, see e.g. Saichev & Zaslavski (1997). In order to properly introduce this kind of fractional derivative we need to consider a peculiar approach to fractional calculus different from the Riemann-Liouville one, already treated in Appendix A. This approach is indeed based on the so-called Riesz potentials (or integrals), that we prefer to consider later. At first, let us see how things become highly transparent by using an heuristic argument, originally due to Feller (1952). The idea is to start from the positive definite differential operator A := − ÷ κ2 = |κ|2 , (6.3) whose symbol is |κ|2 , and form positive powers of this operator as pseudo- differential operators by their action in the Fourier-image space, i.e. Aα/2 := = |κ|α α > 0 . (6.4) Thus the operator −Aα/2 can be interpreted as the required fractional derivative, i.e. Aα/2 ≡ − d d|x|α , α > 0 . (6.5) We note that the operator just defined must not be confused with a power of the first order differential operator d for which the symbol is −iκ . After the above considerations it is straightforward to obtain the Fourier image of the Green function of the Cauchy problem for the space-fractional diffusion equation. In fact, applying the Fourier transform to the equation (6.1), subject to the initial condition u(x, 0+;α) = δ(x) , and accounting for (6.2), we obtain Gc(x, t;α) = Gc(|x|, t;α) ÷ Ĝc(k, t;α) = e−D t |κ| , 0 < α ≤ 2 . (6.6) We easily recognize that the Fourier transform of the Green function corresponds to the canonic form of a symmetric stable distribution of index of stability α and scaling factor γ = (Dt)1/α , see (B.8). Therefore we have Gc(x, t;α) = pα(x; 0, γ, 0) , γ = (Dt)1/α . (6.7) For α = 1 and α = 2 we easily obtain the explicit expressions of the corresponding Green functions since in these cases they correspond to the Cauchy and Gauss distributions, Gc(x, t; 1) = x2 + (D t)2 , (6.8) see (B.5), and Gc(x, t; 2)) = 2/(4D t) , (6.9) in agreement with (2.6). We easily recognize that (D t)1/α (6.10) is the similarity variable for the space-fractional diffusion equation, in terms of which we can express the Green function for any α ∈ (0, 2] . Indeed, we recognize that Gc(x, t;α) = (D t)1/α qα(η; 0) , (6.11) where qα(η; 0) denotes the symmetric stable distribution of order α with Feller-type characteristic function, see (B.14-15). Now we can express the Green function using the Feller series expansions (B.21-22) with θ = 0 . We obtain: for 0 < α < 1 , qα(η; 0) = − Γ(nα + 1) , (6.12a) for 1 < α ≤ 2 , qα(η; 0) = (−1)m Γ[(2m + 1)/α] (2m)! η2m . (6.12b) In the limiting case α = 1 the above series reduce to geometrical series and therefore are no longer convergent in all of C . In particular, they represent the expansions of the function q1(η; 0) = 1/[π(1+η 2)] , convergent for η > 1 and 0 < η < 1 , respectively. We also note that for any α ∈ (0, 2] the functions qα(η; 0) exhibit at the origin the value qα(0; 0) = Γ(1/α)/(π α) , and at the queues, excluding the Gaussian case α = 2 , the algebraic asymptotic behaviour, as η → ∞ , qα(η; 0) ∼ Γ(α + 1) sin η−(α+1) , 0 < α < 2 . (6.13) In Figure 3, as an example, we compare versus x , at fixed t , the fundamental solutions of the Cauchy problem with different α (α = 1/2 , 1 , 3/2 , 2 ). We consider the range −6 ≤ x ≤ +6 and assume D = t = 1 . -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 Figure 3: The Cauchy problem for the simmetric space-fractional diffusion equation. The fundamental solutions versus x : plate a) α = 1/2 (continuous line), α = 1 (dashed line); plate b) α = 3/4 (continuous line), α = 2 (dashed line). Let us now express more properly our operator (6.4) (with symbol |κ|α) as inverse of a suitable integral operator Iα whose symbol is |κ|−α . This operator can be found in the approach by Marcel Riesz to Fractional Calculus, see e.g. Samko, Kilbas & Marichev (1987-1993) and Rubin (1996). We recall that for any α > 0 , α 6= 1 , 3 , 5 , . . . and for a sufficiently well- behaved function φ(x) , x ∈ R , the Riesz integral or Riesz potential Iα and its image in the Fourier domain read Iα φ(x) := 2Γ(α) cos(πα/2) |x − ξ|α−1 φ(ξ) dξ ÷ φ̂(κ) . (6.14) On its turn, the Riesz potential can be written in terms of two Weyl integrals Iα± according to Iα φ(x) = 2 cos(πα/2) Iα+φ(x) + I −φ(x) , (6.15) where  Iα+ φ(x) := (x − ξ)α−1 φ(ξ) dξ , Iα− φ(x) := (ξ − x)α−1 φ(ξ) dξ . (6.16) Then, at least in a formal way, the space-fractional derivative (6.2) turns out to be defined as the opposite of the (left) inverse of the Riesz fractional integral, i.e. d|x|α φ(x) := −I−α φ(x) = − 2 cos(πα/2) I−α+ φ(x) + I − φ(x) . (6.17) Notice that (6.14) and (6.17) become meaningless when α is an integer odd number. However, for our range of interest 0 < α ≤ 2 , the particular case α = 1 can be singled out since the corresponding Green function is already known, see (6.8). Thus, excluding the case α = 1 , our space-fractional diffusion equation (6.1) can be re-written, x ∈ R , t ∈ R+0 , as = −D I−α u , u = u(x, t;α) , 0 < α ≤ 2 , α 6= 1 , (6.18) where the operator I−α is defined by (6.16-17). Here, in order to evaluate the fundamental solution of the Cauchy problem, interpreted as a probability density, we propose a numerical approach, original as far as we know, based on a (symmetric) random walk model, discrete in space and time, see also Gorenflo & Mainardi (1998a), Gorenflo & Mainardi (1998b) and Gorenflo, De Fabritiis & Mainardi (1999). We shall see how things become highly transparent, in that we properly generalize the classical random-walk argument of the standard diffusion equation to our space-fractional diffusion equation (6.18). So doing we are in position to provide a numerical simulation of the related (symmetric) stable distributions in a way analogous to the standard one for the Gaussian law. The essential idea is to approximate the left inverse operators I−α± by the Grünwald-Letnikov scheme, on which the reader can inform himself in the treatises on fractional calculus, see e.g. Oldham & Spanier (1974), Samko, Kilbas & Marichev (1987-1993), Miller & Ross (1993), or in the recent review article by Gorenflo (1997). If h denotes a ”small” positive step-length, these approximating operators read ± φ(x) := (−1)k φ(x ∓ kh) . (6.19) Assume, for simplicity, D = 1 , and introduce grid points xj = j h with h > 0 , j ∈ Z , and time instances tn = n τ with τ > 0 , n ∈ N0 . Let there be given probabilities pj,k ≥ 0 of jumping from point xj at instant tn to point xk at instant tn+1 and define probabilities yj(tn) of the walker being at point xj at instant tn. Then, by yk(tn+1) = pj,k uj(tn) , pj,k = pj,k = 1 , (6.20) with pj,k = pk,j , a symmetric random walk (more precisely a symmetric random jump) model is described. With the approximation yj(tn) ≈ ∫ (xj+h/2) (xj−h/2) u(x, tn) dx ≈ hu(xj , tn) , (6.21) and introducing the ”scaling parameter” 2 | cos(απ/2)| , (6.22) we have solved yj(tn+1) − yj(tn) = − hI−α yj(tn) , (6.23) for yj(tn+1) . So we have proved to have a consistent (for h → 0) symmetric random walk approximation to (6.18) by taking i) for 0 < α < 1 , 0 < µ ≤ 1/2 , −α yj(tn) = µ + yj(tn) + hI − yj(tn) pj,j = 1 − 2µ , pj,j±k = µ )∣∣ , k ≥ 1 ; (6.24) ii) for 1 < α ≤ 2 , 0 < µ ≤ 1/(2α) ,   −α yj(tn) = µ + yj+1(tn) + hI − yj−1(tn) pj,j = 1 − 2µ α , pj,j±1 = µ pj,j±k = µ )∣∣∣ , k ≥ 2 . (6.25) We note that our random walk model is not only symmetric, but also homogeneous, the transition probabilities pj,j±k not depending on the index In the special case α = 2 we recover from (6.25) the well-known three-point approximation of the heat equation, because pj,j±k = 0 for k ≥ 2 . This means that for approximation of common diffusion only jumps of one step to the right or one to the left or jumps of width zero occur, whereas for 0 < α < 2 (α 6= 1) arbitrary large jumps occur with power-like decaying probability, as it turns out from the asymptotic analysis for the transition probabilities given in (6.24-25). In fact, as k → ∞ , one finds pj,j+k ∼ (τ/hα) Γ(α + 1) sin k−(α+1) , 0 < α < 2 . (6.26) This result thus provides the discrete counterpart of the asymptotic behaviour of the long power-law tails of the symmetric stable distributions, as foreseen by (6.13) when 0 < α < 2 . 7 Conclusions We have treated two generalizations of the standard, one-dimensional, diffusion equation, namely, the time-fractional diffusion equation and the symmetric space-fractional diffusion equation. For these equations we have derived the fundamental solutions using the transform methods of Fourier and Laplace, and exhibited their connections to extremal and symmetric stable probability densities, evolving on time or variable in space. For the symmetric space-fractional diffusion equation we have presented a stationary (in time), homogeneous (in space) symmetric random walk model, discrete in space and time, the step-lengths of the spatial grid and the time lapses between transitions properly scaled. In the limit of infinitesimally fine discretization this model (based on the Grünwald-Letnikov approximation to fractional derivatives) is consistent with the continuous diffusion process, i.e. convergent if interpreted as a difference scheme in the sense of numerical analysis2. From the mathematical viewpoint the field of such ”fractional” general- izations is fascinating as there several mathematical disciplines meet and come to a fruitful interplay: e.g. probability theory and stochastic processes, 2Further generalizations have been considered by us and our collaborators in other papers, in which we have given a derivation of discrete random walk models related to more general space-time fractional diffusion equations. For a comprehensive analysis, see Gorenflo et al. (2002). Readers interested to the fundamental solutions of these fractional diffusion equations are referred to the paper by Mainardi et al. (2001) where analytical expressions and numerical plots are found. integro-differential equations, transform theory, special functions, numerical analysis. As one may take from our References, one can observe that since some decades there is an ever growing interest in using the concepts of fractional calculus among physicists and economists. Among economists we like to refer the reader to a collection of papers on the topic of ”Fractional Differencing and Long Memory Processes”, edited by Baillie & King (1996). Appendix A: The Riemann-Liouville Fractional Calculus Fractional calculus is the field of mathematical analysis which deals with the investigation and applications of integrals and derivatives of arbitrary order. The term fractional is a misnomer, but it is retained following the prevailing use. This appendix is mostly based on the recent review by Gorenflo & Mainardi (1997). For more details on the classical treatment of fractional calculus the reader is referred to Erdélyi (1954), Oldham & Spanier (1974), Samko et al. (1987-1993) and Miller & Ross (1993). According to the Riemann-Liouville approach to fractional calculus, the notion of fractional Integral of order α (α > 0) is a natural consequence of the well known formula (usually attributed to Cauchy), that reduces the calculation of the n−fold primitive of a function f(t) to a single integral of convolution type. In our notation the Cauchy formula reads Jnf(t) := fn(t) = (n − 1)! (t − τ)n−1 f(τ) dτ , t > 0 , n ∈ N , (A.1) where N is the set of positive integers. From this definition we note that fn(t) vanishes at t = 0 with its derivatives of order 1, 2, . . . , n − 1 . For convention we require that f(t) and henceforth fn(t) be a causal function, i.e. identically vanishing for t < 0. In a natural way one is led to extend the above formula from positive integer values of the index to any positive real values by using the Gamma function. Indeed, noting that (n − 1)! = Γ(n) , and introducing the arbitrary positive real number α , one defines the Fractional Integral of order α > 0 : Jα f(t) := (t − τ)α−1 f(τ) dτ , t > 0 , α ∈ R+ , (A.2) where R+ is the set of positive real numbers. For complementation we define J0 := I (Identity operator), i.e. we mean J0 f(t) = f(t) . Furthermore, by Jαf(0+) we mean the limit (if it exists) of Jαf(t) for t → 0+ ; this limit may be infinite. We note the semigroup property JαJβ = Jα+β , α , β ≥ 0 , which implies the commutative property JβJα = JαJβ , and the effect of our operators Jα on the power functions Jαtγ = Γ(γ + 1) Γ(γ + 1 + α) tγ+α , α ≥ 0 , γ > −1 , t > 0 . (A.3) These properties are of course a natural generalization of those known when the order is a positive integer. Introducing the Laplace transform by the notation L {f(t)} :=∫∞ −st f(t) dt = f̃(s) , s ∈ C , and using the sign ÷ to denote a Laplace transform pair, i.e. f(t) ÷ f̃(s) , we note the following rule for the Laplace transform of the fractional integral, Jα f(t) ÷ f̃(s) , α ≥ 0 , (A.4) which is the generalization of the case with an n-fold repeated integral. After the notion of fractional integral, that of fractional derivative of order α (α > 0) becomes a natural requirement and one is attempted to substitute α with −α in the above formulas. However, this generalization needs some care in order to guarantee the convergence of the integrals and preserve the well known properties of the ordinary derivative of integer order. Denoting by Dn with n ∈ N , the operator of the derivative of order n , we first note that Dn Jn = I , Jn Dn 6= I , n ∈ N , i.e. Dn is left-inverse (and not right-inverse) to the corresponding integral operator Jn . In fact we easily recognize from (A.1) that Jn Dn f(t) = f(t) − f (k)(0+) , t > 0 . (A.5) As a consequence we expect that Dα is defined as left-inverse to Jα. For this purpose, introducing the positive integer m such that m − 1 < α ≤ m , one defines the Fractional Derivative of order α > 0 : Dα f(t) := Dm Jm−α f(t) , m − 1 < α ≤ m , m ∈ N , (A.6) namely Dα f(t)= Γ(m − α) (t − τ)α+1−m , m − 1 < α < m, f(t) , α = m. (A.6′) Defining for complementation D0 = J0 = I , then we easily recognize that Dα Jα = I , α ≥ 0 , and Dα tγ = Γ(γ + 1) Γ(γ + 1 − α) tγ−α , α ≥ 0 , γ > −1 , t > 0 . (A.7) Of course, these properties are a natural generalization of those known when the order is a positive integer. Note the remarkable fact that the fractional derivative Dα f is not zero for the constant function f(t) ≡ 1 if α 6∈ N . In fact, (A.7) with γ = 0 teaches us that Dα1 = Γ(1 − α) , α ≥ 0 , t > 0 . (A.8) This, of course, is ≡ 0 for α ∈ N, due to the poles of the gamma function in the points 0,−1,−2, . . .. We now observe that an alternative definition of fractional derivative, originally introduced by Caputo (1967) (1969) in the late sixties and adopted by Caputo and Mainardi (1971) in the framework of the theory of Linear Viscoelasticity, is Dα∗ f(t) := J m−α Dm f(t) m − 1 < α ≤ m , m ∈ N , (A.9) namely D ∗α f(t) = Γ(m − α) f (m)(τ) (t − τ)α+1−m dτ , m − 1 < α < m, f(t) , α = m. (A.9′) This definition is of course more restrictive than (A.6), in that requires the absolute integrability of the derivative of order m. Whenever we use the operator Dα∗ we (tacitly) assume that this condition is met. We easily recognize that in general Dα f(t) := Dm Jm−α f(t) 6= Jm−α Dm f(t) := Dα∗ f(t) , (A.10) unless the function f(t) along with its first m − 1 derivatives vanishes at t = 0+. In fact, assuming that the passage of the m-derivative under the integral is legitimate, one recognizes that, for m − 1 < α < m and t > 0 , Dα f(t) = Dα∗ f(t) + Γ(k − α + 1) f (k)(0+) , (A.11) and therefore, recalling the fractional derivative of the power functions (A.7), f(t) − f (k)(0+) = Dα∗ f(t) . (A.12) The alternative definition (A.9) for the fractional derivative thus incorpo- rates the initial values of the function and of its integer derivatives of lower order. The subtraction of the Taylor polynomial of degree m − 1 at t = 0+ from f(t) means a sort of regularization of the fractional derivative. In particular, according to this definition, the relevant property for which the fractional derivative of a constant is still zero can be easily recognized, i.e. Dα∗ 1 ≡ 0 , α > 0 . (A.13) We now explore the most relevant differences between the two fractional derivatives (A.6) and (A.9). We agree to denote (A.9) as the Caputo fractional derivative to distinguish it from the standard Riemann-Liouville fractional derivative (A.6). We observe, again by looking at (A.7), that Dαtα−1 ≡ 0 , α > 0 , t > 0 . From above we thus recognize the following statements about functions which for t > 0 admit the same fractional derivative of order α , with m − 1 < α ≤ m , m ∈ N , Dα f(t) = Dα g(t) ⇐⇒ f(t) = g(t) + α−j , (A.14) Dα∗ f(t) = D ∗ g(t) ⇐⇒ f(t) = g(t) + m−j . (A.15) In these formulas the coefficients cj are arbitrary constants. For the two definitions we also note a difference with respect to the formal limit as α → (m − 1)+ ; from (A.6) and (A.9) we obtain respectively, Dα f(t) → Dm J f(t) = Dm−1 f(t) ; (A.16) Dα∗ f(t) → J Dm f(t) = Dm−1 f(t) − f (m−1)(0+) . (A.17) We now consider the Laplace transform of the two fractional derivatives. For the standard fractional derivative Dα the Laplace transform, assumed to exist, requires the knowledge of the (bounded) initial values of the fractional integral Jm−α and of its integer derivatives of order k = 1, 2, . . . ,m−1 . The corresponding rule reads, in our notation, Dα f(t) ÷ sα f̃(s) − Dk J (m−α) f(0+) sm−1−k , (A.18) where m − 1 < α ≤ m . The Caputo fractional derivative appears more suitable to be treated by the Laplace transform technique in that it requires the knowledge of the (bounded) initial values of the function and of its integer derivatives of order k = 1, 2, . . . ,m− 1 , in analogy with the case when α = m . In fact, by using (A.4) and noting that Jα Dα∗ f(t) = J α Jm−α Dm f(t) = Jm Dm f(t) = f(t) − f (k)(0+) (A.19) we easily prove the following rule for the Laplace transform, Dα∗ f(t) ÷ sα f̃(s) − f (k)(0+) sα−1−k , m − 1 < α ≤ m . (A.20) Indeed, the result (A.20), first stated by Caputo (1969) by using the Fubini-Tonelli theorem, appears as the most ”natural” generalization of the corresponding result well known for α = m . Gorenflo and Mainardi (1997) have pointed out the major utility of the Caputo fractional derivative in the treatment of differential equations of fractional order for physical applications. In fact, in physical problems, the initial conditions are usually expressed in terms of a given number of bounded values assumed by the field variable and its derivatives of integer order, no matter if the governing evolution equation may be a generic integro-differential equation and therefore, in particular, a fractional differential equation3. We now analyze the most simple differential equations of fractional order, including those which, by means of fractional derivatives, generalize the well- known ordinary differential equations related to relaxation and oscillation 3We note that the Caputo fractional derivative was so named after the book by Podlubny (1999). It coincides with that introduced, independently and a few later, by Dzherbashyan and Nersesyan (1968) as a regularization of the Riemann-Liouville fractional derivative. Nowadays, some Authors refer to it as the Caputo-Dzherbashyan fractional derivative. The prominent role of this fractional derivative in treating initial value problems was recognized in interesting papers by Kochubei (1989), (1990). phenomena. Generally speaking, we consider the following differential equation of fractional order α > 0 , Dα∗ u(t) = D u(t) − u(k)(0+) = −u(t) + q(t) , t > 0 , (A.21) where u = u(t) is the field variable and q(t) is a given function. Here m is a positive integer uniquely defined by m − 1 < α ≤ m , which provides the number of the prescribed initial values u(k)(0+) = ck , k = 0, 1, 2, . . . ,m−1 . Implicit in the form of (A.21) is our desire to obtain solutions u(t) for which the u(k)(t) are continuous. In particular, the cases of fractional relaxation and fractional oscillation are obtained for 0 < α < 1 and 1 < α < 2 , respectively The application of the Laplace transform through the Caputo formula (A.20) yields ũ(s) = sα−k−1 sα + 1 sα + 1 q̃(s) . (A.22) Now, in order to obtain the Laplace inversion of (A.22), we need to recall the Mittag-Leffler function of order α > 0 , Eα(z) . This function, so named from the great Swedish mathematician who introduced it at the beginning of this century, is defined by the following series and integral representation, valid in the whole complex plane, Eα(z) = Γ(αn + 1) σα−1 e σ σα − z dσ , α > 0 . (A.23) Here Ha denotes the Hankel path, i.e. a loop which starts and ends at −∞ and encircles the circular disk |σ| ≤ |z|1/α in the positive sense. It turns out that Eα(z) is an entire function of order ρ = 1/α and type 1 . The Mittag-Leffler function provides a simple generalization of the expo- nential function, to which it reduces for α = 1 . Particular cases from which elementary functions are recovered, are = cosh z , E2 = cos z , z ∈ C , (A.24) E1/2(±z1/2) = ez 1 + erf (±z1/2) = ez erfc (∓z1/2) , z ∈ C , (A.25) where erf (erfc) denotes the (complementary) error function. defined as erf (z) := du , erfc (z) := 1 − erf (z) , z ∈ C . A noteworthy property of the Mittag-Leffler function is based on the following duplication formula Eα(z) = Eα/2(+z 1/2) + Eα/2(−z1/2) . (A.26) In (A.25-26) we agree to denote by z1/2 the main branch of the complex root of z . The Mittag-Leffler function is connected to the Laplace integral through the equation ∫ ∞ e−u Eα (u α z) du = 1 − z α > 0 . (A.27) The integral at the L.H.S. was evaluated by Mittag-Leffler who showed that the region of its convergence contains the unit circle and is bounded by the line Re z1/α = 1 . The above integral is fundamental in the evaluation of the Laplace transform of Eα (−λ tα) with α > 0 and λ ∈ C . In fact, putting in (A.27) u = st and uα z = −λ tα with t ≥ 0 and λ ∈ C , we get the Laplace transform pair Eα (−λ tα) ÷ sα + λ , Re s > |λ|1/α . (A.28) Then, using (A.28), we put for k = 0, 1, . . . ,m − 1 , uk(t) := J keα(t) ÷ sα−k−1 sα + 1 , eα(t) := Eα(−tα) , (A.29) and, from inversion of the Laplace transforms in (A.22), we find u(t) = ck uk(t) − q(t − τ)u′0(τ) dτ . (A.30) In particular, the formula (A.30) encompasses the solutions for α = 1 , 2 , since e1(t) = exp(−t) , e2(t) = cos t . When α is not integer, namely for m − 1 < α < m , we note that m − 1 represents the integer part of α (usually denoted by [α]) and m the number of initial conditions necessary and sufficient to ensure the uniqueness of the solution u(t). Thus the m functions uk(t) = J keα(t) with k = 0, 1, . . . ,m−1 represent those particular solutions of the homogeneous equation which satisfy the initial conditions +) = δk h , h, k = 0, 1, . . . ,m − 1 , and therefore they represent the fundamental solutions of the fractional equation (A.21), in analogy with the case α = m . Furthermore, the function uδ(t) = −u′0(t) = −e′α(t) represents the impulse-response solution. The Mittag-Leffler function of order less than one turns out to be related through the Laplace integral to another special function of Wright type, denoted by M(z, ν) with 0 < ν < 1 , following the notation introduced by Mainardi (1994, 1995). Since this function turns out to be relevant in the general framework of fractional calculus with special regard to stable probability distributions, we are going to summarize its basing properties. For more details on this function, see Mainardi (1997), Appendix A. Let us first recall the more general Wright function Wλ,µ(z) , z ∈ C , with λ > −1 and µ > 0 . This function, so named from the British mathematician who introduced it between 1933 and 1941, is defined by the following series and integral representation, valid in the whole complex plane, Wλ,µ(z) = n! Γ(λn + µ) eσ + zσ −λ dσ , (A.31) where Ha denotes the Hankel path. It is possible to prove that the Wright function is entire of order 1/(1+λ) , hence of exponential type if λ ≥ 0 . The case λ = 0 is trivial since W0,µ(z) = e z/Γ(µ) . The case λ = −ν , µ = 1 − ν with 0 < ν < 1 provides the function M(z, ν) of special interest for us. Specifically, we have M(z; ν) := W−ν,1−ν(−z) = W−ν,0(−z) , 0 < ν < 1 , (A.32) and therefore from (A.31-32) M(z; ν) = (−z)n−1 (n − 1)! Γ(ν n) sin (ν n π) eσ − zσ , 0 < ν < 1 . (A.33) In the series representation we have used the reflection formula for the Gamma function, Γ(x) Γ(1−x) = π/ sin πx . Explicit expressions of M(z; ν) in terms of simpler known functions are expected in particular cases when ν is a rational number. Relevant cases are ν = 1/2 , 1/3 for which M(z; 1/2) = − z2/4 , (A.34) M(z; 1/3) = 32/3 Ai z/31/3 , (A.35) where Ai denotes the Airy function. When the argument is real and positive, i.e. z = r > 0 , the existence of the Laplace transform of M(r; ν) is ensured by the asymptotic behaviour, as derived by Mainardi & Tomirotti (1995), as r → +∞ , M(r/ν; ν) ∼ a(ν) r(ν − 1/2)/(1 − ν) exp −b(ν) r1/(1 − ν) , (A.36) where a(ν) = 1/ 2π (1 − ν) , b(ν) = (1 − ν)/ν . It is an instructive exercise to derive the Laplace transform by interchanging the Laplace integral with the Hankel integral in (A.33) and recalling the integral representation (A.23) of the Mittag-Leffler function. We obtain the Laplace transform pair M(r; ν) ÷ Eν(−s) , 0 < ν < 1 . (A.37) For ν = 1/2 , (A.37) with (A.25) and (A.34) provides the result, see e.g. Doetsch (1974), M(r; 1/2) := − r2/4 ÷ E1/2(−s) := exp erfc (s) . (A.38) It would be noted that, since M(r, ν) is not of exponential order, transforming term-by-term the Taylor series of M(r; ν) yields a series of negative powers of s , which represents the asymptotic expansion of Eν(−s) as s → ∞ in a certain sector around the real axis. We also note that (A.37) with (A.23) allows us to compute the moments of any real order δ ≥ 0 of M(r; ν) in the positive real axis. We obtain r δ M(r; ν) dr = Γ(δ + 1) Γ(νδ + 1) , δ ≥ 0 . (A.39) When δ is integer we note that the moments are provided by the derivatives of the Mittag-Leffler function in the origin, i.e. rn M(r; ν) dr = lim (−1)n Eν(−s) = Γ(n + 1) Γ(νn + 1) , (A.40) where n = 0, 1, 2, . . . . The normalization condition 0 M(r; ν) dr = Eν(0) = 1 is recovered for n = 0 . The relation with the Mittag-Leffler function stated in (A.40) can be extended to the moments of non integer order if we replace the ordinary derivative, of order n, with the corresponding fractional derivative, of order δ 6= n, in the Caputo sense. Another exercise on the function M concerns the inversion of the Laplace transform exp(−sν) , either by the complex integral formula or by the formal series method. We obtain the Laplace transform pair M (1/rν ; ν) ÷ exp (−sν) , 0 < ν < 1 . (A.41) For ν = 1/2 , (A.41) with (A.34) provides the known result, see e.g. Doetsch (1974), 2 r3/2 M(1/r1/2; 1/2) := π r3/2 exp [− 1/(4r)] ÷ exp −s1/2 . (A.42) We recall that a rigorous proof of (A.41) was formerly given by Pollard (1946), based on a formal result by Humbert (1945). The Laplace transform pair was also obtained by Mikusiński (1959) and, albeit unaware of the previous results, by Buchen & Mainardi (1975) in a formal way. Appendix B: The Stable Probability Distributions The stable distributions are a fascinating and fruitful area of research in probability theory; furthermore, nowadays, they provide valuable models in physics, astronomy, economics, and communication theory. The general class of stable distributions was introduced and given this name by the French mathematician Paul Lévy in the early 1920’s, see Lévy (1924, 1925). The inspiration for Lévy was the desire to generalize the celebrated Central Limit Theorem, according to which any probability distribution with finite variance belongs to the domain of attraction of the Gaussian distribution. Formerly, the topic attracted only moderate attention from the leading experts, though there were also enthusiasts, of whom the Russian mathematician Alexander Yakovlevich Khintchine should be mentioned first of all. The concept of stable distributions took full shape in 1937 with the appearance of Lévy’s monograph, see Lévy (1937-1954), soon followed by Khintchine’s monograph, see Khintchine (1938). The theory and properties of stable distributions are discussed in some classical books on probability theory including Gnedenko & Kolmogorov (1949-1954), Lukacs (1960-1970), Feller (1966-1971), Breiman (1968-1992), Chung (1968-1974) and Laha & Rohatgi (1979). Also treatises on fractals devote particular attention to stable distributions in view of their properties of scale invariance, see e.g. Mandelbrot (1982) and Takayasu (1990). Sets of tables and graphs have been provided by Mandelbrot & Zarnfaller (1959), Fama & Roll (1968), Bo’lshev & Al. (1968) and Holt & Crow (1973). Only recently, monographs devoted solely to stable distributions and related stochastic processes have been appeared, i.e. Zolotarev (1983-1986), Janicki & Weron (1994), Samorodnitsky & Taqqu (1994), Uchaikin & Zolotarev (1999). We now can cite the paper by Mainardi, Luchko & Pagnini (2001) where the reader can find (convergent and asymptotic) representations and plots of the symmetric and non-symmetric stable densities generated by fractional diffusion equations. Stable distributions have three exclusive properties, which can be briefly summarized stating that they 1) are invariant under addition, 2) possess their own domain of attraction, and 3) admit a canonic characteristic function. Let us now illustrate the above properties which, providing necessary and sufficient conditions, can be assumed as equivalent definitions for a stable distribution. We recall the basic results without proof. A random variable X is said to have a stable distribution P (x) = Prob {X ≤ x} if for any n ≥ 2 , there is a positive number cn and a real number dn such X1 + X2 + . . . + Xn = cn X + dn , (B.1) where X1,X2, . . . Xn denote mutually independent random variables with common distribution P (x) with X . Here the notation = denotes equality in distribution, i.e. means that the random variables on both sides have the same probability distribution. When mutually independent random variables have a common distribution [shared with a given random variable X], we also refer to them as independent, identically distributed (i.i.d) random variables [independent copies of X]. In general, the sum of i.i.d. random variables becomes a random variable with a distribution of different form. However, for independent random variables with a common stable distribution, the sum obeys to a distribution of the same type, which differs from the original one only for a scaling (cn) and possibly for a shift (dn). When in (B.1) the dn = 0 the distribution is called strictly stable. It is known, see Feller (1966-1971), that the norming constants in (B.1) are of the form cn = n 1/α with 0 < α ≤ 2 . (B.2) The parameter α is called the characteristic exponent or the index of stability of the stable distribution. We agree to use the notation X ∼ Pα(x) to denote that the random variable X has a stable probability distribution with characteristic exponent α . We simply refer to P (x) , p(x) := dP/dx (probability density function = pdf) and X as α-stable distribution, density, random variable, respectively. The definition (B.1) with the theorem (B.2) can be stated in an alternative version that needs only two i.i.d. random variables. see also Lukacs (1960- 1970). A random variable X is said to have a stable distribution if for any positive numbers A and B, there is a positive number C and a real number D such that AX1 + B X2 = C X + D , (B.3) where X1 and X2 are independent copies of X . Then there is a number α ∈ (0, 2] such that the number C in (B.3) satisfies Cα = Aα + Bα . For a strictly stable distribution (B.3) holds with D = 0 . This implies that all linear combinations of i.i.d. random variables obeying to a strictly stable distribution is a random variable with the same type of distribution. A stable distribution is called symmetric if the random variable −X has the same distribution. Of course, a symmetric stable distribution is necessarily strictly stable. Noteworthy examples of stable distributions are provided by the Gaussian (or normal) law (with α = 2) and by the Cauchy-Lorentz law (α = 1). The corresponding pdf are known to be pG(x;σ, µ) := e−(x − µ) 2/(2σ2) , x ∈ R , (B.4) where σ2 denotes the variance and µ the mean, and pC(x; γ, δ) := (x − δ)2 + γ2 , x ∈ R , (B.5) where γ denotes the semi-interquartile range and δ the ”shift”. Another (equivalent) definition states that stable distributions are the only distributions that can be obtained as limits of normalized sums of i.i.d. random variables. A random variable X is said to have a domain of attraction,i.e. if there is a sequence of i.i.d. random variables Y1, Y2, . . . and sequences of positive numbers {γn} and real numbers {δn}, such that Y1 + Y2 + . . . Yn d⇒X . (B.6) The notation d⇒ denotes convergence in distribution. It is clear that the previous definition (B.1) yields (B.6), e.g. , by taking the Yis to be independent and distributed like X . The converse is easy to show, see Gnedenko & Kolmogorov (1949-1954). Therefore we can alternatively state that a random variable X is said to have a stable distribution if it has a domain of attraction. When X is Gaussian and the Yis are i.i.d. with finite variance, then (B.6) is the statement of the ordinary Central Limit Theorem. The domain of attraction of X is said normal when γn = n 1/α ; in general, γn = n1/α h(n) where h(x) , x > 0 , is a slow varying function at infinity, that is, lim h(ux)/h(x) = 1 for all u > 0 , see Feller (1971). The function h(x) = log x , for example, is slowly varying at infinity. Another definition specifies the canonic form that the characteristic function (cf) of a stable distribution of index α must have. Recalling that the cf is the Fourier transform of the pdf , we use the notation p̂α(κ) := 〈exp (iκX)〉 ÷ pα(x) . We first note that a stable distribution is also infinitely divisible, i.e. for every positive integer n its cf can be expressed as the nth power of some cf . In fact, using the characteristic function, the relation (B.1) is transformed into [p̂α(κ)] n = p̂α(cn κ) e idnκ . (B.7) The functional equation (B.7) can be solved completely and the solution is known to be p̂α(κ;β, γ, δ) = exp {iδκ − γα |κ|α [1 + i (sign κ)β ω(|κ|, α)]} , (B.8) where ω(|κ|, α) = tan (α π/2) , if α 6= 1 , −(2/π) log |κ| , if α = 1 . (B.9) Consequently a random variable X is said to have a stable distribution if there are four real parameters α, β, γ, δ with 0 < α ≤ 2 , −1 ≤ β ≤ +1 , γ > 0 , such that its characteristic function has the canonic form (B.8-9). Then we write pα(x;β, γ, δ)÷ p̂α(κ;β, γ, δ) and X ∼ Pα(x;β, γ, δ) , so partly following the notation of Holt & Crow (1973) and Samorodnitsky & Taqqu (1994). We note in (B.8-9) that β appears with different signs for α 6= 1 and α = 1 . This minor point has been the source of great confusion in the literature, see Hall (1980) for a discussion. The presence of the logarithm for α = 1 is the source of many difficulties, so this case has often to be treated separately. The cf (B.8-9) turns out to be a useful tool for studying α-stable distri- butions and for providing an interpretation of the additional parameters, β (skewness parameter), γ (scale parameter) and δ (shift parameter), see Samorodnitsky & Taqqu (1994). When α = 2 the cf refers to the Gaussian distribution with variance σ2 = 2 γ2 and mean µ = δ ; in this case the value of the skewness parameter β is not specified because tan π = 0 , and one conventionally takes β = 0 . One easily recognizes that a stable distribution is symmetric if and only if β = δ = 0 and is symmetric about δ if and only if β = 0 . Stable distributions with extremal values of the skewness parameter are called extremal. One can prove that all the extremal stable distributions with 0 < α < 1 are one-sided, the support being R+0 if β = −1 , and R 0 if β = +1 . For the stable distributions Pα(x;β, γ, δ) we now consider the asymptotic behaviour of the tail probabilities, T+(λ) := Prob {X > λ} and T−(λ) := Prob {X < −λ} , as λ → ∞ . For the Gaussian case α = 2 the result is well known, see e.g. Feller (1957), α = 2 : T±(λ) ∼ 1 2/(4γ2) , λ → ∞ . (B.10) Because of the above exponential decay all the moments of the corresponding pdf turn out to be finite, which is an exclusive property of this stable distribution. For all the other stable distributions the singularity of the characteristic function in the origin is responsible for the algebraic decay of the tail probabilities as indicated below, see e.g. Samorodnitsky & Taqqu (1994), 0 < α < 2 : lim λα T±(λ) = Cα γ α (1 ∓ β)/2 , (B.11) where x−α sin x dx 1 − α Γ(2 − a) cos (απ/2) , if α 6= 1 , 2/π , if α = 1 . (B.12) We note that for extremal distributions (β = ±1) the above algebraic decay holds true only for one tail, the left one if β = +1 , the right one if β = −1 . The other tail is either identically zero if 0 < α < 1 (the distribution is one-sided !), or exhibits an exponential decay if 1 ≤ α < 2 . Because of the algebraic decay we recognize that 0 < α < 2 : |x|>λ pα(x;β, γ, δ) dx = O(λ −α) , (B.13) so the absolute moments of a stable non-Gaussian pdf turn out to be finite if their order ν is 0 ≤ ν < α and infinite if ν ≥ α . We are now convinced that the Gaussian distribution is the unique stable distribution with finite variance. Furthermore, when α ≤ 1 , the first absolute moment 〈|X|〉 is infinite as well, so we need to use the median to characterize the expected value. There is however a fundamental property shared by all the stable distributions that we like to point out: for any α the stable pdf are unimodal and indeed bell-shaped, i.e. their n-th derivative has exactly n zeros, see Gawronski (1964). We now come back to the cf of a stable distribution, in order to provide for α 6= 1 and δ = 0 a simpler canonic form which allow us to derive convergent and asymptotic power series for the corresponding pdf . We first note that the two parameters γ and δ in (B.8), being related to a scale transformation and a translation, are not so essential since they do not change the shape of distributions. If we take γ = 1 and δ = 0 , we obtain the so-called standardized form of the stable distribution and X ∼ Pα(x;β, 1, 0) is referred to as the α-stable standardized random variable. Furthermore, we can choose the scale parameter γ in such a way to get from (B.8-9) the simplified canonic form used by Feller (1952, 1966-1971) and Takayasu (1990) for strictly stable distributions (δ = 0) with α 6= 1 , which reads in an ad hoc notation, q̂α(κ; θ) := eiκ y pα(y; θ) dy = exp −|κ|α e±i θ π/2 , (B.14) where the symbol ± takes the sign of κ . This canonic form, that we refer to as the Feller canonic form, is derived from (B.8-9) if in addition to α 6= 1 and δ = 0 we require γα = cos , tan = β tan . (B.15) Here θ is the skewness parameter instead of β and its domain is restricted in the following region (depending on α) |θ| ≤ α , if 0 < α < 1 , 2 − α , if 1 < α < 2 . (B.16) Thus, when we use the Feller canonic form for strictly stable distributions with index α 6= 1 and skewness θ , we implicitly select the scale parameter γ (0 < γ ≤ 1), which is related to α , β and θ by (B.15). Specifically, the random variable Y ∼ Qα(y; θ) turns out to be related to the standardized random variable X ∼ Pα(x;β, 1, 0) by the following relations Y = X/γ , pα(x;β, 1, 0) = γ qα(y = γx; θ) , (B.17) with  γ = [cos (θπ/2)]1/α , θ = (2/π) arctan [β tan (απ/2)] , tan (θπ/2) tan (απ/2) (B.18) We recognize that qα(y, θ) = qα(−y,−θ) , so the symmetric stable distributions are obtained if and only if θ = 0 . We note that for the symmetric stable distributions we get the identity between the standardized and the Lévy canonic forms, since in (B.18) β = θ = 0 implies γ = 1 . A particular but noteworthy case is provided by p2(x; 0, 1, 0) = q2(y; 0) , corresponding to the Gaussian distribution with variance σ2 = 2 . The extremal stable distributions, corresponding to β = ±1 , are now obtained for θ = ±α if 0 < α < 1 , and for θ = ∓(2 − α) if 1 < α < 2 ; for them the scaling parameter turns out to be γ = [cos (|α|π/2)]1/α . It may be an instructive exercise to carry out the inversion of the Fourier transform when α = 1/2 and θ = −1/2 . In this case we obtain the analytical expression for the corresponding extremal stable pdf , known as the (one-sided) Lévy- Smirnov density, q1/2(y;−1/2) = y−3/2 e−1/(4y) , y ≥ 0 . (B.19) The standardized form for this distribution can be easily obtained from (B.19) using (B.17-18) with α = 1/2 and θ = −1/2 . We get γ = [cos (−π/4)]2 = 1/2 , β = −1 , so p1/2(x;−1, 1, 0) = q1/2(x/2;−1/2) = x−3/2 e−1/(2x) , (B.20) where x ≥ 0 , in agreement with Holt & Crow (1973) [§2.13, p. 147]. Feller (1952) has obtained from (B.14) the following representations by convergent power series for the stable distributions valid for y > 0 , with 0 < α < 1 (negative powers), qα(y; θ) = (−y−α)n Γ(nα + 1) (θ − α) , (B.21) 1 < α ≤ 2 (positive powers), qα(y; θ) = (−y)n Γ(n/α + 1) (θ − α) . (B.22) The values for y < 0 can be obtained from (B.21-22) using the identity qα(−y; θ) = qα(y;−θ) , y > 0 . As a consequence of the convergence in all of C of the series in (B.21-22) we recognize that the restrictions of the functions y qα(y; θ) on the two real semi-axis turn out to be equal to certain entire functions of argument 1/|y|α for 0 < α < 1 and argument |y| for 1 < α ≤ 2 . It has be shown, see e.g. Bergström (1952), Chao Chung-Jeh (1953), that the two series in (B.21-22) provide also the asymptotic (divergent) expansions to the stable pdf with the ranges of α interchanged from those of convergence. From (B.21-22) a relation between stable pdf with index α and 1/α can be derived as noted in Feller (1966-1971). Assuming 1/2 < α < 1 and y > 0 , we obtain q1/α(y −α; θ) = qα(y; θ ∗) , θ∗ = α(θ + 1) − 1 . (B.23) A quick check shows that θ∗ falls within the prescribed range, |θ∗| ≤ α , provided that |θ| ≤ 2 − 1/α . We now consider two particular cases of the Feller series (B.21-22), of particular interest for us, which turn out to be related to the entire function of Wright type, M(z; ν) with 0 < ν < 1 , reported in Appendix A. These cases correspond to the following extremal distributions Φ1(y) := qα(y;−α) , y > 0 , 0 < α < 1 , (B.24) Φ2(y) := qα(y;α − 2) , y > 0 , 1 < α ≤ 2 , (B.25) for which the Feller series (B.21-22) reduce to Φ1(y) = (−1)n−1 y−αn−1 Γ(nα + 1) sin (nπα) , y > 0 , (B.26) Φ2(y) = (−1)n−1 yn−1 Γ(n/α + 1) , y > 0 . (B.27) In fact, recalling the series representation of the general Wright function, Wλ,µ(z) with λ > −1 , µ > 0 , see (A.31), and the definition of the function M(z; ν) with 0 < ν < 1 , see (A.32-33), we recognize that Φ1(y) = W−α,0(−y−α) = M(y−α;α) , y > 0 , (B.28) Φ2(y) = W−1/α,0(−y) = M(y; 1/α) , y > 0 . (B.29) We would like to remark that the above relations with the Wright functions have been noted also by Engler (1997). It is worth to point out that, whereas Φ1(y) totally represents the one- sided stable pdf qα(y;−α) , 0 < α < 1 , with support in R+0 , Φ2(y) is the restriction on the positive axis of qα(y;α− 2) , 1 < α ≤ 2 , whose support is all of R . Since the function M(z; ν) turns out to be normalized in R+0 , see (A.39-40), we also note Φ1(y) dy = 1 ; Φ2(y) dy = 1/α . (B.30) Using the results (A.41) and (A.37) we can easily evaluate the Laplace transforms of Φ1(y) and Φ2(y) , respectively. We obtain L[Φ1(y)] = Φ̃1(s) = exp (−sα) , 0 < α < 1 , (B.31) L[Φ2(y)] = Φ̃2(s) = E1/α (−s) , 1 < α ≤ 2 , (B.32) where E1/α(·) denotes the Mittag-Leffler function of order 1/α , see (A.23). It is an instructive exercise to derive the asymptotic behaviours of Φ1(y) and Φ2(y) as y → 0+ and y → +∞ . By using the expressions (B.28−29) in terms of the function M and recalling the series and asymptotic representations of this function, see (A.33) and (A.36), we obtain Φ1(y) = y−(2−α)/[2(1−α)] e−c1 y −α/(1−α) , as y → 0+ , Γ(1 − α) y−α−1 [1 + O (y−α)] , as y → +∞ , (B.33) Φ2(y) = Γ(1 − 1/α) [1 + O (y)] , as y → 0+ , y(2−α)/[2(α−1)] e−c2 y α/(α−1) , as y → +∞ , (B.34) where c1 , c2 are positive constants depending on α . We note that the exponential decay is found for Φ1(y) as y → 0+ but as y → +∞ for Φ2(y) . Explicit expressions for stable pdf can be derived form those for the function M(z; ν) when ν = 1/2 and ν = 1/3 , given in Appendix A, see (A.34- 35). Of course the ν = 1/2 expression can be used to recover the well- known (symmetric) Gaussian distribution q2(y; 0) accounting for (B.29), and the (one-sided) Lévy distribution q1/2(y;−1/2), see (B.19), accounting for (B.28). The ν = 1/3 expression provides, accounting for (B.28), q1/3(y;−1/3) = 3−1/3 y−4/3 Ai (3y)−1/3 y−3/2 K1/3 (B.35) where Ai denotes the Airy function and K1/3 the modified Bessel function of the second kind of order 1/3 . The equivalence between the two expressions in (B.35) can be proved in view of the relation, see Abramowitz & Stegun (1965-1972) [(10.4.14)], Ai (z) = . (B.36) The case α = 1/3 has also been discussed by Zolotarev (1983-1986), who has quoted the corresponding expression of the pdf in terms of K1/3 . A general representation of all stable distributions (thus including the extremal distributions above considered) in terms of special functions has been only recently achieved by Schneider (1986). In his remarkable (but almost ignored) article, Schneider has established that all the stable distributions can be characterized in terms of a general class of special functions, the so-called Fox H functions, so named after Charles Fox (1961). For details on Fox H functions, see e.g. the books Mathai & Saxena (1978), Srivastava & Al. (1982) and the most recent paper by Kilbas and Saigo (1999). These functions are expressed in terms of special integrals in the complex-plane, the Mellin-Barnes integrals4. 4The names refer to the two authors, who in the first 1910’s developed the theory of these integrals using them for a complete integration of the hypergeometric differential equation. However, as pointed out in the the Bateman Project Handbook on High Transcendental Functions, see Erdelyi (1953), these integrals were first used by S. Pincherle in 1888. For a revisited analysis of the pioneering work of Pincherle (1853-1936, Professor of Mathematics at the University of Bologna from 1880 to 1928) we refer to the paper by Mainardi and Pagnini (2003). References [1] Abramowitz, M. and Stegun, I.A. (Editors) : Handbook of Mathematical Functions, Dover, New York 1965. [reprint, 1972] [2] Baillie, R.T. and King, M.L. (Editors) : Fractional Differencing and Long Memory Processes, Journal of Econometrics, 73 1-324 (1996). [3] Bergström, H. : On some expansions of stable distribution functions, Ark. Mat., 2 375-378 (1952). [4] Bol’shev, L.N., Zolotarev, V.M., Kedrova, E.S. and Rybinskaya, M.A. : Tables of cumulative functions of on-sided stable distributions, Theor. Probability Appl., 15 299-309 (1968). [5] Bouchaud, J.-P and Potters, M. : Theory of Financial Risk: from Sta- tistical Physics to Risk Management, Cambridge Univ. Press, Cambridge 2000. [English enlarged version of Théories des Risques Financiers, CEA, Aléa, Saclay 1997.] [6] Breiman, L. : Probability, SIAM, Philadelphia 1992. [The SIAM edition is an unabridged, corrected republication of the 1st edn, Addison-Wesley, Reading Mass. 1968] [7] P.W. Buchen, and Mainardi, F. : Asymptotic expansions for transient viscoelastic waves, J. de Mécanique, 14 597-608 (1975). [8] Caputo, M. : Linear models of dissipation whose Q is almost frequency independent, Part II., Geophys. J. R. Astr. Soc., 13 529-539 (1967). [9] Caputo, M.: Elasticità e Dissipazione, Zanichelli, Bologna 1969. [10] Caputo, M. and Mainardi, F. : Linear models of dissipation in anelastic solids, Rivista del Nuovo Cimento (Ser. II), 1 161-198 (1971). [11] Chung-Jeh, Chao : Explicit formula for the stable law of distribution, Acta Math. Sinica, 3 177-185 (1953). [in Chinese with English summary] [12] Chung, K.L. : A Course in Probability Theory, 2nd edn, Academic Press, New York 1974. [1st edn, Harcourt Brace Jowvanovich, 1968] [13] Dzherbashyan, M.M. and Nersesyan, A.B. : Fractional derivatives and the Cauchy problem for differential equations of fractional order. Izv. Acad. Nauk Armjanskvy SSR, Matematika, 3 3–29 (1968). [In Russian] [14] Engler, H. : Similarity solutions for a class of hyperbolic integrodiffer- ential equations, Differential Integral Eqns, 10 815-840 (1997). [15] Erdélyi, A. (Editor) : Higher Transcendental Functions, Bateman Project, McGraw-Hill, New York 1953; Vol. 1, Ch. 1, §1.19, p. 49. [16] Erdélyi, A. (Editor) : Higher Transcendental Functions, Bateman Project, McGraw-Hill, New York 1955; Vol. 3, Ch. 18, pp. 206-227. [17] Erdélyi, A. (Editor) : Tables of Integral Transforms, Bateman Project McGraw-Hill, New York 1954; Vol. 2, Ch. 13, pp. 181-212. [18] Fama, E. and Roll, R. : Some properties of symmetric stable distributions, J. Amer. Statist. Assoc., 63 817-836 (1968). [19] Feller, W. : On a generalization of Marcel Riesz’ potentials and the semi-groups generated by them, Meddelanden Lunds Universitets Matematiska Seminarium (Comm. Sém. Mathém. Université de Lund), Tome suppl. dédié a M. Riesz, Lund (1952), pp. 73-81. [20] Feller, W. : An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn, Wiley, New York 1968. [1st edn, 1957] [21] Feller, W. : An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn, Wiley, New York 1971. [1st edn. 1966] [22] Fox, C. : The G and H functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc., 98 395-429 (1961). [23] Fujita, Y. : Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math., 27 309-321, 797-804 (1990). [2 papers] [24] Gawronski, W. : On the bell-shape of stable distributions, Annals of Probability, 12 230-242 (1984). [25] Gnedenko, B.V. and Kolmogorov, A.N. : Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, Mass. 1954. [English Transl. from the Russian edition 1949, with notes by K.L. Chung, revised 1968] [26] Gorenflo, R. : Fractional calculus: some numerical methods, in Carpinteri, A. and Mainardi, F. (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, CISM Courses and Lectures # 378, Springer Verlag, Wien 1997, pp. 277-290. [Reprinted in www.fracalmo.org] [27] Gorenflo, R. and Mainardi, F. : Fractional calculus: integral and differential equations of fractional order, in Carpinteri, A. and Mainardi, F. (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, CISM Courses and Lectures # 378, Springer Verlag, Wien 1997, pp. 223- 276. [Reprinted in www.fracalmo.org] [28] Gorenflo, R. and Mainardi, F. : Fractional calculus and stable probability distributions, Archives of Mechanics, 50 377-388 (1998a). [29] Gorenflo, R. and Mainardi, F. : Random walk models for space- fractional diffusion processes, Fractional Calculus and Applied Analysis, 1 167-190 (1998b). [30] Gorenflo, R. De Fabritiis, G. and Mainardi, F. : Discrete random walk models for symmetric Lévy-Feller diffusion processes, Physica A, 269 79– 89 (1999). [31] Gorenflo, R., Mainardi, F., Moretti, D., Pagnini, G. and Paradisi, P. : Discrete random walk models for space-time fractional diffusion, Chemical Physics, 284 521-541 (2002). Special issue on Strange Kinetics Guest Editors: R. Hilfer, R. Metzler, A. Blumen, J. Klafter. [E-print arXiv:cond-mat/0702072] [32] Hall, P. : A comedy of errors: the canonical form for a stable characteristic function, Bull. London Math. Soc., 13 23-27 (1980). [33] Holt, D.R. and Crow, E.L. : Tables and graphs of the stable probability density functions, J. Res. Nat. Bureau Standards, 77B 143-198 (1973). [34] Humbert, P. : Nouvelles correspondances symboliques, Bull. Sci. Mathém. (Paris, II ser.), 69 121-129 (1945). [35] Janicki, A. and Weron, A. : Simulation and Chaotic Behavior of α- Stable Stochastic Processes, Marcel Dekker, New York 1994. [36] Khintchine, A.Y. : Limit Laws for Sums of Independent Variables, ONTI, Moscow 1938 [in Russian] [37] Kilbas, A.A. and Saigo, M. : On the H functions, Journal of Applied Mathematics and Stochastic Analysis, 12 191-204 (1999). [38] Kochubei, A.N. : A Cauchy problem for evolution equations of fractional order, Differential Equations, 25 967–974 (1989). [English translation from the Russian Journal Differentsial’nye Uravneniya] [39] Kochubei, A.N. : Fractional order diffusion, Differential Equations, 26 485–492 (1990). [English translation from the Russian Journal Differentsial’nye Uravneniya] [40] Laha, R.G. and Rohatgi, V.K. : Probability Theory, Wiley, New York 1979. [41] Lévy, P. : Théorie des erreurs. La Loi de Gauss et les lois exceptionelles, Bull. Soc. Math. France, 52 49-85 (1924). [42] Lévy, P. : Calcul des probabilités, Gauthier-Villars, Paris 1925: Part II, Chap. 6. [43] Lévy, P. : Théorie de l’addition des variables aléatoires, 2nd edn, Gauthier-Villars, Paris 1954. [1st edn, 1937] [44] Lukacs, E. : Characteristic Functions, 2nd edn, Griffin, London 1970. [1st edn, 1960] [45] Mainardi, F. : On the initial value problem for the fractional diffusion- wave equation, in S. Rionero and T. Ruggeri (Eds), Waves and Stability in Continuous Media, World Scientific, Singapore 1994, pp. 246-251. [46] Mainardi, F. : The time fractional diffusion-wave equation, Radiofisika, 38 20-36 (1995). [English Translation: Radiophysics & Quantum Electronics] [47] Mainardi, F. : Fractional relaxation-oscillation and fractional diffusion- wave phenomena, Chaos, Solitons & Fractals, 7 1461-1477 (1996). [48] Mainardi, F. : Fractional calculus: some basic problems in continuum and statistical mechanics, in Carpinteri, A. and Mainardi, F. (Eds), Fractals and Fractional Calculus in Continuum Mechanics, CISM Courses and Lectures # 378, Springer-Verlag, Wien 1997, pp. 291-348. [Reprinted in www.fracalmo.org] [49] Mainardi, F., Luchko, Yu. and Pagnini, G. : The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus and Applied Analysis, 4 153-192 (2001). [E-print arXiv:cond-mat/0702419] [50] Mainardi, F. and Pagnini, G. : Salvatore Pincherle, the pioneer of the Mellin-Barnes integrals, Jour. Computational and Applied Mathematics, 153 331-342 (2003). [51] Mainardi, F. and Tomirotti, M. : On a special function arising in the time fractional diffusion-wave equation, in P. Rusev, I. Dimovski and V. Kiryakova (Eds), Transform Methods and Special Functions, Sofia 1994, Science Culture Technology, Singapore 1995, pp. 171-183. [52] Mainardi, F. and Tomirotti, M. : Seismic pulse propagation with constant Q and stable probability distributions, Annali di Geofisica, 40 1311-1328 (1997). [53] Mandelbrot, B.B. and Zarnfaller, F. : Five place tables of certain stable distributions, Technical Report RC-421, IBM Thomas J. Watson Research Center, Yorktown Heights, New York, Dec 31, 1959. [54] Mandelbrot, B.B. : The Fractal Geometry of Nature, Freeman, San Francisco 1982. [55] Mandelbrot, B.B. : Fractals and Scaling in Finance, Springer-Verlag, New York 1997. [56] Mantegna, R.N. and Stanley, H.E. : An Introduction to Econophysics, Cambridge University Press, Cambridge 2000. [57] Mathai, A.M. and Saxena, R.K. : The H-function with Applications in Statistics and Other Disciplines, Wiley Eastern Ltd, New Delhi 1978. [58] Mikusiński, J. : On the function whose Laplace transform is exp (−sαλ) , Studia Math., 18 191-198 (1959). [59] Miller, K.S. and Ross, B. : An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York 1993. [60] Oldham, K.B. and Spanier, J. : The Fractional Calculus, Academic Press, New York 1974. [61] Pincherle, S. : Sulle funzioni ipergeometriche generalizzate, Atti R. Accademia Lincei, Rend. Cl. Sci. Fis. Mat. Nat. (4), 4, 694-700, 792-799 (1888). [62] Podlubny, I. : Fractional Differential Equations, Academic Press, San Diego 1999. [Mathematics in Science and Engineering, Vol. 198] [63] Pollard, H. : The representation of exp (−xλ) as a Laplace integral, Bull. Amer. Math. Soc., 52, 908-910 (1946). [64] Prüss, J. : Evolutionary Integral Equations and Applications, Birkhäuser, Basel 1993. [65] Rubin, B. : Fractional Integrals and Potentials, Pitman Monographs and Surveys in Pure and Appl. Mathematics # 82, Longman, London 1996. [66] Saichev, A.I and Zaslavsky, G.M. : Fractional kinetic equations: solutions and applications, Chaos, 7 753-764 (1997). [67] Samko, S.G., Kilbas, A.A. and Marichev, O.I. : Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam 1993. [Engl. Transl. from the Russian edition, 1987] [68] Samorodnitsky, G. and Taqqu, M.S. : Stable non-Gaussian Random Processes, Chapman & Hall, New York 1994. [69] Schneider, W.R. : Stable distributions: Fox function representation and generalization, in S. Albeverio, G. Casati and D. Merlini (Eds), Stochastic Processes in Classical and Quantum Systems, Lecture Notes in Physics # 262, Springer Verlag, Berlin 1986, 497-511. [70] Schneider, W.R. and Wyss, W. : Fractional diffusion and wave equations, J. Math. Phys., 30 134-144 (1989). [71] Srivastava, H.M., Gupta, K.C. and Goyal, S.P. : The H-Functions of One and Two Variables with Applications, South Asian Publ., New Delhi 1982. [72] H. Takayasu, H. : Fractals in the Physical Sciences, Manchester Univ. Press, Manchester and New York 1990. [73] Uchaikin, V.V. and Zolotarev, V.M. : Chance and Stability. Stable Distributions and their Applications, VSP, Utrecht 1999. [Series ”Modern Probability and Statistics”, No 3] [74] Zolotarev, V.M. : One-dimensional stable distributions, Amer. Math. Soc., Providence, R.I. 1986. [English Transl. from the Russian edition, 1982] Introduction The standard diffusion equation The time-fractional diffusion equation The Cauchy problem for the time-fractional diffusion equation The Signalling problem for the time-fractional diffusion equation The Cauchy problem for the symmetric space-fractional diffusion equation Conclusions

0704.0321

Fabrication of half metallicity in a ferromagnetic metal

Fabrication of half metallicity in a ferromagnetic metal Kalobaran Maiti∗ Department of Condensed Matter Physics and Materials’ Science, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai - 400 005, INDIA (Dated: August 15, 2021) We investigate the growth of half metallic phase in a ferromagnetic material using state-of-the-art full potential linearized augmented plane wave method. To address the issue, we have substituted Ti at the Ru-sites in SrRuO3, where SrRuO3 is a ferromagnetic material. Calculated results establish Ti4+ valence states (similar to SrTiO3), which was predicted experimentally. Thus, Ti substitution dilutes the Ru-O-Ru connectivity, which is manifested in the calculated results in the form of significant band narrowing leading to finite gap between t2g and eg bands. At 75% substitution, a large gap (> 2eV) appears at the Fermi level, ǫF in the up spin density of states, while the down spin states contributes at ǫF characterizing the system a half-metallic ferromagnet. The t2g − eg gap can be tailored judiciously by tuning Ti concentrations to minimize thermal effects, which is often the major bottleneck to achieve high spin polarization at elevated temperatures in other materials. This study, thus, provides a novel but simple way to fabricate half-metallicity in ferromagnetic materials, which are potential candidates for spin-based technology. PACS numbers: 85.70.Ay, 75.30.-m, 71.70.Ch, 71.15.Ap The search of half metallic ferromagnetic materials has seen an explosive growth in the recent times due to its potential technological applications. In these materials, the electronic density of states (DOS) at the Fermi level, ǫF corresponds to only one kind of spin, while the other spin density of states exhibit an energy gap at ǫF . Thus, in the polarized condition, electronic conduction strongly depends on the spin of the charge carriers; the material is insulating for one kind of spin and metallic for the other. This unique property makes them ideal candidates for the development of spin-based electronics. Various theoretical studies predicted half metallicity in Heusler alloys [1], double perovskites [2], manganates [3], CrO2 [4], graphene nanoribbons [5] etc. However, experimen- tal studies on very few materials such as manganates [3] and CrO2 [4], etc. exhibit half metallicity at low temper- atures. Thermal fluctuations often lead to a reduction in spin polarization at elevated temperatures [6] making it difficult for technological applications. In this study, we investigate the evolution of the elec- tronic density of states in SrRu1−xTixO3 as a function of x. SrRuO3 is a ferromagnetic metal with Curie tem- perature of 165 K. Spin polarization at ǫF is found to be negative in the ferromagnetic ground state [7, 8]. SrTiO3, on the other hand, is a band insulator. Various experi- mental studies [9, 10] suggest (4+) valence state of Ti in the intermediate compositions (similar to SrTiO3), which corresponds to 3d0 electronic configuration. Thus, in ad- dition to disorder effect, Ti substitution leads to a dilu- tion of Ru-O-Ru connectivity. Transport measurements in SrRu1−xTixO3 exhibit a range of novel phase tran- sitions involving disorder induced correlated metal, An- derson insulator, correlated insulator and band insulators [11] for different values of x. Using ab initio calculations, we find that Ti substitu- tion at Ru-sites in ferromagnetic SrRuO3 leads to half FIG. 1: (color online) Crystal structure of SrRu0.25Ti0.75O3. In order to obtain the structure of SrRuTiO3, we replaced Ti2 by Ru, and all the Ti and Ru sites are made equivalent. metallicity. Here, reduced Ru-O-Ru connectivity due to Ti-substitution leads to significant narrowing of Ru 4d band and thus, the up spin band moves below ǫF . In- terestingly, the energy gap between t2g and eg bands can be tuned by Ti-concentration. 75% substituted sample exhibits gap as high as 2 eV. Experimental realization of such method on different systems would provide a new direction in the search of HMFs for spin-based technol- The electronic density of states of SrRu1−xTixO3 for x = 0.0, 0.5, 0.75 and 1.0 were calculated using state- of-the-art full potential linearized augmented plane wave method (FLAPW) within the local spin density approxi- mations (LSDA) using WIEN2K software [12]. The crystal structure of SrTiO3 is cubic with the lattice constant, a = 3.905 Å. SrRuO3 possesses close to cubic structure with small orthorhombic distortion. This is manifested clearly by the similar density of states (DOS) of SrRuO3 in real structure vis-a-vis in the equivalent cubic structure [7]. Ti-substitution in SrRuO3 leads the system towards cu- http://arxiv.org/abs/0704.0321v1 �� �� �� � � � � !" /0 12 34 5 6 7 8 9 :; ?@A BCDE ^_` ab cd efg lmn op qr stu z{| } ~� ��� ��� �� �� ¡¢£¤¥¦ §¨©ª FIG. 2: (color online) (a) TDOS, (b) Ti 3d PDOS, (c) Ru 4d PDOS, (d) O 2p PDOS and (e) Sr 4d PDOS of SrRu1−xTixO3. Thin and thick solid lines represent DOS corresponding to x = 0.5 and 0.75, respectively. bic structure. Thus, we have considered cubic structure for all the calculations in this study. A typical unit cell for SrRu0.25Ti0.75O3 is shown in Fig. 1. There are 8 for- mula units in the unit cell constructed by doubling the lattice constant of SrTiO3. In order to preserve cubic symmetry, three types of Ti are considered occupying corners (Ti1), edge centers (Ti2) and face centered posi- tions (Ti3). The body centered position is occupied by Ru. There are three non-equivalent oxygens; O1 forms the octahedra around Ti1-sites, O2 forms the octahedra around Ru-sites and the rest of the oxygen positions are occupied by O3. Thus, the connectivity between Ru-sites occurs via Ru-O2 bondings. The muffin-tin radii (RMT ) for Sr, Ru, Ti and O were set to 1.16 Å 0.95 Å 0.95 Å and 0.74 Å respectively. The convergence for different calcu- lations were achieved considering 512 k points within the first Brillouin zone. The error bar for the energy conver- gence was set to < 0.25 meV per formula unit. In every case, the charge convergence was achieved to be less than 10−3 electronic charge. In Fig. 2, We show the total DOS calculated for SrRu1−xTixO3 (x = 0.5 and 0.75) and the partial DOS obtained by projecting the eigenstates onto the Ti 3d, Ru 4d, O 2p and Sr 4d states. The figure exhibits 5 distinctly separable features. The energy region -1.5 eV to -5 eV is primarily contributed by O 2p partial DOS with negligible contributions from other electronic states. Thus, these contributions are characterized due to the non-bonding O 2p states. Sr 4d partial DOS shown in Fig. 2(e) appear above 5 eV. The peak appears to shift towards higher energy with increasing x. This can be ·¸ ¹º »¼ ½¾ ¿ À Á  à ÄÅ ÑÒÓÔÕ ×ØÙÚÛ ÝÞ ßà áâãäå íîïðñ óô õö ÷øùúûü ýþÿ� � ��� �� �� ����� 89:;< FIG. 3: (color online) (a) TDOS, (b) Ti 3d and Ru 4d PDOS, (c) O 2p PDOS and (d) Sr 4d PDOS of SrTiO3 and SrRuO3. Dashed line represent Sr 4d PDOS rescaled by 20 times. understood by comparing the same in the end members, SrTiO3 and SrRuO3 as demonstrated in Fig. 3. Sr 4d states appear at much higher energies in SrTiO3 com- pared to that in SrRuO3. One reason for such a large shift may be related to the shift of the Fermi level to the top of the O 2p band in SrTiO3. However, the shift of Sr 4d band in the intermediate compositions, where the Fermi level is pinned by the occupancy of the Ru 4d band, indicates that the Madelung potential at Sr-sites increases with the increase in Ti concentrations. Ti 3d partial DOS appears 2 eV above the Fermi level. This clearly demonstrates that the occupancy of Ti 3d states is essentially zero and hence correspond to Ti4+ valency. Such valence states was predicted in the x-ray photoemission spectra [9]. This study provides evidence of such effect theoretically within the effective single par- ticle approach itself. The width of the Ti 3d t2g band is significantly small in x = 0.5 sample (∼ 0.65 eV), which increases to 1.5 eV in x = 0.75 sample and 2.5 eV at x = 1.0 (see Fig. 3). Ru 4d partial DOS exhibit three regions. The narrow and intense feature between the energy range -1.6 to 0.5 eV correspond to the electronic states having t2g sym- metry. The electronic states above 1.8 eV appears due to Ru 4d states having eg symmetry. Notably, the O 2p states also contribute in all the three energy regions. Thus, DOS appearing below -5 eV can be attributed to the Ru 4d - O 2p bonding states having a large O 2p character, and the energy region above -1.5 eV are the anti-bonding states having primarily Ru 4d character. Most interestingly, both the compounds exhibit metallic ground state. However, the t2g bandwidth, W reduces significantly with the increase in x. While W is close to 2.6 eV in SrRuO3, it is about 1.7 eV for x = 0.5 and 0.54 eV for x = 0.75. Such reduction in W is understand- able as Ti-substitution leads to a significant reduction in the hopping interaction strength due to the reduced degree of Ru-O-Ru connectivity. This is clearly evident in Fig. 1; if we assume homogeneous distribution of Ru and Ti atoms in the solid, all the RuO6 octahedra are separated by TiO6 octahedra at x = 0.5. At x = 0.75, the number of Ru-[O-Ti-O]-Ru connectivity reduces to half of that at x = 0.5. Subsequently, U/W (U = local Coulomb interactions strength) will increase significantly and presumably play a role in the transport properties in these compositions [11]. In order to understand the bonding of Ru 4d electronic states with various O 2p states, we compare the Ru 4d t2g and eg bands with the 2p bands corresponding to O1, O2 and O3 for x = 0.75 and 0.5 sample in Fig. 4(a) and 4(b), respectively. All the oxygens are equivalent in the x = 0.5 sample. The energy distribution of O2 2p partial DOS is almost identical in Fig. 4(a) to that observed in Ru 4d partial DOS. This is expected as the RuO6 octahedra is formed by O2 atoms only. The width of the O2 2p band is significantly larger than that of O1 and O3. The most interesting observation is that the t2g and eg bands are separated by a distinct energy gap. This gap is already visible in Ru 4d partial DOS of x = 0.5 sample in Fig. 4(b) and is absent in SrRuO3 as shown in Fig. 3 and in the literature as well [7, 13]. We calculate the crystal field splitting of the Ru 4d band by measuring the separation of the center of gravity of the Ru 4d t2g and eg bands as shown in Fig. 4 by closed circles in both the compositions. It is evident that crystal field splitting, ∆ remains almost the same (∼ 2.1 eV) in both the compositions and is very close to 2 eV found in SrRuO3. Thus, the large energy gap between the t2g and eg bands appears purely due to the band narrowing. Such effect has strong implication in the magnetic phase as described below. It is already well established that the magnetic ground state can be exactly described by these band structure calculations [7, 14, 15, 16]. Thus, we have calculated the ground state energies for ferromagnetic arrangement of moments of the constituents using local spin-density ap- proximations. Interestingly, the eigen energy for the fer- romagnetic ground state in x = 0.5 sample is 5.67 meV/fu lower than the lowest eigen energy for the non-magnetic solution. This is higher than 1.2 meV/fu observed in SrRuO3 in real structure and significantly smaller than 30.4 meV/fu observed in the equivalent cubic struc- ture of SrRuO3. This energy difference between the non-magnetic and magnetic solutions increases to 33.95 meV/fu in x = 0.75. All these results suggest that the stability of the ferromagnetic ground state increases with the decrease in the degree of charge delocalization of the CD EF GH I J K L M TU VW XY Z [ \ ] ^ bcd ef gh ij kl mn op qrs tu vwxyz{|} ²³´ µ¶ ·¸¹º»¼ ½¾¿À FIG. 4: (color online) Ru 4d partial DOS with t2g and eg symmetry are compared with the O 2p partial DOS in (a) SrRu0.25Ti0.75O3 and (b) SrRu0.5Ti0.5O3. ÇÈ É Ê Ë Ì ÍÎ Ï Ð Ñ Ò Ö× ØÙ ÚÛ Ü Ý Þ ß àá âã äå æ ç è é î ï ð ñ ò ó ô õ ùú ûü ýþ ÿ �� �� �� � �� � �� ���� �� �� '()* +,-. 23 4567 [\]^_ `abc ghij kl mnop qr st ���� ���� �� ���� ���� Energy (eV) ª«¬­ ®¯°± FIG. 5: (color online) Up and down spin density of stated corresponding to (a) Ru 4d in SrRu0.5Ti0.5O3, (b) O 2p in SrRu0.5Ti0.5O3, (c) Ru 4d in SrRu0.25Ti0.75O3, and (d) O 2p in SrRu0.25Ti0.75O3. This figure demonstrates that band narrowing in Ru 4d band leads to a gap in the up spin channel leading to half metallicity. valence electrons. The spin magnetic moment centered at Ru-sites is found to be about 0.6 µB in x = 0.5 sample. Inter- estingly, magnetic moment at the interstitial electronic states is significantly large (∼ 0.36 µB). The moment at the O sites is about 0.05 µB. The Ti sites also ex- hibit very small moment (∼ -0.03 µB). Thus the total magnetic moment of the solid becomes 1.24 µB per Ru- atom. This is very similar to that observed (1.2 µB) in SrRuO3. The magnetic moments increase significantly with the increase in x. The moments at Ru site becomes 0.88 µB in x = 0.75 sample. The moments of the intersti- tial states and 2p states at O2 sites also enhance to 0.66 µB and 0.066 µB, respectively. Thus, the total moment turns out to be 1.99 µB, which is very close to the spin only value of 2 µB corresponding to Ru 4t 2g electronic configuration. It is to note here that although the local moment of the highly extended 4d states is significantly smaller than the spin only value as opposed to the case in 3d transition metal oxides [15], Ru 4d moment induces a large degree of polarization in the interstitial and O 2p electrons. These results evidently suggest applicability of Stoner description to capture magnetic properties of these systems. In order to investigate the exchange splitting and the character of density of states in the vicinity of ǫF , we plot the spin-resolved DOS corresponding to Ru 4d and O 2p partial DOS in Fig. 5. In the x = 0.5 sample, both the up and down spin states contribute at ǫF and the exchange splitting is found to be about 0.47 eV. This is again very similar to the case in SrRuO3 [7]. The exchange splitting increases to 0.65 eV in x = 0.75 sample as shown in the figure. Interestingly, the up spin band moves significantly below ǫF and the contributions at ǫF appears only due to the down spin states indicating a half-metallic behav- ior. No contribution of the up spin states observed in the total density of states (not shown here). Considering the paucity of half-metallic materials for various tech- nological applications, achieving half metallicity in the ferromagnetic SrRuO3 by Ti-substitution is remarkable. It is believed that the half metallicity can be achieved via strong d − d hybridization in Heusler alloys involv- ing two transition metal elements in the compound [17]. In transition metal oxides, often doping of large amount of electrons or holes leads to a shift of the Fermi level towards the energy gap of one spin channel leading to half metallicity [3]. The primary difficulty to use these systems in technological applications is the loss of half metallicity at elevated temperatures, where thermal ex- citations leads to significant mixing of various spin chan- nels due to small energy gap at ǫF [6]. In the present case, mechanism to achieve half metallicity is simple and easily achievable experimentally. The most important aspect is that the energy gap between t2g and eg bands can be tailored judiciously by tuning the composition to minimize thermal effects. In summary, we investigate the possibility of fabricat- ing half metallicity by Ti-substitution at the Ru-sites in a ferromagnetic material, SrRuO3. The calculated re- sults using FLAPW method within the local spin density approximations reveal tetravalency of Ti in all the com- positions consistent with the experimental predictions. The Ru 4d band exhibit significant narrowing with the increase in Ti-substitution; the crystal field splitting re- mains almost the same across the whole series. Thus, an energy gap develops between the t2g and eg bands, which gradually grows with the increase in x. Conse- quently, the up spin density of states exhibit an energy gap at the Fermi level, while the down spin states still contribute leading to half metallicity. Most interestingly, the t2g − eg gap can be engineered by tuning x and thus spin mixing effects due to thermal excitations can be min- imized. This study thus provide a novel but simple way to fabricate half metallicity in ferromagnetic materials, which are potential candidates for spin based technol- ogy. Experimental realization of this method would help both chemists and physicists to cultivate new materials. In addition, this study demonstrates that effective sin- gle particle approaches provide a remarkable description of the electronic properties of these systems, which are predicted experimentally. ∗ Electronic mail: [email protected] [1] R.A. de Groot, F.M. Mueller, P.G. van Engen, and K.H.J. Buschow, Phys. Rev. Lett. 50, 2024-2027 (1983), [2] K.-I. Kobayashi, T. Kimura, H. Sawada, K. Terakura, and Y. Tokura, Nature 395, 677-680 (1998). [3] J.H. Park et al., Nature 392, 794-796 (1998). [4] R.S. Keizer, S.T.B. Goennenwein, T.M. Klapwijk, G. Miao, G. Xiao, and A. Gupta, Nature 439, 825-827 (2006). [5] Y.-W. Son, M.L. Cohen, and S.G. Louie, Nature 444, 347-349 (2006). [6] M. Ležaić, Ph. Mavropoulos, J. Enkovaara, G. Bihlmayer, and S. Blügel, Phys. Rev. Lett. 97, 026404 (2006). [7] K. Maiti, Phys. Rev. B 73, 235110 (2006). [8] D.C. Worledge and T.H. Geballe, Phys. Rev. Lett. 85, 5182 (2000). [9] J. Kim, J.-Y. Kim, B.-G. Park, and S.-J. Oh, Phys. Rev. B 73, 235109 (2006). [10] S. Ray, D.D. Sarma, and R. Vijayaraghavan, Phys. Rev. B 73, 165105 (2006). [11] K.W. Kim, J.S. Lee, T.W. Noh, S.R. Lee, and K. Char, Phys. Rev. B 71, 125104 (2005). [12] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, and J. Luitz, WIEN2k, An Augmented Plane Wave + Lo- cal Orbitals Program for Calculating Crystal Properties (Karlheinz Schwarz, Techn. Universität Wien, Austria), 2001. ISBN 3-9501031-1-2. [13] D.J. Singh, J. Appl. Phys. 79, 4818-4820 (1996). [14] N. Hamada, H. Sawada, I. Solovyev, and K. Terakura, Physica B 237-238, 11-13 (1997). [15] D.D. Sarma, N. Shanthi, S.R. Barman, N. Hamada, H. Sawada, and K. Terakura, Phys. Rev. Lett. 75, 1126 (1995). [16] K. Maiti, Phys. Rev. B 73, 115119 (2006). [17] I. Galanakis, P.H. Dederichs, and N. Papanikolaou, Phys. Rev. B 66, 134428 (2002); ibid, 66, 174429 (2002).

0704.0322

Emergence of spatiotemporal chaos driven by far-field breakup of spiral waves in the plankton ecological systems

Emergence of spatiotemporal chaos driven by far-field breakup of spiral waves in the plankton ecological systems Quan-Xing Liu,1 Gui-Quan Sun,1 Bai-Lian Li,2 and Zhen Jin1, ∗ Department of Mathematics, North University of China, Taiyuan, Shan’xi 030051, People’s Republic of China Ecological Complexity and Modeling Laboratory, Department of Botany and Plant Sciences, University of California, Riverside, CA 92521-0124, USA (Dated: October 25, 2018) Alexander B. Medvinsky et al [A. B. Medvinsky, I. A. Tikhonova, R. R. Aliev, B.-L. Li, Z.-S. Lin, and H. Malchow, Phys. Rev. E 64, 021915 (2001)] and Marcus R. Garvie et al [M. R. Garvie and C. Trenchea, SIAM J. Control. Optim. 46, 775-791 (2007)] shown that the minimal spatially extended reaction-diffusion model of phytoplankton-zooplankton can exhibit both regular, chaotic behavior, and spatiotemporal patterns in a patchy environment. Based on that, the spatial plankton model is furtherly investigated by means of computer simulations and theoretical analysis in the present paper when its parameters would be expected in the case of mixed Turing-Hopf bifurcation region. Our results show that the spiral waves exist in that region and the spatiotemporal chaos emerge, which arise from the far-field breakup of the spiral waves over large ranges of diffusion coefficients of phytoplankton and zooplankton. Moreover, the spatiotemporal chaos arising from the far-field breakup of spiral waves does not gradually involve the whole space within that region. Our results are confirmed by means of computation spectra and nonlinear bifurcation of wave trains. Finally, we give some explanations about the spatially structured patterns from the community level. PACS numbers: 87.23.Cc, 82.40.Ck, 82.40.Bj, 92.20.jm Keywords: Spiral waves; Spatio-temporal pattern; Plankton dynamics; Reaction-diffusion system I. INTRODUCTION There is a growing interest in the spatial pattern dy- namics of ecological systems [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. However, many mechanisms of the spatio- temporal variability of natural plankton populations are not known yet. Pronounced physical patterns like ther- moclines, upwelling, fronts and eddies often set the frame for the biological process. Measurements of the underwa- ter light field are made with state-of-the-art instruments and used to calculate concentrations of phytoplankton biomass (as chlorophyll) as well as other forms of organic matter. Very high diffusion of the marine environment would prevent the formation of any stable patch spatial distribution with much longer life-time than the typical time of biodynamics. Meanwhile, in addition to very changeable transient spatial patterns, there also exist other spatial patterns in marine environment, much more stable spatial structure associated with ocean fronts, spa- tiotemporal chaos [10, 11, 14], cyclonic rings, and so called meddies [15]. In fact, it is significant to create the biological basis for understanding spatial patterns of plankton [16]. For instance, the impact of space on the persistence of enriched ecological systems was proved in laboratory experiments [17]. Recently, it has been shown both in laboratory experiments [18] and theoreti- cally [14, 19, 20, 21] that the existence of a spatial struc- ture makes a predator-prey system less prone to extinc- ∗Corresponding author; Electronic address: [email protected] tion. This is due to the temporal variations of the density of different sub-populations can become asynchronous and the events of local extinction can be compensated due to re-colonization from other sites in the space [22]. During a long period of time, all the spiral waves have been widely observed in diverse physical, chemical, and biological systems [23, 24, 25, 26]. However, a quite lim- ited number of documents [11, 12, 27, 28, 29] concern the spiral wave pattern and its breakup in the ecological systems. The investigation of transition from regular patterns to spatiotemporally chaotic dynamics in spatially ex- tended systems remains a challenge in nonlinear sci- ence [14, 23, 30, 31]. In a nonlinear ecology system, the two most commonly seen patterns are spiral waves and turbulence (spatio-temporal chaos) for the level of the community [32]. It has been recently shown that sponta- neous spatiatemoporal pattern formation is an instrinsic property of a predator-prey system [11, 14, 33, 34, 35, 36] and spatiotemporal structures play an important role in ecological systems. For example, spatially induced speci- ation prevents the extinction of the predator-prey mod- els [11, 12, 37]. So far, plankton patchiness has been ob- served on a wide range of spatial temporal scales [38, 39]. There exist various, often heuristic explanations of the spatial patterns phenomenon for these systems. It should be noted that, although conclusive evidence of ecological chaos is still to be found, there is a growing number of indications of chaos in real ecosystems [40, 41, 42, 43]. Recently developed models show that spatial self- structuring in multispecies systems can meet both cri- teria and provide a rich substrate for community-level http://arxiv.org/abs/0704.0322v3 mailto:[email protected] section and a major transition in evolution. In present paper, the scenario in the spatially extended plankton ecological system is observed by means of the numeri- cal simulation. The system has been demonstrated to exhibit regular or chaostic, depending on the initial con- ditions and the parameter values [10, 29]. We find that the far-field breakup of the spiral wave leads to complex spatiotemporal chaos (or a turbulentlike state) in the spa- tially extended plankton model (1). Our results show that regular spiral wave pattern shifts into spatiotempo- ral chaos pattern by modulating the diffusion coefficients of the species. II. MODEL In this paper we study the spatially extended nutrient- phytoplankton-zooplankton-fish reaction-diffusion sys- tem. Following Scheffer’s minimal approach [44], which was originally formulated as a system of ordinary diff- ential equation (ODEs) and later developed models [10, 11, 29, 45, 46], as a further investigation, we study a two-variable phytoplankton and zooplankton model on the level of the community to describe pattern formation with the diffusion. The dimensionless model is written = rp(1 − p)− 1 + bp h+ dp∇ 2p, (1a) 1 + bp h−mh− f n2 + h2 + dh∇ 2h, (1b) where the parameters are r, a, b, m, n, dp, dh, and f which refer to work in Refs. [10, 11]. The explana- tion of model (1) relates to the nutrient-phytoplankton- zooplankton-fish ecological system [see Refs. [10, 29, 44] for details]. The local dynamics are given by g1(p, h) = rp(1− p)− 1 + bp h, (2a) g2(p, h) = 1 + bp h−mh− f n2 + h2 . (2b) From the earlier results [45] about non-spatial system of model (1) by means of numerical bifurcation analysis show that the bifurcation and bistability can be found in the system (1) when the parameters are varied within a realistic range. For the fixed parameters (see the caption of Fig. 1 and 2), we can see that the f controls the dis- tance from Hopf bifurcation. For larger f , there exists only one stable steady state. As f is decreased further, the homogeneous steady state undergoes a saddle node bifurcation (SN), that is fSN = 0.658. In this case, a stable and an unstable steady state become existence. Moreover, the bistability will emerge when the parame- ter f lies the interval fSN > f > fc = 0.445 (this value is more than the Hopf onset, fH = 0.3397). There are three steady states: with these kinetics A and C are linearly stable while B is unstable. Outside this interval, the sys- tem (1) has unique nontrivial equilibrium. Recent stud- ies [11, 29] shown that the systems (1) can well-develop the spiral waves in the oscillation regime, but where the authors only consider the special case, i.e., dp = dh. A few important issue have not yet been properly addressed such as the spatial pattern if dp 6= dh. Here we report the result that emergence of spatiotem- poral chaos due to breakup in the system under the dh 6= dp case. We may now use the f and diffusion ratio, ν = dh/dp, as control parameters to evaluate the region for the spiral wave. Turing instability in reaction-diffusion can be recast in terms of matrix sta- bility [47, 48]. Such with the help of Maple software assistance algebra computing, we obtain the parameters space (f, ν) bifurcation diagrams of the spiral waves as showing Fig. 2, in which two lines are plotted, Hopf line (solid) and Turing lines (dotted) respectively. In domain I, located above all three bifurcation lines, the homo- geneous steady states is the only stable solution of the system. Domain II are regions of homogeneous oscilla- tion in two dimensional spaces [49]. In domain III, both Hopf and Turing instabilities occur, (i.e., mixed Turing- Hopf modes arise), in which the system generally pro- duces the phase waves. Our results show that the system has spiral wave in this regions. One can see that a Hopf bifurcation can occur at the steady when the parameter f passes through a critical values fH while the diffusion coefficients dp = dh = 0 and the bifurcation periodic so- lutions are stable. From our analysis (see Fig. 2), one could also see that the diffusion can induce Turing type instability for the spatial homogeneous stable periodic solutions and the spatially extended model (1) exhibit spatio-temporal chaos patterns. These spatial pattern formation arise from interaction between Hopf and Tur- ing modes, and their subharmonics near hte codimension- two Hopf-Turing bifucation point. Special, it is interest- ing that spiral wave and travelling wave will appear when the parameters correspond to the Turing-Hopf bifurca- tion region III in the spatially extended model (1), i.e., the Turing instability and Hopf bifurcation occur simul- taneously. III. NUMERICAL RESULTS The simulation is done in a two-dimensional (2D) Cartesian coordinate system with a grid size of 600×600. The fourth order Runger-Kutta integrating method is applied with a time step ∆t = 0.005 time unit and a space step ∆x = ∆y = 0.20 length unit. The results remain the same when the reaction-diffusion equations were solved numerically in one and two spatial dimen- sions using a finite-difference approximation for the spa- tial derivatives and an explicit Euler method for the time integration. Neumann (zero-flux) boundary conditions FIG. 1: The sketch map for the bistability and the Hopf bi- furcation in the system (2) with r = 5.0, a = 5.0, b = 5.0, m = 0.6, and n = 0.4. The black curve is the g1(p, h). The colored curves are g2(p, h) with different values of f . The red curve: f = 0.3; the blue: f = 0.445; the green: f = 0.5; and the cyan: f = 0.658. 5 10 15 Turing instability FIG. 2: The sketch map of parameter space (f, ν) bifurcation diagrams for the spatially extended system (1) with r = 5.0, a = 5.0, b = 5.0, m = 0.6, dp = 0.05, and n = 0.4. were emmployed in our simulation. The diffusion terms in Eqs. (1a) and (1b) often describe the spatial mixing of species due to self-motion of the organism. The typi- cal diffusion coefficient of plankton patterns dp is about 0.05, based on the parameters estimatie of Refs [50, 51] using the relationship between turbulent diffusion and the scale of the space in the sea. In the previous stud- ies [10, 11, 29, 45, 46], the authors provided a valueable insight into the role of spatial pattern for the system (1) if dp = dh. From the biological meaning, the diffusion coefficients should satisfy dh ≥ dp. However, in nature waters it is turbulent diffusion that is supposed to domi- nate plankton mixing [52], when dh < dp is allowed. The other reason for choosing such parameter is that it is well- known new patterns, such as Turing patterns, can emerge in reaction-diffusion systems in which there is an imbal- ance between the diffusion coefficients dp and dh [23, 53]. Therefore, we set ν = dh/dp, and investigated whether a spiral wave would break up into complex spatiotemporal chaos when the diffusion ratio was varied. Throughout this paper, we fix dp = 0.05 and dh is a control parameter. In the following, we will show that the dynamic behav- ior of the spiral wave qualitatively change as the control parameter dh increases from zero, i.e., the diffusion ra- tio ν increases from zero, to more than one. For large ν (ν > 1), the outwardly rotating spiral wave is com- pletely stable everywhere, and fills in the space when the proper parameters are chosen, as shown in Fig. 3(A). Fig- ure 3(A) shows a series of snapshots of a well-developed single spiral wave formed spontaneously for the variable p in system (1). The spiral is initiated on a 600×600 grid by the cross-field protocol (the initial distribution chosen in the form of allocated “constant-gradient” perturbation of the co-existence steady state) and zero boundary con- ditions are employed for simulations in the two dimen- sions. From Fig. 3(A) we can see that the well-developed spiral waves are formed firstly by the evolution. Inside the domain, new waves emerge, but are evolved by the spiral wave growing from the center. The spiral wave can steadily grow and finally prevail over the whole do- main (a movie illustrating the dynamical evolution for this case [54] [partly movie−1, movie−2, and movie−3 for dh = 0.2]). Fig. 3(B) shows that the spiral wave first break up far away from the core center and even- tually relatively large spiral fragments are surrounded by a ‘turbulent’ bath remain. The size of the surviv- ing part of the spiral does not shrink when dh is further decreasing until finally dh equals to 0, which is different from phenomenon that is observed previous in the two- dimensional space Belousov-Zhabotinsky and FitzHugn- Nagumo oscillatory system [30, 31, 55, 56, 57], in which the breakup gradually invaded the stable region near the core center, and finally the spiral wave broke up in the whole medium. Figure 3(C) is the time sequences (ar- bitrary units) of the variables p and h at an arbitrary spatial point within the spiral wave region, from which we can see that the spiral waves are caused by the ac- cepted as “phase waves” with substantially group veloc- ity, phase velocity and sinusoidal oscillation rather than the relaxational oscillation with large amplitude. This breakup scenario is similar to the breakup of rotating spiral waves observed in numerical simulation in chemi- cal systems [30, 31, 55, 56, 57], and experiments in BZ systems [58, 59], which shows that spiral wave breakup in these systems was related to the Eckhaus instability and more important, the absolute instability. The corresponding trajectories of the spiral core and the spiral arm (far away from the core center) at y = 300 are shown in Fig. 4, respectively. From Fig. 4, we can see that the spiral core is not completely fixed, but oscil- lates with a large amplitude. However, as dh decreases to a critical value, an unstable modulation develops in 200 220 240 260 280 300 (D) t (arb. units) FIG. 3: Well developed spiral waves and some properties of them. The figures show simulations of the system (1) with r = 5, a = 5, b = 5, m = 0.6, n = 0.4, dp = 0.05, and f = 0.3. (A)Well developed spiral waves shown at subsequent snapshot in time, dh = 0.2. (B) Far-field breakup of the spiral waves shown at subsequent snapshot in time, dh = 0.002. The white (black) areas correspond to maximum (minimum) values of p [Additional movie format available from Ref. [54]]. (C) Oscillations of the variable p and h at an arbitrary spatial point within the regular spiral wave region for both scenarios. Each figure is ran the long time until it spatial patterns are unchange. regions which is far away from the spiral core (cf. the middle column of the Fig. 4). These oscillations eventu- ally grow large enough to cause the spiral arm far away from the core to breakup into complex multiple spiral waves, while the core region remains stable (the corre- sponding movie can be viewed in the online supplemen- tal in Ref. [54] [partly movie−1 and movie−2, and for dh = 0.02]). Figures 3(B) and 4(B) show the dynamic behavior for dh = 0.02, i.e., ν = 0.4. The regular tra- jectories far away from the core are now the same as the region of the spatial chaos (cf. the middle column of the Fig. 4). It is shown that an decrease in the diffusion ra- tio ν which leads to population oscillations of increasing amplitude (cf. the left column of the Fig. 4). In the tradition explain that the minimum value of the popula- tion density decreases and population extinction becomes more probable due to stochastic environmental perturba- tions. However, from the spatial evolution of system (1) (see Fig. 3), the temporal variations of the density of different sub-population can become asynchronous and the events of local extinction can be compensated due to re-colonization (or diffusion) from other sites. FIG. 4: The corresponding trajectories (from left to right) for locations (300, 300), (250, 300), and (50, 300) respectively. The parameters in (A), and (B) were the same as these in Fig. 3(A) and (B), respectively. Furthermore, it is well known that the basic arguments in spiral stability analysis can be carried out by reducing the system to one dimensional space [30, 31, 55, 56, 57]. Here we show some essential properties of the spiral breakup resulting from the numerical simulation. In the next section we will give the theoretical computation by using the eigenvalue spectra. In this model, it is worth noting that we do not neglect the oscillation of the dy- namics in the core as shown in Fig. 4 due to the system exhibiting spatial periodic wave trains when the model is simulated in one-dimensional space. Breakup occurs first far away from the core (the source of waves). The spiral wave breaks towards the core until it gets to some constant distance and then the surviving part of the spi- ral wave stays stable. These minimal stable wavelengths are called λmin. So the one-parameter family may be described by a dispersion curve λ(dh) (see Fig. 5). The minimal stable wavelength λmin of the spiral wave are shown in Fig. 5 coming from the simulation in two di- mensional space. The results of Fig. 5 can be interpreted as follows: the minimal stable wavelengths decrease with respect to the decrease of dh but eventually stay at a relative constant value, which is that the stable spiral waves are always existing for a larger region values of dh. Space-time plots at different times are shown in Fig. 6 for two different dh, i.e., different ν, which display the time evolution of the spiral wave along the cross section in the two-dimensional images of Fig. 3(A) and (B). As shown in Fig. 6(A) and (B) for dh = 0.2 and dh = 0.02 respectively, the waves far away from the core display unstable modulated perturbation due to convective in- stability [30, 31, 55, 56, 57], but this perturbation is gradually advected to the left and right sides, and finally disappears. The instability manifests itself to produce the wave train breakup several waves from the far-field, as shown in Figs. 6(B). FIG. 5: Dependence of the wavelength λmin on the parameter dh for the system (1) with r = 5.0, a = 5.0, b = 5.0, m = 0.6, dp = 0.05, and n = 0.4. Note the log scale for dh. IV. SPECTRA AND NONLINEAR BIFURCATION OF THE SPIRAL WAVE In this section, we concentrate on the linear stabil- ity analysis of spiral wave by using the spectrum the- ory [56, 60, 61, 62, 63]. From the results in Refs. [56, 62] we know that the absolute spectrum must be computed numerically for any given reaction-diffusion systems. In practice, such computations only require discretization in one-dimensional space and compare with computing eigenvalues of the full stability problem on a large do- main due to the spiral wave exhibitting traveling waves in the plane (see Fig. 6 about the space-time graphes). For spiral waves on the unbounded plane, the essential FIG. 6: Space-time plots of variable p for different time and dh. The parameters in (A), and (B) are the same as those in Fig. 3(A) and (B), respectively. spectrum is also required to compute, since it determined only by the far-field wave trains of the spiral. The lin- ear stability spectrum consists of point eigenvalues and the essential spectrum that is a continuous spectrum for spiral waves. For sake of simplicity, the Eqs. (1a) and (1b) can been written as following = dp∇ 2p+ g1(p, h), (3a) = dh∇ 2h+ g2(p, h). (3b) Suppose that (p∗, h∗) are a solutions and refer to them as steady spirals of Eq. (3) that rotate rigidly with a constant angular velocity ω, and that are asymptotically periodic along rays in the plane. In a coratating coordi- nate frame, using the standardized analysis method for the spiral waves [62, 63], the Eq. (3) is given by = dp∇ ρ,θp+ ω + g1(p ∗, h∗), (4a) = dh∇ ρ,θh+ ω + g2(p ∗, h∗), (4b) where (ρ, θ) denote polar coordinates, spirals waves are relative equilibria, then the statianry solutions p∗(ρ, θ) and h∗(ρ, θ) both are 2π-periodic functions with θ = ϕ− ωt. In Eqs. (4a) and (4b) the operator∇2ρ,θ denotes ∂ρρ+ A. Computation of spiral spectra Next, we commpute the leading part of its linear stabil- ity spectrum for the system (4). Consider the linearized evolution equation in the rotating frame, the eigenvalue problem of Eqs. (4a) and (4b) associated with the planar spiral solutions p∗(ρ, θ) and h∗(ρ, θ) are given by ρ,θp+ ω ∗, h∗)p+ gh1 (p ∗, h∗)h = λp, (5a) ρ,θh+ ω ∗, h∗)p+ gh2 (p ∗, h∗)h = λh, (5b) where g 1 , · · · , g 2 denote the derivatives of the nonlin- ear functions and g 1(p, h) = r(1 − p) − rp − (1+bp)2 , gh1 (p, h) = − 2(p, h) = − abph (1+bp)2 , and gh2 (p, h) = −m− 2fnh n2+h2 + 2fnh (n2+h2)2 . We shall ignore isolated eigenvalues that belong to the point spectrum, instabilities caused by point eigenvalues lead to mean- deringor drifting waves, or to an unstable tip motionin in excitable media and oscillation media [56, 64, 65, 66]. This phenomenon is not shown in the present paper. In- stead, we focus on the continuous spectrum that is re- sponsible for the spiral wave breakup in the far field (see Fig. 3(b)). By the results in Ref. [62], it turns out that the boundary of the continuous spectrum depends only on the limiting equation for ρ → ∞. Thus, we have that λ is the boundary of the continuous spectrum if, and only if the limiting equation ρ,ρp+ ω ∗, h∗)p+ gh1 (p ∗, h∗)h = λp, (6a) ρ,ρh+ ω ∗, h∗)p+ gh2 (p ∗, h∗)h = λh, (6b) have solutions p(ρ, θ) and h(ρ, θ) for (ρ, θ) ∈ R+× [0, 2π], which are bounded but does not decay as ρ → ∞. Since spiral waves are rotating waves in the plane, the wave train solutions have the form as u(t, x, y) = u(ρ, ϕ− ωt) for an appropriate wave numbers k and temporal fre- quency ω, where we assume that u is 2π-periodic in its argument so that u(ξ) = u(ξ + 2π) for all ξ and u = (p, h)T. Spiral waves converge to wave trains u(ρ, ϕ − ωt) → uwt(kρ + ϕ − ωt) for ρ → ∞, which are corresponding to asymptotically Archimedean in the two-dimensional space. Assume that k 6= 0 and ω 6= 0, and in this case, we can pass from the theoretical frame ρ to the comoving frame ξ = kρ+ϕ−ωt (ξ ∈ R) in which the eigenvalue equation (6) becomes 2∇2ξ,ξp+ωpξ+g 1(uwt(ξ))p+g 1 (uwt(ξ))h = λp, (7a) 2∇2ξ,ξh+ ωhξ + g 2(uwt(ξ))p + g 2 (uwt(ξ))h = λh.(7b) Indeed, any nontrivial solution u(ξ) = (p(ξ), h(ξ))T cor- responding to the linearization eigenvalue problem (7) give a solution U(ρ, ·) of the eigenvalue problem for the temporal period map of (3) in the corotating frame via U(ρ, ·) = eλtu(kρ− ωt), U(ρ, T ) = eλTu(kρ− 2π). We write the equations (7) as the first-order systems = p1, = h1, = k−2d−1p µp− ωp1 − g 1(uwt(ξ))p− g 1 (uwt(ξ))h = k−2d−1 µh− ωh1 − g 2(uwt(ξ))p− g 2 (uwt(ξ))h in the radial variable ρ. Then the spatial eigenvalues or spatial Floquet exponents are deternined as the roots of the Wronskian A(λ, k) := 0 0 1 0 0 0 0 1 (λ− g 1(uwt(ξ))) − gh1 (uwt(ξ)) − 2(uwt(ξ)) (λ− gh2 (uwt(ξ))) 0 − where k ∈ R. The function U(ρ, ·) = eλteikρu0(kρ− ωt) satisfies the equation (3) when the spatial and temporal exponents ik and λ satisfy the complex dispersion rela- tion det(A(λ, k) − ik) = 0 for λ ∈ C. We call the ik in spectrum of A(λ, k) as spatial eigenvalues or spatial Floquet exponents. The stability of the spiral waves state (p∗, h∗) on the plane is determined by the essential spectrum given by Σess = {λ ∈ C; det(A(λ, k) − ik) = 0 for some k ∈ R}. Now, we compute the continuous spectrum with the equation (9) that are parameterized by the wave num- ber k. For each λ, there are infinitely many stable and unstable spatial eigenvalues. We plot λ in the complex plane associated spatial spectrum, see Fig. 7. By the ex- plaination of Sandstede et al [60], one would know that if the real part of the essentail spectra is positive, then the associated eigenmodes grow exponentially toward the boundary, i.e., they correspond to a far-field instability. Note that we find the essentail spectra are not sensitive to temporal frequency, ω. Re(λ) K30 K20 K10 0 Im(λ) Re(λ) K0.8 K0.6 K0.4 K0.2 0 0.2 Im(λ) FIG. 7: The essentail spectra of wave trains are obtained by using the algorithms outlined in Refs. [60, 61]. The param- eters of (A) and (B) are corresponding to the values used in the simulations of Fig. 3(A) and (B). B. Existence and properties of wave trains Suppose that a reaction-diffusion system on the one- dimensional space such that the variables equal to a homogeneous stationary solution. If the homogeneous steady-state destabilizes, then its linearization accommo- dates waves of the form ei(kx−ωt) for certain values k and ω. Typically, near the transition to instability, small spa- tially periodic travelling waves arise for any wave number close to kc, which is the critical wavenumber. Their wave speed is approximately equal to ωc , where ωc is corre- sponding to kc. In present paper, we focus exclusively on the situation where ωc = 0 and kc 6= 0. The bifurcation with ωc = 0 and kc 6= 0 is known as the Turing bifur- cation, and the bifurcating spatially periodic steady pat- terns are often referred to as Turing patterns. Another class of moved patterns will appear when the instabilities modulated by Hopf-Turing bifurcation, which is resem- ble a travelling waves. Moreover, the common feature of the spiral waves in one-dimensional space mentioned above is the presence of wave trains which are spatially periodic travelling waves of the form pwt(kx−ωt; k) and hwt(kx − ωt; k), where pwt(φ; k) and hwt(φ; k) are 2π- periodic about φ. Typically, the spatial wavenumber k and the temporal frequency ω are related via the non- linear dispersion relation ω = ω(k) so that the phase velocity is given by . (12) A second quantity related to the nonlinear dispersion relation is the group velocity, cg = , of the wave train which also play a central role in the spiral waves. The group velocity cg gives the speed of propagation of small localized wave-package perturbations of the wave train [67]. Here, we are only concerned the existence of travelling wave solution. In fact, the spiral waves move at a constant speed outward from the core (see Fig. 6), so that they have the mathematical form p(x, t) = P (z), and h(x, t) = H(z) where z = x−cpt. Substituting these solution forms into Eq. (3) gives the ODEs + g1(P,H) = 0, (13a) + g2(P,H) = 0. (13b) Here, we investigate numerically the existence, speed and wavelength of travelling wave patterns. Our ap- proach is to use the bifurcation package Matcont 2.4 [68] to study the pattern ODEs (13). To do this, the most natural bifurcation parameters are the wave speed cp and f , but they give no information about the stability of travelling wave as solutions of the model PDEs (3). Our starting point is the homogeneous steady state of Eq. (13) with in the domain III of Fig. 2. The typical bi- furcation diagrams are illustrated in Fig. 8, which shows that steady spatially peroidic travelling waves exist for the larger values of the speed cp, but it is unstable for small values of cp. The changes in stability occur via Hopf bifurcation, from which a branch of periodic orbits emanate. Note that here we use the terms “stable” and “unstable” as referring to the ODEs system (13) rather than the model PDEs. Fig. 8(B) illustrates the max- imun stable wavelength against the bifucation parame- ter, speed cp, and the small amplitudes have very long wavelength. It is known that cp = , hence the tavelling wave solution exist when the cp 6= 0, i.e., k 6= 0, ω 6= 0. Using Matcont 2.4 package, it is possible to track the lo- cus of the Hopf bifurcation points and the Limit point (fold) bifurcation in a parameter plane, and a typical ex- ample of this for the cp-f and cp-dh plane are illustrated in Fig. 9. The travelling wave solutions exist for values of cp and f lying in left of Hopf bifurcation locus (see Fig. 9(A)). The same structure about the cp-dh plane is shown in Fig. 9(B). These reuslts confirm our previ- ous analysis coming from the algebra computation (see Fig. 2) and the numerical results (see Fig. 6). V. CONCLUSIONS AND DISCUSSION We have investigated a spatially extended plank- ton ecological system within two-dimensional space and 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Speed, c Hopf bifurcation point FIG. 8: Typical bifurcation diagrams for the pattern ODEs (13). (A) The spatially periodic travelling waves of system (3) is existence. The changes in stability occur via Hopf bifurcation, from which a branch of periodic orbits em- anate. Thus unstable travelling waves appear. (B) Maxi- mum stable wavelength along the bifurcation parametercp, i.e., k 6= 0, ω 6= 0. The parameter values in (A) and (B) are the same as Fig. 3(A). found that its spatial patterns exhibit spiral waves dy- namics and spatial chaos patterns. Specially, the sce- nario of the spatiotemporal chaos patterns arising from the far-field breakup is observed. Our research is based on numerical analysis of a kinematic mimicking the dif- fusion in the dynamics of marine organisms, coupled to a two component plankton model on the level of the com- munity. By increasing (decreasing) the diffusion ratio of the two variables, the spiral arm first broke up into a turbulence-like state far away from the core center, but which do not invade the whole space. From the previous studies in the Belousov-Zhabotinsky reaction, we know the reason causing this phenomenon can be illuminated theoretically by the M. Bär and L. Brusch [30, 31], as well as by using the spectrum theory that poses by B. Sandstede, A. Scheel et al [56, 60, 61, 69]. The far-field breakup can be verified in field observation and is useful to understand the population dynamics of oceanic ecolog- ical systems. Such as that under certain conditions the interplay between wake (or ocean) structures and bio- logical growth leads to plankton blooms inside mesoscale hydrodynamic vortices that act as incubators of primary production. From Fig. 3 and corresponding the movies, we see that spatial peridic bloom appear in the phyto- plankton populations, and the details of spatial evolution of the distribution of the phytoplankton population dur- ing one bloom cycle, respectively. In Ref. [70], the authors study the optimal control of the model (1) from the spatiotemporal chaos to spiral waves by the parameters for fish predation treated as a multiplicative control variable. Spatial order emerges in a range of spatial models of multispecies interactions. Un- surprisingly, spatial models of multispecies systems often 0 0.5 1 1.5 2 2.5 Speed, c Locus of Hopf bifurcation points 0.5 1 1.5 2 Speed, c Locus of Hopf bifurcation points FIG. 9: An illustration of the variations in parameter space of the pattern ODEs (13). We plot the loci of Hopf bifurcation points. (A) f − cp planes; (B) dh − cp planes. The parameter values in (A) and (B) are the same as Fig. 3(A). manifests very different behaviors from their mean-field counterparts. Two important general features of spatial models of multispecies systems are that they allow the possibility of global persistence in spite of local extinc- tions and so are usually more stable than their mean-field equivalents, and have a tendency to self-organzie spa- tially or regular spatiotemporal patterns [70, 71]. The spatial structures produces nonrandom spatial patterns such as spiral waves and spatiotemporal chaos at scales much larger than the scale of interaction among individ- uals level. These structures are not explicitly coded but emerge from local interaction among individuals and lo- cal diffusion. As we know that plankton plays an important role in the marine ecosystem and the climate, because of their participation in the global carbon and nitrogen cycle at the base of the food chain [72]. From the review [73], a recently developed ecosystem model incorporates differ- ent phytoplankton functional groups and their competi- tion for light and multiple nutrients. Simulations of these models at specific sites to explore future scenarios sug- gest that global environmental change, including global- warming-induced changes, will alter phytoplankton com- munity structure and hence alter global biogeochemical cycles [74]. The coupling of spatial ecosystem model to global climate raises again a series of open questions on the complexity of model and relevant spatial scales. So the study of spatial model with large-scale is more impor- tant in the ecological system. Basing on numerical simu- lation on the spatial model, we can draft that the oceanic ecological systems show permanent spiral waves and spa- tiotemporal chaos in large-scale over a range of parame- ter values dh, which indicates that periodically sustained plankton blooms in the local area. As with all areas of evolutionary biology, theoretical development advances more quickly than does empiraical evidence. The most powerful empirical approach is to conduct experiments in which the spatial pattern can be measured directly, but this is difficulties in the design. However, we can in- directly measured these phenomenona by the simulation and compared with the satellite pictures. For example, the spatiotemporal chaos patterns agree with the per- spective observation of the Fig. 3 in Ref. [73]. Also, some satellite imageries [http://oceancolor.gsfc.nasa.gov] have displayed spiral patterns that represent the phytoplank- ton [the chlorophyll] biomass and thus demonstrated that plankton patterns in the ocean occur on much broader scales and therefore mechanisms thought diffusion should be considered. Acknowledgments This work is supported by the National Natural Sci- ence Foundation of China under Grant No. 10471040 and the Natural Science Foundation of Shan’xi Province Grant No. 2006011009. [1] R. E. Amritkar and Govindan Rangarajan. Spatially synchronous extinction of species under external forcing. Phys. Rev. Lett., 96(25):258102, 2006. [2] Andrzej Pekalski and Michel Droz. Self-organized packs selection in predator-prey ecosystems. Phys. Rev. E, 73(2):021913, 2006. [3] Y.-Y. H. Sayama, M. A. M. de Aguiar, and M. Baranger. Interplay between turing pattern formation and do- main coarsening in spatially extended population models. FORMA, 18:19, 2003. [4] E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak, and E. Meron. Ecosystem engineers: From pat- tern formation to habitat creation. Phys. Rev. Lett., 93(9):098105, 2004. [5] Mark J Washenberger, Mauro Mobilia, and Uwe C Täuber. Influence of local carrying capacity restrictions on stochastic predator&ndash;prey models. J. Phys.: Cond. Matt., 19(6), 2007. [6] Mauro Mobilia, Ivan Georgiev, and Uwe Täuber. Phase transitions and spatio-temporal fluctuations in stochastic lattice lotkacvolterra models. J. Stat. Phys., 128(1):447– 483, 2007. [7] Bernd Blasius, Amit Huppert, and Lewi Stone. Com- plex dynamics and phase synchronization in spatially ex- tended ecological systems. Nature, 399(6734):354–359, 1999. [8] J. von Hardenberg, E. Meron, M. Shachak, and Y. Zarmi. Diversity of vegetation patterns and desertification. Phys. Rev. Lett., 87(19):198101, 2001. [9] A. Provata and G. A. Tsekouras. Spontaneous formation of dynamical patterns with fractal fronts in the cyclic lattice lotka-volterra model. Phys. Rev. E, 67(5):056602, 2003. [10] Alexander B. Medvinsky, Irene A. Tikhonova, Rubin R. Aliev, Bai-Lian Li, Zhen-Shan Lin, and Horst Malchow. Patchy environment as a factor of complex plankton dy- namics. Phys. Rev. E, 64(2):021915, 2001. [11] Alexander B. Medvinsky, Sergei V. Petrovskii, Irene A. Tikhonova, Horst Malchow, and Bai-Lian Li. Spatiotem- poral complexity of plankton and fish dynamics. SIAM Review, 44:311–370, 2002. [12] W. S. C. Gurney, A. R. Veitch, I. Cruickshank, and G. McGeachin. circles and spirals: population persis- tence in a spatially explicit predatorcprey model. Ecol- ogy, 79(7):2516–2530, 1998. [13] J. D. Murray. Mathematical biology. Interdisciplinary applied mathematics. Springer, New York, 3rd edition, 2002. [14] Sergei Petrovskii, Bai-Lian Li, and Horst Malchow. Tran- sition to spatiotemporal chaos can resolve the paradox of enrichment. Ecological Complexity, 1:37–47, 2004. [15] Laurence Armi, Dave Hebert, Neil Oakey, James Price, Philip L. Richardson, Thomas Rossby, and Barry Rud- dick. The history and decay of a mediterranean salt lens. Nature, 333(6174):649–651, 1988. [16] Esa Ranta, Veijo Kaitala, and Per Lundberg. The Spa- tial Dimension in Population Fluctuations. Science, 278(5343):1621–1623, 1997. [17] L S Luckinbill. The effects of space and enrichment on a predator-prey system. Ecology, 55:1142–1147, 1974. [18] M Holyoak. Effects of nurient enrichment on predator- prey metapopulation dynamics. J Anim. Ecol., 69:985– 997, 2000. [19] Vincent A.A. Jansen. Regulation of predator-prey sys- tems through spatial interactions: a possible solution to the paradox of enrichment. Oikos, 74:384390, 1995. [20] Vincent A.A. Jansen and Alun L. Lloyd. Local stabil- ity analysis of spatially homogeneous solutions of multi- patch systems. J. Math. Biol., 41:232252, 2000. [21] Vincent A.A. Jansen. The dynamics of two diffusively coupled predatorprey populations. Theor. Popul. Biol., 59:119131, 2001. [22] J. C. Allen, W. M. Schaffer, and D. Rosko. Chaos reduces species extinction by amplifying local population noise. Nature, 364:229232, 1993. [23] M. C. Cross and P. C. Hohenberg. Pattern formation outside of equilibrium. Rev. Mod. Phys., 65(3):851, 1993. [24] Kyoung J. Lee, Raymond E. Goldstein, and Edward C. Cox. Resetting wave forms in dictyostelium territories. Phys. Rev. Lett., 87(6):068101, 2001. [25] Satoshi Sawai, Peter A. Thomason, and Edward C. Cox. An autoregulatory circuit for long-range self-organization http://oceancolor.gsfc.nasa.gov in dictyostelium cell populations. Nature, 433(7023):323– 326, 2005. [26] Arthur T. Winfree. Varieties of spiral wave behavior: An experimentalist’s approach to the theory of excitable media. Chaos, 1(3):303–334, 1991. [27] V. N. Biktashev, J. Brindley, A. V. Holden, and M. A. Tsyganov. Pursuit-evasion predator-prey waves in two spatial dimensions. Chaos, 14(4):988–994, 2004. [28] M Garvie. Finite-difference schemes for reaction-diffusion equation modeling predato-prey interactions in matlab. Bull. Math. Biol., 69:931–956, 2007. [29] Marcus R. Garvie and Catalin Trenchea. Optimal con- trol of a nutrient-phytoplankton-zooplankton-fish sys- tem. SIAM J. Contr. and Opti., 46(3):775–791, 2007. [30] Markus Bär and Lutz Brusch. Breakup of spiral waves caused by radial dynamics: Eckhaus and finite wavenum- ber instabilities. New Journal of Physics, 6:5, 2004. [31] Markus Bär and Michal Or-Guil. Alternative scenar- ios of spiral breakup in a reaction-diffusion model with excitable and oscillatory dynamics. Phys. Rev. Lett., 82(6):1160–1163, 1999. [32] Craig R Johnson and Maarten C Boerlijst. Selection at the level of the community: the importance of spatial structure. Trends Ecol. and Evol., 17:83–90, 2002. [33] M Pascual. Diffusion-induced chaos in a spatial preda- torprey system. Proc. R. Soc. Lond. B, 251:17, 1993. [34] J. A. Sherratt, B. T. Eagan, and M. A. Lewis. Oscil- lations and chaos behind predatorprey invasion: math- ematical artifact or ecological reality? Phil. Trans. R. Soc. Lond. B, 352:2138, 1997. [35] S. V. Petrovskii and H. Malchow. Wave of chaos: new mechanism of pattern formation in spatio-temporal pop- ulation dynamics. Theor. Popul. Biol., 59:157174, 2001. [36] J. A. Sherratt. Periodic travelling waves in cyclic preda- torprey systems. Ecol. Lett., 4:3037, 2001. [37] N. J. Savill and P. Hogeweg. Spatially induced speciation prevents extinction: the evolution of dispersal distance in oscillatory predator-prey models. Proc. R. Soc. Lond. B, 265(1390):25–32, 1998. [38] E. R. Abraham. The generation of plankton patchiness by turbulent stirring. Nature, 391:577–580, 1998. [39] Carol L. Folt and Carolyn W. Burns. Biological drivers of zooplankton patchiness. Nature, 14:300–305, 1999. [40] M Scheffer. Should we expect strange attractors behind plankton dynamics and if so, should we bother? J. Plank- ton Res., 13:1291–1305, 1991. [41] I. Hanski, P. Turchin, E. Korplmakl, and H. Henttonen. Population oscillations of boreal rodents: regulation by mustelid predators leads to chaos. Nature, 364:232235, 1993. [42] S. Ellner and P. Turchin. Chaos in a noisy world: new methods and evidence from time-series analysis. Am. Nat., 145:343375, 1995. [43] B. Dennis, R. A. Desharnais, J. M. Cushing, S. M. Hen- son, and R. F. Costantino. Estimating chaos and com- plex dynamics in an insect population. Ecol. Monogr., 71:277303, 2001. [44] M Scheffer. Fish and nutrients interplay determines algal biomass: A minimal model. Oikos, 62:271–282, 1991. [45] H. Malchow. Spatio-temporal pattern formation in non- linear non-equilibrium plankton dynamics. Procc. R. Soc. Lond. B, 251:103, 1993. [46] M. Pascual. Diffusion-induced chaos in a spatial predator-prey system. Procc. R. Soc. Lond. B, 251:1–7, 1993. [47] Arnd Scheel. Radialy symmetric patterns of reaction- diffusion systems. Mem. Amer. Math. Soc., 165:86, 2003. [48] R A Satnoianu, M Menzinger, and P K Maini. Turing instabilities in general system. J. Math. Biol., 41:493– 512, 2000. [49] Quan-Xing Liu, Bai-Lian Li, and Zhen Jin. Resonant pat- terns and frequency-locked induced by additive noise and periodically forced in phytoplankton-zooplankton sys- tem, 2007. [50] Sven Erik Jørgensen and G. Bendoricchio. Fundamentals of ecological modelling. Developments in environmental modelling; 21. Elsevier, Amsterdam; New York, 3rd edi- tion, 2001. [51] Akira Okubo. Diffusion and ecological problems: mathe- matical models. Biomathematics; v. 10. Springer-Verlag, Berlin; New York, 1980. [52] George Sugihara and Robert M. May. Nonlinear forecast- ing as a way of distinguishing chaos from measurement error in time series. Nature, 344(6268):734–741, 1990. [53] A. M. Turing. The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of Lon- don. Series B, Biological Sciences, 237(641):37–72, 1952. [55] Fagen Xie, Dongzhu Xie, and James N. Weiss. Inwardly rotating spiral wave breakup in oscillatory reaction- diffusion media. Phys. Rev. E, 74(2):026107, 2006. [56] Björn Sandstede and Arnd Scheel. Absolute versus convective instability of spiral waves. Phys. Rev. E, 62(6):7708–7714, 2000. [57] S. M. Tobias and E. Knobloch. Breakup of spiral waves into chemical turbulence. Phys. Rev. Lett., 80(21):4811– 4814, 1998. [58] Q. Ouyang and J. M. Flesselles. Transition from spirals to defect turbulence driven by a convective instability. Nature, 379(6561):143–146, 1996. [59] Qi Ouyang, H. L. Swinney, and G. Li. Transition from spirals to defect-mediated turbulence driven by a doppler instability. Phys. Rev. Lett., 84(5):1047–1050, 2000. [60] Björn Sandstede and Arnd Scheel. Curvature effects on spiral spectra: Generation of point eigenvalues near branch points. Phys. Rev. E, 73:016217, 2006. [61] Jens D.M. Rademacher, Björn Dandstede, and Arnd Scheel. Computing absolute and essential spectra using continuation. Physics D, 229:166–183, 2007. [62] P. Wheeler and D. Barkley. Computation of Spiral Spec- tra. SIAM J Appl. Dynam. Syst., 2006. [63] Björn Sandstede and Arnd Scheel. Absolute and convec- tive instabilities of waves on unbounded and large bound domians. Physics D, 145:233–277, 2000. [64] Dwight Barkley. Linear stability analysis of rotat- ing spiral waves in excitable media. Phys. Rev. Lett., 68(13):2090–2093, 1992. [65] Dwight Barkley. Euclidean symmetry and the dynamics of rotating spiral waves. Phys. Rev. Lett., 72(1):164–167, 1994. [66] Igor Aranson, Lorenz Kramer, and Andreas Weber. Core instability and spatiotemporal intermittency of spiral waves in oscillatory media. Phys. Rev. Lett., 72(15):2316– 2319, 1994. [67] Björn Sandstede and Arnd Scheel. Defects in oscillatory media: Toward a classification. SIAM J. Appl. Dynam. Syst., 3(1):1–68, 2004. [68] A Dhooge, W Govaerts, Yu A Kuznetsov, W Mestrom, A M Riet, and B Sautois. Matcont and Cl-Matcont: Con- tinuation toolboxes in Matlab. Utrecht University, The Netherlands, 2006. [69] Paul Wheeler and Dwight Barkley. Computation of spiral spectra. SIAM J. Appl. Dynam. Syst., 5:157–177, 2006. [70] Maarten C. Boerlijst. The Geometry of Ecological Inter- actions: Simplifying Spatial Complexity, chapter Spirals and spots: Novel Evolutionary Phenomena through spa- tial self-structuring, pages 171–182. Cambridge Univer- sity Press, 2000. [71] Ulf Dieckmann, Richard Law, and Johan A J Metz. The Geometry of Ecological Interactions: Simplifying Spatial Complexity. Cambridge University Press, 2000. [72] J. Duinker and G. Wefer. Das co2-problem und die rolle des ozeans. Naturwissenschaften, 81(6):237–242, 1994. [73] M. Pascual. Computational ecology: From the complex to the simple and back. Plos Comput. Biol., 1(2):101– 105, 2005. [74] E. Litchman, C. A. Klausmeier, J. R. Miller, O. M. Schofield, and P. G. Falkowski. Multi-nutrient, multi- group model of present and future oceanic phytoplankton communities. Biogeosciences, 3(4):585–606, 2006.

0704.0323

General sequential quantum cloning

General Sequential Quantum Cloning Gui-Fang Dang and Heng Fan Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China. (Dated: November 4, 2018) Some multipartite quantum states can be generated in a sequential manner which may be im- plemented by various physical setups like microwave and optical cavity QED, trapped ions, and quantum dots etc. We analyze the general N to M (N ≤ M) qubits Universal Quantum Cloning Machine (UQCM) within a sequential generation scheme. We show that the N to M sequential UQCM is available. The case of d-level quantum states sequential cloning is also presented. PACS numbers: 03.67.Mn, 03.65.Ud, 52.50.Dv Quantum entanglement plays a key role in quantum computation and quantum information [1]. Multipartite entangled states arise as a resource for quantum infor- mation processing tasks such as the well known quantum teleportation[2], quantum communication [3, 4], clock synchronization [5] etc. In general it is extremely dif- ficult to generate experimentally multipartite entangled states through single global unitary operations. In this sense, the sequential generation of the entangled states appears to be promising. Actually most of the quantum computation networks are designed to implement quan- tum logic gates through a sequential procedure [6]. Re- cently sequential implementing of quantum information processing tasks has been attracting much attention. It is pointed out that photonic multiqubit states can be generated by letting a source emit photonic qubits in a sequential manner [7]. The general sequential generation of entangled multiqubit states in the realm of cavity QED was systematically studied in Refs.[8, 9]. It is also shown that the class of sequentially generated states is identical to the matrix-product-state (MPS) which is very useful in study of spin chains of condensed matter physics [10]. On the other hand, much progress has already been made in the past years in studying quantum cloning ma- chines, for reviews see, for example, Refs.[11, 12, 13]. And various quantum cloning machines have been im- plemented experimently by polarization of photons [14, 15, 16, 17, 18],nuclear spins in Nuclear Magnetic Reso- nance [19, 20], etc. However, these experiments are for 1 to 2 (one qubit input and two-qubit output) or 1 to 3 cloning machines. The more general case will be much difficult. There are some schemes proposed for the gen- eral quantum cloning machines which are not in a sequen- tial manner, see for example, [21, 22]. Recently a 1 to M sequential universal quantum cloning is proposed [23] by using the cloning transformation presented in Ref.[24]. Since it is in a sequential procedure, potentially it re- duces the difficult in implementing this quantum cloning machine. However, as is well known the collective quan- tum cloning machine (the N identical input states are cloned collectively to M copies) is better than the quan- tum cloning machine which can only deal with the in- dividual input(only one input is copied to several copies each time). We know that the general N to M cloning transformation is also available in Refs.[24, 25]. Then a natural question arise is that whether the general N to M sequential cloning machine is possible. In this Letter, we will present the general sequential universal quantum cloning machine. The 1 to M cloning transformations used in Ref.[23] was proposed by Gisin and Massar in Ref.[24]. And the N toM UQCM was also presented in Ref.[24]. However, to use the method proposed in Refs.[8, 23] to find the se- quential cloning machine, the input state |Φ〉⊗N should be expanded in computational basis {|0〉, |1〉}. The ex- plicit quantum cloning transformations with this kind of input were proposed by Fan et al in Ref.[25]. In this Let- ter, based on the result of Ref.[25], the general sequential UQCM will be presented. As presented in Refs.[8, 23], the sequential generation of a multiqubit state is like the following. Let HA be a D-dimensional Hilbert space which acts as the ancil- lary system, and a single qubit (e.g., a time-bin qubit) is in a two-dimensional Hilbert space HB. In every step of the sequential generation of a multiqubit state, a uni- tary time evolution will be acting on the joint system HA ⊗HB. We assume that each qubit is initially in the state |0〉 which is like a blank or an empty state and will not be written out in the formulas. So the unitary time evolution is written in the form of an isometry V : HA → HA⊗HB, where V = i,α,β V α,β |α, i〉〈β|, each V i is a D×D matrix, and the isometry condition takes the i=0 V i†V i = 1. By applying successively n oper- ations of V (not necessarily the same) on an initial ancil- lary state |φI〉 ∈ HA, we obtain |Ψ〉 = V [n]...V [2]V [1]|φI〉. The generated n qubits are in general an entangled state, but the last step qubit-ancilla interaction can be chosen so as to decouple the final multiqubit entangled state from the auxiliary system, so the sequentially generated state is |ψ〉 = i1...in=0 〈φF |V [n]in ...V [1]i1 |φI〉|in, ..., i1〉, (1) where |φF 〉 is the final state of the ancilla. This is the MPS. It was proven that any MPS can be sequentially generated [8]. http://arxiv.org/abs/0704.0323v2 Suppose there are N identical pure quantum states |Φ〉⊗N = (x0|0〉+x1|1〉)⊗N need to be cloned toM copies, where |x0|2 + |x1|2 = 1. We know that the input state can be represented by a basis in symmetric subspace. |Φ〉⊗N = xN−m0 x CmN |(N −m)0,m1〉, (2) where |(N − m)0,m1〉 denotes the symmetric and nor- malized state with (N −m) qubits in the state |0〉 and m qubits in the state |1〉, and we have CmN = N !/(N−m)!m! in standard notation. So if we find the quantum cloning transformations for all states in symmetric subspace, we can clone N pure states to M copies. The UQCM with input in symmetric subspace can be written as [25], |(N −m)0,m1〉 → |ΦmM 〉, (3) where |ΦmM 〉 = βmj |(M −m− j)0, (m+ j)1〉 ⊗Rj ,(4) βmj = M−N−j M−m−jC (m+j) /CN+1M+1, (5) where Rj are the ancillary states of the cloning machine and are orthogonal with each other for different j. For a sequential quantum cloning machine in this Letter, we choose a realization Rj ≡ |(M −N − j)1, j0〉 for the an- cilla states. This UQCM is optimal in the sense that the fidelity between single qubit output state reduced density operator ρoutreduced and the single input |Φ〉 is op- timal. The optimal fidelity is F = 〈Φ|ρoutreduced|Φ〉 = (MN +M + N)/M(N + 2), see Refs.[11, 12, 13] for re- views and the references therein. A realization of this UQCM with photon stimulated emission can be found in Ref.[22] which is not in a sequential manner. We next show that this general N to M UQCM can be generated through a sequential procedure. The basic idea is to show that the final state of the cloning, |ΦmM 〉 in (4), can be expressed in its MPS form. As shown in Ref.[8], any MPS can be sequentially gen- erated. We shall follow the method, for example, as in Refs.[23, 26]. By Schmidt decomposition, we first ex- press the quantum state |ΦmM 〉 as a bi-partite state across 1 : 2... cut, |ΦmM 〉 = λ 1 |0〉|φ [2...(2M−N)] 1 〉+ λ 2 |1〉|φ [2...(2M−N)] Γ[1]i1α1 λ |i1〉|φ[2...(2M−N)]α1 〉, (6) where Γ α1 = δα1,1,Γ α1 = δα1,2, and λ α1 are eigen- values of the first qubit reduced density operator, and we find λ ∑M−m−1 k=−m β M−1/C M , λ ∑M−m−1 k=−m β mk+1C M−1/C m+k+1 M . To correspond with the MPS in (1), we can define V [1]i1 α1 = Γ [1]i1 α1 . Suc- cessively by Schmidt decomposition, the quantum state |ΦmM 〉 in (4) is divided into a bi-partite state with the first n qubits as one part, and the rest as another part, where 1 < n ≤M − 1. We find |ΦmM 〉 = j+1|(n− j)0, j1〉|φ [(n+1)...(2M−N)] j+1 〉, (7) when 1 < n ≤M−N+m,n′ = n; whenM−N+m< n ≤ M − 1, n′ =M −N +m, λ[n]j+1 are eigenvalues of the first n qubits reduced density operator of |ΦmM 〉. According to the results in Eqs.(4,5), we can obtain, j+1 = M−m−n m(j+k) Cm+kM−n m+j+k . (8) And we also have |φ[(n+1)...(2M−N)]j+1 〉 = M−m−n β2m(j+k) × (m+k) m+j+k |(M − n−m− k)0, (m+ k)1〉 ⊗Rj+k. By induction and a concise formula, we have |Φn...(2M−N)]j+1 〉 αn,in [n]in (j+1)αn λ[n]αn |in〉|φ [(n+1)...(2M−N)] [n−1] |0〉|φ[(n+1)...(2M−N)]j+1 〉 +|1〉|φ[(n+1)...(2M−N)]j+2 〉 , (9) where we denote (j+1)αn = δ(j+1)αn n−1/(λ [n−1] n), (10) (j+1)αn = δ(j+2)αn n−1/(λ [n−1] n ). (11) Still we define that V [n]inαnαn−1 = Γ [n]in αn−1αn λ[n]αn . (12) It is thus in the MPS representation. We can further con- sider other cases including the ancilla state of the cloning machine represented as Rj (Note it is not the ancilla state in the MPS representation). We can find that the out- put state of the general UQCM can be expressed as MPS as in form (1). So it can be created sequentially. The explicit results are summarized in the appendix. We have shown that the output states of the general UQCM in (4,5) are MPS’s and thus can be generated sequentially. The sequential matrices V [n] of course de- pend on the input |(N−m)0,m1〉 which are W-like states and are generally multiqubit entangled. For later con- venience, we denote V (m) to express that it depends on input state for different m. By a straightforward method, the sequential cloning operation, i.e., the iso- metrices, depending on different input may take the form m |(N − m)0,m〉〈(N − m)0,m1| ⊗ V (m). However, this operation may need a single global unitary opera- tor which involves N -qubit entangled states except for m = 0,m = N . This contradicts with our aim that each operation should be divided into sequential unitary oper- ators in a quDit (quantum state in D-dimensional space) times qubit system. Here we can use a scheme like the following: the ancillary state interacts with each qubit according to the (N + 1) × D-dimensional isometrices CmN |0〉〈0|⊗N−m⊗|1〉〈1|⊗m⊗V (m) sequentially, here a whole normalization factor is omitted. We know that the operation |0〉〈0|⊗N−m ⊗ |1〉〈1|⊗m acts on each qubit individually. Thus this scheme reduces the com- plexity of the operation. This finishes our general se- quential UQCM for the case of qubit. In case N = 1, we recover the result of Ref.[23] for 1 to M cloning. We should remark that similar as the case of sequen- tial 1 to M UQCM in Ref.[23], for the general sequential UQCM, the minimal dimension D of the ancillary state grows linearly at most with M −N/2 + 1 for even N or M − (N − 1)/2 for odd N . Next we will consider a more general case that the se- quential cloning machine is about the quantum state in d- dimensional Hilbert space. We will use the d-dimensional UQCM proposed by Fan et al in Ref. [25]. This UQCM is a generalization of the cloning machine proposed in Ref.[24] and we can use this UQCM to study its sequen- tial form for d-dimensional case. An arbitrary d-dimensional pure state takes the form |Φ〉 = i=0 xi|i〉 with i=0 |xi|2 = 1. N identical pure states can be expanded in terms of state in symmet- ric subspace |Φ〉⊗N = m1!...md! xm10 ...x d−1|~m〉, where |~m〉 ≡ |m1, ...,md〉 is a symmetric state with mi states of |i − 1〉, and also mi should satisfy a relation i=1mi = N . The cloning transformations with states in symmetric subspace can be written as |~m〉 → |Φ~mM 〉 = |~m+~j〉 ⊗ |~j〉, (13) i=1 C mi+ji CM−NM+d−1 where ~j should satisfy i ji = M − N . This cloning machine is optimal and the corresponding fidelity of a single quantum state between input and output is F = (N(d+M) +M −N) /(d+N)M . As for qubit system, we next show that the output states for all symmetric states input can be expressed as the sequential form. We consider the case 1 < n ≤ M − 1, and the state |Φ~mM 〉 is a bipartite state across 1...n : (n+ 1)... cut, |Φ~mM 〉 = |~j〉|φ[(n+1)...(M+1)] 〉 (15) where ~m(~j−~m+~k) i=1 C ji+ki , (16) |φ[(n+1)...(M+1)] ~m(~j−~m+~k) i=1 C ji+ki |~k〉|~j − ~m+ ~k〉/λ[n] . (17) By the same procedure as that of qubit case, we can obtain the following |φ[n...(M+1)] [n]in λ[n]αn |in〉|φ (n+1)...(M+1)] 〉. (18) Then we have [n]in = δαn(~j+~ein+1) jin+1 + 1 [n−1] . (19) Still we can define V [n]in αnαn−1 = Γ [n]in αn−1αnλ αn , and thus we can find that each state |Φ~mM 〉 is a MPS and thus can be sequentially generated. The detailed result of this part will be presented elsewhere [27]. In conclusion, we show that the generalN toM univer- sal quantum cloning machine can be implemented by a se- quential manner. Since the sequential generation of mul- tipartite state can be implemented in various physical se- tups such as microwave and optical cavity QED, trapped ions and quantum dots etc. This general sequential quan- tum cloning machine may be implemented much easier than the single global implementation scheme. This re- duces dramatically the complexity in implementing the general UQCM. We also show that for d-dimensional quantum state, the sequential UQCM is also available. Besides the universal cloning machine, the 1 toM phase- covariant quantum cloning machine can also be sequen- tially implemented. It will be interesting to consider sim- ilarly the generalN toM phase-covariant cloning and the economic phase-covariant cloning. The sequential asym- metric quantum cloning machine may also be an inter- esting topic. Acknowledgements: HF was supported by ”Bairen” program, NSFC and ”973” program (2006CB921107). Appendix.–The explicit form of matrices V are pre- sented as: V [n]0αnαn−1 = δαnαn−1 × ∑M−m−n k=−m X m+αn−1−1+k ∑M−m−n+1 k=−m X M−n+1 m+αn−1−1+k V [n]0αnαn−1 = δαnαn−1+1 × ∑M−m−n k=−m X m+αn−1+k ∑M−m−n+1 k=−m X M−n+1 m+αn−1−1+k where notations X = β2m(αn−1−1+k), X ′ = β2m(αn−1+k) are used. For case 1 < n ≤ M − N + m,αn−1 = 1, ..., n;αn = 1, ..., (n+1), and for caseM−N+m < n ≤ M − 1, αn−1, αn = 1, ..., (M −N +m+1). We can check that the above defined V satisfies the isometry condition V [n]in V [n]in = 1. Similarly we have V [M ]0αMαM−1 = δαMαM−1 × M−1−1−m) M−1−1−m) M−1−1 M−1−m) V [M ]1αMαM−1 = δαM (αM−1+1) × M−1−m) M−1−1−m) M−1−1 M−1−m) where 0 ≤ m ≤ N −m,αM−1, αM = 1, 2, ..., (M − N + m+ 1). For case concerning about ancilla state of the UQCM, assume 1 ≤ l ≤M −N , we have V [M+l]0αM+lαM+l−1 = δαM+l(αM+l−1−1) × αM+l−1 −m− 1 M −N − l + 1 V [M+l]1αM+lαM+l−1 = δαM+lαM+l−1 × M −N − l − αM+l−1 +m+ 1 M −N − l + 1 (1) For (m+ 1) ≤ αM+l ≤ (M −N +m− l+ 1), (m+ 2) ≤ αM+l−1 ≤ (M −N +m− l+ 2), [M+l]0 αM+lαM+l−1 = δαM+l(αM+l−1−1) αM+l−1−m−1 M−N−l+1 . For αM+l = (M −N +m− l + 2), 1 ≤ αM+l−1 ≤ (M −N +m+ 1), V [M+l]0αM+lαM+l−1 = 0. Otherwise [M+l]0 αM+lαM+l−1 = δαM+lαM+l−1 (2) For (m+ 1) ≤ αM+l, αM+l−1 ≤ (M −N +m− l + 1), V [M+l]1αM+lαM+l−1 = δαM+lαM+l−1 M−N−l−αM+l−1+m+2 M−N−l+1 . For αM+l = (M −N +m− l + 2), 1 ≤ αM+l−1 ≤ (M −N +m+ 1), [M+l]0 αM+lαM+l−1 = 0. Otherwise V [M+l]0 αM+lαM+l−1 = δαM+lαM+l−1 [1] C. H. Bennett and D. P. DiVincenzo, Nature 404, 247 (2000). [2] C. H. Bennett, G.Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. Wootters, Phys. Rev. Lett. 70, 1895 (1993). [3] D. Gottesman and I. Chuang, Nature 402, 390 (1999). [4] R. Raussendorf and Hans J. Briegel, Phys. Rev. Lett. 86, 5188 (2000). [5] V. Giovannetti, S. Lloyd, L. Maccone, Nature 412, 417 (2001). [6] A. Barenco, et al, Phys. Rev. A 52, 3457 (1995). [7] C. Saavedra, K. M. Gheri, T. Torma, J. I. Cirac, and P. Zoller, Phys. Rev. A 61, 062311 (2000). [8] C. Schon, E. Solano, F. Verstraete, J. I. Cirac, and M. M. Wolf, Phys. Rev. Lett. 95, 110503 (2005). [9] C. Schon, K. Hammerer, M. M. Wolf, J. I. Cirac, and E. Solano, quant-ph/0612101, [10] A. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett. 59, 799 (1987). [11] V. Scarani, S. Iblisdir, N. Gisin, A. Acin, Rev. Mod. Phys. 77, 1225 (2005). [12] N. J. Cerf, J. Fiurasek, Progress in Optics 49, 455 (Else- vier 2006). [13] H. Fan, Topics in Applied Physics 102, 63 (2006). [14] A. Lamas-Linares, C. Simon, J. C. Howell, and D. Bouwmeester, Science 296, 712 (2002). [15] F. De Martini, V. Bužek, F. Sciarrino, and C. Sias, Na- ture 419 815 (2002). [16] D. Pelliccia, V. Schettini, F. Sciarrino, C. Sias, and F. De Martini, Phys. Rev. A 68, 042306 (2003). [17] M. T. M. Irvine, A. Lamas Linares, M. J. A. de Dood, and D. Bouwmeester, Phys. Rev. Lett. 92, 047902 (2004). [18] M. Ricci, F. Sciarrino, C. Sias, and F. De Martini, Phys. Rev. Lett. 92, 047901 (2004). [19] H. K. Cummins, C. Jones, A. Furze, N. F. Soffe, M. Mosca, J. M. Peach, and J. A. Jones, Phys. Rev. Lett. 88, 187901 (2002). [20] J. F. Du, et al, Phys. Rev. Lett. 94, 040505 (2005). [21] C. Simon, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 84, 2993 (2000). [22] H. Fan, G. Weihs, K. Matsumoto, and X. B. Wang, Phys. Rev. A 67, 022317 (2003). [23] Y. Delgado, L. Lamata, J. Leon, D. Salgado, and E. Solano, Phys. Rev. Lett. , quant-ph/0607105. [24] N. Gisin, and S. Massar, Phys. Rev. Lett. 79, 2153 (1997). [25] H. Fan, K. Matsumoto, and M. Wadati, Phys. Rev. A 64, 064301 (2001). [26] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003). [27] G. F. Dang and H. Fan, in preparation. http://arxiv.org/abs/quant-ph/0612101 http://arxiv.org/abs/quant-ph/0607105

0704.0324

On the pseudospectrum of elliptic quadratic differential operators

ON THE PSEUDOSPECTRUM OF ELLIPTIC QUADRATIC DIFFERENTIAL OPERATORS Karel Pravda-Starov University of California, Berkeley Abstract. We study the pseudospectrum of a class of non-selfadjoint differential operators. Our work consists in a detailed study of the microlocal properties, which rule the spectral stability or instability phenomena appearing under small pertur- bations for elliptic quadratic differential operators. The class of elliptic quadratic differential operators stands for the class of operators defined in the Weyl quantiza- tion by complex-valued elliptic quadratic symbols. We establish in this paper a simple necessary and sufficient condition on the Weyl symbol of these operators, which en- sures the stability of their spectra. When this condition is violated, we prove that it occurs some strong spectral instabilities for the high energies of these operators, in some regions which can be far away from their spectra. We give a precise geo- metrical description of them, which explains the results obtained for these operators in some numerical simulations giving the computation of “false eigenvalues” far from their spectra by algorithms for eigenvalues computing. Key words. Spectral instability, pseudospectrum, semiclassical quasimodes, non- selfadjoint operators, non-normal operators, condition (Ψ), subellipticity. 2000 AMS Subject Classification. 35P05, 35S05. 1. Introduction 1.1. Miscellaneous facts about pseudospectrum. In recent years, there has been a lot of interest in studying the pseudospectrum of non-selfadjoint operators. The study of this notion has been initiated by noticing that for certain problems of sci- ence and engineering involving non-selfadjoint operators, the predictions suggested by spectral analysis do not match with the numerical simulations. This fact lets thinking that in some cases the only knowledge of the spectrum of an operator is not enough to understand sufficiently its action. To supplement this lack of information contained in the spectrum, some new subsets of the complex plane called pseudospectra have been defined. The main idea about the definition of these new subsets is that it is interesting to study not only the points where the resolvent of an operator is not de- fined, i.e. its spectrum, but also where this resolvent is large in norm. This explains the following definition of the ε-pseudospectrum σε(A) of a matrix or an operator A, σε(A) = z ∈ C, ‖(zI −A)−1‖ ≥ 1 for any ε > 0, if we write by convention that ‖(zI − A)−1‖ = +∞ for every point z belonging to the spectrum σ(A) of the operator. Let us mention that there exists an abundant literature about this notion of pseu- dospectrum. We refer here for the definition and some general properties of pseu- dospectra to the paper [15] of L.N. Trefethen. Let us also point out the more recently published book [16], which draws up a wide all-round view of this topic and gives a lot of illustrations. According to the previous definition, studying the pseudospectra of an operator is exactly studying the level lines of the norm of its resolvent. What is interesting in studying such level lines is that it gives some information about the spectral stability http://arxiv.org/abs/0704.0324v1 of the operator. Indeed, pseudospectra can be defined in an equivalent way in term of spectra of perturbations of the operator. For instance, we have for any A ∈ Mn(C), σε(A) = {z ∈ C, z ∈ σ(A +B) for some B ∈ Mn(C) with ‖B‖ ≤ ε}. It follows that a complex number z belongs to the ε-pseudospectrum of a matrix A if and only if it belongs to the spectrum of one of its perturbations A+B with ‖B‖ ≤ ε. More generally, if A is a closed unbounded linear operator with a dense domain on a complex Hilbert space H , the result of Roch and Silbermann in [13] gives that σε(A) = B∈L(H), ‖B‖L(H)≤ε σ(A +B), where L(H) stands for the set of bounded linear operators on H . From this second description, we understand the interest in studying such subsets if we want for example to compute numerically some eigenvalues of an operator. Indeed, we start to do it by discretizing this operator. This discretization and inevitable round-off errors will generate some perturbations of the initial operator. Eventually, algorithms for eigenvalues computing will determine the eigenvalues of a perturbation of the initial operator, i.e. a value in a ε-pseudospectrum of the initial operator but not necessarily a spectral one. This explains why it is important in such numerical computations to understand if the ε-pseudospectra of studied operators contain more or less deeply their spectra. Let us first notice that this study is a priori non-trivial only for non-selfadjoint operators, or more precisely for non-normal operators. Indeed, we have for a normal operator A an exact expression of the norm of its resolvent given by the following classical formula (see for example (V.3.31) in [8]), (1.1.1) ∀z 6∈ σ(A), ‖(zI −A)−1‖ = 1 z, σ(A) where d z, σ(A) stands for the distance between z and the spectrum of the operator, when A is a closed unbounded linear operator with a dense domain on a complex Hilbert space. This formula proves that the resolvent of a normal operator cannot blow up far from its spectrum. It ensures the stability of its spectrum under small perturbations because the ε-pseudospectrum is exactly equal in this case to the ε- neighbourhood of the spectrum (1.1.2) σε(A) = z ∈ C : d z, σ(A) Nevertheless it is well-known that this formula (1.1.1) is no more true for non-normal operators. For such operators, it can occur that their resolvents are very large in norm far from their spectra. This induces that the spectra of these operators can be very unstable under small perturbations. To illustrate this fact, let us consider the case of the rotated harmonic oscillator and the following numerical computation of its spectrum. The rotated harmonic oscillator is a simple example of elliptic quadratic differential operator Hc = D x + cx 2, Dx = i −1∂x, with c = eiπ/4. The numerical computation is performed on the matrix discretization (HcΨi,Ψj)L2(R) 1≤i,j≤N where N is an integer taken equal to 100 and (Ψj)j∈N∗ stands for the basis of L composed by Hermite functions. The black dots appearing on this computation stand for the numerically computed eigenvalues. We can notice on this numerical simulation that the computed low energies are very close to theoretical ones since the spectrum Figure 1. Computation of some level lines of the norm of the resol- vent ‖(Hc− z)−1‖ = ε−1 for the rotated harmonic oscillator Hc with c = eiπ/4. The right column gives the corresponding values of log10 ε. 0 20 40 60 80 100 120 140 160 dim = 100 of the rotated harmonic oscillator is only composed of eigenvalues regularly spaced out on the half-line eiπ/8R∗+, σ(Hc) = {eiπ/8(2n+ 1) : n ∈ N}. However we notice that it is no more true for the high energies. It occurs for them some strong spectral instabilities, which lead to the computation of “false eigenvalues” far from the half-line eiπ/8R∗+. Let us mention that some comparable computations can be found in [3]. In this paper, we are interested in studying when and how this kind of phenomena occurs in the class of elliptic quadratic differential operators. 1.2. Elliptic quadratic differential operators. We study here the class of elliptic quadratic differential operators. It is the class of pseudodifferential operators defined in the Weyl quantization (1.2.1) q(x, ξ)wu(x) = (2π)n ei(x−y).ξq (x+ y u(y)dydξ, by some symbols q(x, ξ), where (x, ξ) ∈ Rn×Rn and n ∈ N∗, which are some complex- valued elliptic quadratic forms i.e. complex-valued quadratic forms verifying (1.2.2) (x, ξ) ∈ Rn × Rn, q(x, ξ) = 0 ⇒ (x, ξ) = (0, 0). Let us first notice that since the symbols of these operators are some quadratic forms, these are only some differential operators, which are a priori non-selfadjoint because their Weyl symbols are complex-valued. As mentioned before, the rotated harmonic oscillator is an example of such an operator since we have D2x + e iθx2 = (ξ2 + eiθx2)w, 0 < θ < π, if Dx = i −1∂x. This operator is a very simple example of non-selfadjoint operator for which we have noticed on the previous numerical simulation that it occurs some strong spectral instabilities under small perturbations for its high energies. These phenomena have been studied in several recent works. We can mention in particular the works of L.S. Boulton [1], E.B. Davies [3], K. Pravda-Starov [10] and M. Zworski [18], which have given a good understanding of these phenomena. A question, which has been at the origin of this work, has been to study if these phenomena peculiar to the rotated harmonic oscillator are representative, or not, of what occurs more generally in the class of elliptic quadratic differential operators in every dimension. We have tried to answer to the following questions: - Does it always occur some strong spectral instabilities under small perturba- tions for the high energies of these operators ? - If it is not the case, is it possible to give a necessary and sufficient condition on the Weyl symbols of these operators, which ensures their spectral stability ? - Can we precisely describe the geometry, which separates the regions of the resolvent sets where the resolvents of these operators blow up in norm from the ones where one keeps a control on their sizes ? To understand these spectral stability or instability phenomena, we need to study the microlocal properties, which rule these phenomena in the class of elliptic quadratic differential operators. Let us mention that it is M. Zworski who first underlined in [18] the close link between these questions of spectral instabilities and some results of microlocal analysis about the solvability of pseudodifferential operators. 1.3. Semiclassical pseudospectrum. To answer to these previous questions, it is interesting to use a semiclassical setting and to study a notion of pseudospectrum in this new setting. We define for a semiclassical family (Ph)0<h≤1 of operators on L2(Rn), with a domain D, the following notions of semiclassical pseudospectra. Definition 1.3.1. For all µ ≥ 0, the set Λscµ (Ph) = z ∈ C : ∀C > 0, ∀h0 > 0, ∃ 0 < h < h0, ‖(Ph − z)−1‖ ≥ Ch−µ is called semiclassical pseudospectrum of index µ of the semiclassical family (Ph)0<h≤1. The semiclassical pseudospectrum of infinite index is defined by Λsc∞(Ph) = Λscµ (Ph). With this definition, the points in the complement of the semiclassical pseudospectrum of index µ are the points of the complex plane where we have the following control of the resolvent’s norm for sufficiently small values of the semiclassical parameter h, (1.3.1) ∃C > 0, ∃h0 > 0, ∀ 0 < h < h0, ‖(Ph − z)−1‖ < Ch−µ. To prove the existence of semiclassical pseudospectrum of index µ, we will study the question of existence of semiclassical quasimodes (1.3.2) ∀C > 0, ∀h0 > 0, ∃ 0 < h < h0, ∃uh ∈ D, ‖uh‖L2(Rn) = 1 and ‖Phuh − zuh‖L2(Rn) ≤ Chµ, in some points z of the resolvent set, which can be considered as some “almost eigen- values” in O(hµ) in the semiclassical limit. Let us notice that the definition chosen here for the notions of semiclassical pseudospectra differ from the one given in [5] for a semiclassical pseudodifferential operator. In fact, we have chosen a definition for semiclassical pseudospectra inspired by the remark made p.388 in [5], because this definition only depends on the properties of the semiclassical operator rather than on its symbol. The interest of working in a semiclassical setting is a matter of geometry. We can explain this choice by the fact that it is easier for an elliptic quadratic differential oper- ator q(x, ξ)w to describe the geometry of semiclassical pseudospectra of its associated semiclassical operator (q(x, hξ)w)0<h≤1, than to describe directly the geometry of its ε-pseudospectra. The semiclassical setting is particularly well-adapted for the study of elliptic quadratic differential operators because there exists a simple link between this semiclassical setting and the quantum one. Indeed, using that the symbols of these operators are some quadratic forms q, we obtain from the change of variables, y = h1/2x with h > 0, the following identity between the quantum operator q(x, ξ)w and its associated semiclassical operator (q(x, hξ)w)0<h≤1, (1.3.3) q(x, ξ)w − z q(y, hη)w − z if z ∈ C. This identity allows to get some information about the resolvent’s norm behaviour of the quantum operator q(x, ξ)w − z if we have some information about semiclassical pseudospectra for its associated semi- classical operator. Let us mention for example that if a non-zero complex number z belongs to the semiclassical pseudospectrum of infinite index of the operator (q(x, hξ)w)0<h≤1, the identity (1.3.3) induces that the resolvent’s norm of the quantum operator blows up along the half-line zR+ with a rate faster than any polynomials (1.3.4) ∀N ∈ N, ∀C > 0, ∀η0 ≥ 1, ∃η ≥ η0, ‖ q(x, ξ)w − zη )−1‖ ≥ CηN , and this, even if this half-line zR+ does not intersect the spectrum of the opera- tor q(x, ξ)w. Conversely, in the case where z 6∈ Λscµ q(y, hη)w , z 6= 0 and 0 ≤ µ ≤ 1, the identity (1.3.3) shows that we can find some positive constants C1 and C2 such that the resolvent of the operator q(x, ξ)w remains bounded in norm in some regions of the resolvent set of the shape (1.3.5) u ∈ C : |u| ≥ C1, d(∆, u) ≤ C2|proj∆u|1−µ ∩ C \ σ q(x, ξ)w where ∆ = zR+ and proj∆u stands for the orthogonal projection of u on the closed half-line ∆. Indeed, we obtain from (1.3.1) and (1.3.3) that ∃C > 0, ∃η0 ≥ 1, ∀η ≥ η0, q(x, ξ)w − ηeiargz ∥ < Cηµ−1, which induces that for all v ∈ D q(x, ξ)w and η ≥ η0, q(x, ξ)w − ηeiargz L2(Rn) ≥ C−1η1−µ‖v‖L2(Rn), q(x, ξ)w stands for the domain of the operator q(x, ξ)w . Then, we can find a constant η̃0 ≥ 1 such that if z̃ belongs to u ∈ C : |u| ≥ η̃0, d(eiargzR+, u) ≤ 2−1C−1|projeiargzR+u| ∩ C\σ q(x, ξ)w |projeiargzR+ z̃| ≥ η0. This induces using the previous estimates and the triangular inequality that if z̃ belongs to u ∈ C : |u| ≥ η̃0, d(eiargzR+, u) ≤ 2−1C−1|projeiargzR+u| ∩ C\σ q(x, ξ)w we have for all v ∈ D q(x, ξ)w q(x, ξ)w − z̃ q(x, ξ)w − projeiargzR+ z̃ eiargzR+, z̃ ‖v‖L2 ≥ 2−1C−1|projeiargzR+ z̃| 1−µ‖v‖L2 ≥ 2−1C−1η1−µ0 ‖v‖L2, because µ ≤ 1. This last estimate shows that the resolvent of the operator q(x, ξ)w is bounded in norm by 2Cη 0 on the set u ∈ C : |u| ≥ η̃0, d(eiargzR+, u) ≤ 2−1C−1|projeiargzR+u| ∩ C\σ q(x, ξ)w We notice that depending directly on the value of the index µ, 0 ≤ µ < 1, the previous set contains more or less deeply in its interior the half-line {u ∈ C : |u| ≥ η̃0, u ∈ zR+}. This fact explains why in the following we will precise carefully the index of the semiclassical pseudospectrum to which a point does not belong when there is no semiclassical pseudospectrum of infinite index in that point. 2. Statement of the results 2.1. Some notations and some preliminary facts about elliptic quadratic differential operators. Let us begin by giving some notations and recalling known results about elliptic quadratic differential operators. Let q be a complex-valued elliptic quadratic form q : Rnx × Rnξ → C (x, ξ) 7→ q(x, ξ), with n ∈ N∗, i.e. a complex-valued quadratic form verifying (1.2.2). The numerical range Σ(q) of q is defined by the subset in the complex plane of all values taken by this symbol (2.1.1) Σ(q) = q(Rnx × Rnξ ), and the Hamilton map F ∈ M2n(C) associated to the quadratic form q is uniquely defined by the identity (2.1.2) q (x, ξ); (y, η) (x, ξ), F (y, η) , (x, ξ) ∈ R2n, (y, η) ∈ R2n, where q stands for the polar form associated to the quadratic form q and σ is the symplectic form on R2n, (2.1.3) σ (x, ξ), (y, η) = ξ.y − x.η, (x, ξ) ∈ R2n, (y, η) ∈ R2n. Let us first notice that this Hamilton map F is skew-symmetric with respect to σ. This is just a consequence of the properties of skew-symmetry of the symplectic form and symmetry of the polar form (2.1.4) ∀X,Y ∈ R2n, σ(X,FY ) = q(X ;Y ) = q(Y ;X) = σ(Y, FX) = −σ(FX, Y ). Under this assumption of ellipticity, the numerical range of a quadratic form can only take some very particular shapes. It is a consequence of the following result proved by J. Sjöstrand (Lemma 3.1 in [14]), Proposition 2.1.1. Let q : Rnx × Rnξ → C a complex-valued elliptic quadratic form. If n ≥ 2, then there exists z ∈ C∗ such that Re(zq) is a positive definite quadratic form. If n = 1, the same result is fulfilled if we assume besides that Σ(q) 6= C. This proposition shows that the numerical range of an elliptic quadratic form can only take two shapes. The first possible shape is when Σ(q) is equal to the whole complex plane. This case can only occur in dimension n = 1. The second possible shape is when Σ(q) is equal to a closed angular sector with a top in 0 and an opening strictly lower than π. Figure 2. Shape of the numerical range Σ(q) when Σ(q) 6= C. Σ(zq) Indeed, if Σ(q) 6= C, using that the set Σ(q) is a semi-cone tq(x, ξ) = q( tξ), t ∈ R+, (x, ξ) ∈ R2n, because q is a quadratic form, we have Σ(q) = R+z if z is the non-zero complex number given by the proposition 2.1.1 and I is the compact interval I = 1 + i Im(zq)(K), where K is the following compact subset of R2n, (x, ξ) ∈ R2n : Re(zq)(x, ξ) = 1 The compactness of K is a direct consequence of the fact that Re(zq) is a positive definite quadratic form. Elliptic quadratic differential operators define some Fredholm operators (see Lemma 3.1 in [6] or Theorem 3.5 in [14]), (2.1.5) q(x, ξ)w + z : B → L2(Rn), where B is the Hilbert space (2.1.6) u ∈ L2(Rn) : xαDβxu ∈ L2(Rn) if |α+ β| ≤ 2 with the norm ‖u‖2B = |α+β|≤2 ‖xαDβxu‖2L2(Rn). The Fredholm index of the operator q(x, ξ)w + z is independent of z and is equal to 0 if n ≥ 2. In the case where n = 1, this index can take the values −2, 0 or 2. More precisely, this index is always equal to 0 if Σ(q) 6= C. In the following, we will always assume that Σ(q) 6= C. Under this assumption, J. Sjöstrand has proved in the theorem 3.5 in [14] (see also Lemma 3.2 and Theorem 3.3 in [6]) that the spectrum of an elliptic quadratic differential operator q(x, ξ)w : B → L2(Rn), is only composed of eigenvalues with finite multiplicity (2.1.7) σ q(x, ξ)w λ∈σ(F ), −iλ∈Σ(q)\{0} rλ + 2kλ (−iλ) : kλ ∈ N where F is the Hamilton map associated to the quadratic form q and rλ is the dimen- sion of the space of generalized eigenvectors of F in C2n belonging to the eigenvalue λ ∈ C. Let us notice that the spectra of these operators is always included in the numerical range of their Weyl symbols. To end this review of preliminary properties of elliptic quadratic differential oper- ators, let us underline that the property of normality in this class of operators can be easily checked by computing the Poisson bracket of the real part and the imaginary part of their symbols (2.1.8) {Re q, Im q} = ∂Re q ∂Im q ∂Re q ∂Im q Proposition 2.1.2. An elliptic quadratic differential operator q(x, ξ)w : B → L2(Rn), n ∈ N∗, is normal if and only if the quadratic form defined by the Poisson bracket of the real part and the imaginary part of its symbol is equal to zero (2.1.9) ∀(x, ξ) ∈ R2n, {Re q, Im q}(x, ξ) = 0. Proof of Proposition 2.1.2. This proposition is a direct consequence of the composition formula in Weyl calculus (see Theorem 18.5.4 in [7]), which induces that the Weyl symbol of the commutator [qw, (qw)∗] = [qw, qw] = −2i[(Re q)w, (Im q)w], is equal to −2i(Re q ♯ Im q − Im q ♯ Re q) = −2{Re q, Im q}, because Re q and Im q are some quadratic forms. The notation Re q ♯ Im q stands for the Weyl symbol of the operator obtained by composition (Req)w(Imq)w. � Remark. Let us notice that the symplectic invariance of the Poisson bracket (see (21.1.4) in [7]), (2.1.10) {(Re q) ◦ χ, (Im q) ◦ χ} = {Re q, Im q} ◦ χ, if χ stands for a linear symplectic transformation of R2n, implies that the condition (2.1.9) is symplectically invariant. 2.2. Statement of the main results. Let us consider an elliptic quadratic differ- ential operator q(x, ξ)w : B → L2(Rn). We know from (2.1.7) that the spectrum of this operator is contained in the numerical range of its symbol Σ(q). The following proposition gives a first localization of the regions where the resolvent can blow up in norm and where spectral instabilities can occur. Proposition 2.2.1. Let q : Rn × Rn → C, n ∈ N∗, be a complex-valued elliptic quadratic form. We have ∀z 6∈ Σ(q), q(x, ξ)w − z ∥ ≤ 1 z,Σ(q) where d z,Σ(q) stands for the distance from z to the numerical range Σ(q). This result shows that the resolvent of an elliptic quadratic differential operator cannot blow up in norm far from the numerical range of its symbol. We are now going to study what kind of phenomena can occur in this particular set. There are two cases to separate according to the property of normality or non-normality of the operator. 2.2.1. Case of a normal operator. Let us consider a normal elliptic quadratic differ- ential operator q(x, ξ)w : B → L2(Rn). Let us recall that according to the proposition 2.1.2 this property of normality is exactly equivalent to the fact that ∀(x, ξ) ∈ R2n, {Re q, Im q}(x, ξ) = 0. In this case, we have the classical formula (1.1.1) for its resolvent’s norm (2.2.1) ∀z 6∈ σ q(x, ξ)w q(x, ξ)w − z z, σ(q(x, ξ)w) which induces that the ε-pseudospectrum of this operator is exactly equal to the ε-neighbourhood of its spectrum q(x, ξ)w z ∈ C : d z, σ(q(x, ξ)w) , ε > 0. This classical formula (2.2.1) ensures that the resolvent cannot blow up in norm far from the spectrum and induces that the spectrum of such an operator is stable under small perturbations. Example 1. The operator (2.2.2) q1(x, ξ) w = −(1 + i)∂2x1 − ∂ + 4(−1 + i)x1∂x1 + 2(−1 + i)x2∂x1 + 6ix2∂x2 + 2ix1∂x2 + (6 + 5i)x 1 + (11 + i)x 2 + (10 + 4i)x1x2 − 2 + 5i, is an example of a normal elliptic quadratic differential operator. Its spectrum is given q1(x, ξ) (2k1 + 1) + (2k2 + 1) 4 : (k1, k2) ∈ N2 Figure 3. Spectrum and a ε-pseudospectrum of the operator q1(x, ξ) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Σ(q1) Example 2. Let us notice that when the numerical range Σ(q) is reduced to a closed half-line, the elliptic quadratic differential operator q(x, ξ)w is always normal since {Re q, Im q} = |z|2{Re(z−1q), Im(z−1q)} = 0, if z ∈ C∗ is chosen such that Im(z−1q) = 0. In fact, the operator q(x, ξ)w can in this particular case be reduced after a conjugation by a unitary operator on L2(Rn) to the operator + x2j), where λj > 0 for all j = 1, ..., n. Figure 4. Example of a normal elliptic quadratic differential operator. 2.2.2. Case of a non-normal operator. Let us consider a non-normal elliptic quadratic differential operator q(x, ξ)w : B → L2(Rn), n ∈ N∗. We assume in the following that the numerical range Σ(q) is distinct from the whole complex plane (2.2.3) Σ(q) 6= C. As mentioned in the section 2.1, this additional assumption is always fulfilled in dimension n ≥ 2. It only excludes some very particular one-dimensional elliptic quadratic differential operators (see the remark following the proposition 2.2.2 for more precision about these operators). Under this additional assumption, the numerical range Σ(q) is always a closed angular sector with a top in 0 and a positive opening strictly lower than π. 2.2.2.a. On the pseudospectrum at the interior of the numerical range. Let us consider the associated semiclassical elliptic quadratic differential operator (q(x, hξ)w)0<h≤1. We can build in every point of the interior of the numerical range Σ̊(q) some semi- classical quasimodes. Theorem 2.2.1. If the elliptic quadratic differential operator q(x, ξ)w : B → L2(Rn), n ∈ N∗, is non-normal and verifies Σ(q) 6= C then for all z ∈ Σ̊(q) and N ∈ N, there exist h0 > 0 and a semiclassical family (uh)0<h≤h0 ∈ S(Rn) such that ‖uh‖L2(Rn) = 1 and ‖q(x, hξ)wuh − zuh‖L2(Rn) = O(hN ) when h → 0+. This result induces the existence of semiclassical pseudospectrum of infinite index in every point of the interior of the numerical range Σ̊(q). According to (1.3.4), this result in the semiclassical setting induces that the resol- vent’s norm of the quantum operator q(x, ξ)w blows up fastly along all the half-lines belonging to the interior of the numerical range Σ̊(q), (2.2.4) ∀z ∈ Σ̊(q), ∀N ∈ N, ∀C > 0, ∀η0 ≥ 1, ∃η ≥ η0, ‖ q(x, ξ)w − zη )−1‖ ≥ CηN . We deduce from (2.1.7) that as soon as an elliptic quadratic differential operator is non-normal its resolvent blows up in norm in some regions of the resolvent set far from its spectrum. This fact induces that the high energies of such an operator are very unstable under small perturbations as we have already noticed on the numerical computation performed for the rotated harmonic oscillator. It follows that in the class of elliptic quadratic differential operators1 the property of spectral stability is exactly equivalent to the property of normality: σ(q(x, ξ)w) is stable under ⇔ q(x, ξ)w is a normal ⇔ {Re q, Im q} = 0. small perturbations operator By spectral stability, we mean here that the resolvent of these operators cannot blow up in norm far from their spectra. Let us add that it is not very surprising to have this property of spectral stability under the assumption of normality, but it is worth 1If we exclude the one-dimensional particular cases previously mentioned. noticing that as soon as this property is violated, it occurs in this class of operators some strong spectral instabilities under small perturbations for their high energies. Examples. The two following operators (2.2.5) q2(x, ξ) w = −∂2x1 − 2∂ + 4ix2∂x2 + 2x 1 + (4 + i)x 2 + 4x1x2 + 2i (2.2.6) q3(x, ξ) w = −(1 + i)∂2x1 − 2∂ + 4(−1 + i)x1∂x1 + 2(1− i)x2∂x1 − 4ix1∂x2 + (9 + 4i)x21 + (2 + i)x 2 − 4(1 + i)x1x2 − 2 + 2i, are some examples of non-normal elliptic quadratic differential operators. 2.2.2.b. On the pseudospectrum at the boundary of the numerical range. Let us now study what occurs on the boundary of the numerical range ∂Σ(q) for a non-normal elliptic quadratic differential operator q(x, ξ)w : B → L2(Rn). Let us mention that we always assume that Σ(q) 6= C. Under these assumptions, the boundary of the numerical range is composed of the union of the origin 0 and two half-lines ∆1 and ∆2, (2.2.7) ∂Σ(q) = {0} ⊔∆1 ⊔∆2, that we can write (2.2.8) ∆1 = z1R + and ∆2 = z2R + with z1, z2 ∈ ∂Σ(q) \ {0}. We need to define a notion of order for the symbol q(x, ξ) on these two half-lines ∆j , j = 1, 2. Let us begin by recalling the classical definition of the order k(x0, ξ0) of a symbol p(x, ξ) at a point (x0, ξ0) ∈ R2n (see section 27.2, chapter 27 in [7]). This order k(x0, ξ0) is an element of the set N ∪ {+∞} defined by (2.2.9) k(x0, ξ0) = sup j ∈ Z : pI(x0, ξ0) = 0, ∀ 1 ≤ |I| ≤ j where I = (i1, i2, ..., ik) ∈ {1, 2}k, |I| = k and pI stands for the iterated Poisson brackets pI = Hpi1Hpi2 ...Hpik−1 pik , where p1 and p2 are respectively the real and the imaginary part of the symbol p, p = p1 + ip2. The order of a symbol q at a point z is then defined as the maximal order of the symbol p = q − z at every point (x0, ξ0) ∈ R2n verifying p(x0, ξ0) = q(x0, ξ0)− z = 0. Let us underline that the symplectic invariance of the Poisson bracket (2.1.10) induces the same property for the order of a symbol at a point. Since here the symbol q is a quadratic form, all the iterated Poisson brackets are also some quadratic forms. This property of degree two homogeneity of these Poisson brackets induces that the symbol q has the same order at every point of each half-line ∆j , j = 1, 2. This allows to define the order of the symbol q on the half-line ∆j by defining this order by this common value. Let us mention that this order can be finite or infinite. Examples. One can easily check that the Weyl symbol ξ2 + eiθx2, 0 < θ < π, of the rotated harmonic oscillator has an order equal to 2 on the both half-lines R∗+ and eiθR∗+, which composes the boundary of its numerical range. The symbol q2 of the operator defined in (2.2.5) has an order equal to 2 on iR∗+ and to 6 on R Σ(q2) = {z ∈ C : Re z ≥ 0, Im z ≥ 0}. On the other hand, we can verify that the symbol q3 of the operator defined in (2.2.6) is of infinite order on the half-line R∗+ and has an order equal to 2 on e iπ/4R∗+, Σ(q3) = {0} ∪ {z ∈ C∗ : 0 ≤ arg z ≤ π/4}. In the case where the symbol is of finite order on a half-line ∆j , j = 1, 2, we have the following result. Theorem 2.2.2. If the Weyl symbol q(x, ξ) of a non-normal elliptic quadratic differ- ential operator is of finite order kj on the half-line ∆j , j ∈ {1, 2}, ∆j ⊂ ∂Σ(q) \ {0}, then this order is necessary even and there is no semiclassical pseudospectrum of index kj/(kj + 1) on ∆j for the associated semiclassical operator ∆j ⊂ C \ Λsckj/(kj+1) q(x, hξ)w Remark. Let us mention that we can more precisely establish that in dimension n ≥ 1, the order kj is an even integer verifying 2 ≤ kj ≤ 4n− 2. This result is proved in [12]. By rephrasing this result in a quantum setting, it follows from (1.3.5) and (2.1.7) that when the symbol q of a non-normal elliptic quadratic differential operator q(x, ξ)w is of finite order kj on a half-line ∆j , j ∈ {1, 2}, ∆j ⊂ ∂Σ(q) \ {0}, then the resolvent of this operator remains bounded in norm in a set of the following (2.2.10) u ∈ C : |u| ≥ C1, d(∆j , u) ≤ C2|proj∆ju| where C1 and C2 are some positive constants. As we will see in its proof, this absence of semiclassical pseudospectrum is linked to some properties of subellipticity. Let us just underline for the moment that the index kj/(kj + 1), which appears in this result is exactly equal to the loss appearing in the subelliptic estimate hidden behind this result. About the case of infinite order, the situation is much more complicated. Never- theless, we can first notice in this case that we cannot expect to prove a stronger result than an absence of semiclassical pseudospectrum of index 1. Indeed, we can easily check on the example of the operator q3(x, ξ) w defined in (2.2.6) that its spectrum is given by q3(x, ξ) (2k1 + 1) 2 + (2k2 + 1)3 8 : (k1, k2) ∈ N2 We recall that the spectrum of this operator is only composed of eigenvalues and that its symbol is of infinite order on R∗+. It follows from the structure of the spectrum and (1.3.5) that if there is no semiclassical pseudospectrum of infinite index in a point of the half-line R∗+, there is necessary no semiclassical pseudospectrum of index µ with an index µ ≥ 1. In fact, we can prove by using a result of exponential decay in time for the norm of contraction semigroups generated by elliptic quadratic differential operators (see [12]) that there is never some semiclassical pseudospectrum of index 1 on all these half-lines of infinite order. Let us mention that this result of exponential decay will not be proved here but it will be explained in the following how it induces the absence of semiclassical pseudospectrum of index 1. 2.2.3. About the geometry of ε-pseudospectra for elliptic quadratic differential opera- tors. Let us now explain what are the consequences of these results on the geometry of ε-pseudospectra for elliptic quadratic differential operators. Let us begin by con- sidering the one-dimensional case which is a bit particular. In dimension n = 1, an elliptic quadratic differential operator can be reduced after a similitude and a conju- gation by a unitary operator to the harmonic oscillator or to the rotated harmonic oscillator. Proposition 2.2.2. Let us consider q : R×R → C a complex-valued elliptic quadratic form such that Σ(q) 6= C. For all h > 0, there exist a unitary operator (more precisely a metaplectic operator) Uh on L 2(R), which is an automorphism of the spaces S(R) and B, z ∈ C∗ and θ ∈ [0, π[ such that ∀h > 0, q(x, hξ)w = zUh (hDx) 2 + eiθx2 U−1h . Remark. In the case where Σ(q) = C, an elliptic quadratic differential operator q(x, ξ)w can be reduced after a similitude and a conjugation by a unitary operator on L2(Rn) to the operator defined in the Weyl quantization by the symbol (ξ + ix)(ξ + ηx) with η ∈ C, Im η > 0, (ξ − ix)(ξ + ηx) with η ∈ C, Im η < 0, depending on the value of its Fredholm index, which is equal to −2 in the first case and to 2 in the second one. As we will see in the following, this proposition allows us to reduce the study of a one-dimensional non-normal elliptic quadratic differential operator verifying Σ(q) 6= C, to the one of the rotated harmonic oscillator Hθ = D x + e iθx2, 0 < θ < π. Let us mention that the previous results (Theorem 2.2.1 and Theorem 2.2.2) were already known in the particular case of the rotated harmonic oscillator. Indeed, the existence of semiclassical quasimodes inducing the presence of semiclassical pseu- dospectrum of infinite index in every point of the interior of the numerical range for the associated semiclassical operator, is a direct consequence of a result proved by E.B. Davies in [4] (Theorem 1). About the absence of semiclassical pseudospectrum of index 2/3 on the boundary of the numerical range, this result has been proved for the rotated harmonic oscillator in [10]2. As proved in [10], this absence of semiclassical pseudospectrum allows to give a proof of a conjecture stated by L.S. Boulton in [1]. It deals with the geometry of ε- pseudospectra for the rotated harmonic oscillator. Let us now recall some facts about this conjecture and some results proved by L.S. Boulton in [1]. 2Let us recall that the value of the order is equal to 2 in this case. L.S. Boulton has first proved (Theorem 3.3 in [1]) that the resolvent of the rotated harmonic oscillator blows up in norm along all a family of curves of the following form η 7→ bη + eiθηp, where b and p are some positive constants verifying 1/3 < p < 3, (2.2.11) Hθ − (bη + eiθηp) ∥ → +∞ when η → +∞. On the other hand, he also proved that the resolvent of this operator remains bounded in norm on two half-stripes parallel to the half-lines R+ or e iθR+. More precisely, he proved that there exist some positive constants d and Md such that (2.2.12) sup , 0≤b≤d Hθ − (η + ib) ∥ ≤ Md, (2.2.13) sup , 0≤b≤d Hθ − eiθ(η − ib) ∥ ≤ Md. These bounds provide some information about the shape of ε-pseudospectra of the operator Hθ. Indeed, L.S. Boulton has proved using these results that for all suffi- ciently small value of the positive parameter ε, the ε-pseudospectra of the rotated harmonic oscillator is contained in the shaded set appearing on the following figure. The eigenvalues appear on this figure marked by some ⋄. Figure 5. A first localization of the ε-pseudospectra of the rotated harmonic oscillator. More precisely, L.S. Boulton proved that for all 0 < δ < 1 and m ∈ N, there exists a positive constant ε0 such that for all 0 < ε < ε0, (2.2.14) σε(Hθ) ⊂ {z ∈ C : |z − λn| < δ} ∪ λm+1 − δeiθ/2 + Sθ where λn = e iθ/2(2n+ 1), n ∈ N Sθ = {z ∈ C∗ : 0 ≤ arg z ≤ θ} ∪ {0}. In fact, in view of some numerical calculations performed by E.B. Davies in [3], L.S. Boulton has conjectured that the index p = 1/3 appearing in (2.2.11) is the critical one in the following sense: Let us consider 0 < p < 1/3, 0 < δ < 1 and m ∈ N. If bm,p and E are some positive constants verifying bm,pE + e iθEp = λm and ∀η > E, arg zη < θ/2, where zη = bm,pη + e iθηp, let us set Ωm,p = |zη|eiα ∈ C : η ≥ E, arg zη ≤ α ≤ arg(zηeiθ) L.S. Boulton has conjectured the following result. Boulton’s conjecture. There exists ε0 > 0 such that for all 0 < ε < ε0, (2.2.15) σε(Hθ) ⊂ {z ∈ C : |z − λn| < δ} ∪ Ωm,p. The absence of semiclassical pseudospectrum of index 2/3 on the boundary of the numerical range ∂Σ(q)\{0} for the rotated harmonic oscillator3 given by the theorem 2.2.2 shows that this index 1/3 is actually the critical one. Indeed, we can deduce (2.2.15) from (2.2.10) (see [10] for more details) since here kj = 2, j ∈ {1, 2}. As we will see, this theorem 2.2.2 is a consequence of a subelliptic estimate for gen- eral semiclassical pseudodifferential operators proved by N. Dencker, J. Sjöstrand and M. Zworski in [5] (Theorem 1.4). In the particular case of the rotated harmonic oscil- lator, a more elementary proof of this result using only some non-trivial localization scheme in the frequency variable is given in [10]. Let us notice that this inclusion (2.2.15) allows to give a sharp description of the ε- pseudospectra of the rotated harmonic oscillator, which is optimal in view of (2.2.11). Figure 6. Shape of the ε-pseudospectra of the rotated harmonic oscillator. By coming back to the case of an arbitrary dimension n ≥ 1, let us finally underline that using the theorem 2.2.2, we can give similar descriptions of the ε-pseudospectra for non-normal elliptic quadratic differential operators, to the one given by L.S. Boul- ton for the rotated harmonic oscillator, when the symbols of these operators are of finite order on the two open half-lines, which compose the boundary of their numerical ranges. The only difference with the particular case of the rotated harmonic oscillator is that the critical indices, which appear in this description can be different. Indeed, 3The order of the rotated harmonic oscillator’s symbol is equal to 2 on ∂Σ(q) \ {0}. these critical indices depend directly according to (2.2.10) on the order of the symbols on the two half-lines composing the boundary of their numerical ranges. We refer the reader to [10] for more details about the way of getting from (2.2.10) such descriptions of ε-pseudospectra. 3. The proofs of the results Before giving the proofs of the results stated in the previous section, let us begin by recalling the symplectic invariance property of the Weyl quantization (see Theorem 18.5.9 in [7]). This symplectic invariance is actually the most important property of the Weyl quantization. For every affine symplectic transformation χ of R2n, there exists a unitary trans- formation U on L2(Rn), uniquely determined apart from a constant factor of modulus 1, such that U is an automorphism of the spaces S(Rn), B and S ′(Rn), where B is the Hilbert space defined in (2.1.6), and (3.0.1) (a ◦ χ)(x, ξ)w = U−1a(x, ξ)wU, for all a ∈ S ′(R2n). The operator U is a metaplectic operator associated to the affine symplectic transformation χ. This symplectic invariance of the Weyl quantization induces the same property for the semiclassical pseudospectra of elliptic quadratic differential operators in the sense that if q : Rnx × Rnξ → C, is a complex-valued elliptic quadratic form and χ is a linear symplectic transformation of R2n, we have for all µ ∈ [0,∞], (3.0.2) Λscµ (q ◦ χ)(x, hξ)w = Λscµ q(x, hξ)w To prove this fact, let us begin by noticing that for all a ∈ S ′(R2n) and h > 0, we U−1h a(x, ξ) wUh = a(h −1/2x, h1/2ξ)w, where Uhf(x) = h n/4f(h1/2x), since according to the proof of Theorem 18.5.9 in [7], Uh is a metaplectic operator associated to the linear symplectic transformation (x, ξ) 7→ (h−1/2x, h1/2ξ). Let us now consider the case where the symbol a is a quadratic form. The homogeneity property of such a symbol implies that ∀h > 0, a(h−1/2x, h1/2ξ) = 1 a(x, hξ), ∀h > 0, U−1h a(x, ξ) wUh = a(x, hξ)w. If q : Rnx × Rnξ → C is a complex-valued elliptic quadratic form and χ is a linear symplectic transformation of R2n, we can notice that (q ◦ χ)(x, hξ)w , h > 0, is actually an elliptic quadratic differential operator since the symbol q◦χ is an elliptic quadratic form. Let z ∈ C and U be a metaplectic operator associated to the linear symplectic transformation χ. Using that U and Uh are some automorphisms of the Hilbert space B and (3.0.3) U−1h U −1Uhq(x, hξ) wU−1h UUh = U −1hq(x, ξ)wUUh = hU−1h (q ◦ χ)(x, ξ) wUh = (q ◦ χ)(x, hξ)w , we obtain that U−1h U q(x, hξ)w − z U−1h UUh = (q ◦ χ)(x, hξ)w − z Using finally that U−1h U −1Uh is a unitary transformation of L 2(Rn), this identity implies that (q ◦ χ)(x, hξ)w − z q(x, hξ)w − z which proves (3.0.2). In the following, this property of symplectic invariance will allow us to reduce certain symbols to some normal forms by choosing new symplectic coordinates. We can now begin to prove the results stated in the previous section. Let us start by the proof of the proposition 2.2.1. Proof of Proposition 2.2.1. If the numerical range is equal to the whole complex plane, there is nothing to prove. If Σ(q) 6= C, we have seen in the previous section that the numerical range is necessary a closed angular sector with a top in 0 and an opening strictly lower than π. Let us consider z 6∈ Σ(q) and denote by z0 its orthogonal projection on the non- empty closed convex set Σ(q). According to the shape of the numerical range, it follows that z0 belongs to its boundary and that we can find a complex number z1 ∈ C∗, |z1| = 1 such that Σ(z1q) ⊂ z ∈ C : Re z ≥ 0 (3.0.4) z1z ∈ z ∈ C : Re z < 0 z,Σ(q) = d(z1z, iR). Using now that the operator i[Im(z1q)] w is formally skew-selfadjoint, we obtain that for all u ∈ S(Rn), z1q(x, ξ) wu− z1zu, u L2(Rn) = d(z1z, iR)‖u‖2L2(Rn) + z1q(x, ξ) L2(Rn) .(3.0.5) Then, since the quadratic form Re(z1q) is non-negative, we deduce from the symplectic invariance of the Weyl quantization and the theorem 21.5.3 in [7] that there exists a metaplectic operator U such that z1q(x, ξ) = U−1 + x2j) + j=k+1 with k, l ∈ N and λj > 0 for all j = 1, ..., k. By using that U is a unitary operator on L2(Rn), we obtain that the quantity z1q(x, ξ) L2(Rn) ‖DxjUu‖2L2(Rn) + ‖xjUu‖ L2(Rn) j=k+1 ‖xjUu‖2L2(Rn), is non-negative. Then, we can deduce from the Cauchy-Schwarz inequality, (3.0.4) and (3.0.5) that for all u ∈ S(Rn), z,Σ(q) ‖u‖L2(Rn) ≤ |z1| ‖q(x, ξ)wu− zu‖L2(Rn). Finally, using the density of the Schwartz space S(Rn) in B and the fact that |z1| = 1, we obtain that ∀z 6∈ Σ(q), q(x, ξ)w − z ∥ ≤ 1 z,Σ(q) since according to (2.1.7), σ q(x, ξ)w ⊂ Σ(q). � We now consider the one-dimensional case, which is a bit particular. 3.1. The one-dimensional case. In dimension n = 1, we can reduce the study of complex-valued elliptic quadratic forms to exactly three normal forms after a simili- tude and a real linear symplectic transformation. Lemma 3.1.1. Let q : Rx × Rξ → C be a complex-valued elliptic quadratic form in dimension 1. Then, there exists a linear symplectic transformation χ of R2 such that the symbol q ◦ χ is equal to one of the following normal forms: (i) α(ξ2 + eiθx2) with α ∈ C∗, 0 ≤ θ < π. (ii) α(ξ + ix)(ξ + ηx) with α ∈ C∗, η ∈ C, Im η > 0. (iii) α(ξ − ix)(ξ + ηx) with α ∈ C∗, η ∈ C, Im η < 0. In the two last cases (ii) and (iii), the numerical range Σ(q) is equal to the whole complex plane, Σ(q) = C. Proof of Lemma 3.1.1. Let q : R2 → C be a complex-valued elliptic quadratic form. Let us first consider the case where Σ(q) 6= C. We deduce from the proposition 2.1.1 that we can reduce our study to the case where Re q is a positive definite quadratic form. Then, using Lemma 18.6.4 in [7], we can find a real linear symplectic transfor- mation to reduce the quadratic form Re q to the normal form λ(x2 + ξ2), with λ > 0. It follows that there exist some real constants a, b and c such that q(x, ξ) = λ x2 + ξ2 + i(ax2 + 2bxξ + cξ2) Then, we can choose an orthogonal matrix P ∈ O(2,R) diagonalizing the real sym- metric matrix associated to the quadratic form ax2 + 2bxξ + cξ2, with λ1, λ2 ∈ R. If P ∈ O(2,R) \ SO(2,R), we have if σ0 is the matrix with determinant equal to −1, and P̃ = Pσ0. It follows that we can always diagonalize the real symmetric matrix associated to the quadratic form λ−1Im q by conjugating it by an element of SO(2,R). Since the symplectic group is equal in dimension 1 to the group SL(2,R), we can after a linear symplectic transformation of R2 reduce the quadratic form q to x2 + ξ2 + i(γ1x 2 + γ2ξ = α(ξ2 + reiθx2), where γ1, γ2 ∈ R, α ∈ C∗, r > 0 and θ ∈] − π, π[. Let us notice that the elliptic- ity of q actually implies that θ 6≡ π[2π]. Finally, using the real linear symplectic transformation (x, ξ) 7→ (r−1/4x, r1/4ξ), we get a symbol of type (i), αr1/2(ξ2 + eiθx2), if 0 ≤ θ < π. If −π < θ < 0, we need to use besides the real linear symplectic transformation (x, ξ) 7→ (ξ,−x) to obtain a symbol of type (i), 2 eiθ(ξ2 + e−iθx2). Let us now assume that Σ(q) = C. Since the dimension is equal to 1, we can factor the symbol q on C as a polynomial function of degree 2 in the variable ξ. Thus, according to the dependence in the variable x of the polynomial function’s coefficients, we can find some complex numbers λ1, λ2 and α ∈ C∗ such that q(x, ξ) = α(ξ − λ1x)(ξ − λ2x). The ellipticity assumption for the quadratic form q induces that Im λj 6= 0, if j = 1, 2. Using now the linear symplectic transformation (x, ξ) 7→ (x, ξ + Re λ1x), we can assume that (3.1.1) q(x, ξ) = α(ξ − irx)(ξ + bx), with r ∈ R∗ and Im b 6= 0. Let us now check that the assumption Σ(q) = C induces that r Im b < 0. Since (ξ − irx)(ξ + bx) = ξ2 + (b − ir)xξ − irbx2, the condition Σ(q) = C implies that for all (v, w) ∈ R2, there exists a solution (x0, ξ0) ∈ R2 of the system (3.1.2) ξ2 +Re b xξ + r Im b x2 = v xξ(Im b− r) − r Re b x2 = w. Let us first notice that the second equation of (3.1.2) is fulfilled for all w ∈ R only if Im b 6= r. If w 6= 0, it follows from the second equation of (3.1.2) that x0 6= 0 and (3.1.3) ξ0 = w + r Re b x20 (Im b− r)x0 Let us consider the case where v = 0. Using (3.1.3) and the first equation of (3.1.2), we obtain that (w + r Re b x20) 2 +Re b (Im b− r)x20(w + r Re b x20) + r Im b (Im b− r)2x40 = 0. We can rewrite this equation as fw(X0) = 0 if we set X0 = x 0 and (3.1.4) fw(X) = r Im b (Re b)2 + (Im b− r)2 X2 + w Re b (Im b+ r)X + w2. Thus, the condition Σ(q) = C implies that there exists for all w 6= 0, a non-negative solution X0 of the equation fw(X0) = 0. Since the quantity r Im b is assumed to be non-zero, we first study the case where r Im b > 0. In this case, since (3.1.5) f ′w(X) = 2r Im b (Re b)2 + (Im b− r)2 X + w Re b (Im b+ r) 2r Im b (Re b)2 + (Im b− r)2 because Im b 6= r, we have (3.1.6) ∀X ∈ R+, fw(X) ≥ fw(0) = w2 > 0, if w 6= 0 and − w Re b (Im b+ r) 2r Im b (Re b)2 + (Im b− r)2 ) ≤ 0. The estimate (3.1.6) shows that if r Im b > 0, the equation fw(X) = 0 has no non- negative solution for all value of the parameter w 6= 0. This proves that the condition Σ(q) = C induces that r Im b < 0. Using the linear symplectic transformation (x, ξ) 7→ (|r|−1/2x, |r|1/2ξ), we obtain the normal forms (ii) and (iii), α|r|(ξ + ix)(ξ + ηx) with Im η > 0 and α|r|(ξ − ix)(ξ + ηx) with Im η < 0, where η = |r|−1b. Finally, we can easily check that the numerical ranges of the normal forms (ii) and (iii) are actually equal to the whole complex plane C. � Let us notice that the proposition 2.2.2 and the remark following its statement are some direct consequences of the symplectic invariance property of the Weyl quanti- zation (see (3.0.3)) and the previous lemma. We can add that as proved after the lemma 3.1 in [6], the Fredholm indices of the one-dimensional elliptic quadratic dif- ferential operators with symbols of type (i), (ii) and (iii) are respectively equal to 0, −2 and 2. As we have mentioned in the previous section, the results of Theorem 2.2.1 and Theorem 2.2.2 are already known in the particular case of the rotated harmonic oscil- lator. The existence of semiclassical quasimodes inducing the presence of semiclassical pseudospectrum of infinite index in every point of the interior of the numerical range for the associated semiclassical operator, is a direct consequence of a result proved by E.B. Davies in [4] (Theorem 1) and; the absence of semiclassical pseudospectrum of index 2/3 on the boundary of the numerical range has been proved for the ro- tated harmonic oscillator in [10]4. As we have previously mentioned (see (2.1.10) and (3.0.2)), the property of non-normality, the order of symbols and the semiclassical pseudospectra of elliptic quadratic differential operators are symplectically invariant. These properties allow us to reduce by any real linear symplectic transformations the symbols of the elliptic quadratic differential operators that we consider in our proof of the theorem 2.2.1 and the theorem 2.2.2. By using the lemma 3.1.1, we deduce from the results of the theorem 2.2.1 and the theorem 2.2.2 proved for the rotated harmonic oscillator that they are therefore also fulfilled by all non-normal one-dimensional el- liptic quadratic differential operators with a numerical range different from the whole complex plane. We now consider the multidimensional case. As we will see in the following, there is a real jump of complexity between the one-dimensional case and the multidimensional one. This jump is among other things a consequence of the complexity increase of symplectic geometry in dimension n ≥ 2 and the larger diversity appearing in the class of elliptic quadratic differential operators. 4Let us recall that the value of the order is equal to 2 in this case. 3.2. Case of dimension n ≥ 2. We only need to study the case of a non-normal elliptic quadratic differential operator (3.2.1) q(x, ξ)w : B → L2(Rn), in dimension n ≥ 2. Let us recall that in this case, the numerical range Σ(q) is a closed angular sector with a top in 0 and a positive opening strictly lower than π, and that the proposition 2.1.2 gives that (3.2.2) ∃(x0, ξ0) ∈ R2n, {Re q, Im q}(x0, ξ0) 6= 0. Let us begin by studying what occurs at the interior of the numerical range Σ̊(q). 3.2.1. On the pseudospectrum at the interior of the numerical range. To prove the existence of semiclassical quasimodes for the associated semiclassical operator given by the theorem 2.2.1, we need a first purely algebraic step to characterize the points belonging to the interior of the numerical range. Let us consider the following decomposition of the numerical range (3.2.3) Σ(q) = à ⊔ B̃, where (3.2.4) à = z ∈ Σ(q) : ∃(x0, ξ0) ∈ R2n, z = q(x0, ξ0), {Re q, Im q}(x0, ξ0) 6= 0 (3.2.5) B̃ = z ∈ Σ(q) : z = q(x0, ξ0) ⇒ {Re q, Im q}(x0, ξ0) = 0 The next section is devoted to give a geometrical description of these two sets. We establish using purely algebraic arguments that (3.2.6) à = Σ̊(q) and B̃ = ∂Σ(q). This result is a consequence of the geometry induced by the quadratic setting to which the studied symbols belong. Let us begin by noticing that the symplectic invariance of the Poisson bracket (2.1.10) induces the same property for the sets à and B̃. We can therefore use some real linear symplectic transformation to reduce the symbol q. Since {Re(zq), Im(zq)} = |z|2{Re q, Im q}, we deduce from this symplectic invariance, from the proposition 2.1.1 and the lemma 18.6.4 in [7] that after a similitude, we can reduce our study to the case where (3.2.7) Re q(x, ξ) = j + x with λj > 0 for all j = 1, ..., n. 3.2.1.a. Geometrical description of the sets à and B̃. We begin by proving the fol- lowing inclusion (3.2.8) ∂Σ(q) ⊂ B̃. Let us consider z ∈ ∂Σ(q) and (x0, ξ0) ∈ R2n such that z = q(x0, ξ0). This is possible because the numerical range is a closed angular sector. If z = 0, the ellipticity property of q implies that (x0, ξ0) = (0, 0) and {Re q, Im q}(x0, ξ0) = 0, because this Poisson bracket is also a quadratic form. This proves that z ∈ B̃. If z ∈ ∂Σ(q) \ {0}, let us consider the global solution Y of the linear Cauchy problem (3.2.9) Y ′(t) = HRe q Y (t) Y (0) = (x0, ξ0), associated to the Hamilton vector field of the symbol Re q, HRe q = (∂Re q − ∂Re q It is actually a linear Cauchy problem since Re q is a quadratic form. Setting f(t) = Im q Y (t) a direct computation gives that f ′(0) = {Re q, Im q}(x0, ξ0). If f ′(0) 6= 0, we could find t0 6= 0 such that |f(t0)| > |f(0)| = |Im z|. Since Y is the flow associated to the Hamilton vector field of Re q, the quadratic form Re q is constant under it. It follows that for all t ∈ R, Y (t) = Re q Y (0) = Re z and provides a contradiction because, since z ∈ ∂Σ(q) \ {0}, this would imply in view of the shape of the numerical range Σ(q) (see Figure 7) that Y (t0) 6∈ Σ(q). It follows that the Poisson bracket {Re q, Im q}(x0, ξ0) is necessary equal to 0 and Figure 7. q(Y (t that z ∈ B̃. This ends the proof of the inclusion (3.2.8). Let us now assume that (3.2.10) ∂Σ(q) ⊂ B̃, ∂Σ(q) 6= B̃. In this case, we could find (3.2.11) z ∈ B̃ \ ∂Σ(q). Let us first notice that z is necessary non-zero since 0 ∈ ∂Σ(q), and that Re z > 0, since from (3.2.7), (3.2.12) Σ(q) \ {0} ⊂ {z ∈ C∗ : Re z > 0}. The fact that z belongs to the set B̃ implies that (3.2.13) Re q(x, ξ) = Re z Im q(x, ξ) = Im z =⇒ {Re q, Im q}(x, ξ) = 0. We also know that there exists at least one solution to the system appearing in the left-hand-side of (3.2.13). Since from (3.2.7), the quadratic form Re q is positive definite, we can simultaneously reduce the quadratic forms Re q and Im q by finding an isomorphism P of R2n such that in the new coordinates y = P−1(x, ξ), (3.2.14) Re q(Py) = y2j and Im q(Py) = j with α1 ≤ ... ≤ αn. Let us now consider the following quadratic form (3.2.15) p(y) = {Re q, Im q}(Py). We get from (3.2.13) and (3.2.14) that (3.2.16) j=1 y j = Re z j=1 αjy j = Im z =⇒ p(y) = 0. Let us underline that the isomorphism P is not a priori a symplectic transformation and that it does not preserve the Poisson bracket {Re q, Im q}. We consider the two following sets (3.2.17) E1 = y ∈ R2n : r(y) = 0 where (3.2.18) r(y) = (3.2.19) E2 = y ∈ R2n : p(y) = 0 The next lemma gives a first inclusion between these two sets E1 and E2. Lemma 3.2.1. We have (3.2.20) E1 ⊂ E2. Proof of Lemma 3.2.1. Let y ∈ E1. If y = 0 then y belongs to E2 since from (3.2.15), p is a quadratic form in the variable y. If y 6= 0, we set y2j > 0 and ∀j = 1, ..., 2n, ỹj = We recall from (3.2.12) that z ∈ B̃ \ ∂Σ(q) implies that Re z > 0. Then, since, on one hand ỹ2j = Re z, and that, on the other hand, we have from (3.2.17) and (3.2.18) that αj ỹ y2j = Im z, because y ∈ E1, we deduce from (3.2.16) and the homogeneity of degree 2 of the quadratic form p that p(ỹ) = p(y) = 0. According to (3.2.19), this proves that y ∈ E2 and ends the proof of the lemma 3.2.1.� Then, we can notice from (3.2.14) that the boundary of the numerical range ∂Σ(q) is given by (3.2.21) (1 + iα1)R+ ∪ (1 + iαn)R+. Since the numerical range Σ(q) is a closed set, the assumption z ∈ B̃ \ ∂Σ(q) ⊂ Σ(q) \ ∂Σ(q) = Σ̊(q), induces from (3.2.21) that ∈]α1, αn[. This implies that the signature (r1, s1) of the quadratic form r defined in (3.2.18) fulfills (3.2.22) (r1, s1) ∈ N∗ × N∗ and r1 + s1 ≤ 2n. Thus, we can assume after a new labeling that (3.2.23) r(y) = a1y 1 + ...+ ar1y − ar1+1y2r1+1 − ...− ar1+s1y r1+s1 with aj > 0 for all j = 1, ..., r1+ s1. It follows from (3.2.17) and (3.2.23) that in these new coordinates, the set E1 is the direct product of a proper cone C of R r1+s1 and R2n−r1−s1 , (3.2.24) E1 = C × R2n−r1−s1 . Figure 8. We are now going to prove that the two sets E1 and E2 are equal (3.2.25) E1 = E2. Let us reason by the absurd by assuming that it is not the case. Then, we could find from the lemma 3.2.1, (3.2.26) y0 ∈ E2 \ E1, y0 = (y′0, y′′0 ) with y′0 ∈ Rr1+s1 , y′′0 ∈ R2n−r1−s1 . We deduce from (3.2.24) that y′0 6∈ C. Let us now recall an elementary geometrical fact that we will use several times. This fact is that the intersection of a real line and a real quadric surface is reduced to either 0, 1 or 2 points, or the line is completely contained in the quadric surface. We first begin by proving that (3.2.27) Rr1+s1 × {y′′ = y′′0} ⊂ E2. Indeed, let us consider the affine subspace F = {y ∈ R2n : y = (y′, y′′) ∈ Rr1+s1 × R2n−r1−s1 , y′′ = y′′0}. We identify for more simplicity the space F to the space Rr1+s1 . We agree to say that a point x′0 of R r1+s1 belongs to the set E2 to mean that the point (x 0 ) belongs to the set E2. With this convention, it is sufficient for proving the inclusion (3.2.27) to consider some particular lines of Rr1+s1 , containing the point y′0 defined in (3.2.26) and, which have an intersection with the cone C in at least two other different points u′0 and v 0 (see Figure 9). These lines are necessary contained in the quadric surface E2 because from the lemma 3.2.1, E1 ⊂ E2, and that there are at least three different points of intersection between these lines and the quadric surface E2, (u′0, y 0 ) ∈ C × R2n−r1−s1 = E1 ⊂ E2, (v′0, y′′0 ) ∈ C × R2n−r1−s1 = E1 ⊂ E2, and (y′0, y 0 ) ∈ E2. Thus, we prove that the shaded disc appearing on the figure 10 is completely contained in the set E2. By using the cone structure of the set E2, we can deduce that all the interior of the cone C (see Figure 11) is contained in E2. Then, using again other particular intersections with some lines as on the figure 12, we deduce from our identification of the space F to Rr1+s1 that the inclusion (3.2.27) is fulfilled. Figure 9. We now prove that under these conditions, we have the identity (3.2.28) E2 = R Indeed, let us consider (ỹ′0, ỹ 0 ) ∈ R2n = Rr1+s1 × R2n−r1−s1 . If ỹ′0 ∈ C, then (ỹ′0, ỹ 0 ) ∈ E2, Figure 10. These three points belong to E2. The line D is contained in E2. Figure 11. because from (3.2.20) and (3.2.24), (ỹ′0, ỹ 0 ) ∈ E1 and E1 ⊂ E2. If, on the other hand ỹ′0 6∈ C, we can choose a point u ∈ Rr1+s1 different from ỹ′0 such that u 6∈ C, and such that the line containing ỹ′0 and u in R r1+s1 , has an intersection with C in at least two other different points v and w (see Figure 13). Thus, we can find some distinct real numbers t1, t2 ∈ R \ {0, 1} such that v = (1− t1)ỹ′0 + t1u ∈ C and w = (1− t2)ỹ′0 + t2u ∈ C. Considering now the line (1− t)(ỹ′0, ỹ′′0 ) + t(u, y′′0 ) : t ∈ R we can notice that this real line contains at least three different points of E2: (v, (1 − t1)ỹ′′0 + t1y′′0 ), (w, (1 − t2)ỹ′′0 + t2y′′0 ) and (u, y′′0 ). Indeed, this is a consequence of the fact that v and w belong to C, and from (3.2.20), (3.2.24) and (3.2.27). Thus, the line D is contained in the quadric surface E2. This implies that (ỹ′0, ỹ 0 ) ∈ D ⊂ E2. To sum up, we have proved that if the two sets E1 and E2 are different then the set E2 is equal to R 2n. This fact induces in view of (3.2.19) that the quadratic form p is identically equal to zero. By coming back to the first coordinates (x, ξ) = Py, it Figure 12. Figure 13. follows from (3.2.15) that the quadratic form {Re q, Im q} is also identically equal to zero, which contradicts (3.2.2). This proves the identity (3.2.25), E1 = E2. With this fact, we can resume our first reasoning by the absurd, which assume in (3.2.11) the existence of a point z ∈ B̃ \ ∂Σ(q). Let us now consider y0 6∈ E1 = E2. This is possible according to (3.2.2), (3.2.15) and (3.2.19). We deduce from (3.2.17) and (3.2.19) that r(y0) and p(y0) are non-zero. By considering λ ∈ R∗ such that p(y0) = λr(y0) (3.2.29) r̃(y) = p(y)− λr(y), it follows from (3.2.17), (3.2.19), (3.2.25) and (3.2.29) that (3.2.30) E1 ⊂ {y ∈ R2n : r̃(y) = 0}. This inclusion (3.2.30) is strict since r̃(y0) = 0 and y0 6∈ E1. By using now exactly the same reasoning as the one previously described to prove (3.2.25), about the intersections of real lines and quadric surfaces, we prove that the quadratic form r̃ is necessary identically equal to zero. Then, it follows from (3.2.29) (3.2.31) p = λr. By coming back to the first coordinates (x, ξ) = Py, we get using (3.2.14), (3.2.15), (3.2.18) and (3.2.31) that for all (x, ξ) ∈ R2n, (3.2.32) {Re q, Im q}(x, ξ) = λ Im q(x, ξ)− Im z Re q(x, ξ) Let us now consider (x0, ξ0) ∈ R2n such that q(x0, ξ0) ∈ ∂Σ(q) \ {0}. This is possible since the numerical range Σ(q) is a closed angular sector with a top in 0 and a positive opening. We deduce from (3.2.5) and (3.2.8) that we necessarily have {Re q, Im q}(x0, ξ0) = 0. This induces from (3.2.32) that (3.2.33) Im q(x0, ξ0) = Re q(x0, ξ0), because λ ∈ R∗. Since according to the shape of the numerical range Σ(q) and (3.2.12), q(x0, ξ0) ∈ ∂Σ(q) \ {0} ⊂ {z ∈ C : Re z > 0}, the identity (3.2.33) proves that the point z also belongs to the set ∂Σ(q), but it contradicts the initial assumption z ∈ B̃ \ ∂Σ(q). Finally, this ends our reasoning by the absurd and proves (3.2.6). 3.2.1.b. Existence of semiclassical quasimodes at the interior of the numerical range. To prove the existence of semiclassical quasimodes for the associated semiclassical operator (q(x, hξ)w)0<h≤1, in every point of the numerical range’s interior (Theorem 2.2.1), we use an existence result of semiclassical quasimodes for general pseudodifferential operators violating the condition (Ψ)5. Let us mention that this result generalizes the two existence results of semiclassical quasimodes given by E.B. Davies, in the case of Schrödinger operators (Theorem 1 in [4]), and by M. Zworski in [17] and [18], for pseudodifferential operators. This existence result of semiclassical quasimodes can be stated as follows. Let us consider a semiclassical symbol P (x, ξ;h) in S(〈(x, ξ)〉m, dx2 + dξ2) with m ∈ R+, 〈(x, ξ)〉2 = 1 + x2 + ξ2, 5The definition of the condition (Ψ) is recalled below. where S(〈(x, ξ)〉m, dx2 + dξ2) stands for the following symbol class S(〈(x, ξ)〉m, dx2 + dξ2) = a(x, ξ;h) ∈ C∞(Rnx × Rnξ ,C) : ∀α ∈ N2n, sup 0<h≤1 ‖〈(x, ξ)〉−m∂αx,ξa(x, ξ;h)‖L∞(R2n) < +∞ with a semiclassical expansion (3.2.34) P (x, ξ;h) ∼ hjpj(x, ξ), where for all j ∈ N, pj is a symbol of the class S(〈(x, ξ)〉m, dx2 + dξ2) independent from the semiclassical parameter h. Let z ∈ C, we assume that there exists a function q0 ∈ C∞b (R2n,C), where C∞b (R 2n,C) stands for the set of bounded complex-valued functions on R2n with all derivatives bounded, and a bicharacteristic curve, t ∈ [a, b] 7→ γ(t), of the real part Re(q0(p0 − z)) of the symbol q0(p0 − z), with a < b, such that (3.2.35) ∀t ∈ [a, b], q0 6= 0 and q0(γ(a)) p0(γ(a))− z > 0 > Im q0(γ(b)) p0(γ(b))− z Theorem 3.2.1. Under these assumptions (3.2.34) and (3.2.35), for all open neigh- bourhood V of the compact set γ([a, b]) in R2n and for all N ∈ N, there exist h0 > 0 and (uh)0<h≤h0 a semiclassical family in S(Rn) such that ‖uh‖L2(Rn) = 1, FS (uh)0<h≤h0 ⊂ V and ‖P (x, hξ;h)wuh − zuh‖L2(Rn) = O(hN ), when h → 0+. The notation FS (uh)0<h≤h0 stands for the frequency set of the semiclassical fam- ily (uh)0<h≤h0 defined as the complement in R 2n of the set composed by the points (x0, ξ0) ∈ R2n, for which there exists a symbol χ0(x, ξ;h) ∈ S(1, dx2 + dξ2) such that χ0(x0, ξ0;h) = 1 and ‖χ0(x, hξ;h)wuh‖L2(Rn) = O(h∞), when h → 0+. This existence result of semiclassical quasimodes is an adaptation in a semiclassical setting of the proof given by L. Hörmander in [7] for proving that the condition (Ψ) is a necessary condition for the solvability of a pseudodifferential operator (Theorem 26.4.7 in [7]). The existence of this result has been first mentioned in [5]. A complete proof of this adaptation in a semiclassical setting is given in [11]. This result shows that when the principal symbol p0−z of the symbol P−z violates the condition (Ψ), there exists in this point z some semiclassical quasimodes inducing the presence of semiclassical pseudospectrum of infinite index for the semiclassical operator P (x, hξ;h)w. Condition (Ψ). A complex-valued function p ∈ C∞(R2n,C) fulfills the condition (Ψ) if there is no complex-valued function q ∈ C∞(R2n,C) such that the imaginary part Im(qp) of the function qp changes sign from positive values to negative ones along an oriented bicharacteristic of the symbol Re(qp) on which the function q does not vanish. By using the characterization given in the previous section for the interior of the numerical range Σ̊(q) (see (3.2.4) and (3.2.6)), we are now going to prove that the principal symbol q(x, ξ) − z of the semiclassical operator q(x, hξ)w − z, violates the condition (Ψ) for all z in Σ̊(q). This violation of the condition (Ψ) will induce in view of the theorem 3.2.1 that for all z ∈ Σ̊(q) and N ∈ N, we can find a semiclassical quasimode (uh)0<h≤h0 ∈ S(Rn), with h0 > 0, verifying ‖uh‖L2(Rn) = 1 and ‖q(x, hξ)wuh − zuh‖L2(Rn) = O(hN ) when h → 0+, which will end the proof of Theorem 2.2.1. Let us consider z ∈ Σ̊(q). We are now going to prove that there is actually a violation of the condition (Ψ) for the symbol q − z. According to (3.2.4) and (3.2.6), there are two cases to separate. Case 1. Let us assume that there exists (x0, ξ0) ∈ R2n such that (3.2.36) z = q(x0, ξ0), {Re(q − z), Im(q − z)}(x0, ξ0) = {Re q, Im q}(x0, ξ0) < 0. By considering the solution of the following Cauchy problem (3.2.37) Y ′(t) = HRe q Y (t) Y (0) = (x0, ξ0), we define the following function (3.2.38) f(t) = Im q Y (t) − Im q(x0, ξ0). As mentioned before, (3.2.37) is a linear Cauchy problem. It follows that its solution Y is global and that the function f is well-defined on R. A direct computation using (3.2.37) and (3.2.38) gives that for all t ∈ R, (3.2.39) f ′(t) = {Re q, Im q} Y (t) Since from (3.2.36), (3.2.37), (3.2.38) and (3.2.39), f(0) = 0, f ′(0) = {Re q, Im q}(x0, ξ0) < 0 and HRe q−Re z = HRe q, we deduce in this first case that the imaginary part of the function q − z changes sign, at the first order, from positive values to negative ones along the oriented bicharacteristic Y of the symbol Re q−Re z. This proves that the symbol q − z actually violates the condition (Ψ). Case 2. Let us now assume that there exists (x0, ξ0) ∈ R2n such that (3.2.40) z = q(x0, ξ0), {Re(q − z), Im(q − z)}(x0, ξ0) = {Re q, Im q}(x0, ξ0) > 0. We consider as in the previous case, the global solution Y of the Cauchy problem (3.2.37) and the function f defined in (3.2.38). Since from (3.2.37), (3.2.38), (3.2.39) and (3.2.40), (3.2.41) f(0) = 0, f ′(0) = {Re q, Im q}(x0, ξ0) > 0, we deduce this time that the imaginary part of the function q − z also changes sign, at the first order, along the oriented bicharacteristic Y of the symbol Re q − Re z. Nevertheless, this change of sign is done in the “wrong” way. It is a change of sign from negative values to positive ones, which does not induce directly a violation of the condition (Ψ). To check that there is actually a violation of the condition (Ψ) in this second case, we need to study more precisely the behaviour of the function Im q − Im z along this bicharacteristic Y . We deduce from (3.2.41) that there exists ε > 0 such that ∀t ∈ [−ε, ε], f ′(t) > 0, which induces that (3.2.42) f(ε) > 0 and f(−ε) < 0, since from (3.2.41), f(0) = 0. By using the following lemma, we obtain that for all δ > 0, there exists a time t0(δ) > ε such that (3.2.43) |Y t0(δ) − Y (−ε)| < δ. Figure 14. q(Y (�")) z = q(Y (0)) q(Y (")) Lemma 3.2.2. If Y (t) = (x(t), ξ(t)) is the C∞(R,R2n) function solving the linear system of ordinary differential equations Y ′(t) = HRe q Y (t) where Re q is the symbol defined in (3.2.7), then we have ∀t0 ∈ R, ∀ε > 0, ∀M > 0, ∃T1 > M, ∃T2 > M, |Y (t0)− Y (t0 + T1)| < ε and |Y (t0)− Y (t0 − T2)| < ε. Proof of Lemma 3.2.2. If Y (t0) = (a1, ..., an, b1, ..., bn) ∈ R2n, we deduce from (3.2.7) that the function Y (t) = (x(t), ξ(t)) solves the following Cauchy problem ∀j = 1, ..., n, x′j(t) = 2λjξj(t) ξ′j(t) = −2λjxj(t) xj(t0) = aj ξj(t0) = bj. It follows that for all j = 1, ..., n and t ∈ R, (3.2.44) xj(t) = bj sin 2(t− t0)λj + aj cos 2(t− t0)λj ξj(t) = bj cos 2(t− t0)λj − aj sin 2(t− t0)λj Setting βj = λj/π for all j = 1, ..., n, we need to study two different cases. Case 1: ∀j ∈ {1, ..., n}, βj ∈ Q. In this case, the function Y is periodic and the result of Lemma 3.2.2 is obvious. Case 2: (β1, ..., βn) 6∈ Qn. In this second case, we use the following classical result of rational approximation: ∀ε > 0, ∀(θ1, ..., θn) ∈ Rn \Qn, ∃p1, ..., pn ∈ Z, ∃q ∈ N∗ such 0 < sup j=1,...,n If 0 < ε1 < 1/2, we can therefore find some integers p1,1, ..., p1,n ∈ Z and qε1 ∈ N∗ such that 0 < sup j=1,...,n |qε1βj − p1,j | < ε1. j=1,...,n |qε1βj − p1,j | > 0, using again this result of rational approximation, we can find some other integers p2,1, ..., p2,n ∈ Z and qε2 ∈ N∗ such that 0 < sup j=1,...,n |qε2βj − p2,j | < ε2. By using this process, we build some sequences (pm,j)m∈N∗ of Z for j = 1, ..., n, (εm)m∈N∗ of R + and (qεm)m∈N∗ of N ∗ such that for all m ≥ 2, (3.2.45) 0 < sup j=1,...,n |qεmβj − pm,j | < εm = j=1,...,n ∣qεm−1βj − pm−1,j (3.2.46) 0 < εm < The elements of the sequence (qεm)m∈N∗ are necessary two by two different. Indeed, if qεk = qεl for k < l, this would imply according to (3.2.45) and (3.2.46) that ∀j = 1, ..., n, |pk,j − pl,j| ≤ |qεkβj − pk,j |+ |qεlβj − pl,j | < εk + εl < 1, because 0 < ε1 < 1/2, which would induce that ∀j = 1, ..., n, pk,j = pl,j because pk,j and pl,j are some integers; and would contradict (3.2.45) because 0 < sup j=1,...,n |qεlβj − pl,j | < εl ≤ j=1,...,n |qεkβj − pk,j |. Since the sequence (qεm)m∈N∗ is composed of integers two by two different, we can assume after a possible extraction that qεm → +∞ when m → +∞. We deduce from (3.2.44), (3.2.45) and (3.2.46) that Y (t0 + qεm) → Y (t0) when m → +∞. Then, considering (β̃1, ..., β̃n) = (−β1, ...,−βn), we obtain by using the same method a sequence (q̃εm)m∈N∗ of integers such that q̃εm → +∞ and Y (t0 − q̃εm) → Y (t0) when m → +∞. This ends the proof of Lemma 3.2.2. � Since from (3.2.42), f(−ε) < 0, we deduce from (3.2.38) and (3.2.43) that there exists t0 > ε such that f(t0) is arbitrarily close to f(−ε). It follows in particular that we can find t0 > ε such that f(t0) < 0. Since from (3.2.42), f(ε) > 0 and f(t0) < 0, we deduce from (3.2.38) and (3.2.40) that the function t 7→ Im q Y (t) − Im z, changes sign from positive values to negative ones on the interval [ε, t0]. This proves that the imaginary part of the function q−z actually changes sign from positive values to negative ones along the oriented bicharacteristic Y of the symbol Re q−Re z; and that the symbol q − z also violates in this second case the condition (Ψ). This ends the proof of Theorem 2.2.1. 3.2.1.c. Another proof for the existence of semiclassical quasimodes. In the following lines, we give another proof for the existence of semiclassical quasimodes in some points of the numerical range’s interior. The result proved in this section is weaker than the one given by the theorem 2.2.1, since we prove the existence of semiclassical quasimodes in every point of the numerical range’s interior without a finite number of particular half-lines. Let us consider a non-normal elliptic quadratic differential operator (3.2.47) q(x, ξ)w : B → L2(Rn), in dimension n ≥ 2. We assume, as before, that (3.2.7) is fulfilled. Using that the quadratic form Re q is positive definite, we can simultaneously reduce the two quadratic forms Re q and Im q by choosing an isomorphism P of R2n such that in the new coordinates y = P−1(x, ξ), (3.2.48) r1(y) = Re q(Py) = y2j , r2(y) = Im q(Py) = with α1 ≤ ... ≤ αn. Let us study when the differential forms dr1(y) and dr2(y) are linearly dependent on R i.e. when there exist (λ, µ) ∈ R2 \ {(0, 0)} such that (3.2.49) λdr1(y) + µdr2(y) = 0. It follows from (3.2.48) and (3.2.49) that for all j = 1, ..., 2n, (3.2.50) (λ+ µαj)yj = 0. If y 6= 0, then there exists j0 ∈ {1, ..., 2n} such that yj0 6= 0. This implies that (3.2.51) λ+ µαj0 = 0. We deduce from (3.2.50) and (3.2.51) that yj = 0 if αj 6= αj0 . Thus, we obtain that if z ∈ Σ̊(q) \ (1 + iα1)R + ∪ ... ∪ (1 + iαn)R∗+ then the differential forms dRe q and dImq are linearly independent on R in every point of the set q−1(z). Figure 15. (1 + i� (1 + i� (1 + i� (1 + i� (1 + i� Let us consider such a point z ∈ Σ̊(q) \ (1 + iα1)R + ∪ ... ∪ (1 + iαn)R∗+ Since the dimension n ≥ 2, we can apply the lemma 3.1 in [5] (see also the lemma 8.1 in [9]). It follows that for any compact, connected component Γ of q−1(z), we have (3.2.52) {Re q, Im q}(ρ)λq,z(dρ) = 0, where λq,z stands for the Liouville measure on q −1(z), λq,z ∧ dRe q ∧ dIm q = The set q−1(z) is a non-empty submanifold of codimension 2 in R2n. We deduce from (3.2.4) and (3.2.6) that there exist (x0, ξ0) ∈ q−1(z) such that (3.2.53) {Re q, Im q}(x0, ξ0) 6= 0. Then, it follows from (3.2.52) and (3.2.53) that there necessary exists (x̃0, ξ̃0) ∈ q−1(z) such that (3.2.54) {Re q, Im q}(x̃0, ξ̃0) < 0. Under this condition (3.2.54), we can use the reasoning given in the first studied case (see (3.2.36)) to prove that the imaginary part of the function q−z changes sign, at the first order, from positive values to negative ones along an oriented bicharacteristic of the symbol Re q−Re z. This induces that the symbol q−z violates the condition (Ψ); and we can conclude by using the theorem 3.2.1. Let us mention that we can also directly use the existence result of semiclassical quasimodes given by M. Zworski in [17] and [18]. This second proof gives the existence of semiclassical quasimodes in every point belonging to the set Σ̊(q) \ (1 + iα1)R + ∪ ... ∪ (1 + iαn)R∗+ 3.2.2. On the pseudospectrum at the boundary of the numerical range. In this section, we give a proof of the theorem 2.2.2. Let us consider a non-normal elliptic quadratic differential operator q(x, ξ)w : B → L2(Rn), in dimension n ≥ 1. We assume that Σ(q) 6= C, and that its Weyl symbol q(x, ξ) is of finite order kj on a half-line ∆j , j ∈ {1, 2} (See the definition given in (2.2.9)), which composes the boundary of its numerical range (3.2.55) ∂Σ(q) = {0} ⊔∆1 ⊔∆2. As we have already done several times, we can reduce our study to case where (3.2.7) is fulfilled. Proof of Theorem 2.2.2. Let us consider the following symbol belonging to the C∞b (R 2n,C) space, composed of bounded complex-valued functions on R2n with all derivatives bounded (3.2.56) r(x, ξ) = q(x, ξ) − z 1 + x2 + ξ2 with z ∈ ∆j . Setting Σ̃(r) = r(R2n), we can first notice that z ∈ ∂Σ(q) \ {0} ⇒ 0 ∈ ∂Σ̃(r). Let us also notice that the symbol r fulfills the principal-type condition in 0. Indeed, if (x0, ξ0) ∈ R2n was such that r(x0, ξ0) = 0 and dr(x0, ξ0) = 0, we would get from (3.2.56) that (3.2.57) dq(x0, ξ0) = 0. Since from (3.2.7) and (3.2.57), we have dRe q(x0, ξ0) = 2 (x0)jdxj + (ξ0)jdξj this would imply that (x0, ξ0) = (0, 0), q(x0, ξ0) = 0, because q is a quadratic form and that λj > 0 for all j = 1, ..., n. On the other hand, since r(x0, ξ0) = 0, we get from (3.2.56) that q(x0, ξ0) = z 6= 0 because z ∈ ∆j ⊂ ∂Σ(q) \ {0}, which induces a contradiction. It follows that the symbol r actually fulfills the principal-type condition in 0. Let us notice that, since symbol q is of finite order kj in z, this induces in view of (3.2.56) that the symbol r is also of finite order kj in 0. On the other hand, we deduce from (3.2.7) and (3.2.56) that the set {(x, ξ) ∈ R2n : r(x, ξ) = 0} = {(x, ξ) ∈ R2n : q(x, ξ) = z}, is compact. Under these conditions, we can apply the theorem 1.4 in [5], which proves that the integer kj is even and gives the existence of positive constants h0 and C1 such that (3.2.58) ∀ 0 < h < h0, ∀u ∈ S(Rn), ‖r(x, hξ)wu‖L2(Rn) ≥ C1h kj+1 ‖u‖L2(Rn). Remark. We did not check the dynamical condition (1.7) in [5], because this assump- tion is not necessary for the proof of Theorem 1.4. Indeed, this proof only use a part of the proof of lemma 4.1 in [5] (a part of the second paragraph), where this condition (1.7) is not needed. By using some results of symbolic calculus given by Theorem 18.5.4 in [7] and (3.2.56), we can write (3.2.59) r(x, hξ)w(1 + x2 + h2ξ2)w = q(x, hξ)w − z + hr1(x, hξ)w + h2r2(x, hξ)w , (3.2.60) r1(x, ξ) = −ix (x, ξ) + iξ (x, ξ) (3.2.61) r2(x, ξ) = − (x, ξ) − 1 (x, ξ). We can easily check from (3.2.56) that these functions r1 and r2 belong to the space C∞b (R 2n,C), and we deduce from the Calderón-Vaillancourt theorem that there exists a positive constant C2 such that for all u ∈ S(Rn) and 0 < h ≤ 1, (3.2.62) ‖r1(x, hξ)wu‖L2 ≤ C2‖u‖L2 and ‖r2(x, hξ)wu‖L2 ≤ C2‖u‖L2. It follows from (3.2.58), (3.2.59), (3.2.62) and the triangular inequality that for all u ∈ S(Rn) and 0 < h < h0, kj+1 ‖(1 + x2 + h2ξ2)wu‖L2(Rn) ≤ ‖r(x, hξ)w(1 + x2 + h2ξ2)wu‖L2(Rn) ≤ ‖q(x, hξ)wu− zu‖L2(Rn) + C2h(1 + h)‖u‖L2(Rn). Since from the Cauchy-Schwarz inequality, we have for all u ∈ S(Rn) and 0 < h ≤ 1, ‖u‖2L2(Rn) ≤ ‖u‖2L2(Rn) + ‖xu‖2L2(Rn) + ‖hDxu‖2L2(Rn) (1 + x2 + h2ξ2)wu, u L2(Rn) ≤ ‖(1 + x2 + h2ξ2)wu‖L2(Rn)‖u‖L2(Rn), we obtain that for all u ∈ S(Rn) and 0 < h < h0, (3.2.63) C1h kj+1 ‖u‖L2(Rn) ≤ ‖q(x, hξ)wu− zu‖L2(Rn) + C2h(1 + h)‖u‖L2(Rn). Since kj ≥ 1, we deduce from (3.2.63) that there exist some positive constants h′0 and C3 such that for all 0 < h < h 0 and u ∈ S(Rn), ‖q(x, hξ)wu− zu‖L2(Rn) ≥ C3h kj+1 ‖u‖L2(Rn). Using that the Schwartz space S(Rn) is dense in B and that the operator q(x, hξ)w + z, is a Fredholm operator of index 0, we obtain that for all 0 < h < h′0, q(x, hξ)w − z ∥ ≤ C−13 h kj+1 , which ends the proof of Theorem 2.2.2. � About the case of infinite order, the situation is much more complicated. As mentioned before, we cannot expect to prove a stronger result than an absence of semiclassical pseudospectrum of index 1, but we can actually prove that there is never some semiclassical pseudospectrum of index 1 on every half-line of infinite order, by using a result of exponential decay in time for the norm of contraction semigroups generated by elliptic quadratic differential operators proved in [12]. The result proved in [12] shows that the norm of a contraction semigroup ‖etq(x,ξ) ‖L(L2), t ≥ 0, generated by an elliptic quadratic differential operator q(x, ξ)w with a Weyl symbol verifying Re q ≤ 0, ∃(x0, ξ0) ∈ R2n, Re q(x0, ξ0) 6= 0, decreases exponentially in time (3.2.64) ∃M,a > 0, ∀t ≥ 0, ‖etq(x,ξ) ‖L(L2) ≤ Me−at. Let us consider a non-normal elliptic quadratic differential operator q(x, ξ)w : B → L2(Rn), in dimension n ≥ 1 such that Σ(q) 6= C. We explain in the following lines how (3.2.64) allows to prove that there is never some semiclassical pseudospectrum of index 1 on any open half-lines composing the boundary of the numerical range ∂Σ(q) \ {0}. Let z ∈ ∂Σ(q)\{0}. Since the numerical range Σ(q) is a closed angular sector with a top in 0 and a positive opening strictly lower than π, we can find ε ∈ {±1} such (3.2.65) Re(εiz−1q) ≤ 0, ∃(x0, ξ0) ∈ R2n, Re(εiz−1q)(x0, ξ0) 6= 0. Using the theorem 2.8 in [2], we obtain that for all η ∈ R, q(x, ξ)w − ηz = − iz−1ε εiη − εiz−1q(x, ξ)w = − iz−1ε e−iεηsesεiz −1q(x,ξ)wds.(3.2.66) It follows from (3.2.64) and (3.2.65) that for all η ∈ R, q(x, ξ)w − ηz ∥ ≤ |z|−1 ‖esεiz −1q(x,ξ)w‖L(L2)ds ≤ |z|−1 Me−asds = |z|−1M < +∞, which proves the absence of semiclassical pseudospectrum of index 1 on the half-line zR∗+. We can actually use the theorem 2.8 in [2] because iR ⊂ C \ σ εiz−1q(x, ξ)w Indeed, if it was not the case, we would deduce from (2.1.7) that there exists u0 ∈ B \ {0} and λ0 ∈ R such that εiz−1q(x, ξ)wu0 = iλ0u0. Since from (3.2.65), the quadratic form −Re(εiz−1q) is non-negative, we deduce from the symplectic invariance of the Weyl quantization and the theorem 21.5.3 in [7] that there exists a metaplectic operator U such that (3.2.67) − εiz−1q(x, ξ) = U−1 + x2j ) + j=k+1 with k, l ∈ N and λj > 0 for all j = 1, ..., k. By using that U is a unitary operator on L2(Rn), we obtain that 0 = − Re(iλ0u0, u0)L2 = − Re εiz−1q(x, ξ)wu0, u0 εiz−1q(x, ξ) u0, u0 ‖DxjUu0‖2L2 + ‖xjUu0‖2L2 j=k+1 ‖xjUu0‖2L2 , which induces that u0 = 0, because from (3.2.65) and (3.2.67), k + l ≥ 1. It follows from (2.1.7) that there exists ε0 > 0 such that εiz−1q(x, ξ)w ⊂ {z ∈ C : Re z ≤ −ε0}. References [1] L.S.Boulton, Non-self-adjoint harmonic oscillator semigroups and pseudospectra, J. Operator Theory, 47, 413-429 (2002). [2] E.B.Davies, One-Parameter Semigroups, Academic Press, London (1980). [3] E.B.Davies, Pseudospectra, the harmonic oscillator and complex resonances, Proc. R. Soc. Lond. A, 455, 585-599 (1999). [4] E.B.Davies, Semi-classical states for non-self-adjoint Schrödinger operators, Comm. Math. Phys., 200, 35-41 (1999). [5] N.Dencker, J.Sjöstrand, M.Zworski, Pseudospectra of Semiclassical (Pseudo-)Differential Op- erators, Comm. Pure Appl. Math., 57, 384-415 (2004). [6] L.Hörmander, A Class of Hypoelliptic Pseudodifferential Operators with Double Characteristics, Math. Ann., 217, 165-188 (1975). [7] L.Hörmander, The analysis of linear partial differential operators (vol. I,II,III,IV), Springer Verlag (1985). [8] T.Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin (1980). [9] A.Melin, J.Sjöstrand, Determinants of pseudodifferential operators and complex deformations of phase space, Methods Appl. Anal., 9, no.2, 177-237 (2002). [10] K.Pravda-Starov, A complete study of the pseudo-spectrum for the rotated harmonic oscillator, J. London Math. Soc. (2) 73, 745-761 (2006). [11] K.Pravda-Starov, Etude du pseudo-spectre d’opérateurs non auto-adjoints, PhD Thesis of the University of Rennes 1, France (2006). [12] K.Pravda-Starov, Contraction semigroups of elliptic quadratic differential operators, preprint (2007). [13] S.Roch, B.Silbermann, C∗-algebra techniques in numerical analysis, J. Oper. Theory 35, 241- 280 (1996). [14] J.Sjöstrand, Parametrices for pseudodifferential operators with multiple characteristics, Ark. för Mat., 12, 85-130 (1974). [15] L.N.Trefethen, Pseudospectra of linear operators, Siam Review 39, 383-400 (1997). [16] L.N.Trefethen, M.Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press (2005). [17] M.Zworski, A remark on a paper of E.B.Davies, Proc. Am. Math. Soc., 129, 2955-2957 (2001). [18] M.Zworski, Numerical linear algebra and solvability of partial differential equations, Comm. Math. Phys., 229, 293-307 (2002). Department of Mathematics, University of California, Evans Hall, Berke- ley, CA 94720, USA E-mail address: [email protected] 1. Introduction 1.1. Miscellaneous facts about pseudospectrum 1.2. Elliptic quadratic differential operators 1.3. Semiclassical pseudospectrum 2. Statement of the results 2.1. Some notations and some preliminary facts about elliptic quadratic differential operators 2.2. Statement of the main results 3. The proofs of the results 3.1. The one-dimensional case 3.2. Case of dimension n 2 References

0704.0325

Fluctuation-dissipation relation on a Melde string in a turbulent flow, considerations on a "dynamical temperature"

8 Fluctuation-dissipation relation on a Melde string in a turbulent flow, considerations on a “dynamical temperature”. V Grenard, N B Garnier and A Naert. Université de Lyon, Laboratoire de Physique, École Normale Supérieure de Lyon, 46 Allée d’Italie, 69364 Lyon Cedex 07, France. E-mail: [email protected] PACS numbers: 05.70.Ln PACS numbers: 05.40.-a PACS numbers: 05.20.Jj Abstract. We report on measurements of the transverse fluctuations of a string in a turbulent air jet flow. Harmonic modes are excited by the fluctuating drag force, at different wave-numbers. This simple mechanical probe makes it possible to measure excitations of the flow at specific scales, averaged over space and time: it is a scale-resolved, global measurement. We also measure the dissipation associated to the string motion, and we consider the ratio of the fluctuations over dissipation (FDR). In an exploratory approach, we investigate the concept of effective temperature defined through the FDR. We compare our observations with other definitions of temperature in turbulence. From the theory of Kolmogorov (1941), we derive the exponent −11/3 expected for the spectrum of the fluctuations. This simple model and our experimental results are in good agreement, over the range of wave-numbers, and Reynolds number accessible (74000 ≤ Re ≤ 170000). 1. Introduction Turbulent flows exhibit a notoriously complex and unpredictable dynamics: they present a huge number of degrees of freedom, and their dynamics are both far from equilibrium and dissipative [1, 2, 3]. The kinetic energy injected at large scale by shear instability mecanisms is dissipated into heat by the molecular viscosity at small scales. That is, dissipation and injection scales are distinct. Therefore, a transport process through scales is necessary for a flow to be stationary. It is suspected that instability mechanisms associated with non-linearities generate harmonics, therefore transfering energy to smaller scales almost without dissipation. An equivalent picture would consist in vortices stretching each other in such a way that a non-zero energy transfer occurs toward smaller scales. This picture of cascade process was first proposed by Richardson [4]. The cascade stops approximately in the range of scales where the viscosity becomes efficient to damp velocity gradients. In the late thirties, Kolmogorov derived from this idea a phenomenological theory accounting for the fluctuations of various observables in fully developed turbulence [5]. In the present work, we are http://arxiv.org/abs/0704.0325v3 Measurements of a dynamical temperature in turbulence. 2 neither concerned by the large (energy injection) scales, nor by the small (dissipation) scales, but by the intermediate range. In this intermediate inertial range, we study the transport process through scales, expected to be universal. Instead of scale l, one often refers to the wave-number k = 2π/l. The control parameter of the flow is the Reynolds number: Re = V L , where L is the macroscopic scale of the flow (integral scale, or correlation length), V is a characteristic shear velocity at large scale, and ν is the kinematic viscosity of the fluid. It is also the mean ratio of the inertial by the dissipative contribution of the forcing over a fluid particle. Interesting predictions were derived by Kolmogorov (1941), that we use in the following. Especially, the range of scales over which fluctuations occur scales as Re3/4. The prediction for the exponent of the power spectral density as 〈|ṽ|2〉 ∝ k−5/3 is among the most famous successes of this theory [1, 2, 3]. Our experimental system is discribed in detail in the next section. It is a thin string held by its ends at constant tension across a turbulent flow. To formalize briefly, it is an oscillator with multiple resonances, coupled to a particular ’thermostat’: the turbulent flow. This string is used to probe the inertial range of a flow of high enough Reynolds numbers. The device is ’calibrated’ by measuring the average (complex) response to an external perturbation, and then used to measure the free fluctuations caused by turbulence alone. Measurement of the displacement r(t) caused by the turbulent forcing f(t) is performed with small piezoelectric transducers. We measure the average response, i.e. the displacement on one end caused by a known broad band forcing on the other end. Then, measurements of the displacement on one end alone give information on the forcing fluctuations. Our study goes a step forward, in an exploratory way. Knowing the average response function of the string and measuring r(t), we invoque a version of the Fluctuation-Dissipation Theorem extended out of equilibrium, to define an effective temperature of the turbulent flow. This effective temperature happends to be scale-dependant. In this work, fully developped turbulence is addressed from the point of view of statistical mechanics. We first recall one important break-through: the statement of the Fluctuation-Dissipation Theorem (FDT). Consider a pair of conjugate variables (displacement r and force f) of a small system in thermal contact with a large heat reservoir. In the present case the small system is the string, coupled to the turbulent flow which is the reservoir. Displacement r and force f are conjugate in the sense that their product is the work exerted by the flow on the string. The theorem originates from the idea that spontaneous fluctuations r(t) should have the same statistical properties as the relaxation of r(t) after the removal of an external forcing perturbation. The main hypothesis needed to derive this theorem are: – linear response between f and r, – thermal equilibrium between the system under consideration and the thermostat, – thermal equilibrium of the thermostat itself. The response function Hr,f is such that: r(t) = Hx,f (t − t ′)f(t′)dt′. Equivalently it can be written in the Fourier space as: r̃(ω) = H̃r,f f̃(ω). Under some hypothesis, the fluctuations of r (its 2-times correlation function) are linked by a very simple relation with the dissipative response of the system to a perturbation of the conjugate variable f (imaginary part of the average response function). It is simply proportional, and the coefficient is nothing but the temperature multiplied by the Boltzman constant: kBT [6]. The validity of the hypothesis has to be discussed in each case. If they are satisfied, the correlation function of the spontaneous fluctuations is proportional to the response function, i.e. the factor is unique and constant. Moreover, this factor Measurements of a dynamical temperature in turbulence. 3 is the same for all couples of conjugate variables, and this factor is kBT , where T is the temperature of the system. The Boltzman constant kB ≃ 1.38 10 −23JK−1 is an universal constant. This relation can be expressed in spectral variables: 〈|r̃(ω)|2〉 = 2 kBT Im[H̃r,f (ω)]. (1) In this expression of the FDT, 〈|r̃(ω)|2〉 is the power spectral density of the fluctuations of the displacement r, as H̃r,f(ω) is the response function on r to the conjugate variable f . Because the string is very thin, the drag is purely viscous. It is therefore proportional to the velocity, which is in quadrature with the displacement. The dissipation is therefore proportional to the imaginary part of the average response function: Im[H̃]. In the perspective of constructing a non-equilibrium thermodynamics, the FDT has been reconsidered by L. Cugliandolo and J. Kurchan, while investigating amorphous materials relaxing after a thermal quench through the glass transition [7, 8]. We present in the following an exploratory approach of the question of turbulent fluctuations using their extended formalism. The Fluctuation-Dissipation Ratio (FDR) can be rewritten: ω 〈r̃(ω)2〉 Im[H̃r,f(ω)] = 2 kBTeff.(ω), (2) where the temperature is replaced by an ’effective’ temperature Teff., function of frequency ω. The frequency dependence of Teff. expresses the fact that different degrees of freedom are not at equilibrium with each other, resulting in internal energy fluxes. In other words, in our system, each (independent) mode of the string couples to (non-independent) scale of the flow. As the flow is stationary, we average our measurements on time, and finally obtain the frequency dependance of Teff. as defined by equation 2. Measurements of the fluctuations of the string give Fourier components of the excitation of the flow. We measure independently the fluctuations, and the complex average response function to a specified excitation, in a way discussed below. We propose to analyse these measurements with the criteria discussed above. The paper is organised as follows. The next section describes the experimental setup, turbulent flow properties, and the setting of the string. General properties of a vibrating Melde string are also discussed. The measurements are shown in section 3: response, fluctuations, and the Fluctuation Dissipation Ratio of this system. In section 4, we derive from Kolmogorov’s theory a simple scaling model for the fluctuations of the drag, and therefore the FDR, which accounts for the exponent observed in the whole range of accessible Re. The section 5 is devoted to a discussion of our results, especially in comparison to several definitions of temperature in turbulence proposed in the literature. 2. The Melde string and the experimental setup The experimental setup is sketched in Fig. 1. A turbulent air jet originates from a nozzle of diameter 5 cm. The flow facility we used is thoroughly described in [9]. A thin stainless steel string of length 60 cm is located 2 m downstream the nozzle, perpendicular to the axis of the flow. At this distance, the length of the string is about the diameter of the turbulent jet. The displacement of the string is measured using piezoelectric multi-layer ceramics at each end of the string. A piezo Measurements of a dynamical temperature in turbulence. 4 is deformed by a voltage. Reciprocally, if the ceramic in compressed, a voltage is generated. The relation between voltage and deformation is linear, and the frequency response is almost flat in the frequency range we consider here. It can be used as actuator or sensor. We have two piezos, one on each end of the string. The two different measurements we perform are the following. 1) complex response function: one (input) piezo is feeded with a white noise voltage through a power amplifier. The source is that of a HP3562A signal analyser. Standing transverse waves appear in the string, weakly perturbed by the turbulent fluctuations. Mecanical displacement on the other end is transformed into a voltage by the other (output) piezo. It must be amplified, and both input and output voltages are recorded synchronously with a 24 bits A/D converter. The acquisition frequency is 50 kHz. We call response the time averaged ratio of the voltage amplitudes on input and output piezos, recorded simultaneously. Voltages in and out are proportional respectively to the displacement and the constraint (on the piezos). The dimension of the actual response is the inverse of a stiffness, as what we measure is the ratio of voltages. Dimentional prefactors are omited for simplicity, as they are constant for the same setup (string and transducers). The diameter of the string is 100 µm, less than the viscous scale of the flow which is about η ≃ 170 µm at the largest Re accessible. The equation of motion of the PIEZOS STAND Figure 1. Eperimental setup: the thin steel wire is pulled across a turbulent air jet by a 4 Kg weight on a rigid stand. Piezoelectric transducers are in mecanical contact with the wire at each end. undamped and unforced string is a linear wave equation. Its solutions with fixed ends are standing waves r(x, t) = A cos(ωn t − knx), where A is the amplitude, t is time and x is position along the wire. The discrete wave numbers are kn = n , where L Measurements of a dynamical temperature in turbulence. 5 is the length of the string and n is a positive integer. In a first approximation, the waves are not dispersive: ωn = c kn, where c is the phase velocity. T is the tension of the string and µ its mass per unit length, c = T/µ ≃ 300 m/s. With a 4 kg weight on one end, the string’s fundamental frequency is f0 = 344 Hz. Dissipation is mainly due to friction on air, and causes little dispersion. More precise treatment would require terms of dissipation in the wire itself and in the piezoelectric transducers that fix the ends. We neglect this, as the amplitude remains small (a few tens of micrometers) if compared to the length of the ceramic pile (3mm), or even the wire diameter (100µm). The possible coupling with compression wave is not relevant, as the range of frequency is distinct. (Compression wave speed in steel is a few thousands of m/s, larger than what we consider here: c ≃ 300 m/s.) When this wire is immersed into the turbulent flow, the resonant modes are excited by the drag forcing. The quantities measured are averaged along the wire. They are therefore global in space but local in scale, or more precisely in Fourier-space. The vortices at scale l are expected to excite modes of wave-number k = 2π/l. In that sense, the string is acting like a mechanical spectrometer, almost exactly like a Fabry-Perot interferometer. 3. Measurements Modulus of the response function is plotted in Fig. 2. It shows that the resonance peaks are indeed very narrow, ensuring a very precise selection of wave-numbers: the quality factor is approximately Q ≃ 4000. The imaginary part of the response function is giving the dissipation. The width of the peaks in the modulus is also Figure 2. Modulus of the response function versus the harmonic number, at Re = 154000. The abscissa is given in non-dimensional coordinates, normalised by the fundamental frequency. linked to the dissipation, as well as the damping time after a perturbation. We used in the following the measurement of the imaginary part of the response, but checked that these different methods coincide. Only the resonant frequencies are considered in this study, as they are much more sensitive to the velocity fluctuations. This is Measurements of a dynamical temperature in turbulence. 6 especially important at large k, as the kinetic energy of the flow is small. Spectrum of the fluctuation excited by the turbulent drag is shown in Fig. 3. Fluctuations resonance peaks are clearly identified. Spurious vibrations are visible, mainly caused by the vibrations of the stand. Because the peaks are very thin, long acquisitions are necessary, as well as large windows for the FFT calculations (150000 points), in order to achieve a sufficient resolution (0.33Hz). The protocol we used to find the resonance frequencies, the value of the amplitude of fluctuations, and imaginary part of the response, is the following. Resonance frequency is obtained by spline smoothing each peak around the maximum amplitude of the response. Then, imaginary part is measured after being also smoothed. The amplitude of the fluctuations peaks are collected on the spectrum, after local smoothing around the maxima. One can see the Figure 3. Spectrum of the resonance modes of the string excited by turbulent drag fluctuations, at Re = 154000. FDR in Fig. 4, called kBTeff., for several values of Re. Uncertainties on this ratio have multiple origins. Errors indicated by the size of the symbols are those coming from the determination of the resonance frequencies. Spurious vibrations of the stand are difficult to handle: we perform measurements of response and fluctuations in the same conditions, to reduce its influence on the ratio. We believe the scattering of the points in Fig. 4 comes mainly from the weakening of signal/noise ratio for large frequencies, simply because there is less energy in the flow at large k, especially at small Re. The only possible escape on this point is to improve the coupling between the string and the sensors. The wave-number has been rescaled with the internal viscous scale η ∝ Re−3/4. The ordinates have been rescaled by an estimated number of degrees of freedom: (L/η)3 ∝ Re9/4. These Re scalings are both usual consequences from Kolmogorov’s theory. In other words, the “thermal energy” kBTeff. that the FDR is representing in the framework of Cugliandolo et al ’s theory, is given per degree of freedom. Assuming the number of degrees of freedom is the total number of particles of size η in the total volume is usual, but crude. A more realistic description should involve correlations between them, reducing this number. However, all the curves collapse to a single power-law with this scaling. The exponent is discussed in the Measurements of a dynamical temperature in turbulence. 7 Figure 4. Spectrum of the FDR, labelled as thermal agitation per degree of freedom. Axis are rescaled with proper Reynolds number dependence, between 74000 and 170000. The size of the symbols represents the uncertainty in the determination of the maxima of the peaks. The solid line is a k−11/3 power-law given as an eye guide. following section. Please note that the equipartition of energy at equilibrium would require this spectrum to be constant. There is no equilibrium between the Fourier modes, because of the energy flux through scales. Moreover, they are not independent, and probably not Gaussian. There is no reason to expect equipartition. Considering a kinematik temperature as poportional to the kinetic energy, like in the kinetic theory of gases, it would be: T ∝ 〈ṽ2〉. And, because of Kolmogorov’s theory it would scale as k−5/3. The dependance we observe with our definition is much steeper. 4. Scaling law Because the susceptibility of the string is very high at resonance, the half-wave-length modes nλ/2 match with velocity structures of scale l (n is an integer). Therefore, the wave number of the standing wave in the string k = n 2π/λ is the same as k = 2π/l. The necessary condition for this matching is resonance. It also ensures that velocities of the string and fluid equalise, which is crucial for the following argument. Displacement is proportional to the drag forcing, itself proportional to velocity, as drag is viscous: the string diameter-based Reynolds number is small (about 10). The Melde string is not dispersive: ω = 2πf = ck, c being the wave velocity. Therefore, the displacement is r = v/ω = v/(ck), and its power spectrum is: 〈r̃(ω)2〉 = 〈ṽ(ω)2〉(ck)−2 ∝ k−11/3. Because the viscous dissipation at each resonance is proportional to frequency, the FDR of Eq. 2 is simply proportional to c k 〈r̃(ω)2〉 ∝ k−11/3. Following Eq. 2, an effective “thermal agitation” defined by the FDR would be: kBTeff. ∝ k −11/3, in the inertial range of fully developed turbulence. This exponent is compatible with the spectrum we measured, as can be seen in Fig. Measurements of a dynamical temperature in turbulence. 8 5. Discussion Theoretical characterisation of turbulence in terms of temperature were proposed in the past by several authors. The temperatures as defined by T. M. Brown [10] and B. Castaing [11] do not depend on k throughout the inertial range. The qualitative idea is that the cascade transport process is efficient enough to equalise a quantity they call temperature. In another model invoking an extremum principle, B. Castaing proposed a definition of temperature, which might depend on scale [12]. In any case, none of these theories invoke the FDR. On different basis, R. Robert and J. Sommeria proposed a definition of temperature [13], only valid for 2D turbulence. It is not expected to apply in a 3D flow. Now, let’s consider our experimental results from the perspective of the three points of reflexion we proposed in the first section, in relation with the FDT. 1- Linear response: as we mentioned, the coupling between the string and the flow is purely viscous. Therefore, drag force is proportional to velocity: f(t) = γ v(t), γ being a friction coefficient. It is also the time-derivative of the position f(t) = γ ω r(t). Response is linear in r, but the coefficient depends on frequency. 2- Are fluctuations and dissipation proportional ? As we have seen, the measurements of the FDR are consistent with a k−11/3 scaling, it is definitely not constant with respect to k. As our system is out of equilibrium but stationary, there is no time evolution like the relaxation of glasses. 3- Setting a string in a turbulent flow allows to perform measurements on a couple of conjugate force-displacement variables. We have no other set of observables to compare with, for now. We may ask whether what we measure is actually a temperature, in a dynamical sense. If one assumes that each mode of the string is a harmonic oscillator, and that a harmonic oscillator at equilibrium with a bath gives the temperature of this bath through the FDR, then equilibrium between modes of the string and modes of the flow means the temperature is equal: measurements give the temperature of the flow at this corresponding scale. Such interpretation still rely on the assumption that FDR on the oscillator gives the temperature of the oscilaror: this is our working hypothesis. By equilibrium between modes of the string and the flow, we mean a ’no-flux’ condition on energy. This is ensured by the high susceptibility of the string at resonance. In other words, the probe and the reservoir are in equilibrium with each other for each k, but equilibium is obviously not expected between one scale and another. We have performed measurements on a turbulent flow, coupling to it a set of harmonic oscillators: a Melde string. At equilibrium with the flow, in the sense that each mode of the string couples with the fluid at scale l = πc/ω. It gives informations much like a spectrometer, even though the flow itself is strongly out of equilibrium. This is true, of course, as long as the response of the string is fast enough compared to the frequencies of the velocity fluctuations. The displacement spectra are recorded at different values of Re, as well as the complex response of the string over an excitation (contributions of all the standing waves). The matching of the string’s modes and hydrodynamic structures, what we call equilibrium between the string and the flow, is still a questionable working hypothesis. However, drawing inspiration from Cugliandolo et al ’s theory of non-equilibrium temperature based on the FDR, we measured the Fluctuation over Dissipation Ratio of our string in a turbulent flow, for different values of Re. The FDR, multiplied by an appropriate power of the Reynolds number exhibits a unique power law, when Reynolds number is between 74000 and 170000. The exponent is consistent with a Measurements of a dynamical temperature in turbulence. 9 value −11/3 given by a very simple model derived from Kolmogorov 1941 theory. Acknowledgments We acknowledge B. Castaing, E. Leveque, P. Borgnat, F. Delduc, S. Ciliberto, E. Bertin, and K. Gawedzki for many discussions. We also thank V. Bergeron, T. Divoux, and V. Vidal for corrections on the manuscript and for many discussions. Thanks to F. Dumas for his help in the construction of positioning devices. As this system became a teaching experiment, several students contributed to this study as part of their graduate lab-course. They are gratefully acknowledged: A. Louvet, G. Bordes, I. Dossmann, J. Perret, C. Cohen, and M. Mathieu. We also thank the guitar maker D. Teyssot, from Lyon, who gently gave us his thinnest E strings. [1] L.D. Landau and E.M. Lifshitz. Course of Theoretical Physics: Fluid mechanics. Mir, 1971. [2] A.S. Monin and A.M. Yaglom. Statistical fluid mechanics. MIT Press, Cambridge, 1975. [3] U. Frisch. Turbulence: the legacy of A.N. Kolmogorov. Cambridge Univ. Press., 1995. [4] L.F. Richardson. Weather prediction by numerical process. Cambridge Univ. Press, 1922. [5] A.N. Kolmogorov. C. R. Acad. Sci. U.S.S.R., 30, 1941. [6] M. Toda R. Kubo and N. Hashitsume. Statistical Physics II: Nonequilibrium Statistical Mechanics, volume II. Springer, 1985. [7] L. Cugliandolo and J. Kurchan. Phys. Rev. Lett., 71, 1993. [8] J. Kurchan L. Cugliandolo and L. Peliti. Phys. Rev. E, 55, 1997. [9] P. Marcq and A. Naert. Phys. of Fluids, 13, 2001. [10] T.M. Brown. J. Phys. I, 15, 1982. [11] B. Castaing. J. Phys. II, 6, 1996. [12] B. Castaing. J. Phys. II, 50, 1989. [13] J. Sommeria and R. Robert. J. Fluid Mech., 229, 1991. Introduction The Melde string and the experimental setup Measurements Scaling law Discussion

0704.0326

On generalized entropy measures and pathways

ON GENERALIZED ENTROPY MEASURES AND PATHWAYS A.M. MATHAI Department of Mathematics and Statistics, McGill University, Montreal, Canada H3A 2K6, and Centre for Mathematical Sciences, Pala Campus, Arunapuram P.O., Pala-686 574, Kerala, India H.J. HAUBOLD Office for Outer Space Affairs, United Nations, Vienna International Centre, P.O. Box 500, A-1400 Vienna, Austria and Centre for Mathematical Sciences, Pala Campus, Arunapuram P.O., Pala-686 574, Kerala, India Abstract. Product probability property, known in the literature as statistical independence, is examined first. Then generalized entropies are introduced, all of which give generalizations to Shannon entropy. It is shown that the nature of the recursivity postulate automatically determines the logarithmic functional form for Shannon entropy. Due to the logarithmic nature, Shannon entropy naturally gives rise to additivity, when applied to situations having product probability property. It is argued that the natural process is non-additivity, important, for example, in statistical mechanics (Tsallis 2004, Cohen 2005), even in product probability property situations and additivity can hold due to the involvement of a recursivity postulate leading to a logarithmic function. Generalized entropies are introduced and some of their properties are exam- ined. Situations are examined where a generalized entropy of order α leads to pathway models, exponential and power law behavior and related differential equations. Connection of this entropy to Kerridge’s measure of “inaccuracy” is also explored. 1. Introduction Mathai and Rathie (1975) consider various generalizations of Shannon en- tropy (Shannon, 1948), called entropies of order α, and give various properties, including additivity property, and characterization theorems. Recently, Mathai and Haubold (2006, 2006a) explored a generalized entropy of order α, which is connected to a measure of uncertainty in a probability scheme, Kerridge’s (Kerridge, 1961) concept of inaccuracy in a scheme, and pathway models that are considered in this paper. As defined in Mathai and Haubold (2006, 2006a) the entropy Mk,α(P ) is a non-additive entropy and his measure M∗k,α(P ) is an additive entropy. It is also shown that maximization of the continuous analogue of Mk,α(P ), denoted by Mα(f), gives rise to various functional forms for f , depending upon the types of constraints on f . http://arxiv.org/abs/0704.0326v2 Occasionally, emphasis is placed on the fact that Shannon entropy satisfies the additivity property, leading to extensivity. It will be shown that when the product probability property (PPP) holds then a logarithmic function can give a sum and a logarithmic function enters into Shannon entropy due to the assumption introduced through a certain type of recursivity postulate. The concept of statistical independence will be examined in Section 1 to illustrate that simply because of PPP one need not expect additivity to hold or that one should not expect this PPP should lead to extensivity. The types of non- extensivity, associated with a number of generalized entropies, are pointed out even when PPP holds. The nature of non-extensivity that can be expected from a multivariate distribution, when PPP holds or when there is statistical independence of the random variables, is illustrated by taking a trivariate case. Maximum entropy principle is examined in Section 2. It is shown that optimization of measures of entropies, in the continuous populations, under selected constraints, leads to various types of models. It is shown that the generalized entropy of order α is a convenient one to obtain various probability models. Section 3 examines the types of differential equations satisfied by the various special cases of the pathway model. 1.1. Product probability property (PPP) or statistical independence of events Let P (A) denote the probability of the event A. If the definition P (A∩B) = P (A)P (B) is taken as the definition of independence of the events A and B then any event A ∈ S, and S the sure event are independent. But A is contained in S and then the definition of independence becomes inconsistent with the common man’s vision of independence. Even if the trivial cases of the sure event S and the impossible event φ are deleted, still this definition becomes a resultant of some properties of positive numbers. Consider a sample space of n distinct elementary events. If symmetry in the outcomes is assumed then we will assign equal probabilities 1 each to the elementary events. Let C = A ∩B. If A and B are independent then P (C) = P (A)P (B). Let P (A) = , P (B) = , P (C) = ⇒ nz = xy, x, y, z = 1, 2, ..., n− 1, z < x, y (1) deleting S and φ. There is no solution for x, y, z for a large number of n, for example, n = 3, 5, 7. This means that there are no independent events in such cases and it sounds strange from a common man’s point of view. The term “independence” of events is a misnomer. This property should have been called product probability property or PPP of events. There is no reason to expect the information or entropy in a joint distribution to be the sum of the information contents of the marginal distributions when the PPP holds for the distributions, that is when the joint density or probability function is a product of the marginal densities or probability functions. We may expect a term due to the product probability to enter into the expression for the entropy in the joint distribution in such cases. But if the information or entropy is defined in terms of a logarithm, then naturally, logarithm of a product being the sum of logarithms, we can expect a sum coming in such situations. This is not due to independence or due to the PPP of the densities but due to the fact that a functional involving logarithm is taken thereby a product has become a sum. Hence not too much importance should be put on whether or not the entropy on the joint distribution becomes sum of the entropies on marginal distributions or additivity property when PPP holds. 1.2. How is logarithm coming in Shannon’s entropy? Several characterization theorems for Shannon entropy and its various gen- eralizations are given in Mathai and Rathie (1975. Modified and refined versions of Shannon’s own postulates are given as postulates for the first theorem charac- terizing Shannon entropy in Mathai and Rathie (1975). Apart from continuity, symmetry, zero-indifference and normalization postulates the main postulate in the theorem is a recursivity postulate, which in essence says that when the PPP holds then the entropy will be a weighted sum of the entropies, thus in effect, assuming a logarithmic functional form. The crucial postulate is stated here. Consider a multinomial population P = (p1, ..., pm), pi > 0, i = 1, ...,m, p1 + ... + pm = 1, that is, pi = P (Ai), i = 1, ...,m, A1 ∪ ... ∪ Am = S, Ai ∩ Aj = φ, i 6= j. If any pi can take a zero value also then zero-indifferent postulate, namely that the entropy remains the same when an impossible event is incorporated into the scheme, is to be added. Let Hn(p1, ..., pn) denote the entropy to be defined. Then the crucial recursivity postulate says that Hn(p1, ..., pm−1, pmq1, .., pmqn−m+1) = Hm(p1, ..., pm) + pmHn−m+1(q1, ..., qn−m+1) (2) i=1 pi = 1, ∑n−m+1 i=1 qi = 1. This says that if the m-th event Am is par- titioned into independent events P (Am ∩ Bj) = P (Am)P (Bj) = pmqj , j = 1, ..., n − m + 1 so that pm = pmq1 + ... + pmqn−m+1 then the entropy Hn(·) becomes a weighted sum. Naturally, the result will be a logarithmic function for the measure of entropy. There are several modifications to this crucial recursivity postulate. One suggested by Tverberg is that n−m+ 1 = 2 and q1 = q, q2 = 1− q, 0 < q < 1 and H2(q, 1 − q) is assumed to be Lebesgue integrable in 0 ≤ q ≤ 1. Again a characterization of Shannon entropy is obtained. In all the characterization theorems for Shannon entropy this recursivity property enters in one form or the other as a postulate, which in effect implies a logarithmic form for the entropy measure. Shannon entropy Sk has the following form: Sk = −A pi ln pi, pi > 0, i = 1, ..., k, p1 + ...+ pk = 1, (3) where A is a constant. If any pi is assumed to be zero then 0 ln 0 is to be interpreted as zero. Since the constant A is present, logarithm can be taken to any base. Usually the logarithm is taken to the base 2 for ready application to binary systems. We will take logarithm to the base e. 1.3. Generalization of Shannon entropy Consider again a multinomial population P = (p1, ..., pk), pi > 0, i = 1, ..., k, p1 + ... + pk = 1. The following are some of the generalizations of Shannon entropy Sk. Rk,α(P ) = i=1 p , α 6= 1, α > 0, (4) (Rényi entropy of order α of 1961) Hk,α(P ) = i=1 p i − 1 21−α − 1 , α 6= 1, α > 0 (5) (Havrda-Charvát entropy of order α of 1967) Tk,α(P ) = i=1 p i − 1 , α 6= 1, α > 0 (6) (Tsallis entropy of 1988) Mk,α(P ) = i=1 p i − 1 , α 6= 1, −∞ < α < 2 (7) (entropic form of order α) M∗k,α(P ) = i=1 p , α 6= 1, −∞ < α < 2, (8) (additive entropic form of order α). When α → 1 all the entropies of order α described above in (4) to (7) go to Shannon entropy Sk. Rk,α(P ) = lim Hk,α(P ) = lim Tk,α(P ) = lim Mk,α(P ) = lim M∗k,α(P ) = Sk. Hence all the above measures are called generalized entropies of order α. Let us examine to see what happens to the above entropies in the case of a joint distribution. Let pij > 0, i = 1, ...,m, j = 1, ..., n such that j=1 pij = 1. This is a bivariate situation of a discrete distribution. Then the entropy in the joint distribution, for example, Mm,n,α(P,Q) = j=1 p ij − 1 . (10) If the PPP holds and if pij = piqj , p1 + ... + pm = 1, q1 + ... + qn = 1, pi > 0, i = 1, ...,m, qj > 0, j = 1, ..., n and if P = (p1, ..., pm), Q = (q1, ..., qn) (α− 1)Mm,α (P ) Mn,α(Q) = i − 1 j − 1 j + 1 = Mm,n,α(P,Q) −Mm,α(P )−Mn,α(Q). Therefore Mm,n,α(P,Q) = Mm,α(P ) +Mn,α(Q) + (α− 1)Mm,α(P )Mn,α(Q). (11) If any one of the above mentioned generalized entropies in (4) to (8) is written as Fm,n,α(P,Q) then we have the relation Fm,n,α(P,Q) = Fm,α(P ) + Fn,α(Q) + a(α)Fm,α(P )Fn,α(Q). (12) where a(α) = 0 (Rényi entropy Rk,α(P )) = 21−α − 1 (Havrda-Charvát entropy Hk,α(P )) = 1− α (Tsallis entropy Tk,α(P )) = α− 1 (entropic form of order α, i.e., Mk,α(P )) = 0 (additive entropic form of order α, i.e., M∗k,α(P )). (13) When a(α) = 0 the entropy is called additive and when a(α) 6= 0 the entropy is called non-additive. As can be expected, when a logarithmic function is involved, as in the cases of Sk(P ), Rk,α(P ),M k,α(P ), the entropy is additive and a(α) = 0. 1.4. Extensions to higher dimensional joint distributions Consider a trivariate population or a trivariate discrete distribution pijk > 0, i = 1, ...,m, j = 1, ..., n, k = 1, ..., r such that k=1 pijk = 1. If the PPP holds mutually, that is, pair-wise as well as jointly, which then will imply that pijk = piqjsk, pi = 1, qj = 1, sk = 1, P = (p1, ..., pm), Q = (q1, ..., qn), S = (s1, ..., sr). Then proceeding as before, we have for any of the measures described above in (4) to (8), calling it F (·), Fm,n,r,α(P,Q, S) = Fm,α(P ) + Fn,α(Q) + Fr,α(S) + a(α)[Fm,α(P )Fn,α(Q) +Fm,α(P )Fr,α(S) + Fn,α(Q)Fr,α(S)] +[a(α)]2Fm,α(P )Fn,α(Q)Fr,α(S) (14) where a(α) is the same as in (13). The same procedure can be extended to any multivariable situation. If a(α) = 0 we may call the entropy additive and if a(α) 6= 0 then the entropy is non-additive. 1.5. Crucial recursivity postulate Consider the multinomial population P = (p1, ..., pk), pi > 0, i = 1, ..., k, p1+ ... + pk = 1. Let the entropy measure to be determined through appropriate postulates be denoted by Hk(P ) = Hk(p1, ..., pk). For k = 2 let f(x) = H2(x, 1− x), 0 ≤ x ≤ 1 or x ∈ [0, 1]. (15) If another parameter α is to be involved in H2(x, 1−x) then we will denote f(x) by fα(x). From (5) to (7) it can be seen that the generalized entropies of order α of Havrda-Charvát (1967), Tsallis (1988, 2004) and Shannon (1948) entropy satisfy the functional equation fα(x) + bα(x)fα = fα(y) + bα(x)f for x, y ∈ [0, ) with x+ y ∈ [0, 1], with the boundary condition fα(0) = fα(1) (17) where bα(x) = 1− x (Shannon entropy Sk(P )) = (1− x)α (Harvda-Charvát entropy Hk,α(P )) = (1− x)α (Tsallis entropy Tk,α(P )) = (1− x)2−α (entropic form of order α, i.e., Mk,α(P )). (18) Observe that the normalizing constant at x = 1 is equal to 1 for Hk,α(P ) and it is different for other entropies. Thus equations (6),(7),(8), with the appropriate normalizing constants fα( ), can give characterization theorems for the various entropy measures. The form of bα(x) is coming from the crucial recursivity postulate, assumed as a desirable property for the measures. 1.6. Continuous analogues In the continuous case let f(x) be the density function of a real random variable x. Then the various entropy measures, corresponding to the ones in (4) to (8) are the following: Rα(f) = [f(x)]αdx , α 6= 1, α > 0 (19) (Rényi entropy of order α) Hα(f) = 21−α − 1 [f(x)]αdx− 1 , α 6= 1, α > 0 (20) (Havrda-Charvát entropy of order α) Tα(f) = [f(x)]αdx− 1 , α 6= 1, α > 0, (21) (Tsallis entropy of order α) Mα(f) = [f(x)]2−αdx− 1 , α 6= 1, α < 2 (22) (entropic form of order α) M∗α(f) = [f(x)]2−αdx , α 6= 1, α < 2 (23) (additive entropic form of order α). As expected, Shannon entropy in this case is given by S(f) = −A f(x) ln f(x)dx (24) where A is a constant. Note that when PPP (product probability property) or statistical indepen- dence holds then in the continuous case also we have the property in (12) and (14) and then non-additivity holds for the measures analogous to the ones in (3),(5),(6),(7) with a(α) remaining the same. Since the steps are parallel a separate derivation is not given here. 2. Maximum Entropy Principle If we have a multinomial population P = (p1, ..., pk), pi > 0, i = 1, ..., k, p1+ ...+ pk = 1 or the scheme P (Ai) = pi, A1 ∪ ... ∪ Ak = S, P (S) = 1, Ai ∩ Aj = φ, i 6= j then we know that the maximum uncertainty in the scheme or the minimum information from the scheme is obtained when we cannot give any preference to the occurrence of any particular event or when the events are equally likely or when p1 = p2 = ... = pk = . In this case, Shannon entropy becomes, Sk(P ) = Sk( , ..., ) = −A = A ln k (25) and this is the maximum uncertainty or maximum Shannon entropy in this scheme. If the arbitrary functional f is to be fixed by maximizing the entropy then in (19) to (21) we have to optimize [f(x)]αdx for fixed α, over all functional f , subject to the condition f(x)dx = 1 and f(x) ≥ 0 for all x. For applying calculus of variation procedure we consider the functional U = [f(x)]α − λ[f(x)] where λ is a Lagrangian multiplier. Then the Euler equation is the following: = 0 ⇒ αfα−1 − λ = 0 ⇒ f = = constant. (26) Hence f is the uniform density in this case, analogous to the equally likely situation in the multinomial case. If the first moment E(x) = xf(x)dx is assumed to be a given quantity for all functional f then U will become the following for (19) to (21). U = [f(x)]α − λ1[f(x)]− λ2xf(x) and the Euler equation leads to the power law. That is, = 0 ⇒ αfα−1 − λ1 − λ2x = 0 ⇒ f = c1 . (27) By selecting c1, λ1, λ2 appropriately we can create a density out of (27). For α > 1 and λ2 > 0 the right side in (27) increases exponentially. If α = q > 1 and = q − 1 then we have Tsallis’ q-exponential function from the right side of (27). If α > 1 and λ2 = −(α−1) then (27) can produce a density in the category of a type-1 beta. From (27) it is seen that the form of the entropies of Havrda- CharvátHk,α(P ) and Tsallis Tk,α(P ) need special attention to produce densities (Ferri et al. 2005). However, Tsallis has considered a different constraint on E(x). If the density f(x) is replaced by its escort density, namely, µ[f(x)]α where µ−1 = [f(x)]αdx and if the expected value of x in this escort density is assumed to be fixed for all functional f then the U of (26) becomes U = fα − λ1f + µλ2xf = 0 ⇒ αfα−1[1 + µλ2x] = λ1 ⇒ f = (1+λ3x) f = λ∗1[1 + λ3x] where λ3 is a constant and λ 1 is the normalizing constant. If λ3 is taken as λ3 = α− 1 then f = λ∗1[1 + (α− 1)x] α−1 . (28) Then (28) for α > 1 is Tsallis statistics (Tsallis 2004, Cohen 2005). Then for α < 1 also by writing α − 1 = −(1 − α) one gets the case of Tsallis statistics for α < 1 (Ferri et al. 2005). These modifications and the consideration of escort distribution are not necessary if we take the generalized entropy of order α. Thus if we consider Mα(f) and if we assume that the first moment in f(x) itself is fixed for all functional f then the Euler equation gives (2− α)f1−α − λ1 + λ2x = 0 ⇒ f = λ̄ and for λ2 = 1− α we have Tsallis statistics (Tsallis 2004, Cohen 2005) f = λ̄[1− (1− α)x] 1−α (29) coming directly, where λ̄ is the normalizing constant. Let us start with Mα(f) of (20) under the assumptions that f(x) ≥ 0 for all f(x)dx = 1, xδf(x)dx is fixed for all functional f and for a specified δ > 0, f(a) is the same for all functional f , f(b) is the same for all functional f , for some limits a and b, then the Euler equation becomes (2 − α)f1−α − λ1 − λ2x δ = 0 ⇒ f = c1[1 + c 1−α . (30) If c∗1 is written as −s(1− α), s > 0 then we have, writing f1 for f , f1 = c1[1− s(1 − α)x 1−α , δ > 0, α < 1, 0 ≤ x ≤ [s(1− α)] where 1 − s(1 − α)xδ > 0. For α < 1 or −∞ < α < 1 the right side of (31) remains as a generalized type-1 beta model with the corresponding normalizing constant c1. For α > 1, writing 1 − α = −(α − 1) the model in (31) goes to a generalized type-2 beta form, namely, f2 = c2[1 + s(α− 1)x α−1 . (32) When α → 1 in (31) or in (32) we have an extended or stretched exponential form, f3 = c3e . (33) If c∗1 in (30) is taken as positive then (30) for α < 1, α > 1, α → 1 will be increasing exponentially. Hence all possible forms are available from (30). The model in (31) is a special case of the distributional pathway model and for a discussion of the matrix-variate pathway model see Mathai (2005). Special cases of (31) and (32) for δ = 1 are Tsallis statistics (Gell-Mann and Tsallis, 2004; Ferri et al. 2005). Instead of optimizing Mα(f) of (22) under the conditions that f(x) ≥ 0 for all x, f(x)dx = 1 and xδf(x)dx is fixed, let us optimize under the following conditions: f(x) ≥ 0 for all x, f(x)dx < ∞ and the following two moment-like expressions are fixed quantities for all functional f , x(γ−1)(1−α)f(x)dx = fixed , x(γ−1)(1−α)+δf(x)dx = fixed. Then the Euler equation becomes (2− α)f1−α −λ1x (γ−1)(1−α) − λ2x (γ−1)(1−α)+δ = 0 ⇒ f = c xγ−1[1 + c∗xδ] and for c∗ = −s(1 − α), s > 0, we have the distributional pathway model for the real scalar case, namely f(x) = c xγ−1[1− s(1− α)xδ ] 1−α , δ > 0, s > 0 (34) where c is the normalizing constant. For α < 1, (34) gives a generalized type-1 beta form, for α > 1 it gives a generalized type-2 beta form and for α → 1 we have a generalized gamma form. For α > 1, (34) gives the superstatistics of Beck (2006) and Beck and Cohen (2003). For γ = 1, δ = 1, (34) gives Tsallis statistics (Tsallis 2004, Cohen 2005). Densities appearing in a number of physical problems are seen to be special cases of (34), a discussion of which may be seen from Mathai and Haubold (2006a). For example, (34) for δ = 2, γ = 3, α → 1, x > 0 is the Maxwell-Boltzmann density; for δ = 2, γ = 1, α → 1,−∞ < x < ∞ is the Gaussian density; for γ = δ, α → 1 is the Weibull density. For γ = 1, δ = 2, 1 < q < 3 we have the Wigner function W (p) giving the atomic moment distribution in the framework of Fokker-Planck equation, see Douglas, Bergamini, and Renzoni (2006) where W (p) = z−1q [1− β(1 − q)p 1−q , 1 < q < 3. (35) Before closing this section we may observe one more property for Mα(f). As an expected value Mα(f) = E[f(x)]1−α − 1 . (36) But Kerridge’s (Kerridge, 1961) measure of “inaccuracy” in assigning q(x) for the true density f(x), in the generalized form is Hα(f : q) = (21−α − 1) E[q(x)]α−1 − 1 , (37) which is also connected to the measure of directed divergence between q(x) and f(x). In (37) the normalizing constant is 21−α−1, the same factor appearing in Havrda-Charvt́ entropy. With different normalizing constants, as seen before, (36) and (37) have the same forms as an expected value with q(x) replaced by f(x) in (36). Hence Mα(f) can also be looked upon as a type of directed divergence or “inaccuracy” measure. 3. Differential Equations The functional part in (34), for a more general exponent, namely g(x) = = xγ−1[1− s(1 − α)xδ] 1−α , α 6= 1, δ > 0, β > 0, s > 0 (38) is seen to satisfy the following differential equation for γ 6= 1 which defines the differential pathway. g(x) = (γ − 1)xγ−1[1− s(1− α)xδ] −sβδxδ+γ−1[1− s(1− α)xδ] (1−α) . (39) Then for δ = (γ−1)(α−1) , γ 6= 1, α > 1 we have g(x) = (γ − 1)g(x)− sβδ[g(x)]1− (1−α) β (40) = (γ − 1)g(x)− sδ[g(x)]α (41) for β = 1, γ 6= 1, δ = (γ − 1)(α− 1), α > 1. For γ = 1, δ = 1 in (38) we have g(x) = −s[g(x)]η, η = 1− (1 − α) = −s[g(x)]α for β = 1. (43) Here (43) is the power law coming from Tsallis statistics (Gell-Mann and Tsallis, 2004). Acknowledgement The authors would like to thank the Department of Science and Technology, Government of India, New Delhi, for the financial assistance for this work under project No. SR/S4/MS:287/05 which enabled this collaboration possible. 4. References Beck, C. (2006). Stretched exponentials from superstatistics. Physica A, 365, 96-101. Beck, C. and Cohen, E.G.D. (2003). Superstatistics. Physica A, 322, 267-275. Cohen, E.G.D. (2005). Boltzmann and Einstein: Statistics and dynamics - An unsolved problem. Pramana, 64, 635-643. Douglas, P., Bergamini, S., and Renzoni, F. (2006). Tunable Tsallis distribution in dissipative optical lattices. Physical Review Letters, 96, 110601-1-4. Ferri, G.L., Martinez, S., and Plastino, A. (2005). Equivalence of the four versions of Tsallis’s statistics. Journal of Statistical Mechanics: Theory and Experiment, PO4009. Gell-Mann, M. and Tsallis, C. (Eds.) (2004). Nonextensive Statistical Mechan- ics: Interdisciplinary Applications. Oxford University Press, Oxford. Havrda, J. and Charvát, F. (1967). Quantification method of classification pro- cedures: Concept of structural α-entropy. Kybernetika, 3, 30-35. Kerridge, D.F. (1961). Inaccuracy and inference. Journal of the Royal Statisti- cal Society Series B, 23, 184-194. Mathai, A.M. (2005). A pathway to matrix-variate gamma and normal densi- ties. Linear Algebra and Its Applications, 396, 317-328. Mathai, A.M. and Haubold, H.J. (2006). Pathway model, Tsallis statistics, su- perstatistics and a generalized measure of entropy. Physica A , 375), 110-122. Mathai,A.M. and Haubold, H.J. (2006a). On generalized distributions and path- ways. arXiv:cond-mat/0609526v2. Mathai, A.M. and Rathie, P.N. (1975). Basic Concepts in Information Theory and Statistics: Axiomatic Foundations and Applications, Wiley Halstead, New York and Wiley Eastern, New Delhi. Rényi, A. (1961). On measure of entropy and information. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 1960, University of California Press, 1961, Vol. 1, 547-561. Shannon, C.E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27, 379-423, 547-561. Tsallis, C. (1988). Possible generalization of Boltzmann-Gibbs statistics. Jour- nal of Statistical Physics, 52, 479-487. Tsallis, C. (2004). What should a statistical mechanics satisfy to reflect nature?, Physica D, 193, 3-34. http://arxiv.org/abs/cond-mat/0609526

0704.0327

Evolution of a band insulating phase from a correlated metallic phase

Evolution of a band insulating phase from a correlated metallic phase Kalobaran Maiti,∗ Ravi Shankar Singh, and V.R.R. Medicherla Department of Condensed Matter Physics and Materials’ Science, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai - 400 005, INDIA (Dated: October 30, 2018) We investigate the evolution of the electronic structure in SrRu1−xTixO3 as a function of x using high resolution photoemission spectroscopy, where SrRuO3 is a weakly correlated metal and SrTiO3 is a band insulator. The surface spectra exhibit a metal-insulator transition at x = 0.5 by opening up a soft gap. A hard gap appears at higher x values consistent with the transport properties. In contrast, the bulk spectra reveal a pseudogap at the Fermi level, and unusual evolution exhibiting an apparent broadening of the coherent feature and subsequent decrease in intensity of the lower Hubbard band with the increase in x. Interestingly, the first principle approaches are found to be sufficient to capture anomalous evolutions at high energy scale. Analysis of the spectral lineshape indicates strong interplay between disorder and electron correlation in the electronic properties of this system. PACS numbers: 71.10.Hf, 71.20.-b, 71.30.+h The investigation of the role of electron correlation in various electronic properties is a paradigmatic problem in solid state physics. Numerous experimental and the- oretical studies are being performed on correlated elec- tron systems revealing exotic phenomena such as high temperature superconductivity, giant magnetoresistance etc. Electron correlation essentially localizes the valence electrons leading the system towards insulating phase. Correlation induced insulators, known as Mott insulators are characterized by a gapped electronic excitations in a system where effective single particle approaches provide a metallic ground state. The band insulators represent insulating phase described within the single particle ap- proaches. Strikingly, some recent theoretical studies re- veal a correlation induced metallic ground state in a band insulator using ionic Hubbard model [1, 2, 3, 4]. Such un- usual transition has been observed in two dimensions by tuning effective electron correlation strength, U/W (U = electron-electron Coulomb repulsion strength, W = bandwidth) and the local potential, ∆. In order to realize such effect experimentally, we in- vestigate the evolution of the electronic structure in SrRu1−xTixO3 as a function of x, where the end mem- bers, SrRuO3 and SrTiO3 are correlated ferromagnetic metal and band insulator, respectively. Ti remains in tetravalent state in the whole composition range having no electron in the 3d band[5, 6]. Thus, in addition to the introduction of disorder in the Ru-O sublattice, Ti- substitution at the Ru-sites dilutes Ru-O-Ru connectiv- ity leading to a reduction in Ru 4d bandwidth, W and hence, U/W will increase. Transport measurements[7] exhibit plethora of novel phases such as correlated metal (x ∼ 0.0 ), disordered metal (x ∼ 0.3), Anderson insu- lator (x ∼ 0.5), soft Coulomb gap insulator (x ∼ 0.6), disordered correlated insulator (x ∼ 0.8), and band insu- lator (x = 1.0). In this study, we have used high resolution photoemis- sion spectroscopy to probe the density function in the vicinity of the Fermi level, ǫF and at higher energy scale as well. Considering the fact that escape depth of the photoelectrons is small, we have extracted the surface and bulk spectra in every case by varying the surface sensitivity of the technique. The surface spectra exhibit signature of disorder at lower x values in SrRu1−xTixO3, a metal-insulator transition exhibiting a soft gap at ǫF for x = 0.5 and a hard gap for higher x. The bulk spec- tra, on the other hand, reveal an unusual spectral weight transfer and signature of a pseudogap at ǫF at higher x. Photoemission measurements were performed using Gammadata Scienta analyzer, SES2002 and monochro- matized photon sources. The energy resolution for x- ray photoemission (XP) and He II photoemission mea- surements were set at 300 meV and 4 meV, respectively. High quality samples of SrRu1−xTixO3 with large grain size were prepared following solid state reaction route using high purity ingredients[8] followed by a long sin- tering (for about 72 hours) at the final preparation tem- perature. Sharp x-ray diffraction patterns reveal single phase in each composition with no signature of impurity feature. Magnetic measurements using a high sensitiv- ity vibrating sample magnetometer exhibit distinct fer- romagnetic transition at each x up to x = 0.6 studied, as also evidenced by the Curie-Weiss fits in the param- agnetic region. The fits provide an estimation of effec- tive magnetic moment (µ = 2.8 µB , 2.54 µB, 2.45 µB, 2.18 µB, 2.19 µB, 1.95 µB and 1.93 µB) and Curie tem- perature (θP = 164 K, 156.6 K, 150.6 K, 145.3 K, 139 K, 138.6 K and 100 K) for x = 0.0, 0.15, 0.2, 0.3, 0.4, 0.5 and 0.6, respectively. The values of µ and θP for SrRuO3 are observed to be the largest among those available in the literature and corresponds to well characterized sin- gle crystalline materials[9]. In Fig. 1(a), we show the XP valence band spectra exhibiting 4 distinct features marked by A, B, C and D. The features C and D appear beyond 2.5 eV and have large O 2p character as confirmed experimentally http://arxiv.org/abs/0704.0327v1 �� � � � � � � � � � �� �α ()*+,- :;< = > ? @ A BCD RSTUVWX YZ[\]^ _`ab cd ef gh ij p q rst �� �α �� �� FIG. 1: (color online) (a) XP valence band spectra of SrRu1−xTixO3 for various values of x. Solid line represents the O 2p part for x = 0.6. (b) Ru 4d spectra after the sub- traction of the O 2p contributions as shown in (a). (c) Ru 4d band obtained from He II spectra. by changing photoemission cross-sections [10] and theo- retically by band structure calculations [11]. The peaks A and B appear primarily due to the photoemission from electronic states having Ru 4d character. The O 2p part remains almost the same in the whole composition range as expected. While Ru 4d intensity gradually diminishes with the decrease in Ru-concentrations, the lineshape of Ru 4d band exhibits significant redistribution in inten- sity. In order to bring out the clarity, we delineate the Ru 4d band by subtracting O 2p contributions. The sub- tracted spectra, normalized by integrated intensity un- der the curve, exhibit two distinct features as evident in Fig. 1(b). The feature A corresponds to the delocalized electronic density of states (DOS) observed in ab initio results and is termed as coherent feature. The feature B, absent in the ab initio results[11], is often attributed to the signature of correlation induced localized electronic states forming the lower Hubbard band and is known as incoherent feature. The increase in x leads to a de- crease in intensity of A and subsequently, the intensity of B grows gradually. Since the bulk sensitivity of valence electrons at 1486.6 eV photon energy is high (∼ 60%), the spectral evolution in Fig. 1(b) manifests primarily the changes in the bulk electronic structure. In order to discuss the effect due to the surface elec- tronic structure, we show the Ru 4d contributions ex- tracted from the He II spectra in Fig. 1(c), where the surface sensitivity is about 80%. Interestingly, all the spectra are dominated by the peak at higher binding en- ergies (> 1 eV) corresponding to the surface electronic £ ¤ ¥ ²³´ µ¶· ¸¹º ÄÅÆ ÇÈÉ ÊËÌ Ö × Ø åæç èéê ëìí îïð ñòó ôõö÷ �� � �� � � �� ���� ����� ( ) *+, - . /01 23 45 GHIJKLM NOPQRS TUVW X Y Z[\ ]^ _` a b cde iε j ε ~� �� � � ��� � � ��� �ε � ε  ¡ ¢α ° ±²³ ÀÁÂÃÄÅÆ ÇÈÉÊËÌ ÍÎÏÐ ÑÒÓÔÕÖ× ØÙÚÛÜÝ Þßàá â ã äåæ ç è éêë ìí îα �� � ������� ���� & '()* FIG. 2: (color online) S(ǫ) obtained from (a) He II and (b) XP spectra of SrRu1−xTixO3. (b) S(ǫ) in (a) are plotted as a function of (c) |ǫ− ǫF | 0.5 (d) and |ǫ− ǫF | 1.25. S(ǫ) obtained from (e) XP and (f) He II spectra of Ca1−xSrxRuO3. structure as reported in the case of SrRuO3 and the co- herent feature intensity corresponds essentially to the bulk electronic structure[10, 12]. The coherent feature intensity reduces drastically with the increase in x and becomes almost negligible at x = 0.6. This can be vi- sualized clearly in the spectral density of states (SDOS) obtained by symmetrizing (S(ǫ) = I(ǫ) + I(−ǫ); I(ǫ) = photoemission spectra, ǫ = binding energy) the He II and XP spectra. The SDOS corresponding to He II spectrum of SrRuO3 shown in Fig. 2(a) exhibits a sharp dip at ǫF , which increases gradually with the increase in x. The SDOS corresponding to XP spectra in Fig. 2(b), how- ever, exhibits a peak in SrRuO3 presumably due to large resolution broadening and intense coherent feature. This peak loses its intensity and becomes almost flat for x = 0.15 and 0.2. Further increase in x leads to a pseudogap at ǫF , which gradually increases with the increase in x. Both these results clearly indicate gradual depletion of SDOS at ǫF with the increase in Ti-substitution. The effect of resolution broadening of 4 meV in the He II spectra is not significant in the energy scale shown in the figure. The electron and hole lifetime broadening is also negligible in the vicinity of ǫF . Thus, S(ǫ) in Fig. 2(a) provide a good testing ground to investigate evolu- tion of the spectral lineshape at ǫF . The lineshape of S(ǫ) in Fig. 2(a) exhibits significant modification with the increase in x. We, thus, replot S(ǫ) as a function of |ǫ − ǫF | α for various values of α. Two extremal cases representing α = 0.5 and 1.25 are shown in Fig. 2(c) and 2(d), respectively. It is evident that S(ǫ) of SrRuO3 ex- hibit a straight line behavior in Fig. 2(c) suggesting sig- nificant role of disorder in the electronic structure. The influence of disorder can also be verified by substitutions at the A-sites in the ABO3 structure. This has been ver- ified by plotting SDOS obtained from the XP and He II spectra of Ca1−xSrxRuO3 in Fig. 2(e) and 2(f), respec- tively. Here, the electronic properties of the end mem- bers, SrRuO3 and CaRuO3 are known to be strongly in- fluenced by the disorder[13]. Substitution of Sr at the Ca-sites is expected to enhance the disorder effect. The lineshape of S(ǫ) in both Fig. 2(e) and 2(f) remains al- most the same across the whole composition range. Such disorder induced spectral dependence is consistent with the observations in other systems[14, 15] as well. Interestingly, the lineshape modifies significantly with the increase in x and becomes 1.25 in the 60% Ti substi- tuted sample. Ti substitution introduces defects in the Ru-O network, where Ti4+ having no d-electron, does not contribute in the valence band. Thus, in addition to the disorder effects, the reduced degree of Ru-O-Ru con- nectivity leads to a decrease in bandwidth, W , which in turn enhances U/W . In systems consisting of localized electronic states in the vicinity of ǫF , a soft Coulomb gap opens up due to electron-electron Coulomb repulsion; in such a situation, the ground state is stable with respect to single-particle excitations, when SDOS is characterized by (ǫ − ǫF ) 2-dependence [16, 17]. Here, gradual increase in α with the increase in x in the intermediate compo- sitions is curious and indicates strong interplay between correlation effect and disorder in this system. The extraction of surface and bulk spectra requires both the XP and He II spectra collected at significantly different surface sensitivities. Thus, we broaden the He II spectra upto 300 meV and extract the surface and bulk spectra analytically using the same parameters as used before for CaSrRuO3 system[10]. The surface spectra shown in Fig. 3(b) exhibit a gradual decrease in coherent feature intensity with the increase in x and subsequently, the feature around 1.5 eV becomes intense, narrower and slightly shifted towards higher binding energies. The de- crease in intensity at ǫF is clearly visible in the sym- metrized spectra, S(ǫ) shown in Fig. 3(d). Interestingly, S(ǫ) of x = 0.5 sample exhibits a soft gap at ǫF and a hard gap appears in S(ǫ) corresponding to higher x. This spectral evolution is remarkably consistent with the transport properties[7]. These results corresponding to 2- dimensional surface states presumably have strong impli- cation in realizing recent theoretical predictions[1, 2, 3, 4] and the bulk properties of this system. The picture is strikingly different in the bulk spectra where the electronic structure is 3-dimensional. The bulk spectrum of SrRuO3 exhibits an intense and sharp coher- ent feature in the vicinity of ǫF and the incoherent fea- ture appears around 2 eV. The enhancement of U/W due to Ti substitution is expected to increase the incoherent feature intensity. In sharp contrast, the intensity of the + , - . / 0 1 2 345 678 9:; <=>? OPQ RST UVW XYZ[ ���������� ¦§¨©ª«¬­®¯° ±²³´µ¶· ¸¹º»¼½¾¿À ÓÔÕÖ×ØÙ ÚÛÜÝÞß àáâã óôõö÷øù úûüýþÿ� ������ ���� FIG. 3: (color online) Extracted (a) bulk and (b) surface spectra of SrRu1−xTixO3 for various values of x. The SDOS obtained from bulk and surface spectra are shown in (c) and (d), respectively. 2 eV feature reduces significantly and the coherent fea- ture becomes broad. In addition, the bulk spectra of all the intermediate compositions appear very similar. The symmetrized bulk spectra shown in Fig. 3(c) exhibit a small lowering of intensity at ǫF with the increase in x. Since, U is weak in these highly extended 4d systems[10, 12], a perturbative approach may be use- ful to understand the role of electron correlation in the spectral lineshape. We have calculated the bare density of states (DOS) for SrRuO3 and SrRu0.5Ti0.5O3 using state-of-the-art full potential linearized augmented plane wave method[11, 18]. The self energy and spectral func- tions were calculated using this t2g partial DOS as done before[19]. The real and imaginary parts of the self en- ergy are shown in Fig. 4(a) and 4(b), and the spectral functions for different U values are shown in Fig. 4(c) and 4(d) for SrRuO3 and SrRu0.5Ti0.5O3, respectively. The increase in U leads to a spectral weight transfer outside the LDA DOS width creating the lower and upper Hub- bard bands. Subsequently, the total width of the LDA DOS diminishes gradually. While these results exhibit similar scenario as that observed in the most sophisti- cated calculations using dynamical mean field theory, the separation between the lower and upper Hubbard bands is significantly larger than the corresponding values of U . It is important to note here that the band structure cal- culations include the electron-electron interaction term within the local density approximations. The perturba- tion calculations in the present case essentially provide an estimation of the correction in U already included in the effective single particle Hamiltonian. In order to compare with the experimental spectra, the � � � � � � !" #$ % & ' ( )* +,- . / 0 12 34 5 6 7 89 :; < = > ? CDEFG H I JKL abcdefg hijklm nopq � � ��� ����� � � ��� �������� ����� ¡¢ µ ¶ ·¸¹ º»¼½¾¿ÀÁ ÂÃÄÅÆÇÈÉ á â ãäå æçèé êëìíîï ôõö÷ø ����� � � ��� &'()*+, -./012 3456 FIG. 4: (color online) Real and imaginary parts of the self energy of (a) SrRuO3 and (b) SrRu0.5Ti0.5O3 obtained by second order perturbation method following the method of Treglia et al.[19]. Spectral functions for various values of U of (c) SrRuO3 and (d) SrRu0.5Ti0.5O3. Calculated experimen- tal spectra for different values of U of (e) SrRuO3 and (f) SrRu0.5Ti0.5O3. calculated spectral functions are convoluted by Fermi- Dirac distribution function and the gaussian representing the resolution broadening of 300 meV. The comparison is shown in Figs. 4(e) and 4(f). Interestingly, the spectral shape corresponding to U = 0.6 ± 0.1 exhibits remark- able representation of the experimental bulk spectra in both the cases. These results clearly establish that per- turbative approaches and local description of the corre- lation effects are sufficient to capture electronic structure of these weakly correlated systems. The overall narrow- ing of the valence band observed in the substituted com- pounds are essentially a single particle effect and can be attributed to the reduced degree of Ru-O-Ru connectiv- ity in these systems. While the high energy scale features are reproduced remarkably well within this picture, the occurrence of a pseudogap at ǫF with increasing x (not visible in Fig.4 due to large energy scale) suggests in- creasing role of disorder. In summary, the high resolution spectra of SrRuO3 ex- hibit signature of disorder in the vicinity of the Fermi level. Introduction of the Ti4+ sublattice within the Ru4+ sublattice provides a paradigmatic example, where the charge density near Ti4+ sites is close to zero and each Ru4+ site contributes 4 electrons in the valence band. Such large charge fluctuation leads to a significant change in spectral lineshape and a dip appears at ǫF (pseudo- gap). Interestingly, the effects are much stronger in the two dimensional (surface) electronic structure leading to a soft gap at 50% substitution and eventually a hard gap appears. Bulk electronic structure (3-dimensional), how- ever, remains less influenced. A theoretical understand- ing of these effects needs consideration of strong disorder in addition to the electron correlation effects. ∗ Corresponding author: [email protected] [1] A. Fuhrmann, D. Heilmann, and H. Monien, Phys. Rev. B 73, 245118 (2006). [2] S.S. Kancharla and E. Dagotto, Phys. Rev. Lett. 98, 016402 (2007). [3] Arti Garg, H.R. Krishnamurthy, and Mohit Randeria, Phys. Rev. Lett. 97, 046403 (2006). [4] N. Paris, K. Bouadim, F. Hebert, G.G. Batrouni, and R.T. Scalettar, Phys. Rev. Lett. 98, 046403 (2007). [5] J. Kim, J.-Y. Kim, B.-G. Park, and S.-J. Oh, Phys. Rev. B 73, 235109 (2006), M. Abbate, J.A. Guevara, S.L. Cuffini, Y.P. Mascarenhas, and E. Morikawa, Eur. Phys. J. B 25, 203 (2002). [6] S. Ray, D.D. Sarma, and R. Vijayaraghavan, Phys. Rev. B 73, 165105 (2006). [7] K.W. Kim, J.S. Lee, T.W. Noh, S.R. Lee, and K. Char, Phys. Rev. B 71, 125104 (2005). [8] R.S. Singh and K. Maiti, Solid State Commun, 140, 188 (2006). [9] G. Cao, S. McCall, M. Shepard, J.E. Crow, and R.P. Guertin, Phys. Rev. B 56, 321 (1997). [10] K. Maiti and R.S. Singh, Phys. Rev. B 71, 161102(R) (2005). [11] K. Maiti, Phys. Rev. B 73, 235110 (2006). [12] M. Takizawa, D. Toyota, H. Wadati, A. Chikamatsu, H. Kumigashira, A. Fujimori, M. Oshima, Z. Fang, M. Lipp- maa, M. Kawasaki, and H. Koinuma, Phys. Rev. B 72, 060404(R) (2005). [13] K. Maiti, R.S. Singh, and V.R.R. Medicherla, Europhys. Lett. (in print); Condmat/0604648. [14] B.L. Altshuler and A.G. Aronov, Solid State Commun. 30, 115 (1979). [15] D.D. Sarma et al., Phys. Rev. Lett. 80, 4004 (1998). [16] A.L. Efros and B.I. Shklovskii, J. Phys. C: Solid State Phys. 8, L49 (1975). [17] J.G. Massey and M. Lee, Phys. Rev. Lett. 75, 4266 (1995). [18] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, and J. Luitz, WIEN2k, An Augmented Plane Wave + Lo- cal Orbitals Program for Calculating Crystal Properties (Karlheinz Schwarz, Techn. Universität Wien, Austria), 2001. ISBN 3-9501031-1-2. [19] G. Treglia et. al., J. Physique 41, 281 (1980); ibid, Phys. Rev. B 21, 3729 (1980); D.D. Sarma et al., Phys. Rev. Lett. 57, 2215 (1986).

0704.0328

Electroweak phase transitions in the MSSM with an extra $U(1)'$

Electroweak phase transitions in the MSSM with an extra U (1)′ S.W. Ham(1), E.J. Yoo(2), and S.K. Oh(1,2) (1) Center for High Energy Physics, Kyungpook National University, Daegu 702-701, Korea (2) Department of Physics, Konkuk University, Seoul 143-701, Korea Abstract We investigate the possibility of electroweak phase transition in the minimal supersymmetric standard model (MSSM) with an extra U(1)′. This model has two Higgs doublets and a singlet, in addition to a singlet exotic quark superfield. We find that at the one-loop level this model may accommodate the electroweak phase transitions that are strongly first-order in a reasonably large region of the parameter space. In the parameter region where the phase transitions take place, we observe that the lightest scalar Higgs boson has a smaller mass when the strength of the phase transition becomes weaker. Also, the other three heavier neutral Higgs bosons get more large masses when the strength of the phase transition becomes weaker. http://arxiv.org/abs/0704.0328v1 I. INTRODUCTION The baryon asymmetry of the universe can be dynamically generated during the evolution of the universe, if the mechanism of baryogenesis satisfies the three Sakharov conditions [1]. The three Sakharov conditions are: the presence of baryon number violation, the violation of both C and CP, and a deviation from thermal equilibrium. It is known that the universe can escape out of the thermal equilibrium by means of electroweak phase transition, which should be strongly first-order in order to ensure sufficient deviation from thermal equilibrium to generate the baryon asymmetry that is observed today. However, it has been already recognized that the Standard Model (SM) has some difficulty to realize the desired electroweak phase transition. The present experimental lower bound on the mass of the SM Higgs boson does not allow the electroweak phase transition to be strongly first-order [2, 3]. The electroweak phase transition is weakly first-order or higher order in the SM. Thus, the SM is inadequate to generate sufficient baryon asymmetry. Moreover, the amount CP violation in the Cabibbo-Kobayashi-Maskawa (CKM) matrix is too small to account for the baryon asymmetry of the observed universe [4]. Consequently, new physical models beyond the SM have extensively been studied for the possibility of reasonable explanation of the baryon asymmetry of the universe. Espe- cially, the low energy supersymmetric models have been studied widely within the context of electroweak baryogenesis [5-7]. The simplest supersymmetric model that includes the SM is the minimal supersymmetric standard model (MSSM), which possesses in its su- perpotential the µ term that accounts for the mixing between two Higgs doublets. The µ parameter, which has the mass dimension, causes some problem with respect to its energy scale [8]. Several possibilities have been investigated in the literature to solve the so-called µ problem [9-12]. Introducing an additional U(1)′ to the MSSM is one of the plausible explanations for the µ problem of the MSSM. The MSSM with an extra U(1)′ can not only solve the µ problem but we will show that it can also overcome the difficulties that the SM encounters when the SM tries to satisfy the Sakharov conditions. This model can accommodate sufficient CP violation, because it possesses other sources of CP violation besides the CKM matrix. It is possible to realize the explicit CP violation in this model by means of complex CP phases arising from the soft SUSY breaking terms [12]. Then, it is the purpose of this paper to show that this model indeed allows the strongly first-order electroweak phase transitions such that it can successfully explain the baryo- genesis. The characteristics of the electroweak phase transitions are determined essen- tially by the temperature-dependent part of the Higgs potential. We construct the full temperature-dependent Higgs potential at the one-loop level, and examine if the elec- troweak phase transition may be strongly first-order. Two methods are employed for the construction of the temperature-dependent Higgs potential. One method assumes that the critical temperature at which the electroweak phase transition occurs is relatively high, thus the temperature-dependent effective potential is approximated by retaining only terms proportional to T 2, whereas the other method carries out numerically exact integrations of the temperature-dependent effective potential. The thermal effects of par- ticles whose masses are comparatively smaller than the critical temperature are included at the one-loop level in the former method, whereas the particle content is different in the latter method. Either way, we obtain almost the same physical results. Unlike the MSSM, this model allows a strongly first-order electroweak phase transition in a wide region of the parame- ter space, and the first-order electroweak phase transition can be strong enough without requiring a light stop quark. An interesting behavior of this model with respect to the strongly first-order electroweak phase transition is that the mass of the lightest neutral Higgs boson becomes larger when the phase transition gets stronger. On the other hand, the masses of the other three neutral Higgs bosons become smaller when the phase tran- sition gets stronger. II. ZERO TEMPERATURE The MSSM with an extra U(1)′ accommodates in its Higgs sector two Higgs doublets H1 = (H 1 , H 1 ), H2 = (H 2 , H 2 ), and one Higgs singlet, S. In terms of these Higgs fields, the relevant part of the superpotential of this model may be written as W ≈ htQH2t R + hbQH1b R + hkSDLD̄R − λSH ǫH2 , (1) where we take into account only the third generation: tcR and b R are, respectively, the right-handed singlet top and bottom quark superfields, DR is the right-handed singlet exotic quark (a vector-like down quark) superfield, Q is the left-handed SU(2) doublet quark superfield of the third generation, and DL is the left-handed singlet exotic quark superfield. Further, ht, hb and hk are, respectively, the dimensionless Yukawa coupling coefficients of top, bottom, and exotic quark superfields, and ǫ is an antisymmetric 2× 2 matrix with ǫ12 = 1. From the superpotential, at zero temperature, we can construct the Higgs potential at the tree level, which may be read as V0 = VF + VD + VS , (2) where VF = |λ| 2[(|H1| 2 + |H2| 2)|S|2 + |HT 1~σH1 +H 2~σH2) (|H1| 2 − |H2| (Q̃1|H1| 2 + Q̃2|H2| 2 + Q̃3|S| 2)2 , VS = m 2 +m2 2 +m2 |S|2 − [λAλ(H ǫH2)S +H.c.] , (3) where ~σ denotes the three Pauli matrices, g1, g2, and g are the U(1), SU(2), and U(1)′ gauge coupling constants, respectively, Q̃1, Q̃2, and Q̃3 are the U(1) ′ hypercharges of H1, H2, and S, respectively, and m i (i = 1, 2, 3) are the soft SUSY breaking masses. In the Higgs potential, λ and Aλ may in general be complex numbers. However, they will be assumed to be real in the subsequent discussions, as we do not consider CP violation in the Higgs sector. The soft masses are also assumed to be real, without loss of generality, and they are eventually eliminated by imposing minimum conditions with respect to the neutral Higgs fields, The gauge invariance of the superpotential under of U(1)′ requires that the three U(1)′ hypercharges should satisfy Q̃1 + Q̃2 + Q̃3 = 0. The above Higgs potential at the tree level would allow the three neutral Higgs fields , and S to develop the vacuum expectation values (VEVs) v1(0), v2(0), and s(0), respectively. Remark that these VEVs are obtained at zero temperature. However, for simplicity, we omit the temperature dependence of these VEVs until next section where we take into account the finite temperature effect. The tree-level Higgs potential should now be corrected by the radiative one-loop effects. In SUSY models, the radiative corrections due to the top and stop quarks contribute most dominantly to the tree-level Higgs sector. Besides, if tanβ = v2/v1 is very large, the radiative corrections due to the bottom and sbottom quarks should also be included since they become no longer negligible. Furthermore, the radiative corrections due to the exotic quark and squark may be important since the Yukawa coupling of the exotic quark to the singlet field S can be large at the electroweak scale [11]. Therefore, we take into account all the contributions from the top, bottom, exotic quark sector to the tree-level Higgs potential. The one-loop radiative corrections are evaluated by the effective potential method [13]. We assume that the squark masses are degenerate. Ignoring the mixings in the masses of the squarks [14], the one-loop effective potential is given by l=t,b,k + log m̃2 +M2l , (4) where t, b, and k, respectively are top, bottom, and exotic quark fields including the corresponding squark fields, Mt = ht|H2|, Mb = hb|H1|, Mk = hk|S| are the field- dependent quark masses, and m̃ is the soft SUSY breaking mass, which is assumed that m̃ = 1000 GeV ≫ mq (q= t, b, or k). The Higgs sector of the present model consists of six physical Higgs bosons: a pair of charged Higgs boson, one neutral pseudoscalar Higgs boson, and three neutral scalar Higgs bosons. The tree-level mass of the charged Higgs boson is given by m2C± = m W − λ 2v2 + 2λAλs sin 2β , (5) where v = v21 + v 2 = 175 GeV and m W = g 2/2 is the squared mass of the W boson. At the tree level, the mass of the charged Higgs boson might be either smaller or larger than the W boson mass. The tree-level mass of the neutral pseudoscalar Higgs boson is given by m2A = 2λAλv sin 2α , (6) where tanα = (v/2s) sin 2β implies the splitting between the electroweak symmetry break- ing scale and the extra U(1)′ symmetry breaking scale. Note that these tree-level masses of both the neutral pseudoscalar and the charged Higgs bosons do not receive any radiative corrections, because the squark masses are degenerate. The tree-level squared masses of the three neutral scalar Higgs bosons are considerably affected by the radiative corrections. Their squared masses at the one-loop level are given as the eigenvalues of the 3×3 one-loop level mass matrix, whose elements may be written M11 = m Z cos 2 β + 2g v2 cos2 β +m2A sin 2 β cos2 α + fa(m M22 = m Z sin 2 β + 2g v2 sin2 β +m2A cos 2 β cos2 α + fa(m t ) , M33 = 2g 2 +m2A sin 2 α + fa(m M12 = g Q̃1Q̃2v 2 sin 2β + (λ2v2 −m2Z/2) sin 2β −m A cos β sin β cos 2 α , M13 = 2g 1 Q̃1Q̃3vs cos β + 2λ 2vs cosβ −m2A sin β cosα sinα , M23 = 2g Q̃2Q̃3vs sin β + 2λ 2vs sinβ −m2A cos β cosα sinα , (7) where m2Z = (g )v2/2 is the squared mass of the Z boson, and the function fa(m is defined as 3h2qm m̃2 +m2q 4h2qm m̃2 +m2q (m̃2 +m2q) . (8) We assume that the masses of three scalar Higgs bosons Si are sorted such that mS1 ≤ mS2 ≤ mS3 . III. FINITE TEMPERATURE Now, let us study the temperature dependence of the Higgs potential in order to inves- tigate the nature of the electroweak phase transition in the MSSM with an extra U(1)′. We evaluate VT , the temperature-dependent part of the Higgs potential at the one-loop level, using the effective potential method. It is given as [15] l=B,F dx x2 log 1± exp x2 +m2l (φi)/T , (9) where B and F stand for bosons (t̃, b̃, and k̃) and fermions (t, b, and k), and nt = nb = nk = −12 and nt̃ = nb̃ = nk̃ = 12. The negative sign is for bosons and the positive sign is for fermions. Thus, the full Higgs potential at finite temperature at the one-loop level is given by V (T ) = V0 + V1 + VT (10) For numerical analysis, we need to set the values of the relevant parameters of the model. As in the previous section, the soft SUSY breaking mass is set as m̃ = 1000 GeV. The quark masses are set as mt = 175 GeV, mb = 4 GeV, and mk = 400 GeV. From these values, mq̃ = m̃2 +m2q (q = t, b, k) yield the squark masses as mt̃ = 1015 GeV, = 1000 GeV, and m = 1077 GeV. Some caution should be taken for setting the values of Q̃i (i=1, 2, 3), the U(1) hypercharges of the Higgs doublets and the Higgs singlet. In the MSSM with an extra U(1)′, the extra neutral gauge boson mass (mZ′) and the mixing angle (αZZ′) between the two neutral gauge bosons (Z,Z ′) may impose strong constraints on the parameter values. For our numerical analysis, mZ′ is estimated to be larger than 600 GeV, and αZZ′ smaller than 2 × 10−3, for tan β = 3 and s(T = 0) = 500 GeV. Besides, as recent research has suggested [10], we impose the constraint of Q̃1Q̃2 > 0. Further, the U(1) ′ gauge invariance condition requires that Q̃3 = −(Q̃1 + Q̃2). In this paper, we define new charges Qi = g 1Q̃i since Q̃i appear always together with . Then, one may establish the allowed area in the (Q1, Q2)-plane by imposing the above constraints. For tanβ = 3 and s(T = 0) = 500 GeV, the result is shown in Fig. 1, where the small area near the point (Q1, Q2) = (-1, 0) and the upper right corner of Fig. 1 are the allowed areas. The hatched region is the excluded area. There are two specific points in Fig. 1, marked by a star (∗) and a cross (+). The values of Q1 and Q2 at the star-marked point correspond to the ν-model of E6 gauge group realizations [11]. We would take the values of Q1 and Q2 at the cross-marked point, namely, (Q1, Q2) = (-1, -0.1), and hence Q3 =1.1. With these parameter values at hand, we would investigate the possibility of the strongly first-order electroweak phase transition by using two different ways. The first method is to retain only the dominant T 2-proportional part from the high-temperature approximation of VT , and to take account only those particles whose masses are relatively small [6]. The second method is to perform the integration in VT in numerically exact way, and to consider only the contributions of top, bottom, and exotic quarks and squarks. 1. Method A Let us start with the high temperature approximation of VT , which is expressed as [3] VT ≈ − i=t,b,k T 2m2i (φi) m4i (φi) m2i (φi) cFT 2 i=t̃,b̃,k̃ T 2m2i (φi) Tm3i (φi) m4i (φi) m2i (φi) cBT 2 , (11) where log cF = 2.64 and log cB = 5.41. It is known that in the SM the high temperature approximation is consistent with the exact integration of VT within 5 % at temperature T for mF/T < 1.6 and mB/T < 2.2, where mF and mB are respectively the fermion mass and the boson mass that participate in the potential. We select those terms that are proportional to T 2 in the above expression, which become most dominant at high temperature. Thus, we assume that the temperature at which the electroweak phase transition takes place is sufficiently high. We also assume that the U(1) and SU(2) gaugino masses M1 and M2 in the chargino and neutralino sectors are very much larger than the other mass parameters. We take into account the thermal effects due to the Higgs bosons, W , Z, and the extra U(1) gauge boson in the boson sector, and t, b, k quarks, the lighter chargino, and the three light neutralinos in the fermion sector, because their masses are relatively small as compared with temperature, similarly to the analyses of previous articles [6]. Explicitly, the T 2 terms in the high temperature approximation of VT can be expressed as + 4m2 + 2m2 + (2g2 + 6g2 + 6λ2)(|H1| 2 + |H2| 2) + 12λ2|S|2 + 12g 2 + Q̃2 2 + Q̃2 |S|2) + 2g Q̃1Q̃2(|H1| 2 + |H2| Q̃2Q̃3(|H2| 2 + |S|2) + 2g Q̃1Q̃3(|H1| 2 + |S|2) 1 (Q̃1 + Q̃2)(Q̃1|H1| 2 + Q̃2|H2| 2 + Q̃3|S| +6(h2t |H2| 2 + h2b |H1| 2 + h2k|S| . (12) Now, the neutral scalar Higgs fields develop the temperature-dependent VEVs, v1(T ), v2(T ), and s(T ), which we will simply denote v1, v2, and s, respectively. In terms of these temperature-dependent VEVs, the vacuum at finite temperature is defined as the minimum of V (T ) as 〈V (v1, v2, s, T )〉 = 〈V0〉+ 〈V1〉+ 〈VT 〉 , (13) where 〈V0〉 = m g21 + g )2 + λ2(v2 s2 + v2 − 2λAλv1v2s+ (Q̃1v + Q̃2v + Q̃3s 2)2 , 〈V1〉 = fb(m t ) + fb(m b) + fb(m 〈VT 〉 = + 4m2 + 2m2 + (2g2 + 6g2 + 6λ2)(v2 ) + 12λ2s2 + 12g + Q̃2 + Q̃2 s2) + 2g Q̃1Q̃2(v 1 Q̃2Q̃3(v 2 + s 2) + 2g 1 Q̃1Q̃3(v 1 + s 1 (Q̃1 + Q̃2)(Q̃1v 1 + Q̃2v 2 + Q̃3s 2) + 6(h2tv 2 + h 1 + k . (14) In the above expressions, the function fb is defined as + log m̃2 +m2q , (15) and the soft SUSY breaking masses at the one-loop level are given as cos 2β − λ2(s(0)2 + v(0)2 sin2 β) + λAλs(0) tanβ Q̃1(Q̃1v(0) 2 cos2 β + Q̃2v(0) 2 sin2 β + Q̃3s(0) 2)− fc(m b(0)) cos 2β − λ2(s(0)2 + v(0)2 cos2 β) + λAλs(0) cotβ Q̃2(Q̃1v(0) 2 cos2 β + Q̃2v(0) 2 sin2 β + Q̃3s(0) 2)− fc(m t (0)) = − λ2v(0)2 + 2s(0) v(0)2Aλ sin 2β Q̃3(Q̃1v(0) 2 cos2 β + Q̃2v(0) 2 sin2 β + Q̃3s(0) 2)− fc(m k(0)) , (16) where v1(0), v2(0), and s(0) are the VEVs evaluated at zero temperature in the preceding section, tan β = v2(0)/v1(0), v(0) = v1(0)2 + v2(0)2 = 175 GeV, and the function fc is defined as 3h2qm 2 + 2 log m̃2 +m2q m̃2 +m2q . (17) Now, let us determine the critical temperature at which the electroweak phase tran- sition takes place. In our analysis, the critical temperature is defined by a temperature at which 〈V (T )〉 has two distinct minima with equal value, that is, a pair of degenerate vacua. In order to have a pair of degenerate vacua, the potential 〈V (T )〉 should satisfy the minimum condition of 0 = 2m2 s− 2λAλv1v2 + 2λ 1 Q̃3s(Q̃1v 1 + Q̃2v 2 + Q̃3s 2) + 2h2kmkfc(m s[24λ2 + 24g + 20g Q̃3(Q̃1 + Q̃2) + 12k 2] , (18) which is obtained by calculating the first derivative of the full effective potential at the finite temperature with respect to s. For given parameter values at given temperature, one may solve the above minimum condition to express s in terms of the other two VEVs, v1 and v2. Then, by substituting s into 〈V (v1, v2, s, T )〉, one may obtain 〈V (v1, v2, T )〉 which depends only on v1 and v2. By inspecting the shape of 〈V (v1, v2, T )〉 on the (v1, v2)-plane for given parameter values at given temperature, we may determine whether it possess a pair of degenerate vacua or In Fig. 2, the equipotential contours of 〈V (v1, v2, T )〉 are plotted on the (v1, v2)- plane, where the parameter values are set as tanβ = 3, λ = 0.8, s(0) = 500 GeV, mA = 1830 GeV, and the temperature is set as T = 100 GeV, which is actually the critical temperature Tc. One can easily spot two distinct minima of 〈V (v1, v2, T )〉 on the (v1, v2)-plane, namely, one at (0, 0) and the other at (275, 640) GeV. The phase of the state is symmetric at the minimum point (0, 0) on the (v1, v2)-plane, whereas it is broken at (275, 640) GeV. The electroweak phase transition may take place from (0, 0) to (275, 640) GeV on the (v1, v2)-plane, which is evidently discontinuous and therefore it is first-order. The distance on the (v1, v2)-plane between the two minima of 〈V (v1, v2, T )〉, defined as vc, determines the strength of the electroweak phase transition. The electroweak phase transition is said to be strong if vc/Tc > 1, and weak otherwise. In Fig. 2, the distance is calculated to be (275− 0)2 + (640− 0)2 = 696 (GeV) . (19) In Fig. 2, the strength of the electroweak phase transition is about vc/Tc = 6.9, which definitely tells that the electroweak phase transition is a strong one. Therefore, the particular parameter values set for Fig. 2 yields an electroweak phase transition which is first-order as well as strong. Note that vc does not depend on s, that is, we need not to know the values of s at the two minima to calculate vc. Actually, vc is the VEV at the broken phase. The masses of the neutral scalar Higgs bosons at zero temperature for the parameter values of Fig. 2 are obtained as mS1 = 56 GeV, mS2 = 807 GeV, and mS3 = 1827 GeV. We repeat the above job of analysis, varying the values of the relevant parameters. We find that there are a large number of sets of parameter values that allow strongly first-order electroweak phase transitions. Thus, the MSSM with an extra U(1)′ may accommodate TABLE 1: Some sets of λ and mA that allow strongly first-order electroweak phase transitions in the MSSM with an extra U(1)′, obtained by Method A. The values of other parameters are fixed as tanβ = 3, s(0) = 500 GeV, m̃ = 1000 GeV, and Tc = 100 GeV. The pair of numbers in the third column are the coordinates of the broken-phase minimum of 〈V (v1, v2, T )〉. The coordinates of its symmetric-phase minimum is (0, 0) for all sets. The three numbers in the fourth column are the masses of S1, S2, and S3, respectively. The number in the last column is the strength of the first-order electroweak phase transition. λ mA (GeV) (v1, v2) (GeV) mS1 , mS2 , mS3 (GeV) vc/Tc 0.1 478 (1750, 1650) 120, 524, 792 26 0.2 675 (1400, 1500) 118, 674, 796 23 0.3 900 (1200, 1400) 112, 786, 908 18 0.4 1109 (870, 1200) 104, 792, 1112 15 0.5 1306 (600, 1000) 93, 796, 1307 12 0.6 1486 (430, 850) 82, 800, 1485 8 0.7 1660 (340, 700) 70, 803, 1658 7 0.8 1830 (275, 640) 56, 807, 1827 6.9 the desired phase transitions for a wide region in its parameter space. Some of the results are listed in Table 1, where tanβ = 3, s(0) = 500 GeV, and T = 100 GeV are fixed as the values set in Fig. 2, whereas λ and mA have different values. The set of numbers in the last row of Table 1 is the numerical result of Fig. 2. Every set of numbers in each row of Table 1 gives 〈V (v1, v2, T )〉 a pair of degenerate minima, the minimum of symmetric phase at (0, 0) on the (v1, v2)-plane, and the one of broken phase at a different point on the (v1, v2)-plane as given in Table 1. The electroweak phase transition is strongly first-order. One may easily observe in Table 1 that, as the value of λ increases, a larger value of mA allow desired phase transitions. On the other hand, the strength of the phase transition is reinforced if the value of λ decreases. The masses of the neutral scalar Higgs bosons exhibit some interesting behavior. For a larger value of mA, both S2 and S3 have also larger masses whereas S1 has a smaller mass. The tendency is that the strength of the phase transition is reinforced if mS1 increases and if mA, mS2 , and mS3 decrease. In the SM, the strength of the first order electroweak phase transition decreases if its single Higgs boson mass is increased. Also, in the MSSM, we have a weaker phase transition if the lighter one of its two scalar Higgs bosons has a larger mass. In this regard, the tendency of our model is opposite to those of the SM or the MSSM. One can see that this strange behavior also occurs in some parameter region of a non-minimal SUSY model, as shown in Fig. 3 of Ref. [7]. 2. Method B The second method evaluates VT by exact integration to obtain the temperature-dependent full potential V (T ) at one-loop level, where the thermal effects of top, bottom, and exotic quarks and squarks are taken into account. The thermal effects of the gauge bosons can be a help for strengthening the first-order electroweak phase transition, but we would omit them, since the strength of the phase transition is already strong enough. This method starts with the exact integral expression for 〈VT 〉 after replacing the neutral Higgs fields by their VEVs as 〈VT 〉 = − l=t,b,k dx x2 log 1− exp m2l (v1, v2, s) l=t̃,b̃,k̃ dx x2 log 1 + exp m̃2 +m2l (v1, v2, s)  ,(20) which is different from 〈VT 〉 of Method A, while 〈V0〉 and 〈V1〉 are the same as those of Method A. From the full 〈V (T )〉 = 〈V0〉 + 〈V1〉 + 〈VT 〉, we obtain a minimum condition for degenerate vacua as 0 = 2m2 s− 2λAλv1v2 + 2λ )s+ 2g Q3s(Q̃1v + Q̃2v + Q̃3s + 2h2kmkfc(m dx x2 2h2ks exp(− x2 +m2k/T x2 +m2k/T 1 + exp(− x2 +m2k/T dx x2 2h2ks exp(− x2 + (m̃2 +m2k)/T x2 + (m̃2 +m2k)/T 1 + exp(− x2 + (m̃2 +m2k)/T ] , (21) where mk depends only on s and is independent from v1 and v2. Solving the above minimum condition is harder than solving the corresponding mini- mum condition of Method A. Nevertheless, we can solve it by using the bisection method to express s in terms of the other parameters. Then, eliminating s from 〈V (T )〉, we can obtain the expression for 〈V (v1, v2, T )〉 which depends only on v1 and v2. Subsequent steps of numerical analysis are the same as the previous method. In Fig. 3, equipotential contours of 〈V (v1, v2, T )〉 obtained by the present method is plotted on the (v1, v2)-plane, where the parameter values are set slightly different from the previous method: tan β = 3, λ = 0.8, s(0) = 500 GeV, mA = 1780 GeV, and T = 100 GeV. The shape of the equipotential contours of Fig. 3 is almost the same as that of Fig. 2. One can see that there are two distinct minima in Fig. 3, just like Fig. 2: one at (0, 0), and the other at (165, 440) GeV on the (v1, v2)-plane, indicating that the phase transition is first order. The strength of the first-order phase transition is strong, since vc/Tc = 4.7. The masses of the three scalar Higgs bosons are evaluated at zero temperature as mS1 = 82 GeV, mS2 = 804 GeV, and mS3 = 1777 GeV. TABLE 2: Some sets of λ and mA that allow strongly first-order electroweak phase transitions in the MSSM with an extra U(1)′, obtained by Method B. Other descriptions are the same as Table 1. λ mA GeV (v1B, v2B) GeV mSi GeV vc/Tc 0.1 462 (1600, 1600) 121, 468, 791 22 0.2 663 (1400, 1400) 118, 662, 795 19 0.3 885 (1100, 1100) 113, 785, 894 15 0.4 1095 (800, 1200) 106, 792, 1098 14 0.5 1287 (680, 990) 97, 796, 1288 12 0.6 1457 (400, 750) 91, 799, 1456 8 0.7 1620 (300, 600) 86, 801, 1618 6 0.8 1780 (165, 440) 82, 804, 1777 4.7 Comparing Fig. 3 with Fig. 2, one may safely remark that Method A and Method B lead qualitatively the same results. Either method, whether 〈VT 〉 is calculated by direct integration or is simplified by high-temperature approximation, and whether the participating particles at the one-loop level are somewhat exhaustive or selective, we find that the MSSM with and extra U(1)′ allows strongly first-order electroweak phase transitions for certain region in its parameter space. We repeat the numerical analysis by varying the parameter values. and some of the results are listed in Table 2. Like in Table 1, tan β = 3, s(0) = 500 GeV, and T = 100 GeV are fixed, whereas λ and mA are varied. The set of numbers in the last row of Table 2 is the numerical result of Fig. 3. Comparing Table 2 with Table 1, one may notice that the numbers are slightly different from each other but the general behavior of the two tables is exactly the same. IV. DISCUSSIONS AND CONCLUSIONS We investigate the MSSM with an extra U(1)′ if it could accommodate strongly first- order electroweak phase transitions to provide sufficient baryon asymmetry, for reasonable masses of scalar Higgs bosons. To do so, we need the temperature-dependent part of the Higgs potential at the one-loop level. Explicitly, its expression is obtained by two complementary methods: Method A employs high-temperature approximation and retains only the most dominant T 2 terms, and takes into account the thermal effects at the one- loop level of various participating particles. On the other hand, method B performs numerical integrations, and the thermal effects of top, bottom, and exotic quarks and squarks are accounted for. Both methods lead us to essentially the same conclusion: the strongly first-order electroweak phase transition is possible in the MSSM with an extra U(1)′, for a wide region in its parameter space. The masses of the scalar Higgs bosons are obtained within reasonably acceptable ranges. Accordingly, we may expect that the MSSM with an extra U(1)′ can explain the baryon asymmetry of the universe. We remark that the MSSM with an extra U(1)′ exhibits an interesting behavior with respect to the correlation between the strength of the phase transition and the Higgs boson masses. The MSSM with an extra U(1)′ is opposite to the SM or to the MSSM in the sense that the mass of the lightest scalar Higgs boson increases when the strength of the strongly first-order electroweak phase transition becomes stronger. In the SM, its single Higgs boson has a larger mass when the strength of the first order electroweak phase transition decreases. In the MSSM, we also have a larger mass for the lighter one of its two scalar Higgs bosons when the phase transition becomes weaker. ACKNOWLEDGMENTS This research is supported by KOSEF through CHEP. The authors would like to acknowledge the support from KISTI (Korea Institute of Science and Technology Infor- mation) under ”The Strategic Supercomputing Support Program” with Dr. Kihyeon Cho as the technical supporter. The use of the computing system of the Supercomputing Center is also greatly appreciated. [1] A.D. Sakharov, JETP Lett. 5, 24 (1967). [2] V.A. Kuzmin, V.A. Rubakov, and M.E. Shaposhnikov, Phys. Lett. B 155, 36 (1985); M.E. Shaposhnikov, JETP Lett. 44, 465 (1986); Nucl. Phys. B 287, 757 (1987); Nucl. Phys. B 299, 797 (1988); L. McLerran, Phys. Rev. Lett. 62, 1075 (1989); N. Turok and J. Zadrozny, Phys. Rev. Lett. 65, 2331 (1990); Nucl. Phys. B 358, 471 (1991); L. McLerran, M.E. Shaposhnikov, N. Turok, and M. Voloshin, Phys. Lett. B 256, 451 (1991); M. Dine, P. Huet, R. S. Singleton Jr., and L. Susskind, Phys. Lett. B 257, 351 (1991); A. I. Bochkarev, S. V. Kuzmin, and M. E. Shaposhnikov, Phys. Lett. B 244, 257 (1990); Mod. Phys. Lett. A 2, 417 (1987); P. Arnold and O. Espinosa, Phys. Rev. D 47, 3546 (1993); Z. Fodor and A. Hebecker, Nucl. Phys. B 432, 127 (1994); K. Kajantie, M. Laine, K. Rummukainen, and M. Shaposhnikov, Phys. Rev. Lett. 77, 2887 (1996); A. G. Cohen, D. B. Kaplan, and A. E. Nelson, Annu. Rev. Nucl. Part. Sci. 43, 27 (1993); M. Trodden, Rev. Mod. Phys. 71, 1463 (1999); A. Riotto and M. Trodden, Annu. Rev. Nucl. Part. Sci. 49, 35 (1999); F. Csikor and Z. Fodor, and J. Heitger, Phys. Rev. Lett. 82, 21 (1999); F. Csikor, Z. Fodor, and J. Heitger, Phys. Rev. Lett. 82, 21 (1999). [3] G.W. Anderson and L.J. Hall, Phys. Rev. D 45, 2685 (1992). [4] S. Barr, G. Segre, and A. Weldon, Phys. Rev. D 20, 2494 (1979); G.R. Farrar and M.E. Shaposhnikov, Phys. Rev. D 50, 774 (1994). [5] M. Carena, M. Quiros, and C.E.M Wagner, Phys. Lett. B 380, 81 (1996); Nucl. Phys. B 524, 3 (1998); B. de Carlos and J. R. Espinosa, Nucl. Phys. B 503, 24 (1997); M. Laine and K. Rummukainen, Phys. Rev. Lett. 80, 5259 (1998); Nucl. Phys. B 535, 423 (1998); J.M. Cline and G.D. Moore, Phys. Rev. Lett. 81, 3315 (1998); A.T. Davies, C.D. Froggatt, and R.G. Moorhouse, Phys. Lett. B 372, 88 (1996); S.J. Huber and M. G. Schmidt, Eur. Phys. J. C 10, 473 (1999); A. Menon, D.E. Morrissey, and C.E.M. Wagner, Phys. Rev. D 70, 035005 (2004); S.W. Ham, S.K. Oh, and D. Son, Phys. Rev. D 71, 015001 (2005); J. Kang, P. Langacker, T. Li, and T. Liu, Phys. Rev. Lett. 94, 061801 (2005). [6] M. Pietroni, Nucl. Phys. B 402, 27 (1993); M. Bastero-Gil, C. Hugonie, S. F. King, D. P. Roy, and S. Vempati, Phys. Lett. B 489, 359 (2000); S.W. Ham, S.K. Oh, C.M. Kim, E.J. Yoo, and D. Son, Phys. Rev. D 70, 075001 (2004). [7] S.J. Huber and M.G. Schmidt, Nucl. Phys. B 606, 183 (2001). [8] J.E. Kim and H.P. Nilles, Phys. Lett. B 138, 150 (1984). [9] J.L. Hewett and T.G. Rizzo, Phys. Rep. 183, 193 (1989); A. Leike, Phys. Rep. 317, 143 (1999); M. Cvetic and P. Langacker, Phys. Rev. D 54, 3570 (1996); M. Cvetic, D. A. Demir, J. R. Espinosa, L. Everett, and P. Langacker, Phys. Rev. D 54, 3570 (1996); D.A. Demir and N.K. Pak, Phys. Rev. D 57, 6609 (1998); Y. Daikoku and D. Suematsu, Phys. Rev. D 62, 095006 (1998); H. Amini, New J. Phys. 5, 49 (2003). [10] M. Cvetic, D.A. Demir, J.R. Espinosa, L.L. Everett, and P. Langacker, Phys. Rev. D 56, 2861 (1997); Erratum-ibid. D 58, 119905 (1998). [11] S.F. King, S. Moretti, and R. Nevzorov, Phys. Rev. D 73, 035009 (2006); Phys. Lett. B 634, 278 (2006). [12] D.A. Demir and L.L. Everett, Phys. Rev. D 69, 015008 (2004); S.W. Ham, E.J. Yoo, and S.K. Oh, hep-ph/0703041. [13] S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 (1973). [14] Y. Okada, M. Yamaguchi, and T. Yanagida, Prog. Theor. Phys. 85, 1 (1991). [15] L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974). http://arxiv.org/abs/hep-ph/0703041 FIGURE CAPTION FIG. 1. : The allowed area in the (Q1, Q2)-plane. For tanβ = 3 and s(T = 0) = 500 GeV, the small area near the point (Q1, Q2) = (-1, 0) and the upper right corner are the allowed areas, whereas the hatched region is the excluded area. There are two specific points, marked by a star (∗) and a cross (+). The values of Q1 and Q2 at the star-marked point correspond to the ν-model of E6 gauge group realizations. The values of Q1 and Q2 at the cross-marked point are (Q1, Q2) = (-1, -0.1), and hence Q3 =1.1. In our discussions, we choose this point. FIG. 2. : The plot of the equipotential contours of 〈V (v1, v2, T )〉 on the (v1, v2)-plane, obtained by Method A. The parameter values are set as tan β = 3, λ = 0.8, s(0) = 500 GeV, mA = 1830 GeV, and the temperature is set as T = 100 GeV, which is actually the critical temperature Tc. Notice two distinct minima of 〈V (v1, v2, T )〉 on the (v1, v2)-plane: (0, 0) where the phase of the state is symmetric, and (275, 640) GeV, where the phase of the state is broken. The electroweak phase transition may take place from (0, 0) to (275, 640) GeV on the (v1, v2)-plane, which is evidently discontinuous and therefore it is first order. The distance between the two minima is vc = 696 GeV, indicating that the strength of the first-order phase transition is strong (vc/Tc > 1). The masses of the three scalar Higgs bosons are obtained as mS1 = 56 GeV, mS2 = 807 GeV, and mS3 = 1827 FIG. 3. : The plot of the equipotential contours of 〈V (v1, v2, T )〉 on (v1, v2)-plane, ob- tained by Method B. The parameter values are set as tan β = 3, λ = 0.8, s(0) = 500 GeV, mA = 1780 GeV, and Tc = 100 GeV. The coordinates of two minima are: (0, 0) and (165, 440) GeV. The distance between the two minima is vc = 470 GeV, thus the electroweak phase transition between the two minima is strongly first-order. The masses of the three scalar Higgs bosons are obtained as mS1 = 82 GeV, mS2 = 804 GeV, and mS3 = 1777 GeV. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 FIG. 1: The allowed area in the (Q1, Q2)-plane. For tanβ = 3 and s(T = 0) = 500 GeV, the small area near the point (Q1, Q2) = (-1, 0) and the upper right corner are the allowed areas, whereas the hatched region is the excluded area. There are two specific points, marked by a star (∗) and a cross (+). The values of Q1 and Q2 at the star-marked point correspond to the ν-model of E6 gauge group realizations. The values of Q1 and Q2 at the cross-marked point are (Q1, Q2) = (-1, -0.1), and hence Q3 =1.1. In our discussions, we choose this point. 0 50 100 150 200 250 300 350 400 V1 (GeV) V2 (GeV) FIG. 2: The plot of the equipotential contours of 〈V (v1, v2, T )〉 on the (v1, v2)-plane, obtained by Method A. The parameter values are set as tan β = 3, λ = 0.8, s(0) = 500 GeV, mA = 1830 GeV, and the temperature is set as T = 100 GeV, which is actually the critical temperature Tc. Notice two distinct minima of 〈V (v1, v2, T )〉 on the (v1, v2)-plane: (0, 0) where the phase of the state is symmetric, and (275, 640) GeV, where the phase of the state is broken. The electroweak phase transition may take place from (0, 0) to (275, 640) GeV on the (v1, v2)-plane, which is evidently discontinuous and therefore it is first order. The distance between the two minima is vc = 696 GeV, indicating that the strength of the first-order phase transition is strong (vc/Tc > 1). The masses of the three scalar Higgs bosons are obtained as mS1 = 56 GeV, mS2 = 807 GeV, and mS3 = 1827 0 25 50 75 100 125 150 175 200 225 250 V1 (GeV) V2 (GeV) FIG. 3: The plot of the equipotential contours of 〈V (v1, v2, T )〉 on (v1, v2)-plane, obtained by Method B. The parameter values are set as tan β = 3, λ = 0.8, s(0) = 500 GeV, mA = 1780 GeV, and Tc = 100 GeV. The coordinates of two minima are: (0, 0) and (165, 440) GeV. The distance between the two minima is vc = 470 GeV, thus the electroweak phase transition between the two minima is strongly first-order. The masses of the three scalar Higgs bosons are obtained as mS1 = 82 GeV, mS2 = 804 GeV, and mS3 = 1777 GeV. INTRODUCTION ZERO TEMPERATURE FINITE TEMPERATURE Method A Method B DISCUSSIONS AND CONCLUSIONS

0704.0329

Solutions of fractional reaction-diffusion equations in terms of the H-function

arXiv:0704.0329v2 [math.PR] 7 Aug 2007 SOLUTIONS OF FRACTIONAL REACTION-DIFFUSION EQUATIONS IN TERMS OF THE H-FUNCTION H.J. HAUBOLD Office for Outer Space Affairs, United Nations, Vienna International Centre P.O. Box 500, A-1400, Vienna, Austria and Centre for Mathematical Sciences, Pala Campus Arunapuram P.O., Pala-686 574, Kerala, India A.M .MATHAI Department of Mathematics and Statistics, McGill University Montreal, Canada H3A 2K6 and Centre for Mathematical Sciences, Pala Campus Arunapuram P.O., Pala-686 574, Kerala, India R.K. SAXENA Department of Mathematics and Statistics, Jai Narain Vyas University Jodhpur-342004, India Abstract. This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation associated with the Caputo derivative as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the Laplace and Fourier transforms in closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by many authors, notably by Mainardi et al. (2001, 2005) for the fundamental solution of the space-time fractional diffusion equation, and Saxena et al. (2006a, b) for fractional reaction- diffusion equations. The advantage of using Riesz-Feller derivative lies in the fact that the solution of the fractional reaction-diffusion equation containing this derivative includes the fundamental solution for space-time fractional diffusion, which itself is a generalization of neutral fractional diffusion, space-fractional diffusion, and time-fractional diffusion. These specialized types of diffusion can be interpreted as spatial probability density functions evolving in time and are expressible in terms of the H-functions in compact form. 1 Introduction The review of the theory and applications of reaction-diffusion systems is con- tained in many books and articles. In recent work authors have demonstrated the depth of mathematics and related physical issues of reaction-diffusion equa- tions such as nonlinear phenomena, stationary and spatio-temporal dissipative pattern formation, oscillations, waves etc. (Frank, 2005; Grafiychuk, Datsko, http://arxiv.org/abs/0704.0329v2 and Meleshko, 2006, 20076). In recent time, interest in fractional reaction- diffusion equations has increased because the equation exhibits self-organization phenomena and introduces a new parameter, the fractional index, into the equa- tion. Additionally, the analysis of fractional reaction-diffusion equations is of great interest from the analytical and numerical point of view. The objective of this paper is to derive the solution of an unified model of reaction-diffusion system (14), associated with the Caputo derivative and the Riesz-Feller derivative. This new model provides the extension of the models discussed earlier by Mainardi, Luchko, and Pagnini (2001), Mainardi, Pagnini, and Saxena (2005), and Saxena, Mathai, and Haubold (2006a). The present study is in continuation of our earlier work, Haubold and Mathai (1995, 2000) and Saxena, Mathai, and Haubold (2006a, 2006b). 2 Results Required in the Sequel In view of the results J−1/2(x) = cosx. (1) and (Mathai and Saxena, 1978, p. 49), the cosine transform of the H-function is given by tρ−1cos(kt)Hm,np,q (ap,Ap) (bq,Bq) dt (2) n+1,m q+1,p+2 (1−bq,Bq),( (ρ,µ),(1−ap,ap),( , (3) where Re[ρ + µmin1≤j≤m( )] > 0, Re[ρ+ µmax1≤j≤n ] < 0, |argα| < 1 πΩ, Ω > k > 0 and Ω = j=1 Bj − j=m+1 Bj + j=1 Aj − j=n+1 Aj . The Riemann-Liouville fractional integral of order ν is defined by (Miller and Ross, 1993, p. 45; Kilbas et al., 2006) t N(x, t) = (t − u)ν−1N(x, u)du, (4) where Re(ν) > 0. The following fractional derivative of order α > 0 is introduced by Caputo (1969; see also Kilbas et al., 2006) in the form t f(x, t) = Γ(m − α) f (m)(x, τ)dτ (t − τ)α+1−m , m − 1 < α ≤ m, Re(α) > 0, m ∈ N. ∂mf(x, t) , if α = m. (5) where ∂ f(x, t) is the mth partial derivative of f(x,t) with respect to t. The Laplace transform of the Caputo derivative is given by Caputo (1969; see also Kilbas et al., 2006) in the form L {0D t f(x, t); s} = s αF (x, s)− sα−r−1f (r)(x, 0+), (m− 1 < α ≤ m). (6) Following Feller (1952, 1971), it is conventional to define the Riesz-Feller space-fractional derivative of order α and skewness θ in terms of its Fourier transform as F {xD θ f(x); k} = −Ψ α(k)f ∗(k), (7) where Ψθα(k) = |k| αexp[i(signk) ], 0 < α ≤ 2, |θ| ≤ min {α, 2 − α} . (8) When θ = 0, then (8) reduces to F {xD 0 f(x); k} = −|k| α, (9) which is the Fourier transform of the Weyl fractional operator, defined by xf(t) = Γ(n − µ) f(u)du (t − u)µ−n+1 . (10) This shows that the Riesz-Feller operator may be regarded as a generalization of the Weyl operator. Further, when θ = 0, we have a symmetric operator with respect to x that can be interpreted as 0 = − This can be formally deduced by writing −(k)α = −(k2)α/2. For 0 < α < 2 and |θ| ≤ min {α, 2 − α}, the Riesz-Feller derivative can be shown to possess the following integral representation in the x domain: θ f(x) = Γ(1 + α) sin[(α + θ)π/2] f(x + ξ) − f(x) + sin[(α − θ)π/2] f(x − ξ) − f(x) . (12) Finally, we need the following property of the H-function (Mathai and Sax- ena, 1978) Hm,np,q (ap,ap) (bq ,Bq) Hm,np,q (ap,Ap/δ) (bq,Bq/δ , (δ > 0). (13) 3 Unified Fractional Reaction-Diffusion Equa- In this section, we will investigate the solution of the reaction-diffusion equation (14) under the initial conditions (15). The result is given in the form of the following Theorem. Consider the unified fractional reaction-diffusion model t N(x, t) = ηxD θ N(x, t) + Φ(x, t), (14) where η, t > 0, x ∈ r; α, θ, β are real parameters with the constraints 0 < α ≤ 2, |θ| ≤ min(α, 2 − α), 0 < β ≤ 2, and the initial conditions N(x, 0) = f(x), Nt(x, 0) = g(x) ); for x ∈ R, |x|→±∞ N(x, t) = 0, t > 0. (15) Here Nt(x, 0) means the first partial derivative of N(x, t) with respect to t evaluated at t = 0, η is a diffusion constant and Φ(x, t) is a nonlinear function belonging to the area of reaction-diffusion. Further xD θ is the Riesz-Feller space-fractional derivative of order α and asymmetry θ. 0D t is the Caputo time-fractional derivative of order β. Then for the solution of (14), subject to the above constraints, there holds the formula N(x, t) = f∗(k)Eβ,1(−ηt βΨθα(k))exp(−ikx)dk (16) tg∗(k)Eβ,2(−ηk αtβΨθα(k))exp(−ikx)dk ξβ−1dξ Φ∗(k, t − ξ)Eβ,β(−ηk αtβΨθα(k))exp(−ikx)dk. In equation (16) and the following, Eα,β(z) denotes the generalized Mittag- Leffler function (Saxena, Mathai, and Haubold, 2004; Berberan-Santos, 2005; Chamati and Tonchev, 2006). Proof. If we apply the Laplace transform with respect to the time variable t, Fourier transform with respect to space variable x, and use the initial conditions (15) and the formula (7), then the given equation transforms into the form ∼(k, s) − sβ−1f∗(k) − sβ−2g∗(k) = −ηΨθα(k)N ∼(k, s) + Φ ∼(k, s), where according to the conventions followed , the symbol ∼ will stand for the Laplace transform with respect to time variable t and * represents the Fourier transform with respect to space variable x. Solving for N ∼ , it yields ∼(k, s) = f∗(k)sβ−1 sβ + ηΨθα(k) g∗(k)sβ−2 sβ + ηΨθα(k) sβ + ηΨθα(k) . (17) On taking the inverse Laplace transform of (17) and applying the formula a + sα = tα−βEα,α−β+1(−at α), (18) where Re(s) > 0, Re(α) > 0, Re(α − β) > −1; it is seen that N∗(k, t) = f∗(k)Eβ,1(−ηt βΨθα(k)) + g ∗(k)tEβ,2(−ηt βΨθα(k)) Φ∗(k, t − ξ)ξβ−1Eβ,β(−ηΨ α(k)ξ β)dξ. (19) The required solution (16) is now obtained by taking the inverse Fourier trans- form of (19). This completes the proof of the theorem. 4 Special Cases When g(x) = 0, then by the application of the convolution theorem of the Fourier transform to the solution (16) of the theorem, it readily yields Corollary 1. The solution of the fractional reaction-diffusion equation N(x, t) − η N(x, t) = Φ(x, t), x ∈ R, t > 0, η > 0, (20) with initial conditions N(x, 0) = f(x), Nt(x, 0) = 0 for x ∈ R, 1 < β ≤ 2, x→±∞ N(x, t) = 0, (21) where η is a diffusion constant and Φ(x, t) is a nonlinear function belonging to the area of reaction-diffusion, is given by N(x, t) = G1(x − τ, t)f(τ)dτ (t − ξ)β−1dξ G2(x − τ, t − ξ)Φ(τ, ξ)dτ, (22) where α − θ G1(x, t) = exp(−ikx)Eβ,1(−η|t β |Ψθα(k))dk (23) η1/αtβ/α (1,1/α),(β,β/α),(1,ρ) (1,1/α),(1,1),(1,ρ) , (α > 0) G2(x, t) = exp(−ikx)Eβ,β(−ηt βΨθα(k))dk η1/αtβ/α (1,1/α),(β,β/α),(1,ρ) (1,1/α),(1,1),(1,ρ) , (α > 0). (24) In deriving the above results, we have used the inverse Fourier transform formula F−1[Eβ,γ(−ηt βΨαθ (k)); x] = 3,3 [ η1αtβ/α (1,1/α),(γ,β/α),(1,ρ) (1,1/α),(1,1),(1,ρ) ], (25) where Re(β) > 0, Re(γ) > 0, which can be established by following a procedure similar to that employed by Mainardi, Luchko, and Pagnini (2001). Next , if we set f(x) = δ(x), Φ = 0, g(x) = 0, where δ(x) is the Dirac delta-function, then we arrive at the following interesting result given by Mainardi, Pagnini, and Saxena (2005). Corollary 2. Consider the following space-time fractional diffusion model ∂βN(x, t) = η xD θ N(x, t), η > 0, x ∈ R, 0 < β ≤ 2, (26) with the initial conditions N(x, t = 0) = δ(x), Nt(x, 0) = 0, x→±∞ N(x, t) = 0 where η is a diffusion constant and δ(x) is the Dirac delta-function. Then for the fundamental solution of (26) with initial conditions, there holds the formula N(x, t) = 3.3 [ (ηtβ)1/α (1,1/α),(1,β/α),(1,ρ) (1,1/α),(1,1),(1,ρ) ], (27) where ρ = α−θ Some interesting special cases of (26) are enumerated below. (i) We note that for α = β, Mainardi, Pagnini, and Saxena (2005) have shown that the corresponding solution of (26), denoted by Nθα, which we call as the neutral fractional diffusion, can be expressed in terms of elementary function and can be defined for x > 0 as Neutral fractional diffusion: 0 < α = β < 2; θ ≤ min {α, 2 − α} , Nθα(x) = xα−1sin[(π/2)(α − θ)] 1 + 2xαcos[(π/2)(α − θ)] + x2α . (28) The neutral fractional diffusion is not studied at length in the literature. Next we derive some stable densities in terms of the H-functions as special cases of the solution of the equation (26) (ii) If we set β = 1, 0 < α < 2; θ ≤ min {α, 2 − α}then (26) reduces to space fractional diffusion equation, which we denote by Lθα(x) is the fundamental solution of the following space-time fractional diffusion model: ∂N(x, t) = η xD θ N(x, t), η > 0, x ∈ R, (29) with the initial conditions N(x, t = 0) = δ(x), limx→±∞N(x, t) = 0,, where η is a diffusion constant and δ(x) is the Dirac-delta function. Hence for the solution of (29) there holds the formula Lθα(x) = α(ηt)1/α (ηt)1/α (1,1),(ρ,ρ) ),(ρ,ρ) , 0 < α < 1, |θ| ≤ α, (30) where ρ = α−θ . The density represented by the above expression is known as α-stable Lévy density. Another form of this density is given by Lθα(x) = α(ηt)1/α (ηt)1/α (1− 1 ),(1−ρ,ρ) (0,1),(1−ρ,ρ) , 1 < α < 2, |θ| ≤ 2 − α, (iii) Next, if we take α = 2, 0 < β < 2, θ = 0, then we obtain the time fractional diffusion, which is governed by the following time fractional diffusion model: ∂βN(x, t) N(x, t), η > 0, x ∈ R, 0 < β ≤ 2, (32) with the initial conditions N(x, t = 0) = δ(x), Nt(x, 0) = 0, x→±∞ N(x, t) = 0 where η is a diffusion constant and δ(x) is the Dirac delta-function, whose fundamental solution is given by the equation N(x, t) = (ηtβ)1/2 (1,β/2) (1,1) . (33) (iv) Further, if we set α = 2, β = 1 and θ → 0 then for the fundamental solution of the standard diffusion equation N(x, t) = η N(x, t), (34) with initial condition N(x, t = 0) = δ(x), limx→±∞N(x, t) = 0, (35) there holds the formula N(x, t) = η1/2t1/2 (1,1/2) (1,1) = (4πηt)−1/2exp[− ], (36) which is the classical Gaussian density. For further details of these special cases based on the Green function, one can refer to the paper by Mainardi, Luchko, and Pagnini (2001) and Mainardi, Pagnini, and Saxena (2005). Remark. Fractional order moments and the asymptotic expansion of the solu- tion (27) are discussed by Mainardi, Luchko, and Pagnini (2001). Finally, for β = 1/2 in (14), we arrive at Corollary 3. Consider the following fractional reaction-diffusion model t N(x, t) = ηxD θ N(x, t) + Φ(x, t), (37) where η, t > 0, x ∈ R; α, θ are real parameters with the constraints 0 < α ≤ 2, |θ| ≤ min(α, 2 − α), and the initial conditions N(x, 0) = f(x), for x ∈ R, limx→±∞N(x, t) = 0. (38) Here η is a diffusion constant and Φ(x, t) is a nonlinear function belonging to the area of reaction-diffusion. Further xD θ is the Riesz-Feller space fractional derivative of order α and asymmetry θ and D t is the Caputo time-fractional derivative of order 1/2. Then for the solution of (37), subject to the above constraints, there holds the formula N(x, t) = f∗(k)E1/2,1(−ηt βΨθα(k))exp(−ikx)dk (39) ξ−1/2dξ Φ∗(kct − ξ)E 1 (−ηkαt1.2Ψθα(k))exp(−ikx)dk. If we set θ = 0 in (39), then it reduces to the result recently obtained by the authors (2006a) for the fractional reaction-diffusion equation. 5 References Berberan-Santos, M.N. (2005). Properties of the Mittag-Leffler relaxation func- tion, Journal of Mathematical Chemistry, 38, 629-635. Caputo, M. (1969). Elasticita e Dissipazione, Zanichelli, Bologna. Chamati, H. and Tonchev, N.S. (2006). Generalized Mittag-Leffler functions in the theory of finite-size scaling for systems with strong anisotropy and/or long-range interaction, Journal of Physics A: Mathematical and General, 39, 469-478. Feller, W. (1952). On a generalization of Marcel Riesz’ potentials and the semi-groups generated by them, Meddeladen Lund Universitets Matematiska Seminarium (Comm. Sém. Mathém. Université de Lund ), Tome suppl. dédié a M. Riesz, Lund, 73-81. Feller, W. (1966). An Introduction to Probability Theory and its Applications, Vol. II, John Wiley and Sons, New York. Frank, T.D. (2005). Nonlinear Fokker-Planck Equations: Fundamentals and Applications, Springer, Berlin Heidelberg New York. Grafiychuk, V., Datsko, B., and Meleshko, V. (2006). Mathematical model- ing of pattern formation in sub- and superdiffusive reaction-diffusion systems, arXiv:nlin.AO/06110005 v3. Grafiychuk, V., Datsko, B., and Meleshko, V. (2007). Nonlinear oscillations and stability domains in fractional reaction-diffusion systems, arXiv:nlin.PS/0702013 Haubold, H.J. and Mathai, A.M. (2000). The fractional kinetic equation and thermonuclear functions, Astrophysics and Space Science, 273, 53-63. Haubold, H.J. and Mathai, A.M. (1995). A heuristic remark on the periodic variation in the number of solar neutrinos detected on Earth, Astrophysics and Space Science, 228, 113-124. Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applica- tions of Fractional Differential Equations, Elsevier, Amsterdam. Mainardi, F., Luchko, Y., and Pagnini, G. (2001). The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus and Applied Analysis. 4, 153-192. Mainardi, F., Pagnini, G., and Saxena, R.K. (2005). Fox H-functions in frac- tional diffusion, Journal of Computational and Applied Mathematics 178, 321- Mathai, A.M. and Saxena, R.K. (1978). The H-function with Applications in Statistics and Other Disciplines, John Wiley and Sons, New York, London, and Sydney. Miller, K.S. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York. Saxena, R.K., Mathai, A.M., and Haubold, H.J. (2004). On fractional kinetic equations, Astrophysics and Space Science, 282, 281-287. Saxena, R.K., Mathai, A.M., and Haubold, H.J. (2006a). Fractional reaction- diffusion equations, Astrophysics and Space Science, 305, 289-296. Saxena, R.K., Mathai, A.M., and Haubold, H.J. (2006b). Reaction-diffusion systems and nonlinear waves, Astrophysics and Space Science, 305, 297-303. Yu, R. and Zhang, H. (2006). New function of Mittag-Leffler type and its ap- plication in the fractional diffusion-wave equation, Chaos, Solitons and Fractals 30, 946-955.

0704.0330

Random Matrix Theory at Nonzero $\mu$ and $T$

Random Matrix Theory at Nonzero µ and T Kim Splittorff1,∗) and Jacobus Johannes Maria Verbaarschot1,2 ,3 ,∗∗) 1 The Niels Bohr Institute, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark 2 Niels Bohr International Academy, Blegdamsvej 17, DK-2100, Copenhagen Ø 3 Department of Physics and Astronomy, SUNY, Stony Brook, New York 11794 We review applications of random matrix theory to QCD at nonzero temperature and chemical potential. The chiral phase transition of QCD and QCD-like theories is discussed in terms of eigenvalues of the Dirac operator. We show that for QCD at µ 6= 0, which has a sign problem, the discontinuity in the chiral condensate is due to an alternative to the Banks-Casher relation. The severity of the sign problem is analyzed in the microscopic domain of QCD. §1. Introduction Starting from its introduction in nuclear physics by Wigner,1) random matrix theories have been applied to a wide range of problems ranging from the physics of proteins2) to quantum gravity (see3), 4) for a historical review). Three reasons for the ubiquity of random matrix theory come to mind. First, eigenvalues of large ran- dom matrices have universal properties determined by symmetries. Second, random matrices are models for disorder present in many physical systems. Third, random matrix theories have a topological expansion which is important for applications to quantum field theory. One of the attractive features of random matrix theory is that analytical information can be obtained for complex systems which otherwise only can be studied experimentally or numerically. In this review we discuss applications of random matrix theory to QCD at nonzero temperature and chemical potential. Since the order parameter for the chiral phase transition5), 6) and the deconfining phase transition7), 8) are determined by the infrared behavior of the eigenvalues of the Dirac operator, these eigenvalues are essential for the phase transitions in QCD. Remarkably, the distribution of the smallest Dirac eigenvalues is given by universal functions9)–13) that depend only on one or two parameters, the chiral condensate and the pion decay constant. This offers an alternative way to measure these constants on the lattice.14)–22) §2. Random Matrix Theory in QCD Chiral Random Matrix Theory (chRMT) is a theory with the global symmetries of QCD, but matrix elements of the Dirac operator replaced by random numbers9), 10) iW † m , P (W ) ∼ e−NTrW †W . (2.1) ∗) e-mail address: [email protected] ∗∗) e-mail address: [email protected] http://arxiv.org/abs/0704.0330v1 2 K.Splittorff and J.J.M. Verbaarschot This random matrix model has the global symmetries and topological properties of QCD. It is confining in the sense that only color singlets have a nonzero expecta- tion value. It is now well understood that fluctuations of low-lying eigenvalues of the Dirac operator are described by chRMT (see23)–28) for lectures and reviews). Philosphically, this is important because of the realization that chaotic motion dom- inates the dynamics of quarks at low energy. Practically, this is important because we can use powerful random matrix techniques to calculate physical observables. The condition for the applicability of chRMT is that the Compton wavelength of Goldstone bosons associated with the mass scale z of these eigenvalues is much larger than the size of the box. With the squared mass of the associated Goldstone boson given by 2zΣ/F 2π , this condition reads ≪ Λ2. (2.2) The second condition is necessary to factorize the partition function into a contribu- tion from the lightest degrees of freedom and all heavier degrees of freedom. These two conditions determine the microscopic domain of QCD. We stress that z is a scale in the Dirac spectrum so that, for sufficiently large volumes, we always have eigenval- ues in the domain (2.2) where eigenvalues fluctuate according to chRMT. This can be shown rigorously from the following two observations.30), 31) First, the infrared Dirac spectrum follows from a (partially quenched) chiral Lagrangian determined by chiral symmetry, and the inequality (2.2) is the condition for factorization of the partition function into a factor containing the constant modes and another factor containing the nonzero momentum modes. Second, the factor with the constant modes is equal to the large N limit of chiral random matrix theory. In32), 33) the condition (2.2) was imposed on the quark masses and was the bases for a systematic expansion of the chiral Lagrangian known as the ǫ expansion. One feature that underlies universal properties of eigenvalues is that they be- have as repulsive confined charges. This follows from the joint probability distri- bution ∼ k<l(λ )2 exp(−N ). It can be shown that eigenvalues correlations at the micrsocopic scale are universal.34) The reason is spontaneous symmetry breaking and a mass gap so that they can be described in terms of a chiral Lagrangian. 2.1. Chiral Random Matrix Theory at µ 6= 0 and T 6= 0 A nonzero temperature does not change the fluctuating behavior of the Dirac eigenvalues provided that chiral symmetry remains broken. However, a transition to a different universality class takes place at the critical temperature. A random matrix model that reproduces this universal behavior of QCD is obtained by replacing the off-diagonal elements in (2.1) by35) iW → iW + t, iW † → iW † − t with t = diag(−πT, πT ). (2.3) This model has been studied elaborately in the literature (see e.g.35)–40)). A nonzero chemical potential can be introduced analogously to the quark mass. The requirement is that the small µ behaviour of the QCD partition function should Random Matrix Theory 3 0.0 1.0 2.0 3.0 4.0 2µ/mπ m=0.10 m=0.05 m=0.01 Fig. 1. Lattice results for Nc = 2 (taken from 55)) and phase quenched QCD with Nc = 3 (taken from56)) be reproduced by the random matrix partition function. This achieved by modifying (2.1) by41) iW → iW + µ, iW † → iW † + µ, (2.4) resulting in a nonhermitean Dirac operator with eigenvalues scattered in the complex plane. The prescription (2.4) is not unique. A random matrix model that has had a strong impact on recent developments is defined by42) iW → iW + µH, iW † → iW † + µH with H† = H, (2.5) where H is drawn from a Gaussian ensemble of random matrices. This model is in the same universality class as (2.4) but is technically simpler since it can be worked out by means of the complex orthogonal polynomial method.42)–46) There are other types of random matrix models that have been applied to QCD. For example models with random gauge fields such as the Eguchi-Kawai model47) or its 2-dimensional version.48) QCD in 1 dimension49), 50) is a random matrix model as well, with universally fluctuating Dirac eigenvalues. Also models with random Wilson loops51), 52) have attracted significant interest. §3. Phases of QCD and RMT QCD-like theories with charged Goldstone bosons have a critical chemical poten- tial equal to mπ/2. The phase transition to the Bose condensed phase can therefore be described completely in terms of a chiral Lagragian. At the mean field level,53) the kinetic terms of this chiral Lagrangian do not contribute, so that these results can also be obtained from chiral random matrix theory. Indeed, the static part of the chiral Lagrangian53), 54) F 2πµ 2Tr[U,B][U †, B]− ΣTr(MU +MU †). (3.1) can also be obtained from the large N limit of the models (2.4) or (2.5). 4 K.Splittorff and J.J.M. Verbaarschot Tricritial point 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Fig. 2. QCD phase diagram in the µTm-space (taken from58)) In Fig. 1 we display lattice results for QCD with Nc = 2 55) and phase quenched QCD.56) They show an impressive agreement with the results from (3.1) given by the solid curves in both figures. 3.1. Schematic RMT Phase Diagram The phase transition in QCD with Nc = 3 at µc = mN/3 cannot be analyzed by means of chiral Lagrangians. Because of the sign problem lattice studies are not possible either. In such situation there is long tradition to analyze the same problem in a much simpler theory in the hope of obtaining at least a qualitative understanding of the problem. For example, one dimensional QCD,49), 50) or more recently, super Yang-Mills theory and AdS-CFT duality,57) been explored as toy models for QCD. We will use random matrix theory at T 6= 0 and µ 6= 0, introduced in (2.3) and (2.4) to obtain a qualitive understanding of the QCD phase diagram. Lattice QCD simulations show that the chiral phase transition at µ = 0 is of second order or a steep cross-over. At T = 0 we expect a first order phase transition at µc = mN/3. It is natural that the first order line ends in a critical end point or joins the second order critical line at the tricritical point (see Fig. 3.1, left). This is indeed what is observed in random matrix theory58), 59) (see Fig. 3.1, right). A similar phase diagram has also been obtained from the NJL model.60)–62) Another scenario that was discovered in RMT is the splitting of the first order line into two at nonzero isospin chemical potential.63) This behavior was also found in a NJL model64), 65) but might not be stable against flavor mixing interactions.66) §4. Dirac Spectrum in Theories Without a Sign Problem Since the spectrum of the Dirac operator determines the chiral condensate, phase transitions in QCD can be understood in terms of its spectral flow. In this section we discuss theories with a positive fermion determinant such as QCD with two colors and phase quenched QCD, where a probabilistic interpretation of the eigenvalue density is possible. The relation between chiral symmetry breaking and Dirac spectra is much more complicated when the fermion determinant is complex and its discussion will be postponed to the next section. The spectrum of an anti-Hermitean Dirac operator is purely imaginary with an eigenvalue density that is proportional to the volume. If chiral symmetry is broken spontaneously, the chiral condensate becomes discontinuous across the imaginary axis in the thermodynamic limit. Chiral symmetry is restored if such discontinuity Random Matrix Theory 5 mm m m m m T < Tc µ = 0 T > Tc µ = 0 T < Tc µ < µc T < Tc µ = µc T < Tc µ > µc T > Tc µ > µc Fig. 3. Critical behavior of the Dirac spectrum. µc = mπ/2 for T = 0 and increases with T . is absent for example by the formation of a gap in the Dirac spectrum, see eg.71) . For µ 6= 0, the Dirac spectrum broadens into a strip of width 4µ2F 2π/Σ.49), 67) The chemical potential becomes critical when the quark mass hits the edge of this strip. At this point the chiral condensate starts rotating into a pion condensate. Chiral symmetry restoration takes place when a gap forms at zero. A schematic picture of the critical behavior of Dirac eigenvalues is shown in Fig. 3 and the spectral flow of the Dirac eigenvalues with respect to increasing µ and T is summarized in Fig. 4. One conclusion from this behavior is that Tc(µ) is a concave function of µ, and that µc(T ) is a convex function of T . The spectral flow discussed in this section is supported by lattice simulations at T 6= 0 and µ 6= 0 (See Fig. 5) 4.1. Dirac spectrum in the µ-plane We could equally well have diagonalized the Dirac operator in a representation where µγ0 is proportional to the identity, det(D +m+ µγ0) = det(γ0(D +m) + µ). (4.1) These eigenvalues are relevant to the baryon number density. A gap in the spectrum develops at m 6= 0 (see Fig. 6), and the chemical potential becomes critical, µ = mπ/2 when it hits the inner edge of the domain of eigenvalues. Increasing µ Increasing T Fig. 4. Spectral flow of the Dirac spectrum (left) and phase diagram (right) with respect to µ and T in phase quenched QCD and QCD with two colors. 6 K.Splittorff and J.J.M. Verbaarschot 1 1.5 2 2.5 b=0.35 b=0.3525 b=0.355 b=0.3575 b=0.36 1.76(t-0.93) 0.0 0.1 0.2 0.3 β=5.5 β=5.66 β=5.71 β=5.75 β=5.9 Fig. 5. Temperature and chemical potential dependence of Dirac eigenvalues. From left to right taken from.70), 72)–74) 4.2. Quenched Lattice QCD Dirac Spectra at µ 6= 0 Small Dirac eigenvalues at µ 6= 0 have been computed in quenched QCD. The analytical formulas for the average density of the small Dirac eigenvalues are avail- able.68), 69) They were first derived68) by exploiting the Toda lattice hierarchy in the flavor index. Comparisons of random matrix predictions68) for the radial spectral density and lattice QCD results75), 76) are shown in the left panel of Fig. 7. In other cases, such as the overlap Dirac operator77) and QCD with Nc = 2, 78) a similar degree of agreement was found. Both the spectral density and two-point correlations can be derived from the Lagrangian (3.1), i.e. they are determined by two param- eters, Fπ and Σ. This can be exploited to extract these low-energy constants. For example, Fπ and Σ were determined 19), 21) (see also20)) from the correlators shown in the two right panels of Fig. 7. §5. Chiral Symmetry Breaking at µ 6= 0 The full QCD partition function at µ 6= 0 which is the average of det(D +m+ µγ0) = |det(D +m+ µγ0)|eiθ, θ 6= 0, (5.1) has properties which are drastically different from the phase quenched partition function where the phase factor is absent. In particular, µc = mN/3 instead of mπ/2, so that the free energy remains µ-independent until µ = mN/3. For µ < mN/3 the Fig. 6. Eigenvalues of γ0(D + m) for a random matrix Dirac operator at m = 0 (left), m 6= 0 (middle) (both taken from79)), and lattice QCD at m 6= 0 (right, taken from49)). Random Matrix Theory 7 —– Splittorff-Verbaarschot-2004 —– Wettig-2004 0 2 4 6 8 −0.15 −0.05 V = 8 10000 configs µisoFπV = 0.159 1.27 1.37 1.47 π/2 1.67 1.77 1.87 angle (θ) lattice: 6 , µa = 0.006 fit: µFV = 0.14 Fig. 7. The radial spectral density for (left, taken from75), 76)) and two-point correlations (middle taken from19) and right taken from21)). chiral condensate remains discontinuous at m = 0, whereas the chiral condensate of the phase quenched theory approaches zero for m → 0 (see Fig. 5). The only difference between the phase quenched partition function and the full QCD partition function is the phase of the fermion determinant. We conclude that the phase factor is responsible for the discontinuity of the chiral condensate. How can this happen if for each configuration the support of the spectrum is approximately the same? This problem known as the “Silver Blaze Problem”80) was solved in.6) 5.1. Unquenched Spectral Density The spectral density for QCD with dynamical fermions is given by ρNf (λ) = 〈 δ2(λ− λk)detNf (D +m+ µγ0)〉. (5.2) Because of the phase of the fermion determinant, this density is in general complex and can be decomposed as ρNf (λ) = ρNf=0(λ) + ρU (λ). The chiral condensate can then be decomposed as ΣNf (m) = ΣNf=0(m) +ΣU (m), so that the discontinuity in Σ(m) is due to ρU . Asymptotically it behaves as ρU ∼ e µ2F 2V e iIm(λ)ΣV and vanishes outside an ellips starting at Re(λ) = m (see Fig. 9).6) In the right part of this figure we show the real part of the spectral density for QCD with one flavor at nonzero chemical potential. Scatter plot of Dirac eigenvalues ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� quark mass m Support of spectrum Chiral condensate condensate Quenched chiral in full QCD µ2F 2 Σ(m) = 1 Fig. 8. Chiral condensate of quenched and full QCD. 8 K.Splittorff and J.J.M. Verbaarschot Dirac spectrum for Full QCD. ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� Oscillating Region quark mass m -1000100 0.001 0.002 2F 2µ2 µ2F 2 Fig. 9. Support (left) and real part (right, taken from27)) of Dirac spectral density for QCD with Nf = 1 and µ 6= 0. This result explains the mechanism of chiral symmetry breaking at nonzero chemical potential. The phase of the fermion determinant rotates the pion conden- sate back into a chiral condensate, but it does so in an unexpected way.6) The same mechanism is at play for 1d QCD at µ 6= 0.82) §6. Phase of the Fermion Determinant The magnitude of the sign problem can be measured by means of the expectation value of the phase factor of the fermion determiant which can be defined in two ways 〈e2iθ〉Nf = det(D + µγ0 +m) det∗(D + µγ0 +m) detNf (D + µγ0 +m) , 〈e2iθ〉1+1∗ = ZNf=2 Z1+1∗ The average 〈· · · 〉 is with respect to the Yang-Mills action. The sign problem is managable when the average phase factor remains finite in the thermodynamic limit. In the microscopic domain it is possible to obtain exact analytical expressions for the average phase factor by exploiting the equivalence between QCD and RMT in this domain. For µ < mπ/2 the free energy of both QCD and phase quenched QCD are independent of µ. This does not imply that the average phase factor is µ-independent. The µ-dependence originates from the charged Goldstone bosons with mass mπ ± 2µ, and for Nf flavors the mean field result83), 84) for 〈exp(2iθ)〉 reads (1 − 4µ2/m2π)Nf+1. The exact result for the average phase factor for Nf = 2 is shown in Fig. 10 (right), where lattice results85) are also shown (left). The exact result has an essential singularity at µ = 0, but its thermodyanmic limit agrees with the mean result. 0 0.5 1 1.5 2µ/mπ mΣV = 4 mΣV >> 1 Fig. 10. Average phase factor. Lattice QCD results are shown left (taken from85)) and the exact microscopic result83) is shown right. Random Matrix Theory 9 §7. Conclusions The equivalence of chiral random matrix theory and QCD has been exploited succesfully to derive a host of analytical results. Among others, eigenvalue fluctua- tions predicted by chRMT have been observed in lattice simulations, the phases of QCD can be understood in terms of spectral flow, observables can be extracted from the fluctuations of the smallest eigenvalues, the sign problem is not serious when the quark mass is outside the domain of the eigenvalues, and mean field results can be obtained from random matrix theory. Summarizing, chiral random matrix theory is a powerful tool for analyzing the infrared domain of QCD. Acknowledgements The YITP is thanked for its hospitality. G. Akemann, J. Osborn and P.H. Damgaard are acknowledged for valuable discussions. This work was supported by US DOE Grant No. DE-FG-88ER40388 (JV), the Villum Kann Rasmussen Foun- dation (JV), the Danish National Bank (JV) and the Carslberg Foundation (KS). References 1) E.P. Wigner, Proc. Cam. Phil. Soc. 47 (1951) 790. 2) M. Sener and K. Schulten, Phys. Rev. E 65, 031916 (2002). 3) T. Guhr, A. Muller-Groeling and H. A. Weidenmuller, Phys. Rept. 299, 189 (1998). 4) P. J. Forrester, N. C. Snaith and J. J. M. Verbaarschot, J. Phys. A 36, R1 (2003). 5) T. Banks and A. Casher, Nucl. Phys. B 169, 103 (1980). 6) J. C. Osborn, K. Splittorff and J. J. M. Verbaarschot, Phys. Rev. Lett. 94, 202001 (2005). 7) C. Gattringer, Phys. Rev. Lett. 97, 032003 (2006). 8) F. Synatschke, A. Wipf and C. Wozar, arXiv:hep-lat/0703018. 9) E. V. Shuryak and J. J. M. Verbaarschot, Nucl. Phys. A 560, 306 (1993). 10) J. J. M. Verbaarschot, Phys. Rev. Lett. 72, 2531 (1994). 11) J. J. M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70, 3852 (1993). 12) S. M. Nishigaki, P. H. Damgaard and T. Wettig, Phys. Rev. D 58, 087704 (1998). 13) P. H. Damgaard and S. M. Nishigaki, Phys. Rev. D 63, 045012 (2001). 14) M. E. Berbenni-Bitsch et al. , Nucl. Phys. Proc. Suppl. 63, 820 (1998). 15) P. H. Damgaard et al. Phys. Lett. B 495, 263 (2000) 16) T. DeGrand, R. Hoffmann, S. Schaefer and Z. Liu, Phys. Rev. D 74, 054501 (2006). 17) H. Fukaya et al. [JLQCD Collaboration], arXiv:hep-lat/0702003. 18) C. B. Lang, P. Majumdar and W. Ortner, arXiv:hep-lat/0611010. 19) P. Damgaard, U. Heller, K. Splittorff and B. Svetitsky, Phys. Rev. D 72, 091501 (2005). 20) P. Damgaard, U. Heller, K. Splittorff, B. Svetitsky and D. Toublan, Phys. Rev. D 73, 105016 (2006). 21) J. C. Osborn and T. Wettig, PoS LAT2005, 200 (2006) [arXiv:hep-lat/0510115]. 22) G. Akemann, P. H. Damgaard, J. C. Osborn and K. Splittorff, Nucl. Phys. B 766, 34 (2007). 23) M. A. Stephanov, J. J. M. Verbaarschot and T. Wettig, arXiv:hep-ph/0509286. 24) J. J. M. Verbaarschot and T. Wettig, Ann. Rev. Nucl. Part. Sci. 50, 343 (2000). 25) J. J. M. Verbaarschot, arXiv:hep-th/0502029. 26) M. A. Nowak, arXiv:hep-ph/0112296. 27) K. Splittorff, PoS LAT2006 023, arXiv:hep-lat/0610072. 28) G. Akemann, arXiv:hep-th/0701175. 29) J. J. M. Verbaarschot, Phys. Lett. B 368, 137 (1996). 30) J. C. Osborn, D. Toublan and J. J. M. Verbaarschot, Nucl. Phys. B 540, 317 (1999). 31) P. Damgaard, J. Osborn, D. Toublan and J. Verbaarschot, Nucl. Phys. B 547, 305 (1999). 10 K.Splittorff and J.J.M. Verbaarschot 32) J. Gasser and H. Leutwyler, Phys. Lett. B 188, 477 (1987). 33) H. Leutwyler and A. Smilga, Phys. Rev. D 46, 5607 (1992). 34) G. Akemann, P. H. Damgaard, U. Magnea and S. Nishigaki, Nucl. Phys. B 487, 721 (1997). 35) A. D. Jackson and J. J. M. Verbaarschot, Phys. Rev. D 53, 7223 (1996). 36) T. Wettig, H. A. Weidenmueller and A. Schaefer, Nucl. Phys. A 610, 492C (1996). 37) M. A. Stephanov, Phys. Lett. B 375, 249 (1996). 38) A. D. Jackson, M. K. Sener and J. J. M. Verbaarschot, Nucl. Phys. B 479, 707 (1996). 39) M. A. Nowak, G. Papp and I. Zahed, Phys. Lett. B 389, 341 (1996). 40) R. A. Janik, M. A. Nowak, G. Papp and I. Zahed, Phys. Lett. B 446, 9 (1999). 41) M. A. Stephanov, Phys. Rev. Lett. 76, 4472 (1996). 42) J. C. Osborn, Phys. Rev. Lett. 93, 222001 (2004). 43) G. Akemann and A. Pottier, J. Phys. A 37, L453 (2004). 44) Y.V. Fyodorov, B. Khoruzhenko and H.J. Sommers, Ann. Inst. Henri Poincaré: Phys. Theor. 68, 449 (1998). 45) G. Akemann, Phys. Rev. Lett. 80, 072002 (2002); J. Phys. A: Math. Gen. 36, 3363 (2003). 46) M. C. Bergere, arXiv:hep-th/0311227; M. C. Bergere, arXiv:hep-th/0404126. 47) T. Eguchi and H. Kawai, Phys. Rev. Lett. 48, 1063 (1982). 48) D. J. Gross and E. Witten, Phys. Rev. D 21, 446 (1980). 49) P. E. Gibbs, Preprint PRINT-86-0389-GLASGOW, 1986. 50) N. Bilic and K. Demeterfi, Phys. Lett. B 212, 83 (1988). 51) B. Durhuus and P. Olesen, Nucl. Phys. B 184, 461 (1981). 52) A. Dumitru et al., Phys. Rev. D 70, 034511 (2004). 53) J.B. Kogut et al., Nucl. Phys. B 582, 477 (2000). 54) J.B. Kogut, M.A. Stephanov and D. Toublan, Phys. Lett. B 464, 183 (1999). 55) S. Hands et al., ZEur. Phys. J. C 17, 285 (2000). 56) J. B. Kogut and D. K. Sinclair, Phys. Rev. D 66, 034505 (2002). 57) G. Policastro, D. T. Son and A. O. Starinets, Phys. Rev. Lett. 87, 081601 (2001). 58) M. Halasz et al., Phys. Rev. D 58, 096007 (1998). 59) B. Vanderheyden and A. D. Jackson, Phys. Rev. D 62, 094010 (2000). 60) A. Barducci et al. Phys. Rev. D 41, 1610 (1990). 61) J. Berges and K. Rajagopal, Nucl. Phys. B 538, 215 (1999). 62) R. A. Janik, M. A. Nowak, G. Papp and I. Zahed, Nucl. Phys. A 642, 191 (1998). 63) B. Klein, D. Toublan and J. J. M. Verbaarschot, Phys. Rev. D 68, 014009 (2003). 64) A. Barducci, R. Casalbuoni, G. Pettini and L. Ravagli, Phys. Rev. D 72, 056002 (2005). 65) D. N. Walters and S. Hands, Nucl. Phys. Proc. Suppl. 140, 532 (2005). 66) M. Frank, M. Buballa and M. Oertel, Phys. Lett. B 562, 221 (2003). 67) D. Toublan and J. J. M. Verbaarschot, Int. J. Mod. Phys. B 15, 1404 (2001). 68) K. Splittorff and J. J. M. Verbaarschot, Nucl. Phys. B 683, 467 (2004). 69) G. Akemann, Nucl. Phys. B 730, 253 (2005). 70) R. Narayanan and H. Neuberger, Nucl. Phys. B 696, 107 (2004). 71) F. Farchioni et al. Phys. Rev. D 62, 014503 (2000). 72) P. Damgaard, U. Heller, R. Niclasen and K. Rummukainen, Nucl. Phys. B 583, 347 (2000). 73) I. Barbour et al., Nucl. Phys. B 275, 296 (1986); 74) S. Muroya, A. Nakamura, C. Nonaka and T. Takaishi, Prog. Theor. Phys. 110, 615 (2003). 75) T. Wettig, private communication. 76) G. Akemann and T. Wettig, Phys. Rev. Lett. 92, 102002 (2004) [Ibid. 96, 029902 (2006)]. 77) J. Bloch and T. Wettig, Phys. Rev. Lett. 97, 012003 (2006). 78) G. Akemann et al., Nucl. Phys. Proc. Suppl. 140, 568 (2005). 79) M. Halasz, J. Osborn, M. Stephanov and J. Verbaarschot, Phys. Rev. D 61, 076005 (2000). 80) T. D. Cohen, Phys. Rev. Lett. 91, 222001 (2003); arXiv:hep-ph/0405043. 81) G. Akemann, J. Osborn, K. Splittorff and J. Verbaarschot, Nucl. Phys. B 712, 287 (2005). 82) L. Ravagli and J.J.M. Verbaarschot, in preparation. 83) K. Splittorff and J. J. M. Verbaarschot, Phys. Rev. Lett. 98, 031601 (2007). 84) K. Splittorff and J. J. M. Verbaarschot, arXiv:hep-lat/0702011. 85) D. Toussaint, Nucl. Phys. Proc. Suppl. 17, 248 (1990).

0704.0331

Symmetries by base substitutions in the genetic code predict 2' or 3' aminoacylation of tRNAs

Microsoft Word - MS737.rtf Manuscript submitted as a Letter to the Editor. Title: Symmetries by base substitutions in the genetic code predict 2’ or 3’ aminoacylation of tRNAs. Authors: Jean-Luc Jestina, Christophe Souléb Addresses: aUnité de Chimie Organique, URA 2128 CNRS Département de Biologie Structurale et Chimie, Institut Pasteur 25 rue du Dr. Roux, 75724 Paris 15, France email: [email protected] (corresponding author) tel +33 1 4438 9496; fax +33 1 4568 8404 bInstitut des Hautes Etudes Scientifiques, CNRS 35 route de Chartres, 91440 Bures-sur-Yvette, France email: [email protected] Key words : Mutation; degeneracy; aminoacyl-tRNA synthetase; codon; symmetry breaking. Understanding why the genetic code is the way it is, has been the subject of numerous models and still remains largely a challenge (Freeland et al., 2000; Sella and Ardell, 2006). Associations between codons and amino acids were suggested to rely on RNA- amino acid interactions (Raszka and Mandel, 1972; Yarus, 1998). Closely related codons were put in correspondence with closely related amino acids within their biosynthetic pathways (Wong, 2005). Codons have also been grouped into systems characterized by interlocked thermodynamic cycles (Klump, 2006). Evolutionary models that minimise the number of the most frequent mutations provide a rationale for the fact that transitions at the third base of codons are mostly neutral mutations (Goldberg and Wittes, 1966). Similarly, minimization of the deleterious effects of sequence-dependent single-base deletions catalyzed by DNA polymerases provides a rationale for the assignment of stop signals to codons (Jestin and Kempf, 1997). While in-frame stop codons are strictly selected against, out-of-frame stop codons minimize the costs of ribosomal slippages (Seligmann and Pollock, 2004). In this context, the frequencies of codons were found to be highly dependent on the reading frame and highlighted a symmetrical codon pattern (Koch and Lehmann, 1997). As the genetic code is quasi-universal among living organisms, models do not need to be time- dependent, even though time-dependent models have been suggested (Bahi and Michel, 2004; Rodin and Rodin, 2006; Sella and Ardell, 2006). Symmetries in the genetic code are of special interest as they may highlight underlying organization principles of the code. A supersymmetric model for the evolution of the genetic code was proposed: successive breaking of these symmetries would provide an evolutive scenario for the decomposition into sets of synonymous codons (Hornos and Hornos, 1993; Bashford et al., 1997). When the amino acids are mapped to the vertices of a 28-gon, three two-fold symmetries were identified for three subsets of the cognate aminoacyl-tRNA synthetases (Yang, 2004). This letter reports complete sets of two-fold symmetries between partitions of the universal genetic code. By substituting bases at each position of the codons according to a fixed rule, it happens that properties of the degeneracy pattern or of tRNA aminoacylation specificity are exchanged. First the set of sixty-four codons of the genetic code was partitionned in two groups of thirty-two codons depending on whether the third base of triplets is necessary or not to define unambiguously an amino acid or a stop signal (property 1). Rumer reported a symmetry by base substitutions that alters property 1 (Rumer, 1966) . The substitutions exchanging T and G as well as A and C are applied to all three codon bases and are called Rumer’s transformation. If the third base is necessary to define an amino acid, then the symmetrical codon by Rumer’s transformation does not require the third base of codons to be defined so as to define unambiguously the amino acid. Conversely, if the third base does not have to be defined so as to define unambiguously an amino acid, then the symmetrical codon by Rumer’s transformation requires the third base to be given so as to define unambiguously the amino acid. More recently, one of the authors reported a symmetry that leaves unchanged property 1 (Jestin, 2006): this symmetry consists in applying to the first base of codons the substitutions exchanging G and C as well as T and A. For example, GCN codons coding for alanine are exchanged into CCN codons coding for proline; for GCN and CCN codons, the third base does not have to be defined so as to define unambiguously the amino acid. Here we report a third symmetry that alters property 1 (Fig.1). This symmetry is obtained by applying successively the two symmetries described above. It consists in applying the substitution exchanging A and G as well as C and T (a transition) to the first base in the codon, the substitution exchanging A and C as well as G and T (a transversion) to the second base in the codon, and the substitution exchanging A and C as well as G and T (a transversion) in the third base of the codon. We show further that the only other symmetries exchanging both groups into each other are obtained by combining the previous ones with a symmetry acting only on the third base of the codons (here we do not include the substitution on the second base which exchanges A and C when fixing G and T). This can be seen by counting the number of occurrences of A, C, G, and T as first, second or third base in a codon of each group. The result is given in Table 1. These symmetries are valid for the standard genetic code and for other genetic codes such as the vertebrate mitochondrial genetic code which has a higher degree of symmetry of its degeneracy pattern as noted earlier (Lehmann, 2000; Jestin, 2006). In addition to the existence of Rumer’s transformation, Shcherbak discussed the following Rumer’s rule (Shcherbak, 1989), which can be read off Table 1: the ratio R = C+G/T+A of the number of occurrences of C and G by the number of occurrences of T and A in positions 1, 2 and 3 is equal to 3, 3 and 1 respectively in codons of the first group (and hence it is 1/3, 1/3 and 1 for codons of the second group). Similarly, the ratio P = T+C/A+G is 1, 3 and 1 in positions 1, 2 and 3 of the first group of codons. Secondly, we considered another grouping of codons of the genetic code depending on whether the amino acids are acylated by amino acyl-tRNA synthetases at the 2’ or at the 3’ hydroxyl group of the tRNA’s last ribose (property 2) (Sprinzl and Cramer, 1975; Arnez and Moras, 1994). This classification of amino acyl-tRNA synthetases is very similar to the one based on sequence homology and on structural considerations (Eriani et al., 1990; Cusack, 1997). Class I synthetases contain HIGH and KMSKS consensus sequences, which are absent from class II amino acyl tRNA synthetases. At the structural level, class I synthetases also contain a Rossman fold, a domain that binds nucleotides, unlike class II synthetases. Class I enzymes catalyse acylation at the 2’ hydroxyl group of the tRNA while class II enzymes generally catalyse acylation at the 3’ hydroxyl group of the tRNA. PheRS as a class II enzyme that catalyses acylation at the tRNA’s 2’ hydroxyl group is therefore an exception. The case of cysteinyl-tRNACys synthetase (CysRS) is ambiguous and was investigated recently. CysRS is a class I synthetase, but establishes contacts with the major groove of the acceptor stem of the tRNACys as commonly found for class II enzymes. The enzyme from Escherichia coli is able to catalyse the acylation reaction at both 2’ and 3’ hydroxyl groups of the tRNACys. The 2’ acylation is about one order of magnitude faster than the 3’ acylation when catalysed by E. coli cysteinyl-tRNA synthetase in vitro (Shitivelband and Hou, 2005). The following classification was then used for 2’ acylated amino acids (Ile, Leu, Met, Val, Trp, Tyr, Arg, Gln, Glu, Phe) and for 3’ acylated amino acids (His, Pro, Ser, Thr, Asn, Asp, Lys, Ala, Gly). To the class of 2’ acylated amino acids we also added the stop signals, a choice partially justified by the fact that two stop codons of the mitochondrial code of vertebrates code for the 2’ acylated amino acid Arg in the universal code. Note that if cysteine were not in the class 3’, or if a stop signal was not in the class 2’, symmetries could not be identified. If cysteine is assigned to the class 2’ as suggested by the previous paragraph, the symmetries are broken. Loss of the symmetries might have occurred during the evolution of aminoacyl-tRNA synthetases and might be associated to the late appearance of this amino acid in the genetic code (Brooks and Fresco, 2002). When considering molecular properties such as polarity, volume and hydrophobicity, no statistical differences were noted between class 2’ and class I on one hand, class 3’ and class II on the other hand (Table 3). There exist two symmetries by base substitutions that exchange the class 2’ with the class 3’ of the corresponding codon groups (cf. Fig.2). They consist in applying the substitution exchanging A and C as well as G and T (a transversion) to the first base of the codon, the substitution exchanging A and G as well as C and T (a transition) to the second base of the codon, and the substitution exchanging A and C as well as G and T or A and T as well as C and G (a transversion) to the third base of the codon. These two symmetries differ by the substitution exchanging A and G as well as C and T in the third position. They are not related to those depicted in Figures 4 and 5 (Yang, 2004) as Yang’s three symmetries act only on three subsets of amino acids whereas the symmetries described herein are valid for the whole codon table. There are no other symmetries by base substitutions between the two classes 2’ and 3’, as can be seen by counting the occurrences of A, C, G and T in each class and each position (Table 2). Note also the following analog of the Rumer’s rule: both the ratio R = C+G / T+A and the ratio Q = A+C / G+T are equal to 1, 1/3, 1 in positions 1, 2, 3 respectively in the class 2’ (and 1, 3, 1 in the class 3’). In this letter we have described new symmetries by base substitutions in the genetic code for partitions concerning the codon degeneracy level or the tRNA-aminoacylation class. Several evolutionary models have been proposed concerning tRNAs and their aminoacyl-tRNA synthetases (Martinez Gimenez and Tabares Seisdedos, 2002; Klipcan and Safro, 2004; Chechetkin, 2006; Di Giulio, 2006). Newly introduced amino acids may well have been selected to minimize the deleterious effects of mistranslations, and possibly according to their molecular volumes (Torabi et al., 2006). A unique serie of binary divisions of the codon table was recently noted: when the same differentiation rule was applied at each division, the class I / class II pattern arose consistently (Delarue, 2007). Aminoacyl-tRNA synthetases are likely to have evolved by gene duplication and mutation of primordial synthetases within each class, as evidenced by sequence homology (Woese et al., 2000). Consistently, the symmetries highlighted in this manuscript require three base substitutions per codon, which are unlikely to happen, thereby shedding some light on the duplication and divergence mechanism of evolution among the two classes of aminoacyl-tRNA synthetases. Acknowledgements : We thank H. Epstein, E. Yeramian, D. Moras, B. Prum and J. Perona for their help. References : Arnez, J. G., Moras, D. 1994. Aminoacyl-tRNA synthetase tRNA recognition. Oxford, IRL Press 61-81. Bahi, J. M., Michel, C. J. 2004. A stochastic gene evolution model with time dependent mutations. Bull. Math. Biol. 66, 763-778. Bashford, J. D., Tsohantjis, I., Jarvis, P. D. 1997. Codon and nucleotide assignments in a supersymmetric model of the genetic code. Phys. Lett. A 233, 481-488. Brooks, D. J., Fresco, J. R. 2002. Increased frequency of cysteine, tyrosine, and phenylalanine residues since the last universal ancestor. Mol. Cell. Proteomics 1, 125-131. Chechetkin, V. R. 2006. Genetic code from tRNA point of view. J. Theor. Biol. 242, 922- 934. Cusack, S. 1997. Aminoacyl-tRNA synthetases. Curr. Opin. Struct. Biol. 7, 881-889. Delarue, M. 2007. An asymmetric underlying rule in the assignment of codons. RNA 13, 161-169. Di Giulio, M. 2006. The non-monophyletic origin of the tRNA molecule and the origin of genes only after the evolutionary stage of the last universal common ancestor. J. Theor. Biol. 240, 343-352. Di Giulio, M., Capobianco, M. R., Medugno, M. 1994. On the optimization of the physicochemical distances between amino acids in the evolution of the genetic code. J. Theor. Biol. 168, 43-51. Eriani, G., Delarue, M., Poch, O., Gangloff, J., Moras, D. 1990. Partition of tRNA synthetases into two classes based on mutually exclusive sets of sequence motifs. Nature 347, 203-206. Freeland, S. J., Knight, R. D., Landweber, L. F., Hurst, L. D. 2000. Early fixation of an optimal genetic code. Mol. Biol. Evol. 17, 511-518. Goldberg, A. L., Wittes, R. E. 1966. Genetic code: aspects of organization. Science 153, 420-424. Hornos, J. E. M., Hornos, Y. M. M. 1993. Algebraic model for the evolution of the genetic code. Phys. Rev. Lett. 71, 4401-4404. Jestin, J. L. 2006. Degeneracy in the genetic code and its symmetries by base substitutions. C. R. Biol. 329, 168-171. Jestin, J. L., Kempf, A. 1997. Chain-termination codons and polymerase-induced frameshift mutations. FEBS Letters 419, 153-156. Klipcan, L., Safro, M. 2004. Amino acid biogenesis, evolution of the genetic code and aminoacyl-tRNA synthetases. J. Theor. Biol. 228, 389-396. Klump, H. H. 2006. Exploring the energy landscape of the genetic code. Arch. Biochem. Biophys. 453, 87-92. Koch, A. J., Lehmann, J. 1997. About a symmetry of the genetic code. J. Theor. Biol. 189, 171-174. Kyte, J., Doolittle, R. F. 1982. A simple method for displaying the hydropathic character of a protein. J. Mol. Biol. 157, 105-132. Lehmann, J. 2000. Physico-chemical constraints connected with the coding properties of the genetic system. J. Theor. Biol. 202, 129-144. Martinez Gimenez, J. A., Tabares Seisdedos, R. 2002. On the dimerization of the primitive tRNAs: implications in the origin of genetic code. J. Theor. Biol. 217, 493- 498. Raszka, M., Mandel, M. 1972. Is there a physical chemical basis for the present genetic code? J. Mol. Evol. 2, 38-43. Rodin, S. N., Rodin, A. S. 2006. Origin of the genetic code: first aminoacyl-tRNA synthetases could replace isofunctional ribozymes when only the second base of codons was established. DNA Cell Biol. 25, 365-375. Rumer, Y. B. 1966. About the codon's systematization in the genetic code. Proc. Acad. Sci. USSR 167, 1393-1394. Seligmann, H., Pollock, D. D. 2004. The ambush hypothesis: hidden stop codons prevent off-frame gene reading. DNA Cell Biol. 23, 701-705. Sella, G., Ardell, D. H. 2006. The coevolution of genes and genetic codes: Crick's frozen accident revisited. J. Mol. Evol. 63, 297-313. Shcherbak, V. I. 1989. Rumer's rule and transformation in the context of the co-operative symmetry of the genetic code. J. Theor. Biol. 139, 271-276. Shitivelband, S., Hou, Y. M. 2005. Breaking the stereo barrier of amino acid attachment to tRNA by a single nucleotide. J. Mol. Biol. 348, 513-521. Sprinzl, M., Cramer, F. 1975. Site of aminoacylation of tRNAs from Escherichia coli with respect to the 2'- or 3'-hydroxyl group of the terminal adenosine. Proc. Natl. Acad. Sci. USA 72, 3049-3053. Torabi, N., Goodarzi, H., Najafabadi, H. S. 2006. The case of an error minimizing set of coding amino acids. J. Theor. Biol. in press. Woese, C. R., Olsen, G. J., Ibba, M., Soll, D. 2000. Aminoacyl-tRNA synthetases, the genetic code, and the evolutionary process. Microbiol. Mol. Biol. Rev. 64, 202-236. Wong, J. T. 2005. Coevolution theory of the genetic code at age thirty. Bioessays 27, 416- Yang, C. M. 2004. On the 28-gon symmetry inherent in the genetic code intertwined with aminoacyl-tRNA synthetases--the Lucas series. Bull. Math. Biol. 66, 1241-1257. Yarus, M. 1998. Amino acids as RNA ligands: a direct-RNA-template theory for the code's origin. J. Mol. Evol. 47, 109-117. Figure Legends : Figure 1 Exchange of Group I (codons for which the third base does not have to be defined to specify the amino acid) into Group II (codons for which the third base must be defined to specify unambiguously the amino acid or the stop signal) by the transformation (AG/CT for the first base, GT/AC for the second and third bases). N=A,T,G or C; H=A,T or C; Y=T or C; R=A or G. Figure 2 Exchange of the classes 2’ and 3’ by the transformation (AC/GT on the first base, AG/CT on the second base, AC/GT on the third base). The special case of cysteine is labelled by an asterisk and discussed in the text. Table I Number of occurences of the bases A, C, G and T at each position within the codon in each group. Table II Number of occurences of the bases A, C, G and T at each position within the codon in each class. Table III Statistical t-values computed from the data on hydrophobicity (Kyte and Doolittle, 1982), molecular volume and polarity (Di Giulio et al., 1994) comparing the class 2’ with class I, and the class 3’ with class II. These values are below the threshold of significance given in the Student’s table. A C G T Base 1 Group I 4 12 12 4 Group II 12 4 4 12 ____________________________ Base 2 Group I 0 16 8 8 Group II 16 0 8 8 ____________________________ Base 3 Group I 8 8 8 8 Group II 8 8 8 8 Table 1 A C G T Base 1 Class 2’ 6 10 6 10 Class 3’ 10 6 10 6 _____________________________ Base 2 Class 2’ 8 0 8 16 Class 3’ 8 16 8 0 _____________________________ Base 3 Class 2’ 10 6 10 6 Class 3’ 6 10 6 10 Table 2 Class 2’ / Class I Class 3’ / Class II Hydrophobicity 0.07 0.11 Polarity 0.017 0.019 Volume 0.57 0.45 Table 3

0704.0333

Optical properties of the Holstein-t-J model from dynamical mean-field theory

Optical properties of theHolstein-t-Jmodel fromdynamicalmean-field theory E. Cappelluti a,b,∗, S. Ciuchi c, S. Fratini d aDipartimento di Fisica, Università “La Sapienza”, P.le A. Moro 2, 00185 Rome, Italy bSMC Research Center and ISC, INFM-CNR, v. dei Taurini 19, 00185 Rome, Italy cINFM and Dipartimento di Fisica, Università dell’Aquila, via Vetoio, I-67010 Coppito-L’Aquila, Italy dInstitut Néel - CNRS & Université Joseph Fourier, BP 166, F-38042 Grenoble Cedex 9, France Abstract We employ dynamical mean-field theory to study the optical conductivity σ(ω) of one hole in the Holstein-t-J model. We provide an exact solution for σ(ω) in the limit of infinite connectivity. We apply our analysis to Nd2−xCexCuO4. We show that our model can explain many features of the optical conductivity in this compounds in terms of magnetic/lattice polaron formation. Key words: magnetic/lattice polarons, spin fluctuations, optical conductivity, cuprates. PACS: 71.10.Fd, 71.38.-k, 78.20.Bh, 75.30.Ds. The problem of a single hole in the t-J model interact- ing also with the lattice degrees of freedom has attracted recently a notable interest in connection with the physical properties of the underdoped high-T cuprates [1,2,3,4]. An important issue in this regime is the formation of lattice or magnetic polarons (or both of them) and their mutual interaction. Along this line, the one-particle properties (as the effective mass, spectral function, etc.) have been widely investigated with different techniques. Much less effort has been however paid to the study of the optical properties. On the analytical ground, the definition of the optical con- ductivity (OC) in the single hole is a delicate matter which needs particular care even for the pure t-J or Holstein model [5,6]. On the other hand, numerical calculations on clusters are limited by finite size effects [7]. As a general rule, thus, the choice of a particular theoretical approach depends on which property is under examination and on its feasibility to investigate it. In this paper we summarize the main results of our work based on the dynamical mean-field theory (DMFT). Tech- nical details will be presented in a forthcoming longer pub- lication [8]. In the infinite coordination number limit z → ∞, we provide an exact solution for σ(ω) as a functional of the local one-particle Green’s function at finite temper- ature. It should be stressed that, due to the classical treat- ment of the magnetic background, the DMFT solution for ∗ Corresponding author. Tel: (+39) 06-49937453 fax: (+39) 06- 49937440 Email address: [email protected] (E. Cappelluti). 0 1 2 3 4 Ref. [7] this work λ=1, J/t=0.4, ω Fig. 1. Comparison between the optical conductivity σ(ω) obtained by our DMFT solution and Lanczos diagonalization in two dimen- sions on a finite cluster (Ref. [7]). z → ∞ is purely local so that it cannot describe the coher- ent propagation of holes due to the spin fluctuations, nor the metallic Drude-like peak in σ(ω). On the other hand, the local properties (as the average number of phonons, size of the magnetic polaron, etc.) are well captured by this ap- proach, [9] as well as the incoherent contributions to the OC. We can explicitly show this feature by comparing in Fig. 1 our DMFT results with numerical calculations using Lanczos diagonalization for a single hole in the 2DHolstein- t-J model on a 10 cluster [7]. The remarkably good agreement of the overall shape as- sesses the feasibility of our approach to investigate the in- coherent contributions to the finite frequency OC. This is- sue is particularly important in light of the intensive de- bate about the origin of the mid-infrared (MIR) band in the underdoped high-T cuprates. Different interpretations for this feature have been discussed in the literature, involving Preprint submitted to Elsevier 29 October 2018 http://arxiv.org/abs/0704.0333v1 charge/spin fluctuations, stripe ordering, and other mecha- nisms. This spread of differentmechanisms reflects the pres- ence in this doping regime of several actors, which makes it difficult to isolate each effect from the others. A simpler and ideal situation is the case of electron-doped cuprates, as Nd2−xCexCuO4. In these compounds, the long-range anti- ferromagnetic (AF) order extents up to x ≃ 0.14, so that the low doping regime x . 0.1 we are interested in, lies well within the AF phase. On the experimental side, in addi- tion, a detailed and exhaustive study of the optical conduc- tivity as a function of temperature T and of the doping x was recently provided in Ref. [10]. In that work the authors showed that the low doping OC spectra are characterized at low temperature by a MIR pseudogap, with an absorp- tion band edge which varies from EMIR ≃ 0.5− 0.6 for x = 0.05 to EMIR ≃ 0.3− 0.4 for x = 0.1, and is barely distin- guishable for x = 0.125. Quite interestingly, increasing the temperature leads to a filling of the pseudogap, rather than a closing of it. Also remarkable is the temperature depen- dence of the MIR spectral weight which does not present any signature at the long-range Néel temperature TN but rather a kink to a higher “pseudogap” temperature T ∗. We show here that our approach is able to describe all these features, and in particular the MIR band edge, in terms of an optical gap due to the formation of a mag- netic/lattice polaron. We define T ∗ as the temperature where the size of the spin polaron becomes larger than the AF correlation length, that is the maximum tempera- ture where an injected charge actually probes the magnetic background. In this perspective we can identify T ∗ with the mean field Néel temperature of our model, which represents the temperature above which the system is described by a paramagnetic state (rather than the onset of long range order). From Ref. [10] we get for instance T ∗ = 440 K at x = 0.05 and T ∗ = 200 K at x = 0.125. Using the Curie- Weiss relation T ∗ = J/4 we estimate respectively J = 152 meV (J/t = 0.126) and J = 69 meV (J/t = 0.057). Note that such values of J do not represent the bare exchange interaction but rather the effective spin-exchange coupling which is reduced by hole doping. We also set ω0 = 84 meV, consistent with the energy window of the optical phonons in the cuprates. The electron-phonon (el-ph) coupling con- stant is fixed to λ = 0.75 in order to reproduce the exper- imental MIR band edge ≈ 0.5− 0.6 eV in the optical con- ductivity at x = 0.05, and we assume λ to be independent of the doping x. Note that with these choices no more free adjustable parameters remain. In Fig. 2 we show the temperature evolution of the MIR optical conductivity for the representative cases x = 0.05 and x = 0.125 (note that in order to compare with the ex- perimental data of Ref. [10] the tail of a Drude-peak should be superimposed). Most remarkable is the behavior of σ(ω) at low temperature, which shows a well defined gap for x = 0.05 while no gap is found for x = 0.125. This fea- ture reflects the formation of the lattice polaron and its in- terplay with the spin degrees of freedom. While the el-ph coupling λ = 0.75 alone is not strong enough at x = 0.125 0 0.5 1 ω [eV] 0.5 1 1.5 ω [eV] 0 200 400 T [K] x=0.05 x=0.125 T=50K T=440K T=540K T=540K T=340K T=50K T=190K Fig. 2. Temperature dependence of the optical conductivity σ(ω) for x = 0.05 and x = 0.125. Solid lines are used for T ≤ T ∗, dashed lines for T > T ∗. Inset: loss of the MIR spectral weight ∆Neff , as defined in Ref. [10], as function of T for x = 0.05 (filled circles) and x = 0.125 (empty squares). Arrows mark the corresponding T ∗. (J/t = 0.057) to establish a spin/lattice polaron, the lo- calization effects induced by the larger exchange coupling J/t = 0.126 at x = 0.05 favor the lattice polaron forma- tion. This leads thus to the opening of an optical gap in σ(ω) (this key point will be extensively discussed in a forth- coming publication[8]). Increasing T reduces the localiza- tion effects induced by the magnetic ordering. This makes the positive interplay with the el-ph coupling less effective, leading to a progressive filling of the pseudogap. Note that this effect disappears in the disordered magnetic case for T > T ∗, and further increasing of T leads to a reduction of the MIR optical conductivity which is spread on a larger energy window. This is reflected in the characteristic tem- perature behavior of the MIR spectral weight ∆Neff , as de- fined in Ref. [10], which presents a kink at T ∗ (inset of Fig. 2)[11]. References [1] A.S. Mishchenko and and N. Nagaosa, Phys. Rev. Lett. 93 (2004) 0236402; Phys. Rev. B 73 (2006) 092502. [2] O. Rösch and O. Gunnarsson, Phys. Rev. Lett. 92 (2004) 146403; Eur. Phys. J. B 43 (2005) 11. [3] O. Gunnarsson and O. Rösch, Phys. Rev. B 73 (2006) 174521. [4] P. Prelovšek, R. Zeyher, and P. Horsch, Phys. Rev. Lett. 96 (2006) 086402. [5] M.P.H. Stumpf and D.E. Logan, Eur.Phys.J.B, 8 (1999) 377. [6] S. Fratini and S. Ciuchi, Phys. Rev. B 74 (2006) 075101. [7] B. Bäuml et al., Phys. Rev. B 58 (1998) 3663. [8] E. Cappelluti, S. Ciuchi and S. Fratini, in preparation (2007). [9] E. Cappelluti and S. Ciuchi, Phys. Rev. B 66 (2002) 165102. [10] Y. Onose et., Phys. Rev. B 69 (2004) 024504. [11] Since we do not find any isosbestic point in our calculations, we use the experimental energy windows of Ref. [10] to define ∆Neff , namely ωmin = 0.12 eV, ωmax = 0.42 eV for x = 0.05 and ωmax = 0.21 eV for x = 0.125. References

0704.0334

A Multiphilic Descriptor for Chemical Reactivity and Selectivity

Microsoft Word - LA_Multiphilic_3-4-7.doc A Multiphilic Descriptor for Chemical Reactivity and Selectivity J. Padmanabhan1,2, R. Parthasarathi2, M. Elango2, V. Subramanian2,*, B. S. Krishnamoorthy1,3, S. Gutierrez-Oliva4, A. Toro-Labbé4,*, D. R. Roy1 and P. K. Chattaraj1,* 1Department of Chemistry, Indian Institute of Technology, Kharagpur 721302, India. 2Chemical Laboratory, Central Leather Research Institute, Adyar, Chennai 600 020, India. 3School of Chemistry, Bharathidasan University, Tiruchirappalli-620 024, India. 4Laboratorio de Química Teórica Computacional (QTC), Facultad de Química, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile. Abstract In line with the local philicity concept proposed by Chattaraj et al. (Chattaraj, P. K.; Maiti, B.; Sarkar, U. J. Phys. Chem. A. 2003, 107, 4973) and a dual descriptor derived by Toro-Labbé and coworkers (Morell, C.; Grand, A.; Toro-Labbé, A. J. Phys. Chem. A. 2005, 109, 205), we propose a multiphilic descriptor. It is defined as the difference between nucleophilic (ωk+) and electrophilic (ωk-) condensed philicity functions. This descriptor is capable of simultaneously explaining the nucleophilicity and electrophilicity of the given atomic sites in the molecule. Variation of these quantities along the path of a soft reaction is also analyzed. Predictive ability of this descriptor has been successfully tested on the selected systems and reactions. Corresponding force profiles are also analyzed in some representative cases. Also, to study the intra- and intermolecular reactivities another related descriptor namely, the nucleophilicity excess ( ∓ gωΔ ) for a nucleophile, over the electrophilicity in it has been defined and tested on all-metal aromatic compounds. *Authors for correspondence: E-mail: [email protected], [email protected], [email protected], 1. Introduction The understanding of chemical reactivity and site selectivity of the molecular systems has been effectively handled by the conceptual density functional theory (DFT).1 Chemical potential, global hardness, global softness, electronegativity and electrophilicity are global reactivity descriptors, highly successful in predicting global chemical reactivity trends. Fukui function (FF) and local softness are extensively applied to probe the local reactivity and site selectivity. The formal definitions of all these descriptors and working equations for their computation have been described. 1-4 Various applications of both global and local reactivity descriptors in the context of chemical reactivity and site selectivity have been reviewed in detail.3 Parr et al. introduced the concept of Electrophilicity (ω) as a global reactivity index similar to the chemical hardness and chemical potential. 5 This new reactivity index measures the stabilization in energy when the system acquires an additional electronic charge ΔN from the environment. The electrophilicity is defined as ημω 2/2= (1) In Eq. (1), μ ≈ -(I+A)/2 and η ≈ (I-A)/2 are the electronic chemical potential and the chemical hardness of the ground state of atoms and molecules, respectively, approximated in terms of the vertical ionization potential (I) and electron affinity (A). The electrophilicity is a descriptor of reactivity that allows a quantitative classification of the global electrophilic nature of a molecule within a relative scale. 5 Fukui Function (FF) 6 is one of the widely used local density functional descriptors to model chemical reactivity and site selectivity and is defined as the derivative of the electron density ρ ( r ) with respect to the total number of electrons N in the system, at constant external potential ν ( r ) acting on an electron due to all the nuclei in the system [ ] [ ] )()()()( rvN Nrrvrf ∂∂== ρδδμ . (2) The condensed FF are calculated using the procedure proposed by Yang and Mortier,7 based on a finite difference method )()1( NqNqf kkk −+= + for nucleophilic attack (3a) )1()( −−=− NqNqf kkk for electrophilic attack (3b) [ ] 2)1()1( −−+= NqNqf kkok for radical attack (3c) where kq is the electronic population of atom k in a molecule. Chattaraj et al.8 have introduced the concept of generalized philicity. It contains almost all information about hitherto known different global and local reactivity and selectivity descriptors, in addition to the information regarding electrophilic/nucleophilic power of a given atomic site in a molecule. It is possible to define a local quantity called philicity associated with a site k in a molecule with the help of the corresponding condensed- to- atom variants of FF, αkf as αα ωω kk f= (4) where (α= +, - and 0) represents local philic quantities describing nucleophilic, electrophilic and radical attacks respectively. Eq. (4) predicts that the most electrophilic site in a molecule is the one providing the maximum value of ωk+. When two molecules react, which one will act as an electrophile (nucleophile) will depend on, which has a higher (lower) electrophilicity index. This global trend originates from the local behavior of the molecules or precisely at the atomic site(s) that is(are) prone to electrophilic (nucleophilic) attack. Recently the usefulness of electrophilicity index in elucidating the toxicity of polychlorinated biphenyls, benzidine and chlorophenol has been assessed in detail. 9-11 In addition to the knowledge of global softness (S), which is the inverse of hardness, 12 different local softnesses 13 used to describe the reactivity of atoms in molecule, can be defined as k ks Sf α α= (5) where (α= +, - and 0) represents local softness quantities describing nucleophilic, electrophilic and radical attacks respectively. Based on local softness, relative nucleophilicity (sk- /sk+) and relative electrophilicity (sk+ /sk-) indices have also been defined and their usefulness to predict reactive sites also been addressed to.14 It has been established that the quantum chemical model selected to derive wave function; population scheme used to obtain the partial charges and basis set employed in the molecular orbital calculations are important parameters, which significantly influence the FF values. 15-18 The condensed philicity summed over a group of relevant atoms is defined as the “group philicity”. It can be expressed as19 αα ωω where n is the number of atoms coordinated to the reactive atom, αωk is the local electrophilicity of the atom k, and ωgα is the group philicity obtained by adding the local philicity of the nearby bonded atoms. In this study19 the group nucleophilicity index (ωg+) of the selected systems is used to compare the chemical reactivity trends. Toro-Labbé et al20 have recently proposed a dual descriptor (Δf ( r )), which is defined as the difference between the nucleophilic and electrophilic Fukui functions and is given by, Δf(r) = [ (f +(r) - (f - (r) ] (7) If Δf(r) > 0, then the site is favored for a nucleophilic attack, whereas if Δf (r) < 0, then the site may be favored for an electrophilic attack. The associated dual local softness have also been defined as,19 Δsk = S (fk+ - fk-) = (sk+ - sk-) (8) It is defined as the condensed version of Δf (r) multiplied by the molecular softness S. 2. Multiphilic Descriptor In the light of the local philicity concept proposed by Chattaraj et al.8 and the dual descriptor derived by Toro-Labbé and coworkers,20 we propose a multiphilic descriptor using the unified philicity concept, which can concurrently characterize both nucleophilic and electrophilic nature of a chemical species. It is defined as the difference between the nucleophilic and electrophilic condensed philicity functions. It is an index of selectivity towards nucleophilic attack, which can as well characterize an electrophilic attack and is given by,21 Δωk = [ωk+ - ωk- ] = ω [Δƒk] (9) where Δƒk is the condensed-to-atom variant-k of Δƒ(r) (eq 7). If Δωk > 0, then the site k is favored for a nucleophilic attack, whereas if Δωk < 0, then the site k may be favored for an electrophilic attack. Because FFs are positive (0 < ƒk < 1), -1 < Δƒk < 1, and the normalization condition for Δωk is 0=Δ=Δ ∑∑ k fωω (10) Although Δωk and Δfk will contain the same intramolecular reactivity information the former is expected to be a better intermolecular descriptor because of its global information content. We may analyze the nature of ( )rωΔ in terms of that22 of ( )f rΔ as follows: [ ]( )( ) ωω ⎛ ⎞∂∂⎛ ⎞ = ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠ ∂ ∂⎛ ⎞ ⎛ ⎞ = +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠ ( ) ( ) f r f r = + Δ⎜ ⎟∂⎝ ⎠ ( ) ( ) f r r = + Δ⎜ ⎟∂⎝ ⎠ ( ) ( ) r f r ⎡ ⎤∂ ∂⎛ ⎞ ⎛ ⎞ Δ = −⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦ The multiphilicity descriptor, ( )rωΔ is a measure of the difference between local and global (modulated by ( )f r ) reactivity variations associated with the electron acceptance/ removal. Incidentally, the variation of ω∂⎛ ⎞ ⎜ ⎟∂⎝ ⎠ across the periodic table is similar to that of μ.23 2v vN N ⎡ ⎤∂ ∂⎛ ⎞ =⎜ ⎟ ⎢ ⎥∂ ∂⎝ ⎠ ⎣ ⎦ 24 vv N μ μ μ η η η η ⎛ ⎞∂ ∂⎛ ⎞ = −⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ = − μ γ μ = − = − Since γ is generally very small,24 ω∂⎛ ⎞ ⎜ ⎟∂⎝ ⎠ is expected to follow the μ trend. Problems associated with the definition of η and the discontinuity25 in E as a function of N will be present in the ( )f rΔ definition and the discontinuity in ( )rρ . Similar type of differentiation has also been attempted by other research workers.26 Also, to study the intra- and intermolecular reactivities another related descriptor namely, nucleophilicity excess ( ∓ gωΔ ) for a nucleophile, over the electrophilicity (net nucleophilicity) in it is defined as ( )+−+− −=−=Δ ggggg ffωωωω ∓ (11) where )( ωω and )( ωω are the group philicities of the nucleophile in the molecule due to electrophilic and nucleophilic attacks respectively. It is expected that the nucleophilicity excess ( ∓ gωΔ ) for a nucleophile should always be positive whereas it will provide a negative value for an electrophile in a molecule. In the present study, we use both the multiphilicity descriptor and nucleophilicity excess to probe the nature of attack/reactivity at a particular site in the selected systems. 3. Computational Details The geometries of HCHO, CH3CHO, CH3COCH3, C2H5COC2H5, CH2=CHCHO CH3CH=CHCHO, NH2OH, CH3ONH2, CH3NHOH, OHCH2CH2NH2, CH3SNH2, CH3NHSH, SHCH2CH2NH2 and all-metal aromatic molecules, viz., MAl4– (M=Li, Na, K and Cu) are optimized by B3LYP/6-311+G** as available in the GAUSSIAN 98 package.27 Various reactivity and selectivity descriptors such as chemical hardness, chemical potential, softness, electrophilicity and the appropriate local quantities employing natural population analysis (NPA)28, 29 scheme are calculated. HPA scheme (Stockholder Partitioning Scheme) 30 as implemented in the DMOL3 package 31 has also been used to calculate the local quantities employing BLYP/DND method. For all-metal aromatic molecules, ∆SCF method has been utilized to compute the ionization potential (IP) and electron affinity (EA) according to the equations (I=EN-1 - EN, A=EN - EN+1, where I and A are obtained from total electronic energy calculations on the N-1, N, N+1-electron systems at the neutral molecule geometry). 4. Results and Discussion A series of carbonyl compounds is selected in the present study to probe the usefulness of the multiphilicity descriptor (Figure 1). A comparison with various other descriptors and the recently derived dual descriptor is also probed. Due to bipolar nature of C=O bond, both nucleophilic and electrophilic attacks are possible at C and O sites. It is noted that the rate of nucleophilic addition on the carbonyl compound be reduced by electron donating alkyl groups and enhanced by electron withdrawing ones. 32 Recently, we have studied a set of these carbonyl compounds in the light of philicity and group philicity.19 The global molecular properties of the selected series of carbonyl compounds are presented in Table 1. Various local quantities for particular sites of the selected systems are listed in Table 2 and Table 3. Selected compounds are grouped into two sets namely, nonconjugated and α, β-conjugated carbonyl compounds. For the nonconjugated carbonyl compounds, the carbon atom (C1) bearing the carbonyl group is expected to be the most reactive site towards a nucleophilic attack. Table 2 lists the values of local reactivity descriptors using B3LYP/6-311+G** method for NPA derived charges of the selected molecules. NPA derived local quantities predict the expected maximum value for carbonyl carbon (C1) of all the selected molecules for fk+, sk+ and ωk+. But sk+/sk- is unable to provide the maximum value for C1 atom due to negative FF values. One important point to note is that among the descriptors fk+, sk+, ωk+ and sk+/sk-, + value is capable of providing a clear distinction between carbonyl carbon (C1) and the oxygen site for nucleophilic attack. Since, HPA derived charges generally provide non-negative FF values, we also made use of it for local reactivity analysis on carbonyl compounds. HPA derived local reactivity descriptors also predict the expected maximum value for C1 atom in the case of HCHO and CH3CHO but fails to predict for CH3COCH3 and C2H5COC2H5, where oxygen atom is shown to be prone towards nucleophilic attack. Nevertheless, the fk+ value of oxygen is almost same as that of carbonyl carbon (C1), thus making it difficult to make a clear decision on the electrophilic behavior of these atoms. Under these situation, dual descriptors Δf (r), Δs k and multiphilic descriptor Δω (r), give a helping hand. All these quantities provide a clear difference between nucleophilic and electrophilic attacks at a particular site with their sign. That is, they provide positive value for site prone for nucleophilic attack and a negative value at the site prone for electrophilic attack. The advantage of multiphilic descriptor Δω (r) is that they provide higher value in terms of magnitude compared to other dual descriptors. For instance, values of Δf(r), Δsk and Δω(r) for nucleophilic (electrophilic) attack at carbonyl carbon (oxygen) site of CH3CHO are 1.06 (-0.93), 0.17 (-0.15), 3.03 (-2.65) respectively for NPA derived charges. Almost the same trend is followed in the case of HPA derived charges. The second group of compounds namely, α, β-conjugated carbonyl is elaborately studied in the recent past because of the presence of two reactive centers.33 The first reactive site is the carbon (C1) of the carbonyl, and the second is the carbon in the β position (C6). In such a case, the β carbon is activated because of the withdrawing mesomeric effect of the adjacent carbonyl group. As seen from Table 2 and Table 3, NPA derived charges give a maximum value for fk+ to carbonyl carbon whereas HPA derived charges provide maximum fk+ value to the β carbon atom (C6) in the case of CH2=CHCHO molecule. For CH3CH=CHCHO, NPA (HPA) provide maximum fk+ value of 0.44 (0.17) to carbonyl carbon (C1) compared to the β carbon site of 0.34 (0.16). This ambiguous behavior may be due to the dependence of local reactivity descriptors on the selection of basis set and population schemes. Further oxygen site shows high value for fk+ and other local descriptors, making it difficult to predict the proper electrophilic site. Even now Δω (r) exhibits high positive value on both carbons that are supposed to be electrophilic and a high negative value on the oxygen site disclosing clearly its nucleophilic character compared to other dual descriptors. Also it can be noted from Tables 2 and 3 that, even for molecules with more than one reactive sites, Δω (r) is capable of making a clear distinction among them in terms of their magnitude. That is, for molecules 6 and 7 having two reactive sites as carbon (C1) of the carbonyl and the carbon in the β position (C6), our descriptors are capable of distinctly identifying the stronger site (electrophilic/nucleophilic). Optimized structures along with atom numbering for the selected set of amines are presented in Figure 2. Global and local reactivity properties of the selected set of amines calculated using B3LYP/6-311+g** and BLYP/DND methods are presented in Tables 4 to 6. Global reactivity trend based on ω, is given by B3LYP/6-311+g** method (Table 4) (i) CH3ONH2 > OHCH2CH2NH2 > CH3NHOH > NH2OH (ii) CH3NHSH > SHCH2CH2NH2 > CH3SNH2 BLYP/DND method (Table 4) (i) CH3ONH2 > OHCH2CH2NH2 > NH2OH > CH3NHOH (ii) CH3NHSH > SHCH2CH2NH2 > CH3SNH2 Though both the methods show variation in reactivity trend for oxygen containing systems, trends related to sulfur containing systems are same. Based on NPA and HPA charge derived multiphilic descriptor at nitrogen site (∆ωN), following reactivity trend has been obtained, NPA (Table 5) (1) OHCH2CH2NH2 > CH3NHOH > NH2OH > CH3ONH2 (2) CH3NHSH > SHCH2CH2NH2 > CH3SNH2 HPA (Table 6) (1) OHCH2CH2NH2 > CH3ONH2 > NH2OH > CH3NHOH (2) CH3NHSH > SHCH2CH2NH2 > CH3SNH2 It may be noted that trends are same as ω for sulfur containing systems, but shows variations with respect to oxygen containing systems for both NPA and HPA charge derived ∆ωN. So for as the intramolecular reactivity trends are concerned, site with maximum negative value of ∆ωk is the most preferred site for electrophilic attack. Chemical intuition suggests that N site is more prone towards electrophilic attack. Table 7 lists the site with maximum negative value for ∆ωk for the selected set of amines. It is seen that with a few exception, N site is predicted as the most preferred site for electrophilic attack. Further in order to test ∆ωk along intrinsic reaction coordinate (IRC), we consider a cope rearrangement of hexa-1,5-diene. This is an example of [3,3] sigmatropic reaction. Figure 3 provides the optimized geometrical structures with atom numbering for the reactant, transition state and product calculated using B3LYP/6-31G* level of theory. Table 8 gives the global reactivity parameters of the reactant, transition state and product. As expected, hardness is minimum (2.48 eV) and the corresponding electrophilicity index is maximum (1.57 eV) at the transition state. Variation of global reactivity parameter along the IRC path is presented in Table 9 and Figure 4 (a-b). Variation of energy (E) and ω along IRC path is given in Figure 5a. It is seen that both E and ω are maximum around the transition state indicating it as the most unstable structure along the IRC path. Figure 5 b provides the variation of hardness (η) and polarizability (α) along the IRC path. An inverse relationship exists between them. That is, η reaches a minimum whereas α becomes maximum at the transition state as expected. Variation of multiphilic descriptor (∆ωk) along IRC for the important atomic sites (C1 and C3/ C6 and C11) is presented in Figure 5. In going from reactant to product, C1 and C3 (C6 and C11) sites change their nature and become more prone towards electrophilic attack (nucleophilic attack) at the product side. This change in the nature of attack takes place around the transition state. In studying the importance of nucleophilicity excess ( ∓ gωΔ ) descriptor, a careful analysis on the electronic structure, property and reactivity of all-metal aromatic compounds, viz., MAl4– (M=Li, Na, K and Cu) is performed. The four membered aluminum unit Al4 present in all the molecules may be considered as a single unit. This unit can easily take part in charge transfer process with the M (≡Li, Na, K, Cu) atom in those complexes. Figure 6 shows the various stable isomers of MAl4–. The C4v isomer of the MAl4– is reported as energetically most stable, least polarizable and hardest.34, 35 Table 10 presents the group philicity (ωg+, ωg–) values of the Al42– nucleophile and M+ (M=Li, Na, K, Cu) electrophile in the MAl4– isomers. It is found that in all MAl4– isomers the nucleophilicity of the Al42– aromatic unit overwhelms its electrophilic trend (i.e. +− gg ωω ) and therefore gωΔ is positive, whereas the electrophilicity of M + dominates over its nucleophilicity (i.e. gg ωω ) and therefore gωΔ is negative as expected. It is important to note that gωΔ of Al42– is maximum in the case of most stable C4v isomer of the MAl4– molecule. The order of the ∓ gωΔ value of Al4 2– nucleophile in MAl4–, vvv CCC ∞24 , i.e. stabilization of an MAl4– isomer (except in KAl4–) increases its nucleophilicity and accordingly can be used as a better molecular cathode. It is also important to note that the nucleophilicity of the Al42– unit in MAl4– (C4v) increases as K Cu Na Li≺ ≺ ≺ according to the respective nucleophilicity excess values. Standard expressions1-5 for ∆N and ∆E in terms of group electronegativity and group hardness will provide additional insights into the electron transfer process. Variation of kωΔ along the IRC of three selected reactions, 36 viz., a) a thermoneutral reaction: Fa– + CH3-Fb → Fa-CH3 + Fb–, b) an endothermic reaction: HNO → HON, c) an exothermic reaction: H2OO → HOOH is provided in figures 7 (a) – 7(c). For the thermoneutral reaction, both the Fa– (bond making) and Fb– (bond breaking) are nucleophilic. The net nucleophilicity of the Fa– atom is more than that of the Fb– atom along the IRC from reactant side to TS and the situation is reversed for the IRCs pertaining to the TS to product side. For the endothermic reaction, the net nucleophilicity of O (bond making) is higher than that of N (bond breaking) along the IRC. In the case of exothermic reaction, the O1 (bond making) atom is more electrophilic than its nucleophilic activity. Moreover, its Fukui function values as calculated through Mulliken Population Analysis (MPA) scheme become negative in some cases. For the thermoneutral reaction kωΔ is minimum at the transition state. For other two reactions, kωΔ does not always follow the trend that the IRC corresponding to the minimum value of kω ± (if not zero) is in accordance with the Hammond’s postulate.36 Figures 8 (a) – 8 (c) provide the profiles for the corresponding reaction forces.37 Apart from the important points corresponding to the reactant (R), the transition state (TS) and the product (P) there exists two other important points associated with the configurations having the force maximum (Fmax) and the force minimum (Fmin). The zeroes, maxima and minima of the reaction force define key points along the reaction coordinate, which divide it into three reaction regions that are identified through vertical dashed lined in Figure 8. The first stage, in the reactant region, tends to be preparative in nature with emphasis in structural effects such as rotation, bond stretching, angle bending, etc., that will facilitate subsequent steps. The transition state region is mostly characterized by electronic rearrangements whereas the product region is mainly associated to structural relaxation necessary to reach the products. We have shown that analyzing a chemical reaction in terms of these regions can provide significant insight into its mechanism and the roles played by external factors, such as external potentials and solvents.37, 38 Partition of the activation energies in terms of the work done in going from i) R to Fmin: W1, ii) Fmin to TS: W2, iii) TS to Fmax: W3 and iv) Fmax to P: W4 gives the activation energy for the forward reaction (Ef#) as (W1+W2) and that of the reverse reaction (Er#) as -(W3+W4). Therefore the reaction energy (∆E0) becomes (Ef# – Er# = W1+W2+W3+W4). These values are provided in Table 11. As expected ∆E0 is zero, negative and positive for the thermoneutral, exothermic and endothermic reactions respectively. The skew-symmetric nature of the force profile for the thermoneutral reaction suggests that A=W1+W4 and B=W2+W3 would be zero. Similarly A, B would be positive (negative) for the endo(exo)thermic reactions. The transition state at the IRC=0 configuration lies at the middle between Fmax and Fmin configurations for the thermoneutral reaction whereas it lies towards the Fmin(Fmax) configurations for the exo(endo)thermic reaction, a signature of the Hammond postulate via reaction force. Similar values of W1 and W2 (see Table 11) together with the changes observed in the nucleophilicity along the reaction coordinate for the thermoneutral SN2 substitution and for the exothermic reaction H2OO → HOOH indicate that structural and electronic reordering show up at the very beginning of the reaction, 37,38 through a sharp decrease of the nucleophilicity, this change practically ceases at the transition state of the exothermic reaction to reach the product value. It is interesting to note that in both cases the lowering of nucleophilicity of the key atoms from the reactants (Δω(Fa/Fb) ~ 0.014; Δω(O1) ~ 0.14) to the transition state (Δω(Fa/Fb) ~ 0.004; Δω(O1) ~ 0.0) requires a similar amount of energy (9.54 kcal/mol and 7.39 kcal/mol, respectively). It can be observed in Table 11 that for the thermoneutral reaction W1>W2 indicating that the preparation step requires more energy than the transition to product step. On the other hand, the W2 values for the thermoneutral and exothermic reactions are quite close to each other and the work W1 associated to the preparation step in the thermoneutral reaction is larger than that of the exothermic reaction, this indicates that in the SN2 reaction the structural reordering of the CH3 group to reach the D3h structure at the transition state is the key transformation that involve most of the activation energy. In the endothermic HNO → HON reaction the small changes of nucleophilicity together with large values of W1 and W2 indicates that the reaction is mainly driven by the structural reordering in the preparation step. 5. Conclusions A multiphilicity descriptor (Δωk) is proposed and tested in this work. It is shown that, Δωk helps in identifying the electrophilic/nucleophilic nature of a specific site within a molecule. A comparison between different local reactivity descriptors is carried out on a set of carbonyl compounds. Also a selected set of amines is analyzed using Δωk. Further, we also consider a cope rearrangement of hexa-1,5-diene to test the variation of Δωk along IRC path. It is seen that Δωk presents a clear distinction between electrophilic and nucleophilic sites within a molecule in terms of their magnitude and sign. Hence they reveal the fact that multiphilic descriptor can effectively be used in characterizing the electrophilic/nucleophilic nature of a given site in a molecule. Also the importance of nucleophilicity excess ( ∓ gωΔ ) descriptor on the reactivity of all-metal aromatic compounds, viz., MAl4– (M=Li, Na, K and Cu) is successfully analyzed. Important insight into three different types of reactions, viz., a) thermoneutral, b) endothermic and c) exothermic are obtained through the analysis of the multiphilic descriptor profiles within the reaction regions defined by reaction force along the reaction path. The results discussed so far clearly show the importance of the selected descriptors, namely, multiphilic descriptor and nucleophilicity excess in analyzing the overall reactivity trends in molecular systems. Acknowledgment: PKC and DRR thank BRNS, Mumbai for financial assistance. JP and BSK thank the IIT Kharagpur for providing the facilities required for a summer project. JP also thanks the UGC for selecting him to carryout his Ph.D. work under FIP. ATL and SGO wish to thank financial support from FONDECYT, grant N° 1060590, FONDAP through project N° 11980002 (CIMAT) and Programa Bicentenario en Ciencia y Tecnología (PBCT), Proyecto de Inserción Académica N° 8. ATL is also indebted to the John Simon Guggenheim Foundation for a fellowship. References (1) Parr, R.G.; Yang, W. Density Functional Theory of Atoms and Molecules, Oxford University Press: Oxford, 1989. (2) Pearson, R. G. Chemical Hardness - Applications from Molecules to Solids, VCH- Wiley: Weinheim, 1997. (3) Geerlings, P.; De Proft, F.; Langenaeker, W. Chem. Rev. 2003, 103, 1793. (4) Special Issue of J. Chem. Sci. on Chemical Reactivity, 2005, Vol. 117, Guest Editor: Chattaraj, P. K. (5) Parr, R. G.; Szentpaly, L. V.; Liu, S. J. Am. Chem. Soc. 1999, 121, 1922. Chattaraj, P. K.; Sarkar, U.; Roy, D. R. Chem. Rev. 2006, 106, 2065. (6) Parr, R. G.; Yang, W. J. Am. Chem. Soc., 1984, 106, 4049. Fukui, K. Science 1987, 218, 747. Ayers, P. W.; Levy, M. Theor. Chem. Acc. 2000, 103, 353. (7) Yang, W.; Mortier, W. J. J. Am. Chem. Soc. 1986, 108, 5708. (8) Chattaraj, P. K. ; Maiti, B. ; Sarkar, U. J. Phys. Chem. A. 2003, 107, 4973. (9) Parthasarathi, R.; Padmanabhan, J.; Subramanian, V.; Maiti, B.; Chattaraj, P. K. J. Phys. Chem. A. 2003, 107, 10346. (10) Parthasarathi, R.; Padmanabhan, J.; Subramanian, V.; Maiti, B.; Chattaraj, P. K. Current Sci. 2004, 86, 535. (11) Padmanabhan, J.; Parthasarathi, R.; Subramanian, V.; Chattaraj, P. K. Chem. Res. Tox. 2006, 19, 356. (12) Yang, W.; Parr, R. G. Proc. Natl. Acad. Sci. U.S.A. 1985, 82, 6723. (13) Lee, C.; Yang, W.; Parr, R. G. J. Mol. Struct. (Theochem). 1988, 163, 305. (14) Bulat, F.A.; Chamorro, E.; Fuentealba, P.; Toro-Labbé, A. J. Phys. Chem. A 2004, 108, 342. (15) Langenaeker, W.; De Proft, F.; Geerlings, P. J. Mol. Struct. (Theochem). 1996, 362, 175. (16) De Proft, F.; Martin, M. L. J. ; Geerlings, P. Chem. Phys. Lett. 1996, 256, 400. (17) Contreras, R. ; Fuentealba, P. ; Galván, M. ; Pérez, P. Chem. Phys. Lett. 1999, 304, 405. (18) Thanikaivelan, P.; Padmanabhan, J.; Subramanian, V.; Ramasami, T. Theo. Chem. Acc. 2002, 107, 326. (19) Parthasarathi, R.; Padmanabhan, J.; Elango, M.; Subramanian, V.; Chattaraj, P. K. Chem. Phys. Lett. 2004, 394, 225. (20) Morell, C.; Grand, A.; Toro-Labbé, A. J. Phys. Chem. A. 2005, 109, 205. Morell, C.; Grand, A.; Toro-Labbé, A. Chem. Phys. Lett. 2006, 425, 342. (21) J. Padmanabhan, R. Parthasarathi, V. Subramanian, P. K. Chattaraj, J. Phys. Chem. A 110 (2006) 2739. (22) Morell, C.; Grand, A.; Toro-Labbe, A. Chem. Phys. Lett. 2006, 425, 342. (23) Chamorro, E.; Chattaraj, P. K.; Fuentealba, P. J. Phys. Chem. A 2003, 107, 7068. (24) Fuentealba, P.; Parr, R. G. J. Chem. Phys. 1991, 94, 5559. (25) Perdew, J. P.; Parr, R. G.; Levy, M.; Balduz, J. L., Jr. Phys. Rev. Lett. 1982, 49, 1691. (26) Ayers, P. W.; Morell, C.; De Proft, F.; Geerlings, P., unpublished work. (27) Gaussian 98, Revision A.5, Gaussian Inc., Pittsburgh, PA, 1998. (28) Reed, A. E.; Weinhold, F. J Chem Phys. 1983, 78, 4066. (29) Reed A E.; Weinstock, R. B.; Weinhold, F. J Chem Phys. 1985, 83, 735. (30) Hirshfeld, F. L. Theor. Chim. Acta. 1977, 44, 129. (31) DMOL3, Accelrys, Inc. San Diego, California. (32) March, J. Advanced Organic Chemistry: Reactions, Mechanisms and Structure, Wiley & Sons: New York. 1998. (33) Patai, S.; Rappoport, Z. In The Chemistry of Alkenes; Interscience Publishers: London, 1964; p 469. Wong, S. S.; Paddon-Row, M. N.; Li, Y.; Houk, K. N. J. Am. Chem. Soc. 1990, 112, 8679. Langenaeker, W.; Demel, K.; Geerlings, P. J. Mol. Struct.: THEOCHEM 1992, 259, 317. Dorigo, A. E.; Morokuma, K. J. Am. Chem. Soc. 1989, 111, 6524. (34) (a) Li, X.; Kuznetsov, A. E.; Zhang, H.-F.; Boldyrev, A. I.; Wang, L.-S. Science 2001, 291, 859. (b) Kuznetsov, A.; Birch, K.; Boldyrev, A. I.; Li, X.; Zhai, H.; Wang, L.-S. Science 2003, 300, 622. (35) Chattaraj, P. K.; Roy, D. R.; Elango, M.; Subramanian, V. J. Phys. Chem. A 2005, 109, 9590. Roy, D. R.; Chattaraj, P. K.; Subramanian, V. Ind. J.Chem. A 2006, 45A, 2369. Bulat, F.A.; Toro-Labbé, A. J. Phys. Chem. A 2003, 107, 3987. (36) Chattaraj, P. K.; Roy, D. R. J. Phys. Chem. A 2006, 110, 11401. Chattaraj, P. K.; Roy, D. R. J. Phys. Chem. A 2005, 109, 3771. (37) Toro-Labbé, A. J. Phys. Chem. A 1999, 103, 4398. Jaque, P.; Toro-Labbé, A. J. Phys. Chem. A 2000, 104, 995. Martínez, J.; Toro-Labbé, A. Chem. Phys. Lett. 2004, 392, 132. Herrera, B.; Toro-Labbé, A. J. Chem. Phys. 2004, 121, 7096. Toro-Labbé, A.; Gutiérrez-Oliva, S.; Concha, M. C.; Murray, J. S.; Politzer, P. J. Chem. Phys. 2004, 121, 4570. Gutiérrez-Oliva, S.; Herrera, B.; Toro-Labbé, A.; Chermette, H. J. Phys. Chem. A 2005, 109, 1748. (38) Politzer, P.; Burda, J. V.; Concha, M. C.; Lane, P.; Murray, J. S. J. Phys. Chem. A 2006, 110, 756. Rincón, E.; Jaque, P.; Toro-Labbé, A. J. Phys. Chem. A 2006, 110, 9478. Burda, J. V.; Toro-Labbé, A.; Gutiérrez-Oliva, S.; Murray, J. S.; Politzer, P. J. Phys. Chem. A 2007, in press. TABLE 1: Calculated Global Reactivity Properties of the Selected Molecules using B3LYP/6-311+g** and BLYP/DND method. η μ ω S η μ ω S Molecules B3LYP/6-311+g** (eV) BLYP/DND (eV) HCHO 2.960 -4.707 3.742 0.169 1.942 -4.260 4.673 0.258 CH3CHO 3.115 -4.224 2.864 0.161 2.096 -3.791 3.425 0.238 CH3COCH3 3.144 -3.910 2.432 0.159 2.133 -3.456 2.800 0.234 C2H5COC2H5 3.153 -3.799 2.288 0.159 2.151 -3.367 2.635 0.233 CH2=CHCHO 2.503 -4.904 4.805 0.200 1.545 -4.413 6.303 0.324 CH3CH=CHCHO 2.542 -4.631 4.217 0.197 1.593 -4.132 5.359 0.314 TABLE 2: Calculated Local Reactivity Properties of the Selected Molecules using B3LYP/6-311+g** method for NPA derived charges. Molecule fk - Δfk +- fk HCHO C 0.8323 -0.1722 0.1406 -0.0291 -4.8331 3.1146 -0.6444 1.0045 0.1697 3.7591 O 0.0399 0.9409 0.0067 0.1589 0.0424 0.1494 3.5211 -0.9010 -0.1522 -3.3718 CH3CHO C1 0.8178 -0.2416 0.1313 -0.0388 -3.3856 2.3419 -0.6917 1.0593 0.1700 3.0337 O 0.0072 0.9320 0.0012 0.1496 0.0077 0.0206 2.6691 -0.9250 -0.1484 -2.6485 CH3COCH3 C1 0.3142 -0.2916 0.0500 -0.0464 -1.0772 0.7640 -0.7092 0.6058 0.0964 1.4732 O -0.2540 0.9286 -0.0404 0.1477 -0.2734 -0.6170 2.2582 -1.1820 -0.1881 -2.8755 C2H5COC2H5 C1 0.3064 -0.2944 0.0486 -0.0467 -1.0408 0.7011 -0.6736 0.6007 0.0953 1.3746 O -0.2650 0.8751 -0.0420 0.1388 -0.3024 -0.606 2.0025 -1.1400 -0.1807 -2.6080 CH2=CHCHO C6 0.2789 0.2070 0.0557 0.0413 1.3472 1.3402 0.9944 0.0719 0.0144 0.3458 C1 0.4355 -0.2288 0.0870 -0.0457 -1.9033 2.0926 -1.0995 0.6643 0.1327 3.1921 O -0.0560 0.9265 -0.0112 0.1851 -0.0605 -0.2700 4.4518 -0.9830 -0.1963 -4.7213 CH3CH=CHCHO C6 0.3437 0.0926 0.0676 0.0182 3.7143 1.4494 0.3904 0.2511 0.0494 1.0590 C1 0.4408 -0.2365 0.0867 -0.0465 -1.8642 1.8592 -0.9973 0.6773 0.1332 2.8566 O -0.0670 0.9281 -0.0132 0.1825 -0.0721 -0.2820 3.9142 -0.9950 -0.1957 -4.1964 TABLE 3: Calculated Local Reactivity Properties of the Selected Molecules using BLYP/DND method for HPA derived charges. Molecule fk - Δfk +- fk HCHO C 0.3973 0.2373 0.1023 0.0611 1.6744 1.8563 1.1088 0.1600 0.0412 0.7476 O 0.3010 0.4232 0.0775 0.1090 0.7113 1.4064 1.9774 -0.1222 -0.0315 -0.5710 CH3CHO C1 0.2998 0.1642 0.0715 0.0391 1.8267 1.0268 0.5624 0.1356 0.0324 0.4644 O 0.2708 0.3782 0.0646 0.0902 0.7165 0.9275 1.2953 -0.1074 -0.0256 -0.3678 CH3COCH3 C1 0.2108 0.1154 0.0494 0.0271 1.8262 0.5902 0.3231 0.0954 0.0223 0.2671 O 0.2359 0.3499 0.0553 0.0820 0.6742 0.6605 0.9797 -0.1140 -0.0267 -0.3192 C2H5COC2H5 C1 0.1346 0.0990 0.0313 0.0230 1.3598 0.3547 0.2609 0.0356 0.0083 0.0938 O 0.1449 0.2873 0.0337 0.0668 0.5045 0.3818 0.7570 -0.1424 -0.0331 -0.3752 CH2=CHCHO C1 0.1780 0.1357 0.0577 0.0440 1.3117 1.1219 0.8553 0.0423 0.0137 0.2666 C6 0.2062 0.1253 0.0668 0.0406 1.6457 1.2997 0.7898 0.0809 0.0262 0.5099 O 0.1797 0.3414 0.0582 0.1106 0.5264 1.1326 2.1518 -0.1620 -0.0524 -1.0191 CH3CH=CHCHO C6 0.1592 0.1114 0.0500 0.0350 1.4291 0.8532 0.5970 0.0478 0.0150 0.2562 C1 0.1741 0.1095 0.0547 0.0344 1.5900 0.9330 0.5868 0.0646 0.0203 0.3462 O 0.1739 0.2450 0.0546 0.0769 0.7098 0.9319 1.3130 -0.0710 -0.0223 -0.3810 TABLE 4: Calculated Global Reactivity Properties of the Selected Molecules using B3LYP/6-311+g** and BLYP/DND method. η μ ω S η μ ω S Molecules B3LYP/6-311+g** (eV) BLYP/DND (eV) NH2OH 3.869 -3.553 1.632 0.129 3.411 -1.399 0.287 0.147 CH3ONH2 3.630 -3.738 1.925 0.138 3.549 -3.053 1.313 0.141 CH3NHOH 3.482 -3.392 1.652 0.144 3.229 -1.308 0.265 0.155 OHCH2CH2NH2 3.343 -3.507 1.840 0.150 3.348 -2.689 1.080 0.149 CH3SNH2 3.050 -3.331 1.819 0.164 2.447 -1.750 0.626 0.204 CH3NHSH 3.148 -3.629 2.092 0.159 2.466 -3.596 2.622 0.203 SHCH2CH2NH2 3.135 -3.417 1.862 0.159 2.521 -1.843 0.674 0.198 TABLE 5: Calculated Local Reactivity Properties of the Selected Molecules using B3LYP/6- 311+g** method for NPA derived charges. Molecule fk - sk - Δfk +- fk NH2OH N 0.1870 0.4140 0.0274 0.0607 2.2139 0.0536 0.1187 -0.2270 -0.0333 -0.0651 O 0.2390 0.2300 0.0350 0.0337 0.9623 0.0685 0.0659 0.0090 0.0013 0.0026 CH3ONH2 C 0.0870 0.0680 0.1410 1.3130 0.0123 0.0096 0.7816 0.1142 0.0893 0.0190 N 0.1500 0.3510 0.0211 0.0495 2.3400 0.1969 0.4608 -0.2010 -0.0283 -0.2639 O 0.0720 0.1740 0.0101 0.0245 2.4167 0.0945 0.2284 -0.1020 -0.0144 -0.1339 CH3NHOH C 0.0470 0.0740 0.0073 0.0115 1.5745 0.0124 0.0196 -0.0270 -0.0042 -0.0071 N 0.1200 0.3390 0.0186 0.0525 2.8250 0.0318 0.0898 -0.2190 -0.0339 -0.0580 O 0.2100 0.1770 0.0325 0.0274 0.8429 0.0556 0.0469 0.0330 0.0051 0.0087 OHCH2CH2NH2 C1 0.0540 0.0330 0.0081 0.0049 0.6111 0.0583 0.0356 0.0210 0.0031 0.0227 C2 0.0400 0.0610 0.006 0.0091 1.5250 0.0432 0.0659 -0.0210 -0.0031 -0.0227 N 0.0630 0.3470 0.0094 0.0518 5.5079 0.0680 0.3746 -0.2840 -0.0424 -0.3066 O 0.1400 0.1010 0.0209 0.0151 0.7214 0.1511 0.1090 0.0390 0.0058 0.0421 CH3SNH2 C 0.0550 0.0640 0.0112 0.0131 1.1636 0.0344 0.0400 -0.0090 -0.0018 -0.0056 N 0.1490 0.0820 0.0305 0.0168 0.5503 0.0932 0.0513 0.0670 0.0137 0.0419 S 0.3580 0.5510 0.0732 0.1126 1.5391 0.2239 0.3447 -0.1930 -0.0394 -0.1207 CH3NHSH C 0.0530 0.0540 0.0107 0.0110 1.0189 0.1390 0.1416 -0.0010 -0.0002 -0.0026 N 0.1310 0.1740 0.0266 0.0353 1.3282 0.3434 0.4562 -0.0430 -0.0087 -0.1127 S 0.4530 0.4420 0.0919 0.0896 0.9757 1.1876 1.1588 0.0110 0.0022 0.0288 SHCH2CH2NH2 C1 0.0780 0.0410 0.0155 0.0081 0.5256 0.0525 0.0276 0.0370 0.0073 0.0249 C2 0.0290 0.0250 0.0058 0.0050 0.8621 0.0195 0.0168 0.0040 0.0008 0.0027 N 0.0380 0.1270 0.0075 0.0252 3.3421 0.0256 0.0856 -0.0890 -0.0177 -0.0600 S 0.3890 0.4710 0.0772 0.0934 1.2108 0.2621 0.3173 -0.0820 -0.0163 -0.0552 TABLE 6 Calculated Local Reactivity Properties of the Selected Molecules using BLYP/DND method for HPA derived charges. Molecule fk - sk - Δfk +- fk NH2OH N 0.1837 0.9327 0.0237 0.1205 5.0777 0.2997 1.5218 -0.7490 -0.0970 -1.2220 O -0.0770 0.5114 -0.0100 0.0661 -6.6170 -0.1261 0.8344 -0.5890 -0.0760 -0.9610 CH3ONH2 C 0.5410 0.0819 0.0746 0.0113 0.1513 1.0412 0.1576 0.4592 0.0633 0.8837 N -0.1510 0.2534 -0.0210 0.0349 -1.6740 -0.2913 0.4877 -0.4050 -0.0560 -0.7790 O -0.1790 0.9011 -0.0250 0.1242 -5.0267 -0.3450 1.7342 -1.0800 -0.1490 -2.0790 CH3NHOH C 0.4598 0.1677 0.0660 0.0241 0.3647 0.7598 0.2771 0.2921 0.0419 0.4827 N -0.0580 0.7950 -0.0080 0.1142 -13.725 -0.0957 1.3136 -0.8530 -0.1220 -1.4090 O -0.2690 0.4537 -0.0390 0.0651 -1.6855 -0.4448 0.7497 -0.7230 -0.1040 -1.1940 OHCH2CH2NH2 C1 0.1186 0.0254 0.0177 0.0038 0.2140 0.2181 0.0467 0.0932 0.0139 0.1715 C2 0.4003 0.1067 0.0599 0.0160 0.2666 0.7365 0.1964 0.2936 0.0439 0.5401 N -0.3040 0.9520 -0.0450 0.1424 -3.1337 -0.5589 1.7514 -1.2560 -0.1880 -2.3100 O -0.3340 0.5965 -0.0500 0.0892 -1.7842 -0.6151 1.0974 -0.9310 -0.1390 -1.7120 CH3SNH2 C 0.0667 0.3358 0.0100 0.0502 5.0377 0.1226 0.6178 -0.2690 -0.0400 -0.4950 N -0.297 0.4790 -0.044 0.0717 -1.6119 -0.5467 0.8813 -0.7760 -0.1160 -1.4280 S 0.3671 0.6485 0.0549 0.0970 1.7667 0.6753 1.1931 -0.2810 -0.0420 -0.5180 CH3NHSH C 0.1715 0.1732 0.0256 0.0259 1.0100 0.3154 0.3186 -0.0020 -0.0003 -0.0030 N -0.225 0.9064 -0.0340 0.1356 -4.0267 -0.4141 1.6676 -1.1320 -0.1690 -2.0820 S 0.3479 0.2249 0.0520 0.0336 0.6465 0.6400 0.4137 0.1230 0.01840 0.2262 SHCH2CH2NH2 C1 0.0117 0.2268 0.0017 0.0339 19.432 0.0215 0.4172 -0.2150 -0.0320 -0.3960 C2 0.1651 0.0876 0.0247 0.0131 0.5309 0.3037 0.1612 0.0774 0.0116 0.1425 N -0.292 0.7628 -0.0440 0.1141 -2.6164 -0.5364 1.4035 -1.0540 -0.1580 -1.9400 S 0.1064 0.5646 0.0159 0.0845 5.3089 0.1957 1.0388 -0.4580 -0.0690 -0.8430 TABLE 7: Atomic site with maximum value for multiphilic descriptor (∆ωk) for the selected set of amines. site with maximum value for ∆ωk molecule NPA HPA NH2OH N N CH3ONH2 O N CH3NHOH N N OHCH2CH2NH2 N N CH3SNH2 N S CH3NHSH N N SHCH2CH2NH2 N N TABLE 8: Global reactivity descriptors calculated at B3LYP/6-31G* level of theory. Species η (eV) (eV) (eV) Reactant 3.64 -2.89 1.15 Transition State 2.48 -2.79 1.57 Product 3.64 -2.89 1.15 TABLE 9: Global reactivity descriptors along the intrinsic reaction coordinate calculated at B3LYP/6-31G* level of theory. Points along (Hartrees) (eV) (eV) (eV) (a.u.) 1 -234.5673091 2.65 -2.7825 1.46 64.94 2 -234.5661087 2.63 -2.7827 1.47 65.21 3 -234.5649450 2.61 -2.7828 1.49 65.47 4 -234.5638273 2.59 -2.7836 1.50 65.74 5 -234.5627655 2.57 -2.7836 1.51 65.98 6 -234.5617681 2.55 -2.7843 1.52 66.22 7 -234.5608445 2.54 -2.7843 1.53 66.42 8 -234.5600030 2.53 -2.7851 1.54 66.63 9 -234.5592516 2.51 -2.7852 1.54 66.80 10 -234.5585980 2.50 -2.7859 1.55 66.96 11 -234.5580104 2.50 -2.7857 1.56 67.07 12 -234.5575677 2.49 -2.7866 1.56 67.20 13 -234.5575677 2.49 -2.7866 1.56 67.20 14 -234.5580104 2.50 -2.7857 1.56 67.07 15 -234.5585980 2.50 -2.7859 1.55 66.96 16 -234.5592516 2.51 -2.7852 1.54 66.80 17 -234.5600030 2.53 -2.7851 1.54 66.63 18 -234.5608445 2.54 -2.7843 1.53 66.42 19 -234.5617681 2.55 -2.7843 1.52 66.22 20 -234.5627655 2.57 -2.7836 1.51 65.98 21 -234.5638273 2.59 -2.7836 1.50 65.74 22 -234.5649450 2.61 -2.7830 1.49 65.47 23 -234.5661087 2.63 -2.7827 1.47 65.21 24 -234.5673092 2.65 -2.7825 1.46 64.94 TABLE 10: Group Philicity ( + gω ) Values for Nucleophilic and Electrophilic Attacks Respectively for the Ionic Units of Different Isomers of LiAl4–, NaAl4–, KAl4– and CuAl4–. Isomers Ionic Unit gωΔ Al42– 0.0070 0.0095 0.0025 LiAl4– (C∞v) Li+ 0.0063 0.0037 -0.0025 Al42– 1.3E-05 0.0055 0.0055 LiAl4– (C2v) Li+ 0.0068 0.0013 -0.0055 Al42– -0.0372 0.2965 0.3338 LiAl4– (C4v) Li+ 0.4055 0.0718 -0.3338 Al42– 0.0070 0.0102 0.0032 NaAl4– (C∞v) Na+ 0.0074 0.0042 -0.0032 Al42– -0.0001 0.0078 0.0079 NaAl4– (C2v) Na+ 0.0096 0.0017 -0.0079 Al42– -0.0073 0.1024 0.1097 NaAl4– (C4v) Na+ 0.1301 0.0204 -0.1097 Al42– 0.0044 0.0095 0.0051 KAl4– (C∞v) K+ 0.0106 0.0054 -0.0051 Al42– 0.0023 0.0101 0.0078 KAl4– (C2v) K+ 0.0118 0.0039 -0.0078 Al42– 0.0008 0.0066 0.0057 KAl4– (C4v) K+ 0.0078 0.0021 -0.0057 Al42– 0.0031 0.0036 0.0006 CuAl4– (C∞v) Cu+ 0.0014 0.0009 -0.0006 Al42– 0.0036 0.0036 0.0048 CuAl4– (C2v) Cu+ 0.0008 0.0008 -0.0048 Al42– 0.0178 0.0332 0.0154 CuAl4– (C4v) Cu+ 0.0131 -0.0023 -0.0154 TABLE 11: Profiles of the forward activation energy ( #fEΔ ), reverse activation energy ( #rEΔ ) and reaction energy ( 0EΔ ) of a thermoneutral reaction (Fa– + CH3-Fb → Fa--CH3 + Fb–; an endothermic reaction (HNO → HON) and an exothermic reaction (H2OO → HOOH). Reaction #fEΔ ξ1 ξ2 W1 W2 W3 W4 Thermo-neutral B3LYP/6-311++G** 9.54 9.54 0.0 -1.33 1.33 5.42 4.12 -4.12 - 5.42 Endothermic B3LYP/6-311+G** 75.39 34.84 40.55 -0.80 0.60 43.97 31.42 -13,20 - 21.64 Exothermic B3LYP/6-311+G** 7.39 52.85 -45.46 -0.65 0.87 3.93 3.46 - 19.99 - 32.86 Figure 1. Optimized structures with atom numbering for the selected carbonyl compounds. Figure 2. Optimized structures with atom numbering for the selected amine systems. Reactant Transition State Product Figure 3:Optimized geometrical structures calculated using B3LYP/6-31G* level of theory. -234.568 -234.566 -234.564 -234.562 -234.560 -234.558 -234.556 Energy (Hartree) Electrophilicity Index (eV) Intrinsic Reaction Coordinate lectrophilicity Index (eV 2.66 Chemical Hardness (eV) Polarizability (au) Intrinsic Reaction Coordinate olarizability (au) Figure 4 (a-b):Variation of global reactivity descriptors along intrinsic reaction coordinate. -0.008 -0.006 -0.004 -0.002 0.000 0.002 0.004 0.006 Intrinsic Reaction Coordinate C1,C3 sites -0.008 -0.006 -0.004 -0.002 0.000 0.002 0.004 0.006 Intrinsic Reaction Coordinate C6,C11 sites Figure 5 (a-b): Variation of multiphilic descriptor along intrinsic reaction coordinate for the selected atomic sites. MAl4– [C∞v] MAl4– [C2v] MAl4– [C4v] M=Li, Na, K, Cu Figure 6. Optimized structures of various isomers of MAl4– (M ≡ Li, Na, K, Cu). -3 -2 -1 0 1 2 3 -239.704 -239.702 -239.700 -239.698 -239.696 -239.694 -239.692 -239.690 -239.688 -239.686 -3 -2 -1 0 1 2 3 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 Energy Δω (Fa) Δω (F (a) -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 -130.52 -130.50 -130.48 -130.46 -130.44 -130.42 -130.40 -130.38 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 Energy -2 -1 0 1 2 3 -151.60 -151.58 -151.56 -151.54 -151.52 -151.50 -2 -1 0 1 2 3 -0.02 Energy (O2) Δω (O1) (c) Figure 7 (a-c): Profiles of net nucleophilicity (∆ωk) of along the path of the gas phase (a) thermoneutral SN2 substitution: Fa- + CH3-Fb → Fa-CH3 + Fb-, (b) endothermic reaction: HNO → HON and (c) exothermic reaction: H2OO → HOOH. Also shown is the profile of energy. Figure 8 Reaction force profiles along the reaction coordinate for (a) thermoneutral reaction: Fa– + CH3-Fb → Fa--CH3 + Fb–; (b) endothermic reaction: HNO → HON; (c) the exothermic reaction: H2OO → HOOH. The vertical dashed lines define the reaction regions as follows: reactant (left), transition state (middle) and product (right). -4 -2 0 2 4 -2 -1 0 1 2 3 -2 -1 0 1 2 max(a)

0704.0335

Approximation of the distribution of a stationary Markov process with application to option pricing

Approximation of the distribution of a stationary Markov process with application to option pricing Bernoulli 15(1), 2009, 146–177 DOI: 10.3150/08-BEJ142 Approximation of the distribution of a stationary Markov process with application to option pricing GILLES PAGÈS1 and FABIEN PANLOUP2 Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Université Paris 6, Case 188, 4 pl. Jussieu, F-75252 Paris Cedex 5. E-mail: [email protected] Laboratoire de Statistiques et Probabilités, Université Paul Sabatier & INSA Toulouse, 135, Avenue de Rangueil, 31077 Toulouse Cedex 4. E-mail: [email protected] We build a sequence of empirical measures on the space D(R+,R d) of Rd-valued cadlag functions on R+ in order to approximate the law of a stationary R d-valued Markov and Feller process (Xt). We obtain some general results on the convergence of this sequence. We then apply them to Brownian diffusions and solutions to Lévy-driven SDE’s under some Lyapunov-type stability assumptions. As a numerical application of this work, we show that this procedure provides an efficient means of option pricing in stochastic volatility models. Keywords: Euler scheme; Lévy process; numerical approximation; option pricing; stationary process; stochastic volatility model; tempered stable process 1. Introduction 1.1. Objectives and motivations In this paper, we deal with an Rd-valued Feller Markov process (Xt) with semigroup (Pt)t≥0 and assume that (Xt) admits an invariant distribution ν0. The aim of this work is to propose a way to approximate the whole stationary distribution Pν0 of (Xt). More pre- cisely, we want to construct a sequence of weighted occupation measures (ν(n)(ω,dα))n≥1 on the Skorokhod space D(R+,R d) such that ν(n)(ω,F ) n→+∞−→ F (α)Pν0(dα) a.s. for a class of functionals F :D(R+,R d) which includes bounded continuous functionals for the Skorokhod topology. One of our motivations is to develop a new numerical method for option pricing in sta- tionary stochastic volatility models which are slight modifications of the classical stochas- tic volatility models, where we suppose that the volatility evolves under its stationary regime. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2009, Vol. 15, No. 1, 146–177. This reprint differs from the original in pagination and typographic detail. 1350-7265 c© 2009 ISI/BS http://arxiv.org/abs/0704.0335v3 http://isi.cbs.nl/bernoulli/ http://dx.doi.org/10.3150/08-BEJ142 mailto:[email protected] mailto:[email protected] http://isi.cbs.nl/BS/bshome.htm http://isi.cbs.nl/bernoulli/ http://dx.doi.org/10.3150/08-BEJ142 Approximation of the distribution of a stationary Markov process 147 1.2. Background and construction of the procedure This work follows on from a series of recent papers due to Lamberton and Pagès ([12, 13]), Lemaire ([14, 15]) and Panloup ([18, 19, 20]), where the problem of the approximation of the invariant distribution is investigated for Brownian diffusions and for Lévy-driven SDE’s.1 In these papers, the algorithm is based on an adapted Euler scheme with de- creasing step (γk)k≥1. To be precise, let (Γn) be the sequence of discretization times: Γ0 = 0, Γn = k=1 γk for every n≥ 1, and assume that Γn → +∞ when n→+∞. Let (X̄Γn)n≥0 be the Euler scheme obtained by “freezing” the coefficients between the Γn’s and let (ηn)n≥1 be a sequence of positive weights such that Hn := k=1 ηk →+∞ when k→+∞. Then, under some Lyapunov-type stability assumptions adapted to the stochas- tic processes of interest, one shows that for a large class of steps and weights (ηn, γn)n≥1, ν̄n(ω, f) := ηkf(X̄Γk−1) n→+∞−→ f(x)ν0(dx) a.s., (1) (at least)2 for every bounded continuous function f . Since the problem of the approximation of the invariant distribution has been deeply studied for a wide class of Markov processes (Brownian diffusions and Lévy-driven SDE’s) and since the proof of (1) can be adapted to other classes of Markov processes under some specific Lyapunov assumptions, we choose in this paper to consider a general Markov pro- cess and to assume the existence of a time discretization scheme (X̄Γk)k≥0 such that (1) holds for the class of bounded continuous functions. The aim of this paper is then to inves- tigate the convergence properties of a functional version of the sequence (ν̄n(ω,dα))n≥1. Let (Xt) be a Markov and Feller process and let (X̄t)t≥0 be a stepwise constant time discretization scheme of (Xt) with non-increasing step sequence (γn)n≥1 satisfying γn = 0, Γn := n→+∞−→ +∞. (2) Letting Γ0 := 0 and X̄0 = x0 ∈Rd, we assume that X̄t = X̄Γn ∀t ∈ [Γn,Γn+1[ (3) and that (X̄Γn)n≥0 can be simulated recursively. We denote by (Ft)t≥0 and (F̄t)t≥0 the usual augmentations of the natural filtrations (σ(Xs,0≤ s≤ t))t≥0 and (σ(X̄s,0≤ s≤ t))t≥0, respectively. 1Note that computing the invariant distribution is equivalent to computing the marginal laws of the stationary process (Xt) since ν0Pt = ν0 for every t≥ 0. 2The class of functions for which (1) holds depends on the stability of the dynamical system. In particular, in the Brownian diffusion case, the convergence may hold for continuous functions with subexponential growth, whereas the class of functions strongly depends on the moments of the Lévy process when the stochastic process is a Lévy-driven SDE. 148 G. Pagès and F. Panloup For k ≥ 0, we denote by (X̄(k)t )t≥0 the shifted process defined by t := X̄Γk+t. In particular, X̄ t = X̄t. We define a sequence of random probabilities (ν (n)(ω,dα))n≥1 on D(R+,R d) by ν(n)(ω,dα) = ηk1{X̄(k−1)(ω)∈dα}, where (ηk)k≥1 is a sequence of weights. For t ≥ 0, (ν(n)t (ω,dx))n≥1 will denote the se- quence of “marginal” empirical measures on Rd defined by t (ω,dx) = ηk1{X̄(k−1) (ω)∈dx} 1.3. Simulation of (ν(n)(ω,F )) For every functional F :D(R+,R d)→R, the following recurrence relation holds for every n≥ 1: ν(n+1)(ω,F ) = ν(n)(ω,F ) + (F (X(n)(ω))− ν(n)(ω,F )). (4) Then, if T is a positive number and F :D(R+,R d) → R is a functional depending only on the trajectory between 0 and T , (ν(n)(ω,F ))n≥1 can be simulated by the following procedure. Step 0. (i) Simulate (X̄ t )t≥0 on [0, T ], that is, simulate (X̄Γk)k≥0 for k = 0, . . . ,N(0, T ), where N(n,T ) := inf{k ≥ n,Γk+1 − Γn > T } = max{k ≥ 0,Γk − Γn ≤ T }, n≥ 0, T > 0. Note that n 7→N(n, t) is an increasing sequence since (γn) is non-increasing, and that ΓN(n,T ) − Γn ≤ T < ΓN(n,T )+1 − Γn. (ii) Compute F ((X̄ t )t≥0) and ν (1)(ω,F ). Store the values of (X̄Γk) for k = 1, . . . ,N(0, T ). Step n (n≥ 1). (i) Since the values (X̄Γk)k≥0 are stored for k = n, . . . ,N(n− 1, T ), simulate (X̄Γk)k≥0 for k =N(n−1, T )+1, . . . ,N(n,T ) in order to obtain a path of (X̄ on [0, T ]. (ii) Compute F ((X̄ t )t≥0) and use (4) to compute ν (n+1)(ω,F ). Store the values of (X̄Γk) for k = n+ 1, . . . ,N(n,T ). Approximation of the distribution of a stationary Markov process 149 Remark 1. As shown in the description of the procedure, one generally has to store the vector [X̄Γn , . . . , X̄ΓN(n,T) ] at time n. Since (γn) is a sequence with infinite sum that decreases to 0, it follows that the size of this vector increases “slowly” to +∞. For instance, if γn = Cn −ρ with ρ ∈ (0,1), its size is of order nρ. However, it is important to remark that even though the number of values to be stored tends to +∞, that is not always the case for the number of operations at each step. Indeed, since X̄(n+1) is obtained by shifting X̄(n), it is usually possible to use, at step n+ 1, the preceding computations and to simulate the sequence (F (X̄(n)))n≥0 in a “quasi-recursive” way. For instance, such remark holds for Asian options because the associated pay-off can be expressed as a function of an additive functional (see Section 5 for simulations). Before outlining the sequel of the paper, we list some notation linked to the spaces D(R+,R d) and D([0, T ],Rd) of cadlag Rd-valued functions on R+ and [0, T ], respectively, endowed with the Skorokhod topology. First, we denote by d1 the Skorokhod distance on D([0,1],Rd) defined for every α, β ∈D([0,1],Rd) by d1(α,β) = inf t∈[0,1] |α(t)− β(λ(t))|, sup 0≤s<t≤1 λ(t)− λ(s) where Λ1 denotes the set of increasing homeomorphisms of [0,1]. Second, for T > 0, φT :D(R+,R d) 7→D([0,1],Rd) is the function defined by (φT (α))(s) = α(sT ) for every s ∈ [0,1]. We then denote by d the distance on D(R+,R d) defined for every α,β ∈D(R+,Rd) d(α,β) = e−t(1∧ d1(φt(α), φt(β))) dt. (6) We recall that (D(R+,R d), d) is a Polish space and that the induced topology is the usual Skorokhod topology on D(R+,R d) (see, e.g., Pagès [16]). For every T > 0, we set σ(πu,0≤ u≤ s), where πs :D(R+,R d)→Rd is defined by πs(α) = α(s). For a functional F :D(R+,Rd)→ R, FT denotes the functional defined for every α ∈D(R+,Rd) by FT (α) = F (α T ) with αT (t) = α(t ∧ T ) ∀t≥ 0. (7) Finally, we will say that a functional F :D(R+,R d)→R is Sk-continuous if F is contin- uous for the Skorokhod topology on D(R+,R d) and the notation “ =⇒” will denote the weak convergence on D(R+,R In Section 2, we state our main results for a general Rd-valued Feller Markov process. Then, in Section 3, we apply them to Brownian diffusions and Lévy-driven SDE’s. Section 4 is devoted to the proofs of the main general results. Finally, in Section 5, we complete this paper with an application to option pricing in stationary stochastic volatility models. 150 G. Pagès and F. Panloup 2. General results In this section, we state the results on convergence of the sequence (ν(n)(ω,dα))n≥1 when (Xt) is a general Feller Markov process. 2.1. Weak convergence to the stationary regime As explained in the Introduction, since the a.s. convergence of (ν 0 (ω,dx))n≥1 to the invariant distribution ν0 has already been deeply studied for a large class of Markov processes (Brownian diffusions and Lévy driven SDE’s), our approach will be to derive the convergence of (ν(n)(ω,dα))n≥1 toward Pν0 from that of (ν 0 (ω,dx))n≥1 to the invariant distribution ν0. More precisely, we will assume in Theorem 1 that (C0,1): (Xt) admits a unique invariant distribution ν0 and 0 (ω,dx) =⇒ ν0(dx) a.s., whereas in Theorem 2, we will only assume that (C0,2): (ν 0 (ω,dx))n≥1 is a.s. tight on R We also introduce three other assumptions, (C1), (C2) and (C3,ε), regarding the conti- nuity in probability of the flow x 7→ (Xxt ), the asymptotic convergence of the shifted time discretization scheme to the true process (Xt) and the steps and weights, respectively. (C1): For every x0 ∈Rd, ǫ > 0 and T > 0, limsup 0≤t≤T |Xxt −Xx0t | ≥ ǫ = 0. (8) (C2): (X̄t) is a non-homogeneous Markov process and for every n≥ 0, it is possible to construct a family of stochastic processes (Y (n,x) t )x∈Rd such that (i) L(Y (n,x)) D(R+,R = L(X̄(n)|X̄(n)0 = x); (ii) for every compact set K of Rd, for every T ≥ 0, 0≤t≤T |Y (n,x)t −Xxt | n→+∞−→ 0 in probability. (9) (C3,ε): For every n≥ 1, ηn ≤CγnHεn. Remark 2. Assumption (C2) implies, in particular, that asymptotically and uniformly on compact sets of Rd, the law of the approximate process (X̄(n)), given its initial value, is close to that of the true process. If there exists a unique invariant distribution ν0, the second part of (C2) can be relaxed to the following, less stringent, assertion: for all ǫ > 0, there exists a compact set Aǫ ⊂Rd such that ν0(A ǫ)≤ ǫ and such that 0≤t≤T |Y (n,x)t −Xxt | n→+∞−→ 0 in probability. (10) Approximation of the distribution of a stationary Markov process 151 This weaker assumption can some times be needed in stochastic volatility models like the Heston model (see Section 5 for details). The preceding assumptions are all that we require for the convergence of (ν(n)(ω,dα))n≥1 to Pν0 along the bounded Sk-continuous functionals, that is, for the a.s. weak conver- gence on D(R+,R d). However, the integration of non-bounded continuous functionals F :D([0, T ],Rd)→ R will need some additional assumptions, depending on the stability of the time discretization scheme and on the steps and weights sequences. We will sup- pose that F is dominated (in a sense to be specified later) by a function V : Rd → R+ that satisfies the following assumptions for some s≥ 2 and ε < 1. H(s, ε): For every T > 0, (i) sup 0≤t≤T Vs(Y (n,x)t ) ≤CTVs(x), (ii) sup 0 (V)<+∞, (iii) E[V2(X̄Γk−1)]<+∞, ∆N(k,T ) E[Vs(1−ε)(X̄Γk−1)]<+∞, where T 7→CT is locally bounded on R+ and ∆N(k,T ) =N(k,T )−N(k− 1, T ). For every ε < 1, we then set K(ε) = {V ∈ C(Rd,R+),H(s, ε) holds for some s≥ 2}. Remark 3. Apart from assumption (i), which is a classical condition on the finite time horizon control, the assumptions in H(s, ε) strongly rely on the stability of the time discretization scheme (and then, to that of the true process). More precisely, we will see when we apply our general results to SDE’s that these properties are some consequences of the Lyapunov assumptions needed for the tightness of (ν 0 (ω,dx))n≥1. We can now state our first main result. Theorem 1. Assume (C0,1), (C1), (C2) and (C3,ε) with ε ∈ (−∞,1). Then, a.s., for every bounded Sk-continuous functional F :D(R+,R d)→R, ν(n)(ω,F ) n→+∞−→ F (α)Pν0 (dα), (11) where Pν0 denotes the stationary distribution of (Xt) (with initial law ν0). Furthermore, for every T > 0, for every non-bounded Sk-continuous functional F :D(R+,R d)→ R, (11) holds a.s. for FT (defined by (7)) if there exists V ∈ K(ε) and 152 G. Pagès and F. Panloup ρ ∈ [0,1) such that |FT (α)| ≤C sup 0≤t≤T Vρ(αt) ∀α ∈D(R+,Rd). (12) In the second result, the uniqueness of the invariant distribution is not required and the sequence (ν 0 (ω,dx))n≥1 is only supposed to be tight. Theorem 2. Assume (C0,2), (C1), (C2) and (C3,ε) with ε ∈ (−∞,1). Assume that 0 (ω,dx))n≥1 is a.s. tight on R d. We then have the following. (i) The sequence (ν(n)(ω,dα))n≥1 is a.s. tight on D(R+,R d) and a.s., for ev- ery convergent subsequence (nk(ω))n≥1, for every bounded Sk-continuous functional F :D(R+,R d)→R, ν(nk(ω))(ω,F ) n→+∞−→ F (α)Pν∞(dα), (13) where Pν∞ is the law of (Xt) with initial law ν∞ being a weak limits for (ν 0 (ω,dx))n≥1. Furthermore, for every T > 0, for every non-bounded Sk-continuous functional F :D(R+,R d)→R, (13) holds a.s. for FT if (12) is satisfied with V ∈K(ε) and ρ ∈ [0,1). (ii) If, moreover, l≥k+1 |∆ηℓ| n→+∞−→ 0, (14) then ν∞ is necessarily an invariant distribution for the Markov process (Xt). Remark 4. Condition (14) holds for a large class of steps and weights. For instance, if ηn = C1n −ρ1 and γn = C2n −ρ2 with ρ1 ∈ [0,1] and ρ2 ∈ (0,1], then (14) is satisfied if ρ1 = 0 or if ρ1 ∈ (max(0,2ρ2 − 1),1). 2.2. Extension to the non-stationary case Even though the main interest of this algorithm is the weak approximation of the pro- cess when stationary, we observe that when ν0 is known, the algorithm can be used to approximate Pµ0 if µ0 is a probability on R d that is absolutely continuous with respect to ν0. Indeed, assume that µ0(dx) = φ(x)ν0(dx), where φ :R d → R is a continuous non- negative function. For a functional F :D(R+,R d)→ R, denote by Fφ the functional de- fined on D(R+,R d) by Fφ(α) = F (α)φ(α(0)). Then, if ν(n)(ω,dα) (Sk)⇒ Pν0(dα) a.s., we also have the following convergence: a.s., for every bounded Sk-continuous functional F :D(R+,R d)→R, ν(n)(ω,Fφ) n→+∞−→ Fφ(α)Pν0 (dα) = F (α)Pµ0 (dα). Approximation of the distribution of a stationary Markov process 153 3. Application to Brownian diffusions and Lévy-driven SDE’s Let (Xt)t≥0 be a cadlag stochastic process solution to the SDE dXt = b(Xt−) dt+ σ(Xt−) dWt + κ(Xt−) dZt, (15) where b :Rd → Rd, σ :Rd 7→Md,ℓ (set of d× ℓ real matrices) and κ :Rd 7→Md,ℓ are con- tinuous functions with sublinear growth, (Wt)t≥0 is an ℓ-dimensional Brownian motion and (Zt)t≥0 is an integrable purely discontinuous R ℓ-valued Lévy process independent of (Wt)t≥0 with Lévy measure π and characteristic function given for every t≥ 0 by E[ei〈u,Zt〉] = exp ei〈u,y〉 − 1− i〈u, y〉π(dy) Let (γn)n≥1 be a non-increasing step sequence satisfying (2). Let (Un)n≥1 be a sequence of i.i.d. random variables such that U1 =N (0, Iℓ) and let ξ := (ξn)n≥1 be a sequence of independent Rℓ-valued random variables, independent of (Un)n≥1. We then denote by (X̄t)t≥0 the stepwise constant Euler scheme of (Xt) for which (X̄Γn)n≥0 is recursively defined by X̄0 = x ∈Rd and X̄Γn+1 = X̄Γn + γn+1b(X̄Γn) + γn+1σ(X̄Γn)Un+1 + κ(X̄Γn)ξn+1. (16) We recall that the increments of (Zt) cannot be simulated in general. That is why we generally need to construct the sequence (ξn) with some approximations of the true increments. We will come back to this construction in Section 3.2. As in the general case, we denote by (X̄(k))k≥0 and (ν (n)(ω,dα))n≥1 the sequences of associated shifted Euler schemes and empirical measures, respectively. Let us now introduce some Lyapunov assumptions for the SDE. Let EQ(Rd) denote the set of essentially quadratic C2-functions V :Rd → R∗+ such that limV (x) = +∞ as |x| →+∞, |∇V | ≤C V and D2V is bounded. Let a ∈ (0,1] denote the mean reversion intensity. The Lyapunov (or mean reversion) assumption is the following. (Sa): There exists a function V ∈ EQ(Rd) such that: (i) |b|2 ≤CV a, Tr(σσ∗(x)) + ‖κ(x)‖2 |x|→+∞= o(V a(x)); (ii) there exist β ∈R and ρ > 0 such that 〈∇V, b〉 ≤ β − ρV a. From now on, we separate the Brownian diffusions and Lévy-driven SDE cases. 3.1. Application to Brownian diffusions In this part, we assume that κ= 0. We recall a result by Lamberton and Pagès [13]. Proposition 1. Let a ∈ (0,1] such that (Sa) holds. Assume that the sequence (ηn/γn)n≥1 is non-increasing. 154 G. Pagès and F. Panloup (a) Let (θn)n≥1 be a sequence of positive numbers such that n≥1 θnγn < +∞ and that there exists n0 ∈N such that (θn)n≥n0 is non-increasing. Then, for every positive r, θnγnE[V r(X̄Γn−1)]<+∞. (b) For every r > 0, 0 (ω,V r)<+∞ a.s. (17) Hence, the sequence (ν 0 (ω,dx))n≥1 is a.s. tight. (c) Moreover, every weak limit of this sequence is an invariant probability for the SDE (15). In particular, if (Xt)t≥0 admits a unique invariant probability ν0, then for every continuous function f such that f ≤CV r with r > 0, limn→∞ ν(n)0 (ω, f) = ν0(f) a.s. Remark 5. For instance, if V (x) = 1 + |x|2, then the preceding convergence holds for every continuous function with polynomial growth. According to Theorem 3.2 in Lemaire [14], it is possible to extend these results to continuous functions with exponential growth, but it then strongly depends on σ. Further the conditions on steps and weights can be less restrictive and may contain the case ηn = 1, for instance (see Remark 4 of Lamberton and Pagès [13] and Lemaire [14]). We then derive the following result from the preceding proposition and from Theorems 1 and 2. Theorem 3. Assume that b and σ are locally Lipschitz functions and that κ = 0. Let a ∈ (0,1] such that (Sa) holds and assume that (ηn/γn) is non-increasing. (a) The sequence (ν(n)(ω,dα))n≥1 is a.s. tight on C(R+,Rd)3 and every weak limit of (ν(n)(ω,dα))n≥1 is the distribution of a stationary process solution to (15). In par- ticular, when uniqueness holds for the invariant distribution ν0, a.s., for every bounded continuous functional F :C(R+,Rd)→R, ν(n)(ω,F ) n→+∞−→ F (x)Pν0 (dx). (18) (b) Furthermore, if there exists s ∈ (2,+∞) and n0 ∈N such that ∆N(k,T ) is non-increasing and ∆N(k,T ) <+∞, (19) 3C(R+,R d) denotes the space of continuous functions on R+ with values in R d endowed with the topology of uniform convergence on compact sets. Approximation of the distribution of a stationary Markov process 155 then, for every T > 0, for every non-bounded continuous functional F :C(R+,Rd)→ R, (18) holds for FT if the following condition is satisfied: ∃r > 0 such that |FT (α)| ≤C sup 0≤t≤T V r(αt) ∀α ∈ C(R+,Rd). Remark 6. If ηn =C1n −ρ1 and γn =C2n −ρ2 with 0< ρ2 ≤ ρ1 ≤ 1, then for s ∈ (1,+∞), (19) is fulfilled if and only if s > 1/(1− ρ1). It follows that there exists s ∈ (2,+∞) such that (19) holds as soon as ρ1 < 1. Proof of Theorem 3. We want to apply Theorem 2. First, by Proposition 1, assumption (C0,2) is fulfilled and every weak limit of (ν 0 (ω,dx)) is an invariant distribution. Second, it is well known that (C1) and (C2) are fulfilled when b and σ are locally Lispchitz sublinear functions. Then, since (C3,ε) holds with ε = 0, (18) holds for every bounded continuous functional F . Finally, one checks that H(s,0) holds with V := V r (r > 0). It is classical that assumption (a) is true when b and σ are sublinear. Assumption (b) follows from Proposition 1(b). Let θn,1 = ηn/(γnH n) and θn,2 =∆N(n,T )/(γnH n). Using (19) and the fact that (ηn/γn) is non-increasing yields that (θn,1) and (θn,2) satisfy the conditions of Proposition 1 (see (35) for details). Then, (iii) and (iv) of H(s,0) are consequences of Proposition 1(a). This completes the proof. � 3.2. Application to Lévy-driven SDE’s When we want to extend the results obtained for Brownian SDE’s to Lévy-driven SDE’s, one of the main difficulties comes from the moments of the jump component (see Panloup [18] for details). For simplification, we assume here that (Zt) has a moment of order 2p≥ 2, that is, that its Lévy measure π satisfies the following assumption with p≥ 1: (H1p) : |y|>1 π(dy)|y|2p <+∞. We also introduce an assumption about the behavior of the moments of the Lévy measure at 0: (H2q) : |y|≤1 π(dy)|y|2q <+∞, q ∈ [0,1]. This assumption ensures that (Zt) has finite 2q-variations. Since |y|≤1 |y|2π(dy) is finite, this is always satisfied for q = 1. Let us now specify the law of (ξn) introduced in (16). When the increments of (Zt) can be exactly simulated, we denote by (E) the Euler scheme and by (ξn,E) the associated sequence = Zγn ∀n≥ 1. 156 G. Pagès and F. Panloup When the increments of (Zt) cannot be simulated, we introduce some approximated Euler schemes (P) and (W) built with some sequences (ξn,P ) and (ξn,W ) of approximations of the true increment (see Panloup [19] for more detailed presentations of these schemes). In scheme (P), =Zγn,n, where (Z·,n)n≥1 a sequence of compensated compound Poisson processes obtained by truncating the small jumps of (Zt)t≥0: Zt,n := 0<s≤t ∆Zs1{|∆Zs|>un} − t |y|>un yπ(dy) ∀t≥ 0, (20) where (un)n≥1 is a sequence of positive numbers such that un → 0. We recall that n→+∞−→ Z locally uniformly in L2 (see, e.g., Protter [21]). As shown in Panloup [19], the error induced by this approximation is very large when the local behavior of the small jumps component is irregular. However, it is possible to refine this approximation by a Wienerization of the small jumps, that is, by replacing the small jumps by a linear transform of a Brownian motion instead of discarding them (see Asmussen and Rosinski [2]). The corresponding scheme is denoted by (W) with ξn,W satisfying = ξn,P + γnQnΛn ∀n≥ 1, where (Λn)n≥1 is a sequence of i.i.d. random variables, independent of (ξn,P )n≥1 and (Un)n≥1, such that Λ1 =N (0, Iℓ) and (Qn) is a sequence of ℓ× ℓ matrices such that n)i,j = |y|≤uk yiyjπ(dy). We recall the following result obtained in Panloup [18] in our slightly simplified frame- work. Proposition 2. Let a ∈ (0,1], p≥ 1 and q ∈ [0,1] such that (H1p), (H2q) and (Sa) hold. Assume that the sequence (ηn/γn)n≥1 is non-increasing. Then, the following assertions hold for schemes (E), (P) and (W). (a) Let (θn) satisfy the conditions of Proposition 1. Then, n≥1 θnγnE[V p+a−1(X̄Γn−1)]< (b) We have 0 (ω,V p/2+a−1)<+∞ a.s. (21) Hence, the sequence (ν 0 (ω,dx))n≥1 is a.s. tight as soon as p/2+ a− 1> 0. Approximation of the distribution of a stationary Markov process 157 (c) Moreover, if Tr(σσ∗)+ ‖κ‖2q ≤CV p/2+a−1, then every weak limit of this sequence is an invariant probability for the SDE (15). In particular, if (Xt)t≥0 admits a unique invariant probability ν0, for every continuous function f such that f = o(V p/2+a−1), limn→∞ ν 0 (ω, f) = ν0(f) a.s. Remark 7. For schemes (E) and (P), the above proposition is a direct consequence of Theorem 2 and Proposition 2 of Panloup [18]. As concerns scheme (W), a straightforward adaptation of the proof yields the result. Our main functional result for Lévy-driven SDE’s is then the following. Theorem 4. Let a ∈ (0,1] and p≥ 1 such that p/2+ a− 1> 0 and let q ∈ [0,1]. Assume (H1p), (H q) and (Sa). Assume that b, σ and κ are locally Lipschitz functions. If, more- over, (ηn/γn)n≥1 is non-increasing, then the following result holds for schemes (E), (P) and (W). (a) The sequence (ν(n)(ω,dα))n≥1 is a.s. tight on D(R+,R d). Moreover, if Tr(σσ∗) + ‖κ‖2q ≤CV p/2+a−1 or 1 l≥k+1 |∆ηℓ| n→+∞−→ 0, (22) then every weak limit of (ν(n)(ω,dα))n≥1 is the distribution of a stationary process solu- tion to (15). (b) Assume that the invariant distribution is unique. Let ε≤ 0 such that (C3,ε) holds. Then, a.s., for every T > 0, for every Sk-continuous functional F :D(R+,R d)→R, (18) holds for FT if there exist ρ ∈ [0,1) and s≥ 2, such that |FT (α)| ≤C sup 0≤t≤T V (ρ(p+a−1))/s(αt) ∀α ∈D(R+,Rd) and if ∆N(k,T ) s(1−ε) is non-increasing and ∆N(k,T ) s(1−ε) <+∞. (23) Remark 8. In (22), both assumptions imply the invariance of every weak limit of 0 (ω,dx)). These two assumptions are very different. The first is needed in Proposition 2 for using the Echeverria–Weiss invariance criteria (see Ethier and Kurtz [7], page 238, Lamberton and Pagès [12] and Lemaire [14]), whereas the second appears in Theorem 2, where our functional approach shows that under some mild additional conditions on steps and weights, every weak limit is always invariant. For (23), we refer to Remark 6 for simple sufficient conditions when (γn) and (ηn) are some polynomial steps and weights. 158 G. Pagès and F. Panloup 4. Proofs of Theorems 1 and 2 We begin the proof with some technical lemmas. In Lemma 1, we show that the a.s weak convergence of the random measures (ν(n)(ω,dα))n≥1 can be characterized by the convergence (11) along the set of bounded Lipschitz functionals F for the distance d. Then, in Lemma 2, we show with some martingale arguments that if the functional F depends only on the restriction of the trajectory to [0, T ], then the convergence of (ν(n)(ω,F ))n≥1 is equivalent to that of a more regular sequence. This step is fundamental for the sequel of the proof. Finally, Lemma 4 is needed for the proof of Theorem 2. We show that under some mild conditions on the step and weight sequences, any Markovian weak limit of the sequence (ν(n)(ω,dα))n≥1 is stationary. 4.1. Preliminary lemmas Lemma 1. Let (E,d) be a Polish space and let P(E) denote the set of probability measures on the Borel σ-field B(E), endowed with the weak convergence topology. Let (µ(n)(ω,dα))n≥1 be a sequence of random probabilities defined on Ω×B(E). (a) Assume that there exists µ(∞) ∈ P(E) such that for every bounded Lipschitz func- tion F :E→R, µ(n)(ω,F ) n→+∞−→ µ(∞)(F ) a.s. (24) Then, a.s., (µ(n)(ω,dα))n≥1 converges weakly to µ (∞) on P(E). (b) Let U be a subset of P(E). Assume that for every sequence (Fk)k≥1 of Lipschitz and bounded functions, a.s., for every subsequence (µ(φω(n))(ω,dα)), there exists a sub- sequence (µ(φω◦ψω(n))(ω,dα)) and a U -valued random probability µ(∞)(ω,dα) such that for every k ≥ 1, µ(ψω◦φω(n))(ω,Fk) n→+∞−→ µ(∞)(ω,Fk) a.s. (25) Then, (µ(n)(ω,dα))n≥1 is a.s. tight with weak limits in U . Proof. We do not give a detailed proof of the next lemma, which is essentially based on the fact that in a separable metric space (E,d), one can build a sequence of bounded Lipschitz functions (gk)k≥1 such that for any sequence (µn)n≥1 of probability measures on B(E), (µn)n≥1 weakly converges to a probability µ if and only if the convergence holds along the functions gk, k ≥ 1 (see Parthasarathy [22], Theorem 6.6, page 47 for a very similar result). � For every n≥ 0, for every T > 0, we introduce τ(n,T ) defined by τ(n,T ) := min{k ≥ 0,N(k,T )≥ n}=min{k ≤ n,Γk + T ≥ Γn}. (26) Approximation of the distribution of a stationary Markov process 159 Note that for k ∈ {0, . . . , τ(n,T )− 1}, {X̄(k)t ,0≤ t≤ T } is �FΓn -measurable and T − γτ(n,T )−1 ≤ Γn − Γτ(n,T ) ≤ T. Lemma 2. Assume (C3,ε) with ε < 1. Let F :D(R+,R d)→R be a Sk-continuous func- tional. Let (Gk) be a filtration such that F̄Γk ⊂ Gk for every k ≥ 1. Then, for any T > 0: (a) if FT (defined by (7)) is bounded, ηk(FT (X̄ (k−1))−E[FT (X̄(k−1))/Gk−1]) n→+∞−→ 0 a.s.; (27) (b) if FT is not bounded, (27) holds if there exists V :Rd→R+, satisfying H(s, ε) for some s≥ 2, such that |FT (α)| ≤C sup0≤t≤T V(αt) for every α ∈D(R+,Rd); furthermore, ν(n)(ω,FT )<+∞ a.s. (28) Proof. We prove (a) and (b) simultaneously. Let Υ(k) be defined by Υ(k) = FT (X̄ (k)). We have (k−1) −E[Υ(k−1)/Gk−1]) (k−1) −E[Υ(k−1)/Gn]) (29) ηk(E[Υ (k−1)/Gn]−E[Υ(k−1)/Gk−1]). (30) We have to prove that the right-hand side of (29) and (30) tend to 0 a.s. when n→+∞. We first focus on the right-hand side of (29). From the very definition of τ(n,T ), we have that {X̄(k)t ,0≤ t≤ T } is F̄Γn -measurable for k ∈ {0, . . . , τ(n,T )− 1}. Hence, since FT is σ(πs,0≤ s≤ T )-measurable and F̄Γn ⊂ Gn, it follows that Υ(k) is Gn-measurable and that Υ(k) = E[Υ(k)/Gn] for every k ≤ τ(n,T )− 1. Then, if FT is bounded, we derive from (C3,ε) that (k−1) −E[Υ(k−1)/Gn]) ≤ 2‖FT ‖sup k=τ(n,T )+1 k=τ(n,T )+1 H1−εn (Γn − Γτ(n,T )) 160 G. Pagès and F. Panloup ≤ C(T ) H1−εn n→+∞−→ 0 a.s., where we used the fact that (Hn)n≥1 and (γn)n≥1 are non-decreasing and non-increasing sequences, respectively. Assume, now, that the assumptions of (b) are fulfilled with V satisfying H(s, ε) for some s≥ 2 and ε < 1. By the Borel–Cantelli-like argument, it suffices to show that k=τ(n,T )+1 (k−1) −E[Υ(k−1)/Gn]) <+∞. (31) Let us prove (31). Let ak := η (s−1)/s k and bk(ω) := η (k−1) − E[Υ(k−1)/Gn]). The Hölder inequality applied with p̄= s/(s− 1) and q̄ = s yields k=τ(n,T )+1 akbk(ω) k=τ(n,T )+1 )s−1( n k=τ(n,T )+1 ηk|Υ(k−1) −E[Υ(k−1)/Gn]|s Now, since FT (α) ≤ sup0≤t≤T V(α), it follows from the Markov property and from H(s, ε)(i) that E[|FT (X̄(k))|s/F̄Γk ]≤CE 0≤t≤T Vs(X̄(k)t )/F̄Γk ≤CTVs(X̄Γk). Then, using the two preceding inequalities and (C3,ε) yields k=τ(n,T )+1 (k−1) −E[Υ(k−1)/Gn]) k=τ(n,T )+1 )s−1( n k=τ(n,T )+1 ηkE[Vs(X̄Γk−1)] k=τ(n,T )+1 k=τ(n,T )+1 Vs(X̄Γk−1) k=τ(n,T )+1 t∈[0,S(n,T )] Vs(X̄τ(n,T )t ) where S(n,T ) = Γn−1 − Γτ(n,T ) and C does not depend n. By the definition of τ(n,T ), S(n,T )≤ T . Then, again using H(s, ε)(i) yields k=τ(n,T ) (k−1) −E[Υ(k−1)/Gn]) s(1−ε) E[Vs(X̄(τ(n,T )))]. Approximation of the distribution of a stationary Markov process 161 Since n 7→ N(n,T ) is an increasing function, n 7→ τ(n,T ) is a non-decreasing function and Card{n, τ(n,T ) = k}=∆N(k+1, T ) :=N(k+1, T )−N(k,T ). Then, since n 7→Hn increases, a change of variable yields k=τ(n,T )+1 (k−1) −E[Υ(k−1)/Gn]) ∆N(k,T ) s(1−ε) E[Vs(X̄Γk−1)]<+∞, by H(s, ε)(iv). Second, we prove that (30) tends to 0. For every n≥ 1, we let (E[Υ(k−1)/Gn]−E[Υ(k−1)/Gk−1]). (32) The process (Mn)n≥1 is a (Gn)-martingale and we want to prove that this process is L2-bounded. Set Φ(k,n) = E[FT (X̄ (k))/Gn]− E[FT (X̄(k))/Gk]. Since FT is σ(πs,0 ≤ s ≤ T )-measurable, the random variable Φ(k,n) is F̄ΓN(k,T) -measurable. Then, for every i ∈ {N(k,T ), . . . , n}, Φ(k,n) is Gi-measurable so that E[Φ(i,n)Φ(k,n)] =E[Φ(k,n)E[Φ(i,n)/Gi]] = 0. It follows that E[M2n] = E[(Φ(k−1,n)) ] + 2 N(k−1,T )∧n i=k+1 E[Φ(i−1,n)Φ(k−1,n)]. (33) Then, E[M2n] ≤ E[(Φ(k−1,n)) ] + 2 N(k−1,T ) i=k+1 E[Φ(i−1,n)Φ(k−1,n)] H2−εk E[(Φ(k−1,n)) ] (34) H2−εk N(k−1,T ) i=k+1 γi sup E[Φ(i−1,n)Φ(k−1,n)] 162 G. Pagès and F. Panloup where, in the second inequality, we used assumption (C3,ε) and the decrease of i 7→ 1/H1−εi . Hence, if FT is bounded, using the fact that ∑N(k−1,T ) i=k+1 γi ≤ T yields E[M2n]≤C H2−εk H2−ε1 <+∞ (35) since ε < 1. Assume, now, that the assumptions of (b) hold and let FT be dominated by a function V satisfying H(s, ε). By the Markov property, the Jensen inequality and H(s, ε)(i), E[(Φ(k,n)) 0≤t≤T V2(X̄(k)t )/F̄Γk ≤CTE[V2(X̄Γk)]. We then derive from the Cauchy–Schwarz inequality that for every n, k ≥ 1, for every i ∈ {k, . . . ,N(k,T )}, |E[Φ(i,n)Φ(k,n)]| ≤C E[V2(X̄Γi)] E[V2(X̄Γk)]≤C sup t∈[0,T ] E[V2(X̄(k)t )]≤CE[V2(X̄Γk)], where, in the last inequality, we once again used H(s, ε)(i). It follows that E[M2n]≤C H2−εk E[V2(X̄Γk−1)]<+∞, by H(s, ε)(iii). Therefore, (34) is finite and (Mn) is bounded in L 2. Finally, we derive from the Kronecker lemma that ηk(E[FT (X̄ (k−1))/Gn]−E[FT (X̄(k−1))/Gk−1]) n→+∞−→ 0 a.s. As a consequence, supn≥1 ν (n)(ω,FT )<+∞ a.s. if and only if E[FT (X̄ (k−1))/Fk−1]<+∞ a.s. This last property is easily derived from H(s, ε)(i) and (ii). This completes the proof. � Lemma 3. (a) Assume (C1) and let x0 ∈Rd. We then have limx→x0 E[d(Xx,Xx0)] = 0. In particular, for every bounded Lispchitz (w.r.t. the distance d) functional F :D(R+,R R, the function ΦF defined by ΦF (x) = E[F (Xx)] is a (bounded) continuous function on (b) Assume (C2). For every compact set K ⊂Rd, E[d(Y n,x,Xx)] n→+∞−→ 0. (36) Approximation of the distribution of a stationary Markov process 163 Set ΦFn (x) = E[F (Y n,x)]. Then, for every bounded Lispchitz functional F :D(R+,R d)→R, |ΦF (x)−ΦFn (x)| n→+∞−→ 0 for every compact set K ⊂Rd. (37) Proof. (a) By the definition of d, for every α, β ∈D(R+,Rd) and for every T > 0, d(α,β)≤ 1∧ sup 0≤t≤T |α(t)− β(t)| + e−T . (38) It easily follows from assumption (C1) and from the dominated convergence theorem limsup E[d(Xx,Xx0)]≤ e−T for every T > 0. Letting T →+∞ implies that limx→x0 E[d(Xx,Xx0)] = 0. (b) We deduce from (38) and from assumption (C2) that for every compact setK ⊂Rd, for every T > 0, limsup E[d(Y n,x,Xx)]≤ e−T . Letting T →+∞ yields (36). � Lemma 4. Assume that (ηn)n≥1 and (γn) satisfy (C3,ε) with ε < 1 and (14). Then: (i) for every t≥ 0, for every bounded continuous function f :Rd→R, t (ω, f)− ν 0 (ω, f) n→+∞−→ 0 a.s.; (ii) if, moreover, a.s., every weak limit ν(∞)(ω,dα) of (ν(n)(ω,dα))n≥1 is the dis- tribution of a Markov process with semigroup (Qωt )t≥0, then, a.s., ν (∞)(ω,dα) is the distribution of a stationary process. Proof. (i) Let f :Rd →R be a bounded continuous function. Since X̄(k)t = X̄ΓN(k,t) , we t (ω, f)− ν 0 (ω, f) = ηk(f(X̄ΓN(k−1,t))− f(X̄Γk−1)). From the very definition of N(n,T ) and τ(n,T ), one checks that N(k − 1, T )≤ n− 1 if and only if τ(n,T )≥ k. Then, ηkf(X̄Γk−1) = τ(n,t) ηN(k−1,t)+1f(X̄ΓN(k−1,t)) ηkf(X̄Γk−1)1{k−1/∈N({0,...,n},t)}. 164 G. Pagès and F. Panloup It follows that t (ω, f)− ν 0 (ω, f) = τ(n,t) (ηk − ηN(k−1,t)+1)f(X̄ΓN(k−1,t)) τ(n,t)+1 ηkf(X̄ΓN(k−1,t)) ηkf(X̄Γk−1)1{k−1/∈N({0,...,n},t)}. Then, since f is bounded and since ηk1{k−1/∈N({0,...,n},t)} = τ(n,t) ηN(k−1,t)+1 τ(n,t) |ηk − ηN(k−1,t)+1|+ k=τ(n,t)+1 we deduce that |ν(n)t (ω, f)− ν 0 (ω, f)| ≤ 2‖f‖∞ τ(n,t) |ηk − ηN(k−1,t)+1|+ k=τ(n,t)+1 Hence, we have to show that the sequences of the right-hand side of the preceding in- equality tend to 0. On the one hand, we observe that |ηk − ηN(k−1,t)+1| ≤ N(k−1,T )+1 ℓ=k+1 |ηℓ − ηℓ−1| ≤ max ℓ≥k+1 |∆ηℓ| N(k−1,T )+1 Using the fact that ∑N(k−1,T )+1 ℓ=k γℓ ≤ T + γ1 and condition (14) yields τ(n,t) |ηk − ηN(k−1,t)+1| n→+∞−→ 0. On the other hand, by (C3,ε), we have k=τ(n,T )+1 H1−εn k=τ(n,T )+1 H1−εn n→+∞−→ 0 a.s., which completes the proof of (i). Approximation of the distribution of a stationary Markov process 165 (ii) Let Q+ denote the set of non-negative rational numbers. Let (fℓ)ℓ≥1 be an every- where dense sequence in CK(Rd) endowed with the topology of uniform convergence on compact sets. Since Q+ and (fℓ)ℓ≥1 are countable, we derive from (i) that there exists Ω̃⊂Ω such that P(Ω̃) = 1 and such that for every ω ∈ Ω̃, every t ∈Q+ and every ℓ≥ 1, t (ω, fℓ)− ν 0 (ω, fℓ) n→+∞−→ 0. Let ω ∈ Ω̃ and let ν(∞)(ω,dα) denote a weak limit of (ν(n)(ω,dα))n≥1. We have t (ω, fℓ) = ν 0 (ω, fℓ) ∀t ∈Q+ ∀ℓ≥ 1 and we easily deduce that t (ω, f) = ν 0 (ω, f) ∀t ∈R+ ∀f ∈ CK(Rd). Hence, if ν(∞)(ω,dα) is the distribution of a Markov process (Yt) with semigroup (Q t )t≥0, we have, for all f ∈ CK(Rd), Qωt f(x)ν 0 (ω,dx) = f(x)ν 0 (ω,dx) ∀t≥ 0. 0 (ω,dx) is then an invariant distribution for (Yt). This completes the proof. � 4.2. Proof of Theorem 1 Thanks to Lemma 1(a) applied with E =D(R+,R d) and d defined by (6), ν(n)(ω,dα) =⇒ Pν0(dα) a.s.⇐⇒ ν(n)(ω,F ) n→+∞−→ F (x)Pν0 (dx) a.s. (39) for every bounded Lipschitz functional F :D(R+,R d)→ R. Now, consider such a func- tional. By the assumptions of Theorem 1, we know that a.s., (ν 0 (ω,dx))n≥1 converges weakly to ν0. Set Φ F (x) := E[F (Xx)], x ∈Rd. By Lemma 3(a), ΦF is a bounded contin- uous function on Rd. It then follows from (C0,1) that F (X̄ (k−1) n→+∞−→ ΦF (x)ν0(dx) = F (x)Pν0 (dx) a.s. Hence, the right-hand side of (39) holds for F as soon as ηk(F (X̄ (k−1))−ΦF (X̄(k−1)0 )) n→+∞−→ 0 a.s. (40) 166 G. Pagès and F. Panloup Let us prove (40). First, let T > 0 and let FT be defined by (7). By Lemma 2, ηkFT (X̄ (k−1))− 1 ηkE[FT (X̄ (k−1))/F̄Γk−1 ] n→+∞−→ 0 a.s. (41) With the notation of Lemma 3(b), we derive from assumption (C2)(i) that E[FT (X̄ (k−1))/F̄Γk−1 ] = Φ k (X̄ (k−1) Let N ∈N. On one hand, by Lemma 3(b), k (X̄ (k−1) 0 )−ΦFT (X̄ (k−1) 0 ))1{|X̄(k−1) n→+∞−→ 0 a.s. (42) On the other hand, the tightness of (ν 0 (ω,dx))n≥1 on R d yields ψ(ω,N) := sup 0 (ω, (B(0,N) N→+∞−→ 0 a.s. It follows that, a.s., ηk|ΦFTk (X̄ (k−1) 0 )−ΦFT (X̄ (k−1) 0 )|1{|X̄(k−1) ≤ 2‖F‖∞ψ(ω,N) N→+∞−→ 0. Hence, a combination of (42) and (43) yields ∀T > 0 1 k (X̄ (k−1) 0 )−ΦFT (X̄ (k−1) n→+∞−→ 0 a.s. (44) Finally, let (Tℓ)ℓ≥1 be a sequence of positive numbers such that, Tℓ→+∞ when ℓ→+∞. Combining (44) and (41), we obtain that, a.s., for every ℓ≥ 1, limsup ηk(F (X̄ (k−1))−ΦF (X̄(k−1))) ≤ lim sup ηk(F (X̄ (k−1))−FTℓ(X̄(k−1))) + limsup FTℓ (X̄ (k−1) 0 )−ΦF (X̄ (k−1) Approximation of the distribution of a stationary Markov process 167 By the definition of d, |F − FTℓ | ≤ e−Tℓ . Then, a.s., limsup ηk(F (X̄ (k−1))−ΦF (X̄(k−1)0 )) ≤ 2e−Tℓ ∀ℓ≥ 1. Letting ℓ→+∞ implies (40). The generalization to non-bounded functionals in Theorem 1 is then derived from (28) and from a uniform integrability argument. 4.3. Proof of Theorem 2 (i) We want to prove that the conditions of Lemma 1(b) are fulfilled. Since (ν 0 (ω,dx))n≥1 is supposed to be a.s. tight, one can check that for every bounded Lipschitz functional F :D(R+,R d)→R, (40) is still valid. Then, let (Fℓ)ℓ≥1 be a sequence of bounded Lipschitz functionals. There exists Ω̃⊂Ω with P(Ω̃) = 1 such that for every ω ∈ Ω̃, (ν(n)0 (ω,dx))n≥1 is tight and ηk(Fℓ(X̄ (k−1)(ω))−ΦFℓ(X̄(k−1)0 (ω))) n→+∞−→ 0 ∀ℓ≥ 1. (45) Let ω ∈ Ω̃ and let φω :N 7→N be an increasing function. As (ν(φω(n))0 (ω,dx))n≥1 is tight, there exists a convergent subsequence (ν (φω◦ψω(n)) 0 (ω,dx))n≥1. We denote its weak limit by ν∞. Since Φ Fℓ is continuous for every ℓ≥ 1 (see Lemma 3(a)), (φω◦ψω(n)) 0 (ω,Φ n→+∞−→ ν∞(ΦFℓ) = Fℓ(α)Pν∞(dα) ∀ℓ≥ 1. We then derive from (45) that for every ℓ≥ 1 ν(φω◦ψω(n))(ω,Fℓ) n→+∞−→ Fℓ(α)Pν∞(dα). It follows that the conditions of Lemma 1(b) are fulfilled with U = {Pµ, µ ∈ I}, where µ ∈P(Rd),∃ω ∈ Ω̃ and an increasing function φ :N 7→N, µ= lim ν(φ(n))(ω,dα) Hence, by Lemma 1(b), we deduce that (ν(n)(ω,dα))n≥1 is a.s. tight with U -valued limits. Finally, Theorem 2(ii) is a consequence of condition (14) and Lemma 4(ii). 168 G. Pagès and F. Panloup 5. Path-dependent option pricing in stationary stochastic volatility models In this section, we propose a simple and efficient method to price options in stationary stochastic volatility (SSV) models. In most stochastic volatility (SV) models, the volatil- ity is a mean reverting process. These processes are generally ergodic with a unique invariant distribution (the Heston model or the BNS model for instance (see below) but also the SABR model (see Hagan et al. [8]), . . .). However, they are usually considered in SV models under a non-stationary regime, starting from a deterministic value (which usually turns out to be the mean of their invariant distribution). However, the instanta- neous volatility is not easy to observe on the market since it is not a traded asset. Hence, it seems to be more natural to assume that it evolves under its stationary regime than to give it a deterministic value at time 0.4 From a purely calibration viewpoint, considering an SV model in its SSV regime will not modify the set of parameters used to generate the implied volatility surface, although it will modify its shape, mainly for short maturities. This effect can in fact be an asset of the SSV approach since it may correct some observed drawbacks of some models (see, e.g., the Heston model below). From a numerical point of view, considering SSV models is no longer an obstacle, es- pecially when considering multi-asset models (in the unidimensional case, the stationary distribution can be made more or less explicit like in the Heston model; see below) since our algorithm is precisely devised to compute by simulation some expectations of func- tionals of processes under their stationary regime, even if this stationary regime cannot be directly simulated. As a first illustration (and a benchmark) of the method, we will describe in detail the algorithm for the pricing of Asian options in a Heston model. We will then show in our numerical results to what extent it differs, in terms of smile and skew, from the usual SV Heston model for short maturities. Finally, we will complete this section with a numerical test on Asian options in the BNS model where the volatility is driven by a tempered stable subordinator. Let us also mention that this method can be applied to other fields of finance like interest rates, and commodities and energy derivatives where mean-reverting processes play an important role. 4When one has sufficiently close observations of the stock price, it is in fact possible to derive a rough idea of the size of the volatility from the variations of the stock price (see, e.g., Jacod [10]). Then, using this information, a good compromise between a deterministic initial value and the stationary case may be to assume that the distribution µ0 of the volatility at time 0 is concentrated around the estimated value (see Section 2.2 for application of our algorithm in this case). Approximation of the distribution of a stationary Markov process 169 5.1. Option pricing in the Heston SSV model We consider a Heston stochastic volatility model. The dynamic of the asset price process (St)t≥0 is given by S0 = s0 and dSt = St(rdt+ (1− ρ2)vt dW 1t + ρ vt dW dvt = k(θ− vt) dt+ ς vt dW where r denotes the interest rate, (W 1,W 2) is a standard two-dimensional Brownian motion, ρ ∈ [−1,1] and k, θ and ς are some non-negative numbers. This model was introduced by Heston in 1993 (see Heston [9]). The equation for (vt) has a unique (strong) pathwise continuous solution living in R+. If, moreover, 2kθ > ς 2, then (vt) is a positive process (see Lamberton and Lapeyre [11]). In this case, (vt) has a unique invariant probability ν0. Moreover, ν0 = γ(a, b) with a= (2k)/ς 2 and b= (2kθ)/ς2. In the following, we will assume that (vt) is in its stationary regime, that is, that L(v0) = ν0. 5.1.1. Option price and stationary processes Using our procedure to price options in this model naturally needs to express the option price as the expectation of a functional of a stationary stochastic process. Näıve method. (may work) Since (vt)t≥0 is stationary, the first idea is to express the option price as the expectation of a functional of (vt)t≥0: by Itô calculus, we have St = s0 exp rt− 1 vs ds vs dW 1− ρ2 vs dW . (46) Since vs dW s =Λ(t, (vt)) := vt − v0 − kθt+ k vs ds it follows by setting Mt = vs dW s that St =Ψ(t, (vs), (Ms)), (47) where Ψ is given for every t≥ 0, u and w ∈ C(R+,R) by Ψ(t, u,w) = s0 exp rt− 1 u(s) ds + ρΛ(t, u) + 1− ρ2w(t) Then, let F :C(R+,R) → R be a non-negative measurable functional. Conditioning by FW 2T yields E[FT ((St)t≥0)] = E[F̃T ((vt)t≥0)], 170 G. Pagès and F. Panloup where, for every u ∈ C(R+,R), F̃T (u) = E t, u, u(s) dW 1s For some particular options such as the European call or put (thanks to the Black– Scholes formula), the functional F̃ is explicit. In those cases, this method seems to be very efficient (see Panloup [20] for numerical results). However, in the general case, the computation of F̃ will need some Monte Carlo methods at each step. This approach is then very time-consuming in general – that is why we are going to introduce another representation of the option as a functional of a stationary process. General method. (always works) We express the option premium as the expectation of a functional of a two-dimensional stationary stochastic process. This method is based on the following idea. Even though (vt,Mt) is not stationary, (St) can be expressed as a functional of a stationary process (vt, yt). Indeed, consider the following SDE given by dyt =−yt dt+ vt dW dvt = k(θ− vt) dt+ ς vt dW First, one checks that the SDE has a unique strong solution and that assumption (S1) is fulfilled with V (x1, x2) = 1+ x 2. This ensures the existence of an invariant distribu- tion ν̃0 for the SDE (see, e.g., Pagès [17]). Then, since (vt) is positive and has a unique invariant distribution, the uniqueness of the invariant distribution follows. Then, assume that L(y0, v0) = ν̃0. Since (vt,Mt) = (vt, yt − y0 + ys ds), we have, for every positive measurable functional F :C(R+,R)→R, E[FT ((St)t≥0)] = E[FT ((ψ(t, vt,Mt))t≥0)] = Eν̃0 t, vt, yt − y0 + ys ds where Pν̃0 is the stationary distribution of the process (vt, yt). Every option price can then be expressed as the expectation of an explicit functional of a stationary process. We will develop this second general approach in the numerical tests below. Remark 9. The idea of the second method holds for every stochastic volatility model for which (St) can be written as follows: St =Φ t, vt, hi(|vs|) dY is , (50) where, for every i ∈ {1, . . . , p}, hi :R+ →R is a positive function such that hi(x) = o(|x|) as |x| → +∞, (Y it ) is a square-integrable centered Lévy process and (vt) is a mean reverting stochastic process solution to a Lévy driven SDE. Approximation of the distribution of a stationary Markov process 171 In some complex models, showing the uniqueness of the invariant distribution may be difficult. In fact, it is important to note at this stage that the uniqueness of the invariant distribution for the couple (vt, yt) is not required. Indeed, by construction, the local martingale (Mt) does not depend on the choice of y0. It follows that if L(y0, v0) = µ̃, with µ̃ constructed such that L(v0) = ν0, (49) still holds. This implies that it is only necessary that uniqueness holds for the invariant distribution of the stochastic volatility process. 5.1.2. Numerical tests on Asian options We recall that (vt) is a Cox–Ingersoll–Ross process. For this type of processes, it is well known that the genuine Euler scheme cannot be implemented since it does not preserve the non-negativity of the (vt). That is why some specific discretization schemes have been studied by several authors (Alfonsi [1], Deelstra and Delbaen [5] and Berkaoui et al. [4, 6]). In this paper, we consider the scheme studied by the last authors in a decreasing step framework. We denote it by (v̄t). We set v̄0 = x > 0 and v̄Γn+1 = |v̄Γn + kγn+1(θ− v̄Γn) + ς v̄Γn(W −W 2Γn)|. We also introduce the stepwise constant Euler scheme (ȳt) of (yt)t≥0 defined by ȳΓn+1 = ȳΓn − γn+1ȳΓn + v̄Γn(W̃ − W̃ 1Γn), ȳ0 = y ∈R Denote by (v̄ t ) and (ȳ t ) the shifted processes defined by v̄ t := v̄Γk+t and ȳ ȳΓk+t, and let (ν (n)(ω,dα))n≥1 be the sequence of empirical measures defined by ν(n)(ω,dα) = ηk1{(v̄(k−1),ȳ(k−1))∈dα}. The specificity of both the model and the Euler scheme implies that Theorems 1 and 2 cannot be directly applied here. However, a specific study using the fact that (9) holds for every compact set of R∗+ ×R when 2kθ/ς2 > 1+ 2 6/ς (see Theorem 2.2 of Berkaoui et al. [4] and Remark 9) shows that ν(n)(ω,dα) =⇒ Pν̃0(dα) a.s. when 2kθ/ς2 > 1+ 2 6/ς . Details are left to the reader. Let us now state our numerical results obtained for the pricing of Asian options with this discretization. We denote by Cas(ν0,K,T ) and Pas(ν0,K,T ) the Asian call and put prices in the SSV Heston model. We have Cas(ν0,K,T ) = e Ss ds−K 172 G. Pagès and F. Panloup Pas(ν0,K,T ) = e K − 1 Ss ds With the notation of (49), approximating Cas(ν0,K,T ) and Pas(ν0,K,T ) by our proce- dure needs to simulate the sequences (Cnas)n≥1 and (P as)n≥1 defined by Cnas = Ψ(s, v̄(k−1), M̄ (k−1)) ds−K Pnas = K − 1 Ψ(s, v̄(k−1), M̄ (k−1)) ds These sequences can be computed by the method developed in Section 1.3. Note that the specific properties of the exponential function and the linearity of the integral imply that ( Ψ(t, v̄(n−1), M̄ (n−1)) ds) can be computed quasi-recursively. Let us state our numerical results for the Asian call with parameters s0 = 50, r = 0.05, T = 1, ρ= 0.5, θ = 0.01, ς = 0.1, k = 2. We also assume that K ∈ {44, . . . ,56} and choose the following steps and weights: γn = ηn = n −1/3. In Table 1, we first state the reference value for the Asian call price obtained for N = 108 iterations. In the two following lines, we state our results for N = 5.104 and N = 5.105 iterations. Then, in the last lines, we present the numerical results obtained Table 1. Approximation of the Asian call price K 44 45 46 47 48 49 50 Asian call (ref.) 6.92 5.97 5.04 4.12 3.25 2.46 1.78 N = 5 · 104 6.89 6.07 5.07 4.13 3.18 2.49 1.77 N = 5 · 105 6.90 6.02 5.00 4.11 3.24 2.46 1.79 N = 5 · 104 (CP parity) 6.92 5.96 5.04 4.13 3.26 2.46 1.78 N = 5 · 105 (CP parity) 6.92 5.97 5.04 4.12 3.25 2.47 1.78 K 51 52 53 54 55 56 Asian call (ref.) 1.23 0.82 0.53 0.33 0.21 0.12 N = 5 · 104 1.21 0.81 0.51 0.34 0.22 0.11 N = 5 · 105 1.23 0.82 0.53 0.33 0.21 0.13 N = 5 · 104 (CP parity) 1.23 0.82 0.53 0.31 0.21 0.12 N = 5 · 105(CP parity) 1.23 0.82 0.53 0.33 0.21 0.13 Approximation of the distribution of a stationary Markov process 173 using the call-put parity Cas(ν0,K,T )− Pas(ν0, S0,K,T ) = (1− e−rT )−Ke−rT (52) as a means of variance reduction. The computation times for N = 5.104 and N = 5.105 (using MATLAB with a Xeon 2.4 GHz processor) are about 5 s and 51 s, respectively. In particular, the complexity is quasi-linear and the additional computations needed when we use the call-put parity are negligible. 5.2. Implied volatility surfaces of Heston SSV and SV models Given a particular pricing model (with initial value s0 and interest rate r) and its asso- ciated European call prices denoted by Ceur(K,T ), we recall that the implied volatility surface is the graph of the function (K,T ) 7→ σimp(K,T ), where σimp(K,T ) is defined for every maturity T > 0 and strike K as the unique solution of CBS(s0,K,T, r, σimp(K,T )) =Ceur(K,T ), where CBS(s0,K,T, r, σ) is the price of the European call in the Black–Scholes model with parameters s0, r and σ. When Ceur(K,T ) is known, the value of σimp(K,T ) can be numerically computed using the Newton method or by dichotomy if the first method is not convergent. In this last part, we compare the implied volatility surfaces induced by the SSV and SV Heston models where we suppose that the initial value of (vt) in the SV Heston model is the mean of the invariant distribution, that is, we suppose that v0 = θ. 5 We also assume that the parameters are those of (51), except the correlation coefficient ρ. In Figures 1 and 2, the volatility curves obtained when T = 1 are depicted, whereas in Figures 3 and 4, we set the strikeK atK = 50 and let the time vary. These representations show that when the maturity is long, the differences between the SSV and SV Heston models vanish. This is a consequence of the convergence of the stochastic volatility to its stationary regime when T →+∞. The main differences between these models then appear for short maturities. That is why we complete this part by a representation of the volatility curve when T = 0.1 for ρ= 0 and ρ= 0.5 in Figures 5 and 6, respectively. We observe that for short maturities, the volatility smile is more curved and the skew is steeper. These phenomena seem interesting for calibration since one well-known drawback of the standard Heston model is that it can have overly flat volatility curves for short maturities. 5.3. Numerical tests on Asian options in the BNS SSV model The BNS model introduced in Barndorff-Nielsen and Shephard [3] is a stochastic volatility model where the volatility process is a Lévy-driven positive Ornstein–Uhlenbeck process. 5This choice is the most usual in practice. 174 G. Pagès and F. Panloup Figure 1. ρ= 0, K 7→ σimp(K,1). The dynamic of the asset price (St) is given by St = S0 exp(Xt), dXt = (r− 12vt) dt+ vt dWt + ρdZt, ρ≤ 0, dvt = −µvt dt+dZt, µ > 0, Figure 2. ρ= 0.5, K 7→ σimp(K,1). Approximation of the distribution of a stationary Markov process 175 Figure 3. ρ= 0, T 7→ σimp(50, T ). where (Zt) is a subordinator without drift term and Lévy measure π. In the following, we assume that (Zt) is a tempered stable subordinator, that is, that π(dy) = 1{y>0} c exp(−λy) dy, c > 0, λ > 0, α∈ (0,1). As in the Heston model, we want to use our algorithm as a way of option pricing when the stochastic volatility evolves under its stationary regime and test it on Asian options using the method described in detail in Section 5.1. This model does not require a specific Figure 4. ρ= 0.5, T 7→ σimp(50, T ). 176 G. Pagès and F. Panloup Figure 5. ρ= 0, T 7→ σimp(50, T ). discretization and the approximate Euler scheme (P) (see Section 3.2) relative to (vt) can be implemented using the rejection method. In Table 2, we present our numerical results obtained for the following choices of parameters, steps and weights: ρ=−1, λ= µ= 1, c= 0.01, α= 1 , γn = ηn = n −1/3. The computation times forN = 5.104 andN = 5.105 are about 8.5 s and 93 s, respectively. Note that for this model, the convergence seems to be slower because of the approximation of the jump component. Figure 6. ρ= 0.5, T 7→ σimp(50, T ). Approximation of the distribution of a stationary Markov process 177 Table 2. Approximation of the Asian call price in the BNS model K 44 45 46 47 48 49 50 Asian call (ref.) 6.75 5.83 4.93 4.05 3.18 2.35 1.57 N = 5 · 104 6.83 5.91 5.01 4.10 3.22 2.35 1.51 N = 5 · 105 6.78 5.86 4.96 4.06 3.19 2.34 1.52 N = 5 · 104 (CP parity) 6.76 5.85 4.94 4.07 3.20 2.29 1.51 N = 5 · 105 (CP parity) 6.75 5.83 4.93 4.04 3.17 2.32 1.54 K 51 52 53 54 55 56 Asian call (ref.) 0.91 0.55 0.39 0.29 0.23 0.18 N = 5 · 104 0.77 0.46 0.33 0.27 0.22 0.19 N = 5 · 105 0.79 0.48 0.34 0.27 0.21 0.17 N = 5 · 104 (CP parity) 0.79 0.47 0.37 0.27 0.23 0.19 N = 5 · 105(CP parity) 0.83 0.50 0.36 0.28 0.22 0.17 Acknowledgement The authors would like to thank Vlad Bally for interesting comments on the paper. References [1] Alfonsi, A. (2005). On the discretization schemes for the CIR (and Bessel squared) pro- cesses. Monte Carlo Methods Appl. 11 355–384. MR2186814 [2] Asmussen, S. and Rosinski, J. (2001). Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Probab. 38 482–493. MR1834755 [3] Barndorff-Nielsen, O.E. and Shephard, N. (2001). Modelling by Lévy processes for financial economics. In Lévy Processes 283–318. Boston: Birkhäuser. MR1833702 [4] Berkaoui, A., Bossy, M. and Diop, A. (2008). Euler scheme for SDE’s with non-Lipschitz diffusion coefficient: Strong convergence. ESAIM Probab. Statist. 12 1–11. MR2367990 [5] Deelstra, G. and Delbaen, F. (1998). Convergence of discretized stochastic (interest rate) processes with stochastic drift term. Appl. Stochastic Models Data Anal. 14 77–84. MR1641781 [6] Diop, A. (2003). Sur la discrétisation et le comportement à petit bruit d’EDS unidimension- nelles dont les coefficients sont à dérivées singulières. Ph.D. thesis, Univ. Nice Sophia Antipolis. [7] Ethier, S. and Kurtz, T. (1986). Markov Processes, Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: Wiley. MR0838085 [8] Hagan, D., Kumar, D., Lesniewsky, A. and Woodward, D. (2002). Managing smile risk. Wilmott Magazine 9 84–108. [9] Heston, S. (1993). A closed-form solution for options with stochastic volatility with appli- cations to bond and currency options. Review of Financial Studies 6 327–343. [10] Jacod, J. (2008). Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Process. Appl. 118 517–559. MR2394762 http://www.ams.org/mathscinet-getitem?mr=2186814 http://www.ams.org/mathscinet-getitem?mr=1834755 http://www.ams.org/mathscinet-getitem?mr=1833702 http://www.ams.org/mathscinet-getitem?mr=2367990 http://www.ams.org/mathscinet-getitem?mr=1641781 http://www.ams.org/mathscinet-getitem?mr=0838085 http://www.ams.org/mathscinet-getitem?mr=2394762 178 G. Pagès and F. Panloup [11] Lamberton, D. and Lapeyre, B. (1996). Introduction to Stochastic Calculus Applied to Fi- nance. London: Chapman and Hall/CRC. MR1422250 [12] Lamberton, D. and Pagès, G. (2002). Recursive computation of the invariant distribution of a diffusion. Bernoulli 8 367–405. MR1913112 [13] Lamberton, D. and Pagès, G. (2003). Recursive computation of the invariant distribution of a diffusion: The case of a weakly mean reverting drift. Stoch. Dynamics 4 435–451. MR2030742 [14] Lemaire, V. (2007). An adaptive scheme for the approximation of dissipative systems. Stochastic Process. Appl. 117 1491–1518. MR2353037 [15] Lemaire, V. (2005). Estimation numérique de la mesure invariante d’un processus de diffu- sion. Ph.D. thesis, Univ. Marne-La Vallée. [16] Pagès, G. (1985). Théorèmes limites pour les semi-martingales. Ph.D. thesis, Univ. Paris [17] Pagès, G. (2001). Sur quelques algorithmes récursifs pour les probabilités numériques. ESAIM Probab. Statist. 5 141–170. MR1875668 [18] Panloup, F. (2008). Recursive computation of the invariant measure of a SDE driven by a Lévy process. Ann. Appl. Probab. 18 379–426. MR2398761 [19] Panloup, F. (2008). Computation of the invariant measure of a Lévy driven SDE: Rate of convergence. Stochastic Process. Appl. 118 1351–1384. [20] Panloup, F. (2006). Approximation du régime stationnaire d’une EDS avec sauts. Ph.D. thesis, Univ. Paris VI. [21] Protter, P. (1990). Stochastic Integration and Differential Equations. Berlin: Springer. MR1037262 [22] Parthasarathy, K.R. (1967). Probability Measures on Metric Spaces. New York: Academic Press. MR0226684 Received April 2007 and revised March 2008 http://www.ams.org/mathscinet-getitem?mr=1422250 http://www.ams.org/mathscinet-getitem?mr=1913112 http://www.ams.org/mathscinet-getitem?mr=2030742 http://www.ams.org/mathscinet-getitem?mr=2353037 http://www.ams.org/mathscinet-getitem?mr=1875668 http://www.ams.org/mathscinet-getitem?mr=2398761 http://www.ams.org/mathscinet-getitem?mr=1037262 http://www.ams.org/mathscinet-getitem?mr=0226684 Introduction Objectives and motivations Background and construction of the procedure Simulation of ((n)(,F))n1 General results Weak convergence to the stationary regime Extension to the non-stationary case Application to Brownian diffusions and Lévy-driven SDE's Application to Brownian diffusions Application to Lévy-driven SDE's Proofs of Theorems 1 and 2 Preliminary lemmas Proof of Theorem 1 Proof of Theorem 2 Path-dependent option pricing in stationary stochastic volatility models Option pricing in the Heston SSV model Option price and stationary processes Numerical tests on Asian options Implied volatility surfaces of Heston SSV and SV models Numerical tests on Asian options in the BNS SSV model Acknowledgement References

0704.0336

Influence of Phonon dimensionality on Electron Energy Relaxation

Influence of Phonon dimensionality on Electron Energy Relaxation J. T. Karvonen and I. J. Maasilta Nanoscience Center, Department of Physics, P.O. Box 35, FIN-40014 University of Jyväskylä, Finland. We studied experimentally the role of phonon dimensionality on electron-phonon (e-p) interaction in thin copper wires evaporated either on suspended silicon nitride membranes or on bulk substrates, at sub-Kelvin temperatures. The power emitted from electrons to phonons was measured using sensitive normal metal-insulator-superconductor (NIS) tunnel junction thermometers. Membrane thicknesses ranging from 30 nm to 750 nm were used to clearly see the onset of the effects of two- dimensional (2D) phonon system. We observed for the first time that a 2D phonon spectrum clearly changes the temperature dependence and strength of the e-p scattering rate, with the interaction becoming stronger at the lowest temperatures below ∼ 0.5 K for the 30 nm membranes. PACS numbers: 63.22.+m, 63.20.Kr, 85.85.+j It is an established fact that at sub-Kelvin tempera- tures the thermal coupling between conduction electrons and the lattice becomes very weak [1]. This has signifi- cant implications for the operation of low-temperature detectors and coolers [2], or for any solid-state sys- tems where dissipation and cooling are relevant. Low- temperature electron-phonon (e-p) interaction has been studied widely during the past decades, but mostly only for the case in which the phonons are fully three di- mensional (3D) [3, 4, 5, 6]. However, due to signifi- cant advances in fabrication of thin suspended structures, many practical devices and detectors exist in which the phonons are expected to move freely only within the plane of a membrane, forming a quasi-2D system [7]. The question how the two-dimensionality of the phonon modes influences e-p interaction has been addressed the- oretically for certain cases [8, 9, 10], but no clear exper- imental observation of the effect has been reported to date, although several attempts have been made [11, 12]. In this paper, we show for the first time experimen- tally that the electron-phonon interaction clearly changes depending on the dimensionality of the phonons, as ex- pected from theory. E-p coupling was measured with the help of sensitive NIS tunnel junction thermometry [13], for thin Cu wires on suspended silicon nitride (SiNx) membranes with thickness varying from 30 nm to 750 nm, which spans the transition from 2D to 3D phonons. In addition, samples with identical Cu wires on bulk substrates were also measured for comparison. For the thinnest membranes, the e-p interaction was strengthened in comparison with the bulk samples, and its tempera- ture dependence changed significantly, as is predicted by the theory [8, 9, 10]. The change was large enough to give indirect evidence that the dispersive (ω ∼ k2), flex- ural modes of the membrane likely play a major role in the e-p interaction. In the presence of stress-free boundaries, the bulk transversal and longitudinal phonon modes (with sound velocities ct and cl, respectively) couple to each other and form a new set of eigenmodes, which in the case of a suspended membrane are known as the horizontal shear modes (h), and symmetric (s) and antisymmet- ric (a) Lamb modes [14]. The frequencies ω for the h modes are simply ω = ct + (mπ/d)2, where k‖ is the wave vector component parallel to the membrane sur- faces, d is the membrane thickness and the integer m is the branch number. However, the dispersion relations of the s and a Lamb modes cannot be given in a closed an- alytical form, but have to be calculated numerically. The lowest three branches, dominant for thin membranes at low temperatures, have low frequency analytical expres- sions: ωh = ctk‖, ωs = csk‖, and ωa = k2‖, where cs = 2ct − c2t )/c is the effective sound velocity of the s mode, and m⋆ = ~ − c2t )/3c is an effective mass for the a-mode ”particle”. This lowest a- mode with its quadratic dispersion is mostly responsible for the non-trivial behavior of the e-p interaction [9, 10]. Note that already a single free surface affects the modes [15] and the e-p interaction [16], as the bulk modes cou- ple and form another new set of eigenstates, including the surface localized Rayleigh-mode. Thus, the widely observed result for e-p power flow P = ΣV (T 5e − T from a metal volume V with Te the electron and Tp the phonon temperature, is not expected to hold even for thin enough films on bulk substrates. A schematic of the Cu wire samples on suspended sil- icon nitride membranes and the used measuring circuit is shown in Fig. 1. 17 samples were made on either sus- pended membranes or bulk substrates, where nitridized (100) Si wafers with 30, 200 and 750 nm thick low-stress SiNx top layers were used as the substrate for both cases. The suspension of the SiNx membranes (size 600×300 µm2) was achieved by anisotropic backside wet etching of the silicon substate in KOH, and the metallic structures were fabricated using standard e-beam lithography and multi-angle shadow mask evaporation techniques. As the e-p interaction strength is sensitive to the thickness and disorder level of the metal [17], we minimized its effect by evaporating the Cu wires of a specific thickness on all the different substrates simultaneously. Ultrathin Cu layers (t=14-30 nm) were used to strengthen the effect of the thin membranes. The oxide layer forming the tun- nel junction barriers was produced by thermal oxidation of Al. Table I presents the essential dimensions of the http://arxiv.org/abs/0704.0336v2 samples discussed in this paper, measured by scanning electron (SEM) and atomic force (AFM) microscopies. The electron mean free path l was determined from the resistance of the wire at base temperature 60 mK, using the accurately measured dimensions of the wire. TABLE I: Parameters for samples. M= suspended SiNx mem- brane and B= bulk substrate. B6 had an oxidized Si sub- strate. Sample SiNx d Cu t V l τ (0.2K) τ (0.8K) (nm) (nm) [(µm)3] (nm) (µs) (µs) M1 30 14 2.71 5.7 2.6 0.16 B1 30 14 2.46 4.9 7.1 0.030 M2 200 14 2.44 4.6 15.0 0.11 B2 200 18 3.67 4.1 6.4 0.045 M3 30 19 5.50 11.2 2.2 0.30 B3 30 19 4.62 9.8 4.3 0.034 M4 750 22 6.09 10.3 3.1 0.030 B4 750 22 5.87 8.7 3.9 0.013 M5 30 32 6.09 22 1.8 0.31 B5 30 32 5.09 19 2.7 0.038 B6 - 32 7.10 22 1.6 0.031 CuAl Nb/Al FIG. 1: (Color online) A Schematic of the suspended samples and the measuring circuit. Red lines are the normal metal Cu, light gray Al for SINIS-junctions and dark gray Al or Nb for SN-junctions. We used the hot-electron technique [3] to measure the e-p interaction by overheating the electrons by Joule heat power P and measuring the resulting electron tempera- ture Te. All the samples had two electrically isolated Cu normal metal wires next to each other (Fig. 1). The longer wire (L = 500µm) was heated by applying a slowly ramping voltage across the pair of superconducting Nb (or Al) leads in direct metallic contact to Cu, forming SN junctions. These junctions provide excellent electri- cal, but very poor thermal conductance due to Andreev reflection, as the junctions are biased within the super- conducting gap ∆. Thus, due to the lack of outdiffu- sion of electrons and the long length of the wire, input heat is distributed uniformly in the interior of the wire and the electron gas cools dominantly by phonons, in- stead of diffusively [18] or by thermal photons [19]. Since L >> Le−e, the electron-electron scattering length, elec- tron temperature is also well defined without complica- tions from non-equilibrium [20]. In our sample geometry the electron temperature is measured with two additional Al leads forming a NIS tunnel junctions pair (SINIS) in the middle of heated wire, as a function of input Joule power P = IV measured in a four probe configuration. The purpose of the short Cu wire, with additional SI- NIS thermometer on it, is to give an estimate of the local phonon temperature Tp, as the e-p power flow depends on both Te and Tp. The current-biased Al SINIS thermometer is ideally suited to measure temperature below a few Kelvins, [2] due to its high sensitivity (in our DC measurement ∼ 0.1 mK at 0.1 K) and low power dissipation. In addition, for all the data here, the SINIS voltage vs. temperature response follows the BCS theory without fitting param- eters very accurately at least down to ∼ 0.2 K, where typically saturation sets in. This saturation depends on the strength of the e-p interaction (size of thermometer and type of substrate) and the amount of filtering, and thus we conclude that it is most likely caused by external noise heating. For this reason we take the most conser- vative approach and assume that all saturation is caused by it, in which case we can use BCS theory to convert the measured voltage data for all temperatures. Even if the electrons lose their energy overwhelmingly to the phonons in our sample geometry, it is still pos- sible that the measured temperature is not only deter- mined by the e-p interaction. This is because the emitted phonons could be removed so ineffectively from the mem- brane that the phonon transmission becomes a bottleneck for the energy flow. Bulk scattering of phonons at low temperatures is very weak [7], even for thin disordered membranes [21], as is boundary resistance for thin films on bulk substrates [22, 23]. In contrast, almost noth- ing quantitative is known about the boundary resistance between a thin metal film and a thin 2D membrane, or between a thin 2D membrane and a bulk substrate. How- ever, it seems clear that if the combined metal film and membrane thickness is below the thermal wavelength of the phonons, the phonon modes in the two materials are strongly coupled, leading to an effectively non-existent boundary resistance. Hence, if we check that the mem- brane temperature Tp is not too high compared to Te (effective enough hot phonon removal), we can be confi- dent that the measured Te reflects the e-p interaction. Figure 2 shows the main result of the measurements, with Te and Tp plotted vs. the heating power density p = P/V for all membrane thicknesses (30 nm, 200 nm and 750 nm). In addition, data from a few represen- tative bulk samples are shown. Compared to the cor- responding bulk substrate sample (B4), Te of the 750 nm membrane (M4) shows no difference at all, and it effectively behaves as bulk. This is reasonable, because for the 750 nm membrane the estimated dimensional- ity cross-over temperature [24, 25] Tcr = ~ct/(2kBd) is ∼ 30 mK, with ct = 6200 m/s for SiN. The phonon temperatures Tp, however, show a big difference: The 0.1 1 10 100 1000 of M3 T of M2 of M4 T of B1-B6 of M1 of M2 of M4 of B1 and B2 of B4 Heating power density [pW / ( m)3] FIG. 2: (Color online) Measured electron and phonon tem- peratures Te and Tp versus the applied heating power density in log-log-scale. bulk samples show almost no response from the satura- tion value of the thermometer ∼ 190 mK, whereas the membrane phonons heat up measurably, most likely due to the boundary resistance between the membrane and the bulk. Nevertheless, this increase in Tp for all sam- ples is small enough not to influence the e-p interaction. For the 200 nm thick membrane (M2) (Tcr ∼ 110 mK), at low heating power densities [p < 40 pW/(µm)3] the temperature dependence follows the behavior of the bulk sample (B2), although with a difference in the absolute value. This shows that the strength of the e-p coupling weakens compared to the bulk. At higher powers and temperatures (p > 40 pW/(µm)3, where Te > 0.6 K), Te starts to increase more rapidly in the membrane sam- ple, most likely due to the boundary resistance effects. The phonons in the 30 nm thick membrane sample (M1) are expected to be in the 2D limit at low temperatures (Tcr ∼ 0.5K), and a clear sign of this can be seen in Fig. 2 as a strongly different behavior of the measured Te vs. p curve with respect to all other samples. Below ∼ 6 pW/(µm)3 the e-p coupling is notably stronger (Te lower) than in the corresponding bulk (B1) or any other sample, but again at highest temperatures the influence of other effects starts to dominate over the e-p coupling. To study the temperature dependence of the data in Fig. 2 more accurately, we plot the logarithmic deriva- tives d(log p)/d(logTe) in Fig. 3 (a)-(c). For low heat- ing powers (T ne >> T p ) Pe−p ≈ T e , where n is the power law of the e-p interaction, thus in that regime d(log p)/d(logTe) = n. Typically this exponent is n ≈ 5 for thicker (t > 30 nm) metal films on bulk substrates [3, 4, 17], if the disorder in the film is not too strong [26, 27, 28]. From Fig. 3 (a) we first of all see that for the 30 nm membrane sample M1, the difference to the bulk sample B1 is very clear. The M1 data has a 0.1 1 10 100 1000 M1 B1 M2 B2 M4 B4 Heating power density [pW / ( m)3] FIG. 3: (Color online) Numerical logarithmic derivatives of the measured data in Fig. 2. (a) Te data for M1 and B1, (b) Te data for M2 and B2, (c) Te data for M4 and B4. plateau of n ∼ 4.5 between p = 0.1 - 6 pW/(µm)3, while for B1, n continuously decreases from much higher val- ues. Note that the strong increase of d(log p)/d(logTe) below p ∼ 0.1 pW/(µm)3 is caused by the saturation of the Te measurement, and not by the e-p interaction. The point where n starts deviating from n = 4.5 cor- responds to Te ≈ 0.4 K, which is surprisingly consistent with the estimated Tcr ∼ 0.5 K. In contrast, the tempera- ture dependence of the 200 nm membrane (M2) and bulk (B2) samples [Fig. 3 (b)] are identical with each other and with the 30 nm bulk sample (B1), as long as the e-p interaction is dominant (up to 40 pW/(µm)3). The 750 nm membrane (M4) and bulk (B4) samples also give identical values of n [Fig. 3 (c)]. The difference between sample pairs M4,B4 and M2,B2 is caused by the Cu wire thickness, which is expected to influence the temperature dependence strongly [16, 27]. Finally, we discuss the effect of the Cu wire thickness on the measured e-p interaction. The results for the thinnest 30 nm membrane samples, with Cu thickness t = 14,19 and 32 nm are shown in Figs 4 (a) and (c). It is apparent that the metal film thickness has only a minor effect on the e-p interaction on thin membranes, and only influences the boundary resistance in the 3D limit, by increasing its effect for thicker t, as expected. However, for wires on bulk substrates, Figs 4 (b) and (d), the effect of the Cu wire thickness on e-p interaction is more profound. The thinner the Cu film, the more its temperature dependence deviates from n = 5, which, for comparison, is observed for a more typical t = 32 nm Cu wire on oxidized Si (B6). This behavior is qualita- 0.1 1 10 100 1000 0.1 1 10 100 1000 (a) (b) Heating power density [pW /( m)3] FIG. 4: (Color online) (a) Te versus p = P/V for 30 nm mem- brane samples M1,M3,M5. (b) Te versus p for bulk samples, from top to bottom B1 (top), B3, B5 and B6 (bottom). (c) d(log p)/d(log T ) of the data in (a). (d) d(log p)/d(log T ) of the data in (b). From top to bottom: green line B1 (top), magenta B3, blue B5, Red B6 (bottom). In (d) noise has been filtered to help the eye. tively consistent with the predicted effect of the surface phonon modes [16], but could also depend on the disor- der, as the thickening of the film increases the mean free path l (Table I) and pushes the sample closer to the clean limit. An apparent exponent as high as ∼ 7 could pos- sibly be explained by the combination of strong disorder and surface modes, but again, detailed theory is lacking. In conclusion, we have obtained the first clear evidence that the electron-phonon interaction at low tempera- tures changes quite significantly when the phonon modes become two-dimensional. To quantify the effects, the electron thermal relaxation times τ = γV Te/(dP/dTe), where γ = 100 J/K2m3 for Cu, are presented in Table I for all the samples at two temperatures Te = 0.2 and 0.8 K. At Te < 0.5 K, the thinnest membranes can have a a factor 2-3 strengthening effect, whereas at higher tem- peratures the thermal relaxation from membranes can be an order of magnitude weaker compared to bulk samples. The membrane close to transition region (d=200 nm) was shown to have a weaker (∼ factor of two) e-p interaction strength than the bulk samples. Thinning the metal film on bulk substrates also leads to a sizeable weakening of the e-p interaction. The observed power law exponent for the 2D limit is consistent with n ≈ 4.5, and is much smaller than the corresponding bulk exponent n = 6..7. A reduction by more than a factor one gives indirect evi- dence of the importance of the flexural, dispersive Lamb- modes for the membrane electron-phonon interaction, in agreement with theory [9, 10]. Discussions with T. Kühn and A. Sergeev and tech- nical assistance by H. Niiranen are acknowledged. This work was supported by the Academy of Finland project Nos. 118665 and 118231, and by the Finnish Academy of Sciences and Letters (J.T.K.). [1] V. F. Gantmakher, Rep. Prog. Phys. 37, 317 (1974). [2] F. Giazotto et al., Rev. Mod. Phys. 78, 217 (2006). [3] M. L. Roukes et al., Phys. Rev. Lett. 55, 422 (1985). [4] F. C. Wellstood, C. Urbina, and J. Clarke, Phys. Rev. B 49, 5942 (1994). [5] M. Kanskar and M. N. Wybourne, Phys. Rev. Lett. 73, 2123 (1994). [6] D. R. Schmidt, C. S. Yung, and A. N. Cleland, Phys. Rev. B 69, 140301 (2004). [7] A. N. Cleland, Foundations of Nanomechanics, Springer, Berlin (2003). [8] D. Belitz and S. Das Sarma, Phys. Rev. B 36, 7701 (1987). [9] K. Johnson, M. N. Wybourne and N. Perrin, Phys. Rev. B 50, 2035 (1994). [10] B. A. Glavin et al., Phys. Rev. B 65, 205315 (2002). [11] J. F. DiTusa et al., Phys. Rev. Lett. 68, 1156 (1992). [12] Y. K. Kwong et al., J. Low Temp. Phys. 88, 261 (1992). [13] J. M. Rowell and D. C. Tsui, Phys. Rev. B 14, 2456 (1976). [14] B. A. Auld, Acoustic Fields and Waves in Solids, 2nd. Ed., Robert E. Krieger Publishing, Malabar, 1990. [15] M. A. Geller, Phys. Rev. B 70, 205421 (2004). [16] S.-X. Qu, A. N. Cleland and M. R. Geller, Phys. Rev. B 72, 224301 (2005). [17] J. T. Karvonen, L. J. Taskinen and I. J. Maasilta, J. Low Temp. Phys. 146, 213 (2007). [18] C. Hoffmann, F. Lefloch, and M. Sanquer, Eur. Phys. J. B 29, 629 (2002). [19] M. Meschke, W. Guichard, and J. P. Pekola, Nature 444, 187 (2006). [20] H. Pothier et al., Phys. Rev. Lett. 79, 3490 (1997). [21] T. Kühn et al., arXiv:0705.1936. [22] E. T. Swartz and R. O. Pohl, Rev. Mod. Phys. 61, 605 (1989). [23] In this work, a maximum 1-5 % effect for Te at 1K for t = 15..30 nm. [24] T Kühn et al., Phys. Rev. B 70, 125425 (2004). [25] T. Kühn and I. J. Maasilta, Nucl. Instrum. Methods Phys. Res. A 559, 724 (2006); cond-mat/0702542. [26] A. Schmid, Z. Phys. 259, 421 (1973); in Localization, Interaction and Transport Phenomena, Springer 1985. [27] M. Yu. Reizer and A. V. Sergeev, Zh. Eksp. Teor. Fiz. 90, 1056 (1986) [Sov. Phys. JETP 63, 616 (1986)]; A. Sergeev and V. Mitin, Phys. Rev. B 61, 6041 (2000). [28] L. J. Taskinen and I. J. Maasilta, Appl. Phys. Lett. 89, 143511 (2006). http://arxiv.org/abs/0705.1936 http://arxiv.org/abs/cond-mat/0702542

0704.0337

Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains

Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains François Golse ∗ † Ecole Polytechnique, CMLS 91128 Palaiseau Cedex, France Alex Mahalov ‡and Basil Nicolaenko § Department of Mathematics and Statistics Arizona State University Tempe, AZ 85287-1804, USA Abstract A class of three-dimensional initial data characterized by uniformly large vorticity is considered for the 3D incompressible Euler equations in bounded cylindrical domains. The fast singular oscillating limits of the 3D Euler equations are investigated for parametrically resonant cylin- ders. Resonances of fast oscillating swirling Beltrami waves deplete the Euler nonlinearity. These waves are exact solutions of the 3D Euler equations. We construct the 3D resonant Euler systems; the latter are countable uncoupled and coupled SO(3;C) and SO(3;R) rigid body systems. They conserve both energy and helicity. The 3D resonant Eu- ler systems are vested with bursting dynamics, where the ratio of the enstrophy at time t = t∗ to the enstrophy at t = 0 of some remarkable orbits becomes very large for very small times t∗; similarly for higher norms Hs, s ≥ 2. These orbits are topologically close to homoclinic cycles. For the time intervals where Hs norms, s ≥ 7/2 of the limit resonant orbits do not blow up, we prove that the full 3D Euler equa- tions possess smooth solutions close to the resonant orbits uniformly in strong norms. Key-Words: Incompressible Euler Equations, Rotating Fluids, Rigid Body Dynamics, Enstrophy Bursts MSC: 35Q35, 76B03, 76U05 ∗[email protected] †and Laboratoire J.-L. Lions, Université Paris Diderot-Paris 7 ‡[email protected] §[email protected] http://arxiv.org/abs/0704.0337v1 1 Introduction The issues of blowup of smooth solutions and finite time singularities of the vorticity field for 3D incompressible Euler equations are still a major open problem. The Cauchy problem in 3D bounded axisymmetric cylindrical domains is attracting considerable attention: with bounded, smooth, non- axisymmetric 3D initial data, under the constraints of conservation of bounded energy, can the vorticity field blow up in finite time? Outstanding numeri- cal claims for this have recently been disproven [Ke], [Hou1], [Hou2]. The classical analytical criterion of Beale-Kato-Majda [B-K-M] for non-blow up in finite time requires the time integrability of the L∞ norm of the vorticity. DiPerna and Lions [Li] have given examples of global weak solutions of the 3D Euler equations which are smooth (hence unique) if the initial conditions are smooth (specifically in W1,p(D), p > 1). However, these flows are really 2-Dimensional in x1, x2, 3-components flows, independent from the third co- ordinate x3. Their examples [DiPe-Li] show that solutions (even smooth ones) of the 3D Euler equations cannot be estimated in W1,p for 1 < p <∞ on any time interval (0, T ) if the initial data are only assumed to be bounded inW1,p. Classical local existence theorems in 3D bounded or periodic domains by Kato [Ka], Bourguignon-Brézis [Bou-Br] and Yudovich [Yu1], [Yu2] require some minimal smoothness for the initial conditions (IC), e.g., in Hs(D), s > 5 The classical formulation for the Euler equations is ∂tV+ (V · ∇)V = −∇p, ∇ ·V = 0, (1.1) V ·N = 0 on ∂D, (1.2) where ∂D is the boundary of a bounded, connected domain D, N the normal to ∂D, V(t, y) = (V1, V2, V3) the velocity field, y = (y1, y2, y3), and p is the pressure. The equivalent Lamé form [Ar-Khe] ∂tV + curlV ×V +∇ = 0, (1.3) ∇ ·V = 0, (1.4) ∂tω + curl(ω ×V) = 0, (1.5a) ω = curlV, (1.5b) implies conservation of Energy: E(t) = |V(t, y)|2 dy. (1.6) The helicity Hel(t) [Ar-Khe], [Mof], is conserved: Hel(t) = V · ω dy, (1.7) for D = R3 and when D is a periodic lattice. Helicity is also conserved for cylindrical domains, provided that ω·N = 0 on the cylinder’s lateral boundary at t = 0 (see [M-N-B-G]). From the theoretical point of view, the principal difficulty in the analysis of 3D Euler equations is due to the presence of the vortex stretching term (ω · ∇)V in the vorticity equation (1.5a). The equations (1.3) and (1.5a) are equivalent to: ∂tω + [ω,V] = 0, (1.8) where [a, b] = curl (a × b) is the commutator in the infinite dimensional Lie algebra of divergence-free vector fields [Ar-Khe]. This point of view has led to celebrated developments in Topological Methods in Hydrodynamics [Ar-Khe], [Mof]. The striking analogy between the Euler equations for hydrodynamics and the Euler equations for a rigid body (the latter associated to the Lie Algebra of the Lie group SO(3,R)) had already been pointed out by Moreau [Mor1]; Moreau was the first to demonstrate conservation of Helicity (1961) [Mor2]. This has led to extensive speculations to what extent/in what cases are the solutions of the 3D Euler equations “close” to those of coupled 3D rigid body equations in some asymptotic sense. Recall that the Euler equations for a rigid body in R3 is: mt + ω ×m = 0, m = Aω, (1.9a) mt + [ω,m] = 0, (1.9b) where m is the vector of angular momentum relative to the body, ω the angular velocity in the body and A the inertia operator [Ar1], [Ar-Khe]. The Russian school of Gledzer, Dolzhansky, Obukhov [G-D-O] and Vishik [Vish] has extensively investigated dynamical systems of hydrodynamic type and their applications. They have considered hydrodynamical models built upon generalized rigid body systems in SO(n,R), following Manakhov [Man]. Inspired by turbulence physics, they have investigated “shell” dynamical sys- tems modeling turbulence cascades; albeit such systems are flawed as they only preserve energy, not helicity. To address this, they have constructed and studied in depth n-dimensional dynamical systems with quadratic homoge- neous nonlinearities and two quadratic first integrals F1, F2. Such systems can be written using sums of Poisson brackets: i2,...,in ǫi1i2...inpi4...in − ∂F1 , (1.10) where constants pi4...in are antisymmetric in i4, ..., in. A simple version of such a quadratic hydrodynamic system was introduced by Gledzer [Gl1] in 1973. A deep open issue of the work by the Gledzer- Obukhov school is whether there exist indeed classes of I.C. for the 3D Cauchy Euler problem (1.1) for which solutions are actually asymptotically close in strong norm, on arbitrary large time intervals to solutions of such hydro- dynamic systems, with conservation of both energy and helicity. Another unresolved issue is the blowup or global regularity for the “enstrophy” of such systems when their dimension n→ ∞. This article reviews some current new results of a research program in the spirit of the Gledzer-Obukhov school; this program builds-up on the re- sults of [M-N-B-G] for 3D Euler in bounded cylindrical domains. Following the original approach of [B-M-N1]-[B-M-N4] in periodic domains, [M-N-B-G] prove the non blowup of the 3D incompressible Euler equations for a class of three-dimensional initial data characterized by uniformly large vorticity in bounded cylindrical domains. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to some 2D manifold. The initial vortex stretching is large. The approach of proving regularity is based on investigation of fast singular oscillating limits and nonlinear averaging methods in the context of almost periodic functions [Bo-Mi], [Bes], [Cor]. Harmonic analysis tools based on curl eigenfunctions and eigenvalues are crucial. One establishes the global regularity of the 3D limit resonant Euler equations without any restriction on the size of 3D initial data. The resonant Euler equations are characterized by a depleted nonlin- earity. After establishing strong convergence to the limit resonant equations, one bootstraps this into the regularity on arbitrary large time intervals of the solutions of 3D Euler Equations with weakly aligned uniformly large vorticity at t = 0. [M-N-B-G] theorems hold for generic cylindrical domains, for a set of height/radius ratios of full Lebesgue measure. For such cylinders, the 3D limit resonant Euler equations are restricted to two-wave resonances of the vorticity waves and are vested with an infinite countable number of new con- servation laws. The latter are adiabatic invariants for the original 3D Euler equations. Three-wave resonances exist for a nonempty countable set of h/R (h height, R radius of the cylinder) and moreover accumulate in the limit of vanishingly small vertical (axial) scales. This is akin to Arnold tongues [Ar2] for the Mathieu-Hill equations and raises nontrivial issues of possible sin- gularities/lack thereof for dynamics ruled by infinitely many resonant triads at vanishingly small axial scales. In such a context, the 3D resonant Euler equations do conserve the energy and helicity of the field. In this review, we consider cylindrical domains with parametric resonances in h/R and investigate in depth the structure and dynamics of 3D resonant Euler systems. These parametric resonances in h/R are proven to be non- empty. Solutions to Euler equations with uniformly large initial vorticity are expanded along a full complete basis of elementary swirling waves (T2 in time). Each such quasiperiodic, dispersive vorticity wave is a quasiperiodic Beltrami flow; these are exact solutions of 3D Euler equations with vorticity parallel to velocity. There are no Galerkin-like truncations in the decomposition of the full 3D Euler field. The Euler equations, restricted to resonant triplets of these dispersive Beltrami waves, determine the “resonant Euler systems”. The basic “building block” of these (a priori ∞-dimensional) systems are proven to be SO(3;C) and SO(3;R) rigid body systems: U̇k + (λm − λn)UmUn = 0 U̇m + (λn − λk)UnUk = 0 U̇n + (λk − λm)UkUm = 0 (1.11) These λ’s are eigenvalues of the curl operator in the cylinder, curlΦ±n = ±λnΦ±n ; the curl eigenfunctions are steady elementary Beltrami flows, and the dispersive Beltrami waves oscillate with the frequencies ± h , n3 ver- tical wave number (vertical shear), 0 < ǫ < 1. Physicists [Ch-Ch-Ey-H] have computationally demonstrated the physical impact of the polarization of Bel- trami modes Φ± on intermittency in the joint cascade of energy and helicity in turbulence. Another “building block” for resonant Euler systems is a pair of SO(3;C) or SO(3;R) rigid bodies coupled via a common principal axis of inertia/mo- ment of inertia: ȧk = (λm − λn)Γaman (1.12a) ȧm = (λn − λk)Γanak (1.12b) ȧn = (λk − λm)Γakam + (λk̃ − λm̃)Γ̃ak̃am̃ (1.12c) ȧm̃ = (λn − λk̃)Γ̃anak̃ (1.12d) ȧk̃ = (λm̃ − λn)Γ̃am̃an, (1.12e) where Γ and Γ̃ are parameters in R defined in Theorem 4.10. Both reso- nant systems (1.11) and (1.12) conserve energy and helicity. We prove that the dynamics of these resonant systems admit equivariant families of homo- clinic cycles connecting hyperbolic critical points. We demonstrate bursting dynamics: the ratio ||u(t)||2Hs/||u(0)||2Hs , s ≥ 1 can burst arbitrarily large on arbitrarily small times, for properly chosen para- metric domain resonances h/R. Here ||u(t)||2Hs = 2s|un(t)|2 . (1.13) The case s = 1 is the enstrophy. The “bursting” orbits are topologically close to the homoclinic cycles. Are such dynamics for the resonant systems relevant to the full 3D Euler equations (1.1)-(1.8)? The answer lies in the following crucial “shadowing” Theorem 2.10. Given the same initial conditions, given the maximal time interval 0 ≤ t < Tm where the resonant orbits of the resonant Euler equations do not blow up, then the strong norm Hs of the difference between the exact Euler orbit and the resonant orbit is uniformly small on 0 ≤ t < Tm, provided that the vorticity of the I.C. is large enough. Paradoxically, the larger the vortex streching of the I.C., the better the uniform approximation. This deep result is based on cancellation of fast oscillations in strong norms, in the context of almost periodic functions of time with values in Banach spaces (Section 4 of [M-N-B-G]). It includes uniform approximation in the spaces Hs, s > 5/2. For instance, given a quasiperiodic orbit on some time torus Tl for the resonant Euler systems, the exact solutions to the Euler equations will remain ǫ-close to the resonant quasiperiodic orbit on a time interval 0 ≤ t ≤ maxTi, 1 ≤ i ≤ l, Ti elementary periods, for large enough initial vorticity. If orbits of the resonant Euler systems admit bursting dynamics in the strong norms Hs, s ≥ 7/2, so do some exact solutions of the full 3D Euler equations, for properly chosen parametrically resonant cylinders. 2 Vorticity waves and resonances of elemen- tary swirling flows We study initial value problem for the three-dimensional Euler equations with initial data characterized by uniformly large vorticity: ∂tV+ (V · ∇)V = −∇p, ∇ ·V = 0, (2.1) V(t, y)|t=0 = V(0) = Ṽ0(y) + e3 × y (2.2) where y = (y1, y2, y3), V(t, y) = (V1, V2, V3) is the velocity field and p is the pressure. In Eqs. (1.1) e3 denotes the vertical unit vector and Ω is a constant parameter. The field Ṽ0(y) depends on three variables y1, y2 and y3. Since curl(Ω e3 × y) = Ωe3, the vorticity vector at initial time t = 0 is curlV(0, y) = curlṼ0(y) + Ωe3, (2.3) and the initial vorticity has a large component weakly aligned along e3, when Ω >> 1. These are fully three-dimensional large initial data with large initial 3D vortex stretching. We denote by Hsσ the usual Sobolev space of solenoidal vector fields. The base flow Vs(y) = e3 × y, curlVs(y) = Ωe3 (2.4) is called a steady swirling flow and is a steady state solution (1.1)-(1.4), as curl(Ωe3×Vs(y)) = 0. In (2.2) and (2.3), we consider I.C. which are an arbi- trary (not small) perturbation of the base swirling flow Vs(y) and introduce V(t, y) = e3 × y + Ṽ(t, y), (2.5) curlV(t, y) = Ωe3 + curlṼ(t, y), (2.6) ∂tṼ + curlṼ× Ṽ +Ωe3 × Ṽ + curlṼ×Vs(y) +∇p′ = 0, ∇ · Ṽ = 0, (2.7) Ṽ(t, y)|t=0 = Ṽ0(y). (2.8) Eqs. (2.1) and (2.7) are studied in cylindrical domains C = {(y1, y2, y3) ∈ R3 : 0 < y3 < 2π/α, y21 + y22 < R2} (2.9) where α and R are positive real numbers. If h is the height of the cylinder, α = 2π/h. Let Γ = {(y1, y2, y3) ∈ R3 : 0 < y3 < 2π/α, y21 + y22 = R2}. (2.10) Without loss of generality, we can assume that R = 1. Eqs. (2.1) are consid- ered with periodic boundary conditions in y3 V(y1, y2, y3) = V(y1, y2, y3 + 2π/α) (2.11) and vanishing normal component of velocity on Γ V ·N = Ṽ ·N = 0 on Γ; (2.12) where N is the normal vector to Γ. From the invariance of 3D Euler equations under the symmetry y3 → −y3, V1 → V1, V2 → V2, V3 → −V3, all results in this article extend to cylindrical domains bounded by two horizontal plates. Then the boundary conditions in the vertical direction are zero flux on the vertical boundaries (zero vertical velocity on the plates). One only needs to restrict vector fields to be even in y3 for V1, V2 and odd in y3 for V3, and double the cylindrical domain to −h ≤ y3 ≤ +h. We choose Ṽ0(y) in H s(C), s > 5/2. In [M-N-B-G], for the case of “non- resonant cylinders”, that is, non-resonant α = 2π/h, we have established regularity for arbitrarily large finite times for the 3D Euler solutions for Ω large, but finite. Our solutions are not close in any sense to those of the 2D or “quasi 2D” Euler and they are characterized by fast oscillations in the e3 direction, together with a large vortex stretching term ω(t, y) · ∇V(t, y) = ω1 , t ≥ 0 with leading component V(t, y) ≫ 1. There are no assumptions on oscillations in y1, y2 for our solutions (nor for the initial condition Ṽ0(y)). Our approach is entirely based on sturying fast singular oscillating limits of Eqs. (1.1)-(1.5a), nonlinear averaging and cancelation of oscillations in the nonlinear interactions for the vorticity field for large Ω. This has been devel- oped in [B-M-N2], [B-M-N3], and [B-M-N4] for the cases of periodic lattice domains and the infinite space R3. It is well known that fully three-dimensional initial conditions with uni- formly large vorticity excite fast Poincaré vorticity waves [B-M-N2], [B-M-N3], [B-M-N4], [Poi]. Since individual Poincaré wave modes are related to the eigenfunctions of the curl operator, they are exact time-dependent solutions of the full nonlinear 3D Euler equations. Of course, their linear superposition does not preserve this property. Expanding solutions of (2.1)-(2.8) along such vorticity waves demonstrates potential nonlinear resonances of such waves. First recall spectral properties of the curl operator in bounded, connected domains: Proposition 2.1 ([M-N-B-G]) The curl operator admits a self-adjoint ex- tension under the zero flux boundary conditions, with a discrete real spectrum λn = ±|λn|, |λn| > 0 for every n and |λn| → +∞ as |n| → ∞. The corre- sponding eigenfunctions Φ±n curlΦ±n = ±|λn|Φ n (2.13) are complete in the space U ∈ L2(D) : ∇ ·U = 0 and U ·N|∂D = 0 and U dz = 0 (2.14) Remark 2.2 In cylindrical domains, with cylindrical coordinates (r, θ, z), the eigenfunctions admit the representation: Φn1,n2,n3 = (Φr,n1,n2,n3(r),Φθ,n1,n2,n3(r),Φz,n1,n2,n3(r)) e in2θeiαn3z, (2.15) with n2 = 0,±1,±2, ..., n3 = ±1,±2, ... and n1 = 0, 1, 2, .... Here n1 indexes the eigenvalues of the equivalent Sturm-Liouville problem in the radial coor- dinates, and n = (n1, n2, n3). See [M-N-B-G] for technical details. From now on, we use the generic variable z for any vertical (axial) coordinate y3 or x3. For n3 = 0 (vertical averaging along the axis of the cylinder), 2-Dimensional, 3-component solenoidal fields must be expanded along a complete basis for fields derived from 2D stream functions: curl(φne3), φne3 , φn = φn(r, θ), −△φn = µnφn, φn|∂Γ = 0, and curlΦn = curl(φne3), µnφne3 a, be3 denotes a 3-component vector whose horizontal projection is a and vertical projection is be3. Let us explicit elementary swirling wave flows which are exact solutions to (2.1) and (2.7): Lemma 2.3 For every n = (n1, n2, n3), the following quasiperiodic (T time) solenoidal fields are exact solution of the full 3D nonlinear Euler equa- tions (2.1): V(t, y) = e3 × y + exp( Jt)Φn(exp(− Jt)y) exp(±i αΩt), (2.16) n3 is the vertical wave number of Φn and exp( Jt) the unitary group of rigid body rotations: 0 −1 0 1 0 0 0 0 0  , eΩJt/2 = cos(Ωt ) − sin(Ωt sin(Ωt ) cos(Ωt 0 0 1  . (2.17) Remark 2.4 These fields are exact quasiperiodic, nonaxisymmetric swirling flow solutions of the 3D Euler equations. For n3 6= 0, their second components Ṽ(t, y) = exp( Jt)Φn(exp(− Jt)y) exp(± in3 αΩt) (2.18) are Beltrami flows (curlṼ×Ṽ ≡ 0) exact solutions of (2.7) with Ṽ(t = 0, y) = Φn(y). Ṽ(t, y) in Eq. (2.18) are dispersive waves with frequencies Ω and n3α|λn|Ω, where α = 2π . Moreover, each Ṽ(t, y) is a traveling wave along the cylinder’s axis, since it contains the factor iαn3(±z ± Note that n3 large corresponds to small axial (vertical) scales, albeit 0 ≤ α|n3/λn| ≤ 1. Proof of Lemma 2.3. Through the canonical rigid body transformation for both the field V(t, y) and the space coordinates y = (y1, y2, y3): V(t, y) = e+ΩJt/2U(t, e−ΩJt/2y) + Jy, x = e−ΩJt/2y, (2.19) the 3D Euler equations (2.1), (2.2) transform into: ∂tU+ (curlU+Ωe3)×U = −∇ (|x1|2 + |x2|2) + , (2.20) ∇ ·U = 0, U(t, x)|t=0 = U(0) = Ṽ0(x), (2.21) For Beltrami flows such that curlU×U ≡ 0, these Euler equations (2.20)- (2.21) in a rotating frame reduce to: ∂tU+Ωe3 ×U+∇π = 0, ∇ ·U = 0, which are identical to the Poincaré-Sobolev nonlocal wave equations in the cylinder [M-N-B-G], [Poi], [Sob], [Ar-Khe]: ∂tΨ+Ωe3 ×Ψ+∇π = 0, ∇ ·Ψ = 0, (2.22) curl2Ψ−Ω2 ∂ Ψ = 0, Ψ ·N|∂D = 0. (2.23) It suffices to verify that the Beltrami flows Ψn(t, x) = Φn(x) exp ±iαn3|λn|Ωt where Φ±n (x) and ±|λn| are curl eigenfunctions and eigenvalues, are exact solutions to the Poincaré-Sobolev wave equation, in such a rotating frame of reference. Remark 2.5 The frequency spectrum of the Poincaré vorticity waves (solu- tions to (2.22)) is exactly ±iαn3|λn|Ω, n = (n1, n2, n3) indexing the spectrum of curl. Note that n3 = 0 (zero frequency of rotating waves) corresponds to 2-Dimensional, 3-Components solenoidal vector fields. We now transform the Cauchy problem for the 3D Euler equations (2.1)- (2.2) into an infinite dimensional nonlinear dynamical system by expanding V(t, y) along the swirling wave flows (2.16)-(2.18): V(t, y) = e3 × y (2.24a) + exp n=(n1,n2,n3) un(t) exp (2.24b) V(t = 0, y) = e3 × y + Ṽ0(y) (2.24c) Ṽ0(y) = n=(n1,n2,n3) un(0)Φn(y), (2.24d) where Φn denotes the curl eigenfunctions of Proposition 2.1 if n3 6= 0, and curl(φne3), φne3 if n3 = 0 (2D case, Remark 2.2). As we focus on the case where helicity is conserved for (2.1)-(2.2), we consider the class of initial data Ṽ0 such that [M-N-B-G]: curlṼ0 ·N = 0 on Γ, where Γ is the lateral boundary of the cylinder. The infinite dimensional dynamical system is then equivalent to the 3D Euler equations (2.1)-(2.2) in the cylinder, with n = (n1, n2, n3) ranging over the whole spectrum of curl, e.g.: k3+m3=n3 k2+m2=n2 × < curlΦk ×Φm,Φn > uk(t)um(t) (2.25) curlΦ±k = ±λkΦ k if k3 6= 0, curlΦk = curl(φke3), µkφke3 if k3 = 0 (2D, 3-components, Remark 2.2), similarly for m3 = 0 and n3 = 0. The inner product < , > denotes the L2 complex-valued inner product in D. This is an infinite dimensional system of coupled equations with quadratic nonlinearities, which conserve both the energy E(t) = |un(t)|2 and the helicity Hel(t) = ±|λn| |u±n (t)|2. The quadratic nonlinearities split into resonant terms where the exponential oscillating phase factor in (2.25) reduces to unity and fast oscillating non- resonant terms (Ω >> 1). The resonant set K is defined in terms of vertical wavenumbers k3,m3, n3 and eigenvalues ±λk, ±λm, ±λn of curl: K = {± k3 = 0, n3 = k3 +m3, n2 = k2 +m2}. (2.27) Here k2,m2, n2 are azimuthal wavenumbers. We shall call the “resonant Euler equations” the following ∞-dimensional dynamical system restricted to (k,m, n) ∈ K: (k,m,n)∈K < curlΦk ×Φm,Φn > ukum = 0, (2.28a) un(0) ≡< Ṽ0,Φn >, (2.28b) here curlΦ±k = ±λkΦ k if k3 6= 0, curlΦk = curl(φke3), µkφke3 if k3 = 0; similarly for m3 = 0 and n3 = 0 (2D components, Remark 2.2). If there are no terms in (2.28a) satisfying the resonance conditions, then there will be some modes for which Lemma 2.6 The resonant 3D Euler equations (2.28) conserve both energy E(t) and helicity Hel(t). The energy and helicity are identical to that of the full exact 3D Euler equations (2.1)-(2.2). The set of resonances K is studied in depth in [M-N-B-G]. To summarize, K splits into: (i ) 0-wave resonances, with n3 = k3 = m3 = 0; the corresponding reso- nant equations are identical to the 2-Dimensional, 3-Components Euler equations, with I.C. Ṽ0(y1, y2, y3) dy3. (ii) Two-Wave resonances, with k3m3n3 = 0, but two of them are not null; the corresponding resonant equations (called “catalytic equations”) are proven to possess an infinite, countable set of new conservation laws [M-N-B-G]. (iii) Strict three-wave resonances for a subset K∗ ⊂ K. Definition 2.7 The set K∗ of strict 3 wave resonances is: = 0, k3m3n3 6= 0, n3 = k3 +m3, n2 = k2 +m2 (2.29) Note that K∗ is parameterized by h/R, since α = 2π parameterizes the eigen- values λn, λk, λm of the curl operator. Proposition 2.8 There exist a countable, non-empty set of parameters h which K∗ 6= ∅. Proof. The technical details, together with a more precise statement, are postponed to the proof of Lemma 3.7. Concrete examples of resonant ax- isymmetric and helical waves are discussed in [Mah] ( cf. Figure 2 in the article). Corollary 2.9 Let Ṽ0(y1, y2, y3) dy3 = 0, i.e. zero vertical mean for the I.C. Ṽ0(y) in (2.2), (2.8), (2.24d) and (2.28b). Then the resonant 3D Euler equations are invariant on K∗: (k,m,n)∈K∗ λk < Φk ×Φm,Φn > ukum = 0, k3m3n3 6= 0, (2.30a) un(0) =< Ṽ0,Φn > (2.30b) (where Ṽ0 has spectrum restricted to n3 6= 0). Proof. This is an immediate corollary of the “operator splitting” Theorem 3.2 in [M-N-B-G]. � We shall call the above dynamical systems the “strictly resonant Euler system”. This is an ∞-dimensional Riccati system which conserves Energy and Helicity. It corresponds to nonlinear interactions depleted on K∗. How do dynamics of the resonant Euler equations (2.28) or (2.30) approx- imate exact solutions of the Cauchy problem for the full Euler equations in strong norms? This is answered by the following theorem, proven in Section 4 of [M-N-B-G]: Theorem 2.10 Consider the initial value problem V(t = 0, y) = e3 × y + Ṽ0(y), Ṽ0 ∈ Hsσ, s > 7/2 for the full 3D Euler equations, with ||Ṽ0||Hs ≤M0s and curlṼ0 ·N = 0 on Γ. • Let V(t, y) = Ω e3 × y + Ṽ(t, y) denote the solution to the exact Euler equations. • Let w(t, x) denote the solution to the resonant 3D Euler equations with Initial Condition w(0, x) ≡ w(0, y) = Ṽ0(y). • Let ||w(t, y)||Hsσ ≤Ms(TM ,M s ) on 0 ≤ t ≤ TM , s > 7/2. Then, ∀ǫ > 0, ∃ Ω∗(TM ,M0s , ǫ) such that, ∀Ω ≥ Ω∗: Ṽ(t, y)− exp un(t)e −i n3 on 0 ≤ t ≤ TM , ∀β ≥ 1, β ≤ s− 2. Here || · ||Hβ is defined in (1.13). The 3D Euler flow preserves the condition curlṼ0 · N = 0 on Γ, that is curlV(t, y) · N = 0 on Γ, for every t ≥ 0 [M-N-B-G]. The proof of this “error-shadowing” theorem is delicate, beyond the usual Gronwall differential inequalities and involves estimates of oscillating integrals of almost periodic functions of time with values in Banach spaces. Its importance lies in that solutions of the resonant Euler equations (2.28) and/or (2.30) are uniformly close in strong norms to those of the exact Euler equations (2.1)-(2.2), on any time interval of existence of smooth solutions of the resonant system. The infinite dimensional Riccati systems (2.28) and (2.30) are not just hydro- dynamic models, but exact asymptotic limit systems for Ω ≫ 1. This is in contrast to all previous literature on conservative 3D hydrodynamic models, such as in [G-D-O]. 3 Strictly resonant Euler systems: the SO(3) We investigate the structure and the dynamics of the “strictly resonant Euler systems” (2.30). Recall that the set of 3-wave resonances is: (k,m, n) : ± k3 = 0, k3m3n3 6= 0, n3 = k3 +m3, n2 = k2 +m2 (3.1) From the symmetries of the curl eigenfunctions Φn and eigenvalues λn in the cylinder, the following identities hold under the transformation n2 → −n2, n3 → −n3 Φ(n1,−n2,−n3) = Φ∗(n1, n2, n3) , λ(n1,−n2,−n3) = λ(n1, n2, n3) . (3.2) where ∗ designates the complex conjugate (see Section 3, [M-N-B-G] for details). The eigenfunctions Φ(n1, n2, n3) involve the radial functions Jn2(β(n1, n2, αn3)r) and J (β(n1, n2, αn3)r), with λ2(n1, n2, n3) = β 2(n1, n2, αn3) + α 2n23; β(n1, n2, αn3) are discrete, countable roots of equation (3.30) in [M-N-B-G], obtained via an equivalent Sturm-Liouville radial problem. Since the curl eigenfunctions are even in r → −r, n1 → −n1, we will extend the indices n1 = 1, 2, ...,+∞ to −n1 = −1,−2, ... with the above radial symmetry in mind. Corollary 3.1 The 3-wave resonance set K∗ is invariant under the symme- tries σj , j = 0, 1, 2, 3, where σ0(n1, n2, n3) = (n1, n2, n3), σ1(n1, n2, n3) = (−n1, n2, n3), σ2(n1, n2, n3) = (n1,−n2, n3) σ3(n1, n2, n3) = (n1, n2,−n3) . Remark 3.2 For 0 < i ≤ 3, 0 < j ≤ 3, 0 < l ≤ 3 σ2j = Id, σiσj = −σl if i 6= j and σiσjσl = −Id, for i 6= j 6= l. The σj do preserve the convolution conditions in K∗. We choose an α for which the set K∗ is not empty. We further take the hypothesis of a single triple wave resonance (k,m, n), modulo the symmetries Hypothesis 3.3 K∗ is such that there exists a single triple wave number resonance (n, k,m), modulo the symmetries σj , j = 1, 2, 3 and σj(k) 6= k, σj(m) 6= m, σj(n) 6= n for j = 2 and j = 3. Under the above hypothesis, one can demonstrate that the strictly resonant Euler system splits into three uncoupled systems in C3: Theorem 3.4 Under hypothesis 3.3, the resonant Euler system reduces to three uncoupled rigid body systems in C3: + i(λk − λm)CkmnUkUm = 0 (3.3a) − i(λm − λn)CkmnUnU∗m = 0 (3.3b) − i(λn − λk)CkmnUnU∗k = 0 (3.3c) where Ckmn = i < Φk ×Φm,Φ∗n >, Ckmn real and the other two uncoupled systems obtained with the symmetries σ2(k,m, n) and σ3(k,m, n). The energy and the helicity of each subsystem are conserved: k + UmU m + UnU n) = 0, (λkUkU k + λmUmU m + λnUnU n) = 0. Proof. It follows from U−k = U k , λ(−k) = λ(+k), similarly for m and n; and in a very essential way from the antisymmetry of < Φk ×Φm,Φ∗n >, together with curlΦk = λkΦk. That Ckmn is real follows from the eigenfunc- tions explicited in Section 3 of [M-N-B-G]. � Remark 3.5 This deep structure, i.e. SO(3;C) rigid body systems in C3 is a direct consequence of the Lamé form of the full 3D Euler equations, cf. Eqs. (1.3) and (2.7), and the nonlinearity curlV ×V. The system (3.3) is equivariant with respect to the symmetry operators (z1, z2, z3) → (z∗1 , z∗2 , z∗3), (z1, z2, z3) → (exp(iχ1)z1, exp(iχ2)z2, exp(iχ3)z3) , provided χ1 = χ2 + χ3. It admits other integrals known as the Manley- Rowe relations (see, for instance [We-Wil]). It differs from the usual 3- wave resonance systems investigated in the literature, such as in [Zak-Man1], [Zak-Man2], [Gu-Ma] in that (1) helicity is conserved, (2) dynamics of these resonant systems rigorously “shadow” those of the exact 3D Euler equations, see Theorem 2.10. Real forms of the system (3.3) are found in Gledzer et al. [G-D-O], corre- sponding to the exact invariant manifold Uk ∈ iR, Um ∈ R, Un ∈ R, albeit without any rigorous asymptotic justification. The C3 systems (3.3) with helicity conservation laws are not discussed in [G-D-O]. The only nontrivial Manley-Rowe conservation laws for the resonant sys- tem (3.3), rigid body SO(3;C), which are independent from energy and he- licity, are: (rkrmrn sin(θn − θk − θm)) = 0, where Uj = rj exp(iθj), j = k,m, n, and E1 = (λk − λm)r2n − (λm − λn)r2k, E2 = (λm − λn)r2k − (λn − λk)r2m. The resonant system (3.3) is well known to possess hyperbolic equilibria and heteroclinic/homoclinic orbits on the energy surface. We are interested in rigorously proving arbitrary large bursts of enstrophy and higher norms on arbitrarily small time intervals, for properly chosen h/R. To simplify the presentation, we establish the results for the simpler invariant manifold Uk ∈ iR, and Um, Un ∈ R. Rescale time as: t→ t/Ckmn. Start from the system U̇n + i(λk − λm)UkUm = 0 U̇k − i(λm − λn)UnU∗m = 0 U̇m − i(λn − λk)UnU∗k = 0 (3.4) Assume that Uk ∈ iR and that Um, Un ∈ R: set p = iUk, q = Um and r = Un, as well as λk = λ, λm = µ and λn = ν: then ṗ+ (µ− ν)qr = 0 q̇ + (ν − λ)rp = 0 ṙ + (λ− µ)pq = 0 (3.5) This system admits two first integrals: E = p2 + q2 + r2 (energy) H = λp2 + µq2 + νr2 (helicity) (3.6) System (3.5) is exactly the SO(3,R) rigid body dynamics Euler equations, with inertia momenta Ij = |λj | , j = k,m, n [Ar1]. Lemma 3.6 ([Ar1], [G-D-O]) With the ordering λk > λm > λn, i.e. λ > µ > ν, the equilibria (0,±1, 0) are hyperbolic saddles on the unit energy sphere, and the equilibria (±1, 0, 0), (0, 0,±1) are centers. There exist equivariant families of heteroclinic connections between (0,+1, 0) and (0,−1, 0). Each pair of such connections correspond to equivariant homoclinic cycles at (0, 1, 0) and (0,−1, 0). We investigate bursting dynamics along orbits with large periods, with initial conditions close to the hyperbolic point (0, E(0), 0) on the energy sphere E. We choose resonant triads such that λk > 0, λn < 0, λk ∼ |λn|, |λm| ≪ λk, equivalently: λ > µ > ν, λν < 0, |µ| ≪ λ and λ ∼ |ν|. (3.7) Lemma 3.7 There exist h/R with K∗ 6= ∅, such that λk > λm > λn, λkλn < 0, |λm| ≪ λk and λk ∼ |λn|. Remark 3.8 Together with the polarity ± of the curl eigenvalues, these are 3-wave resonances where two of the eigenvalues are much larger in mod- uli than the third one. In the limit |k|, |m|, |n| ≫ 1, λk ∼ ±|k|, λm ∼ ±|m|, λn ∼ ±|n|, the eigenfunctions Φ have leading asymptotic terms which involve cosines and sines periodic in r, cf. Section 3 [M-N-B-G]. In the strictly resonant equations (2.30), the summation over the quadratic terms becomes an asymptotic convolution in n1 = k1+n1. The resonant three waves in Lemma 3.7 are equivalent to Fourier triads k + m = n, with |k| ∼ |n| and |m| ≪ |k|, |n|, in periodic lattices. In the physics of spectral theory of turbulence [Fri], [Les], these are exactly the triads responsible from transfer of energy between large scales and small scales. These are the triads which have hampered mathematical efforts at proving the global regularity of the Cauchy problem for 3D Navier-Stokes equations in periodic lattices [Fe]. Proof of Lemma 3.7 ([M-N-B-G]) The transcendental dispersion law for 3-waves in K∗ for cylindrical domains, is a polynomial of degree four in ϑ3 = 1/h2: P̃ (ϑ3) = P̃4ϑ 3 + P̃3ϑ 3 + P̃2ϑ 3 + P̃1ϑ3 + P̃0 = 0, (3.8) with n2 = k2 +m2 and n3 = k3 +m3. Then with hk = β2(k1,k2,αk3) , hm = β2(m1,m2,αm3) , hn = β2(n1,n2,αn3) , cf. the radial Sturm-Liouville problem in Section 3, [M-N-B-G], the coefficients of P̃ (ϑ3) are given by: P̃4 = −3, P̃3 = −4(hk + hm + hn), P̃2 = −6(hkhm + hkhn + hmhn), P̃1 = −12hkhmhn, P̃0 = h n + h n + h k − 2(hkhmh2n + hkhnh2m + hmhnh2k). Similar formulas for the periodic lattice domain were first derived in [B-M-N2], [B-M-N3], [B-M-N4]. In cylindrical domains the resonance condition for K∗ is identical to ϑ3 + hk ϑ3 + hm ϑ3 + hn with ϑ3 = , hk = β 2(k)/k23 , hm = β 2(m)/m23, hn = β 2(n)/n23; Eq. (3.8) is the equivalent rational form. From the asymptotic formula (3.44) in [M-N-B-G], for large β: β(n1, n2, n3) ∼ n1π + n2 + ψ, (3.9) where ψ = 0 if lim m2 = 0 (e.g. h fixed, m2/m3 → 0) and ψ = ±π2 if lim m2 = ±∞ (e.g. m2 fixed, h → ∞). The proof is completed by taking leading terms P̃0+ϑ3P̃1 in (3.8), ϑ3 = ≪ 1, and m2 = 0, k2 = O(1), n2 = O(1). � We now state a theorem for bursting of the H3 norm in arbitrarily small times, for initial data close to the hyperbolic point (0, E(0), 0): Theorem 3.9 (Bursting dynamics in H3). Let λ > µ > ν, λν < 0, |µ| ≪ λ and λ ∼ |ν|. Let W (t) = λ6p(t)2 + µ6q(t)2 + ν6r(t)2 the H3-norm squared of an orbit of (3.5). Choose initial data such that: W (0) = λ6p(0)2 + µ6q(0)2 with λ6p(0)2 ∼ 1 W (0) and µ6q(0)2 ∼ 1 W (0). Then there exists t∗ > 0, such W (t) ≥ W (0) where t∗ ≤ 6√ W (0) µ2Ln(λ/|µ|)(λ/|µ|)−1. Remark 3.10 Under the conditions of Lemma 3.7, ≫ 1, whereas µ2(Ln(λ/|µ|))(λ/|µ|)−1 ≪ 1. Therefore, over a small time interval of length O(µ2(Ln(λ/|µ|))(λ/|µ|)−1) ≪ 1, the ratio ||U(t)||H3/||U(0)||H3 grows up to a maximal value O (λ/|µ|)3 ≫ 1. Since the orbit is periodic, the H3 semi- norm eventually relaxes to its initial state after some time (this being a mani- festation of the time-reversibility of the Euler flow on the energy sphere). The “shadowing” theorem 2.10 with s > 7/2 ensures that the full, original 3D Eu- ler dynamics, with the same initial conditions, will undergo the same type of burst. Notice that, with the definition (1.13) of ‖ · ‖Hs , one has ||Ωe3 × y||H3 = ||curl3(Ωe3 × y)||L2 = 0 . Hence the solid rotation part of the original 3D Euler solution does not con- tribute to the ratio ||V(t)||H3/||V(0)||H3 . Theorem 3.11 (Bursting dynamics of the enstrophy). Under the same con- ditions for the 3-wave resonance, let Ξ(t) = λ2p(t)2 + µ2q(t)2 + ν2r(t)2 the enstrophy. Choose initial data such that Ξ(0) = λ2p(0)2 + µ2q(0)2 + ν2r(0)2 with λ2p(0)2 ∼ 1 Ξ(0), µ2q(0)2 ∼ 1 Ξ(0). Then there exists t∗∗ > 0, such that Ξ(t∗∗) ≥ where t∗∗ ≤ 1√ Ln (λ/|µ|) (λ/|µ|)−1 . Remark 3.12 It is interesting to compare this mechanism for bursts with ear- lier results in the same direction obtained by DiPerna and Lions. Indeed, for each p ∈ (1,∞), each δ ∈ (0, 1) and each t > 0, Di Perna and Lions [DiPe-Li] constructed examples of 2D-3 components solutions to Euler equations such ||V(0)||W 1,p ≤ ǫ while ||V(t)||W 1,p ≥ 1/δ . Their examples essentially correspond to shear flows of the form V(t, x1, x2) = u(x2) w(x1 − tu(x2), x2) where u ∈W 1,px2 while w ∈W . Obviously curlV(t, x1, x2) = (∂2 − tu′(x2)∂1)w(x1 − tu(x2), x2) −∂1w(x1 − tu(x2), x2) −u′(x2) Thus, all components in curlV(t, x1, x2) belong to L loc, except for the term −tu′(x2)∂1w(x1 − tu(x2), x2) . For each t > 0, this term belongs to Lp for all choices of the functions u ∈ W 1,px2 and w ∈ W x1,x2 if and only if p = ∞. Whenever p < ∞, DiPerna and Lions construct their examples as some smooth approximation of the situation above in the strong W 1,p topology. In other words, the DiPerna-Lions construction works only in cases where the initial vorticity does not belong to an algebra — specifically to Lp, which is not an algebra unless p = ∞. The type of burst obtained in our construction above is different: in that case, the original vorticity belongs to the Sobolev space H2, which is an algebra in space dimension 3. Similar phenomena are observed in all Sobolev spaces Hβ with β ≥ 2 — which are also algebras in space dimension 3. In other words, our results complement those of DiPerna-Lions on bursts in higher order Sobolev spaces, however at the expense of using more intricate dynamics. We proceed to the proofs of Theorem 3.9 and 3.11. We are interested in the evolution of Ξ = λ2p2 + µ2q2 + ν2r2 (enstrophy) (3.10) Compute Ξ̇ = −2 λ2(µ− ν) + µ2(ν − λ) + ν2(λ− µ) pqr (3.11) ˙(pqr) = −(µ− ν)q2r2 − (ν − λ)r2p2 − (λ− µ)p2q2 (3.12) Using the first integrals above, one has  (3.13) where V an is the Vandermonde matrix V an = 1 1 1 λ µ ν λ2 µ2 ν2 For λ 6= µ 6= ν 6= λ, this matrix is invertible and V an−1 = (λ−µ)(λ−ν) −(µ+ν) (λ−µ)(λ−ν) (λ−µ)(λ−ν) (µ−ν)(µ−λ) −(ν+λ) (µ−ν)(µ−λ) (µ−ν)(µ−λ) (ν−λ)(ν−µ) −(λ+µ) (ν−λ)(ν−µ) (ν−λ)(ν−µ) Hence (λ− µ)(λ − ν) (Ξ− (µ+ ν)H + µνE) (µ− ν)(µ− λ) (Ξ− (ν + λ)H + νλE) (ν − λ)(ν − µ) (Ξ− (λ+ µ)H + λµE) (3.14) so that (µ− ν)q2r2 = − (Ξ− (ν + λ)H + νλE) (Ξ− (λ + µ)H + λµE) (λ− µ)(λ − ν)(µ− ν) (ν − λ)r2p2 = − (Ξ− (λ + µ)H + λµE) (Ξ− (µ+ ν)H + µνE) (λ− µ)(λ − ν)(µ− ν) (λ− µ)p2q2 = − (Ξ− (µ+ ν)H + µνE) (Ξ− (ν + λ)H + νλE) (λ− µ)(λ − ν)(µ− ν) Later on, we shall use the notations x−(λ, µ, ν) = (µ+ ν)H − µνE x0 (λ, µ, ν) = (µ+ λ)H − µλE x+(λ, µ, ν) = (λ+ ν)H − λνE (3.15) Therefore, we find that Ξ satisfies the second order ODE Ξ̈ =− 2Kλ,µ,ν ((Ξ− x−(λ, µ, ν))(Ξ − x0(λ, µ, ν)) +(Ξ− x0(λ, µ, ν))(Ξ − x+(λ, µ, ν)) + (Ξ− x+(λ, µ, ν))(Ξ − x0(λ, µ, ν))) which can be put in the form Ξ̈ = −2Kλ,µ,νP ′λ,µ,ν(Ξ) (3.16) where Pλ,µ,ν is the cubic Pλ,µ,ν(X) = (X − x−(λ, µ, ν))(X − x0(λ, µ, ν))(X − x+(λ, µ, ν)) (3.17) Kλ,µ,ν = λ2(µ− ν) + µ2(ν − λ) + ν2(λ− µ) (λ− µ)(λ− ν)(µ − ν) (3.18) In the sequel, we assume that the initial data for (p, q, r) is such that r(0) = 0 , p(0)(q(0) 6= 0 Let us compute x−(λ, µ, ν) = λνp(0) 2 + µ2q(0)2 + µ(λ− ν)p(0)2 x0 (λ, µ, ν) = λ 2p(0)2 + µ2q(0)2 x+(λ, µ, ν) = λ 2p(0)2 + ν + λ µ2q(0)2 (3.19) We shall also assume that λ > µ > ν , λν < 0 , |µ| ≪ λ and λ ∼ |ν| (3.20) Then Kλ,µ,ν > 0 — in fact Kλ,µ,ν ∼ 2, and Ξ is a periodic function of t such Ξ(t) = x0(λ, µ, ν) , sup Ξ(t) = x+(λ, µ, ν) (3.21) with half-period Tλ,µ,ν = Kλ,µ,ν ∫ x+(λ,µ,ν) x0(λ,µ,ν) −Pλ,µ,ν(x) (3.22) We are interested in the growth of the (squared) H3 norm W (t) = λ6p(t)2 + µ6q(t)2 + ν6r(t)2 (3.23) Expressing p2, q2 and r2 in terms of E, H and Ξ, it is found that λ6(Ξ− x−(λ, µ, ν)) (λ− µ)(λ − ν) µ6(Ξ− x+(λ, µ, ν)) (µ− ν)(µ− λ) ν6(Ξ− x0(λ, µ, ν)) (ν − λ)(ν − µ) (3.24) Hence, when Ξ = x+(λ, µ, ν), then λ6(x+(λ, µ ν) − x−(λ, µ, ν)) (λ− µ)(λ− ν) ν6(x+(λ, µ ν)− x0(λ, µ, ν)) (ν − λ)(ν − µ) λ6(x+(λ, µ ν) − x−(λ, µ, ν)) (λ− µ)(λ− ν) Let us compute x+(λ, µ ν)− x−(λ, µ, ν) = (λ − µ)(λ− ν)p(0)2 + ν + λ µ2q(0)2 & −νλq(0)2 ∼ λ2q(0)2 (3.25) We shall pick the initial data such that W (0) = λ6p(0)6 + µ6q(0)6 with λ6p(0)2 ∼ 1 W (0) and µ6q(0)2 ∼ 1 W (0) (3.26) Hence, when Ξ reaches x+(λ, µ, ν), one has λ8q(0)2 (λ− µ)(λ− ν) µ6(λ − µ)(λ− ν) W (0) ∼ 1 W (0) . (3.27) Hence W jumps from W (0) to a quantity ∼ 1 W (0) in an interval of time that does not exceed one period of the Ξ motion, i.e. 2Tλ,µ,ν . Let us estimate this interval of time. We recall the asymptotic equivalent for the period of an elliptic integral in the modulus 1 limit. Lemma 3.13 Assume that x− < x0 < x+. Then (x− x−)(x− x0)(x+ − x) x+ − x− x+−x0 x+−x− uniformly in x−, x0, and x+ as x+−x0 x+−x− → 1. x+(λ, µ, ν) − x−(λ, µ, ν) λ2q(0)2 ∼ |µ| W (0) x0(λ, µ, ν) − x−(λ, µ, ν) = (λ− µ)(λ − ν)p(0)2 (3.28) so that x+−x0 x+−x− 1− (λ−µ)(λ−ν)p(0) (λ−µ)(λ−ν)p(0)2+(µ(ν+λ)−νλ−µ2)q(0)2 ∼ (λ− µ)(λ− ν)p(0) 2 + (µ(ν + λ)− νλ− µ2)q(0)2 2(λ− µ)(λ − ν)p(0)2 ∼ q(0) 2p(0)2 W (0)/2µ6 W (0)/2λ6 Hence 2Tλ,µ,ν . W (0) ≤ 12√ W (0) (3.29) Conclusion: collecting (3.26), (3.27) and (3.29), we see that the squared H3 norm W varies from W (0) to a quantity ∼ ρ6W (0) in an interval of time . 12√ W (0) µ2 ln ρ . (Here ρ = λ/µ). We now proceed to obtain similar bursting estimates for the enstrophy. We return to (3.21) and (3.22). Pick the initial data so that Ξ(0) = λ2p(0)2 + µ2q(0)2 with λ2p(0)2 ∼ 1 Ξ(0) and µ2q(0)2 ∼ 1 Ξ(0). x+(λ, µ, ν) − x−(λ, µ, ν) = (λ− µ)(λ − ν)p(0)2 + ν + λ µ2q(0)2 ∼ 2λ2p(0)2 + λ2q(0)2 ∼ while x0(λ, µ, ν)− x−(λ, µ, ν) = (λ− µ)(λ− ν)p(0)2 ∼ 2λ2p(0)2 ∼ Ξ(0). Hence, in the limit as ρ = λ/|µ| → +∞, one has 2Tλ,µ,ν ∼ ρ2Ξ(0) 1− Ξ(0)1 ρ2Ξ(0) 2Ξ(0) 1− 2ρ−2 2Ξ(0) And Ξ varies from x0(λ, µ, ν) = Ξ(0) to x+(λ, µ, ν) ∼ ρ2Ξ(0) on an interval of time of length Tλ,µ,ν . � 4 Strictly resonant Euler systems: the case of 3-waves resonances on small-scales 4.1 Infinite dimensional uncoupled SO(3) systems In this section, we consider the 3-wave resonant set K∗ when |k|2, |m|2, |n|2 ≥ , 0 < η ≪ 1, i.e. 3-wave resonances on small scales; here |k|2 = k21 + k22 + k23 , where (k1, k2, k3) index the curl eigenvalues, and similarly for |m|2, |n|2. Recall that k2 + m2 = n2, k3 + m3 = n3 (exact convolutions), but that the sum- mation on k1, m1 on the right hand side of Eqs. (2.30) is not a convolution. However, for |k|2, |m|2, |n|2 ≥ 1 , the summation in k1, m1 becomes an asymptotic convolution. First: Proposition 4.1 The set K∗ restricted to |k|2, |m|2, |n|2 ≥ 1 , ∀η, 0 < η ≪ 1 is not empty: there exist at least one h/R with resonant three waves satisfying the above small scales condition. Proof. We follow the algebra of the exact transcendental dispersion law (3.8) derived in the proof of Lemma 3.7. Note that P̃ (ϑ3) < 0 for ϑ3 = large enough. We can choose hm = β2(m1,m2,αm3) = 0, say in the specific limit → 0, and β(m1,m2, αm3) ∼ m1π+m2 π2 + . Then P̃0 = h n > 0 and P̃ (ϑ3) must possess at least one (transcendental) root ϑ3 = In the above context, the radial components of the curl eigenfunctions in- volve cosines and sines in βr (cf. Section 3, [M-N-B-G]) and the summation in k1, m1 on the right hand side of the resonant Euler equations (2.30) becomes an asymptotic convolution. The rigorous asymptotic convolution estimates are highly technical and detailed in [Fro-M-N]. The 3-wave resonant systems for |k|2, |m|2, |n|2 ≥ 1 are equivalent to those of an equivalent periodic lattice [0, 2π]× [0, 2π]× [0, 2πh], ϑ3 = 1h2 ; the resonant three wave relation becomes: ϑ3 + ϑ1 ϑ3 + ϑ1 ϑ3 + ϑ1 = 0, (4.1a) k +m = n, k3m3n3 6= 0. (4.1b) The algebraic geometry of these rational 3-wave resonance equations has been investigated in depth in [B-M-N3] and [B-M-N4]. Here ϑ1, ϑ2, ϑ3 are periodic lattice parameters; in the small-scales cylindrical case, ϑ1 = ϑ2 = 1 (after rescaling of n2, k2, m2), ϑ3 = 1/h 2, h height. Based on the algebraic geometry of “resonance curves” in [B-M-N3], [B-M-N4], we investigate the resonant 3D Euler equations (2.30) in the equivalent periodic lattices. First, triplets (k,m, n) solution of (4.1) are invariant under the reflec- tion symmetries σ0, σ1, σ2, σ3 defined in Corollary 3.1 and Remark 3.2: σ0 = Id, σj(k) = (ǫi,jki), 1 ≤ i ≤ 3, ǫi,j = +1 if i 6= j, ǫi,j = −1 if i = j, 1 ≤ j ≤ 3. Second the set K∗ in (4.1) is invariant under the homothetic transformations: (k,m, n) → (γk, γm, γn), γ rational. (4.2) The resonant triplets lie on projective lines in the wavenumber space, with equivariance under σj , 0 ≤ j ≤ 3 and γ-rescaling. For every given equivariant family of such projective lines, the resonant curve is the graph of ϑ3 versus , for parametric domain resonances in ϑ1, ϑ2, ϑ3. Lemma 4.2 (p.17, [B-M-N4]). For every equivariant (k,m, n), the resonant curve in the quadrant ϑ1 > 0, ϑ2 > 0, ϑ3 > 0 is the graph of a smooth function ϑ3/ϑ1 ≡ F (ϑ2/ϑ1) intersected with the quadrant. Theorem 4.3 (p.19, [B-M-N4]). A resonant curve in the quadrant ϑ3/ϑ1 > 0, ϑ2/ϑ1 > 0 is called irreducible if: k23 k m23 m n23 n  6= 0. (4.3) An irreducible resonant curve is uniquely characterized by six non-negative algebraic invariants P1, P2, R1, R2, S1, S2, such that P21 ,P22 R21,R22 S21 ,S22 and permutations thereof. Lemma 4.4 (p. 25, [B-M-N4]). For resonant triplets (k,m, n) associated to a given irreducible resonant curve, that is verifying Eq. (4.3), consider the convolution equation n = k +m. Let σi(n) 6= n, ∀i, 1 ≤ i ≤ 3. Then there are no more that two solutions (k,m) and (m, k), for a given n, provided the six non-degeneracy conditions (3.39)-(3.44) in [B-M-N4] for the algebraic invariants of the irreducible curve are verified. For more details on the technical non-degeneracy conditions, see the Ap- pendix. An exhaustive algebraic geometric investigation of all solutions to n = k +m on irreducible resonant curves is found in [B-M-N4]. The essence of the above lemma lies in that given such an irreducible, “non-degenerate” triplet (k,m, n) on K∗, all other triplets on the same irreducible resonant curves are exhaustively given by the equivariant projective lines: (k,m, n) → (γk, γm, γn), for some γ rational , (4.4) (k,m, n) → (σjk, σjm,σjn), j = 1, 2, 3, (4.5) and permutations of k and m in the above. Of course the homothety γ and the σj symmetries preserve the convolution. This context of irreducible, “non- degenerate” resonant curves yields an infinite dimensional, uncoupled system of rigid body SO(3;R) and SO(3;C) dynamics for the 3D resonant Euler equations (2.30). Theorem 4.5 For any irreducible triplet (k,m, n) which satisfy Theorem 4.3, and under the “non-degeneracy” conditions of Lemma 4.4 (cf. Appendix), the resonant Euler equations split into the infinite, countable sequence of uncou- pled SO(3;R) systems: ȧk = Γkmn(λm − λn)aman, (4.6a) ȧm = Γkmn(λn − λk)anak, (4.6b) ȧn = Γkmn(λk − λm)akam, (4.6c) for all (k,m, n) = γ(σj(k ∗), σj(m ∗), σj(n ∗)), γ = ±1,±2,±3..., 0 ≤ j ≤ 3. (4.7) k∗,m∗, n∗ are some relatively prime integer vectors in Z3 characterizing the equivariant family of projective lines (k,m, n); Γkmn = i < Φk ×Φm,Φ∗n >, Γkmn real. Proof. Theorem 4.5 is a simpler version for invariant manifolds of more general SO(3;C) systems. It is a straightforward corollary of Proposition 3.2, Proposition 3.3, Theorem 3.3, Theorem 3.4 and Theorem 3.5 in [B-M-N4]. The latter article did not explicit the resonant equations and did not use the curl-helicity algebra fundamentally underlying this present work. Rigorously asymptotic infinite countable sequences of uncoupled SO(3;R), SO(3;C) sys- tems are not derived via the usual harmonic analysis tools of Fourier modes, in the 3D Euler context. Polarization of curl eigenvalues and eigenfunctions and helicity play an essential role. Corollary 4.6 Under the conditions λn∗ − λk∗ > 0, λk∗ − λm∗ > 0, the resonant Euler systems (4.6) admit a disjoint, countable family of homoclinic cycles. Moreover, under the conditions λn∗ ≫ +1, λm∗ ≪ −1, |λk∗ | ≪ λn∗ , each subsystem (4.6) possesses orbits whose Hs norms, s ≥ 1, burst arbitrarily large in arbitrarily small times. Remark 4.7 One can prove that there exists some Γmax, 0 < Γmax < ∞, such that |Γkmn| < Γmax, for all (k,m, n) on the equivariant projective lines defined by (4.7). Systems (4.6) “freeze” cascades of energy; their total enstro- phy Ξ(t) = (k,m,n)(λ k(t) + λ m(t) + λ n(t)) remains bounded, albeit with large bursts of Ξ(t)/Ξ(0), on the reversible orbits topologically close to the homoclinic cycles. 4.2 Coupled SO(3) rigid body resonant systems We now derive a new resonant Euler system which couples two SO(3;R) rigid bodies via a common principle axis of inertia and a common moment of inertia. This 5-dimensional system conserves energy, helicity, and is rather interesting in that dynamics on its homoclinic manifolds show bursting cas- cades of enstrophy to the smallest scale in the resonant set. We consider the equivalent periodic lattice geometry under the conditions of Proposition 4.1. In Appendix, we prove that for an “irreducible” 3-wave resonant set which now satisfies the algebraic “degeneracy” (A-4), there exist exactly two “prim- itive” resonant triplets (k,m, n) and (k̃, m̃, n), where k, m, k̃, m̃ are relative prime integer valued vectors in Z3: Lemma 4.8 Under the algebraic degeneracy condition (A-4) the irreducible equivariant family of projective lines in K∗ is exactly generated by the follow- ing two “primitive” triplets: n = k +m, k = ak, m = bm, (4.8a) n = k̃ + m̃, k̃ = a′σi(k) + b ′σj(m), (4.8b) that is, n = ak + bm, (4.8c) n = a′σi(k) + b ′σj(m), (4.8d) where σi 6= σj are some reflection symmetries, a, b, a′, b′ are relatively prime integers, positive or negative, and k, m are relatively prime integer valued vectors in Z3, that is: (a, a′) = (b, b′) = (a, b) = (a′, b′) = 1, (k,m) = 1, where ( , ) denotes the Greatest Common Denominator of two integers. All other resonant wave number triplets are generated by the group actions σl, l = 1, 2, 3, and homothetic rescalings (k,m, n) → γ(k,m, n), (k̃, m̃, n) → γ(k̃, m̃, n), (γ ∈ Z) of the “primitive” triplets. Remark 4.9 It can be proven that the set of such coupled “primitive” triplets is not empty on the periodic lattice. The algebraic irreducibility condition of Lemma 4.2 implies that ±k3/|k| = ±k̃3/|k̃| and ±m3/|m| = ±m̃3/|m̃|, which is obviously verified in equations (4.8). Theorem 4.10 Under conditions of Lemma 4.8 the resonant Euler system reduces to a system of two rigid bodies coupled via an(t): ȧk = (λm − λn)Γaman (4.9a) ȧm = (λn − λk)Γanak (4.9b) ȧn = (λk − λm)Γakam + (λk̃ − λm̃)Γ̃ak̃am̃ (4.9c) ȧm̃ = (λn − λk̃)Γ̃anak̃ (4.9d) ȧk̃ = (λm̃ − λn)Γ̃am̃an, (4.9e) where Γ = i < Φk×Φm,Φ∗n >, Γ̃ = i < Φk̃×Φm̃,Φ∗n >. Energy and Helicity are conserved. Theorem 4.11 The resonant system (4.9) possesses three independent con- servation laws: E1 = a2k + (1− α)a2m, (4.10a) E2 = a2n + αa2m + (1 − α̃)a2m̃, (4.10b) E3 = a2k̃ + α̃a m̃, (4.10c) where α = (λm − λk)/(λn − λk), (4.11a) α̃ = (λm̃ − λn)/(λk̃ − λn). (4.11b) Theorem 4.12 Under the conditions λm < λk < λn, (4.12a) λm̃ < λn < λk̃, (4.12b) which imply α < 0, α̃ < 0, the equilibria (±ak(0), 0, 0, 0,±ak̃(0)) are hyper- bolic for |ak̃(0)| small enough with respect to |ak(0)|. The unstable manifolds of these equilibria are one dimensional, and the nonlinear dynamics of system (4.9) are constrained on the ellipse E1 (4.10a) for ak(t), am(t), the hyperbola E3 (4.10c) for ak̃(t), am̃(t), and the hyperboloid E2 (4.10b) for am(t), am̃(t), an(t). Theorem 4.13 Let the 2-manifold E1 ∩E2 ∩E3 be coordinatized by (am, am̃). On this 2-manifold, the resonant system (4.9) is Hamiltonian, and therefore integrable. Its Hamiltonian vector field h is defined by ιhω = Γ(λn − λk) − Γ̃(λn − λk̃) , (4.13) where ιhω designates the inner product of the symplectic 2-form dam ∧ dam̃ akanak̃ (4.14) with the vector field h. Proof of Theorem 4.13: Eliminating ak(t) via E1, an(t) via E2, ak̃(t) via E3, the resonant system (4.9) reduces to: ȧm = ±Γ(λn − λk)(E1 − (1− α)a2m) 2 (E2 − αa2m + (α̃− 1)a2m̃) ȧm̃ = ±Γ̃(λn − λk̃)(E2 − αa m + (α̃− 1)a2m̃) 2 (E3 − α̃a2m̃) after changing the time variable into (E1 − (1− α)a2m) 2 (E2 − αa2m + (α̃− 1)a2m̃) 2 (E3 − α̃a2m̃) 2 ds . On each component of the manifold E1 ∩E2 ∩E3, the following functionals are conserved: H(am, am̃) = ± Γ̃(λn − λk̃) (E1 − (1− α)a2m)1/2 ± Γ(λn − λk) (E3 − α̃a2m̃)1/2 Observe that the system of two coupled rigid bodies (4.9) does not seem to ad- mit a simple Lie-Poisson bracket in the original variables (ak, am, an, am̃, ak̃). Yet, when restricted to the 2-manifold E1 ∩E2 ∩E3 that is invariant under the flow of (4.9), it is Hamiltonian and therefore integrable. This raises the following interesting issue: according to the shadowing Theorem 2.10, the Euler dynamics remains asymptotically close to that of chains of coupled SO(3;R) and SO(3;C) rigid body systems. Perhaps some new information could be obtained in this way. We are currently investigating this question and will report on it in a forthcoming publication [G-M-N]. Already the simple 5-dimensional system (4.9) has interesting dynamical properties, wich we could not find in the existing literature on systems related to spinning tops. Consider for instance the dynamics of the resonant system (4.9) with I.C. topologically close to the hyperbola equilibria (±ak(0), 0, 0, 0,±ak̃(0)). Un- der the conditions of (4.12) and with the help of the integrability Theorem 4.13, it is easy to construct equivariant families of homoclinic cycles at these hyperbolic critical points: Corollary 4.14 The hyperbolic critical points (±ak(0), 0, 0, 0,±ak̃(0)) pos- sess 1-dimensional homoclinic cycles on the cones a2n + (1− α̃)a2m̃ = −αa2m (4.15) with α < 0, α̃ < 0. Note that these are genuine homoclinic cycles, NOT sums of heteroclinic connections. Initial conditions for the resonant system (4.9) are now chosen in a small neighborhood of these hyperbolic critical points, the corresponding orbits are topologically close to these cycles. With the ordering: λm < λk < λn, (4.16a) |λk| ≪ |λm|, |λk| ≪ λn, (4.16b) λm̃ < λn < λk̃, (4.16c) |λm̃| ≪ λk̃, (4.16d) λk̃ ≫ λn, (4.16e) which can be realized with |a | ≫ 1 and | b | ≪ 1 in the resonant triplets (4.8), we can demonstrate bursting dynamics akin to Theorem 3.9 and 3.11 for enstrophy and Hs norms, s ≥ 2. The interesting feature is the maximization of |ak̃(t)| near the turning points of the homoclinic cycles on the cones (4.15). This corresponds to transfer of energy to the smallest scale k̃, λk̃. In a publication in preparation, we investigate infinite systems of the cou- pled rigid bodies equations (4.9). APPENDIX We focus on a resonant wave number triplet (n, k,m) ∈ (Z∗)3 verifying • the convolution relation n = k +m, (A-1) • the resonant 3-wave resonance relation ± n3√ 1 + ϑ2n 2 + ϑ3n ± k3√ 1 + ϑ2k 2 + ϑ3k ± m3√ 1 + ϑ2m 2 + ϑ3m (A-2) • the condition of “non-catalyticity” k3m3n3 6= 0, (A-3) • and the degeneracy condition of [B-M-N4] (see p26) Giri,j(k,m) = kinjml + klmjni = 0, (A-4) where (i, j, l) is a permutation of (1, 2, 3). Then, we know (see lemma 3.5 (2) of [B-M-N4]) that the system of equations (A-3)-(A-4) for the unknown k and m, given the vector n, admits exactly 4 solutions in Z3 × Z3: (k,m), (m, k), (k̃, m̃), (m̃, k̃). Here k and m are the two vectors of the original resonant triplet, whereas k̃ = ασi(k), m̃ = βσj(m) where mikl −mlki mikl +mlki /∈ {0,±1} and β = mlkj −mjkl mlkj +mjkl /∈ {0,±1} and where the symmetries σi and σj are defined by σi : u = (ul)l=1,2,3 → (−1)δilul l=1,2,3 One verifies that σ2i = σ j = Id, σiσj = σjσi = −σl. That is, the group generated by σi and σj is the Klein group Z/2Z× Z/2Z. Let us first write the irrational numbers α and β under the irreducible representation , β = , with a, a′, b, b′ ∈ Z∗ and (a, a′) = (b, b′) = 1, where ( , ) denotes the Greatest Common Denominator of the integer pair. From k̃ ∈ Z3, it follows that a|a′k; but since (a, a′) = 1, the Euclid’s lemma yields that a|k. Similarly, b|m. Now set k ∈ Z3, m = 1 m ∈ Z3. Hence the integer vector n admits the two decompositions n = ak + bm = a′σi(k) + b ′σj(m). Since the function z 7−→ z3√ 1 + ϑ2z 2 + ϑ3z is homogeneous of degree 0, we see that within the resonance condition (A-2) we can replace each vector k,m and n by any colinear vectors - either integer or not. Suppose now that there exists some positive integer d 6= 1 such that d|k; then d|n, so that by setting n, k0 = k, m0 = we finally obtain n0 = ak0 + bk0 = a ′σi(k0) + b ′σj(m0). The triplets (n0, ak0, bm0) and (n0, a ′σi(k0), b ′σj(m0)) further verify from the above remark, the convolution relation (A-1) and the resonance relation (A-2). Hence, without loss of generality, we can assume that the only positive integer d such that d|k and d|m is 1; which we denote by (k,m) = 1. Equivalently, k1Z+ k2Z+ k3Z+m1Z+m2Z+m3Z+ = Z. Finally, suppose there exists some positive integer d 6= 1 such that d|a and d|b. Then d|n; set n, a0 = a, b0 = Observe that Giri,j(a0k, b0m) = Giri,j(ak, bm) = 0. It follows from lemma 3.5 (2) of [B-M-N4] that the vector n0 of the resonant triplet (n0, a0k, b0m) can also be written as n0 = k̂ + m̂ with (n0, k̂, m̂) verifying (A-2). But then n = dn0 = ak + bm = a ′σi(k) + b ′σj(m) = dk̂ + dm̂. From lemma 3.5 (2) of [B-M-N4], (dk̂, dm̂) must coincide with either one of the pairs (a′σi(k), b ′σj(m)), (b ′σj(m), a ′σi(k)). In particular, d|a′k and d|b′m. Since d|a and (a, a′) = 1, we have (d, a′); similarly (d, b′) = 1. But then Euclid’s lemma yields that d|k and d|m, which contradicts the fact that (k,m) = 1. Hence we have proven that (a, b) = 1. In a similar way, one can show that (a′, b′) = 1. Conclusion: It follows from the above study that n ∈ Z∗ admits the two decompositions n = ak + bm = a′σi(k) + b ′σj(m) (a, a′) = (b, b′) = (a, b) = (a′, b′) = 1, (k,m) = 1. The triplets (n, ak, bm) and (n, a′σi(k), b ′σj(m)) both verify the resonant condition (A-2) (from the homogeneity of this condition) as well as the condi- tion of non-catalyticity (A-3). Indeed, aba′b′ 6= 0 and the condition (A-3) on the initial triplet (n, k,m) imply that the reduced triplet (n, k,m) also verifies (A-3)). Finally, the degeneracy condition (A-4) Giri,j(ak, bm) = 0 is verified. Acknowledgments. We would like to thank A.I. Bobenko, C. Bardos and G. Seregin for very useful discussions. The assistance of Dr. B. S. Kim is gratefully acknowledged. A.M. and B.N. acknowledge the support of the AFOSR contract FA9550-05-1-0047. References [Ar1] Arnold, V.I., Mathematical methods of classical mechanics, Springer- Verlag, New York-Berlin, 1978. [Ar2] Arnold, V.I., Small denominators. I. Mappings of the circumference onto itself, Amer. Math. Soc. Transl. Ser. 2, 46 (1965), p. 213-284. [Ar-Khe] Arnold, V.I. and Khesin, B.A., Topological Methods in Hydrody- namics, Applied Mathematical Sciences, 125, Springer, 1997. [B-M-N1] Babin, A., Mahalov, A. and Nicolaenko, B., Global splitting, in- tegrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids, European J. Mechanics B/Fluids, 15 (1996), p. 291-300. [B-M-N2] Babin, A., Mahalov, A. and Nicolaenko, B., Global regularity and integrability of 3D Euler and Navier-Stokes equations for uniformly ro- tating fluids, Asymptotic Analysis, 15 (1997), p. 103–150. [B-M-N3] Babin, A., Mahalov, A. and Nicolaenko, B., Global regularity of 3D rotating Navier-Stokes equations for resonant domains, Indiana Univ. Math. J., 48 (1999), No. 3, p. 1133-1176. [B-M-N4] Babin, A., Mahalov, A. and Nicolaenko, B., 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vortic- ity, Indiana Univ. Math. J., 50 (2001), p. 1-35. [B-K-M] Beale, J.T., Kato, T. and Majda, A., Remarks on the breakdown of smooth solutions for the 3D Euler equations, Commun. Math. Phys., 94 (1984), p. 61-66. [Bes] Besicovitch, A.S., Almost Periodic Functions, Dover, New York, 1954. [Bo-Mi] Bogoliubov, N.N. and Mitropolsky, Y. A., Asymptotic Methods in the Theory of Non-linear Oscillations, Gordon and Breach Science Pub- lishers, New York, 1961. [Bou-Br] Bourguignon, J.P. and Brezis, H., Remark on the Euler equations, J. Func. Anal., 15 (1974), p. 341-363. [Ch-Ch-Ey-H] Chen, Q., Chen, S., Eyink, G.L. and Holm, D.D., Intermit- tency in the joint cascade of energy and helicity, Phys. Rev. Letters, 90 (2003), p. 214503. [Cor] Corduneanu, C., Almost periodic Functions, Wiley-Interscience, New York, 1968. [DiPe-Li] DiPerna, R.J. and Lions, P.L., Ordinary differential equations, Sobolev spaces and transport theory, Invent. Math., 98 (1989), p. 511- [Fe] Fefferman, C.L., Existence and smoothness of the Navier-Stokes equa- tions, The millennium prize problems, Clay Math. Inst., Cambridge, MA (2006), p. 57-67. [Fri] Frisch, U., Turbulence: the legacy of A. N. Kolmogolov, Cambridge University Press, 1995. [Fro-M-N] Frolova, E., Mahalov, A. and Nicolaenko, B., Restricted interac- tions and global regularity of 3D rapidly rotating Navier-Stokes equa- tions in cylindrical domains, Journal of Mathematical Sciences, Springer, to appear. [Gl1] Gledzer, E.B., Systema gidrodinamicheskovo tipa, dopuskayuchaya dva kvadratichnykh integrala dvizheniya, D. A. N. USSR, 209 (1973), No. 5. [G-D-O] Gledzer, E.B., Dolzhanski, F.V. and Obukhov, A.M., Systemi gidro- dinamitcheskovo tipa i ikh primetchnii, Nauka, Moscow, (1987). [G-M-N] Golse, F., Mahalov, A., Nicolaenko, B., in preparation. [Gu-Ma] Guckenheimer, J. and Mahalov, A., Resonant triad interaction in symmetric systems, Physica D, 54 (1992), 267-310. [Hou1] Hou, T.Y., Deng, J. and Yu, X., Geometric properties and nonblowup of 3D incompressible Euler flow, C.P.D.E., 30 (2005), p. 225-243. [Hou2] Hou, T.Y. and Li, R., Dynamic depletion of vortex stretching and non- blowup of the 3D incompressible Euler equations, CALTECH, preprint (2006). [Ka] Kato, T., Nonstationary flows of viscous and ideal fluids in R3, J. Func. Anal., 9 (1972), p. 296-305. [Ke] Kerr, R.M., Evidence for a singularity of the three dimensional, incom- pressible Euler equations, Phys. Fluids, 5 (1993), No. 7, p. 1725-1746. [Les] Lesieur, M., Turbulence in fluids, 2nd edition, Kluwer, Dortrecht, 1990. [Li] Lions, P.L., Mathematical Topics in Fluid Mechanics: Incompressible Models Vol 1, Oxford University Press, 1998. [Mah] Mahalov, A., The instability of rotating fluid columns subjected to a weak external Coriolis force, Phys. Fluids A, 5 (1993), No. 4, p. 891-900. [M-N-B-G] Mahalov, A., Nicolaenko, B., Bardos, C. and Golse, F., Non blow- up of the 3D Euler equations for a class of three-dimensional initial data in cylindrical domains, Methods and Applications of Analysis, 11 (2004), No. 4, p. 605-634. [Man] Manakhov, S.V., Note on the integration of Euler’s equations of the dynamics of a n-dimensional rigid body, Funct. Anal. and Appl., 10 (1976), No. 4, p. 328-329. [Mor1] Moreau, J.J., Une methode de cinematique fonctionelle en hydrody- namicque, C.R. Acad. Sci. Paris, 249 (1959), p. 2156-2158 [Mor2] Moreau, J.J., Constantes d’un ilôt tourbillonaire en fluide parfait barotrope, C.R. Acad. Sci. Paris, 252 (1961), p. 2810-2812 [Mof] Moffatt, H.K., The degree of knottedness of tangled vortex lines, J. Fluid Mech., 106 (1969), p. 117-129. [Poi] Poincaré, H., Sur la précession des corps déformables, Bull. As- tronomique, 27 (1910), p. 321-356. [Sob] Sobolev, S.L., Ob odnoi novoi zadache matematicheskoi fiziki, Izvestiia Akademii Nauk SSSR, Ser. Matematicheskaia, 18 (1954), No. 1, p. 3–50. [Vish] Vishik, S. M., Ob invariantnyh characteristikah kvadratichno- nelineynyh sistem kaskadnovo tipa, D. A. N. USSR, 228 (1976), No. 6, p. 1269-1270. [We-Wil] Weiland, J. and Wilhelmsson, H., Coherent nonlinear interactions of waves in plasmas, Pergamon, Oxford, 1977. [Yu1] Yudovich, V.I., Non stationary flow of an ideal incompressible liquid, Zb. Vych. Mat., 3 (1963), p. 1032-1066 [Yu2] Yudovich, V.I., Uniqueness theorem for the basic nonstationary prob- lem in th dynamics of an ideal incompressible fluid, Math. Res. Letters, 2 (1995), p. 27-38. [Zak-Man1] Zakharov, V.E. and Manakov, S.V., Resonant interactions of wave packets in nonlinear media, Sov. Phys. JETP Lett., 18 (1973), 243-245. [Zak-Man2] Zakharov, V.E. and Manakov, S.V., The theory of resonance in- teraction of wave packets in nonlinear media, Sov. Phys. JETP, 42 (1976), 842-850. Introduction Vorticity waves and resonances of elementary swirling flows Strictly resonant Euler systems: the SO(3) case Strictly resonant Euler systems: the case of 3-waves resonances on small-scales Infinite dimensional uncoupled SO(3) systems Coupled SO(3) rigid body resonant systems

0704.0338

Synergistic Effects of MoDTC and ZDTP on Frictional Behaviour of Tribofilms at the Nanometer Scale

Microsoft Word - S_Bec_ZDTP_MoDTC_Tribology_Letters.doc Synergistic effects of MoDTC and ZDTP on frictional behaviour of tribofilms at the nanometer scale S. Bec1*, A. Tonck1, J.M. Georges1 and G.W. Roper2 1Laboratoire de Tribologie et Dynamique des Systèmes, UMR CNRS 5513, Ecole Centrale de Lyon, 36 av. Guy de Collongue, 69134 Ecully Cedex, France. 2Lubricants Technology Dept., Shell Global Solutions, Shell Research and Technology Centre, Thornton, P. O. Box 1, Chester CH1 3SH, UK. *To whom correspondence should be addressed Abstract The layered structure and the rheological properties of anti-wear films generated in a rolling/sliding contact from lubricants containing zinc dialkyldithiophosphate (ZDTP) and/or molybdenum dialkyldithiocarbamate (MoDTC) additives have been studied by dynamic nanoindentation experiments coupled with a simple modelling of the stiffness measurements. Local nano-friction experiments were conducted with the same device in order to determine the evolution of the friction coefficient as a function of the applied pressure for the different lubricant formulations. For the MoDTC film, the applied pressure in the friction test remains low (<0.5 GPa) and the apparent friction coefficient is high (µ>0.4). For the tribofilms containing MoDTC together with ZDTP, which permits the applied pressure to increase up to a few GPa through some accommodation process, a very low friction domain appears (0.01<µ<0.05), located a few nanometers below the surface of the tribofilm. This low friction coefficient is attributed to the presence of MoS2 planes sliding over each other in a favourable configuration obtained when the pressure is sufficiently high, which is made possible by the presence of ZDTP. Keywords : ZDTP, MoDTC, tribofilm structure, nanoindentation, mechanical properties, nanofriction, low friction. 1. Introduction In addition to zinc dialkyldithiophosphate (ZDTP) additives, extensively used for their exceptional antioxidant and anti-wear properties under boundary conditions in automotive engines, lubricating oils contain several additives, among which there are detergent and dispersant additives whose main role is to keep oil insoluble contaminants and degradation products in suspension, at elevated temperature for the detergent additives, and at low temperatures for the dispersant ones. Organo molybdenum compounds such as molybdenum dithiocarbamate (MoDTC) are also used as friction modifiers for energy saving. However, when used together in formulated oils, additives interact in various ways resulting either in synergies or in adverse effects affecting the oil performance regarding anti-wear and friction behaviour, and modifying the characteristics of the protective surface films generated during friction (tribofilms). A lot of investigations have been conducted to evaluate the performances of additive mixtures and to determine the composition of associated tribofilms. Several factors were identified as playing a role: additive structure [1, 2], additives concentration [3- 6], base oil nature [7, 8], …, or combinations of these parameters. A detailed review on published information on that topic was written by Willermet [9]. Non-chemical parameters such as characteristics of the solid antagonists (hardness, roughness) or test conditions (load, temperature, sliding speed) [3, 10] also might influence the additive interactions. Among this variety of additive interactions, we will focus on that between ZDTP and MoDTC, extensively studied through chemical investigations. All published works agree upon the fact that friction and anti-wear performances of oils are improved when ZDTP and MoDTC are used together. The formation of molybdenum disulphide (MoS2) on the rubbing surfaces has been evidenced by several authors [11, 12]. Using UHV friction tests, coupled with high-resolution TEM observation of wear debris and spectroscopic studies, Grossiord et al. has given evidence for the mechanism of single MoS2 sheet lubrication [13]. The aim of this paper is to enlarge the knowledge of the local mechanical and frictional properties of anti-wear tribofilms to those of films obtained from lubricants containing different additives (ZDTP, MoDTC, detergent/dispersant) or mixtures of additives, in order to explore the ZDTP/MoDTC synergy on a mechanical point of view. The only published results on that topic are the recent papers from Ye et al. who performed AFM observations and nanoindentation measurements on ZDTP and ZDTP + MoDTC tribofilms [14, 15]. In the present study, nanoindentation tests with continuous stiffness measurements were performed on unwashed and solvent-washed tribofilms to determine their mechanical properties. The frictional behaviour of the tribofilms was investigated through local nanofriction experiments, conducted with the same device. The evolution of the friction coefficient as a function of the applied pressure for the different lubricant formulations leading to different tribofilms has been determined. 2. Preliminary results obtained on ZDTP anti-wear tribofilms The structure and the rheological properties of anti-wear films from a zinc dialkyldithiophosphate (ZDTP) solution generated in a rolling/sliding contact, simulating engine valve train conditions, have been studied in detail with analytical and surface force tools and the results have been published by the authors in a previous paper [16]. As preamble to the present paper, only the main points are summarised here. The ZDTP solution was a commercial secondary alkyl ZDTP additive at 0.1% weight of phosphorus in a highly refined base oil. The ZDTP anti-wear films have a complex structure that has been determined by extensive use of surface analytical techniques. It has been shown that the ZDTP films consisted of at least three non-homogeneous layers: on the steel surface, there is a sulphide/oxide layer, which is almost completely covered by a protective phosphate layer, with the addition of a viscous overlayer of ZDTP degradation precipitates (alkyl phosphate precipitates). This latter layer was removed when the film was washed with an alkane solvent. Therefore, the properties of the ZDTP films have been studied both before and after solvent washing with n-heptane. First, sphere/plane squeeze experiments were performed with a surface force apparatus (SFA) on unwashed films, showing that the overlayer of alkyl phosphate precipitates was heterogeneous and discontinuous, with a thickness of about 900 nm. Second, the mechanical properties were obtained from nanoindentation experiments, performed after replacing the sphere by a diamond tip, and coupled with in-situ topographic imaging procedures to measure the contact area. From the indentation experiments, the properties of the films were determined from normal stiffness measurements and through the application of a rheological film model. On the unwashed specimens, the viscous layer of alkyl phosphate precipitates was detected by the indentation tests. It is a very soft layer, mobile under the diamond tip, with a thickness of a few hundred of nanometers, which was in good agreement with that of sphere/plane experiments. It was also shown that indentation experiments removed this overlayer in the proximity of the tip, probably through a shear flow mechanism. This procedure can be compared to a soft "mechanical" sweep and the mechanical properties of ZDTP tribofilms after such a cleaning were found to be similar to those of solvent washed specimens. The solvent washed tribofilms, comprising sulphide and phosphate layers, exhibited an elastoplastic behaviour and, during the loading stage of the indentation, the hardness and the Young's modulus of the phosphate layer increased from their initial values of about 2 GPa for the hardness and between 30 and 40 GPa for the Young's modulus. In particular, the initial hardness of the polyphosphate layer at the beginning of the indentation tests was close to the mean applied pressure during the films generation. This suggested that the layer accommodated the contact pressure in the tribotest or during the loading stage of the indentation, and could thus be regarded as a final and local pressure sensor. The characteristics of the full ZDTP films ensure gradual changes in mechanical properties between the substrate, bonding layers and outer layers with the viscous overlayer serving as the tribofilm's precursor. The properties of these layered films can thus adapt to a wide range of imposed conditions and provide appropriate level of resistance to contact between the metal surfaces. As the severity of loading increases, so too do the resistive forces within the film. This ensures that the shear plane remains located inside the ZDTP protective film, which explains the exceptional efficiency of ZDTP films as anti-wear films. 3. Experimental 3.1. Tribofilms The tribofilms were generated at Shell Research and Technology Centre, Thornton, U.K., with a reciprocating Amsler machine [17] designed to simulate the contact conditions of the cam/follower system in an internal combustion engine valve train. A flat block specimen (8 mm x 8 mm size, 4 mm thick) has a reciprocating motion in loaded contact with a rotating disc. The block and the disc were made in through-hardened EN31 steel. Special care was taken with the roughness of the blocks which were polished until the average roughness was Ra = 0.01 µm. The movement of the block was driven by a crank linked to the motion of the disc axis through a gearbox. The block motion was approximately sinusoidal and at the same frequency as the disc rotation. Load was applied to the contact by a spring arrangement, acting through a roller bearing. The surface in contact with the loading bearing (rear surface of the reciprocating element) was curved to permit self-alignment between the block and the disc. The films were generated at a normal load of 400 N (mean contact pressure of 0.36 GPa), speed of 600 rev/min., block temperature of approximately 100°C for 5 hours. The lubricants consisted of a highly refined base oil with different commercial additives (details of the oil formulation are not relevant to the present work): - MoDTC solution, - ZDTP + MoDTC solution, - ZDTP + MoDTC + detergent/dispersant solution ("full formulation"). The rubbing area on the polished block was typically 5 mm long in the sliding direction. Previous analyses have shown that the composition in the centre of the wear track was reasonably uniform, while the composition within 1 mm of the ends of the wear track could vary significantly. The mechanical measurements on the films with the Surface Force Apparatus have been performed in the central area of the wear track. An additional unworn and polished block was used to obtain reference values for the EN31 steel substrate. To preserve the film structures, the blocks were stored in the base oil (containing predominantly paraffinic hydrocarbons, with very low concentration of polar compounds) immediately after production of the films in the reciprocating Amsler tests and they were immersed again, when not in use. 3.2. Surface Force Apparatus The Ecole Centrale de Lyon Surface Force Apparatus (SFA) used in these experiments has been described in previous publications [18, 19]. The general principle is that a macroscopic spherical body or a diamond tip can be moved toward and away from a planar one (the ZDTP specimen) using the expansion and the vibration of a piezoelectric crystal, along the three directions, Ox, Oy (parallel to the plane surface) and Oz (normal to the plane surface). The plane specimen is supported by double cantilever sensors, measuring quasi-static normal and tangential forces (respectively Fz and Fx). Each of these is equipped with a capacitive sensor. The sensor's high resolution allows a very low compliance to be used for the force measurement (up to 2 x 10-6 m/N). Three capacitive sensors were designed to measure relative displacements in the three directions between the supports of the two solids, with a resolution of 0.01 nm in each direction. Each sensor capacitance was determined by incorporating it in an LC oscillator operating in the range 5 - 12 MHz [20]. 3.3. Tests methodology All the experiments were conducted at room temperature. Preliminary results obtained on anti-wear films from a ZDTP solutions have shown that n-heptane washing damages the film [16]. That is why the blocks were tested first as obtained from the Amsler friction test, without any cleaning and second after washing with n-heptane. The unwashed specimens were mounted on the SFA as taken from the storage base oil. Excess of base oil was simply removed by placing the side of the specimen on absorbing paper, which allowed the surface to be always preserved by an oil film (thickness > 10 µm). Nanoindentation tests The aim of these tests was to determine the elastoplastic properties of the tribofilms (hardness and Young's modulus) and their “mechanical” structure (number of layers and estimation of the thickness of each layer that constitutes the film). The method used to perform nanoindentation experiment with the SFA has already been published in detail [21]. Specific procedures have been developed for the characterisation of ZDTP tribofilms and have been described in previous papers [16, 22]. In this study, the determination of the near surface mechanical properties (first nanometers) was obtained through a specific tip shape calibration, performed on a gold film deposited by magnetron sputtering onto a silicon substrate. This film was very smooth (peak to valley roughness around 1 nm, measured on a scan length of 1 µm) and its hardness was constant versus depth from the surface and until the penetration depth equals the gold film's thickness [21]. For the nanoindentation experiments, a trigonal diamond tip with an angle of 115.12° between edges (Berkovitch type) was used. The indentation tests were performed in controlled displacement mode. The standard set-up included the continuous quasi-static measurements of the resulting normal force Fz versus the normal displacement Z, at a slow penetration speed, generally 0.1 to 0.5 nm/s. It also included the simultaneous measurements of the rheological behaviour (dissipative and conservative or elastic contributions) of the tested surface, thanks to simultaneous small sinusoidal motions at a frequency of 37 Hz, with an amplitude of about 0.2 nm RMS. Furthermore, using the Z feedback in the constant force mode and the tangential displacement of the indenter, the surface topography was imaged before and after the indentation test, with the same diamond tip. This was made practically possible because of the partial elastic recovery during the unloading cycle and hence the geometry of tip and indent were different which was necessary to permit resolution of the indent. For this scanning procedure, a constant normal load of 0.5 µN was typically used. Such in-situ imaging procedure enables the operator to choose precisely the location of the indentation test on the surface and, after the test, to quantify the plastic pile-up around the indent and thus to measure the actual contact area. Rheological film model The elastic properties of the films were very difficult to extract from the indentation tests because of both the influence of the substrate and of the film structure itself. They were obtained through the stiffness measurements, which are global (film+substrate) measurements. To extract the properties of each layer of the film, a simple model has been developed, and its main features are described as follows. The experimental stiffness versus normal displacement curve was identified with the elastic response of a structure composed of one or two homogeneous elastic layers on a substrate (semi-infinite elastic half space) indented by a rigid cylindrical punch of radius a. For such a system, modelled by two springs connected in series [23], the calculated global stiffness (Kz) depends on the reduced Young's modulus of the substrate (Es* with Es*=Es/(1-νs2)), measured on an unworn steel block, over the contact radius (a) and depends also on four unknown parameters which are the reduced Young's modulus (Ef*, Ef*=Ef/(1-νf2)) and the thickness (t) of each layer. For each test, their values were adjusted to obtain a good fit between the measured stiffness curve and the calculated one. This procedure provided the structure (one or two layers), the thickness and the reduced Young's modulus of each layer that constituted the tribofilms. Details are given in a previous paper [16]. Following this model, the global stiffness of a single layer system is given by: 22K t a a E aEz f s π π * * (1) This simple model describes perfectly the behaviour of model systems such as gold layers on a silicon substrate [21]. In the case of tribofilms, deviations may be observed at a critical pressure or at a critical depth from which the experimentally measured stiffness may be found to exceed significantly the theoretical one. This is interpreted as a change in the surface properties due to the applied pressure and appears to be related to a measured hardness increase. Indeed, as the applied pressure can reach values much larger than the initial hardness value of the surface, the resulting plastic flow may induce a small volume reduction and molecular rearrangements which could be sufficient to induce a noticeable change in the mechanical properties. From a threshold pressure value, H0, the stiffness curve was then influenced both by the substrate's elasticity and by the change in mechanical properties. This pressure dependence can be introduced in the model by writing that in the deformed volume of material, when H>H0 (i.e. when the film accommodates the applied pressure through hardness increase), the film modulus Ef* is proportional to the hardness (the ratio Ef*/H remains constant). It gives the following equation: EE = (2) Ef0* is the reduced Young's modulus value, when the applied pressure is equal to or lower than the threshold pressure H0. When necessary, by introducing this effect in our modelling and by adjusting the value of the threshold pressure, we were able to fit correctly the whole stiffness curve. An example of such fit is given on figure 1. The evolution of the film modulus Ef* versus plastic depth can also be extracted from equation 1 using the experimentally measured global (film+substrate) stiffness values Kz and the film's thickness, t, independently of equation 2. This permits a check on whether it is proportional to the hardness as assumed in equation 2. In the example shown figure 2, the calculated Young's modulus of the film (from equation 1 with a film thickness t=25 nm) is found to be proportional to the measured hardness with a mean ratio Ef*/H=16.5, in good agreement with the ratio Ef0*/H0=17/1.05=16.2 obtained from the stiffness fit. Full formulation (ZDTP+MoDTC+detergent/dispersant) Solvent washed tribofilm 0 10 20 30 40 50 Penetration depth (nm) Measured stiffness Calculated stiffness, t=25 nm, Efo*=17 GPa, without pressure accommodation Calculated stiffness, t=25 nm, Efo*=17 GPa, with pressure accommodation, Ho=1.05 GPa Figure 1: Example of application of the rheological film model: measured and calculated global stiffness for a tribofilm obtained from the full formulation (MoDTC + ZDTP + detergent/dispersant). A good fit between the measured and the calculated values is obtained with a single layer system (thickness t=25 nm and reduced Young's modulus Ef0*=17 GPa) and a pressure accommodation effect from a threshold pressure H0=1.05 GPa. t = 25 nmEf0* = 17 GPa H0 = 1.05 GPa Full formulation (ZDTP+MoDTC+detergent/dispersant) Solvent washed tribofilm 0 10 20 30 40 50 60 Plastic depth (nm) Reduced Young's modulus of the tribofilm, Ef* Hardness of the tribofilm, H t = 25 nmEf0* = 17 GPa H0 = 1.05 GPa Full formulation (ZDTP+MoDTC+detergent/dispersant) Solvent washed tribofilm 0 10 20 30 40 50 60 Plastic depth (nm) Reduced Young's modulus of the tribofilm, Ef* Hardness of the tribofilm, H Figure 2: Example of evolution of film's reduced Young's modulus and hardness versus plastic depth, for a tribofilm obtained from the "full formulation" (MoDTC + ZDTP + detergent/dispersant). The Young's modulus of the film is calculated using equation 1 with the measured stiffness values and using only the film's thickness determined from the fit shown figure 1 (t = 25 nm). Nanofriction experiments Nanofriction experiments were conducted on the blocks by moving the diamond tip along Ox direction (parallel to the surface) at low speed (2 to 5 nm/s) along a distance of 0.5 µm. The objective of these tests was to determine how the friction coefficient varies as a function of the applied pressure. The tests were conducted at monitored increasing depth. During the tests, the normal, Fz, and the tangential, Fx, forces were recorded, which allowed us to calculate the apparent friction coefficient µ=Fx/Fz (see example figure 3). Full formulation (ZDTP+MoDTC+detergent/dispersant) Solvent washed tribofilm 0 20 40 60 80 100 120 140 Time (s) µ=Fx/Fz 0 5 10 15 20 25 Penetration depth (nm) Indentation test Nanofriction test Smaller contact area Figure 3: Procedure used for the nanofriction tests. The diamond tip is oriented edge first and the nanofriction tests are conducted at monitored increasing depth. During the test, the normal (Fz) and tangential (Fx) forces are recorded. The friction coefficient µ=Fx/Fz is calculated. Large pile-up was observed in the case of nanofriction with the diamond tip oriented face first, which may induce large uncertainty in the calculation of the contact area. That is why the nanofriction tests were conducted edge first. In these conditions, the estimation of the applied pressure at a given depth was obtained using low load nanoindentation tests, made in the near proximity of the nanofriction tests. Assuming that, at a given depth, the hardness of the tribofilm should be the same for the friction test and for the near indentation test, the contact area, and then the applied pressure, were obtained from the difference between the normal force measured for the two tests at the same depth (see insert on figure 3). Using the in-situ imaging procedure, figure 4 shows an example of an image of the surface of a tribofilm after such a nanofriction test. 100 nm Beginning of the testBeginning of the wear Direction of friction 100 nm Beginning of the testBeginning of the wear Direction of friction Figure 4: Typical image of the surface of a tribofilm after a nanofriction experiment. The image is obtained with the in-situ imaging procedure. 4. Results The first part presents the mechanical properties of the different tribofilms, determined from the nanoindentation experiments. Their structure, one or two layers, and their thickness were deduced from the use of our rheological film model. The results concerning the frictional behaviour of the tribofilms are given in a second part. 4.1. Structure and mechanical properties of the tribofilms MoDTC tribofilms The tribofilm obtained from base oil + MoDTC has been tested without washing and after washing with n-heptane. Even on the solvent washed block, it was not possible to make any local topographic image nor line scanning preliminary to the indentations tests, revealing that the film was very soft and was easily damaged by the diamond tip. Representative hardness curves obtained on the MoDTC tribofilms are shown on figure 5. MoDTC tribofilm 0 40 80 120 160 Plastic depth (nm) Unwashed tribofilm Solvent-washed tribofilm Figure 5: Typical hardness curves obtained on the MoDTC tribofilms. Open symbols correspond to hardness curves obtained on the unwashed film. Black symbols correspond to hardness curves obtained on the solvent-washed film. Very low mechanical properties were measured on the unwashed MoDTC tribofilm. The surface hardness ranged from 0.02 to 0.1 GPa indicating the presence of a very soft overlayer covering the tribofilm. After washing with n-heptane, the indentation tests showed that this overlayer has been removed by the washing procedure. The remaining tribofilm was a soft homogeneous layer, whose hardness was typically in the range 0.4 - 0.5 GPa at the beginning of the tests. Adhesion to the diamond tip was detected at the end of the unloading part of the tests. The film thickness and the structure (number of layers) have been obtained from the stiffness measurements performed during the experiments using the rheological film model. The film appeared to be homogeneous in its thickness, and for most of the tests, its elastic behaviour corresponded to that of a single layer, with constant properties versus depth. The thickness of the film was found to be between 30 and 75 nm. The reduced Young's modulus was typically equal to 7 – 8 GPa. ZDTP + MoDTC tribofilms From optical observation, the ZDTP + MoDTC unwashed film was very thin. This was confirmed by the indentation tests. Prior to any contact, a very soft layer, 60 to 120 nm thick, was detected at the surface of the unwashed film. Indentation tests conducted after scanning or imaging the surface of the unwashed film ("mechanical sweep") showed that the film was spatially heterogeneous. Its thickness and its mechanical properties varied depending on the test location: - In some places, only a very thin layer (a few nanometers thick) with a reduced Young's modulus of 50 GPa covered the work-hardened steel substrate (tests A and B on figure 6). - A thicker layer (15 to 30 nm) with a reduced Young's modulus of 50 – 80 GPa was found in other places (tests C and D on figure 6), sometimes with accommodation pressure effect (threshold pressure H0 = 4.8 GPa). Such layer behaves like the sulphide-oxide layer of the ZDTP tribofilm [16]. - Elsewhere, the structure of the tribofilm was more complex, with a soft layer covering a stiffer one. For example, test E on figure 6 corresponds to a soft layer, 12 nm thick, with properties comparable to those of the MoDTC tribofilm (hardness of 0.2 GPa and reduced Young's modulus of 5 GPa) which covers a stiffer layer, 18 nm thick, with a reduced Young's modulus of 50 GPa. This heterogeneity was confirmed by the indentation tests conducted on the solvent-washed ZDTP + MoDTC tribofilm, where, at least, three different types of film were identified: - In some places, the film behaved like a one layer system, able to accommodate the pressure (pressure threshold 2.8 GPa). Its thickness was between 35 nm and 150 nm. The surface hardness was about 2 – 3 GPa and the reduced Young's modulus was about 55 - 65 GPa. - In other places, the film behaved like a bilayered structure: a surface layer, about 25 nm thick, with properties comparable to those of the MoDTC tribofilm (hardness of 0.3 - 0.4 GPa, reduced Young's modulus of 8 GPa), covers a stiffer layer, 150 nm thick, with a reduced Young's modulus of about 80 GPa. - Elsewhere, the surface film was between 3 and 15 nm thick, with properties comparable to the lower properties measured on the ZDTP tribofilm (hardness about 1 – 1.5 GPa and reduced Young's modulus about 10 GPa). For some tests, this surface film was able to accommodate the pressure, with a pressure threshold of 1 – 1.5 GPa. It covers a stiffer layer, 10 to 55 nm thick, with a reduced Young's modulus varying from 60 to 110 GPa. ZDTP+MoDTC tribofilm Unwashed block 0 20 40 60 80 100 120 Plastic depth (nm) Prior to any contact After imaging - test A After imaging - test B After imaging - test C After imaging - test D After imaging - test E Figure 6: Representative hardness curves obtained on the unwashed ZDTP + MoDTC tribofilm, prior to any contact and after the imaging procedure. The film is spatially heterogeneous in thickness and in mechanical properties. ZDTP + MoDTC + detergent/dispersant tribofilms ("full formulation" tribofilms) Nanoindentation tests performed in fresh areas, prior to any contact showed that, at the surface of the unwashed tribofilm, there was a very soft layer, mobile under the diamond tip, with an apparent thickness of a few hundreds of nanometers. Representative hardness curves obtained on the unwashed block near these initial contacts are shown figure 7. Contrary to the ZDTP + MoDTC tribofilm, the film was found to be spatially homogeneous. Only its thickness was found to vary, depending on the tested area. A very thin softer layer was detected at the surface of the tribofilm, which did not resist to imaging nor scanning, except if the normal load was very low (lower than 0.3 µN). This layer had a hardness value (about 0.3 – 0.4 GPa) comparable to the hardness value of the MoDTC tribofilm. The observed large hardness increase when the load increased also indicated that the tribofilm had a great capability to accommodate the applied pressure. This result was confirmed by the interpretation of the stiffness measurements using the rheological model, which also showed that the tribofilm had a complex structure. At its surface, there was first a layer with a thickness of only a few nanometers (2 nm to 7 nm) and a reduced Young's modulus of 10 - 15 GPa. Then, there was a second layer (thickness between 20 nm and 140 nm) with a higher reduced Young modulus of 65 – 80 GPa. A similar tribofilm was tested after n-heptane washing. It also had a great ability to accommodate the applied pressure. From the stiffness measurements, on most places, the film was found to behave like a film constituted by two layers. The surface layer was thin (5 to 25 nm) with a reduced Young's modulus value in the range 15 – 20 GPa. The thickness of the underlayer was found to vary between 0 (no underlayer, example of figures 1 and 2) and 100 nanometers and its elastic modulus was in the range 110 - 120 GPa. ZDTP + MoDTC + detergent/dispersant Unwashed tribofilm 0 10 20 30 40 50 60 Plastic depth (nm) First test, prior to any contact Without preliminary scanning After scanning or imaging Figure 7: Representative hardness curves obtained on the unwashed ZDTP + MoDTC + detergent/dispersant tribofilm ("full formulation"), prior to any contact and in the region near the first contacts, either without preliminary surface scanning or after scanning/imaging procedure. Figure 8 compares representative hardness curves for all tested tribofilms. For the ZDTP + MoDTC tribofilms, three curves are plotted because of the variety of obtained results revealing the spatial heterogeneity of this tribofilm. A representative hardness curve for the ZDTP tribofilm tested in the same conditions in a previous study [16] has been added for comparison. 0 5 10 15 20 25 30 35 40 Total penetration depth (nm) ZDTP, solvent washed MoDTC, solvent washed ZDTP + MoDTC, unwashed (2 tests) ZDTP + MoDTC, solvent washed Full formulation, unwashed Full formulation, solvent washed Figure 8: Comparison of the hardness curves obtained on the different tribofilms. The hardness curve obtained for a ZDTP anti-wear tribofilm obtained from a previous study is plotted for comparison. 4.2. Nanofriction experiments Nanofriction experiments were conducted on the three preceding tribofilms and also on a ZDTP tribofilm and on a ZDTP + detergent/dispersant tribofilm. In order to simplify the following graphs, only one representative curve was plotted for each tribofilm (or two when it was necessary to illustrate the dispersion when it was significant). Figure 9 shows the evolution of the friction force versus the normal force for the tested tribofilms. For a given formulation, there was very little difference between the results obtained on unwashed and on solvent washed tribofilms at low load, indicating that the solvent washing does not seem to affect the frictional behaviour of the tribofilm. This agrees with the idea that the soft viscous overlayer is supposed to serve as precursor for the tribofilm rather than that it plays a mechanical role during friction. 0 3 6 9 12 15 Normal force, Fz (µN) ZDTP, solvent washed ZDTP + Det/Disp, solvent washed MoDTC, solvent washed ZDTP + MoDTC, solvent washed ZDTP + MoDTC, solvent washed ZDTP + MoDTC, unwashed Full formulation, unwashed Full formulation, solvent washed Figure 9: Friction force (Fx) versus normal force (Fz) during nanofriction tests with increasing penetration depth for different tribofilms. It is also worth noting that the heterogeneity in mechanical properties found on the ZDTP + MoDTC tribofilm also exists in the frictional properties. For this tribofilm, the friction force at low normal loads may be comparable either to the friction force obtained for the ZDTP tribofilm or to the friction force obtained for the "full formulation" tribofilm. Under the present testing conditions, it can be observed that the lower friction forces were obtained for films containing MoDTC together with ZDTP. The higher were obtained for the tribofilm from MoDTC alone. Figure 10 shows the evolution of the friction coefficient versus mean pressure. The existence of low friction coefficient values (0.01<µ<0.05) appears to be related both to the presence of MoDTC additive in the initial lubricant and to the ability for the tribofilm to reach sufficiently high pressure values (1.5 – 3 GPa) during the friction test. Thus, the MoDTC tribofilm, which is not able to resist to the contact pressure by increasing its mechanical properties seems to be ineffective in reducing friction, contrary to the tribofilms containing ZDTP and MoDTC together, which are able to accommodate the contact pressure by increasing their mechanical properties. Nevertheless, both behaviours (high or low friction) were observed for the ZDTP + MoDTC tribofilms. This is certainly due to the spatial heterogeneity of these tribofilms, which behave on some places like ZDTP tribofilms, or elsewhere like "full formulation" tribofilms. It was also observed that tribofilms formed without MoDTC were ineffective in reducing friction even if high contact pressures were reached during the friction tests. 0 1 2 3 4 5 6 Mean pressure P (GPa) ZDTP, solvent washed ZDTP + Det/Disp, solvent washed MoDTC, solvent washed ZDTP + MoDTC, solvent washed ZDTP + MoDTC, solvent washed ZDTP + MoDTC, unwashed Full formulation, unwashed Full formulation, solvent washed Figure 10: Apparent friction coefficient versus mean pressure for the different tested tribofilms. When the evolution of the friction coefficient is plotted versus penetration depth (figure 11), it appears that, when it existed, the low friction coefficient domain was detected a few nanometers below the surface of the tribofilm. It also shows that, for the full formulation, the low friction domain was deeper for the unwashed tribofilm than for the solvent washed one. The unwashed tribofilm appears to be covered by a surface layer with rather bad frictional properties, which can be removed by solvent washing or by "mechanical" sweep (low load scanning procedures for example). 0 2 4 6 8 10 12 14 16 18 20 Penetration depth (nm) ZDTP, solvent washed ZDTP + Det/Disp, solvent washed MoDTC, solvent washed ZDTP + MoDTC, solvent washed ZDTP + MoDTC, solvent washed ZDTP + MoDTC, unwashed Full formulation, unwashed Full formulation, solvent washed Figure 11: Apparent friction coefficient versus penetration depth for the different tested tribofilms. 5. Discussion Because of the inhomogeneous and patchy nature of anti-wear tribofilms and of their low thickness, very few results are published concerning their mechanical properties [24-28]. Moreover, the differences in sample preparation and the diversity of used techniques and experimental procedures render delicate the comparison of the obtained results. For example, the Young’s modulus values given by Aktary et al. for a ZDTP tribofilm [28] are significantly higher that those we measured but one explanation can be that they did not take into account the substrate’s elasticity in their calculations, contrary to what is done in the current study. Or if we attempt to compare our results with those recently published by Ye et al. on ZDTP and ZDTP + MoDTC tribofilms [14, 15], this reveals significant differences. For example, Ye et al. found that both tribofilms possess the same hardness and modulus depth distributions, corresponding to continuously and functionally graded materials, when in the present work, the hardness curves for similar tribofilms did not coincide and the use of our rheological film model allowed us to describe the tribofilms as layered materials with properties adaptable to contact conditions. The hardness and modulus values, respectively 10 GPa at a contact depth of 30 nm and 215 GPa at a depth of 20 nm, that they reported are also significantly higher than those we measured and also higher than those given by Aktary et al. This could be due to differences in sample preparation and also certainly to the use of different methods and assumptions for the treatment of the nanoindentation data. Concerning the frictional behaviour of the tribofilms, the presented nanofriction tests were conducted in unlubricated conditions, at very low speed (2 to 5 nm/s) and the measured nanofriction coefficients corresponded to the friction between the diamond tip and the tribofilm (over its steel substrate). That is why it also seems difficult to compare our values to macroscopic friction coefficient values obtained on classical tribometers. The latter are representative of steel on steel contact in the presence of a tribofilm and are averaged over the whole contact surface. However, our local values are not far from the end of test Amsler macroscopic friction coefficient values published by Pidduck and Smith [25] for ZDTP, ZDTP + detergent/dispersant and ZDTP + friction modifier tribofilms. Moreover, these macroscopic values were found to be proportional, with a factor 0.7, to micro-friction coefficient values measured with Lateral Force Microscopy by the same authors, making them suggest that there may be a link between macro and micro-frictional behaviour of smooth regions of anti-wear tribofilms. Unfortunately, no tribofilm obtained from friction modifier alone were tested in this study, with which we could compare our results. Nevertheless, macroscopic friction coefficient values, in the range 0.10 – 0.14, measured on an alternative ball on plane tribometer were reported by Muraki and Wada [6] for oil containing MoDTC alone. They conclude that such lubricant was ineffective in reducing friction, contrary to the oil containing MoDTC together with ZDTP. More recently, similar high macroscopic friction coefficient values (in the range 0.095 – 0.2) were measured by Unnikrishnan et al. for oil containing MoDTC alone [29]. On the other hand, Grossiord et al. reported very low steady-state friction coefficient (0.04) measured for base oil + MoDTC during SRV friction tests, and a lower steady-state value (0.02) for friction tests in a UHV tribometer, carried out by sliding a macroscopic hemispherical steel pin again a flat covered by a MoDTC tribofilm [13]. From tests carried out in a high frequency reciprocating rig, Graham et al. [30] also reported that, in the absence of ZDTP, MoTDC additives were effective in reducing friction at a combination of high additive concentration and high temperature (up to 0.4% wt. and 200°C). Such diversity of results, certainly partly due to the various tests conditions, makes unreasonable a comparison between the very high nanofriction coefficient measured on the MoDTC tribofilm under the present testing conditions and those published values. As, regarding the literature, the formation of MoS2 was well established for MoDTC containing lubricants, the question is how can we explain such high friction coefficient during the nanofriction tests ? Or what caused the very low friction observed when ZDTP was used together with MoDTC ? From figure 10, the low friction coefficient values (0.01<µ<0.05) were observed for the MoDTC containing lubricants when the contact pressure was in the range 1.5 – 3 GPa (the question of the spatial heterogeneity of the ZDTP + MoDTC tribofilm will be discussed latter). These high pressures were measured for tribofilms able to increase their mechanical properties, thus accommodating the contact conditions, which was demonstrated to be the case for ZDTP anti-wear tribofilms [16]. On the other hand, high pressures were not reached for the soft MoDTC tribofilm. Thus, the easy sliding of the MoS2 sheets could result from a favourable orientation induced by sufficiently high contact pressure values. The ability of MoS2 sheets to orient in a favourable direction was reported by Grossiord et al. [31] and Martin et al. [32], who recently investigated tribochemical interactions between ZDTP, MoDTC and OCB (overbased detergent calcium borate) additives. Using high resolution TEM observations of wear debris, coupled with wear scar micro-spot XPS analysis, they observed perfectly oriented MoS2 sheets, with their basal plane parallel to the flaky wear fragments. Such "mechanical" interpretation of the role of the contact pressure agrees with previous work of Muraki et al. who studied the effect of roller hardness on the rolling sliding characteristics of MoDTC in the presence of ZDTP and concluded that the friction reduction effect increased with higher degree of roller hardness [10]. Yamamoto also reported that a necessary condition for improving the friction and wear characteristics of a lubricant was the formation of surface films composed of iron phosphates with high hardness and Mo-S compounds [11]. Concerning the spatial heterogeneity of the ZDTP + MoDTC tribofilms, it can be worth noting that using high resolution TEM observations of wear debris collected after friction tests, coupled with AES and XPS studies of rubbing surfaces, Grossiord et al. described the ZDTP + MoDTC tribofilm as being composed of a mixture of glassy zinc phosphate zones containing molybdenum, and carbon- rich zones containing zinc and highly-dispersed MoS2 single sheets [13, 33]. The observation that, during the nanofriction tests, the low friction domain was located a few nanometers below the surface also corroborates this interpretation. As the nanofriction tests were conducted at increasing depth, the sufficiently high pressures were obtained after a few nanometers penetration depth inside the MoS2 containing layer (with properties similar to the MoDTC tribofilm), thanks to the presence of the underneath resisting anti-wear layer, whose characteristics are similar to those of the phosphate layer of the ZDTP tribofilm. Finally, combining the results obtained from the nanoindentation and nanofriction experiments, we can propose a possible schematic description of the anti-wear tribofilms obtained from the "full formulation" oil. Some assumptions are also made on what happened during nanofriction tests on such tribofilms (see figure 12 on which for convenient drawing, as the Berkovitch diamond tip is not sharp, it was represented by a flat punch). A soft layer containing non-oriented MoS2 sheets is present at the surface of the tribofilm (layer (a) in figure 12). This layer, 0 to 25 nm thick, has mechanical properties comparable with those of the MoDTC tribofilm (0.3 – 0.5 GPa for the hardness and 3 – 10 GPa for the reduced Young's modulus). Its friction coefficient is rather high. This layer is easily damaged or removed by the diamond tip during imaging or line-scanning procedures. When the contact pressure is sufficiently high, friction induces a favourable orientation of the MoS2 sheets, over a thickness of 1 or 2 nanometers (layer (b) in figure 12), resulting in very low friction coefficient values which combine with the anti-wear efficiency of the tribofilm. Under this layer, there is then an anti-wear layer (layer (c) in figure 12), with properties similar to those of the polyphosphate layer of the ZDTP tribofilm. Then, just over the substrate (noted (e)in figure 12), there is a bonding layer (layer (d) in figure 12) with high mechanical properties (oxides, sulfides). Figure 12: Possible schematic description of the anti-wear tribofilm obtained from the "full formulation" and orientation of the MoS2 planes of the outer layer resulting from a nanofriction tests (for convenient drawing, as the Berkovitch diamond tip is not sharp, it was represented by a flat punch). The thickness of each layer is arbitrary drawn as it varies significantly depending on the tested area (from zero when the layer is not present to a few tens of nanometers). (a) Soft layer containing non-oriented MoS2 sheets, with mechanical properties comparable to those of the MoDTC tribofilm, (b) Layer of favourably frictionally oriented MoS2 sheets with a typical thickness of 1 or 2 nm, (c) Layer with properties similar to those of the polyphosphate layer of the ZDTP tribofilm, (d) Bonding layer with high mechanical properties (oxides, sulfides), (e) Steel substrate. 6. Conclusions Thanks to the combined used of (i) nanoindentation experiments with continuous stiffness measurements coupled with imaging procedures, (ii) a specifically developed rheological film model and (iii) nanofriction tests, synergistic effects of ZDTP and MoDTC on frictional behaviour of anti-wear tribofilms have been evidenced from mechanical considerations. One original feature of this study lies in the characterisation of unwashed anti-wear tribofilms with their full structure preserved. The structure and nanomechanical properties (hardness and reduced Young's modulus) of tribofilms formed with different mixtures of additives (ZDTP, MoDTC, detergent/dispersant) were first determined. Concerning the occurrence of very low friction (0.01<µ<0.05), the contact pressure was found to be a critical parameter. The low friction coefficient values were attributed to a favourable orientation of MoS2 sheets present in the outer layer of the tribofilms formed from MoDTC containing lubricants. Such a favourable orientation occurred only if sufficiently high contact pressure was reached. These high contact pressures were attained when ZDTP was used as oil additive together with MoDTC because one of the main characteristics of ZDTP additives is to form protective anti-wear tribofilms under boundary lubrication, with varying structure and properties with depth, among which is an amazing ability to increase their mechanical properties, thus accommodating the contact conditions. A possible schematic description of the tribofilms containing both ZDTP and MoDTC was deduced and a mechanism was proposed to account for the mechanical synergy that occurs during nanofriction tests on such tribofilms. Aknowledgement The authors thank Shell Research Limited for financial support and permission to publish. References [1] F. G. Rounds, ASLE Transactions, 24 (4) (1980) 431-440. [2] M. Muraki and H. Wada, Tribology International, 35 (2002) 857-863. [3] Z. Yin, M. Kasrai, M. Fuller, G. M. Bancroft, K. Fyfe and K. H. Tan, Wear, 202 (1997) 172-191. [4] Z. Yin, M. Kasrai, G. M. Bancroft, K. Fyfe, M. L. Colaianni and K. H. Tan, Wear, 202 (1997) 192-201. [5] P. A. Willermet, D. P. Dailey, R. O. Carter III, P. J. Schmitz, W. Zhu, J. C. Bell and D. Park, Tribology International, 28 (3) (1995) 163-175. [6] M. Muraki and H. Wada, in Lubricants and Lubrication - Proceedings of Leeds-Lyon 21, D. Dowson et al. , Elsevier, Tribology Series, 30, (1995) 409-422. [7] A. K. Misra, A. K. Mehrotra and R. D. Srivastava, Wear, 31 (2) (1975) 345-357. [8] M. D. Johnson, R. K. Jensen and S. Korcek, SAE Technical Paper Series, Engine Oil Rheology and Tribology (SP-1303) - n°972860, (1997) 37-47. [9] P. A. Willermet, Tribology Letters, 5 (1998) 41-47. [10] M. Muraki, Y. Yanagi and K. Sakaguchi, Japanese Journal of Tribology, 40 (2) (1995) 41-51. [11] Y. Yamamoto, S. Gondo, T. Kamakura and N. Tanaka, Wear, 112 (1986) 79-87. [12] M. Muraki, Y. Yanagi and K. Sakaguchi, Tribology International, 30 (1) (1997) 69-75. [13] C. Grossiord, K. Varlot, J. M. Martin, T. Le Mogne, C. Esnouf and K. Inoue, Tribology International, 31 (12) (1998) 737-743. [14] J. Ye, M. Kano and Y. Yasuda, Tribology Letters, 13 (1) (2002) 41-47. [15] J. Ye, M. Kano and Y. Yasuda, Journal of Applied Physics, 93 (9) (2003) 5113-5117. [16] S. Bec, A. Tonck, J. M. Georges, R. C. Coy, J. C. Bell and G. W. Roper, Proc. R. Soc. Lond. A, 455 (1999) 4181-4203. [17] G. W. Roper and J. C. Bell, Society of Automotive Engineers Fuels and Lubricants Meeting and Exposition, Toronto, Canada, Paper SAE 952473, (1995) [18] A. Tonck, J. M. Georges and J. L. Loubet, J. Colloid Interface Sci., 126 (1988) 150-163. [19] A. Tonck, S. Bec, D. Mazuyer, J. M. Georges and A. A. Lubrecht, Journal of Engineering Tribology - Proc Instn Mech Engrs Part J, 213 (J5) (1999) 353-361. [20] J. M. Georges, A. Tonck, D. Mazuyer, E. Georges, J. L. Loubet and F. Sidoroff, J. Phys. II France, 6 (1996) 57-76. [21] S. Bec, A. Tonck, J. M. Georges, E. Georges and J. L. Loubet, Philosophical Magazine A, Revue CL, 74, (5), (1996) 1061-1072. [22] A. Tonck, S. Bec, J. M. Georges, J. C. Bell, R. C. Coy and G. W. Roper, in Lubrication at the Frontier : The Role of the Interface and Surface Layers in the Thin Films and Boundary Regimes, Proceedings of Leeds-Lyon 25, D. Dowson et al. , Elsevier, Tribology Series, 36, Amsterdam, The Netherlands, (1999) 39-47. [23] S. Bec and A. Tonck, in The Third Body Concept : Interpretation of Tribological Phenomena, Proceedings of Leeds-Lyon 22, G. Dalmaz, D. Dowsen, C.M. Taylor and T.H.C Childs, Elsevier, Tribology Series, 31, Amsterdam, the Netherlands, (1996) 173-184. [24] P. A. Willermet, R. O. Carter III, P. J. Schmitz, M. Everson, D. J. Scholl and W. H. Weber, Lubrication Sci., 9 (4) (1997) 325-348. [25] A. J. Pidduck and S. G.C., Wear, 212 (1997) 254-264. [26] O. L. Warren, J. F. Graham, P. R. Norton, J. E. Houston and T. A. Michalske, Tribology Letters, 4 (1998) 189-198. [27] J. F. Graham, C. McCague and P. R. Norton, Tribology Letters, 6 (1999) 149-157. [28] M. Aktary, M. T. McDermott and G. A. McAlpine, Tribology Letters, 12 (3) (2002) 155- 162. [29] R. Unnikrishnan, M. C. Jain, A. K. Harinarayan and A. K. Mehta, Wear, 252 (2002) 240- 249. [30] J. F. Graham, H. A. Spikes and S. Korcek, Tribology Transactions, 44 (4) (2001) 626- 636. [31] C. Grossiord, J. M. Martin, K. Varlot, B. Vacher and T. Le Mogne, Tribology Letters, 8 (4) (2000) 203-212. [32] J. M. Martin, C. Grossiord, K. Varlot, B. Vacher, T. Le Mogne and Y. Yamada, Lubrication Science, 15 (2) (2003) 119-132. [33] C. Grossiord, J. M. Martin, T. Le Mogne, K. Inoue and J. Igarashi, Journal of Vacuum Science and Technology A, 17 (3) (1999) 884-890.

0704.0339

Lattice Boltzmann inverse kinetic approach for the incompressible Navier-Stokes equations

Lattice Boltzmann inverse kinetic approach for the incompressible Navier-Stokes equations Enrico Fonda1,Massimo Tessarotto1,2 and Marco Ellero3 1Dipartimento di Matematica e Informatica, Università di Trieste, Italy 2Consorzio di Magnetofluidodinamica, Trieste, Italy 3Institute of Aerodynamics, Technical University of Munich, Munich, Germany (Dated: August 18, 2021) In spite of the large number of papers appeared in the past which are devoted to the lattice Boltzmann (LB) methods, basic aspects of the theory still remain unchallenged. An unsolved theo- retical issue is related to the construction of a discrete kinetic theory which yields exactly the fluid equations, i.e., is non-asymptotic (here denoted as LB inverse kinetic theory). The purpose of this paper is theoretical and aims at developing an inverse kinetic approach of this type. In principle infinite solutions exist to this problem but the freedom can be exploited in order to meet important requirements. In particular, the discrete kinetic theory can be defined so that it yields exactly the fluid equation also for arbitrary non-equilibrium (but suitably smooth) kinetic distribution func- tions and arbitrarily close to the boundary of the fluid domain. This includes the specification of the kinetic initial and boundary conditions which are consistent with the initial and boundary conditions prescribed for the fluid fields. Other basic features are the arbitrariness of the ”equi- librium” distribution function and the condition of positivity imposed on the kinetic distribution function. The latter can be achieved by imposing a suitable entropic principle, realized by means of a constant H-theorem. Unlike previous entropic LB methods the theorem can be obtained without functional constraints on the class of the initial distribution functions. As a basic consequence, the choice of the the entropy functional remains essentially arbitrary so that it can be identified with the Gibbs-Shannon entropy. Remarkably, this property is not affected by the particular choice of the kinetic equilibrium (to be assumed in all cases strictly positive). Hence, it applies also in the case of polynomial equilibria, usually adopted in customary LB approaches. We provide different possible realizations of the theory and asymptotic approximations which permit to determine the fluid equations with prescribed accuracy. As a result, asymptotic accuracy estimates of customary LB approaches and comparisons with the Chorin artificial compressibility method are discussed. PACS numbers: 47.27.Ak, 47.27.eb, 47.27.ed 1 - INTRODUCTION - INVERSE KINETIC THEORIES Basic issues concerning the foundations classical hy- drodynamics still remain unanswered. A remarkable as- pect is related the construction of inverse kinetic theo- ries (IKT) for hydrodynamic equations in which the fluid fields are identified with suitable moments of an appropri- ate kinetic probability distribution. The topic has been the subject of theoretical investigations both regarding the incompressible Navier-Stokes (NS) equations (INSE) [1, 2, 3, 4, 5, 6] and the quantum hydrodynamic equations associated to the Schrödinger equation [7]. The impor- tance of the IKT-approach for classical hydrodynamics goes beyond the academic interest. In fact, INSE rep- resent a mixture of hyperbolic and elliptic pde’s, which are extremely hard to study both analytically and nu- merically. As such, their investigation represents a chal- lenge both for mathematical analysis and for computa- tional fluid dynamics. The discovery of IKT [1] provides, however, a new starting point for the theoretical and nu- merical investigation of INSE. In fact, an inverse kinetic theory yields, by definition, an exact solver for the fluid equations : all the fluid fields, including the fluid pres- sure p(r, t), are uniquely prescribed in terms of suitable momenta of the kinetic distribution function, solution of the kinetic equation. In the case of INSE this per- mits, in principle, to determine the evolution of the fluid fields without solving explicitly the Navier-Stokes equa- tion, nor the Poisson equations for the fluid pressure [6]. Previous IKT approaches [2, 3, 4, 5, 7] have been based on continuous phase-space models. However, the inter- esting question arises whether similar concepts can be adopted also to the development of discrete inverse ki- netic theories based on the lattice Boltzmann (LB) the- ory. The goal of this investigation is to propose a novel LB theory for INSE, based on the development of an IKT with discrete velocities, here denoted as lattice Boltzmann inverse kinetic theory (LB-IKT). In this paper we intend to analyze the theoretical foundations and basic proper- ties of the new approach useful to display its relation- ship with previous CFD and lattice Boltzmann methods (LBM) for incompressible isothermal fluids. In particu- lar, we wish to prove that it delivers an inverse kinetic http://arxiv.org/abs/0704.0339v1 theory, i.e., that it realizes an exact Navier-Stokes and Poisson solver. 1a - Motivations: difficulties with LBM’s Despite the significant number of theoretical and nu- merical papers appeared in the literature in the last few years, the lattice Boltzmann method [8, 9, 10, 11, 12, 13, 14] - among many others available in CFD - is prob- ably the one for which a complete understanding is not yet available. Although originated as an extension of the lattice gas automaton [15, 16] or a special discrete form of the Boltzmann equation [17], several aspects re- garding the very foundation of LB theory still remain to be clarified. Consequently, also the comparisons and ex- act relationship between the various lattice Boltzmann methods (LBM) and other CFD methods are made dif- ficult or, at least, not yet well understood. Needless to say, these comparisons are essential to assess the relative value (based on the characteristic computational com- plexity, accuracy and stability) of LBM and other CFD methods. In particular the relative performance of the numerical methods depend strongly on the characteris- tic spatial and time discretization scales, i.e., the minimal spatial and time scale lengths required by each numerical method to achieve a prescribed accuracy. On the other hand, most of the existing knowledge of the LBM’s prop- erties originates from numerical benchmarks (see for ex- ample [18, 19, 20]). Although these studies have demon- strated the LBM’s accuracy in simulating fluid flows, few comparisons are available on the relative computational efficiency of the LBM and other CFD methods [17, 21]. The main reason [of these difficulties] is probably because current LBM’s, rather than being exact Navier-Stokes solvers, are at most asymptotic ones (asymptotic LBM’s), i.e., they depend on one or more infinitesimal parame- ters and recover INSE only in an approximate asymptotic sense. The motivations of this work are related to some of the basic features of customary LB theory representing, at the same time, assets and weaknesses. One of the main reasons of the popularity of the LB approach lays in its simplicity and in the fact that it provides an ap- proximate Poisson solver, i.e., it permits to advance in time the fluid fields without explicitly solving numeri- cally the Poisson equation for the fluid pressure. How- ever customary LB approaches can yield, at most, only asymptotic approximations for the fluid fields. This is because of two different reasons. The first one is the dif- ficulty in the precise definition of the kinetic boundary conditions in customary LBM’s, since sufficiently close to the boundary the form of the distribution function pre- scribed by the boundary conditions is not generally con- sistent with hydrodynamic equations. The second reason is that the kinetic description adopted implies either the introduction of weak compressibility [8, 9, 11, 12, 13, 14] or temperature [22] effects of the fluid or some sort of state equation for the fluid pressure [23]. These assump- tions, although physically plausible, appear unacceptable from the mathematical viewpoint since they represent a breaking of the exact fluid equations. Moreover, in the case of very small fluid viscosity customary LBM’s may become inefficient as a conse- quence of the low-order approximations usually adopted and the possible presence of the numerical instabilities mentioned above. These accuracy limitations at low vis- cosities can usually be overcome only by imposing severe grid refinements and strong reductions of the size of the time step. This has the inevitable consequence of rais- ing significantly the level of computational complexity in customary LBM’s (potentially much higher than that of so-called direct solution methods), which makes them inefficient or even potentially unsuitable for large-scale simulations in fluids. A fundamental issue is, therefore, related to the con- struction of more accurate, or higher-order, LBM’s, ap- plicable for arbitrary values of the relevant physical (and asymptotic) parameters. However, the route which should permit to determine them is still uncertain, since the very existence of an underlying exact (and non- asymptotic) discrete kinetic theory, analogous to the con- tinuous inverse kinetic theory [2, 3], is not yet known. According to some authors [24, 25, 26] this should be linked to the discretization of the Boltzmann equation, or to the possible introduction of weakly compressible and thermal flow models. However, the first approach is not only extremely hard to implement [27], since it is based on the adoption of higher-order Gauss-Hermite quadra- tures (linked to the discretization of the Boltzmann equa- tion), but its truncations yield at most asymptotic the- ories. Other approaches, which are based on ’ad hoc’ modifications of the fluid equations (for example, intro- ducing compressibility and/or temperature effects [28]), by definition cannot provide exact Navier-Stokes solvers. Another critical issue is related to the numerical sta- bility of LBM’s [29], usually attributed to the violation of the condition of strict positivity (realizability condition) for the kinetic distribution function [29, 30]. Therefore, according to this viewpoint, a stability criterion should be achieved by imposing the existence of an H-theorem (for a review see [31]). In an effort to improve the ef- ficiency of LBM numerical implementations and to cure these instabilities, there has been recently a renewed in- terest in the LB theory. Several approaches have been proposed. The first one involves the adoption of entropic LBM’s (ELBM [30, 32, 33, 34] in which the equilibrium distribution satisfies also a maximum principle, defined with respect to a suitably defined entropy functional. However, usually these methods lead to non-polynomial equilibrium distribution functions which potentially re- sult in higher computational complexity [35] and less nu- merical accuracy[36]. Other approaches rely on the adop- tion of multiple relaxation times [37, 38]. However the efficiency, of these methods is still in doubt. Therefore, the search for new [LB] models, overcoming these limita- tions, remains an important unsolved task. 1b - Goals of the investigation The aim of this work is the development of an inverse kinetic theory for the incompressible Navier-Stokes equa- tions (INSE) which, besides realizing an exact Navier- Stokes (and Poisson) solver, overcomes some of the lim- itations of previous LBM’s. Unlike Refs. [2, 3], where a continuous IKT was considered, here we construct a dis- crete theory based on the LB velocity-space discretiza- tion. In such a type of approach, the kinetic description is realized by a finite number of discrete distribution func- tions fi(r, t), for i = 0, k, each associated to a prescribed discrete constant velocity ai and defined everywhere in the existence domain of the fluid fields (the open set Ω×I ). The configuration space Ω is a bounded subset of the Euclidean space R3and the time interval I is a subset of R. The kinetic theory is obtained as in [2, 3] by introduc- ing an inverse kinetic equation (LB-IKE) which advances in time the distribution function and by properly defin- ing a correspondence principle, relating a set of velocity momenta with the relevant fluid fields. To achieve an IKT for INSE, however, also a proper treatment of the initial and boundary conditions, to be satisfied by the kinetic distribution function, must be in- cluded. In both cases, it is proven that they can be de- fined to be exactly consistent - at the same time - both with the hydrodynamic equations (which must hold also arbitrarily close to the boundary of the fluid domain) and with the prescription of the initial and Dirichlet bound- ary conditions set for the fluid fields. Remarkably, both the choice of the initial and equilibrium kinetic distri- bution functions and their functional class remain essen- tially arbitrary. In other words, provided suitable min- imal smoothness conditions are met by the kinetic dis- tributions function, for arbitrary initial and boundary kinetic distribution functions, the relevant moment equa- tions of the kinetic equation coincide identically with the relevant fluid equations. This includes the possibility of defining a LB-IKT in which the kinetic distribution function is not necessarily a Galilean invariant. This arbitrariness is reflected also in the choice of pos- sible ”equilibrium” distribution functions, which remain essentially free in our theory, and can be made for exam- ple in order to achieve minimal algorithmic complexity. A possible solution corresponds to assume polynomial- type kinetic equilibria, as in the traditional asymptotic LBM’s. These kinetic equilibria are well-known to be non-Galilean invariant with respect to arbitrary finite velocity translations. Nevertheless, as discussed in detail in Sec.4, Subsection 4A, although the adoption of Galilei invariant kinetic distributions is in possible, this choice does not represent an obstacle for the formulation of a LB-IKT. Actually Galilean invariance need to be fulfilled only by the fluid equations. The same invariance prop- erty must be fulfilled only by the moment equations of the LB-IKT and not necessarily by the whole LB inverse kinetic equation (LB-IKE). Another significant development of the theory is the formal introduction of an entropic principle, realized by a constant H-theorem, in order to assure the strict pos- itivity of the kinetic distribution function in the whole existence domain Ω× I. The present entropic principle departs significantly from the literature. Unlike previ- ous entropic LBM’s it is obtained without imposing any functional constraints on the class of the initial kinetic distribution functions. Namely without demanding the validity of a principle of entropy maximization (PEM, [39]) in a true functional sense on the form of the distri- bution function. Rather, it follows imposing a constraint only on a suitable set of extended fluid fields, in particu- lar the kinetic pressure p1(r, t).The latter is uniquely re- lated to the actual fluid pressure p(r, t) via the equation p1(r, t) = p(r, t) + Po(t), with Po(t) > 0 to be denoted as pseudo-pressure. The constant H-theorem is therefore obtained by suitably prescribing the function Po(t) and implies the strict positivity. The same prescription as- sures that the entropy results maximal with respect in the class of the admissible kinetic pressures, i.e., it satisfies a principle of entropy maximization. Remarkably, since this property is not affected by the particular choice of the kinetic equilibrium, the H-theorem applies also in the case of polynomial equilibria. We stress that the choice of the entropy functional remains essentially arbitrary, since no actual physical interpretation can be attached to it. For example, without loss of generality it can always be identified with the Gibbs-Shannon entropy. Even pre- scribing these additional properties, in principle infinite solutions exist to the problem. Hence, the freedom can be exploited to satisfy further requirements (for example, mathematical simplicity, minimal algorithmic complex- ity, etc.). Different possible realizations of the theory and comparisons with other CFD approaches are considered. The formulation of the inverse kinetic theory is also use- ful in order to determine the precise relationship between the LBM’s and previous CFD schemes and in particular to obtain possible improved asymptotic LBM’s with pre- scribed accuracy. As an application, we intend to con- struct asymptotic models which satisfy with prescribed accuracy the required fluid equations [INSE] and possi- bly extend also the range of validity of traditional LBM’s. In particular, this permits to obtain asymptotic accuracy estimates of customary LB approaches. The scheme of presentation is as follows. In Sec.2 the INSE problem is recalled and the definition of the extended fluid fields {V, p1} is presented. In Sec. 3 the basic assumptions of previous asymptotic LBM’s are recalled. In.Sec.4 and 5 the foundations of the new inverse kinetic theory are laid down and the integral LB inverse kinetic theory is presented, while in Sec. 6 the entropic theorem is proven to hold for the kinetic distribution function for properly defined kinetic pressure. Finally, in Sec.7 various asymp- totic approximations are obtained for the inverse kinetic theory and comparisons are introduce with previous LB and CFD methods and in Sec. 8 the main conclusions are drawn. 2 - THE INSE PROBLEM A prerequisite for the formulation of an inverse kinetic theory [2, 3] providing a phase-space description of a clas- sical (or quantum) fluid is the proper identification of the complete set of fluid equations and of the related fluid fields. For a Newtonian incompressible fluid, referred to an arbitrary inertial reference frame, these are provided by the incompressible Navier-Stokes equations (INSE) for the fluid fields {ρ,V,p} ∇ ·V = 0, (1) NV = 0, (2) ρ(r,t) = ρo. (3) There are supplemented by the inequalities p(r,t) ≥ 0, (4) ρo > 0. (5) Equations (1)-(3) are defined in a open connected set Ω ⊆ R3 (defined as the subset of R3 where ρ(r,t) > 0) with boundary δΩ, while Eqs. (4) and (5) apply on its closure Ω. Here the notation is standard. Thus, N is the NS operator NV ≡ρo V +∇p+ f − µ∇2V, (6) with D +V · ∇ the convective derivative, f denotes a suitably smooth volume force density acting on the fluid element and µ ≡ νρo > 0 is the constant fluid viscosity. In particular we shall assume that f can be represented in the form f = −∇Φ(r) + f1(r,t) where we have separated the conservative ∇Φ(r) and the non-conservative f1 parts of the force. Equations (1)-(3) are assumed to admit a strong solution in Ω × I, with I ⊂ R a possibly bounded time interval. By assumption {ρ,V,p} are continuous in the closure Ω. Hence if in Ω×I, f is at least C(1,0)(Ω×I), it follows necessarily that {V,p} must be at least C(2,1)(Ω × I). In the sequel we shall impose on {V,p} the initial conditions V(r,to) = Vo(r), (7) p(r, to) = po(r). Furthermore, for greater mathematical simplicity, here we shall impose Dirichlet boundary conditions on δΩ V(·,t)| δΩ = VW (·,t)|δΩ p(·,t)| δΩ = pW (·,t)|δΩ . Eqs.(3) and (7)-(8) define the initial-boundary value problem associated to the reduced INSE (reduced INSE problem). It is important to stress that the previous problem can also formulated in an equivalent way by re- placing the fluid pressure p(r, t) with a function p1(r, t) (denoted kinetic pressure) of the form p1(r, t) = Po + p(r, t), (9) where Po = Po(t) is prescribed (but arbitrary) real func- tion of time and is at least Po(t) ∈ C (1)(I). {V,p1} will be denoted hereon as extended fluid fields and Po(t) will be denoted as pseudo-pressure. 3 - ASYMPTOTIC LBM’S 3A - Basic assumptions As is well known, all LB methods are based on a dis- crete kinetic theory, using a so-called lattice Boltzmann velocity discretization of phase-space (LB discretization). This involves the definition of a kinetic distribution func- tion f, which can only take the values belonging to a finite discrete set {fi(r, t), i = 0, k} (discrete kinetic dis- tribution functions). In particular, it is assumed that the functions fi, for i = 0, k, are associated to a discrete set of k+1 different ”velocities” {ai, i = 0, k} . Each ai is an ’a priori’ prescribed constant vector spanning the vector space Rn (with n = 2 or 3 respectively for the treatment of two- and three-dimensional fluid dynamics),and each fi(r, t) is represented by a suitably smooth real function which is defined and continuous in Ω×I and in particular is at least C(k,j)(Ω× I) with k ≥ 3. The crucial aspect which characterizes customary LB approaches [8, 9, 10, 11, 12, 13, 14, 17, 40, 41] involves the construction of kinetic models which allow a finite sound speed in the fluid and hence are based on the assumption of a (weak) compressibility of the same fluid. This is realized by assuming that the evolution equation (kinetic equation) for the discrete distributions fi(r, t) (i = 1, k), depends at least one (or more) infinitesimal (asymptotic) parameters (see below). Such approaches are therefore denoted as asymptotic LBM’s. They are characterized by a suitable set of assumptions, which typically include: 1. LB assumption #1: discrete kinetic equation and correspondence principle: the first assumption con- cerns the definition of an appropriate evolution equation for each fi(r, t) which must hold (together with all its moment equations) in the whole open set Ω× I. In customary LB approaches it takes the form of the so-called LB-BGK equation [13, 41, 42] L(i)fi = Ωi(fi), (10) where i = 0, k. Here L(i) is a suitable streaming operator, Ωi(fi) = −νc(fi − f i ) (11) (with νc ≥ 0 a constant collision frequency) is known as BKG collision operator (after Bhatba- gar, Gross and Krook [43]) and f i is an ”equi- librium” distribution to be suitably defined. In customary LBM’s it is implicitly assumed that the solution of Eq.(10), subject to suitable initial and boundary conditions exists and is unique in the functional class indicated above. In partic- ular, usually L(i) is either identified with the fi- nite difference streaming operator (see for example [8, 11, 13, 42]), i.e., L(i)fi(r, t) = LFD(i)fi(r, t) ≡ [fi(r+ ai∆t, t+∆t)− fi(r, t)] or with the dif- ferential streaming operator (see for instance [17, 40, 41]) L(i) = LD(i) ≡ + ai · . (12) Here the notation is standard. In particular, in the case of the operator LFD(i), ∆t and c∆t ≡ Lo are appropriate parameters which define respectively the characteristic time- and length- scales associ- ated to the LBM time and spatial discretizations. A common element to all LBM’s is the assump- tion that all relevant fluid fields can be identified, at least in some approximate sense, with appro- priate momenta of the discrete kinetic distribu- tion function (correspondence principle). In par- ticular, for neutral and isothermal incompressible fluids, for which the fluid fields are provided re- spectively by the velocity and pressure fluid fields {Yj(r, t), j = 1, 4} ≡ {V(r, t), p(r, t)} , it is as- sumed that they are identified with a suitable set of discrete velocity momenta (for j = 1, 4) Yj(r, t) = i=0,k Xji(r, t)fi(r, t), (13) where Xji(r, t) (with i = 0, k and j = 1, k) are ap- propriate, smooth real weight functions. In the literature several examples of correspondence prin- ciples are provided, a particular case being provided by the so-called D2Q9 (V, p)-scheme [44, 45] p(r, t) = c2 i=0,k fi = c i=0,k i , (14) V(r,t) = i=1,k aifi = i=1,k i , (15) where k = 8 and c = min {|ai| > 0, i = 0, k} is a characteristic parameter of the kinetic model to be interpreted as test particle velocity. In customary LBM’s the parameter cs = (with D the dimen- sion of the set Ω) is interpreted as sound speed of the fluid. In order that the momenta (14) and (15) recover (in some suitable approximate sense) INSE , however, appropriate subsidiary conditions must be met. 2. LB assumption #2: Constraints and asymptotic conditions: these are based on the introduc- tion of a dimensionless parameter ε, to be consid- ered infinitesimal, in terms of which all relevant parameters can be ordered. In particular, it is required that the following asymptotic orderings [17, 40, 41] apply respectively to the fluid fields ρo,V(r, t), p(r, t), the kinematic viscosity ν = µ/ρo and Reynolds number Re = LV/ν: ρo,V(r, t), p(r, t) ∼ o(ε 0), (16) [1 + o(ε)] ∼ o(εαR), (17) Re ∼ 1/o(ε αR), (18) where αR ≥ 0. Here we stress that the position for ν holds in the case of D2Q9 only, while the generalization to 3D and other LB discretizations. is straightforward. Furthermore, the velocity c and collision frequency νc are ordered so that c ∼ 1/o(εαc), (19) νc ∼ 1/o(ε αν ), (20) ∼ o(εα), (21) with α ≡ αν−αc > 0; the characteristic length and time scales, Lo ≡ c∆t and ∆t for the spatial and time discretization are assumed to scale as ∼ o(εαL), (22) ∼ o(εαt), (23) with αt, αL > 0. Here L and T are the (smallest) characteristic length and time scales, respectively for spatial and time variations of V(r, t) and p(r.t). Imposing also that 1 results infinitesimal at least of order ∼ o(εα) it follows that it must be also αt − αL > 0. These assumptions imply necessarily that the dimension- less parameter M eff ≡ V (Mach number) must be ordered as M eff ∼ O(εαc) (24) (small Mach-number expansion). 3. LB assumption #3: Chapman-Enskog expansion - Kinetic initial conditions, relaxation conditions: it is assumed that the kinetic distribution function fi(r, t) admits a convergent Chapman-Enskog ex- pansion of the form fi = f i + δf i + δ i + .., (25) where δ ≡ εα and the functions f i (j ∈ N) are assumed smooth functions of the form (multi- scale expansion) f i (ro, r1, r2, ..to, t1, t2, ..), where rn = δ nr, tn = δ nt and n ∈ N. In typical LBM’s the parameter δ is usually identified with ε (which requires letting α = 1), while the Chapman-Enskog expansion is usually required to hold at least up to order o(δ2). In addition the initial conditions fi(r, to) = f i (r, to), (26) (for i = 0, k) are imposed in the closure of the fluid domain Ω. It is well known [46] that this position generally (i.e., for non-stationary fluid fields), im- plies the violation of the Chapman-Enskog expan- sion close to t = to, since the approximate fluid equations are recovered only letting δf 0, i.e., assuming that the kinetic distribution func- tion has relaxed to the Chapman-Enskog form (25). This implies a numerical error (in the evaluation of the correct fluid fields) which can be overcome only discarding the first few time steps in the numerical simulation. 4. LB assumption #5: Equilibrium kinetic distribu- tion: a possible realization for the equilibrium dis- tributions f i (i = 0, k) is given by a polynomial of second degree in the fluid velocity [44] i (r, t) = wi [p− Φ(r)] + (27) +wiρo ai ·V ai ·V Here, without loss of generality, the case of the D2Q9 LB discretization will be considered, with wi and ai (for i = 0, 8) denoting prescribed dimension- less constant weights and discrete velocities. Notice that, by definition, f i is not a Galilei scalar. Nev- ertheless, it can be considered approximately in- variant, at least with respect to low-velocity trans- lations which do not violate the low-Mach number assumption (24). 5. LB assumption #6: Kinetic boundary conditions: They are specified by suitably prescribing the form of the incoming distribution function at the bound- ary δΩ. [47, 48, 49, 50, 51, 51, 52, 53, 54, 54, 55, 56, 57, 58, 59]. However, this position is not generally consistent with the Chapman-Enskog solution (25) (see related discussion in Appendix A). As a con- sequence violations of the hydrodynamic equations may be expected sufficiently close to the boundary, a fact which may be only alleviated (but not com- pletely eliminated) by adopting suitable grid refine- ments near the boundary. An additional potential difficulty is related to the condition of strict posi- tivity of the kinetic distribution function [57] which is not easily incorporated into the no-slip boundary conditions [50, 51, 52]. 3B - Computational complexity of asymptotic LBM’s The requirements posed by the validity of these hy- potheses may strongly influence the computational com- plexity of asymptotic LBM’s which is usually associated to the total number of ”logical” operations which must be performed during a prescribed time interval. There- fore, a critical parameter of numerical simulation meth- ods is their discretization time scale ∆t. This is - in turn - related to the Courant number NC = , where V and Lo.denote respectively the sup of the magnitude of the fluid velocity and the amplitudes of the spatial dis- cretization. As is well known ”optimal” CFD simulation methods typically allow Lo ∼ L and a definition of the time step ∆t = ∆tOpt such that NC ∼ V ∆tOpt ∼ 1. In- stead, for usual LBM’s satisfying the low-M eff assump- tion (24), the Courant number is very small since it re- sults NC = M eff Lo ∼ O(εα)Lo . This means that their discretization time scale of ∆t is much smaller than ∆tOpt and reads ∆t ∼M eff ∆tOpt. (28) In addition, depending on the accuracy of the numeri- cal algorithms adopted for the construction of the dis- crete kinetic distribution function, also the ratio Lo sults infinitesimal in the sense Lo ∼ o(εαL), with suitable αL > 0. Finally, we stress that LB approaches based on the adoption of the finite-difference streaming opera- tor LFD(i) are usually only accurate to order o(∆t 2). For them, therefore, the requirement placed by Eq.(28) might be even stronger. This implies that traditional LBM’s may involve a vastly larger computation time than that afforded by more efficient numerical methods. 4 - NEW LB INVERSE KINETIC THEORY (LB-IKT) A basic issue in LB approaches [8, 11, 13, 42] con- cerns the choice of the functional class of the discrete kinetic distribution functions fi (i = 0, k) as well as the related definition of the equilibrium discrete distribution function f i [which appears in the BGK collision opera- tor; see Eq.(11)]. This refers in particular to their trans- formation properties with respect to arbitrary Galilean transformations, and specifically to their Galilei invari- ance with respect to velocity translations with constant velocity. In statistical mechanics it is well known that the ki- netic distribution function is usually assumed to be a Galilean scalar. The same assumption can, in principle, be adopted also for LB models. However, the kinetic distribution functions fi and f i do not necessarily re- quire a physical interpretation of this type. In the se- quel we show that for a discrete inverse kinetic theory it is sufficient that fi and f i be so defined that the mo- ment equations coincide with the fluid equations (which by definition are Galilei covariant). It is sufficient to de- mand that both fi and f i are identified with a ordinary scalars with respect to the group of rotation in R2, while they need not be necessarily invariant with respect to arbitrary velocity translations. This means that fi is in- variant only for a particular subset of inertial reference frames. For example for a fluid which at the initial time moves locally with constant velocity an element of this set can be identified with the inertial frame which in the same position is locally co-moving with the fluid. The adoption of non-translationally invariant discrete distributions fi is actually already well known in LBM and results convenient for its simplicity. This means, manifestly, that in general no obvious physical interpre- tation can be attached to the other momenta of the dis- crete kinetic distribution function. As a consequence, the very definition of the concept of statistical entropy to be associated to the f ′is is essentially arbitrary, as well as the related principle of entropy maximization, typically used for the determination of the equilibrium distribution function f i . Several authors, nevertheless, have investi- gated the adoption of possible alternative formulations, which are based on suitable definitions of the entropy functional and/or the requirement of approximate or ex- act Galilei invariance (see for example [29, 32, 62]). 4A - Foundations of LB-IKT As previously indicated, there are several important motivations for seeking an exact solver based on LBM. The lack of a theory of this type represents in fact a weak point of LB theory. Besides being a still unsolved theoretical issue, the problem is relevant in order to de- termine the exact relationship between the LBM’s and traditional CFD schemes based on the direct discretiza- tion of the Navier–Stokes equations. Following ideas re- cently developed [2, 3, 4, 5, 7], we show that such a theory can be formulated by means of an inverse kinetic theory (IKT) with discrete velocities. By definition such an IKT should yield exactly the complete set of fluid equations and which, contrary to customary kinetic approaches in CFD (in particular LB methods), should not depend on asymptotic parameters. This implies that the inverse ki- netic theory must also satisfy an exact closure condition. As a further condition, we require that the fluid equa- tions are fulfilled independently of the initial conditions for the kinetic distribution function (to be properly set) and should hold for arbitrary fluid fields. The latter re- quirement is necessary since we must expect that the validity of the inverse kinetic theory should not be lim- ited to a subset of possible fluid motions nor depend on special assumptions, like a prescribed range of Reynolds numbers. In principle a phase-space theory, yielding an inverse kinetic theory, may be conveniently set in terms of a quasi-probability, denoted as kinetic distribution func- tion, f(x, t). A particular case of interest (investigated in Refs.[2, 3]) refers to the case in which f(x, t) can actu- ally be identified with a phase-space probability density. In the sequel we address both cases, showing that, to a certain extent, in both cases the formulation of a generic IKT can actually be treated in a similar fashion. This requires the introduction of an appropriate set of consti- tutive assumptions (or axioms). These concern in par- ticular the definitions of the kinetic equation - denoted as inverse kinetic equation (IKE) - which advances in time f(x, t) and of the velocity momenta to be identified with the relevant fluid fields (correspondence principle). However, further assumptions, such as those involving the regularity conditions for f(x, t) and the prescription of its initial and boundary conditions must clearly be added. The concept [of IKT] can be easily extended to the case in which the kinetic distribution function takes on only discrete values in velocity space. In the sequel we consider for definiteness the case of the so-called LB discretization, whereby - for each (r, t) ∈ Ω × I - the kinetic distribution function is discrete, and in particu- lar admits a finite set of discrete values fi(r, t) ∈ R, for i = 0, k, each one corresponding to a prescribed constant discrete velocity ai ∈ R 3 for i = 0, k. 4B - Constitutive assumptions Let us now introduce the constitutive assumptions (ax- ioms) set for the construction of a LB-IKT for INSE, whose form is suggested by the analogous continuous inverse kinetic theory [2, 3]. The axioms, define the ”generic” form of the discrete kinetic equation, its func- tional setting, the momenta of the kinetic distribution function and their initial and boundary conditions, are the following ones: Axiom I - LB–IKE and functional setting. Let us require that the extended fluid fields {V,p1} are strong solutions of INSE, with initial and boundary conditions (7)-(8) and that the pseudo pressure po(t) is an arbitrary, suitably smooth, real function. In particu- lar we impose that the fluid fields and the volume force belong to the minimal functional setting: p1,ΦǫC (2,1)(Ω× I), VǫC(3,1)(Ω× I), (29) (1,0)(Ω× I). We assume that in the set Ω×I the following equation LD(i)fi = Ωi(fi) + Si (30) [LB inverse kinetic equation (LB-IKE)] is satisfied iden- tically by the discrete kinetic distributions fi(r, t) for i = 0, k. Here Ωi(fi) and LD(i) are respectively the BGK and the differential streaming and operators [Eqs.(11) and (12)], while Si is a source term to be defined. We require that KB-IKE is defined in the set Ω× I, so that Ωi(fi) and Si are at least that C (1)(Ω × I) and contin- uous in Ω × I. Moreover Ωi(fi), defined by Eq.(11), is considered for generality and will be useful for compar- isons with customary LB approaches. We remark that the choice of the equilibrium kinetic distribution f the BGK operator remains completely arbitrary. We assume furthermore that in terms of fi the fluid fields {V, p1} are determined by means of functionals of the form MXj [fi] = i=0,8 Xjfi (denoted as discrete velocity momenta). For X = X1, X2 (with X1 = c 2, X2 = these are related to the fluid fields by means of the equa- tions (correspondence principle) p1(r, t)− Φ(r) = c i=0,8 fi = c i=0,8 i , (31) V(r,t)= i=1,8 aifi = i=1,8 i , (32) where c = min {|ai| , i = 1, 8} is the test particle veloc- ity and f i is defined by Eq.(27) but with the kinetic pressure p1 that replaces the fluid pressure p adopted previously [44]. These equations are assumed to hold identically in the set Ω × I and by assumption, fi and i belong to the same functional class of real functions defined so that the extended fluid fields belong to the minimal functional setting (29). Moreover, without loss of generality, we consider the D2Q9 LB discretization. Axiom II - Kinetic initial and boundary conditions. The discrete kinetic distribution function satisfies, for i = 0, k and for all r belonging to the closure Ω, the initial conditions fi(r, to) = foi(r,to) (33) where foi(r,to) (for i = 0, k) is a initial distribution func- tion defined in such a way to satisfy in the same set the initial conditions for the fluid fields p1o(r) ≡ Po(to) + po(r)− Φ(r) = (34) i=0,8 foi(r), Vo(r) = i=1,8 aifoi(r) . (35) To define the analogous kinetic boundary conditions on δΩ, let us assume that δΩ is a smooth, possibly moving, surface. Let us introduce the velocity of the point of the boundary determined by the position vector rw ∈ δΩ, de- fined by Vw(rw(t), t) = rw(t) and denote by n(rw, t) the outward normal unit vector, orthogonal to the bound- ary δΩ at the point rw. Let us denote by f i (rw, t) and f i (rw , t) the kinetic distributions which carry the discrete velocities ai for which there results respectively (ai −Vw) ·n(rw , t) > 0 (outgoing-velocity distributions) and (ai −Vw) · n(rw, t) ≤ 0 (incoming-velocity distribu- tions) and which are identically zero otherwise. We as- sume for definiteness that both sets, for which |ai| > 0, are non empty (which requires that the parameter c be suitably defined so that c > |Vw|). The bound- ary conditions are obtained by prescribing the incom- ing kinetic distribution f i (rw , t), i.e., imposing (for all (rw, t) ∈ δΩ× I) i (rw, t) = f oi (rw , t). (36) Here f oi (rw, t) are suitable functions, to be assumed non-vanishing and defined only for incoming discrete ve- locities for which (ai −Vw)·n(rw , t) ≤ 0. Manifestly, the functions f oi (rw, t) (i = 0, k) must be defined so that the Dirichlet boundary conditions for the fluid fields are identically fulfilled, namely there results p1w(rw, t) = Po(t) + pw(rw, t)− Φ(r) = (37) i=0,k oi (rw, t) + f i (rw, t) Vw(rw, t) = (38) i=1,k oi (rw, t) + f i (rw, t) Here, again, the functions foi(r) and f oi (rw, t) (for i = 0, k) must be assumed suitably smooth. A particular case is obtained imposing identically for i = 0, k foi(r,to) = f i (r, to), (39) oi (rw, t) = f i (rw , t), (40) where the identification with f oi (rw, t) and f oi (rw, t) is intended respectively in the subsets ai ·n(rw, t) > 0 and ai ·n(rw , t) ≤ 0. Finally, we notice that in case Neumann boundary conditions are imposed on the fluid pressure, Eq.(37) still holds provided pw(rw, t) is intended as a calculated value. Axiom III - Moment equations. If fi(r, t), for i = 0, k, are arbitrary solutions of LB- IKE [Eq.(30)] which satisfy Axioms I and II validity of Axioms I and II, we assume that the moment equations of the same LB-IKE, evaluated in terms of the moment op- erators MXj [·] = i=0,8 Xj ·, with j = 1, 2, coincide iden- tically with INSE, namely that there results identically [for all (r, t) ∈ Ω× I] MX1 [Lifi − Ωi(fi)− Si] = ∇ ·V = 0, (41) MX2 [Lifi − Ωi(fi)− Si] = NV = 0. (42) Axiom IV - Source term. The source term is required to depend on a finite num- ber of momenta of the distribution function. It is as- sumed that these include, at most, the extended fluid fields {V,p1} and the kinetic tensor pressure Π = 3 fiaiai − ρoVV. (43) • Furthermore, we also normally require (except for the LB-IKT described in Appendix B) that Si(r, t) results independent of f i (r,t), foi(r) and fwi(rw , t) (for i = 0, k). Although, the implications will made clear in the fol- lowing sections, it is manifest that these axioms do not specify uniquely the form (and functional class) of the equilibrium kinetic distribution function f i (r,t), nor of the initial and boundary kinetic distribution func- tions (33),(36). Thus, both f i (r,t), foi(r,to) and the related distribution they still remain in principle com- pletely arbitrary. Nevertheless, by construction, the initial and (Dirichlet) boundary conditions for the fluid fields are satisfied identically. In the sequel we show that these axioms define a (non-empty) family of parameter- dependent LB-IKT’s, depending on two constant free pa- rameters νc, c > 0 and one arbitrary real function Po(t). The examples considered are reported respectively in the following Sec. 5,6 and in the Appendix B. 5 - A POSSIBLE REALIZATION: THE INTEGRAL LB-IKT We now show that, for arbitrary choices of the distri- butions fi(r,t) and f i (r,t) which fulfill axioms I-IV, an explicit (and non-unique) realization of the LB-IKT can actually be obtained. We prove, in particular, that a pos- sible realization of the discrete inverse kinetic theory, to be denoted as integral LB-IKT, is provided by the source Si = (44) − ai · f1−µ∇ V −∇ ·Π+∇p ≡ S̃i, where wi is denoted as first pressure term. Holds, in fact, the following theorem. Theorem 1 - Integral LB-IKT In validity of axioms I-IV the following statements hold. For an arbitrary particular solution fi and for ar- bitrary extended fluid fields : A) if fi is a solution of LB-IKE [Eq.(30)] the moment equations coincide identically with INSE in the set Ω×I; B) the initial conditions and the (Dirichlet) boundary conditions for the fluid fields are satisfied identically; C) in validity of axiom IV the source term S̃i is non- uniquely defined by Eq.(44). Proof A) We notice that by definition there results identically S̃i = aiS̃i = (46) f−µ∇2V−∇ ·Π+∇p On the other hand, by construction (Axiom I) fi (i = 1, k) is defined so that there results identically i=0 Ωi = 0 and i=0 aiΩi = 0. Hence the momenta MX1 ,MX2 of LB-IKE deliver respectively i=1,8 aifi = 0 (47) i=1,8 aifi + ρoV · ∇V +∇p1 + f−µ∇ V = 0 (48) where the fluid fields V,p1 are defined by Eqs.(31),(32). Hence Eqs.(47) and (48) coincide respectively with the isochoricity and Navier-Stokes equations [(1) and (2)]. As a consequence, fi is a particular solution of LB-IKE iff the fluid fields {V,p1} are strong solutions of INSE. B) Initial and boundary conditions for the fluid fields are satisfied identically by construction thanks to Axiom C) However, even prescribing νc, c > 0 and the real function Po(t), the functional form of the equation can- not be unique The non uniqueness of the functional form of the source term S̃i(r, t) is assumed to be indepen- dent of f i (r,t) [and hence of Eq.(30)] is obvious. In fact, let us assume that S̃i is a particular solution for the source term which satisfies the previous axioms I- IV. Then, it is always possible to add to Si arbitrary terms of the form S̃i + δSi, with δSi 6= 0 which depends only on the momenta indicated above, and gives van- ishing contributions to the first two moment equations, namely MXj [δSi] = i=0,8 XjδSi = 0, with j = 1, 2. To prove the non-uniqueness of the source term Si, it is suf- ficient to notice that, for example, any term of the form δSi = F (r, t), with F (r, t) an arbitrary real function (to be assumed, thanks to Axiom IV, a linear function of the fluid velocity), gives vanishing contribu- tions to the momentaMX1 ,MX2 . Hence S̃i is non-unique. The implications of the theorem are straightforward. First, manifestly, it holds also in the case in which the BGK operator vanishes identically. This occurs letting νc = 0 in the whole domain Ω × I. Hence the inverse kinetic equation holds independently of the specific defi- nition of f i (r,t). An interesting feature of the present approach lies in the choice of the boundary condition adopted for fi(r,t), which is different from that usually adopted in LBM’s [see for example [14] for a review on the subject]. In par- ticular, the choice adopted is the simplest permitting to fulfill the Dirichlet boundary conditions [imposed on the fluid fields]. This is obtained prescribing the functional form of fi(r,t) on the boundary of the fluid domain (δΩ), which is identified with a function foi(r, t). Second, the functional class of fi(r,t), f i (r,t) and of foi(r, t) remains essentially arbitrary. Thus, in particu- lar, the initial and boundary conditions, specified by the same function foi(r, t), can be defined imposing the po- sitions (39),(40). As further basic consequence, f i (r,t) and fi(r,t) need not necessarily be Galilei-invariant (in particular they may not be invariant with respect to ve- locity translations), although the fluid equations must be necessarily fully Galilei-covariant. As a consequence it is always possible to select f i (r,t) and foi(r, t) based on convenience and mathematical simplicity. Thus, be- sides distributions which are Galilei invariant and sat- isfy a principle of maximum entropy (see for example [22, 30, 32, 34, 60, 61]), it is always possible to iden- tify them [i.e., f i (r,t), foi(r, t)] with a non-Galilean in- variant polynomial distribution of the type (27) [mani- festly, to be exactly Galilei-invariant each f i (r,t) should depend on velocity only via the relative velocity ui = ai −V]. We mention that the non-uniqueness of the source term S̃i can be exploited also by imposing that f i (r,t) re- sults a particular solution of the inverse kinetic equation Eq.(30) and there results also foi(r, t) = f i (r,t). In Ap- pendix B we report the extension of THM.1 which is ob- tained by identifying again f i (r,t) with the polynomial distribution (27). 6 - THE ENTROPIC PRINCIPLE - CONDITION OF POSITIVITY OF THE KINETIC DISTRIBUTION FUNCTION A fundamental limitation of the standard LB ap- proaches is their difficulty to attain low viscosities, due to the appearance of numerical instabilities [14]. In numeri- cal simulations based on customary LB approaches large Reynolds numbers is usually achieved by increasing nu- merical accuracy, in particular strongly reducing the time step and the grid size of the spatial discretization (both of which can be realized by means of numerical schemes with adaptive time-step and using grid refinements). Hence, the control [and possible inhibition] of numerical instabilities is achieved at the expense of computational efficiency. This obstacle is only partially alleviated by approaches based on ELBM [22, 30, 32, 34, 60, 61]. Such methods are based on the hypothesis of fulfilling an H- theorem, i.e., of satisfying in the whole domain Ω × I the condition of strict positivity for the discrete kinetic distribution functions. This requirement is considered, by several authors (see for example [26, 29, 62]), an es- sential prerequisite to achieve numerical stability in LB simulations. However, the numerical implementation of ELBM typically induce a substantial complication of the original algorithm, or require a cumbersome fine-tuning of adjustable parameters [22, 37]. 6A - The constant entropy principle and PEM A basic aspect of the IKT’s here developed is the possi- bility of fulfilling identically the strict positivity require- ment by means of a suitable H-theorem which provides also a maximum entropy principle. In particular, in this Section, extending the results of THM.1 and 2, we intend to prove that a constant H-theorem can be established both for the integral and differential LB-IKT’s defined above. The H-theorem can be reached by imposing for the Gibbs-Shannon entropy functional the requirement that for all t ∈ I there results S(f) = − i=0,8 fi ln(fi/wi) = 0, (49) which implies that S(f) is necessarily maximal in a suit- able functional set {f} . The result can be stated as fol- lows: Theorem 2 - Constant H-theorem In validity of THM.1, let us assume that: 1) the configuration domain Ω is bounded; 2) at time to the discrete kinetic distribution functions fi, for i = 0, 8, are all strictly positive in the set Ω. Then the following statements hold: A) by suitable definition of the pseudo pressure Po(t), the Gibbs-Shannon entropy functional S(f) = i=0,8 fi ln(fi/wi) can be set to be constant in the whole time interval I. This holds provided the pseudo- pressure Po(t) satisfies the differential equation (1 + log fi) = (50) ai · ∇fi − Ŝi (1 + log fi) , where Ŝi = Si + B) if the entropy functional S(f) = i=0,8 fi ln(fi/wi) is constant in the whole time interval I the discrete kinetic distribution functions fi are all strictly positive in the whole set Ω× I; C) an arbitrary solution of LB-IKE [Eq.(30)] which satisfies the requirement A) is extremal in a suitable func- tional class and maximizes the Gibbs-Shannon entropy . Proof: A) Invoking Eq.(30), there results ∂S(t) [1 + log fi] = (51) (ai · ∇fi − Si) (1 + log fi) , where Si is the source term, provided by Eq.(44). By direct substitution it follows the thesis. B) If Eq.(50) holds identically in there results ∀t ∈ I, S (t) = S (t0) , which implies the strict positivity of fi, for all i = 0, 8. C) Let us introduce the functional class {f + αδf} = {fi = fi(t) + αδfi(t), i = 0, 8} , (52) where α is a finite real parameter and the syn- chronous variation δfi(t) is defined δfi(t) = dfi(t) ≡ ∂fi(t) dt. Introducing the synchronous variation of the en- tropy, defined by δS (t) = ∂ , with ψ(α) = S (f + αδf) , it follows δS (t) = dt ∂S(t) . (53) Since in validity of Eq.(50) there results ∂S(t) which in view of Eq.(53) implies also δS (t) = 0. It is im- mediately follows that there results necessarily δ2S (t) ≤ 0, i.e., S (t) is maximal. Therefore, the kinetic distribu- tion function which satisfies IKE (Eq.(30)] is extremal in the functional class of variations (52) and maximizes the Gibbs-Shannon entropy functional. 6B - Implications In view of statement B, THM.2 warrants the strict pos- itivity of the discrete distribution functions fi (i = 0, 8) only in the open set Ω × I, while nothing can be said regarding their behavior on the boundary δΩ (on which fi might locally vanish). However, since the inverse ki- netic equation actually holds only in the open set Ω× I, this does not affect the validity of the result. While the precise cause of the numerical instability of LBM’s is still unknown,the strict positivity of the distribution function is usually considered important for the stability of the nu- merical solution [29, 30]. It must be stressed that the nu- merical implementation of the condition of constant en- tropy Eq.(50) should be straightforward, without involv- ing a significant computational overhead for LB simula- tions. Therefore it might represent a convenient scheme to be adopted also for customary LB methods. 7 - ASYMPTOTIC APPROXIMATIONS AND COMPARISONS WITH PREVIOUS CFD METHODS A basic issue is the relationship with previous CFD nu- merical methods, particularly asymptotic LBM’s. Here we consider, for definiteness, only the case of the inte- gral LB-IKT introduced in Sec.5. Another motivation is the possibility of constructing new improved asymptotic models, which satisfy with prescribed accuracy the re- quired fluid equations [INSE], of extending the range of validity of traditional LBM’s and fulfilling also the en- tropic principle (see Sec.6). The analysis is useful in particular to establish on rigorous grounds the consis- tency of previous LBM’s. The connection [with previ- ous LBM’s] can be reached by introducing appropriate asymptotic approximations for the IKT’s, obtained by assuming that suitable parameters which characterize the IKT’s are infinitesimal (or infinite) (asymptotic parame- ters). A further interesting feature is the possibility of constructing in principle a class of new asymptotic LBM’s with prescribed accuracy , i.e., in which the distribution function (and the corresponding momenta) can be de- termined with predetermined accuracy in terms of per- turbative expansions in the relevant asymptotic parame- ters. Besides recovering the traditional low-Mach number LBM’s [17, 21, 40], which satisfy the isochoricity condi- tion only in an asymptotic sense and are closely related to the Chorin artificial compressibility method, it is possible to obtain an improved asymptotic LBM’s which satisfy exactly the same equation. We first notice that the present IKT is characterized by the arbitrary positive parameters νc, c and the initial value Po(to), which enter respectively in the definition of the BGK operator [see (11)], the velocity momenta and equilibrium distribution function f i . Both c and Po(to) must be assumed strictly positive, while, to assure the validity of THM.2, Po(to) must be defined so that (for all i = 0, 8) f i (r,to) > 0 in the closure Ω. Thanks to THM.1.and 2 the new theory is manifestly valid for arbitrary finite value of these parameters. This means that they hold also assuming o(εαν ) , (54) o(εαc) , (55) Po(to) ∼ o(ε 0), (56) where ε denotes a strictly positive real infinitesimal, αν , αc > 0 are real parameters to be defined, while the extended fluid fields {ρ,V, p1} and the volume force f are all assumed independent of ε. Hence, with respect to ε they scale ρo,V,p1, f ∼ o(ε 0). (57) As a result, for suitably smooth fluid fields (i.e., in va- lidity of Axiom 1) and appropriate initial conditions for fi(r, t), it is expected that the first requirement actually implies in the whole set Ω× I the condition of closeness fi(r, t) ∼= f i (r, t) [1 + o(ε)] , consistent with the LB As- sumption #4. To display meaningful comparisons with previous LBM’s let us introduce the further assumption that the fluid viscosity is small in the sense µ ∼ o(εαµ), (58) with αµ ≥ 1 another real parameter to be defined. Asymptotic approximations for the corresponding LB- IKE [Eq.(30)] can be directly recovered by introducing appropriate asymptotic orderings for the contributions appearing in the source term Si = S̃i. Direct inspec- tion shows that these are provided by the (dimensional) parameters M effp,a ≡ , (59) ∣∣∇ ·Π−∇p ∣∣ , (60) ∣∣µ∇2V ∣∣ . (61) The first two M effp,a and M are here denoted respec- tively as (first and second) pressure effective Mach num- bers, driven respectively by the pressure time-derivative and by the divergence of the pressure anisotropy Π−p1. Furthermore, M is denoted as velocity effective Mach number. Physically relevant examples [of asymptotic LBM’s] can be achieved by introducing suitable orderings in terms of the single infinitesimal ε for the parameters M effp,a ,M .We stress that these orderings, in principle, can be introduced without actually introducing restrictions on the fluid fields, i.e., retaining the assump- tion that the extended fluid fields are independent of ε. Interesting cases are provided by the asymptotic order- ings indicated below. 7A - Small effective Mach numbers (Meffp,a ,M An important aspect of LB theory is the possibility of constructing asymptotic LBM’s with prescribed accu- racy with respect to the infinitesimal parameter ε, in the sense that the fluid equations are satisfied at least cor- rect up to terms of order o(εn) included, with n = 1 or 2, namely ignoring error terms of order o(εn+1) or higher. Let us, first, consider the case in which all pa- rametersM effp,a ,M and M are all infinitesimal w.r. to ε (low-effective-Mach numbers). Since the parameters c and νc are free, they can be defined so that that there results c ∼ νc ∼ 1/o(ε) [which implies αc = αν = 1]. This requires M effp,a ∼M ∼ o(ε2). (62) If, we consider a low-viscosity fluid for which the kine- matic viscosity ν = µ/ρo can be assumed of order ε [and hence αµ = 1] it follows that ∼ o(ε2). (63) Thanks to the assumptions (54)-(58) there follows ∇ · Π − ∇p ∼ o(ε) and µ∇2V ∼ o(ε),which implies that the source term S̃i, ignoring corrections of order o(ε becomes S̃i ∼= S̃Ai [1 + o(ε)] , (64) S̃Ai ≡ − ai · f . (65) It is immediate to determine the corresponding moment equations, which read: +∇ ·V = 0, (66) NV = 0+ o(ε2), (67) Formally the first equation can be interpreted as an evo- lution equation for the kinetic pressure p1. Nevertheless, in view of the ordering (62) it actually implies the iso- choricity condition ∇ ·V = 0 + o(ε2). (68) Instead, the second one [Eq.(67)]. due to the asymp- totic approximation (63), reduces to the Euler equation. Therefore in this case the asymptotic approximation (64) is not adequate. To recover the correct Navier-Stokes equation a more accurate approximation is needed, real- ized requiring that the hydrodynamic equations are sat- isfied correct to order o(ε3). A fist possibility is to con- sider a more accurate approximation for the source term. Restoring the pressure and viscous source terms in (64) there results the asymptotic source term S̃Bi ≡ − ai · f1−µ∇ , (69) where in validity of the previous orderings S̃i ∼= S̃Bi [1 + o(ε)] . (70) The corresponding moment equations become therefore ∇ ·V = 0, (71) NV = 0+ o(ε3). (72) It is remarkable that in this case the isochoricity condi- tion is exactly fulfilled, even if the source term is not the exact one. For the sake of reference, it is interesting to mention another possible small-Mach-number ordering. This is obtained imposing for the parameters c and νc , (73) o(ε2) , (74) while requiring for ν = µ/ρo the same constraint adopted by asymptotic LBM’s, namely Eq.(17). In this case one can show that the moment equation (72) is actually satis- fied correct to order o(ε3), while the isochoricity condition is only satisfied to order o(ε2). The following theorem can, in fact, be proven: Theorem 3 - Low effective-Mach-numbers asymptotic approximation In validity of THM.1, let us invoke the following as- sumptions: 1) LB assumptions #3 and #4 for the discrete kinetic distributions fi ( i = 0, 8); 2) the free parameters c and νc are assumed to satisfy the asymptotic orderings (73),(74); 3) the fluid viscosity µ is assumed of order µ ∼ o(ε) 4) the fluid viscosity µ is prescribed so that the kine- matic viscosity ν = µ/ρo is defined in accordance to Eq.(17); 5) the kinetic pressure p1 is assumed slowly varying in the sense ∂ ln p1 ∼ o(ε). (75) It follows that the source term is approximated by Eq.(64) and moment equations are provided by the asymptotic equations: +∇ ·V = 0 + o(ε3), (76) NV = 0+ o(ε3), (77) i.e., the isochoricity and NS equation are recovered re- spectively correct to order o(ε2) and o(ε3). Proof First we notice that the ordering assumptions 2)-5) require M effp,a ∼ o(ε 3) (78) ∼ o(ε2), (79) ∼ o(ε4), (80) which imply at least the validity of Eqs.(64)-(67). The proof of Eqs.(76) and (77) is immediate. In both cases it sufficient to notice that in validity of hypotheses 1)-3) and in terms of a Chapman-Enskog perturbative solution of Eq.(30) there results actually −µ∇2V −∇ ·Π+∇p = O + o(ε3), (81) and hence S̃i reduces to Eq.(64). The predictions of THM.3 are relevant for comparisons and to provide asymptotic accuracy estimates for previ- ous asymptotic LBM’s [see Refs. [17, 21, 40]]. In fact, the asymptotic moment equations (76) and (77) formally coincide with the analogous moment equations predicted by such theories, when the kinetic pressure p is replaced by the fluid pressure p1 (i.e., if the function Po(t) is set identically equal to zero). [17, 21, 40]. Nevertheless, the accuracy of customary LBM’s depends on the properties of the solutions of INSE. In fact, if one assumes ∂ ln p1 ∼ o(ε0) (82) the customary (V, p) asymptotic LBM [17, 21, 40] result actually accurate only to order o(ε2). Therefore, in such case to reach an accuracy of order o(ε3) the approxima- tion (69) must be invoked for the source term. The other interesting feature of Eqs.(76) and (77) is that they provide a connection with the artificial com- pressibility method (ACM) postulated by Chorin [63], previously motivated merely on the grounds of an asymp- totic LBM [21]. In fact, these coincides with the Chorin’s pressure relaxation equation where c can be interpreted as sound speed of the fluid. However - in a sense - this analogy is purely formal and is only due to the neglect of the first pressure source term in Si. It disappears alto- gether in Eq.(71) if we adopt the more accurate asymp- totic source term (69). A further difference is provided by the adoption of the kinetic pressure p1 which replaces the fluid pressure p (used in Chorin approach). We stress that the choice of p1 here adopted, with Po(t) determined by the entropic principle, represents an important differ- ence, since it permits to satisfy everywhere in Ω× I the condition of strict positivity for the discrete kinetic dis- tribution functions. 7B - Finite pressure-Mach number Meffp,a Another possible asymptotic ordering, usually not per- mitted by customary asymptotic LBM’s, is the one in which the test particle velocity is finite, namely c ∼ o(ε0), the viscosity remains arbitrary and is taken of order µ ∼ o(ε0) while again νc is assumed νc ∼ 1/o(ε 2) [i.e., αc = νc = 0, αν = 2]. In this case the pressure Mach M effp,a number results finite, while velocity and the sec- ond pressure Mach numbers are considered infinitesimal, respectively of first and second order in ε, namely M effp,a ∼ o(ε ∼ o(ε), (83) ∼ o(ε2). To obtain the fluid equation with the prescribed accu- racy, say of order o(ε2), it is sufficient to approximate the source term S̃i in terms of S̃i ∼= S 1 + o(ε2) . The set of asymptotic moment equations coincide therefore with Eqs.(71),(72). Again, the isochoricity condition is exactly fulfilled, while in this case the NS equation is accurate only to order o(ε2). 7C - Small effective pressure-Mach numbers (Meffp,a ,M ) and finite velocity-Mach number (M Finally, another interesting case is the one in which the fluid viscosity µ remains finite (strongly viscous fluid), i.e., in the sense µ ∼ o(ε0) [i.e., αµ = 0] while both parameters c and νc are suitably large, and respectively scale as c ∼ 1/o(ε), νc ∼ 1/o(ε 2) [i.e., αc = 1, αν = 2]. Due to assumptions (54)-(58) one obtains ∇ ·Π−∇p ∼ o(ε2) and µ∇2V ∼ o(ε0). It follows that the effective Mach numbers scale respectively as ∼ o(ε3) (84) M effp,a ∼ M ∼ o(ε2), If we impose on µ also the same constraint set by Eq.(17), the customary asymptotic LBM’s can be invoked also in this case. However, since the first pressure and veloc- ity Mach numbers are only second order accurate, the NS equation is recovered to order o(ε2) only. Never- theless, it is possible to recover with prescribed accuracy the fluid equations (71),(72). This is obtained adopt- ing the source term S̃i ∼= S̃Bi [see Eq.(69)]. As a basic consequence, the isochoricity equation is satisfied exactly (hence no meaningful analogy with Chorin’s approach arises), while the NS equation results correct to order o(ε3). These results provide a meaningful extension of the customary asymptotic LBM’s. We stress that the entropic approach here developed holds independently of the asymptotic orderings here considered [for the param- etersM effp,a ,M ]. Thus it can be used in all cases to assure the strict positivity of the discrete distribution function. 8 - CONCLUSIONS In this paper we have presented the theoretical foun- dations of a new phase-space model for incompressible isothermal fluids, based on a generalization of customary lattice Boltzmann approaches.We have shown that many of the limitations of traditional (asymptotic) LBM’s can be overcome. As a main result, we have proven that the LB-IKT can be developed in such a way that it furnishes exact Navier-Stokes and Poisson solvers, i.e., it is - in a proper sense - an inverse kinetic theory for INSE. The theory exhibits several features, in particular we have proven that the integral LB-IKT (see Sec.5): 1. determines uniquely the fluid pressure p(r, t) via the discrete kinetic distribution function without solving explicitly (i.e., numerically) the Poisson equation for the fluid pressure. Although analo- gous to traditional LBM’s, this is interesting since it is achieved without introducing compressibility and/or thermal effects. In particular the present theory does not rely on a state equation for the fluid pressure. 2. is complete, namely all fluid fields are expressed as momenta of the distribution function and all hy- drodynamic equations are identified with suitable moment equations of the LB inverse kinetic equa- tion. 3. allows arbitrary initial and boundary conditions for the fluid fields. 4. is self-consistent : the kinetic theory holds for ar- bitrary, suitably smooth initial conditions for the kinetic distribution function. In other words, the initial kinetic distribution function must remain ar- bitrary even if a suitable set of its momenta are prescribed at the initial time. 5. the associated the kinetic and equilibrium distri- bution functions can always be chosen to belong to the class of non-Galilei-invariant distributions. In particular the equilibrium kinetic distribution can always be identified with a polynomial of second degree in the velocity. 6. is non-asymptotic, i.e., unlike traditional LBM’s it does not depend on any small parameter, in partic- ular it holds for finite Mach numbers. 7. fulfills an entropic principle, based on a constant-H theorem. This theorem assures, at the same time, the strict positivity of the discrete kinetic distri- bution function and the maximization of the as- sociated Gibbs-Shannon entropy in a properly de- fined functional class. Remarkably the constant H- theorem is fulfilled for arbitrary (strictly positive) kinetic equilibria. This includes also the case of polynomial kinetic equilibria. A further remarkable aspect of the theory concerns the choice of the kinetic boundary conditions to be satisfied by the distribution function (Axiom II) and obtained by prescribing the form of the incoming-velocity distribution [see Eq.(36)]. Thanks to Eqs.(34),(35), this requirement [of the LB-IKT] the boundary conditions for the fluid fields are satisfied exactly while the fluid equations are by construction identically fulfilled also arbitrarily close to the boundary. This result, in a proper sense, applies only to Dirichlet boundary conditions for the fluid fields [see Eqs.(8)]. Nevertheless the same approach can be in principle extended to the case of mixed or Neumann boundary conditions for the fluid fields. Moreover, we have shown that a useful implication of the theory is provided by the possibility of constructing asymptotic approximations to the inverse kinetic equa- tion. This permits to develop a new class of asymptotic LBM’s which satisfy INSE with prescribed accuracy, to obtain useful comparisons with previous CFD methods (Chorin’s ACM) and to achieve accuracy estimates for customary asymptotic LBM’s. The main results of the paper are represented by THM’s 1-3, which refer respec- tively to the construction of the integral LB-IKT, to the entropic principle and to construction of the low effective- Mach-numbers asymptotic approximations. For the sake of reference, also another type of LB-IKT, which admits as exact particular solution the polynomial kinetic equi- librium, has been pointed out (THM.1bis). The construction of a discrete inverse kinetic theory of this type for the incompressible Navier-Stokes equations represents an exciting development for the phase-space description of fluid dynamics, providing a new starting point for theoretical and numerical investigations based on LB theory. In our view, the route to more accurate, higher-order LBM’s, here pointed out, will be important in order to achieve substantial improvements in the effi- ciency of LBM’s in the near future. APPENDIX A The basic argument regarding the accuracy of the boundary conditions adopted by customary asymptotic LBM’s is provided by Ref.[46]. In fact. let us assume that on the boundary δΩ the incoming distribution function i (rw , t) is prescribed according to Eqs.(33),(37) and (38), being f oi (rw, t) prescribed suitably smooth func- tions which are non vanishing only only for incoming dis- crete velocities ai for which (ai −Vw) ·n(rw, t) ≤ 0. For definiteness, let us assume that f oi (rw , t) ≡ f i (rw, t) where f i (rw, t) denotes a suitable equilibrium distribu- tion. It follows that suitably close to the boundary the kinetic distribution differs from the Chapman-Enskog so- lution (25). The numerical error can be overcome only discarding the first few spatial grid (close to the bound- ary) in the numerical simulation [46]. APPENDIX B Unlike standard kinetic theory, the distinctive feature of LB-IKT’s is the possibility of adopting a non-Galilei invariant kinetic distribution function (i.e., non-invariant with respect to velocity translations). Here we re- port another example of discrete inverse kinetic theory of this type. Let us modify Axiom IV so that to permit that a particular solution of LB-IKE [Eq.(30)] is pro- vided by fi = f i . Here we identify f i with the (non- Galilei invariant) polynomial kinetic distribution defined by Eq.(27) but with the kinetic pressure p1 that replace the fluid pressure p. In this case one can prove that the source term Si reads Si = S i ≡ S̃i +∆Si, (85) where ∆Si = (ai −V) · ∇V− ai∇·V · ai+ V − ai 3ai ·V + (86) wiρoai · ∇ ai ·V Here N1 ≡ N−ρo , where N is the Navier-Stokes opera- tor (6), namely N1 is the nonlinear operator which acting onV yields N1V = ρoV·∇V+∇ [p1 − Φ (r)]+f1−µ∇ Hence, invoking INSE, ∆Si can also be written in the equivalent form ∆Si = (ai −V) · ∇V− ai∇·V · ai+ V − ai 3ai ·V + (87) wiρoai · ∇ ai ·V The following result holds: Theorem 1bis - Differential LB-IKT In validity of axioms I-IV and the assumption that fi = f i is a particular solution of Eq.(30), the following statements hold: i is a particular solution of LB-IKE [Eq.(30)] if and only if the extended fluid fields {V,p1} are strong solutions of INSE of class (29), with initial and boundary conditions (7)-(8), and arbitrary pseudo pressure po(t) of class C(1)(I). Moreover, for an arbitrary particular solution fi and for arbitrary extended fluid fields : For an arbitrary particular solution fi : B) fi is a solution of LB-IKE [Eq.(30)] if and only if the extended fluid fields {V,p1} are arbitrary strong solutions of INSE of class (29), with initial and boundary conditions (7)-(8), and arbitrary pseudo pressure po(t) of class C(1)(I); C) the moment equations of L-B IKE coincide identi- cally with INSE in the set Ω× I; D) the initial conditions and the (Dirichlet) boundary conditions for the fluid fields are satisfied identically; E) the source term Si is uniquely defined by Eqs.(85),(86); Proof: The proof of propositions A,B, C and D is analogous to that provided in THM.1. Assuming Si = S i , the proof of B follows from straightforward algebra. In fact, letting fi(r, t) = f i (r, t) for all (r, t) ∈ Ω × I in the LB-IKE [Eq.(30)], one finds that Eq.(30) is fulfilled iff the fluid fields satisfy the Navier-Stokes, isochoricity and incompressibility equations (1),(2) and (3). The proof of proposition E can be reached in a similar way. The uniqueness of the source term Si is an immediate conse- quence of the uniqueness of the solutions for INSE. ACKNOWLEDGEMENTS Useful comments and stimulating discussions with K.R. Sreenivasan, Direc- tor, ICTP (International Center of Theoretical Physics, Trieste, Italy) are warmly acknowledged. Research de- veloped in the framework of PRIN Project Fundamen- tals of kinetic theory and applications to fluid dynamics, magnetofluid dynamics and quantum mechanics (MIUR, Ministry for University and Research, Italy), with the support of the Consortium for Magnetofluid Dynamics, Trieste, Italy. [1] M. Ellero and M. Tessarotto, Bull. Am Phys. Soc. 45 (9), 40 (2000). [2] M. Tessarotto and M. Ellero, RGD24 (Italy, July 10-16, 2004), AIP Conf. Proc. 762, 108 (2005). [3] M. Ellero and M. Tessarotto, Physica A 355, 233 (2005). [4] M. Tessarotto and M. Ellero, Physica A 373, 142 (2007); arXiv: physics/0602140. [5] M. Tessarotto and M. Ellero, “On the uniqueness of con- tinuous inverse kinetic theory for incompressible fluids,” in press on AIP Conf. Proc., RGD25 (St. Petersburg, Russia, July 21-28, 2006); arXiv:physics/0611113. [6] M. Tessarotto, M. Ellero, N. Aslan, M. Mond and P. Nicolini, “An exact pressure evolution equation for the incompressible Navier-Stokes equationsInverse”, arXiv:physics/0612072 (2006). [7] M. Tessarotto, M. Ellero and P. Nicolini, Phys.Rev. A 75, 012105 (2007); arXiv:quantum-ph/060691. [8] G.R. McNamara and G. Zanetti, Phys. Rev. Lett. 61, 2332 (1988). [9] F. Higuera, S. Succi and R. Benzi, Europhys. Lett. 9,345 (1989). [10] S. Succi, R. Benzi, and F. Higuera, Physica D 47, 219 (1991). [11] S. Chen, H. Chen, D. O. Martinez, and W. H. Matthaeus, Phys.Rev. Lett. 67, 3776 (1991). [12] R. Benzi et al., Phys. Rep. 222, 145 (1992). [13] H. Chen, S. Chen, and W. Matthaeus, Phys. Rev. A 45, R5339 (1992). [14] S. Succi, The Lattice Boltzmann Equation for Fluid Dy- namics and Beyond (Numerical Mathematics and Scien- tific Computation), Oxford Science Publications (2001). [15] U. Frisch, B. Hasslacher, and Y. Pomeau, Rev.Lett. 56, 1505 (1986). [16] U. Frisch, D. d’Humieres, B. Hasslachaer, P. Lallemand, Y. Pomeau, and J.-P. Rivet, Complex Syst. 1, 649 (1987). [17] X. He and L-S. Luo, Phys. Rev. E 56, 6811 (1997). [18] D. O. Martinez, W. H. Matthaeus, S. Chen, et al., Phys. Fluids 6, 1285 (1994). [19] S. L. Hou, Q. S. Zou, S. Y. Chen, G. Doolen, and A. C. Cogley, J. Comput. Phys. 118, 329 (1995). [20] X. He and G. Doolen, J. Comput. Phys. 134, 306 (1997). http://arxiv.org/abs/physics/0602140 http://arxiv.org/abs/physics/0611113 http://arxiv.org/abs/physics/0612072 [21] X. He, G.D. Doolen, and T. Clark, J. Comp. Physics 179 (2), 439 (2002). [22] S. Ansumali and I. V. Karlin, Phys. Rev. E 65, 056312 (2002). [23] Y. Shi, T. S. Zhao and Z. L. Guo, Phys. Rev.E 73, 026704 2006. [24] X. Shan and X. He, Phys. Rev. Lett. 80, 65 (1998). [25] S. Ansumali, I.V. Karlin, and H. C. Öttinger, Euro- phys.Lett. 63, 798 (2003). [26] S.S. Chikatamarla and I.V. Karlin, Phys. Rev.Lett. 97, 190601 (2006). [27] A. Bardow, I.V. Karlin, and A. A. Gusev, Europhys. Lett. 75, 434 (2006). [28] S. Ansumali and I.V. Karlin, Phys. Rev. Lett. 95, 260605 (2005). [29] S. Succi, I.V. Karlin, and H. Chen, Rev. Mod. Phys. 74, 1203 (2002). [30] B. M. Boghosian, J. Yepez, P.V. Coveney, and A. J. Wag- ner, Proc. R. Soc. A 457, 717 (2001). [31] M. E. McCracken and J. Abraham, Phys. Rev.E 71, 036701 (2005). [32] I.V. Karlin and S. Succi, Phys. Rev. E 58, R4053 (1998). [33] I.V. Karlin, A. N. Gorban, S. Succi, and V. Boffi, Phys.Rev. Lett. 81, 6 (1998). [34] I.V. Karlin, A. Ferrante, and H. C. Öttinger, Europhys. Lett. 47, 182 (1999). [35] W.-A. Yong and L.-S. Luo, Phys. Rev. E 67, 051105 (2003). [36] P. J. Dellar, Europhys. Lett. 57, 690 (2002). [37] P. Lallemand and L.-S. Luo, Phys. Rev. E 61, 6546 (2000). [38] P. Lallemand and L.-S. Luo, Phys. Rev. E 68, 036706 (2003). [39] E.T. Shannon, Phys. Rev. 106, 620 (1957). [40] T. Abe, J. Comput. Phys. 131, 241 (1997). [41] N. Cao, S.Chen, S.Jin, and D. Mart́ınez, Phys. Rev. E 55, R21 (1997). [42] Y.-H. Qian, D. d’Humieres, and P. Lallemand, Euro- phys.Lett. 17, 479 (1992). [43] P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev. A 94,511 (1954). [44] X. He and L.S. Luo, Phys. Rev. E 55, R6333 (1997). [45] Q. Zou and X. He, Phys. Fluids 9, 1951 (1997). [46] P.A.Skordos, Phys.Review E 48, 4823 (1993). [47] R. Comubert, D. d’Humieres, and D. Levermore, Physica D 47, 241 (1991). [48] D.P. Ziegler, J. Stat. Phys. 71, 1171 (1993). [49] I. Ginzbourg and P. M. Adler, J. Phys. II France 4, 191 (1994). [50] A.J.C. Ladd, J. Fluid Mech. 271, 285 (1994). [51] D.R. Noble, S. Chen, J.G. Georgiadis, R.O. Buckius, Phys. Fluids 7 (l), 203 (1995). [52] Q. Zou and H. He, Phys. Fluids 7, 1788 (1996). [53] R.S. Maier, R.S. Bernard, and D.W. Grunau, Phys. Flu- ids 7, 1788 (1996). [54] S. Chen, Daniel Martınez and R.Mei, Phys. Fluids 8(9), 2528 (1996). [55] R. Mei, L.S. Luo, and W. Shyy, J. Comput. Phys. 155, 307 (1999). [56] M. Bouzidi, M. Firadaouss, and P. Lallemand, Phys. Flu- ids 13, 452 (2001). [57] S. Ansumali and I. V. Karlin, Phys. Rev. E 66, 026311 (2002). [58] I. Ginzburg and D.d’Humieres, Phys. Rev. E 68, 066614 (2003). [59] M. Junk and Z. Yang, Phys. Rev. E 72, 066701 (2005). [60] S Ansumali and I. V. Karlin, Phys. Rev. E 62, 7999 (2000). [61] S. Ansumali, I.V. Karlin, and H. C. Ottinger, Europhys. Lett. 63, 798 (2003). [62] B.M. Boghosian, P.J. Love, P. V. Coveney, I.V. Karlin, S.Succi, J.Yepez, Phys.Rev E 68, 025103(R) (2003). [63] A.J. Chorin, J.Comp.Phys. 2, 12 (1967).

0704.0340

Phonon-mediated decay of an atom in a surface-induced potential

Phonon-mediated decay of an atom in a surface-induced potential Fam Le Kien,1,∗ S. Dutta Gupta,1,2 and K. Hakuta1 Department of Applied Physics and Chemistry, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan School of Physics, University of Hyderabad, Hyderabad, India (Dated: August 4, 2021) We study phonon-mediated transitions between translational levels of an atom in a surface-induced potential. We present a general master equation governing the dynamics of the translational states of the atom. In the framework of the Debye model, we derive compact expressions for the rates for both upward and downward transitions. Numerical calculations for the transition rates are performed for a deep silica-induced potential allowing for a large number of bound levels as well as free states of a cesium atom. The total absorption rate is shown to be determined mainly by the bound-to-bound transitions for deep bound levels and by bound-to-free transitions for shallow bound levels. Moreover, the phonon emission and absorption processes can be orders of magnitude larger for deep bound levels as compared to the shallow bound ones. We also study various types of transitions from free states. We show that, for thermal atomic cesium with temperature in the range from 100 µK to 400 µK in the vicinity of a silica surface with temperature of 300 K, the adsorption (free-to-bound decay) rate is about two times larger than the heating (free-to-free upward decay) rate, while the cooling (free-to-free downward decay) rate is negligible. PACS numbers: 34.50.Dy,33.70.Ca I. INTRODUCTION Over the past few years, tight confinement of cold atoms has drawn considerable attention. The interest in this area is motivated not only by the fundamental na- ture of the problem, but also by its potential applications in atom optics and quantum information. A method for microscopic trapping and guiding of individual atoms along a nanofiber has been proposed [1]. Surface–atom quantum electrodynamic effects have constituted another interesting area, where a great deal of work has been carried out. Modification of spontaneous emission of an atom [2] and radiative exchange between two distant atoms [3] mediated by a nanofiber have been investigated. Surface-induced deep potentials have played a major role and have received due attention in recent years. Oria et al. have studied various theoretical schemes to load atoms into such potentials [4, 5]. A rigorous theory of spontaneous decay of an atom in a surface-induced po- tential invoking the density-matrix formalism has been developed [6]. The role of interference between the emit- ted and reflected fields and also the role of transmission into the evanescent modes were identified. Further cal- culations on the excitation spectrum have been carried out [7]. Bound-to-bound transitions were shown to lead to significant effects like a large red tail of the excita- tion spectrum as compared to the weak consequences of free-to-bound transitions. A crucial step in this direction was the experimental observation of the excitation spec- trum and the channeling of the fluorescent photons along the nanofiber [8], opening up avenues for novel quantum information devices. In most of the problems involving surface–atom inter- action, the macroscopic surface is usually kept at room temperature. Thus the pertinent question that can be asked is what would be the effect of heating on the cold atoms. It is understood that transfer of heat to the trapped atoms will lead to a change in the occupation probability of the vibrational levels as well as their co- herence. Phonon-induced changes in the populations of the vibrational levels have been studied by several groups [5, 9, 10]. In a nice and compact treatment based on the dyadic Green function and the Fermi golden rule, Henkel et al. showed that the effects can be very different de- pending on the nature of the atomic/molecular species [9]. The time scales for various species were estimated. It should be stressed that the trap considered by Henkel et al. was not necessarily a surface trap and misses out on many of the aspects of the surface–atom interaction [9]. Based on the assumption that the surface–atom in- teraction can be represented by a Morse potential, the phonon-mediated decay was estimated by Oria et al. [5]. Their estimate was based on the formalism developed by Gortel et al. [10]. However, all the previous theories focus on only the transition rates and thus are not gen- eral enough. In this paper, we present a general density- matrix formalism to calculate the phonon-mediated de- cay of populations as well as the changes in coherence. We derive the relevant master equation for the density matrix of the atom. We emphasize that our density- matrix equation describes the full dynamics of the cou- pling between trapped atoms and phonons and does not assume any particular form of the trapping potential. Under the Debye approximation, we derive compact ex- pressions for the phonon-mediated decay rates. Numer- ical calculations are carried out assuming the potential model considered in [4]. In contrast to the previous work, we include a large number of vibrational levels due to the deep surface–atom potential. We show that there can be significant differences in the decay rates when the initial level is chosen as one of the shallow or deep bound levels. We also calculate and analyze the decay rates for various http://arxiv.org/abs/0704.0340v1 types of transitions from free states. The paper is organized as follows. In Sec. II we de- scribe the model. In Sec. III we derive the basic dynam- ical equations for the phonon-mediated decay processes. In Sec. IV we present the results of numerical calcula- tions. Our conclusions are given in Sec. V. II. DESCRIPTION OF THE MODEL SYSTEM We assume the whole space to be divided into two re- gions, namely, the half-space x < 0, occupied by a nondis- persive nonabsorbing dielectric medium (medium 1), and the half-space x > 0, occupied by vacuum (medium 2). We examine a single atom moving in the empty half- space x > 0. We assume that the atom is in a fixed internal state |i〉 with energy h̄ωi. Without loss of gen- erality, we assume that the energy of the internal state |i〉 is zero, i.e. ωi = 0. We describe the interaction be- tween the atom and the surface. We first consider the surface-induced interaction potential and then add the atom-phonon interaction. A. Surface-induced interaction potential In this subsection, we describe the interaction between the atom and the surface in the case where thermal vi- brations of the surface are absent. The potential en- ergy of the surface–atom interaction is a combination of a long-range van der Waals attraction and a short-range repulsion [11]. Despite a large volume of research on the surface–atom interaction, due to the complexity of sur- face physics and the lack of data, the actual form of the potential is yet to be ascertained [11]. For the purpose of numerical demonstration of our formalism, we choose the following model for the potential [4, 11]: U(x) = Ae−αx − C3 . (1) Here, C3 is the van der Waals coefficient, while A and α determine the height and range, respectively, of the surface repulsion. The potential parameters C3, A, and α depend on the nature of the dielectric and the atom. In numerical calculations, we use the parameters of fused silica, for the dielectric, and the parameters of ground- state atomic cesium, for the atom. The parameters for the interaction between silica and ground-state atomic cesium are theoretically estimated to be C3 = 1.56 kHz µm3, A = 1.6× 1018 Hz, and α = 53 nm−1 [6]. We introduce the notation ϕν(x) for the eigenfunc- tions of the center-of-mass motion of the atom in the potential U(x). They are determined by the stationary Schrödinger equation + U(x) ϕν(x) = Eνϕν(x). (2) Here m is the mass of the atom. In the numerical ex- ample with atomic cesium, we have m = 132.9 a.u. = 2.21 × 10−25 kg. The eigenvalues Eν are the center- of-mass energies of the translational levels of the atom. These eigenvalues are the shifts of the energies of the translational levels from the energy of the internal state |i〉. Without loss of generality, we assume that the center-of-mass eigenfunctions ϕν(x) are real functions, i.e. ϕ∗ν(x) = ϕν(x). In Fig. 1, we show the potential U(x) and the wave functions ϕν(x) of a number of bound levels with en- ergies in the range from −1 GHz to −5 MHz. We also plot the wave function of a free state with energy of about 4.25 MHz. In order to have some estimate about the spa- tial extent of a wave function ϕν(x), we define a crossing point xcross, which corresponds to the rightmost solution of the equation U(x) = Eν . Note that, for shallow lev- els, the wave function generally peaks close to the point xcross. We plot the eigenvalue modulus |Eν | and the cross- ing point xcross in Figs. 2(a) and 2(b), respectively. It is clear from the figure that, for ν in the range from 0 to 300, the eigenvalue varies dramatically from about 158 THz to about 322 kHz, while the wave function extends only up to 170 nm. FIG. 1: Energies and wave functions of the center-of-mass motion of an atom in a surface-induced potential. The pa- rameters of the potential are C3 = 1.56 kHz µm 3, A = 1.6 × 1018 Hz, and α = 53 nm−1. The mass of the atom is m = 2.21 × 10−25 kg. We plot bound levels with energies in the range from −1 GHz to −5 MHz and also a free state with energy of about 4.25 MHz. FIG. 2: Eigenvalue modulus |Eν | (a) and crossing point xcross (b) as functions of the vibrational quantum number ν. The parameters used are as in Fig. 1. We introduce the notation |ν〉 = |ϕν〉 and ων = Eν/h̄ for the state vectors and frequencies of translational lev- els. Then, the Hamiltonian of the atom in the surface- induced potential can be represented in the diagonal form h̄ωνσνν . (3) Here, σνν = |ν〉〈ν| is the population operator for the translational level ν. We emphasize that the summation over ν includes both the discrete (Eν < 0) and continuous (Eν > 0) spectra. The levels ν with Eν < 0 are called the bound (or vibrational) levels. In such a state, the atom is bound to the surface. It is vibrating, or more exactly, moving back and forth between the walls formed by the van der Waals part and the repulsive part of the potential. The levels ν with Eν > 0 are called the free (or continuum) levels. The center-of-mass wave functions of the bound states are normalized to unity. The center-of- mass wave functions of the free states are normalized to the delta function of energy. B. Atom–phonon interaction In this subsection, we incorporate the thermal vibra- tions of the solid into the model. Due to the thermal effects, the surface of the dielectric vibrates. The surface- induced potential for the atom is then U(x− xs), where xs is the displacement of the surface from the mean po- sition 〈xs〉 = 0. We approximate the vibrating potential U(x− xs) by expanding it to the first order in xs, U(x− xs) = U(x) − U ′(x)xs. (4) The first term, U(x), when combined with the kinetic energy p2/2m, yields the Hamiltonian HA [see Eq. (3)], which leads to the formation of translational levels of the atom. The second term, −U ′(x)xs, accounts for the thermal effects in the interaction of the atom with the solid. Note that the quantity F = −U ′(x) is the force of the surface upon the atom. Hence, the force of the atom upon the surface is −F = U ′(x) and, consequently, U ′(x)xs is the work required to displace the surface for a small distance xs. It is well known that, for a smooth surface, the gas atom interacts only with the phonons polarized along the x direction [10]. In the harmonic approximation, we 2MNωq iqR + b†qe −iqR). (5) Here, M is the mass of a particle of the solid, N is the particle number density, ωq and q are the frequency and wave vector of the x-polarized acoustic phonons, re- spectively, R = (0, y, z) is the lateral component of the position vector (x, y, z) of the atom, and bq and b q are the annihilation and creation phonon operators, respec- tively. Without loss of generality, we choose R = 0. Meanwhile, the operator U ′ can be decomposed as U ′ = νν′ σνν′ 〈ν|U ′|ν′〉, where σνν′ = |ν〉〈ν′| is the operator for the translational transition ν ↔ ν′. Hence, the en- ergy term −U ′(x)xs leads to the atom–phonon interac- tion Hamiltonian [10] HI = h̄ S(bq + b q), (6) gνν′σνν′ . (7) Here we have introduced the atom–phonon coupling co- efficients gνν′ = Fνν′√ 2MNh̄ , (8) Fνν′ = − ϕν(x)U ′(x)ϕν′ (x)dx (9) being the matrix elements for the force of the surface upon the atom. We note that Fνν′ = −mω2νν′xνν′ , where xνν′ = 〈ν|x|ν′〉 and ωνν′ = ων −ων′ are the surface–atom dipole matrix element and the translational transition frequency, respectively. Hence, the coupling coefficient gνν′ depends on the dipole matrix element xνν′ and the transition frequency ωνν′ . Since ωνν = 0, we have gνν = We note that the Hamiltonian of the x-polarized acous- tic phonons is given by h̄ωqb qbq. (10) The total Hamiltonian of the atom–phonon system is H = HA +HI +HB. (11) We use the above Hamiltonian to study the phonon- mediated decay of the atom. III. DYNAMICS OF THE ATOM In this section, we present the basic equations for the phonon-mediated decay processes. We derive a general master equation for the reduced density operator of the atom in subsection IIIA, obtain analytical expressions for the relaxation rates and frequency shifts in subsec- tion III B, and calculate the rates and the shifts in the framework of the Debye model in subsection III C. A. Master equation In the Heisenberg picture, the equation for the phonon operator bq(t) is ḃq(t) = −iωqbq(t)− S(t), (12) which has a solution of the form bq(t) = bq(t0)e −iωq(t−t0) − iWq(t). (13) Here, t0 is the initial time and Wq is given by Wq(t) = e−iωq(t−τ)S(τ) dτ. (14) Consider an arbitrary atomic operator O which acts only on the atomic states but not on the phonon states. The time evolution of this operator is governed by the Heisen- berg equation ∂O(t) [HA(t) +HI(t),O(t)], (15) which, with account of Eqs. (6) and (13), yields ∂O(t) [HA(t),O(t)] [S(t),O(t)][bq(t0)e−iωq(t−t0) − iWq(t)] [b†q(t0)e iωq(t−t0) + iW †q(t)][O(t), S(t)]. We assume the initial density of the atom–phonon sys- tem to be the direct product state ρΣ(t0) = ρ(t0)ρB(t0), (17) with the atom in an arbitrary state ρ(t0) and the phonons in a thermal state ρB(t0) = Z −1 exp[−HB(t0)/kBT ]. (18) Here, Z is the normalization constant and T is the tem- perature of the phonon bath. For the initial condition (17), the Bogolubov’s lemma [12], applied to an arbitrary operator Θ(t), asserts the following: 〈Θ(t)bq(t0)〉 = n̄q〈[bq(t0),Θ(t)]〉, (19) where the mean number of phonons in the mode q is given by n̄q = exp(h̄ωq/kBT )− 1 . (20) Let Θ be an atomic operator. We then have the commu- tation relation [bq(t),Θ(t)] = 0, which yields [bq(t0),Θ(t)] = ie iωq(t−t0)[Wq(t),Θ(t)]. (21) Combining Eq. (19) with Eq. (21) leads to 〈Θ(t)bq(t0)〉 = ieiωq(t−t0)n̄q〈[Wq(t),Θ(t)]〉. (22) We perform the quantum mechanical averaging for ex- pression (16) and use Eq. (22) to eliminate the phonon operators bq(t0) and b q(t0). The resulting equation can be written as ∂〈O(t)〉 〈[HA(t),O(t)]〉 n̄q + 1√ 〈[S(t),O(t)]Wq(t) +W †q(t)[O(t), S(t)]〉 〈Wq(t)[O(t), S(t)] + [S(t),O(t)]W †q(t)〉. We note that Eq. (23) is exact. It does not contain phonon operators explicitly. The dependence on the phonon operators is hidden in the time shift of the oper- ator S(τ) in expression (14) for the operator Wq(t). We now show how the dependence of the operator Wq(t) on the phonon operators can be approximately eliminated. We assume that the atom–phonon coupling coefficients gνν′ are small. The use of the zeroth-order approximation σνν′(τ) = σνν′ (t)e iωνν′(τ−t) in the expres- sion for S(τ) [see Eq. (7)] yields S(τ) = gνν′σνν′ (t)e iωνν′(τ−t), (24) which is accurate to first order in the coupling coeffi- cients. Inserting Eq. (24) into Eq. (14) gives Wq(t) = gνν′σνν′(t)δ−(ων′ν − ωq), (25) where δ−(ω) = lim e−i(ω+iǫ)τ dτ δ(ω). (26) Here, in order to take into account the effect of adiabatic turn-on of interaction, we have added a small positive parameter ǫ to the integral and have used the limit t0 → −∞. Introducing the notation gνν′σνν′δ−(ων′ν − ωq), (27) we can rewrite Eq. (23) in the form ∂〈O(t)〉 〈[HA(t),O(t)]〉 (n̄q + 1)〈[S(t),O(t)]Kq(t) +K†q(t)[O(t), S(t)]〉 n̄q〈Kq(t)[O(t), S(t)] + [S(t),O(t)]K†q(t)〉. (28) In order to examine the time evolution of the reduced density operator ρ(t) of the atom in the Schrödinger picture, we use the relation 〈O(t)〉 = Tr[O(t)ρ(0)] = Tr[O(0)ρ(t)], transform to arrange the operator O(0) at the first position in each operator product, and eliminate O(0). Then, we obtain the Liouville master equation ∂ρ(t) = − i [HA, ρ(t)] (n̄q + 1){[Kqρ(t), S] + [S, ρ(t)K†q]} n̄q{[S, ρ(t)Kq] + [K†qρ(t), S]}. (29) Equations (28) and (29) are valid to second order in the coupling coefficients. These equations allow us to study the time evolution and dynamical characteristics of the atom interacting with the thermal phonon bath. We note that Eq. (29) is a particular form of the Zwanzig’s generalized master equation, which can be obtained by the projection operator method [13]. B. Relaxation rates and frequency shifts We use Eq. (29) to derive an equation for the matrix elements ρjj′ ≡ 〈j|ρ|j′〉 of the reduced density operator of the atom. The result is ∂ρjj′ = −iωjj′ρjj′ + (γejj′νν′ + γ jj′νν′)ρνν′ [(γejν + γ jν)ρνj′ + (γ j′ν + γ j′ν)ρjν ], (30) where the coefficients γejj′νν′ = 2π n̄q + 1 gjνgj′ν′ [δ−(ωνj − ωq) + δ+(ων′j′ − ωq)], γejν = 2π n̄q + 1 gjµgνµδ−(ωνµ − ωq) (31) γajj′νν′ = 2π gjνgj′ν′ [δ−(ωj′ν′ − ωq) + δ+(ωjν − ωq)], γajν = 2π gjµgνµδ+(ωµν − ωq) (32) are the decay parameters associated with the phonon emission and absorption, respectively. Here, the nota- tion δ+(ω) = δ −(ω) has been used. Equation (30) describes phonon-induced variations in the populations and coherences of the translational levels of the atom. We analyze the characteristics of the relax- ation processes. For simplicity of mathematical treat- ment, we first consider only transitions from discrete lev- els. The equation for the diagonal matrix element ρjj for a discrete level j can be written in the form (γejjνν + γ jjνν )ρνν − (γejj + γajj + c.c.)ρjj + off-diagonal terms. (33) When the off-diagonal terms are neglected, Eq. (33) re- duces to a simple rate equation. It is clear from Eq. (33) that the rate for the downward transition from an upper level l to a lower level k (k < l) is Rekl = γ kkll = 2π n̄q + 1 g2lkδ(ωlk − ωq), (34) while the rate for the upward transition from a lower level k to an upper level l (l > k) is Ralk = γ llkk = 2π g2lkδ(ωlk − ωq). (35) Equations (34) and (35) are in agreement with the re- sults of Gortel et al. [10], obtained by using the Fermi golden rule. We note that Rekl and R lk with l ≤ k are mathematically equal to zero because they have no phys- ical meaning. For convenience, we introduce the notation Rlk = R lk, R lk, or 0 for l < k, l > k, or l = k, respec- tively. It is clear that the off-diagonal coefficients Rlk with l 6= k are the rates of transitions. However, the di- agonal coefficients Rkk have no physical meaning and are mathematically equal to zero. As seen from Eq. (33), the phonon-mediated depletion rate of a level k is Γkk = 2Re(γ kk + γ kk). The explicit expression for this rate is Γkk = 2π n̄q + 1 g2kµδ(ωkµ − ωq) g2µkδ(ωµk − ωq). (36) We note that Γkk = µk +R µk) = µ Rµk. We can write Γkk = Γ kk + Γ kk, where Γekk = Reµk (37) Γakk = Raµk (38) are the contributions due to downward transitions (phonon emission) and upward transitions (phonon ab- sorption), respectively. In the above equations, the sum- mation over µ can be extended to cover not only the discrete levels but also the continuum levels. Meanwhile, the equation for the off-diagonal matrix element ρlk for a pair of discrete levels l and k can be written in the form ∂ρlk/∂t = −(iωlk + γell + γall + γe∗kk + γa∗kk)ρlk + . . . , or, equivalently, = −i(ωlk +∆lk − iΓlk)ρlk + . . . . (39) Here the frequency shift ∆lk is given by ∆lk = n̄q + 1 ωlµ − ωq ωµk + ωq ωlµ + ωq ωµk − ωq , (40) while the coherence decay rate Γlk is expressed as Γlk = π n̄q + 1 g2lµδ(ωlµ − ωq) + g2kµδ(ωkµ − ωq) g2µlδ(ωµl − ωq) + g2µkδ(ωµk − ωq) When we set l = k in Eq. (40), we find ∆kk = 0. When we set l = k in Eq. (41), we recover Eq. (36). We note that Γlk = µl + R µk + R µl + R µk)/2 = µ(Rµl + Rµk)/2. Comparison between Eqs. (41) and (36) yields the relation Γlk = (Γll +Γkk)/2. We can also write Γlk = Γ lk + Γ lk, where Γ µl + R µk)/2 and Γalk = µl + R µk)/2 are the contributions due to downward transitions (phonon emission) and upward transitions (phonon absorption), respectively. In the above equations, the summation over µ can be extended to cover not only the discrete levels but also the contin- uum levels. We now discuss phonon-mediated transitions from con- tinuum (free) levels. We start by considering free-to- bound transitions. For a continuum level f with energy Ef > 0, the center-of-mass wave function ϕf (x) is nor- malized per unit energy. In this case, the quantity Rνf becomes the density of the transition rate. A free level f can be approximated by a level of a quasicontinuum [14]. A discretization of the continuum can be realized by using a large box of length L with reflecting boundary condi- tions [15]. We label En the energies of the eigenstates in the box and φn(x) the corresponding wave functions. Note that such states are standing-wave states [14, 15]. The relation between a quasicontinuum-state wave func- tion φnf (x), normalized to unity in the box, and the cor- responding continuum-state wave function ϕf (x), nor- malized per unit energy, with equal energies Enf = Ef , is [15] ϕf (x) ∼= ]−1/2 φnf (x) )1/2 ( φnf (x). (42) Consequently, for a single atom initially prepared in the quasicontinuum standing-wave state |nf 〉 = |φnf 〉, the rate for the transition to an arbitrary bound state |ν〉 is approximately given by Gνf = vfRνf , (43) where vf = (2Ef/m)1/2 is the velocity of the atom in the initial standing-wave state |f〉. The phonon-mediated free-to-bound decay rate (adsorption rate) is then given Gνf , (44) where the summation includes only bound levels. It is clear from Eq. (43) that, in the continuum limit L → ∞, the rate Gνf tends to zero. This is because a free atom can be anywhere in free space and therefore the effect of phonons on a single free atom is negligible. In order to get deeper insight into the free-to-bound transition rate density Rνf , we consider a macroscopic atomic ensemble in the thermodynamic limit [14]. Sup- pose that there are N0 atoms in a volume with a large length L and a transverse cross section area S0. Assume that all the atoms are in the same quasicontinuum state |nf 〉 and interact with the dielectric independently. The rate for the transitions of the atoms from the quasicon- tinuum state |nf〉 to an arbitrary bound state |ν〉, defined as the time derivative of the number of atoms in the state |ν〉, is Dνf = N0Gνf . In order to get the rate for the con- tinuum state |f〉, we need to take the thermodynamical limit, where L → ∞ and N0 → ∞ but N0/L remains constant. Then, the rate for the transitions of the atoms from the continuum state |f〉 to an arbitrary bound state |ν〉 is given by Dνf = πh̄ρ0S0vfRνf = 2πh̄NfRνf . Here, ρ0 = N0/LS0 is the atomic number density and Nf = ρ0S0vf/2 is the number of atoms incident into the dielectric surface per unit time. It is clear that the tran- sition rate Dνf is proportional to the incidence rate Nf as well as the transition rate density Rνf . We emphasize that Dνf is a characteristics for a macroscopic atomic en- semble in the thermodynamic limit while Gνf is a mea- sure for a single atom. When the length of the box, L, and the number of atoms, N0, are finite, the dynamics of the atoms cannot be described by the free-to-bound rate Dνf directly. Instead, we must use the transition rate per atom Gνf = Dνf/N0, which depends on the length L of the box that contains the free atoms [see Eq. (43)]. In a thermal gas, the atoms have different velocities and, therefore, different energies. For a thermal Maxwell- Boltzmann gas with temperature T0, the distribution of the kinetic energy Ef of the atomic center-of-mass motion along the x direction is P (Ef ) = πkBT0 e−Ef/kBT0 . (45) The transition rate to an arbitrary bound state |ν〉 is then given by GνT0 = GνfP (Ef ) dEf , i.e. GνT0 = e−Ef/kBT0RνfdEf , (46) where λD = (2πh̄ 2/mkBT0) 1/2 is the thermal de Broglie wavelength. The phonon-mediated free-to-bound decay rate (adsorption rate) is given by GT0 = GνT0 = GfP (Ef ) dEf . (47) In the above equation, the summation over ν includes only bound levels. Note that Eq. (46) is in qualitative agreement with the results of Refs. [5, 14]. It is easy to extend the above results to the case of free-to-free transitions. Indeed, it can be shown that the density of the rate for the transition from a quasicontin- uum state |nf 〉, which corresponds to a free state |f〉, to a different free state |f ′〉 is given by Qf ′f = vfRf ′f . (48) For convenience, we introduce the notation Qef ′f = Qf ′f or 0 for Ef ′ < Ef or Ef ′ ≥ Ef , respectively, and Qaf ′f = Qf ′f or 0 for Ef ′ > Ef or Ef ′ ≤ Ef , respectively. Then, we have Qf ′f = Q f ′f , 0, or Q f ′f for Ef ′ < Ef , Ef ′ = Ef , or Ef ′ > Ef , respectively. The downward (phonon-emission) and upward (phonon-absorption) free-to-free decay rates for the free state |f〉 are given by Qef = Qef ′fdEf ′ (49) Qaf = Qaf ′fdEf ′ , (50) respectively. The total free-to-free decay rate for the free state |f〉 is Qf = Qef +Qaf = Qf ′fdEf ′ . For a thermal gas, we need to replace the transition rate density Qf ′f and the decay rate Qf by Qf ′T0 = Qf ′fP (Ef ) dEf and QT0 = QfP (Ef ) dEf , respec- tively, which are the averages of Qf ′f and Qf , respec- tively, with respect to the energy distribution P (Ef ) of the initial state. Like in the other cases, we have Qf ′T0 = Q f ′T0 +Qaf ′T0 and QT0 = Q +QaT0 , where Qef ′T0 = Qef ′fP (Ef ) dEf , Qaf ′T0 = ∫ Ef′ Qaf ′fP (Ef ) dEf (51) are the downward and upward transition rate densities QeT0 = QefP (Ef ) dEf , QaT0 = QafP (Ef ) dEf (52) are the downward and upward decay rates. The thermal decay ratesQeT0 andQ describe the cooling and heating processes, respectively. It can be easily shown thatQeT0 < QaT0 , Q > QaT0 , and Q = QaT0 when T0 < T , T0 > T , and T0 = T , respectively. The relation Q < QaT0 (QeT0 > Q ), obtained for T0 < T (T0 > T ), indicates the dominance of heating (cooling) of free atoms by the surface. C. Relaxation rates and frequency shifts in the framework of the Debye model In order to get insight into the relaxation rates and frequency shifts, we approximate them using the Debye model for phonons. In this model, the phonon frequency ωq is related to the phonon wave number q as ωq = vq, where v is the sound velocity. Furthermore, the summa- tion over the first Brillouin zone is replaced by an integral over a sphere of radius qD = (6π 2N/V )1/3, where V is the volume of the solid. The Debye frequency and the Debye temperature are given by ωD = vqD and TD = h̄ωD/kB, respectively. For fused silica, we have v = 5.96 km/s, NM/V = 2.2 g/cm3, and M = 9.98× 10−26 kg [16]. Us- ing these parameters, we find qD = 109.29 × 106 cm−1, ωD = 10.4 THz, and TD = 498 K. In order to perform the summation over phonon states in the framework of the Debye model, we invoke the thermodynamic limit, i.e., replace · · · = V |q|≤qD . . . dq = . . . ω2qdωq. (53) Then, for transitions between an upper level l and a lower level k, where 0 < ωlk < ωD, Eqs. (34) and (35) yield Rekl = Mh̄ω3D (n̄lk + 1)ωlkF lk (54) Ralk = Mh̄ω3D n̄lkωlkF lk. (55) Here, n̄lk is given by Eq. (20) with ωq replaced by ωlk. We emphasize that, according to Eqs. (54) and (55), the phonon-emission rate Rekl and the phonon-absorption rate Ralk depend not only on the matrix element Flk of the force but also on the translational transition fre- quency ωlk. The frequency dependences of the transi- tion rates are comprised of the frequency dependences of the mean phonon number n̄lk, the phonon mode den- sity 3Nω2lk/ω D, and the matrix element Flk = −U ′lk = −mω2lkxlk of the force. An additional factor comes from the presence of the phonon frequency in Eq. (5) for the surface displacement and, consequently, in the atom– phonon interaction Hamiltonian (6). It is clear that an increase in the phonon frequency leads to a decrease in the mean phonon number and an increase in the phonon mode density. The matrix element of the force usu- ally first increases and then decreases with increasing phonon frequency. Due to the existence of several com- peting factors, the frequency dependences of the tran- sition rates are rather complicated. They usually first increase and then decrease with increasing phonon fre- quency. We note that, for transitions with ωlk > ωD, we have Rekl = R lk = 0. We conclude this section by noting that the use of Eq. (53) in Eq. (40) yields the frequency shift ∆lk = ∆ lk +∆ lk , (56) where 2Mh̄ω3D F 2lµ ωlµ − ω F 2µk ωµk + ω ωdω (57) Mh̄ω3D ω2lµ − ω2 ω2µk − ω2 n̄ωωdω are the zero- and finite-temperature contributions, re- spectively. In Eq. (58), n̄ω is given by Eq. (20) with ωq replaced by ω. IV. NUMERICAL RESULTS AND DISCUSSIONS In this section, we present the numerical results based on the analytical expressions derived in the previous section for the phonon-mediated relaxation rates of the translational levels of the atom. In particular, we use Eqs. (54) and (55), obtained in the framework of the De- bye model, for our numerical calculations. We consider transitions from bound states as well as free states. The transitions from bound states to other translational lev- els occur in the case where the atom is initially already adsorbed or trapped near the surface. The transitions from free states to other translational levels occur in the processes of adsorbing, heating, and cooling of free atoms by the surface. Due to the difference in physics of the ini- tial situations, we study the transitions from bound and free states separately. A. Transitions from bound states FIG. 3: Phonon-emission rates Reν′ν from the vibrational lev- els (a) ν = 280 and (b) ν = 120 to other levels ν′ as functions of the lower-level energy Eν′ . The arrows mark the initial states. The parameters of the solid are M = 9.98 × 10−26 kg and ωD = 10.4 THz. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 1. FIG. 4: Phonon-absorption rates Raν′ν from the vibrational levels (a) ν = 280 and (b) ν = 120 to other levels ν′ as func- tions of the upper-level energy Eν′ . The left (right) panel in each row corresponds to bound-to-bound (bound-to-free) transitions. The arrows mark the initial states. The param- eters used are as in Fig. 3. The temperature of the phonon bath is T = 300 K. We start from a given bound level and calculate the rates of phonon-mediated atomic transitions, both down- ward and upward. The profiles of the phonon-emission (downward-transition) rate Reν′ν [see Eq. (54)] and the phonon-absorption (upward-transition) rateRaν′ν [see Eq. (55)] are shown in Figs. 3 and 4, respectively. The upper (lower) part of each of these figures corresponds to the case of the initial level ν = 280 (ν = 120), with energy Eν = −156 MHz (Eν = −8.4 THz). The left (right) panel of Fig. 4 corresponds to bound-to-bound (bound-to-free) upward transitions. The temperature of the surface is assumed to be T = 300 K. As seen from Figs. 3 and 4, the transition rates have pronounced localized profiles. Due to the competing effects of the mean phonon num- ber, the phonon mode density, and the matrix element of the force, the transition rates usually first increase and then decrease with increasing phonon frequency. It is clear from a comparison of Figs. 3(a) and 3(b) and also a comparison of Figs. 4(a) and 4(b) that transitions from shallow levels have probabilities orders of magni- tude lower than those from deeper levels. The main rea- son is that the wave functions of the shallow states are spread further away from the surface than those for the deep states. Due to this difference, the effects of the sur- face vibrations are weaker for the shallow levels than for the deep levels. Another pertinent feature that should be noted from the figure is the following: Since transi- tion frequencies involved are large, they may overshoot the Debye frequency ωD = 10.4 THz, leading to a cutoff on the lower (higher) side of the frequency axis for the emission (absorption) curve. In order to see the overall effect of the individual tran- sition rates shown above, we add them up. First we ex- amine the phonon-absorption rates of bound levels. The total phonon-absorption rate Γaνν of a bound level ν is the sum of the individual absorption rates Raµν over all the upper levels µ, both bound and free [see Eq. (38)]. We plot in Fig. 5 the contributions to Γaνν from two types of transitions, bound-to-bound and bound-to-free (des- orption) transitions. The solid curve of the figure shows that the bound-to-bound phonon-absorption rate is large (above 1010 s−1) for deep and intermediate levels. How- ever, it reduces dramatically with increasing ν in the region of large ν and becomes very small (below 10−5 s−1) for shallow levels. Meanwhile, the dashed curve of Fig. 5 shows that the bound-to-free phonon-absorption rate (i.e., the desorption rate) is zero for deep levels, since the energy required for the transition is greater than the Debye energy [5]. However, the desorption rate is sub- stantial (above 105 s−1) for intermediate and shallow lev- els. Thus, the total phonon-absorption rate Γaνν is mainly determined by the bound-to-bound transitions in the case of deep levels and by the bound-to-free transitions in the case of shallow levels. One of the reasons for the dramatic reduction of the bound-to-bound phonon-absorption rate in the region of shallow levels is that the number of up- per bound levels µ becomes small. The second reason is that the frequency of each individual transition becomes small, leading to a decrease of the phonon mode den- sity. The third reason is that the center-of-mass wave functions of shallow levels are spread far away from the surface, leading to a reduction of the effect of phonons on the atom. Unlike the bound-to-bound phonon-absorption rate, the bound-to-free phonon-absorption rate is substantial in the region of shallow levels. This is because the free- state spectrum is continuous and the range of the bound- to-free transition frequency can be large (up to the De- bye frequency ωD = 10.4 THz). The gradual reduction of the bound-to-free phonon-absorption rate in the region of shallow levels is mainly due to the reduction of the time that the atom spends in the proximity of the surface. FIG. 5: Contributions of bound-to-bound (solid curve) and bound-to-free (dashed curve) transitions to the total phonon- absorption rate Γaνν versus the vibrational quantum number ν of the initial level. The parameters used are as in Fig. 3. The temperature of the phonon bath is T = 300 K. The total phonon-emission rate Γeνν [see Eq. (37)] and the total phonon-absorption rate Γaνν [see Eq. (38)] are shown in Fig. 6 by the solid and dashed curves, respec- tively. It is clear from the figure that emission is com- parable to but slightly stronger than absorption. Such a dominance is due to the fact that phonon emission moves the atom to a center-of-mass state closer to the surface while phonon absorption changes the atomic state in the opposite direction (see Figs. 1 and 2). Our results for the rates are in good qualitative agreement with the results of Oria et al., albeit with the Morse potential [5]. We stress that we include a large number of vibrational lev- els as a consequence of the deep silica–cesium potential. Note that the earlier work on this theme involved much fewer levels [5]. FIG. 6: Phonon-emission decay rate Γeνν (solid lines) and phonon-absorption decay rate Γaνν (dashed lines) of a bound level as functions of the vibrational quantum number ν. The inset shows the rates in the linear scale to highlight the dif- ferences in the dissociation limit. The parameters used are as in Fig. 3. The temperature of the phonon bath is T = 300 K. FIG. 7: Same as in Fig. 6 except that T = 30 K. We next study the effect of temperature on the decay rates. The results for the phonon-mediated decay rates for T = 30 K are shown in Fig. 7. In contrast to Fig. 6, the absorption rate is now much smaller than the corre- sponding emission rate for both shallow and deep levels. Thus, while it is difficult to distinguish the two log-scale curves for deep and shallow levels at room temperature (see Fig. 6), they are well resolved at low temperature. B. Transitions from free states We now calculate the rates for transitions from free states to other levels. We first examine free-to-bound transitions, which correspond to the adsorption process. According to Eq. (43), the free-to-bound (more exactly, quasicontinuum-to-bound) transition rate Gνf depends not only on the continuum-to-bound transition rate den- sity Rνf but also on the length L of the free-atom quan- tization box. To be specific, we use in our numerical calculations the value L = 1 mm, which is a typical size of atomic clouds in magneto-optical traps [17]. FIG. 8: Free-to-bound transition rates Gνf for transitions from the free plane-wave states with energies (a) Ef = 2 MHz and (b) Ef = 3.1 THz to bound levels ν as functions of the bound-level energy Eν . The arrows mark the energies of the initial free states. The insets show Gνf on the log scale versus Eν in the range from −200 MHz to −0.2 MHz to highlight the rates to shallow bound levels. The length of the free- atom quantization box is L = 1 mm. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 3. We plot in Fig. 8 the free-to-bound transition rate Gνf [see Eq. (43)] as a function of the vibrational quantum number ν. The upper (lower) part of the figure corre- sponds to the case of the initial-state energy Ef = 2 MHz (Ef = 3.1 THz), which is close to the average ki- netic energy per atom in an ideal gas with temperature T0 = 200 µK (T0 = 300 K). We observe that the free-to- bound transition rate first increases and then decreases with increasing transition frequency ωfν = (Ef − Eν)/h̄. Such behavior results from the competing effects of the mean phonon number, the phonon mode density, and the matrix element of the force, like in the case of bound- to-bound transitions (see Fig. 3). We also see a cut- off of the transition frequency, which is associated with the Debye frequency. Comparison of Figs. 8(a) and 8(b) shows that the transitions from low-energy free states have probabilities orders of magnitude smaller than those from high-energy free states. One of the reasons is that the transition rate Gνf is proportional to the velocity vf = (2Ef/m)1/2 [see Eq. (43)]. The dependence of the transition rate density Rνf on the transition fre- quency ωfν also plays an important role. Because of this, the rates for the transitions from low-energy free states to shallow bound levels are very small [see the inset of Fig. 8(a)]. FIG. 9: Free-to-bound decay rate Gf as a function of the free-state energy Ef . The inset highlights the magnitude and profile of the decay rate for Ef in the range from 0 to 20 MHz. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8. We show in Fig. 9 the free-to-bound decay rate Gf [see Eq. (44)], which is a characteristic of the adsorp- tion process, as a function of the free-state energy Ef . We see that Gf first increases and then decreases with increasing Ef . The increase of Gf with increasing Ef in the region of small Ef (see the inset) is mainly due to the increase in the atomic incidence velocity vf . In this region, we have Gf ∝ vf ∝ Ef [see Eqs. (43) and (44)]. For Ef in the range from 0 to 20 MHz, which is typical for atoms in magneto-optical traps, the maximum value of Gf is on the order of 10 4 s−1 (see the inset of Fig. 9). Such free-to-bound (adsorption) rates are sev- eral orders of magnitude smaller than the bound-to-free (desorption) rates (see the dashed curve in Fig. 5). The decrease of Gf with increasing Ef in the region of large Ef is mainly due to the reduction of the atom–phonon coupling coefficients. FIG. 10: Free-to-bound transition rates GνT0 for transitions from the thermal states with temperatures (a) T0 = 200 µK and (b) T0 = 300 K to bound levels ν as functions of the bound-level energy Eν . The insets show GνT0 on the log scale versus Eν in the range from −200 MHz to −0.2 MHz to high- light the rates to shallow bound levels. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8. FIG. 11: Free-to-bound decay rate GT0 as a function of the atomic temperature T0 in the ranges (a) from 100 µK to 400 µK and (b) from 50 K to 350 K. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8. In a thermal gas, the adsorption process is charac- terized by the transition rate GνT0 [see Eq. (46)] and the decay rate GT0 [see Eq. (47)], which are the av- erages of the free-to-bound transition rate Gνf and the free-to-bound decay rate Gf , respectively, over the free- state energy distribution (45). We plot the free-to-bound transition rate GνT0 and the free-to-bound decay rate GT0 in Figs. 10 and 11, respectively. Comparison be- tween Figs. 10(a) and 9(a) shows that the transition rates from low-temperature thermal states and low-energy free states look quite similar to each other. The reason is that the spread of the energy distribution is not substantial in the case of low temperatures. The spread of the en- ergy distribution is however substantial in the case of high temperatures, leading to the softening of the cut- off frequency effect [compare Fig. 10(b) with Fig. 9(b)]. Figure 11 shows that the free-to-bound decay rate GT0 first increases and then reduces with increasing atomic temperature T0. For T0 in the range from 100 µK to 400 µK, which is typical for atoms in magneto-optical traps, the maximum value of GT0 is on the order of 10 4 s−1 [see Fig. 11(a)]. Such free-to-bound (adsorption) rates are several orders of magnitude smaller than the bound-to- free (desorption) rates (see the dashed curve in Fig. 5). Figure 11(a) shows that, in the region of low atomic tem- perature T0, one has GT0 ∝ T0, in agreement with the asymptotic behavior of Eqs. (46) and (47). FIG. 12: Free-to-free transition rate densities Qf ′f for the upward (solid lines) and downward (dashed lines) transitions from the free states |f〉 with energies (a) Ef = 2 MHz and (b) Ef = 3.1 THz to other free states |f ′〉 as functions of the final- level energy Ef ′ . The arrows mark the energies of the initial free states. The inset in part (a) shows Qf ′f versus Ef ′ in the range from 0 to 4 MHz to highlight the small magnitude of the rate density for downward transitions (dashed line). The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8. We now examine free-to-free transitions, both upward and downward, which corresponding to the heating and cooling processes of free atoms by the surface. We plot in Fig. 12 the free-to-free transition rate density Qf ′f [see Eq. (48)] as a function of the final-level energy Ef ′ . The upper (lower) part of the figure corresponds to the case of the initial-state energy Ef = 2 MHz (Ef = 3.1 THz), which is close to the average kinetic energy per atom in an ideal gas with temperature T0 = 200 µK (T0 = 300 K). The rate densities are shown for the upward (phonon- absorption) and downward (phonon-emission) transitions by the solid and dashed lines, respectively. The fig- ure shows that the free-to-free transition rate density in- creases or decreases with increasing transition frequency if the latter is not too large or is large enough, respec- tively. We also observe a signature of the Debye cutoff of the phonon frequency. Comparison of Figs. 12(a) and 12(b) shows that transitions from low-energy free states have probabilities orders of magnitude smaller than those from high-energy free states. Figure 12(a) and its inset show that, when the energy of the free state is low, the free-to-free downward (cooling) transition rate is very small as compared to the free-to-free upward (heating) transition rate. FIG. 13: Free-to-free upward and downward decay rates Qaf (solid lines) and Qef (dashed lines) as functions of the energy Ef of the initial free state. The insets highlight the magni- tudes and profiles of the decay rates for Ef in the range from 0 to 20 MHz. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8. We show in Fig. 13 the free-to-free upward (phonon- absorption) and downward (phonon-emission) decay rates Qaf [see Eq. (50)] and Q f [see Eq. (49)] as functions of the free-state energy Ef . We observe that Qaf and Qef increase with increasing Ef in the range from 0 to 8 THz. The increase of Qaf with increasing Ef in the region of small Ef (see the left inset) is mainly due to the increase in the atomic incidence velocity vf . In this region, we have Qaf ∝ vf ∝ Ef [see Eqs. (48) and (50)]. The increase of Qef with increasing Ef in the region of small Ef (see the right inset) is due to not only the increase in the atomic incidence velocity vf [see Eq. (48)] but also the increase of the transition rate density Qef ′f and the increase of the integration interval (0, Ef) [see Eq. (49)]. In this region, the dependence of Qef on the energy Ef is of higher order than E3/2f . The left inset of Fig. 13 shows that, for Ef in the range from 0 to 20 MHz, the maximum value of Qaf is on the order of 10 4 s−1. Such free-to-free upward (heating) decay rates are comparable to but about two times smaller than the corresponding free-to-bound (adsorption) decay rates (see the inset of Fig. 9). Meanwhile, the right inset of Fig. 13 shows that, in the region of small Ef , the free-to-free downward (cool- ing) decay rate Qef is very small. FIG. 14: Free-to-free transition rate densities QafT0 for up- ward transitions (solid lines) and QefT0 for downward tran- sitions (dashed lines) from the thermal states with tempera- tures (a) T0 = 200 µK and (b) T0 = 300 K to free levels f as functions of the free-level energy Ef . The inset in part (a) shows the rate densities versus Ef in the range from 0 to 8 MHz to highlight the small magnitude of QefT0 (dashed line). The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8. FIG. 15: Free-to-free decay rates QaT0 (solid lines) and (dashed lines) for upward and downward transitions, re- spectively, as functions of the atomic temperature T0 in the ranges (a) from 100 µK to 400 µK and (b) from 50 K to 350 K. For comparison, the free-to-bound decay rate GT0 is re- plotted from Fig. 11 by the dotted lines. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8. In the case of a thermal gas, the phonon-mediated heat transfer between the gas and the surface is characterized by the free-to-free transition rate densitiesQafT0 andQ [see Eqs. (51)] and the free-to-free decay rates QaT0and QeT0 [see Eqs. (52)]. We plot the free-to-free transition rate densities QafT0 and Q in Fig. 14. Comparison between Figs. 14(a) and 12(a) shows that the transition rate densities from low-temperature thermal states and low-energy free states are quite similar to each other. The spread of the initial-state energy distribution is not substantial in this case. However, the energy spread of the initial state is substantial in the case of high tem- peratures, concealing the cutoff frequency effect [com- pare Fig. 14(b) with Fig. 12(b)]. We display the free- to-free decay rates QaT0 and Q in Fig. 15. The solid and dashed lines correspond to the upward (heating) and downward (cooling) transitions, respectively. For com- parison, the free-to-bound decay rate (adsorption rate) GT0 is re-plotted from Fig. 11 by the dotted lines. We observe that, for T0 in the range from 100 µK to 400 µK [see Fig. 15(a)], the adsorption rate GT0 (dotted line) is about two times larger than the heating rate QaT0 (solid line), while the cooling rate QeT0 (dashed line) is negligi- ble. Figure 15(a) shows that, in the region of low atomic temperatures, one has QT0 ∼= QaT0 ∝ T0, in agreement with the asymptotic behavior of expressions (52). The figure also shows that QeT0 quickly increases with increas- ing atomic temperature T0. The relation Q < QaT0 , obtained for T0 < T , indicates the dominance of heating of cold free atoms by the surface. The substantial mag- nitude of the free-to-bound transition rate GT0 (dotted line) indicates that a significant number of atoms can be adsorbed by the surface. According to Fig. 15(b), the free-to-free downward transition rate QeT0 (dashed line) crosses the upward transition rate QaT0 (solid line) when T0 = T = 300 K, and then becomes the dominant decay rate. The relation QeT0 > Q , obtained for T0 > T , indi- cates the dominance of cooling of hot free atoms by the surface. V. CONCLUSIONS In conclusion, we have studied the phonon-mediated transitions of an atom in a surface-induced potential. We developed a general formalism, which is applicable for any surface–atom potential. A systematic derivation of the corresponding density-matrix equation enables us to investigate the dynamics of both diagonal and off- diagonal elements. We included a large number of vi- brational levels originating from the deep silica–cesium potential. We calculated the transition and decay rates from both bound and free levels. We found that the rates of phonon-mediated transitions between transla- tional levels depend on the mean phonon number, the phonon mode density, and the matrix element of the force from the surface upon the atom. Due to the effects of these competing factors, the transition rates usually first increase and then reduce with increasing transition fre- quency. We focused on the transitions from bound states. Two specific examples, namely, when the initial level is a shallow level also when it can be one of the deep levels have been worked out. We have shown that there can be marked differences in the absorption and emission behav- ior in the two cases. For example, both the absorption and emission rates from the deep bound levels can be sev- eral orders (in our case, six orders) of magnitude larger than the corresponding rates from the shallow bound lev- els. We also analyzed various types of transitions from free states. We have shown that, for thermal atomic ce- sium with temperature in the range from 100 µK to 400 µK in the vicinity of a silica surface with temperature of 300 K, the adsorption (free-to-bound decay) rate is about two times larger than the heating (free-to-free upward de- cay) rate, while the cooling (free-to-free downward decay) rate is negligible. Acknowledgments We thank M. Chevrollier for fruitful discussions. This work was carried out under the 21st Century COE pro- gram on “Coherent Optical Science.” [∗] Also at Institute of Physics and Electronics, Vietnamese Academy of Science and Technology, Hanoi, Vietnam. [1] V. I. Balykin, K. Hakuta, Fam Le Kien, J. Q. Liang, and M. Morinaga, Phys. Rev. A 70, 011401(R) (2004); Fam Le Kien, V. I. Balykin, and K. Hakuta, Phys. Rev. A 70, 063403 (2004). [2] Fam Le Kien, S. Dutta Gupta, V. I. Balykin, and K. Hakuta, Phys. Rev. A 72, 032509 (2005). [3] Fam Le Kien, S. Dutta Gupta, K. P. Nayak, and K. Hakuta, Phys. Rev. A 72, 063815 (2005). [4] E. G. Lima, M. Chevrollier, O. Di Lorenzo, P. C. Se- gundo, and M. Oriá, Phys. Rev. A 62, 013410 (2000). [5] T. Passerat de Silans, B. Farias, M. Oriá, and M. Chevrollier, Appl. Phys. B 82, 367 (2006). [6] Fam Le Kien and K. Hakuta, Phys. Rev. A 75, 013423 (2007). [7] Fam Le Kien, S. Dutta Gupta, and K. Hakuta, e-print quant-ph/0610067. [8] K. P. Nayak, P. N. Melentiev, M. Morinaga, Fam Le Kien, V. I. Balykin, and K. Hakuta, e-print quant-ph/0610136. [9] C. Henkel and M. Wilkens, Europhys. Lett. 47, 414 (1999). [10] Z. W. Gortel, H. J. Kreuzer, and R. Teshima, Phys. Rev. B 22, 5655 (1980). [11] H. Hoinkes, Rev. Mod. Phys. 52, 933 (1980). [12] N. N. Bogolubov, Commun. of JINR, E17-11822, Dubna (1978); N. N. Bogolubov and N. N. Bogolubov Jr., Ele- mentary Particles and Nuclei (USSR) 11, 245 (1980). [13] R. Zwanzig, Lectures in Theoretical Physics, eds. W. E. Brittin, B. W. Downs, and J. Downs (Interscience, New York, 1961) Vol. 3, p. 106; G. S. Agarwal, Progress in Optics, ed. E. Wolf (North-Holland, Amsterdam, 1973) Vol. 11, p. 3; L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, New York, 1995) p. [14] J. Javanainen and M. Mackie, Phys. Rev. A 58, R789 (1998); M. Mackie and J. Javanainen, ibid. 60, 3174 (1999). [15] E. Luc-Koenig, M. Vatasescu, and F. Masnou-Seeuws, Eur. Phys. J. D 31, 239 (2004). [16] See, for example, G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 2001). [17] H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, New York, 1999). http://arxiv.org/abs/quant-ph/0610067 http://arxiv.org/abs/quant-ph/0610136

0704.0341

Infrared Evolution Equations: Method and Applications

Infrared Evolution Equations: Method and Applications B.I. Ermolaev Ioffe Physico-Technical Institute, 194021 St.Petersburg, Russia M. Greco Department of Physics and INFN, University Rome III, Rome, Italy S.I. Troyan St.Petersburg Institute of Nuclear Physics, 188300 Gatchina, Russia It is a brief review on composing and solving Infrared Evolution Equations. They can be used in order to calculate amplitudes of high-energy reactions in different kinematic regions in the double- logarithmic approximation. PACS numbers: 12.38.Cy I. INTRODUCTION Double-logarithmic (DL) contributions are of a special interest among radiative corrections. They are interesting in two aspects: first, in every fixed order of the perturbation theories they are the largest terms among the radiative corrections depending on the total energy and second, they are easiest kind of the corrections to sum up. DL corrections were discovered by V.V. Sudakov in Ref. [1] in the QED context. He showed that DL terms appear from integrations over soft, infrared (IR) -divergent momenta of virtual photons. All-order resummation of such contributions led to their exponentiations. Next important step was done in Refs. [2] where calculation and summation of DL contributions was considered in a systematic way. They found a complementary source of DL terms: soft virtual fermions. This situation appears in the Regge kinematics. The all-order resummations of DL contributions in the Regge kinematic are quite involved and yield more complicated expressions than the Sudakov exponentials. Nonetheless important was the proof of the factorization of bremsstrahlung photons with small k⊥ in the high-energy hadronic reactions found in Ref. [3] and often addressed as the Gribov’s bremsstrahlung theorem. This statement, suggested originally in the framework of the phenomenological QED of hadrons was extended to QCD in Refs. [4]. Calculation in the double-logarithmic approximation (DLA) amplitudes of the fermion-antifermion annihilation in the Regge forward and backward kinematics involves accounting for DL contributions from soft quarks and soft gluons. These reactions in QED and QCD have many common features. The e+e− -annihilation was studied in Refs. [2]. The quark-aniquark annihilation DLA was investigated in Ref. [5]. The method of calculation here was based on factorization of virtual quarks and gluons with minimal k⊥. Generally speaking, the results obtained in Ref. [5] could be obtained with the method of Ref. [2], however the technique of calculations suggested in Ref. [5] was much more elegant and efficient. Although Ref. [5] is about quark scattering only, it contains almost all technical ingredients necessary to compose Infrared Evolution Equations for any of elastic scattering amplitudes. Nevertheless it could not directly be applied to inelastic processes involving emission of soft particles. Such a generalization was obtained in Refs. [4, 6]. The basic idea of the above-mentioned method was suggested by L.N. Lipatov: to investigate evolution with respect to the infrared cut-off. The present, sounding naturally term ”Infrared Evolution Equations” (IREE) for this method was suggested by M. Krawczyk in Ref. [7] where amplitudes for the backward Compton scattering were calculated in DLA. The aim of the present brief review is to show how to compose and solve IREE for scattering amplitudes in different field theories and kinematic regions. The paper is organized as follows: in Sect. II we consider composing IREE in the technically simplest hard kinematics. In Sect. III we consider composing IREE in the forward kinematics and apply it to studying the structure function g1 of the polarized Deep-Inelastic scattering (DIS) at small x. The point is that the commonly used theoretical instrument to study g1 is DGLAP [11]. It collects logarithms of Q 2 to all orders in αs but does not include the total resummation of logarithms of 1/x, though it is important at small x. Accounting for such a resummaton leads to the steep rise of g1 at the small-x region. As is shown in Sect. IV, DGLAP lacks the resummaion but mimics it inexplicitly, through the special choice of fits for the initial parton densities. Invoking such peculiar fits together with DGLAP to describe g1 at x ≪ 1 led to various misconceptions in the literature. They are enlisted and corrected in Sect. V. The total resummaion of the leading logarithms is essential in the region of small x. In the opposite region of large x, DGLAP is quite efficient. It is attractive to combine the resummation with DGLAP. http://arxiv.org/abs/0704.0341v1 The manual for doing it is given in Sect. VI. Finally, Sect. VII is for concluding remarks. II. IREE FOR SCATTERING AMPLITUDES IN THE HARD KINEMATICS From the technical point of view, the hard kinematics, where all invariants are of the same order, is the easiest for analysis. For the simplest, 2 → 2 -processes, the hard kinematics means that the Mandelstamm variables s, t, u obey s ∼ −t ∼ −u . (1) In other words, the cmf scattering angles θ ∼ 1 in the hard kinematics. This kinematics is the easiest because the ladder Feynman graphs do not yield DL contributions here and usually the total resummation of DL contributions leads to multiplying the Born amplitude by exponentials decreasing with the total energy. Let us begin with composing and solving an IREE for the well-known object: electromagnetic vertex Γµ of an elementary fermion (lepton or quark). As is known, Γµ = ū(p2) γµf(q 2)− σµνqν g(q2) u(p1) (2) where p1,2 are the initial and final momenta of the fermion, m stands for the fermion mass and the transfer momentum q = p2−p1. Scalar functions f and g in Eq. (2) are called form factors. Historically, DL contributions were discovered by V. Sudakov when he studied the QED radiative corrections to the form factor f at |q2| ≫ |p21,2|. Following him, let us consider vertex Vµ at |q2| ≫ p21 = p22 = m2 (3) i.e. we assume the fermion to be on–shell and account for DL electromagnetic contributions. We will drop m for the sake of simplicity. A. IREE for the form factor f(q2) in QED Step 1 is to introduce the infrared cut-off µ in the transverse (with respect to the plane formed by momenta p1,2) momentum space for all virtual momenta ki: ki ⊥ > µ (4) where i = 1, 2, ... Step 2 is to look for the softest virtual particle among soft external and virtual particles. The only option we have is the softest virtual photon. Let denote its transverse momenta ≡ k⊥. By definition, k⊥ = min ki ⊥ . (5) Step 3: According to the Gribov theorem, the propagator of the softest photon can be factorized (i.e. it is attached to the external lines in all possible ways) whereas k⊥ acts as a new cut-off for other integrations. Adding the Born contribution fBorn = 1 we arrive at the IREE for f in the diagrammatic form. It is depicted in Fig. 1. IREE in the analytic form are written in the gauge-invariant way, but their diagrammatical writing depends on the gauge. In the present paper we use the Feynman gauge. Applying to it the standard Feynman rules, we write it in the analytic form: f(q2, µ2) = fBorn − e dαdβdk2 − µ2) f(q2, k2 (sαβ − k2 + ıǫ)(−sα+ sαβ − k2 + ıǫ)(sβ + sαβ − k2 + ıǫ) where we have used the Sudakov parametrization k = αp2 + βp1 + k⊥ and denoted s = −q2 ≈ 2p1p2. As f(q2, k2⊥) does not depend on α and β, the DL integration over them can be done with the standard way, so we are left with a simple integral equation to solve: f(q2, µ2) = fBorn − e ln(s/k2 )f(q2, k2 ) . (7) FIG. 1: The IREE for the Sudakov form factor. The letters in the blobs stand for IR cut-off. Differentiation of Eq. (7) over µ2 (more exactly, applying −µ2∂/∂µ2) reduces it to a differential equation ∂f/∂(ln(s/µ2)) = −(e2/8π2) ln(s/µ2)f (8) with the obvious solution f = fBorn exp[−(α/4π) ln2(q2/m2)] (9) where we have replaced µ by m and used α = e2/4π. Eq. (9) is the famous Sudakov exponential obtained in Ref. [1]. B. IREE for the form factor g(q2) in QED Repeating the same steps (see Ref. [8] for detail) leads to a similar IREE for the form factor g: g(q2,m2, µ2) = gBorn(s,m2)− e ln(s/k2 )g(q2,m2, k2 ) (10) where gBorn(s,m2) = −(m2/s)(α/π) ln(s/m2). Solving this equation and putting µ = m in the answer leads to the following relation between form factors f and g: g(s) = −2 , (11) with ρ = s/m2. Combining Eqs. (9,11) allows to write a simple expression for the DL asymptotics of the vertex Γµ: Γµ = ū(p2) σµνqν u(p1) exp[−(α/4π) ln2 ρ] . (12) C. e+e− -annihilation into a quark-antiquark pair Let us consider the e+e− -annihilation into a quark q(p1) and q̄(p2) at high energy when 2p1p2 ≫ p21,2. We consider the channel where the e+e− -pair annihilates into one heavy photon which decays into the q(p1) q̄(p2) -pair: e+e− → γ∗ → q(p1) q̄(p2) . (13) We call this process elastic. In this case the most sizable radiative corrections arise from the graphs where the quark and antiquark exchange with gluons and these graphs look absolutely similar to the graphs for the electromagnetic vertex Γµ considered in the previous subsection. As a result, accounting for the QCD radiative corrections in DLA to the elastic form factors fq, gq of quarks can be obtained directly from Eqs. (9,11) by replacement α → αsCF , (14) with CF = (N 2 − 1)/2N = 4/3. D. e+e− -annihilation into a quark-antiquark pair and gluons In addition to the elastic annihilation (13), the final state can include gluons: e+e− → γ∗ → q(p1) q̄(p2) + g(k1), ..g(kn) . (15) We call this process the inelastic annihilation. The QED radiative corrections to the inelastic annihilation (15) in DLA are absolutely the same as the corrections to the elastic annihilation. On the contrary, the QCD corrections account for gluon exchanges between all final particles. This makes composing the IREE for the inelastic annihilation be more involved (see Ref. [4]). The difference to the considered elastic case appears at Step 2: look for the softest virtual particle among soft external and virtual particles. Indeed, now the softest particle can be both a virtual gluon and an emitted gluon. For the sake of simplicity let us discuss the 3-particle final state, i.e. the process e+e− → γ∗ → q(p1) q̄(p2) + g(k1) . (16) The main ingredient of the scattering amplitude of this process is the new electromagnetic vertex Γ µ of the quark. In DLA, it is parameterized by new form factors F (1) and G(1) Γµ = B1(k1)ū(p2) (1)(q, k1)− σµνqν G(1)(q, k1) u(p1) (17) where (1) corresponds to the number of emitted gluons, q = p1 + p2 and l is the polarization vector of the emitted gluon. The bremsstrahlung factor B1 in Eq. (17) at high energies is expressed through k1 ⊥: ( p2l − p1l . (18) We call F (n), G(n) inelastic form factors. Let us start composing the IREE for F (1). Step 1 is the same like in the previous case. Step 2 opens more options. Let us first choose the softest gluon among virtual gluons and denote its transverse momentum k⊥ The integration over k⊥ runs from µ to s. As µ < k1 ⊥ < s, we have two regions to consider: Region D1 were µ < k1⊥ < k⊥ < s (19) and Region D2 were µ < k⊥ < k1⊥ < s (20) Obviously, the softest particle in Region D1 is the emitted gluon, so it can be factorized as depicted in graphs (b,b’) of Fig. 2. On the contrary, the virtual gluon is the softest in Region D2 were its propagator is factorized as shown in graphs (c,d,d’) of Fig. 2. Adding the Born contribution (graphs (a,a’) in Fig. 2) completes the IREE for F (1) depicted in Fig. 2. Graphs (a-b’) do not depend on µ and vanish when differentiated with respect to µ. Blobs in graphs (c-d’) do not depend on the longitudinal Sudakov variables, so integrations over α, β can be done like in the first loop. After that the differential IREE for F (1) is ∂F (1) CF ln (2p2k1 (2p1k1 F (1) . (21) Solving Eq. (21) and using that (2p1k1)(2p2k1) = sk 1⊥ leads to the expression F (1) = exp CF ln (k21⊥ suggested in Ref. [9] and proved in Ref. [4] for any n. The IREE for the form factor G(n) was obtained and solved in Ref. [8]. It was shown that G(n) = −2∂F (n)/∂ρ . (23) FIG. 2: The IREE for the inelastic quark form factor. E. Exponentiation of Sudakov electroweak double-logarithmic contributions The IREE -method was applied in Ref. [10] to prove exponentiation of DL correction to the electroweak (EW) reactions in the hard kinematics. There is an essential technical difference between the theories with the exact gauge symmetry (QED and QCD) and the EW interactions theory with the broken SU(2) ⊗ U(1) gauge symmetry: only DL contributions from virtual photons yield IR singularities needed to be regulated with the cut-off µ whereas DL contributions involving W and Z -bosons are IR stable because the boson masses MW and MZ act as IR regulators. In Ref. [10] the difference between MW and MZ was neglected and the parameter M & MW ≈ MZ (24) was introduced, in addition to µ, as the second IR cut-off. It allowed to drop masses MW,Z . The IREE with two IR cut-offs was composed quite similarly to Eq. (6), with factorizing one by one the softest virtual photon, Z-boson and W -boson. As a result the EW Sudakov form factor FEW is FEW = exp − α(Q ln2(s/µ2)− SU(2) (Y 21 + Y − α(Q ln2(s/M2) where Q1,2 are the electric charges of the initial and final fermion (with W -exchanges accounted, they may be different), Y1,2 are their hyper-charges and C SU(2) F = (N 2 − 1)/2N , with N = 2. We have used in Eq. (25) the standard notations g and g′ for the SU(2) and U(1) -EW couplings. The structure of the exponent in Eq. (25) is quite clear: the first, µ -dependent term comes from the factorization of soft photons like the exponent in Eq. (9) while other terms correspond to the W and Z -factorization; the factor in the squared brackets is the sum of the SU(2) and U(1) Casimirs, with the photon Casimir being subtracted to avoid the double counting. In the limit µ = M the group factor in the exponent is just the Casimir of SU(2)⊗ U(1). III. APPLICATION OF IREE TO THE POLARIZED DEEP-INELASTIC SCATTERING Cross-sections of the polarized DIS are described by the structure functions g1,2. They appear from the standard parametrization of the spin-dependent part Wµν of the hadronic tensor: Wµν = ıǫµνλρqλ Sρg1(x,Q Sρ − pρ g2(x,Q where p, m and S are the momentum, mass and spin of the incoming hadron; q is the virtual photon momentum; Q2 = −q2; x = Q2/2pq. Obviously, Q2 > 0 and 0 6 x 6 1. Unfortunately, g1,2 cannot be calculated in a straightforward model-independent way because it would involve QCD at long distances. To avoid this problem, Wµν is regarded as a convolution of Φq,g - probabilities to find a polarized quark or gluon and the partonic tensors W̃ (q,g) µν parameterized identically to Eq. (26). In this approach W̃ (q,g) µν involve only QCD at short distances, i.e. the Perturbative QCD while long-distance effects are accumulated in Φq,g. As Φq,g are unknown, they are mimicked by the initial quark and gluon densities δq, δg. They are fixed aposteriori from phenomenological considerations. So, the standard description of DIS is: Wµν ≈ W (q)µν ⊗ δq +W (g)µν ⊗ δg . (27) The standard theoretical instrument to calculate g1 is DGLAP[11] complemented with standard fits[12] for δq, δg. We call it Standard Approach (SA). In this approach g1(x,Q 2) = Cq(x/z)⊗∆q(z,Q2) + Cg(x/z)⊗∆g(z,Q2) (28) where Cq, g are coefficient functions and ∆q(z,Q2), ∆g(z,Q2) are called the evolved (with respect to Q2)quark and gluon distributions. They are found as solutions to DGLAP evolution equations d lnQ2 Pqq∆q + Pqg∆g d lnQ2 Pgq∆q + Pgg∆g where Pab are the splitting functions. The Mellin transforms γab of Pab are called the DGLAP anomalous dimensions. They are known in the leading order (LO) where they are ∼ αs and in the next-to-leading order (NLO), i.e. ∼ α2s. Similarly, Cq,g are known in LO and NLO. Details on this topic can be found in the literature (e.g. see a review [13]). Structure function g1 has the flavor singlet and non-singlet components, g 1 and g 1 . Expressions for g 1 are simpler, so we will use mostly them in the present paper when possible. It is convenient to write g1 in the form of the Mellin integral. In particular, gNS DGLAP1 (x,Q 2) = (e2q/2) CNS(ω)δq(ω) exp γNS(ω, αs(k where µ2 is the starting point of the Q2 -evolution; CNS and γNS are the non-singlet coefficient function and anomalous dimension. In LO γNS(ω,Q ω(1 + ω) + S2(ω) , (31) CLONS(ω) = 1 + 2ω + 1 ω(1 + ω) S1(ω) + S 1(ω)− S2(ω) FIG. 3: The IREE for the non-singlet component of the spin structure function g1. with Sr(ω) = j=1 1/j r . The initial quark and gluon densities in Eq. (30) are defined through fitting experimental data. For example, the fit for δq taken from the first paper in Ref. [12] is δq(x) = Nx−α (1− x)β(1 + γxδ) , (32) with N being the normalization, α = 0.576, β = 2.67, γ = 34.36 and δ = 0.75. DGLAP equations were suggested for describing DIS in the region x . 1, Q2 ≫ µ2 (33) (µ stands for a mass scale, µ ≫ ΛQCD) and there is absolutely no theoretical grounds to apply them in the small-x region, however being complemented with the standard fits they are commonly used at small x. It is known that SA provide a good agreement with available experimental data but the price is invoking a good deal of phenomenological parameters. The point is that DGLAP, summing up leading lnk Q2 to all orders in αs, cannot do the same with leading lnk(1/x). The later is not important in the region (33) where lnk(1/x) ≪ 1 but becomes a serious drawback of the method at small x. The total resummation of DL contributions to g1 in the region x ≪ 1, Q2 ≫ µ2 (34) was done in Refs. [14]. The weakest point in those papers was keeping αs as a parameter, i.e. fixed at an unknown scale. Accounting for the most important part of single-logarithmic contributions, including the running coupling effects were done in Refs. [15]. In these papers µ2 was treated as the starting point of the Q2 -evolution and as the IR cut-off at the same time. The structure function g1 was calculated with composing and solving IREE in the following It is convenient to compose IREE not for g1 but for forward (with |t| . µ2) Compton amplitude M related to g1 as follows: ℑM . (35) It is also convenient to use for amplitude M the asymptotic form of the Sommerfeld-Watson transform: ξ(−)(ω)F (ω,Q2/µ2) (36) where ξ(−)(ω) = [e−ıπω − 1]/2 ≈ −ıπω/2 is the signature factor. The transform of Eq. (36) and is often addressed as the Mellin transform but one should remember that it coincides with the Mellin transform only partly. IREE for Mellin amplitudes F (ω,Q2) look quite simple. For example, the IREE for the non-singlet Mellin amplitude FNS related to gNS1 by Eqs. (35,36) is depicted in Fig. 3. In the Mellin space it takes the simple form: [ω + ∂/∂y]FNS = (1 + ω/2)HNSF NS (37) where y = ln(Q2/µ2). Eq. (37) involves a new object (the lowest blob in the last term in Fig. 3): the non-singlet anomalous dimension HNS accounting for the total resummaton of leading logarithms of 1/x. Like in DGLAP, the anomalous dimension does not depend on Q2 but, in contrast to DGLAP, HNS can be found with the same method. The IREE for it is algebraic: ωHNS = A(ω)CF /8π 2 + (1 + ω/2)H2NS +D(ω)/8π 2 . (38) The system of Eqs. (37,38) can be easily solved but before doing it let us comment on them. The left-hand sides of Eqs. (37,38) are obtained with applying the operator −µ2∂/∂µ2 to Eq. (36). The Born contribution in Fig. 3 does not depend on µ and therefore vanishes. The last term in Fig. 3 (the rhs of Eq. (37)) is the result of a new, t -channel factorization which does not exist in the hard kinematics defined in Eq. (1). In order to compose the IREE for the Compton amplitude M , in accordance with the prescription in the previous section we should first introduce the cut-off µ. Then Step 2 is to tag the softest particles. In the case under discussion we do not have soft external particles. Had the softest particle been a gluon, it could be factorized in the same way like in Sect. II. However, the only option now is to attach the softest propagator to the external quark lines and get ln(t/µ2) = 0 from integration over β (cf Eq. (7)). So, the softest gluon does not yield DL contributions. The other option is to find a softest quark. The softest t -channel quark pair factorizes amplitude M into two amplitudes (the last term in Fig. 3) and yield DL contributions. The IREE for HNS is different: (i) HNS does not depend on Q 2, so there is not a derivative in the lhs of Eq. (37). (ii) The Born term depends on µ and contributes to the IREE (term A in Eq. (37))). (iii) As all external particles now are quarks, the softest virtual particle can be both a quark and gluon. The case when it is the t -channel quark pair, corresponds to the quadratic term in the rhs of Eq. (37). The case of the softest gluon yields the term D, with D(ω) = dηe−ωη ln (ρ+ η [ ρ+ η (ρ+ η)2 + π2 where b = (33− 2nf )/12π and η = ln(µ2/Λ2QCD). The term A in Eq. (37) stands instead of αs. The point is that the standard parametrization αs = αs(Q 2) cannot be used at x ≪ 1 and should be changed (see Ref. [16] for detail). It leads to the replacement αs by A(ω) = η2 + π2 dρe−ωρ (ρ+ η)2 + π2 . (40) Having solved Eqs. (37,38), we arrive at the following expression for gNS1 in the region (34): gNS1 (x,Q 2) = (e2q/2) (1/x)ωCNS(ω)δq(ω) exp HNS(ω)y where the coefficient function CNS(ω) is expressed through HNS(ω): CNS(ω) = ω −HNS(ω) and HNS(ω) is the solution of algebraic equation (43): HNS = (1/2) ω2 −B(ω) where B(ω) = (4πCF (1 + ω/2)A(ω) +D(ω))/(2π 2) . (44) It is shown in Ref. [17] that the expression for g1 in the region x ≪ 1, Q2 . µ2 (45) can be obtained from the expressions obtained in Refs. [15] for g1 in region (34) by the shift Q2 → Q2 + µ20 (46) where µ0 = 1 GeV for the non-singlet g1 and µ0 = 5.5 GeV for the singlet. IV. COMPARISON OF EXPRESSIONS (30) AND (41) FOR gNS1 Eqs. (30) and (41) read that the non-singlet g1 is obtained from δq with evolving it with respect to x (using the coefficient function) and with respect to Q2 (using the anomalous dimension). Numerical comparison of Eqs. (30) and (41) can be done when δq is specified. A. Comparison of small-x asymptotics, neglecting the impact of δq In the first place let us compare the small-x asymptotics of for gNS DGLAP1 and g 1 , assuming that δq does not affect them. In other words, we compare the differencee in the x-evolution at x → 0. Applying the saddle-point method to Eqs. (30) and (41) leads to the following expressions: gNS DGLAP1 ∼ exp ln(1/x) ln ln(Q2/Λ2QCD) gNS1 ∼ (1/x)∆NS(Q2/µ2)∆NS/2 (48) where ∆NS = 0.42 is the non-singlet intercept 1. Expression (47) is the well-known DGLAP asymptotics. Obviously, the asymptotics (48) is much steeper than the DGLAP asymptotics (30). B. Numerical comparison between Eqs. (30) and (41), neglecting the impact of δq A comparison between Eqs. (30) and (41) strongly depends on the choice of δq but also depends on the difference between the coefficient functions and anomalous dimensions. To clarify the latter we choose the simplest form of δq: δq(ω) = Nq . (49) It corresponds to the evolution from the bare quark where δq(x) = Nqδ(1 − µ2/s). Numerical results for R = [gNS1 − gNS DGLAP1 ]/gNS DGLAP1 with δq chosen by Eq. (49) manifest (see Ref. [19] for detail) that R increases when x is decreases. In particular, R > 0.3 at x . 0.05. It means that the total resummation of leading ln (1/x) cannot be neglected at x . 0.05 and DGLAP cannot be used beyond x ≈ 0.05. On the other hand, it is well–known that Standard Approach based on DGLAP works well at x ≪ 0.05. To solve this puzzle, we have to consider the standard fit for δq in more detail. C. Analysis of the standard fits for δq There are known different fits for δq. We consider the fit of Eq. (32). Obviously, in the ω -space Eq. (32) is a sum of pole contributions: δq(ω) = Nη (ω − α)−1 + mk(ω + λk) , (50) with λk > 0, so that the first term in Eq. (50) corresponds to the singular term x −α of Eq. (32) and therefore the small-x asymptotics of fDGLAP is given by the leading singularity ω = α = 0.57 of the integrand in Eq. (50) so that the asymptotics of gNS DGLAP1 (x,Q 2) is not given by the classic exponential of Eq. (47) but actually is the Regge-like: gNS DGLAP1 ∼ C(α)(1/x)α ln(Q2/Λ2)/ ln(µ2/Λ2) )γ(α)/b , (51) with b = (33 − 2nf)/12π. Comparison of Eq. (48) and Eq. (51) demonstrates that both DGLAP and our approach lead to the Regge behavior of g1, though the DGLAP prediction is more singular than ours. Then, they predict 1 The singlet intercept is much greater: ∆S = 0.86. different Q2 -behavior. However, it is important that our intercept ∆NS is obtained by the total resummation of the leading logarithmic contributions and without assuming singular fits for δq whereas the SA intercept α in Eq. (47) is generated by the phenomenological factor x−0.57 of Eq. (32) which makes the structure functions grow when x decreases and mimics in fact the total resummation2. In other words, the role of the higher-loop radiative corrections on the small-x behavior of the non-singlets is, actually, incorporated into SA phenomenologically, through the initial parton densities fits. It means that the singular factors can be dropped from such fits when the coefficient functions account for the total resummation of the leading logarithms and therefore fits for δq become regular in x in this case. They also can be simplified. Indeed, if x in the regular part N (1 − x)β(1 + γxδ) of the fit (32) is not large, all x -dependent terms can be neglected. So, instead of the rather complicated expression of Eq. (32), δq can be approximated by a constant or a linear form δq(x) = N(1 + ax) . (52) with 2 phenomenological parameters instead of 5 in Eq. (32). V. CORRECTING MISCONCEPTIONS The total resummation of lnk(1/x) allows to correct several misconceptions popular in the literature. We list and correct them below. Misconception 1: Impact of non-leading perturbative and non-perturbative contributions on the intercepts of g1 is large. Actually: Confronting our results and the estimates of the intercepts in Refs. [18] obtained from fitting available experimental data manifests that the total contribution of non-leading perturbaive and non-perturbative contributions to the intercepts is very small, so the main impact on the intercepts is brought by the leading logarithms. Misconception 2: Intercepts of g1 should depend on Q 2 through the parametrization of the QCD coupling αs = α(Q Actually: This is groundless from the theoretical point of view and appears only if the the parametrization of the QCD coupling αs = α(k ) is kept in all ladder rungs. It is shown in Ref. [16] that this parametrization cannot be used at small x and should be replaced by the parametrization of Eq. (40). Misconception 3: Initial densities δq(x) and δg(x) are singular but they are defined at x not too small. Later, being convoluted with the coefficient functions, they become less singular. Actually: It is absolutely wrong: Eq. (50) proves that the pole singularity x−α in the fits does not become weaker with the x-evolution. Misconception 4: Fits for the initial parton densities are complicated because they mimic unknown non- perturbative contributions. Actually: Our results demonstrate that the singular factors in the fits mimic the total resummation of lnk(1/x) and can be dropped when the resummation is accounted for. In the regular part of the fits the x -dependence is essential for large x only, so impact of non-perturbative contributions is weak at the small-x region. Misconception 5: Total resummations of lnk(1/x) may become of some importance at extremely small x but not for x available presently and in a forthcoming future. Actually: The efficiency of SA in the available small-x range is based on exploiting the singular factors in the standard fits to mimic the resummations. So, the resummations have always been used in SA at small x in an inexplicit way, through the fits, but without being aware of it. 2 We remind that our estimates for the intercepts ∆NS ,∆S were confirmed (see Refs. [18]) by analysis of the experimental data VI. COMBINING THE TOTAL RESUMMATION AND DGLAP The total resummaton of leading logarithms of x considered in Sect. IV is essential at small-x. When x ∼ 1, all terms ∼ lnk(1/x) in the coefficient functions and anomalous dimensions cannot have a big impact compared to other terms. DGLAP accounts for those terms. It makes DGLAP be more precise at large x than our approach. So, there appears an obvious appeal to combine the DGLAP coefficient functions and anomalous dimensions with our expressions in order to obtain an approach equally good in the whole range of x : 0 < x < 1. The prescription for such combining was suggested in Ref. [19]. Let us, for the sake of simplicity, consider here combining the total resummation and LO DGLAP. The generalization to NLO DGLAP can be done quite similarly. The prescription consists of the following points: Step A: Take Eqs. (31) and replace αs by A of Eq. (40), converting γNS into γ̃NS and C NS into C̃ Step B: Sum up the obtained expressions and Eqs. (42,43): c̃NS = C̃ NS +HS , h̃NS = γ̃NS +HNS . (53) New expressions c̃NS , h̃NS combine the total resummation and DGLAP but they obviously contain the double count- ing: some of the first–loop contributions are present both in Eqs. (31) and in Eqs. (42,43). To avoid the double counting, let us expend Eqs. (42,43) into series and retain in the series only the first loop contributions3: A(ωCF ) NS = 1 + A(ωCF ) . (54) Finally, there is Step C: Subtract the first-loop expressions (54) from Eq. (53)) to get the combined, or ”synthetic” as we called them in Ref. [19], coefficient function cNS and anomalous dimension hNS : cNS = c̃NS − C(1)NS , hNS = h̃NS −H NS . (55) Substituting Eqs. (55) in Eq. (41) leads to the expression for gNS1 equally good at large and small x. This description does not require singular factors in the fits for the initial parton densities. An alternative approach for combining DLA expression for g1 was suggested in Ref. [20]. However, the parametrization of αs in this approach was simply borrowed from DGLAP, which makes this approach be unreliable at small x. VII. CONCLUSION We have briefly considered the essence of the IREE method together with examples of its application to different processes. They demonstrate that IREE is indeed the efficient and reliable instrument for all-orders calculations in QED, QCD and the Standard Model of EW interactions. As an example in favor of this point, let us just remind that there exist wrong expressions for the singlet g1 in DLA obtained with an alternative technique and the exponentiation of EW double logarithms obtained in Ref. [10] had previously been denied in several papers where other methods of all-order summations were used. VIII. ACKNOWLEDGEMENT B.I. Ermolaev is grateful to the Organizing Committee of the Epiphany Conference for financial support of his participation in the conference. [1] V.V. Sudakov. Sov. Phys. JETP 3(1956)65. [2] V.N. Gorshkov, V.N. Gribov, G.V. Frolov, L.N. Lipatov. Yad.Fiz.6(1967)129; Yad.Fiz.6(1967)361. [3] V.N. Gribov. Yad. Fiz. 5(1967)399. 3 For combining the total resummation with NLO DGLAP one more term in the series should be retained [4] B.I. Ermolaev, L.N. Lipatov, V.S. Fadin. Yad. Fiz. 45(1987)817; B.I. Ermolaev. Yad. Fiz. 49(1989)546; M. Chaichian and B. Ermolav. Nucl. Phys. B 451(1995)194. [5] R. Kirschner and L.N. Lipatov. ZhETP 83(1982)488; Nucl. Phys. B 213(1983)122. [6] B.I. Ermolaev and L.N. Lipatov. Yad. Fiz. 47(1988)841; Yad. Fiz. 48(1988)1125; Int. j. Mod. Phys. A 4(1989)3147. [7] B.I. Ermolaev and M. Krawczyk. Proc of Kazimerz Conf on physics of elementary interactions. 1990. [8] B.I. Ermolaev and S.I. Troyan. Nucl. Phys. B 590(2000)521. [9] B.I. Ermolaev and V.S. Fadin. JETP Lett. 33(1981)269. [10] V.S. Fadin, L.N. Lipatov, A. Martin, M. Melles. Phys. Rev. D 61(2000)094002. [11] G. Altarelli and G. Parisi, Nucl. Phys.B126 (1977) 297; V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438; L.N.Lipatov, Sov. J. Nucl. Phys. 20 (1972) 95; Yu.L. Dokshitzer, Sov. Phys. JETP 46 (1977) 641. [12] G. Altarelli, R.D. Ball, S. Forte and G. Ridolfi. Nucl. Phys. B496 (1997) 337; Acta Phys. Polon. B29(1998)1145; E. Leader, A.V. Sidorov and D.B. Stamenov. Phys. Rev. D73 (2006) 034023; J. Blumlein, H. Botcher. Nucl. Phys. B636 (2002) 225; M. Hirai at al. Phys. Rev. D69 (2004) 054021. [13] W.L. Van Neerven. hep-ph/9609243. [14] B.I. Ermolaev, S.I. Manaenkov and M.G. Ryskin. Z. Pyss. C 69(1996)259; J. Bartels, B.I. Ermolaev and M.G. Ryskin. Z. Pyss. C 70(1996)273; Z. Pyss. C 72(1996)627. [15] B.I. Ermolaev, M. Greco, S.I. Troyan. Nucl. Phys.B 571 (2000) 137; Nucl. Phys.B 594 (2001) 71; Phys.Lett.B 579 (2004) [16] B.I. Ermolaev, M. Greco and S.I. Troyan. Phys.Lett.B 522(2001)57. [17] B.I. Ermolaev, M. Greco and S.I. Troyan. hep-ph/0605133. [18] J. Soffer and O.V. Teryaev. Phys. Rev.56( 1997)1549; A.L. Kataev, G. Parente, A.V. Sidorov. Phys.Part.Nucl 34(2003)20; Nucl.Phys.A666(2000)184; A.V. Kotikov, A.V. Lipatov, G. Parente, N.P. Zotov. Eur.Phys.J.C26(2002)51; V.G. Krivohijine, A.V. Kotikov, hep-ph/0108224; A.V. Kotikov, D.V. Peshekhonov hep-ph/0110229. [19] B.I. Ermolaev, M. Greco and S.I. Troyan. Phys.Lett.B B.I. Ermolaev, M. Greco and S.I. Troyan. Phys.Lett.B 622(2005)93. [20] B. Badalek, J. Kwiecinski. Phys. Lett. B 418(1998)229; J. Kwiecinski, B. Ziaja. hep-ph/9802386. http://arxiv.org/abs/hep-ph/9609243 http://arxiv.org/abs/hep-ph/0605133 http://arxiv.org/abs/hep-ph/0108224 http://arxiv.org/abs/hep-ph/0110229 http://arxiv.org/abs/hep-ph/9802386 Introduction IREE for scattering amplitudes in the hard kinematics IREE for the form factor f(q2) in QED IREE for the form factor g(q2) in QED e+e- -annihilation into a quark-antiquark pair e+e- -annihilation into a quark-antiquark pair and gluons Exponentiation of Sudakov electroweak double-logarithmic contributions Application of IREE to the polarized Deep-Inelastic Scattering Comparison of expressions (30) and (41) for g1NS Comparison of small-x asymptotics, neglecting the impact of q Numerical comparison between Eqs. (30) and (41), neglecting the impact of q Analysis of the standard fits for q Correcting misconceptions Combining the total resummation and DGLAP Conclusion Acknowledgement References

0704.0342

Cofibrations in the Category of Frolicher Spaces. Part I

Cofibrations in the Category of Frölicher Spaces: Part I Brett Dugmore Cadiz Financial Strategists (Pty) Ltd, Cape Town, South Africa Email: [email protected] Patrice Pungu Ntumba Department of Mathematics and Applied Mathematics University of Pretoria Hatfield 0002, Republic of South Africa Email: [email protected] Abstract Cofibrations are defined in the category of Frölicher spaces by weak- ening the analog of the classical definition to enable smooth homotopy extensions to be more easily constructed, using flattened unit intervals. We later relate smooth cofibrations to smooth neighborhood deforma- tion retracts. The notion of smooth neighborhood deformation retract gives rise to an analogous result that a closed Frölicher subspace A of the Frölicher space X is a smooth neighborhood deformation retract of X if and only if the inclusion i : A →֒ X comes from a certain subclass of cofibrations. As an application we construct the right Puppe sequence. Subject Classification (2000): 55P05. Key Words:Frölicher spaces, Flattened unit intervals, Smooth neighborhood de- formation retracts, Smooth cofibrations, Cofibrations with FCIP, Puppe se- quence. 1 Preliminaries The purpose of this section is to survey brielfy the notion of Frölicher spaces. Frölicher spaces arise naturally in physics, and do generalize the concept of smooth manifolds. A Frölicher space, or smooth space as initially called by Frölicher and Kriegl [7], is a triple (X, CX ,FX) consisting of a setX , and subsets CX ⊆ XR, FX ⊆ RX such that • FX ◦ CX = {f ◦ c| f ∈ FX , c ∈ CX} ⊆ C∞(R) • ΦCX := {f : X → R| f ◦ c ∈ C∞(R) for all c ∈ CX} = FX http://arxiv.org/abs/0704.0342v1 • ΓFX := {c : R → X | f ◦ c ∈ C∞(R)for all f ∈ FX} = CX Frölicher and Kriegl [7], and Kriegl and Michor [10] are our main reference for Frölicher spaces. The following terminology will be used in the paper: Given a Frölicher space (X, CX ,FX), the pair (CX ,FX) is called a smooth structure; the elements of CX and FX are called smooth curves and smooth functions respec- tively. The topology assumed for a Frölicher space (X, CX ,FX) throughout the paper is the initial topology TF induced by the set FX of functions. When there is no fear of confusion, a Frölicher space (X, CX ,FX) will simply be denoted X . The most natural Frölicher spaces are the finite dimensional smooth manifolds, where if X is such a smooth manifold, then CX and FX consist of all smooth curves R → X and smooth functions X → R. Euclidean finite dimensional smooth manifolds Rn, when viewed as Frölicher spaces, are called Euclidean Frölicher spaces. In the sequel, by Rn, n ∈ N, we mean the Frölicher space Rn, equipped with its usual smooth manifold structure. A Frölicher space X is called Hausdorff if and only if the smooth real-valued functions on X are point-separating, i.e. if and only if TF is Hausdorff. A Frölicher structure (CX ,FX) on a set X is said to be generated by a set F0 ⊆ RX (resp. C0 ⊆ XR) if CX = ΓF0 and FX = ΦΓF0 (resp. FX = ΦC0 and CX = ΓΦC0 ). Note that different sets F0 ⊆ RX on the same set X may give rise to a same smooth structure on X . A set mapping ϕ : X → Y between Frölicher spaces is called a map of Frölicher spaces or just a smooth map if for each f ∈ FY , the pull back f ◦ϕ ∈ FX . This is equivalent to saying that for each c ∈ CX , ϕ ◦ c ∈ CY . For Frölicher spaces X and Y , C∞(X,Y ) will denote the collection of all the smooth maps X → Y . The resulting category of Frölicher spaces and smooth maps is denoted by FRL. Some useful facts regarding Frölicher spaces can be gathered in the following Theorem 1.1 The category FRL is complete (i.e. arbitrary limits exist ), co- complete (i.e. arbitrary colimits exist), and Cartesian closed. Given a collection of Frölicher spaces {Xi}i∈I , let X = i∈I Xi be the set product of the sets {Xi}i∈I and πi : X → Xi, i ∈ I, denote the projection map (xi)i∈I 7→ xi. The initial structure on X is generated by the set {f ◦ πi : f ∈ FXi}. The ensuing Frölicher space (X,ΓF0, ϕΓF0) is called the product space of the family {Xi}i∈I . Clearly, ΓF0 = {c : R → X | if c(t) = (ci(t))i∈I , then ci ∈ CXi for every i ∈ I}. Now, let i∈I Xi be the disjoint union of sets {Xi}i∈I , and ιXi : Xi → i∈I Xi the inclusion map. Place the smooth final structure on i∈I Xi corresponding to the family {ιXi}i∈I . The resulting Frölicher space is called the coproduct of {Xi}i∈I , and denoted i∈I Xi, and Xi = {f : Xi → R| for each i ∈ I, f |Xi ∈ FXi} is the collection of smooth functions for the coproduct. Corollary 1.1 Let X, Y , and Z be Frölicher spaces. Then the following canon- ical mappings are smooth. • ev: C∞(X,Y )×X → Y , (f, x) 7→ f(x) • ins:X → C∞(Y,X × Y ), x 7→ (y 7→ ins(x)(y) = (x, y)) • comp:C∞(Y, Z)× C∞(X,Y ) → C∞(X,Z), (g, f) 7→ g ◦ f • f∗ : C∞(X,Y ) → C∞(X,Z), f∗(g) = f ◦ g, where f ∈ C∞(Y, Z) • g∗ : C∞(Z, Y ) → C∞(X,Y ), g∗(f) = f ◦ g, where g ∈ C∞(X,Z). Given Frölicher spaces X , Y , and Z; in view of the cartesian closedness of the category FRL, the exponential law C∞(X × Y, Z) ∼= C∞(X,C∞(Y, Z)) holds. Because FX = C∞(X,R), it follows by cartesian closedness of FRL that the collection FX can be made into a Frölicher space on its own right. Finally we would like to show how to construct smooth braking functions, following Hirsch [8]. Smooth braking functions are tools that are behind most results in this paper. In [11], it is shown that the function ϕ : R → R given by ϕ(u) = 0 if u ≤ 0 u if u > 0 is smooth. Substituting x2 for u in the above function, one sees that the function ψ : R → R, given by ψ(x) = 0 if x ≤ 0 x2 if u > 0 is smooth. Now, let us construct a smooth function α : R → R with the following properties. Let 0 ≤ a < b. α(t) should satisfy: • α(t) = 0 for t ≤ a, • 0 < α(t) < 1 for a < t < b, • α is strictly increasing for a < t < b, • α(t) = 1 for t ≥ b. Define α : R → [0, 1] by α(t) = γ(x)dx γ(x)dx where γ(x) = ψ(x− a)ψ(b − x). In the sequel, the notation αǫ, 0 < ǫ < , will refer to a smooth braking function with the following properties • αǫ(t) = 0 for t ≤ ǫ, • 0 < αǫ(t) < 1 for ǫ < t < 1− ǫ, • α strictly increasing for ǫ < t < 1− ǫ, • αǫ(t) = 1 for 1− ǫ ≤ t. 2 Basic Constructions of Homotopy Theory in In this section, we define the fundamental notions of homotopy theory in the category FRL, such as the homotopy relation and the mapping cylinder. We begin with an overview of our approach to homotopy in FRL, and then discuss alternate Frölicher structures on the unit interval which are used in this and subsequent sections. 2.1 Our Approach to Homotopy Theory in FRL One might begin investigating homotopy theory in FRL by simply following the homotopy theory of topological spaces, replacing continuous functions with smooth ones. One can certainly define the notion of a homotopy H : I×X → Y between smooth maps H(0,−) and H(1,−) in this way (which we do). One can even get as far as the left Puppe sequence (see [4]), but eventually difficulties begin to arise. Extending functions defined on a subspace of a Frölicher space tends to be a little tricky, and so the definition of a cofibration in FRL is one that needs careful consideration. We envisage to construct the right Puppe sequence in a future paper. To do this we define a slightly weaker notion of cofibration than the notion obtained from topological spaces. In addition, we define the mapping cylinder of a smooth map f : X → Y using not the unit interval, but a modified version called the weakly flattened unit interval, denoted I, which, as one can show, is topologically homeomorphic to the unit interval. This modified structure on the unit interval allows us to show that the inclusion of a space X into the mapping cylinder of f : X → Y is a cofibration (in our weaker sense ). The weakly flattened unit interval is useful, but it also has its drawbacks. It would be ideal to have a single structure on the unit interval that can be used throughout out homotopy theory, but the weakly flattened unit interval is not suitable, because it has the rather restrictive property that a smooth map f : I → I on the usual unit interval often does not define a smooth map f : I → I unless the endpoints of the interval are mapped to the endpoints. This restrictive property means that we only use the flattened unit intervals where they are absolutely necessary. In our future work, we will investigate whether with our modified notions of cofibration and mapping cylinder, Baues’ cofibration axioms are satisfied. 2.2 Flattened Structures on the Unit Interval We define two main Frölicher structures which we call the flattened unit in- terval and the weakly flattened unit interval . Let (CI ,FI) be the subspace structure induced on I by the inclusion I →֒ R. Definition 2.1 The Frölicher space (I, CI,FI), where the structure (CI,FI) is the structure generated by the set F = {f ∈ FI| there exists 0 < ǫ < 14 with f(t) = f(0) for t ∈ [0, ǫ) and f(t) = f(1) for t ∈ (1− ǫ, 1]}, is called the flattened unit interval. It is easy to see that any continuous map c : R → [0, 1] defines a structure curve on I if and only if it is smooth at every point t ∈ R, where c(t) ∈ (0, 1), . We define the left (resp. right) flattened unit interval, denoted by I− (resp. I+), to be the Frölicher space whose underlying set is the unit interval [0, 1], and structure is the structure generated by the structure functions in FI that are constant near 0 (resp. 1). Definition 2.2 The Frölicher space (I, CI,FI), with the structure defined below is called the weakly flattened unit interval. The underlying set is the unit interval; the structure (CI,FI) is generated by the family F = {f ∈ FI | lim f(t) = 0, lim f(t) = 0, n ≥ 1}. We call the property, for all f ∈ F , f(t) = 0, lim f(t) = 0, n ≥ 1, the zero derivative property of f . We shall prove that all structure functions on I have the zero derivative property, in other words, FI = F . To that effect, we need the following lemma. Lemma 2.1 Let c : R → R be a smooth real-valued function at t = t0, and let f : R → R be a smooth real-valued function at t = c(t0). Then, (f ◦ c)(t0) = f (n)(c(t0))(c′(t0))n + terms of the form af (k)(c(t0))(c ′(t0)) m1(c′′(t0)) m2 . . . (c(n−1)(t0)) mn−1 , where k < n and a ∈ R. In addition, if a 6= 0 then at least one ofm2,m3, . . . ,mn−1 is also non-zero. Proof. The proof is done by induction. For the sake of brevity, we call the term f (n)(c(t0))(c ′(t0)) n the primary term for n, and the terms of the form af (k)(c(t0))(c ′(t0)) m1(c′′(t0)) m2 . . . (c(n−1)(t0)) mn−1 the lower order terms for n. The statement is true for n = 1 and for n = 2. Suppose the result is true for n = k. To show that the result holds for n = k + 1, since dtk+1 (f ◦ c)(t0) = (f (k)(c(t0))(c ′(t0)) +terms of the form d (af (j)(c(t0))(c ′(t0)) m1(c′′(t0)) m2 . . . (c(k−1)(t0)) mk−1), where j < k + 1 and a ∈ R, we need only show that (af (j)(c(t0))(c ′(t0)) m1(c′′(t0)) m2 . . . (c(k−1)(t0)) mk−1) gives rise to lower terms for n = k + 1, which is by the way straightforward. � Theorem 2.1 FI = {f ∈ FI | limt→0+ d f(t) = 0 = limt→1− f(t)} =: F Proof. That F ⊆ FI is evident. We must show the reverse inequality. Let 0 < ǫ < 1 , and 0 < M < 1. Consider the function cM : R → R, given by cM (t) = (1− αǫ(|t|))βM (t) + αǫ(|t|), where αǫ : R → R is a smooth braking function as defined in the Preliminaries, and βM : R → R is given by βM (t) = −Mt if t ≤ 0 t if t > 0 It is easily seen that cM is continuous over all R, and smooth over all R except at t = 0. Also note that 0 < cM (t) < 1 for all t ∈ R, and cM (t) = βM (t) = 0 for all 0 ≤ t < ǫ. Now, cM (t) = βM (t) = −M, for −ǫ < t < 0 cM (t) = βM (t) = 1, for 0 < t < ǫ For n > 1, we have cM (t) = βM (t) = 0, for t ∈ (−ǫ, 0) ∪ (0, ǫ). We now show that for cM ∈ ΓF . To this end, let f ∈ F . To show that f ◦ cM : R → R is smooth, it is obvious that we need only concentrate on the point t = 0, because f ◦ c is smooth at every t 6= 0. It follows for t 6= 0, and n ∈ N that Lemma 2.1 applies. But as t → 0, cM (t) → 0+, and so, letting s = cM (t), we have f (j)(cM (t)) = lim f (j)(s) = 0, for all j ∈ N, by the zero derivative property of f . Thus, as t approaches the value 0, the primary term and all the lower order terms of d (f ◦ cM )(t) vanish, and we have shown that f ◦ cM is smooth at t = 0. This implies that f ◦ cM ∈ C∞(R,R) for all f ∈ F . It follows that cM ∈ ΓF . We are now ready to show that FI ⊆ F . To this end, suppose that we are given a structure function f ∈ FI. We shall show that this f has the zero derivative property, and is thus an element of F . Since f ∈ FI, we know that f ◦ c is a smooth real-valued function for every c ∈ ΓF . In particular, f ◦ cM is smooth for all 0 < M < 1. Thus, for any n ∈ N, (f ◦ cM )(t) = lim (f ◦ cM )(t). As t→ 0−, cM (t) → 0+; let us consider the lower order terms for n. Each term of the form af (k)(cM (t))(c M (t)) m1(c′′M (t)) m2 . . . (c (n−1) M (t)) has some term (c (t))mi , for some i > 1, with mi 6= 0. But limt→0− c (t) = 0, if i > 1, and so af (k)(cM (t))(c M (t)) m1(c′′M (t)) m2 . . . (c (n−1) (t))mn−1 = 0. So all the lower order terms fall away, therefore limt→0− (f ◦ cM )(t) = limt→0− f (n)(cM (t))(c′M (t))n = limt→0− f (n)(cM (t))(−M)n = lims→0+ f (n)(s)(−M)n, where s = cM (t). In a similar way one shows that (f ◦ cM )(t) = lim f (n)(s). But f◦cM is smooth, therefore lims→0+ f (n)(s)(−M)n = lims→0+ f (n)(s), which implies that lims→0+ f (n)(s) = 0. We have shown that the zero derivative property of f holds for the left endpoint of the unit interval. To show that the zero derivative property of f holds for the right endpoint of f , note that dM : R → R, dM (t) = 1− cM (t), is a smooth real-valued function with d(0) = 1, and 0 ≤ dM (t) ≤ 1 for all t ∈ R. One can follow a similar procedure to the above, using dM instead of cM to show that lims→1− f (n) = 0. � 2.3 Some Properties of Smooth Functions between the Flattened Unit Intervals One has to be careful when dealing with the various flattened unit intervals. A smooth function f : I → I from the R- Frölicher subspace unit interval I to itself need not define a smooth function f : I → I, for example. Conversely, not every smooth function f : I → I defines a smooth function f : I → I. In particular, we need to be aware of the fact that addition and multiplication of functions when defined between the various flattened unit intervals does not preserve smoothness, as is the case with the usual unit interval. Example 2.1 The function f : I → I, f(t) = 1 t is clearly smooth, but the corresponding function f : I → I, given by the same formula, is not smooth. To see this, let α : R → R be a smooth braking function with the properties that • α(t) = −1, for t < − 3 • α(t) = t, for − 1 < t < 1 • α(t) = 1, for t > 3 Define c : R → I by c(t) = 1− |α(t)|. The curve c is smooth everywhere except at t = 0, where c(0) = 1. However, every generating function f on I is constant near 1, and so the composite f ◦ c is smooth. Thus c is a structure curve on I. Now, f ◦ c : R → I is given by (f ◦ c)(t) = 1 (1 − |α(t)|). Let h : I → R be a structure function with the properties that • h(s) = 0, for s < 1 • h(s) = s, for 1 < s < 3 • h(s) = 1, for 7 Then (h ◦ f ◦ c)(t) = 1 (1− |α(t)|) for t near 0, and is not smooth at t = 0. Thus f does not define a smooth function from I to I. Example 2.2 The function f : I → I, f(t) = t, is smooth, but the corresponding f : I → I, given by the same formula, is not smooth. This follows from the fact that f is smooth on the open interval (0, 1), and a generating function g on I is constant near 0 and 1. On the side, f : I → I is not smooth, because if c : R → I is a structure curve with c(t) = t2 near t = 0, then (f ◦ c)(t) = |t| near t = 0, which is not smooth on I at t = 0. Example 2.3 The functions f, g : I− → I−, given by f(t) = 1 t and g(t) = 1 are both smooth, but the sum f(t) + g(t) = 1 is not smooth. The following lemma follows from the definition of the Frölicher structures on the various flattened unit intervals. Lemma 2.2 Let f : I → I be a smooth function with the properties that f(0) = 0 and f(1) = 1. Then the following maps are smooth: • f : I → I±, • f : I → I, • f : I± → I, • f : I → I, • f : I → I. The function defined in the following example is for later reference. Example 2.4 Let H : I × I− → I− be given by H(t, s) = (1 − α(t))s, where α : R → R is a smooth braking function with the properties that • α(t) = 0 for t < 1 • 0 ≤ α(t) ≤ 1 for all t ∈ R, • α(t) = 1 for t > 3 We show that H is smooth. To see this, let f : I− → R be a generating function on I−. So f is constant near 0. Now, let c : R → I × I− be a structure curve, given by c(v) = (t(v), s(v)). The curve t is a structure curve on I, and so is a smooth real-valued function for all v ∈ R, except possibly when t(v) = 0 or t(v) = 1. Similarly, the curve s is a structure curve on I−, and so is smooth for all v ∈ R except possibly when s(v) = 0. Now consider the composite H ◦ c : R → I−. Clearly, α(t(v)) is smooth for all v, since the only possible points for non-smoothness occur when t(v) = 0 or t(v) = 1, and α(t(v)) is locally constant near these points. Consequently, H ◦ c is smooth everywhere except possibly when s(v) = 0. Now, let’s consider f ◦H ◦ c : R → R; the only possible points for non-smoothness are those in which s is 0, i.e. H◦ = 0. But f is a structure generating function on I−, and so is locally constant near 0. This shows that f ◦H ◦ c is smooth for all v ∈ R, and thus H is smooth. 2.4 Homotopy in FRL and Related Objects Definition 2.3 (1) Let X be a Frölicher space, and x0, x1 ∈ X. We say that x0 is smoothly path-connected to x1 if there is a smooth path c : I → X such that c(0) = x0 and c(1) = x1. We write x0 ≃ x1. The relation ≃ is called smooth homotopy when it is applied to hom-sets. (2) Let f : X → Y be a map of Frölicher spaces. f is called a smooth homotopy equivalence provided there exists a smooth map g : Y → X such that f ◦ g ≃ 1Y and g ◦ f ≃ 1X . One can show that smooth homotopy is a congruence in RFL. In practice, we say that smooth maps f, g : X → Y are smoothly homotopic if there exists a smooth map H : I ×X → Y with H(0,−) = f and H(1,−) = g. If A ⊆ X is subspace of X , then we say that H is a smooth homotopy (rel A) if the map H has the additional property that H(t, a) = a for each t ∈ I and a ∈ A. See Cherenack [5] and Dugmore [6] for more detail regarding smooth homotopy. The notion of deformation retract is fundamental to topological homotopy theory. The following definitions are adapted for smooth homotopy, and will be needed at a later stage. Definition 2.4 Let A ⊆ X be a subspace of a Frölicher space X, and let i : A →֒ X denote the inclusion map. Then • We say that A is a retract of X if there exists a smooth map r : X → A such that ri = 1A. We call r a retraction. • We call A a weak deformation retract of X if the inclusion i is a smooth homotopy equivalence. • The subspace A is called a deformation retract of X if there exists a re- traction r : X → A such that ir ≃ 1X . • The subspace A is called a strong deformation retract of X if there exists a retraction r : X → A such that ir ≃ 1X(relA). Definition 2.5 The mapping cylinder If of f : X → Y is defined by the fol- lowing pushout I ×X // If where i1 : X → I ×X is given by i1(x) = (1, x), for any x ∈ X. We denote the elements of If by [t, x] or [y], where (t, x) ∈ I ×X and y ∈ Y . Replacing I ×X in the above pushout diagram by I×X or I×X, we obtain the flattened mapping cylinder If and weakly flattened mapping cylinder If of f respectively. We use the same notation for elements of these flattened mapping cylinders as described above for the mapping cylinder. There is also a map i0 : X → I ×X , defined by i0(x) = (0, x) for x ∈ X . This induces an inclusion map i′0 : X → If , which identifies X with the Frölicher subspace i′0(X) of If . An inclusion is induced in a similar way for the flattened mapping cylinders. If one identifies {0}×X to a point in the mapping cylinder If of a map f : X → Y , then one obtains the mapping cone Tf of the map f . In a similar fashion, we define the flattened mapping cone Tf and weakly flattened mapping cone Tf of a smooth map f : X → Y . 2.5 Cofibrations in FRL A cofibration is a map i : A→ X for which the problem of extending functions from i(A) to X is a homotopy problem. In other words, if a map f : i(A) → Z can be extended to a map f∗ : X → Z, then so can any map homotopic to f . For topological spaces, the usual definition is phrased in a slightly more restrictive way. The extension of a map g ≃H f , for some homotopy H : I × i(A) → Z, is required to exist at every level of the homotopy simultaneously. In other words, one requires each H(t,−) to be extendable in such a way that the resulting homotopy H∗ : I ×X → Z is continuous. We weaken this definition somewhat, to enable smooth homotopy extensions to be more easily constructed using a flattening at the endpoints of the homo- topy. This enables us to characterize smooth cofibrations in terms of a flattened unit interval, and then later to relate smooth cofibrations to smooth neigh- borhood deformation retracts. Our definition of smooth cofibration, though different from from Cap’s definition, see [1], leads to several classical results as does Cap’s. As pointed out by Cap, the analogue of the classical definition of cofibration would not allow even {0} →֒ I to be a smooth cofibration. So, we have the following Definition 2.6 A smooth map i : A → X is called a smooth cofibration if, corresponding to to every commutative diagram of the form (0,1A) f // Z 66mmmmmmmmmmmmmm there exists a commutative diagram in FRL of the form (0,1X ) ::tttttttttt where G′ : I × A → Z is given by G′(t, a) = G(αǫ(t), a) for some 0 < ǫ < 12 , and each t ∈ I, a ∈ A. The problem of extending a map smoothly from a subspace of a Frölicher space to the whole space is a more difficult problem than simply extending con- tinuously. It is mainly for this reason that the definition of smooth cofibration differs somewhat from the corresponding definition of a topological cofibration. Lemma 2.3 Let i : A → X be a smooth cofibration, then i is an initial mor- phism in FRL. In addition, if A is Hausdorff, then i is injective. So in this case A can be regarded as a subspace of X. Proof. Let us show that every smooth map f : A→ R factors through i, that is for every f ∈ FA, there exists f̃ ∈ FX such that f = f̃ ◦ i. To this end, consider the smooth map G : I × A → R, given by H(t, a) = tf(a). Clearly, 0|A = G(0,−), where 0 : X → R is the constant map 0. It follows that there is map F : I ×X → R such that F ◦ (1× i) = G′. Then, clearly f̃ := F (1,−) has the desired property. The remaining part of the proof of Proposition 3.3, in [1], holds verbatim here as well. � In this paper, we are interested only in cofibrations that are injective. Hence- forth, all cofibrations are assumed to be injective. All topological cofibrations are inclusions, and this result is true for smooth cofibrations too. The proof of the following lemma is essentially the same as the proof given by James [9] for the topological result, although James’s proof is in some sense dual to ours, using path-spaces in place of cartesian products and the adjoint versions of our homotopies. Lemma 2.4 A cofibration i // X is a smooth inclusion. Proof. Let Ii be a mapping cylinder of i, and let j : X → Ii be the standard inclusion map. Consider the smooth map γ : I → I, γ(t) = 1 − t, for all t ∈ I, and the quotient map q : (I ×A) ⊔X → Ii; we have the following commutative diagram (0,1A) j // Ii 66mmmmmmmmmmmmmm where G(t, a) = [(1 − t, a)]. Notice that the map G is smooth. Since i is a cofibration, we have the commutative diagram (0,1X ) ::uuuuuuuuuu where G′(t, a) = G(αǫ(t), a) for some 0 < ǫ < . Define U : X → Ii by U(x) = F (1, x). We have U ◦ i = G′(1,−), where G′(1, a) = [(0, a)], for every a ∈ A. Thus the assignment a 7→ G′(1, a) defines the usual inclusion of A into the mapping cylinder. From this we deduce that U ◦ i is an inclusion, and hence i is an inclusion. � There is an equivalent formulation of definition 2.6, given in the following lemma. Lemma 2.5 A smooth map i // X is a cofibration if and only if, for every smooth map h : (0×X)∪(I−×i(A)) → Z, the following diagram (0×X) ∪ (I− × i(A)) h // I− ×X 77oooooooooooooo where j is the evident inclusion, exists in FRL. Proof. Suppose that the inclusion A // i // X is a smooth cofibration, and suppose that h : (0 × X) ∪ (I− × i(A)) → Z is a smooth map. We have the diagram (0×B) ∪ (I− × i(A)) h // I− ×X We need to fill in a smooth map G : I− × X → Z which makes the resulting diagram commute. To do this, notice that h|I−×i(A) is smooth, and thus the corresponding map h|I × i(A), using the usual unit interval, is also smooth. We have the following diagram (0,1A) h|0×X // Z 66mmmmmmmmmmmmmm where h|0×X(0,−) : X → Z is denoted as h|0×X . The fact that i is a smooth cofibration yields the following FRL-commutative diagram: h0×X // (0,1A) ::tttttttttt where (h|I−×A)′(t, a) = h|I−×A(αǫ(t), a), for some 0 < ǫ < 12 . Now, chose a smooth braking function β : R → R with the following properties. • α(t) = 0 for t < ǫ • α(t) = t for ǫ < t. F may not be smooth on I− × A due to the flattening requirements of the left flattened unit interval. To correct this, set G(t, a) = F (β(t), a). Notice that the insertion of this braking function does not affect the commutativity conditions of G, since the only adjustments to F occur in the first coordinate where the map (h|I−×X)′ is constant. Now, assume the converse, i.e. to every smooth map h : (0 × X) ∪ (I− × i(A)) → Z, corresponds a commutative diagram (0×X) ∪ (I− × i(A)) h // I− ×X 77oooooooooooooo We wish to show that the inclusion i : A → X is a cofibration; so assume we have the following diagram (0,1A) f // Z 66mmmmmmmmmmmmmm There exists the diagram (0,1A) f // Z 66mmmmmmmmmmmmmm where G′(t, a) = G(αǫ(t), a). Our hypothesis allows us to construct the diagram (0×X) ∪ (I− × i(A)) f∪G′ // I− ×X 77oooooooooooooo Note that f ∪ G′ is smooth since αǫ(t) is constant near 0. Since H is smooth on I− ×X it defines a smooth map on I ×X . One can verify that the diagram (0,1X) ::tttttttttt commutes as required. � 3 Smooth Neighborhood Deformation Retracts This section is concerned with the formulation of a suitable notion of smooth neighborhood deformation retract. For topological spaces, the statement that a closed subspace A of X is a neighborhood deformation retract of X is equivalent to the statement that the inclusion i : A →֒ X is a closed cofibration. We show that in the category of Frölicher spaces there is a notion of smooth neighborhood deformation retract that gives rise to an analogous result that a closed Frölicher subspace A of the Frölicher space X is a smooth neighborhood deformation retract of X if and only if the inclusion i : A →֒ X comes from a certain subclass of cofibrations. As an application, we construct the right Puppe sequence. 3.1 SNDR pairs and SDR pairs The definition of ‘smooth neighborhood deformation retract’ that we adopt in this paper is similar to the definition of ‘R-SNDR pair’suggested in [6], but we have modified the definition in order to retain only the essential aspects of ‘first coordinate independence’ defined in [6]. We begin by defining the ‘first coordinate independence property’ of a func- tion on a product of a Frölicher space with I (or I−, I+). Definition 3.1 Let i : A → X be a smooth map, and c : R → X a structure curve on X. Define Λ(c, i) = {t∗ ∈ c−1(i(A))| there exists a sequence {tn} of real numbers with limn→∞ tn = t∗ and each tn ∈ c−1(X − i(A))}. The points in Λ(c, i) are those values in R where the curve ‘enters’ i(A) from X − i(A), or ‘touches’ a point in i(A) whilst remaining in X − i(A) nearby. Now, we are ready to define the ‘first coordinate independence property’ for a structure function on a product. Definition 3.2 Let i : A→ X be a smooth map and suppose f : I×X → R is a structure function on I ×X. Let c : R → I ×X, given by c(s) = (t(s), x(s)) have the following properties • The map x(s) is a structure curve on X. • For all ǫ > 0, t(s) is a smooth real-valued function on R−∪s∗∈Λ(x,i)[s∗ − ǫ, s∗ + ǫ]. If, for every such map c, the composite f ◦ c is a smooth real-valued function, then we say that f : I×X → R has the first independence property (FCIP) with respect to i. Extending the definition, we say that a map g : I × X → Y has the FCIP with respect to i if the composite h ◦ g : I ×X → R has the FCIP with respect to i for every h ∈ FY . Notice that we can formulate a similar definition of the FCIP if we replace I throughout by I− or I+, leaving the rest of the definition unchanged. We will have occasion to use this type of first coordinate independence property in the later part of this work. Note. Let i : A→ X , and suppose that we are given a map g : I×X → Y . Let f : Y → R be a structure function on Y , and suppose that f ◦ g : I ×X → R has the FCIP with respect to i for any such f . Then, given a smooth map h : Y → Z, the composite f ′ ◦ h ◦ g : I×X → R has the FCIP with respect to i for any structure function f ′ on Z. The above note applies equally well if g : I− ×X → Y or g : I+ ×X → Y has the FCIP with respect to i when composed with a smooth function h on Y . Example 3.1 1. For any i : A→ X , the projection onto the second coordinate πX : I×X → X has the FCIP. 2. Let α : R → R be a smooth braking function with the properties that • α(t) = 0 if t < 1 • 0 < α(t) < 1 if 1 ≤ t ≤ 3 • α(t) = 1 if 3 Consider 0 →֒ I−. Let H : I× I− → I− be given by H(t, s) = (1−α(t))s. Then, f ◦H : I× I− → R has the FCIP with respect to the inclusion 0 →֒ I−, for any f ∈ FI− . Definition 3.3 Consider a smooth inclusion i : A →֒ X. Suppose that there exists a smooth map u : X → I, with u−1(0) = i(A). If there exists a smooth map H : I×X → X that satisfies the following properties: • H has the FCIP with respect to i. • H(0, x) = x for all x ∈ X. • H(t, x) = x for all (t, x) ∈ I× i(A). • H(1, x) ∈ i(A) for all x ∈ X with u(x) < 1, then the pair (X,A) is called a smooth neighborhood deformation retract pair, or SNDR pair for short. If, in addition, H is such that H(1 × X) ⊂ i(A), then the pair (X,A) is called a smooth deformation retract pair, or an SDR pair for short. The subspace A is called a smooth neighborhood deformation retract or smooth deformation retract of X if (X,A) is an SNDR pair or SDR pair, respectively. The pair (u,H) is called a representation for the SNDR (or SDR) pair. Example 3.2 1. The pair (X, ∅) is an SNDR pair. A representation is u(x) = 1, H(t, x) = x, for each t ∈ I and x ∈ X . 2. The pair (X,X) is an SNDR pair. A representation is u(X) = 0, H(t, x) = x, for each t ∈ I and x ∈ X . Lemma 3.1 The pair (I−, 0) is an SDR pair. Proof. Let α : R → R be the smooth braking function of Examples 3.1. A representation for (I−, 0) as an SDR pair is (u,H), where u : I− → I and H : I× I− → I− are given by u(s) = s, and H(t, s) = (1 − α(t))s. Clearly, the identity u : I− → I is smooth. And the map H , as shown in Example 2.4, is smooth and clearly has the FCIP with respect to the inclusion, since whenever v approaches a value for which s(v) = 0, one has g((1− α(t(v)))s(v)) = g(0) for v in a neighborhood of this value and g ∈ FI− . � Lemma 3.2 The pair (I, {0, 1}) is an SNDR pair. Proof. A representation (u,H) for the SNDR pair can be given as follows. Define u : I → I to be a bump function such that • u(t) = 0 for t = 0 or t = 1, • u(t) = 1 for t ∈ [ 1 • 0 < u(t) < 1 otherwise, and let β : I → I be a braking function with the properties that β(s) = 0 for 0 ≤ s ≤ 1 , and β(s) = 1 for 3 ≤ s ≤ 1. Let 0 < ǫ 1 , and define H : I× I → I by H(t, s) = (1− αǫ(t))s+ αǫ(t)β(s). It is clear that H(0, s) = s, H(t, 0) = 0, and H(t, 1) = 1. Suppose that u(s) < 1. Then, s ∈ [0, 1 ) ∪ (3 , 1]. This implies that β(s) = 0 or β(s) = 1. We then have H(1, s) = 0 or H(1, s) = 1, which means that H(1, s) ∈ {0, 1} if u(s) < 1. To see that H is smooth, let f : I → R be a generating function for the flattened unit interval. The only possible points of non-smoothness are points where t = 0, 1 and s = 0, 1. The braking function αǫ ensures that H is locally constant in the tb variable whenever t is near 0 or 1, so no problem arises from the t component. When s is near s = 0, we have H(t, s) near 0, and so the generating function f is locally constant. Similarly, when s is near s = 1, we have H(t, s) near 1, and the generating function f is again locally constant. � We now show that the product of SNDR pairs is again an SNDR pair. Theorem 3.1 Let i : A →֒ X and j : B →֒ Y be inclusion mappings. If (X,A) and (Y,B) are SNDR pairs, then so is (X × Y, (X ×B) ∪ (A× Y )). If one of (X,A) or (Y,B) is an SDR pair, then so is the pair (X × Y, (X ×B) ∪ (A× Y )). Proof. Let α : R → I be a smooth braking function with the properties that α(t) = 0 for t ≤ 1 , and α(t) = 1 for t ≥ 3 , and let β : R → R be a smooth increasing braking function with the properties that β(t) = t for t ≤ 1 , and β(t) = 1 for t ≥ 3 . Suppose that (u,H) and (v, J) are representations for the SNDR pairs (X,A) and (Y,B), respectively. Let u : X → I, and v : Y → I be given by u(x) = β(u(x)) and v(y) = β(v(y)) respectively. Define w : X×Y → I by w(x, y) = u(x)v(y). The braking function β ensures smoothness of u and v, and consequently of w. We have w−1(0) = (X × B) ∪ (A × Y ), as required. Define Q : I×X × Y → X × Y as follows . Q(t, x, y) = (H(α(t), x), J(α(t), y)) if u(x) = v(y) = 0 (H(α(t), x), J(α( )α(t), y)) if v(y) ≥ u(x), v(y) > 0, (H(α( )α(t), x), J(α(t), y)) if u(x) ≥ v(y), u(x) > 0. We must show that Q is a smooth map, with the first coordinate independence property with respect to the inclusion (X × B) ∪ (A × Y ) →֒ X × Y . We first consider each part of the definition of Q separately. The first part is clearly smooth. Let us verify that Q is smooth on the second part of its definition; the third part is similar. We need only focus on the component J(α( )α(t), y). Each function making up J(α( )α(t), y) is smooth individually, so we need only pay extra attention to those parts that involve flattened unit intervals, remembering that addition and multiplication on the flattened unit interval need not preserve smoothness, as is the case for the usual unit interval. So let us consider α( ); it is smooth except possibly when approaches 0 or 1, since it is here that structure curves on the flattened unit interval need not be smooth in the usual sense. Clearly, if u(x) approaches 0 and v(y) does not approach 0, then the braking function α ensures that = 0 near such points. If v(y) approaches 0, then u(x) must approach 0 too. This situation is dealt with later. Thus, Q, in part two of the definition, is smooth, and one can show similarly that Q in the third part of the definition is smooth as well. Let us now consider the overlaps of the three parts of the definition of Q. Observe that if u(x) is in a sufficiently small neighborhood of v(y), with u(x) 6= 0 and v(y) 6= 0, then we have α(u(x) ) = 1, and so the second and third parts of the definition of Q coincide here. Thus, it remains only to show that Q is smooth as u(x) and v(y) both approach 0. If Q is smooth in each of its coordinates then it is smooth, so consider the coordinate involving the map J . Let c : R → I×X × Y be a structure that is given by c(s) = (t(s), x(s), y(s)). Then, the map c1 : R → I× Y , given by c1(s) = (α(t(s)), y(s)) if u(x(s)) = v(y(s)) = 0 u(x(s)) v(y(s)) )α(t(s)), y(s)) if v(y(s)) ≥ u(x(s)), v(y(s)) > 0 (α(t(s)), y(s)) if u(x(s)) ≥ v(y(s)), u(x(s)) > 0 is a map satisfying the conditions of Definition 3.2, since its second coordinate is smooth, but its first coordinate may be singular as v(y(s)) ( and hence u(x(s))) approaches 0. Since J has the first coordinate independence property, the map (Joc1)(s) = J(α(t(s)), y(s)) if u(x(s)) = v(y(s)) = 0 u(x(s)) v(y(s)) )α(t(s)), y(s)) if v(y(s)) ≥ u(x(s)), v(y(s)) > 0 J(α(t(s)), y(s)) if u(x(s)) ≥ v(y(s)), u(x(s)) > 0 is smooth. Thus, Q ◦ c is smooth, and since c is arbitrary, Q is smooth. In a similar way, the coordinate of Q involving H can be shown to be smooth. We now verify that Q satisfies the required boundary conditions. When t = 0, all three lines defining Q reduce to (H(0, x), J(0, y)) = (x, y). Let x ∈ A and y ∈ B; then u(x) = v(y) = 0. Therefore, Q reduces to (H(α(t), x), J(α(t), y)) = (x, y). If x ∈ A and y /∈ B, then Q is given by the second part of its definition, which reduces to (H(α(t), x), J(0, y)). The case when x /∈ A and y ∈ B is similar. If t = 1 and 0 < w(x, y) < 1 then either 0 < u(x) < 1 or 0 < v(y) < 1. Suppose that 0 < u(x) < 1. Then either u(x) ≤ v(y) or v(y) < u(x). If u(x) ≤ v(y), then Q is given by the second part of its definition, which reduces to (H(1, x), J(α( , y)) ∈ i(A)× Y . If v(y) < u(x), then the third part of the definition of Q applies and Q reduces to (H(α( ), x), J(1, y)) ∈ X × j(B). Finally, we must show that for any f ∈ FX×Y , f ◦Q has the first coordinate independence property with respect to the inclusion (X×B)∪(A×Y ) →֒ X×Y . To this end, consider a map c : R → I×X×Y , given by c(s) = (t(s), x(s), y(s)). Let {sn} be a sequence of real numbers converging to s∗ with c(sn) ∈ (X×Y )− ((A× Y ) ∪ (X ×B)), and c(s∗) ∈ (A× Y ) ∪ (X ×B). There are three cases to consider. • Suppose that c(s∗) ∈ A×B. Then x(s∗) ∈ A and y(s∗) ∈ B. The fact that H and J have the first coordinate independence property with respect to i and j respectively means that each coordinate of Q is smooth, and so Q is smooth. • Suppose that c(s∗) ∈ A × Y , and that y(s∗) /∈ B. Then at each of the points c(sn), (Q ◦ c)(sn) is given by the second part of the definition of Q, for n large enough. Since x(s∗) ∈ A, the component of Q involving H is smooth, since H has the first coordinate independence property. For any s in a neighborhood of s∗, α( u(x(s)) v(y(s)) ) = 0. Thus, the component of Q involving J is constant for s in a neighborhood of s∗, and so is smooth there. • The case with c(s∗) ∈ X ×B, and x(s∗) /∈ A is similar to the second case above. For the last part of the theorem, suppose that (u,H) represent (X,A) as an SDR pair. If we replace u by u′ = 1 u, then (u′, H) also represent (X,A) as an SDR pair. Making the above constructions now with u′ in place of u, it follows that w(x, y) < 1 for all (x, y) and so Q(1, x, y) ∈ (X × B) ∪ (A × Y ). This completes the proof. � 4 Cofibrations In this section, we show that for a subspace A ⊆ X that is closed in the under- lying topology, the inclusion i : A → X is a cofibration if and only if (X,A) is an SNDR pair. Definition 4.1 Let i : A→ X be a cofibration. We call i a cofibration with FCIP if any homotopy extension can be chosen to have the FCIP with respect to i. Using the equivalent formulation of the notion of cofibration, given by Lemma 2.5, we may restate Definition 4.1 as follows: A cofibration i : A → X is a cofibration with the FCIP if and only if the map G that we may fill in to complete the commutative diagram (0×X) ∪ (I− ×A) h // I− ×X may be chosen to have the FCIP with respect to the inclusion i. We have the following result, which corresponds to a similar topological result. Lemma 4.1 A smooth map i : A → X is a cofibration (with the FCIP) if and only if (0 × X) ∪ (I− × A) is a retract of I− × X, (where the retraction r : I− ×X → (0×X) ∪ (I− ×A) has the FCIP ). Proof. In the one direction, suppose that (0 × X) ∪ (I− × A) is a retract of I− ×X . We wish to complete the following diagram: (0×X) ∪ (I− ×A) h // I− ×X By hypothesis, there exists r : I−×X → (0×X)∪ (I− ×A) such that r ◦ j = 1. Define G = h ◦ r. If r has the FCIP, then so does h ◦ r. Conversely, suppose that i : A → X is a cofibration (with the FCIP). We may find a map r such that the diagram (0×X) ∪ (I− ×A) 1// (0 ×X) ∪ (I− ×A) I− ×X commutes. Thus, r ◦ j = 1. If i is cofibration with the FCIP with respect to i, then r can be chosen to have the FCIP. � The next theorem shows the relationship between cofibrations, retracts and SNDR pairs. Theorem 4.1 Let i : A → X be an inclusion, with A closed in the underlying topology of X. Then the following are equivalent. (1) The pair (X,A) is an SNDR pair. (2) There is a smooth retraction r : I− × X → (0 ×X) ∪ (I− × A) with the FCIP. (3) The map i : A→ X is a cofibration with the FCIP. Proof. To show that (1) and (2) are equivalent, note that the pair (I−×X, (0× X) ∪ (I− × A)) is an SDR pair, as a consequence of Lemma 3.1 and Theorem 3.1. Let (w,Q) be a representation for the pair (I− ×X, (0×X)∪ (I− ×A)) as an SDR pair, and let Q be constructed as in Theorem 3.1. Define r : I− ×X → (0×X) ∪ (I− ×A) by r(t, x) = Q(1, t, x), where (t, x) ∈ I− ×X . We observe that r has the FCIP, since Q has this property, and Q has this property since each of its components has this property. The equivalence of (2) and (3) is Lemma 4.1. We need only show that (2) implies (1). Let r : I−×X → (0×X)∪ (I−×A) be a retraction with the FCIP with respect to i. Define H : I × X → X by H(t, x) = (πX ◦ r)(α(t), x), where πX is the projection onto the second coordinate, and α : R → R is a braking function with the following properties: α(t) = 0 for t ≤ 0, α(t) = 1 for t ≥ 3 , and 0 < α(t) < 1 for 0 < t < 3 . This braking function is necessary to ensure smoothness at the right endpoint of the flattened unit interval I. Smoothness at the left endpoint is already taken care of by the fact that r is defined in terms of the left flattened unit interval. The map H satisfies the following properties: • H has the FCIP since r has this property. • H(0, x) = (πX ◦ r)(0, x) = x, for x ∈ X . • H(t, x) = (πX ◦ r)(α(t), x) = x, for x ∈ A. We now construct u : X → I. Let πI : I×X → I denote the projection onto I. Define a smooth function β : R → R by β(t) = 0 if t ≤ 0 t2 if t > 0. Now, define u : X → I by u(x) = β(α(t) − (πI ◦ r)(1, x)(πI ◦ r)(α(t), x))dt β(α(t))dt It is clear that u is a smooth mapping. We now verify that (u,H) represents (X,A) as an SNDR pair. (1) Let x ∈ A. Clearly, (πI ◦ r)(1, x) = 1 and πI ◦ r)(α(t), x) = α(t), and so β(α(t)− (πI ◦ r)(1, x)(πI ◦ r)(α(t), x))dt = 0. Thus, u(x) = 0, for all x ∈ A. (2) Suppose that x ∈ X−A. Since 0×(X−A) is open in the underlying topology on (0×X)∪ (I− ×A), we may choose an open neighborhood W ⊆ 0× (X −A) of (0, x). Since r is continuous, there is a neighborhood V ⊆ I− ×X such that r(V ) ⊆W ⊆ 0× (X −A). Now, consider the mapping qx : I → I×X , given by qx(t) = (α(t), x), for each x ∈ X . This is clearly smooth. Thus, there exists a neighborhood U ⊆ I− such that qx(U) ⊆ V . In other words, U × {x} ⊆ V . So, we have (πI ◦ r)(α(t), x) = 0, for all t ∈ U . Thus, we have u(x) = β(α(t) − (πI ◦ r)(1, x)(πI ◦ r)(α(t), x))dt + β(α(t))dt β(α(t))dt Combining this with part (1), we deduce that u−1(0) = A. (3) Suppose that x is such that u(x) < 1. There must be a neighborhood U of I such that (πI◦r)(1, x)(πI ◦r)(α(t), x) > 0, for t ∈ U . Thus (πI◦r)(1, x) > 0, but this implies that r(1, x) ∈ I×A, and hence H(1, x) ∈ A. The proof is complete. 5 The Mapping Cylinder In this section we show that the inclusion of X into the flattened mapping cylinder If of a map f : X → Y is a cofibration with the FCIP. Theorem 5.1 Let f : X → Y be a smooth map. Then, the pair (If , X) is an SNDR pair. Proof. Let α : I → R be a smooth braking function with the following proper- ties: α(t) = 0 if 0 ≤ t ≤ 1 , α(t) = 1 if 3 ≤ t ≤ 1, 0 < α(t) < 1, otherwise. Define two more braking functions α1, α2 : I → R as follows: α1(0) = 0, 0 < α1(t) < 1 if 0 < t < 3 , α1(t) = 1 if ≤ t ≤ 1, and α2(t) = 0 if 0 ≤ t ≤ 34 , α2(t) = 1 ≤ t ≤ 1. Now, define u : If → I by u([t, x]) = α1(t) and u([y]) = 1, for (t, x) ∈ I×X and y ∈ Y . Define H : I× If → If by H(s, [t, x]) = [(1 − α(s))t+ α(s)α2(t), x] if (t, x) ∈ I×X H(s, [y]) = [y] if y ∈ Y . That u is smooth comes from the fact that it is smooth when restricted to each component of the coproduct (I×X)⊔Y ; it is thus smooth on the quotient To see that the map H : I × If → If is smooth, note that since we are working in a cartesian closed category, products commute with quotients, i.e. if q is quotient, then so is 1× q, where 1 is an identity map. Thus, we may think of H as being defined on the space (I× I×X) ⊔ (I× Y ) where ∼ is the identification (t, 1, x) = (t, f(x)) for t ∈ I, and x ∈ X . Since H is smooth when restricted to each component of the coproduct (I×I×X)⊔(I×Y ), H is smooth on the quotient I× If . We now verify that (u,H) is a representation for (If , X) as an SNDR pair. • u−1(0) = [0, x] = i0(X). • H(0, [t, x]) = [t, x] and H(0, [y]) = [y]. • H(s, [0, x]) = [0, x]. • If u[t, x] < 1, then t < 3 and so α2(t) = 0. Thus, H(1, [t, x]) = [0, x]. This completes the proof. � Finally, we have the following important corollary. Corollary 5.1 Given any smooth map f : X → Y , the inclusion X →֒ If is a cofibration with the FCIP. 6 The Exact Sequence of a Cofibration Our aim in this section is to show how one can use SNDR pairs to prove the existence of the right exact Puppe sequence. We state the result in Theorem 6.1 and break the proof of the result up into a number of lemmas. We follow the method used by Whitehead [12] for the topological case. Throughout this section we work in the category FRL∗ of pointed Fr—’olicher spaces, and basepoint preserving smooth maps. Theorem 6.1 Let W be an object in FRL∗, and suppose that i : A →֒ X is a cofibration in FRL∗. For any basepoint x0 ∈ A ⊆ X there is a sequence . . . // [ A,W ] Ti,W ] X,W ] A,W ] // . . . . . . // [ A,W ] // [Ti,W ] // [X,W ] // [A,W ] which is an exact sequence in SETS∗, where j : X → Ti is the inclusion dis- cussed in Paragraphe 2.4 and k : Ti → A is the quotient map defined below. It is, in fact, possible to prove that the sequence above is an exact sequence of groups as far as A,W ] and that the morphisms to this point are group homomorphisms, but we shall not do so here. The reduced(flattened)suspension of a pointed Frölicher space X is de- fined as X = (I/{0, 1}) ∧X, where the reduced join is defined as for topological spaces with the identified set taken as basepoint, and with 0 the basepoint of I. In this section, whenever we refer to the suspension of a space , we mean the reduced flattened suspension defined above. Lemma 6.1 If (x,A) is an SNDR pair and p : X → X/A the quotient map, then the sequence i // X p // X/A is right exact. Proof. To show that the given sequence is right exact we must show that for any Frölicher space W the following sequence is exact in SETS: [X/A,W ] // [X,W ] // [A,W ] . It is easy to see that im p∗ ⊆ ker i∗. To see the reverse inclusion, let g : X → W be an element of [X,W ], with g|A ≃ w0 (rel w0), where w0 ∈ W . Since i // X is an SNDR pair, the map i is a cofibration, and so we may extend w0 to a smooth map g ′ : X → W such that g′ ≃ g. But g′ is constant on A, and so there exists a smooth map g1 : X/A → W such that p∗(g1) = g′. This shows that ker i∗ ⊂ im p∗. � Lemma 6.2 For any smooth map f : X → Y , the sequence f // Y l // Tf is right exact, where l is the usual inclusion of Y into the mapping cone; i.e. y 7→ [y] ∈ Tf . Proof. One can show that there is a homotopy commutative diagram i ��@ // Tf where i, j, and l are the usual inclusions, and p is the quotient map that collapses away {0} ×X to a point. Since, by Theorem 5.1, (If , X) is an SNDR pair, it follows from Lemma 6.1 that the sequence i // If p // Tf is right exact. It is fairly easy to show that j : Y → If is a homotopy equivalence. Therefore, the sequence f // Y l // Tf is right exact. � Lemma 6.3 For any smooth map i : A → X, there is an infinite right exact sequence i // X // Ti // . . . i // Tin−2 // Tin−1 // . . . where in, n ≥ 1, are inclusion maps. Proof. The pair (Ti, X) is an SNDR pair. The representation for the pair (If , X) in Theorem 5.1 can be adapted to show this. One iterates the procedure of Lemmas 6.1 and 6.2. � One can easily see that there is an isomorphism between Ti/X and Define q : Ti → A to be the map which identifies X ⊂ Ti to a point, followed by the isomorphism Ti/X → Lemma 6.4 The sequence // Ti is right exact. Proof. As noted above the pair (Ti, X) is an SNDR pair. We have the com- mutative diagram // Ti where p : Ti → Ti/X is the identification map, and q0 : Ti/X → A is an isomorphism. The top line of the diagram is right exact, by Lemma 6.1, and so the sequence // Ti is right exact. � There is a commutative diagram // Ti where q1 is a homotopy equivalence. ( See Whitehead [12] for more details of this map. ) Using commutative diagrams of this form, one can now proceed almost exactly as one does in the topological situation, as in Whitehead [12] for example, to get the following infinite right exact sequence: i // X // Ti // . . . . . . // // . . . The definition of right exactness now gives us the exact sequence of Theorem References [1] A. Cap, K-Theory for Convenient Algebra, Dissertationen, Faculty of Mathematics, University of Vienna, 1993. [2] Cherenack P., Applications of Frölicher Spaces to Cosmology, Ann. Univ. Sci. Budapest 41(1998), 63-91. [3] P. Cherenack , Frölicher versus Differential Spaces: A prelude to Cosmol- ogy, Kluwer Academic Publishers 2000, 391-413. [4] P. Cherenack, The Left Exactness of the Smooth Left Puppe Sequence. In L. Tamassy and J. Szenthe, editors, New Developments in Differential Geometry, (Proceedings of the Colloquium on Differential Geometry, De- brecen, Hungary, July 26-30, 1994), Mathematics and Its Applications. Kluwer Academic Publishers, 1996. [5] P. Cherenack, Smooth Homotopy, Topology with Applications, (18):27-41, 1984. [6] B. Dugmore, The Right Exactness of the Smooth Right Puppe Sequence. Master’s Thesis, University of Cape Town, 1996. [7] A. Frölicher, A. Kriegl, Linear Spaces and Differentiation Theory, J. Wiley and Sons, New York, 1988. [8] M.W. Hirsch, Differential Topology, GTM 33, Springer-Verlag, New York, 1976. [9] I.M. James, General Topology and Homotopy Theory, Springer-Verlag, Berlin, 1984. [10] A. Kriegl, P. Michor, Convenient Settings of Global Analysis, Am. Math. Soc., 1997. [11] Jet Nestruev, Smooth Manifolds and Observables, Springer-Verlag New York, Inc., 2003 [12] G.W. Whitehead, Elements of Homotopy Theory, Springer-Verlag, New York, 1978. Preliminaries Basic Constructions of Homotopy Theory in FRL Our Approach to Homotopy Theory in FRL Flattened Structures on the Unit Interval Some Properties of Smooth Functions between the Flattened Unit Intervals Homotopy in FRL and Related Objects Cofibrations in FRL Smooth Neighborhood Deformation Retracts SNDR pairs and SDR pairs Cofibrations The Mapping Cylinder The Exact Sequence of a Cofibration

0704.0343

Experimental observation of structural crossover in binary mixtures of colloidal hard spheres

Experimental observation of structural crossover in binary mixtures of colloidal hard spheres Jörg Baumgartl1,∗, Roel P.A. Dullens1, Marjolein Dijkstra2, Roland Roth3 and Clemens Bechinger1 12. Physikalisches Institut, Universität Stuttgart, 70550 Stuttgart, Germany 2Soft Condensed Matter Group, Utrecht University, 3584 CC Utrecht, The Netherlands 3Max-Planck-Institut für Metallforschung, 70569 Stuttgart, Germany and Institut für Theoretische und Angewandte Physik, Universität Stuttgart, 70569 Stuttgart, Germany Using confocal-microscopy we investigate the structure of binary mixtures of colloidal hard spheres with size ratio q = 0.61. As a function of the packing fraction of the two particle species, we observe a marked change of the dominant wavelength in the pair correlation function. This behavior is in excellent agreement with a recently predicted structural crossover in such mixtures. In addition, the repercussions of structural crossover on the real-space structure of a binary fluid are analyzed. We suggest a relation between crossover and the lateral extension of networks containing only equally sized particles that are connected by nearest neighbor bonds. This is supported by Monte-Carlo simulations which are performed at different packing fractions and size ratios. PACS numbers: 82.70.Dd, 61.20.-p Most systems in nature and technology are mixtures of differently sized particles. Each distinct particle size introduces another length scale and its competition gives rise to an exceedingly rich phenomenology in compari- son with single-component systems. Already the simplest conceivable multi-component system, i.e. a binary mix- ture of hard spheres, exhibits interesting and complex behavior. Just a few examples include entropy driven formation of binary crystals [1, 2, 3], frustrated crys- tal growth [4], the Brazil nut effect [5], glass-formation [6, 7] and entropic selectivity in external fields [8]. Al- though interaction potentials in atomic systems are more complex than those of hard spheres, the principle of vol- ume exclusion is ubiquitous and thus always dominates the short-range order in liquids [9]. Accordingly, hard spheres form one of the most important and successful model systems in describing fundamental properties of fluids and solids. It has been demonstrated that many of their features can be directly transferred to atomic systems where fundamental mechanisms are often ob- structed by additional material specific effects [10]. Bi- nary hard sphere systems are fully characterized by their size ratio q = σS/σB with σi the diameters of the small (S) and big (B) spheres and the small and big sphere packing fractions ηS , ηB , respectively. The pair-correlation functions, gij(r), are the central measure of structure in fluids; they describe the probabil- ity of finding a particle of size i at distance r from another particle of size j. It is well known that all pair-correlation functions in any fluid mixture with short-ranged inter- actions (not just hard spheres) exhibit the same type of asymptotic decay, which can be either purely (mono- tonic) exponential or exponentially damped oscillatory ([11] and references therein). This prediction, which is valid in all dimensions, suggests that all pair-correlation functions decay with a common wavelength and decay length in the asymptotic limit. For binary hard-sphere mixtures where ηB � ηS or ηS � ηB , this is rather obvi- ous since the system is dominated by either big or small particles. The pair-correlation functions will asymptot- ically oscillate with a wavelength determined either by σB (ηB � ηS) or σS (ηS � ηB). Rather surprising is that the above statement is also valid for all other rela- tive packing fractions where the system is not dominated by particles of a single size ([11, 12]). Accordingly, in the asymptotic limit the (ηS , ηB) phase diagram is divided by a sharp crossover line where the decay lengths of the contributions to gij(r) with the two wavelengths become identical. Below and above this line, however, the pair- correlation function is either determined by the diameter of the small spheres or that of the big spheres [13]. Despite the generic character of structural crossover and the close relationship between structural and me- chanical properties, this effect has not been observed in experiments as the asymptotic limit is difficult to reach in scattering experiments on atomic and molecular liq- uids. However, recent calculations suggest that struc- tural crossover is already detectable at relatively small distances [12]. Because colloidal particles are directly accessible in real space, such systems provide an oppor- tunity to explore the structure of binary fluids and to investigate structural crossover experimentally. As colloidal suspension we used an aqueous binary mixture of small melamin particles (σS = 2.9µm) and big polystyrene spheres (σB = 4.8µm). Addition of salt screens residual electrostatic interactions thus lead- ing to an effective hard sphere system. Since melamin has a higher density (ρM = 1.51g/cm3) than polystyrene (ρP = 1.05g/cm3) the sedimentation velocities are sim- ilar and, hence, we obtain a homogeneous system after mixing. The suspension was contained in a cylindrical sample cell with a silica bottom plate to allow optical imaging with an inverted confocal microscope in reflec- tion mode (Leica TCS SP2). From the images, particle positions were obtained with digital video microscopy [14]. Strong layering at the bottom wall allowed us to image only the first two-dimensional bottom layer of the three-dimensional system. We define the packing fraction Figure 1: Different paths with constant total packing fraction η = ηS + ηB in the (ηS , ηB)- plane. Experimental data (open symbols: η = 0.72, q = 0.61) are sorted into ten bins. The bin size is indicated by the ’error bars’. Closed symbols cor- respond to the MC-simulations (N: η = 0.62, q = 0.4) and (•: η = 0.57, q = 0.5). For convenience all samples are la- beled with numbers increasing in the direction indicated by the arrows. as ηi = πσ2i /4, with ρi the number density of component i. Variation of the relative packing fractions of the par- ticles was achieved by addition of small particles to a suspension of big spheres (Fig.1). Thus, the total pack- ing fraction in the two-dimensional bottom layer remains constant for all samples: η = 0.72. In the following we will refer to the different samples by the sample numbers (No.) as given in Fig.1. Typical snapshots of the system for different packing fractions of big and small particles are shown in Figs.2A- C. The images demonstrate how the structure of the bot- tom layer changes from being rich in small particles (No. 1, Fig.2A) to being rich in big particles (No. 10, Fig.2C). Fig.2B (No. 5) corresponds to about the same number density of small and big spheres. In order to analyze the samples for a possible structural crossover, we calculated the pair correlation function from the determined particle positions. To minimize statistical noise we did not distin- guish between big and small spheres. This is justified be- cause the crossover has been predicted to be visible in all pair-correlation functions and thus also in any linear com- bination [11, 12]. The dominating wavelength in the os- cillations is identified by computing the total correlation function htot(r) = i,j xixjhij(r) = ij xixj [gij(r)−1], with the mole fraction xi = ρi/ i ρi of component i [12]. Fig.2D exemplarily shows ln |htot(r)| for samples No. 1,5, and 9. Note that in this representation the oscillation wavelength is halved. The correlation functions of sam- ples No.1 and 9 clearly oscillate with a single wavelength, respectively, given by ≈ σB/2 and ≈ σS/2. In contrast, sample 5 does not show a dominating wavelength but an interference of different length scales which is typical near the structural crossover. It is important to mention, Figure 2: A-C) Typical snapshots of the bottom layer of a binary mixture observed with a confocal microscope used in reflection mode. The mixtures correspond to sample 10 (A), 5 (B) and 1 (C). The field of view is 40×40µm2. D) Logarithmic plot of the total correlation functions htot(r) for the experi- mental binary mixtures with η = 0.72 ± 0.04. Correlation functions are plotted for sample numbers 1,5 and 9 (compare Fig.1) and are shifted in vertical direction for clarity. The hor- izontal bars correspond to σB/2 and σS/2, respectively. E) Fourier-transforms of htot(r) for the experimental data points (compare Fig. 1). Vertical lines indicate the wave vectors k corresponding to the diameters of the small (S) and big par- ticles (B), respectively. (color online). that this intermediate behavior is only observed for sam- ples No.5 and 6, i.e. only for about 10% of the entire range over which ηB and ηS was varied. The experimen- tally identified crossover-region is in excellent agreement with the theoretically calculated value of ηS ≈ 0.3 at those size ratios, which were determined from the decay of the pair correlation functions calculated within density functional theory in the test particle limit [15]. Fig.2E Figure 3: Visualization of the different bond-types as determined by a Delaunay triangulation: big-big (black), big-small (yellow) and small-small (red). Different plots correspond to the sample numbers as indicated in Fig.1. The field of view is 180× 180µm2. shows the Fourier transforms of htot(r) for all samples where the rather sudden change of the dominating wave- length is seen more clearly [16]. At small and high pack- ing fractions, the correlations are clearly dominated by frequencies corresponding to either small or large parti- cles (vertical lines) while around sample No.5 hardly any dominating frequency is observed. This experimentally confirms structural crossover as well as its occurrence at finite particle distances. So far, structural crossover has been discussed in terms Figure 4: Averaged radii of gyration 〈Rig〉 (normalized to L/2 with L2 the size of the field of view) of networks formed by large (solid symbols) and small particles (open symbols) as a function of the sample number for A) the experimental data, B) the MC-simulations at η = 0.57 and q = 0.5 and, C) the MC-simulations at η = 0.62 and q = 0.4. The correspond- ing packing fraction of small particles ηS is indicated as well. The grey area and the dashed line respectively indicate the crossover as inferred from the correlation functions and from density functional theory. (color online). of pair correlation functions, i.e. spatially averaged quan- tities. Since our experiments naturally provide detailed structural information, we investigate what the reper- cussions are of the structural crossover on the real-space structure. We first subjected a Delaunay triangulation to the set of particle centers and identified nearest-neighbor bonds between big-big (black), big-small (yellow), and small-small (red) particles, respectively (see Fig.3). As observed in Fig.3, sample 1 predominantly consists of big- big bonds which form a large network spreading across the entire field of view. With increasing sample No., i.e. increasing ηS , the number of small-small bonds increases, which leads to fragmentation of the big-big network into smaller, randomly distributed patches. At large sample numbers, the role of big and small particles is inverted and small-small bonds form a network spanning the en- tire area (No.10). Having distinguished between differ- ent bond-types, a natural and well-known measure of the spatial extend of a network formed by ni particles of size i at positions ~xik (k = 1 . . . n i) is given by the radius of gyration Rig = k=1(~x k − ~R 2, with ~Ri0 the cen- troid position of the network. Computing this quantity for all, say N iC , networks formed by connected particles of size finally yields a weighted averaged radius of gyra- tion 〈Rig〉 = m=1 ni(m)R g(m) where N i denote the total number of particles i. We calculated 〈Rig〉 for net- works consisting of connected big or small particles and plotted these values for our experimental data in Fig.4A as a function of the sample number. At small and high sample numbers the quantities saturate while a relatively sharp transition with an intersection point occurs around sample 6. This location is indeed in very good agreement with the crossover transition as determined from the cor- relation functions in Fig.2 and density functional theory (also indicated in Fig. 4A). This suggests that the struc- tural crossover corresponds to a competition between the sizes of networks consisting of connected big or small par- ticles, respectively. As structural crossover is also predicted for other size ratios and packing fractions, we use Monte-Carlo (MC) simulations to test our findings for more dilute systems with size ratios q = 0.5 and q = 0.4. The corresponding paths through the phase diagram (see closed symbols in Fig.1) were obtained from 2-dimensional simulations with a fixed number of particles of about 0 < N < 3000 for both species and box areas of about 1500σ2B employing periodic boundary conditions. From the configurational snapshots we first determined the region of crossover by analyzing htot(r) (the correlation functions are sampled using 104 MC cycles per particle). Then, we performed the above described Delaunay triangulation to calculate 〈Rig〉 for networks of connected big or small particles, re- spectively. The corresponding radii of gyration are plot- ted in Fig.4B and C and show a similar behavior as in the experiment. Again, the intersection points are consistent with the crossover region as inferred from the correlation functions and DFT calculations. Note that the crossover region sensitively depends on the size ratio and packing fractions. Both the experiment and Monte-Carlo sim- ulations show that structural crossover is accompanied by a pronounced change in the typical size of networks consisting of connected big and small particles. By in- troducing small particles into a system of big spheres, connections between big particles are broken and, at the same time, connections between small particles are made. This sensitively affects the typical size of networks con- taining connected, equally-sized particles and thereby the chance of finding another particle with the same size at a relatively large distance. Consequently, the change from 〈RBg 〉 > 〈RSg 〉 to 〈RSg 〉 > 〈RBg 〉 (and vice versa) provides a simple real-space argument why the oscillation wave- length of the gij(r) in the asymptotic limit is either set by σB or σS . We have experimentally demonstrated the structural crossover in a binary colloidal hard sphere system. Fur- thermore, we show that structural crossover is strongly coupled to the size of networks containing connected equally-sized particles only. Going across the structural crossover, the size ratio of such networks comprised by either connected big or small particles is reversed. We be- lieve this real-space configurational picture of structural crossover is not just applicable to binary hard spheres, as structural crossover is a generic feature of mixtures with competing length scales. Moreover, it shows inter- esting similarities with force chains in granular matter [17] and glassy systems [6, 7, 18] of dissimilar sized parti- cles. Therefore, our finding may help to gain more insight into structure-related properties in binary systems at an universal level. ∗Electronic address: [email protected] stuttgart.de [1] P. Bartlett, R. H. Ottewill and P. N. Pusey, Phys. Rev. Lett. 68, 3801 (1992). [2] A. B. Schofield, Phys. Rev. E 64, 51403 (2001). [3] M. D. Eldrige, P. A. Madden and D. Frenkel, Nature 365, 35 (1993). [4] V. W. A. de Villeneuve, R. P. A. Dullens, D. G. A. L. Aarts, E. Groeneveld, J. H. Scherff, W. K. Kegel and H. N. W. Lekkerkerker, Science 309, 1231 (2005). [5] D. C. Hong, P. V. Quinn and S. Luding, Phys. Rev. Lett. 86, 3423 (2001). [6] T. Eckert and E. Bartsch, Phys. Rev. Lett. 89, 125701 (2002). [7] D. N. Perera and P. Harrowell, Phys. Rev. E 59, 5721 (1999). [8] R. Roth and D. Gillespie, Phys. Rev. Lett. 95, 247801 (2005). [9] S. Sastry, T. M. Truskett, P. G. Debenedetti, S. Torquato and F. H. Stillinger, Mol. Phys. 95, 289 (1998). [10] W. Poon, P. Pusey and H. N. W. Lekkerkerker, Physics World April, 27 (1996). [11] C. Grodon, M. Dijkstra, R. Evans and R. Roth, J. Chem. Phys. 121, 7869 (2004). [12] C. Grodon, M. Dijkstra, R. Evans and R. Roth, Mol. Phys. 103, 3009 (2004). [13] For very asymmetric size ratios, i.e. q < 0.3, there can be additional regions in which oscillations at intermediate wavelength can be observed. [14] J. C. Crocker and D. G. Grier, J. Colloid Interface Sci. 179, 298 (1996). [15] R. Roth, R. Evans and S. Dietrich, Phys. Rev. E 62, 5360 (2000). [16] In two dimensions for radial symmetric functions the Fouriertransform becomes a Besseltransform. However, for the identification of the dominant wavelength the usual Fouriertransform, which is numerically easier to handle, predicts equivalent results [17] C. S. O’Hern, S. A. Langer, A. J. Liu and S. R. Nagel, Phys. Rev. Lett. 86, 111 (2001). [18] N. Hoffman, F. Ebert, C. N. Likos, H. Löwen and G. Maret, Phys. Rev. Lett. 97, 078301 (2006). References

0704.0344

The Blazar Spectral Sequence and GLAST

The Blazar Spectral Sequence and GLAST L. Maraschi, G. Ghisellini and F. Tavecchio INAF-Osservatorio Astronomico di Brera, Milano, Italy Abstract. The present status and understanding of the "spectral sequence" of blazars is discussed in the perspective of the upcoming GLAST launch. The vast improvement in sensitivity will allow to i) determine more objectively the "average" gamma-ray properties of classes objects ii) probe more deeply the ratio between accretion power and jet power in different systems. Keywords: Gamma-rays - Relativistic jets - Galaxies: active PACS: 95.85.Pw; 98.54.Cm INTRODUCTION The spectral sequence of blazars (Fossati et al.,1998 al.,1998) was constructed merging three complete blazar samples (two radio selected, one X-ray selected: 2 Jy FSRQ, Wall & Peacock 1985, 1Jy BL Lac, Kuhr et al. 1981, and Slew Survey BL Lac, Elvis et al. 1992), grouping all the objects in radio luminosity bins and averaging monochromatic luminosities of objects within each radio-luminosity bin. The procedure is thus prone to various biases (Maraschi & Tavecchio 2001), in particular the gamma-ray data were largely incomplete. The resulting "sequence" shows that the blazar SEDs are double humped and that the two peaks shift to higher energies with decreasing luminosity. Systematic modelling of the SEDs of individual objects (Ghisellini et al. 1998) yields basically uniform beaming factors and jet parameters varying along the sequence in the sense of an increasing energy density and decreasing electron critical energy at higher luminosities. Thus the "sequence" offers a suggestive indication that the basic spectral properties of blazar jets could be related to the different powers involved and possibly represent an evolutionary sequence in cosmic history (Boettcher and Dermer 2002; Cavaliere and D’Elia 2002). The validity of the sequence concept has been questioned on the basis of deeper and larger blazar surveys (e.g. Giommi et al. 2005, Padovani 2007) which however lack until now the very important gamma-ray data. Here we wish to address two points. The first concerns the validity of the original claim within the presently known bright blazar SEDs, the second concerns an anticipation of the types of blazars that may be detected by GLAST. NEW DATA / NEW SOURCES Given the limited space we will illustrate our points schematically, commenting few representative figures. All the figures will have in the background the double humped lines interpolating the blazar spectral sequence. The latter are just polinomial expressions connecting the average monochromatic luminosities obtained as described above. The SED of a new high redshift FSRQ serendipitously discovered by SWIFT (BAT) J0746+2548 (z=2.979) (Sambruna et al. 2006) is shown in Fig. 1a. Clearly J0746 is extremely luminous and conforms well to the sequence, possibly suggesting a gamma-ray peak at Mev energies. The spectral shape in the gamma-ray band that will be measured by GLAST for a large number of blazars will provide an essential information to constrain the position of the high energy peak of blazar SEDs thus probing the sequence concept. 3C 454.3 is a highly variable FSRQ (z=0.859) already detected in gamma-rays by EGRET. The data for a "normal" state (Tavecchio et al. 2007) are shown in Fig. 1b. This source could be detected with GLAST at 1% the intensity level shown in the figure which is the average of EGRET measurements. The source underwent a strong outburst recently and was observed by SWIFT (BAT) and INTEGRAL up to more than 100 keV (Pian et al. 2006, Giommi et al. 2006). In the latter state the expected gamma-ray flux could have been an order of magnitude brighter than detected by EGRET. A source with an intrinsically similar jet could then be detected in gamma-rays even if the jet was at a larger angle to the line of sight. The thick lines in Fig. 2 represent the model used to describe the "normal" state of 3C 454.3, computed for different viewing angles. The gamma-ray emission could be detected by GLAST up to an angle http://arxiv.org/abs/0704.0344v1 FIGURE 1. Spectral Energy Distribution of the blazars J0746+2548 (left, from Sambruna et al. 2006) and 2251+158 (right, from Tavecchio et al. 2007) overimposed on the curves interpolating the blazar sequence. For 2251+158 we also report the model used to reproduce the data (upper black curve) and the emission expected for a misaligned jet with angles of respectively 6, 8 and 10 degrees (from top to bottom). of 10 degrees to the jet axis. In this case the SED would be significantly different than expected from the sequence, simply because the jet emission is less beamed and less prominent with respect the SED of the accretion disk, included here as a blackbody component plus a Seyfert like X-ray component. The sequence is not expected to extend to objects with jets seen at intermediate angles. The different Doppler factor causes only a linear shift of the peak position but a dramatic change in luminosiy. Fig. 2 is devoted to blazars with lower luminosities. This part of the sequence is populated exclusively by BL Lac objects defined as HBLs due to their SEDs peaking at high energies, in the X-ray and TeV bands. In Fig. 2a the data for the "normal" state of PKS 2155-304 are plotted in green. They are well consistent with the sequence. The multifrequency data obtained during the exceptional TeV flare observed from this source in July August 2006 are also shown (see Foschini et al. 2007). During the outburst the two emission peaks do not appear to shift much in frequency but the luminosities increase by a large factor (for a short time) especially in the TeV band. Thus the high state SED deviates remarkably from the sequence expectations. For these objects, though relatively weak at GeV energies, GLAST observations will be important to define the shape of the high energy peak and its possible evolution during outbursts. Finally, in Fig. 2b we show the data for 1629+4008 (z=0.272), a blazar with an emission peak between the UV and the X-ray band discovered within a survey aimed at finding objects with anomalous properties (Padovani et al. 2002). The SED of this source complies reasonably well with the sequence expectation for an HBL, however this object shows emission lines which is not the case for HBLS. In fact the sequence included only X-ray selected BL Lacs, but no X-ray selected radio-loud objects with emission lines, as no such complete sample was available at the time (see Wolter & Celotti 2001). This source indicates that jets with SEDs peaking at high energies can occur in emission line AGNs. This is a new result, which however does not break the correlations inferred from the sequence, as it occurs in the low luminosity range. The question then is: what distinguishes HBLs from objects like 1629? Why emission lines are completely absent in HBLs but present in 1629 whose jet is of comparable luminosity? According to our ideas (Maraschi 2001, Maraschi & Tavecchio 2003) HBL should accrete at highly subEddington rates, therefore in the radiatively inefficient accretion (RIAF) regime, while 1629, which shows emission lines, should be in the “standard” accretion disk regime, therefore near to its Eddington limit. This in turn implies that this source contains a central black hole of relatively modest mass. From the accretion luminosity, assuming that it corresponds to 0.1 the Eddington luminosity we can infer a mass of 6× 107 solar masses. More direct estimates of the black hole mass are needed to confirm this prediction. FIGURE 2. SEDs of the blazars 2251-304 (left, Foschini et al. 2007) and 1629+4008 (Padovani et al. 2002) overimposed on the blazar sequence interpolations. For PKS 2155-304 a normal state is shown together with optical/X-ray and TeV data during the exceptional outburst of July-August 2006 CONCLUSIONS The few examples discussed above are meant to indicate how the concept of a spectral sequence for blazars, based on averages over limited samples involving only the brightest objects of each class, may be probed by GLAST. In particular, strong emphasis has been put in the past on BL Lac objects, neglecting the X-ray selected counterparts of FSRQ which may also be gamma-ray emitters. GLAST is expected to produce extraordinary advances in this field. It will increase by orders of magnitude the number of objects with measured gamma-ray flux (see Dermer these proceedings) thus allowing to study deeper and differently selected samples. These will certainly contain "mixed" objects in which the jet emission is less prominent in comparison to other AGN properties. The new gamma-ray populations should carry great potential for understanding the link between accretion power and the production of jets in extragalactic objects. REFERENCES 1. Böttcher, M., & Dermer, C. D. 2002, ApJ, 564, 86 2. Cavaliere, A., & D’Elia, V. 2002, ApJ, 571, 226 3. Elvis, M., Plummer, D., Schachter, J., & Fabbiano, G. 1992, ApJS, 80, 257 4. Foschini, L., et al. 2007, ApJ, 657, L81 5. Fossati, G., Maraschi, L., Celotti, A., Comastri, A., & Ghisellini, G. 1998, MNRAS, 299, 433 6. Ghisellini, G., Celotti, A., Fossati, G., Maraschi, L., & Comastri, A. 1998, MNRAS, 301, 451 7. Giommi, P., et al. 2006, A&A, 456, 911 8. Giommi, P., Piranomonte, S., Perri, M., & Padovani, P. 2005, A&A, 434, 385 9. Kuehr, H., Witzel, A., Pauliny-Toth, I. I. K., & Nauber, U. 1981, A&AS, 45, 367 10. Maraschi, L., & Tavecchio, F. 2003, ApJ, 593, 667 11. Maraschi, L. 2001, AIP Conf. Proc. 586: 20th Texas Symposium on relativistic astrophysics, 586, 409 12. Maraschi, L., & Tavecchio, F. 2001, ASP Conf. Ser. 234: X-ray Astronomy 2000, 234, 437 13. Padovani, P., in "The Multi-messenger approach to high energy gamma-ray sources", 2007, in press (astro-ph/0610545) 14. Padovani, P., Costamante, L., Ghisellini, G., Giommi, P., & Perlman, E. 2002, ApJ, 581, 895 15. Pian, E., et al. 2006, A&A, 449, L21 16. Sambruna, R. M., et al. 2006, ApJ, 646, 23 17. Tavecchio, F., et al. 2007, ApJ, in press (astro-ph/0703359) 18. Wall, J. V., & Peacock, J. A. 1985, MNRAS, 216, 173 19. Wolter, A., & Celotti, A. 2001, A&A, 371, 527 http://arxiv.org/abs/astro-ph/0610545 http://arxiv.org/abs/astro-ph/0703359 Introduction New data / new sources Conclusions

0704.0345

A High Robustness and Low Cost Model for Cascading Failures

epl draft A High Robustness and Low Cost Model for Cascading Failures Bing Wang and Beom Jun Kim Department of Physics, BK21 Physics Research Division, and Institute of Basic Science, Sungkyunkwan University, Suwon 440-746, Korea PACS 89.75.Hc – Networks and genealogical trees PACS 05.10.-a – Computational methods in statistical physics and nonlinear dynamics PACS 89.20.Hh – World Wide Web, Internet PACS 89.75.Fb – Structures and organization in complex systems Abstract. - We study numerically the cascading failure problem by using artificially created scale-free networks and the real network structure of the power grid. The capacity for a vertex is assigned as a monotonically increasing function of the load (or the betweenness centrality). Through the use of a simple functional form with two free parameters, revealed is that it is indeed possible to make networks more robust while spending less cost. We suggest that our method to prevent cascade by protecting less vertices is particularly important for the design of more robust real-world networks to cascading failures. The network robustness has been one of the most central topics in the complex network research [1]. In scale-free networks, the existence of hub vertices with high degrees has been shown to yield fragility to intentional attacks, while at the same time the network becomes robust to random failures due to the heterogeneous degree distribu- tion [2–5]. On the other hand, for the description of dy- namic processes on top of networks, it has been suggested that the information flow across the network is one of the key issues, which can be captured well by the betweenness centrality or the load [6]. Cascading failures can happen in many infrastructure networks, including the electrical power grid, Internet, road systems, and so on. At each vertex of the power grid, the electric power is either produced or transferred to other vertices, and it is possible that from some reasons a vertex is overloaded beyond the given capacity, which is the maximum electric power the vertex can handle. The breakdown of the heavily loaded single vertex will cause the redistribution of loads over the remaining vertices, which can trigger breakdowns of newly overloaded ver- tices. This process will go on until all the loads of the remaining vertices are below their capacities. For some real networks, the breakdown of a single vertex is suffi- cient to collapse the entire system, which is exactly what happened on August 14, 2003 when an initial minor distur- bance in Ohio triggered the largest blackout in the history of United States in which millions of people suffered with- out electricity for as long as 15 hours [7]. A number of as- pects of cascading failures in complex networks have been discussed in the literature [8–16], including the model for describing cascade phenomena [8], the control and defense strategy against cascading failures [9, 10], the analytical calculation of capacity parameter [11], and the modelling of the real-world data [12]. In a recent paper [16], the cas- cade process in scale-free networks with community struc- ture has been investigated, and it has been found that a smaller modularity is easier to trigger cascade, which implies the importance of the modularity and community structure in cascading failures. In the research of the cascading failures, the following two issues are closely related to each other and of signif- icant interests: One is how to improve the network ro- bustness to cascading failures, and the other particularly important issue is how to design manmade networks with a less cost. In most circumstances, a high robustness and a low cost are difficult to achieve simultaneously. For exam- ple, while a network with more edges are more robust to failures, in practice, the number of edges is often limited by the cost to construct them. In brevity, it costs much to build a robust network. Very recently, Schäfer et. al. pro- posed a new proactive measure to increase the robustness of heterogeneous loaded networks to cascades. By defin- ing the load dependent weights, the network turns to be more homogeneous and the total load is decreased, which means the investment cost is also reduced [15]. In the present Letter, for simplicity, we try to find a possible way of protecting networks based on the flow along shortest- http://arxiv.org/abs/0704.0345v1 B. Wang B.J. Kim l/lmax This work ML model in Ref.[8] Fig. 1: The capacity c is assigned as c = λ(l)l with the initial load l. The step function λ(l) = 1 + αΘ(l/lmax − β) with two free parameters α and β is used in our model. For comparison, the curve for the Motter-Lai (ML) capacity model in Ref. [8], where λ(l) = constant, is also shown. hop path, first proposed by Motter-Lai [8]. Through the use of our improved capacity model, we numerically exam- ine the cascades in scale-free networks and the electrical power grid network. Since for heterogeneously loaded net- works, overload avalanches can be triggered by the failure of only one of the most loaded vertices, the following re- sults are all based on the removal of one vertex with the highest load. Our results suggest that networks can indeed be made more robust while spending less cost. We first construct the Barabási-Albert (BA) scale-free network [17] of the size N = 5000 with the average degree 〈k〉 ≈ 4 to study the cascading failures. The BA network is characterized by the degree distribution p(k) ∼ k−γ with the degree exponent γ = 3, and it has been shown that the load distribution also exhibits the power-law behavior [6], which means that there exist a few vertices with very large loads. The betweenness centrality for each vertex, defined as the total number of shortest paths passing through it, is used as the measure of the load and computed by using the efficient algorithm [18]. The capacity cv for the vertex v is assigned as cv = λ(lv)lv, (1) where lv is the initial load without failed vertices. Al- though it should be possible to find, via a kind of the variational approach, the optimal functional form of λ(lv) which gives rise to the lower cost and the higher robust- ness (see below for the definitions of the two) we in this work simplify λ(lv) as shown in Fig. 1: λ(lv) = 1 + αΘ(lv/lmax − β), (2) where Θ(x) = 0(1) for x < 0(> 0) is the Heaviside step function, lmax = maxv lv, and we use α ∈ [0,∞) and β ∈ [0, 1] as two control parameters in the model. In Ref. [8] a constant λ has been used (see Fig. 1 for comparison), which corresponds to the limiting case of β = 0 with the identification λ = 1 + α in our model. At the initial time t = 0, the vertex with the highest load is removed from the network, and then new loads for all other vertices are recomputed.1 We then check the failure condition cv < lv(t) for each vertex, and remove all overloaded vertices to get the network at t + 1. The above process continues until all existing vertices fulfill the condition cv > lv(t), and the size of the giant component N ′ at the final stage is measured. The relative size of the cascading failures is conveniently captured by the ratio [8] , (3) which we call the robustness from now on. For networks of homogeneous load distributions, the cascade does not happen and g ≈ 1 has been observed [8]. Also for net- works of scale-free load distributions, one can have g ≈ 1 if randomly chosen vertices, instead of vertices with high loads, are destroyed at the initial stage [8]. In general, one can split, at least conceptually, the to- tal cost for the networks into two different types: On the one hand, there should be the initial construction cost to build a network structure, which may include e.g., the cost for the power transmission lines in power grids, and the cost proportional to the length of road in road networks. Another type of the cost is required to make the given network functioning, which can be an increasing function of the amount of flow and can be named as the running cost. For example, we need to spend more to have big- ger memory sizes and faster network card and so on for the computer server which delivers more data packets. In the present Letter, we assume that the network structure is given, (accordingly the construction cost is fixed), and focus only on the running cost which should be spent in addition to the initial construction cost. Without consideration of the cost to protect vertices, the cascading failure can be made never to happen by assigning extremely high values to capacities. However, in practice, the capacity is severely limited by cost. We expect the cost to protect the vertex v should be an in- creasing function of cv, and for convenience define the cost λ(lv)− 1 /N. (4) It is to be noted that for a given value of α, the original Motter-Lai (ML) capacity model in Ref. [8] has always a higher value of the cost than our model (see Fig. 1). Al- though e = 0 at β = 1, it should not be interpreted as a costfree situation; we have defined e only as a relative measure in comparison to the case of λ(l) = 1 for all ver- tices. For a given network structure, the key quantities to be measured are g(α, β) and e(α, β), and we aim to in- crease g and decrease e, which will eventually provide us 1In real situations of failures, the initial breakdown can happen at any vertex in the network. However, the eventual scale of dam- ages must be greater when a heavily loaded vertex is broken, and accordingly we in this work restrict ourselves to the worst case when the vertex with the highest load is initially broken. A High Robustness and Low Cost Model for Cascading Failures 0.002 0.003 0.004 α =1.00 =0.30 =0.25 =0.20 =0.15 =0.10 0.002 0.003 0.004 (b) α =0.30 =0.25 =0.20 =0.15 =0.10 0 0.2 0.4 0.6 0.8 1 α =0.30 =0.25 =0.20 =0.15 =0.10 Fig. 2: Cascading failures in the BA network of the size N = 5000 and the average degree 〈k〉 ≈ 4, triggered by the removal of a single vertex with the highest load. The robustness g and the cost e in Eqs. (3) and (4) are shown in (a) and (b), respectively, as functions of β at various α values [see Fig. 1 for α and β, the two parameters in the function λ(l) in Eq. (2)]. (c) The relation between e and g at different α’s. Compared with the ML model in Ref. [8], it is clearly shown that the network can be made more robust but with less cost. a way to achieve the high robustness and the low cost at the same time. In Fig. 2(a), we report the robustness g for the BA net- work of the size N = 5000 with the average degree 〈k〉 ≈ 4 as a function of β at α = 0.10, 0.15, 0.20, 0.25, 0.30, and 1.0 (from bottom to top). As β increases further beyond the region in Fig. 2(a), the robustness g is found to de- crease toward zero (not shown here), which is as expected since the larger β makes vertices with larger loads less pro- tected (see Fig. 1). We also skip in Fig. 2 small values of β below approximately 0.001: If β < lmin/lmax, with the minimum load lmin, all vertices are given λ(l) = 1 + α, equivalent to the ML model corresponding to β = 0. It is shown in Fig. 2(a) that for α . 0.30, g first increases and then decreases as β is increased, exhibiting a well- developed maximum gmax at β = β ∗. This is a partic- ularly interesting observation since the network becomes more robust (larger g) by protecting less vertices (larger β). In more detail, the curve for α = 0.20 in Fig. 2(a) shows the maximum gmax ≈ 0.62 (at β ∗ ≈ 0.00133), which is about 3.5 times bigger than g ≈ 0.175 (at β = 0). In other words, the network can be made much more robust by assigning smaller capacities to vertices with less loads. For larger values of α, on the other hand, it is found that gmax occurs at β = 0, which indicates that the above find- ing, i.e., possibility of making network more robust by protecting less vertices, does not hold, as exemplified by the curve for α = 1 in Fig. 2(a). The above observation is closely related with Ref. [9], where it has been found that in order to reduce the size of cascades (or to have a larger g), some of less loaded vertices should be removed just after the initial attack. In reality, however, we believe that the direct application of this strategy of intentional breakdowns is not easy, for cas- cading failures usually propagate across the whole network very soon just after the initial breakdown. In contrast, we propose in this work a way to make the network better prepared to breakdowns, by protecting less vertices. In order to look at the cost benefit of protecting less vertices in a more careful way, we plot in Fig. 2(b) the cost e in Eq. (4) versus β at various values of α. As is expected from Fig. 1, the cost e is shown to be a mono- tonically decreasing (increasing) function of β (α) at fixed α (β). Take again the case with α = 0.20 as an exam- ple with e(β∗) ≈ 0.153 and e(β = 0) = 0.2: It is then concluded that for α = 0.2 one can make the network 3.5 (≈ 0.62/0.175) times more robust while spending only 76.5% (≈ 0.153/0.2) of the original cost. In Fig. 2(c), we use the same data as in Fig. 2(a) and (b), and show the relation between the robustness and the cost for α = 0.10, · · · , 0.30 from bottom to top. For comparison, the values (g,e) for β = 0, corresponding to the ML model, are also displayed as symbols at the end of curves. It is clearly shown that for a given α, one can achieve the higher robustness and the lower cost by tuning β toward the right-most point on each curve. We can also use Fig. 2(c) to choose the most efficient way to get a given robustness g: For example, suppose that g = 0.6 is the required robustness. The vertical line for g = 0.6 crosses several different curves, and one can choose the crossing point which has the lowest cost. We next study the cascading failures in the real net- work structure of the North American power grid of the size N = 4941 [19]. Although the electrical power grid network is a very homogeneous network in terms of the degree distribution, the load distribution, in a sharp con- trast, shows a strong heterogeneity as shown in Fig. 3. In other words, the degree distribution is more like an ex- ponential one, while the load distribution is similar to the power-law form. The broad load distribution can be one of the reasons of the fragility of the power grid to cascading failures [8]. We then apply, the same method as we used above, to the power grid, and obtain g and e as functions of β for B. Wang B.J. Kim 104 105 106 0 5 10 15 20 Fig. 3: The cumulative load distribution of power grid network P (l) in log-log scale. The inset shows the cumulative degree distribution P (k) of the power grid in linear-log scale. 0 0.01 0.02 0.03 0.04 0.05 α =1.0 =0.8 =0.4 =0.2 =0.1 0 0.01 0.02 0.03 0.04 0.05 α =1.0 =0.8 =0.4 =0.2 =0.1 0 0.2 0.4 0.6 0.8 α =1.0 =0.8 =0.4 =0.2 =0.1 Fig. 4: Cascading failures in the electrical power grid of the size N = 4941. (Compare with Fig. 2 for the corresponding plots for the BA network.) The robustness g and the cost e versus β at various α values are shown in (a) and (b), respectively, while (c) is for the relation between e and g. Again, it is shown that one can achieve the higher robustness and the less cost simultaneously, by choosing the right-most point in (c). 0.001 0.002 0.003 0.004 =0.0 =0.2 =0.4 =0.6 =0.8 =1.0 Fig. 5: Cascading failures in the BA network of the size N = 5000 and the average degree 〈k〉 ≈ 4, triggered by the removal of a single vertex with the highest load. Each vertex’s capacity is disturbed with probability ε for α = 0.2. The data are averaged over 20 runs. given values of α. Figure 4 for the cascading failures of the power grid is in parallel to Fig. 2 for the BA network: Fig. 4(a) for g versus β, (b) for e versus β, and (c) for e versus g. There are some quantitative differences between curves for the power grid and the BA network. However, qualitatively speaking, both networks are shown to ex- hibit the following common features: (i) For a given α, the robustness has a maximum gmax at β = β ∗, (ii) e is a monotonically decreasing function of β at a given α, and (iii) there exists a lob-like structure in the g-e plane, which indicates that one can make the network exhibit a higher robustness and a lower cost at the same time than the cor- responding values for the ML model. It is worth mention- ing that the power grid in Fig. 4 can be made to show the higher g and the lower e than the ML model in a broader region of α: Even at α = 1, the power grid can have much better robustness and much less cost in comparison to the ML model. Specifically, at α = 1.0 the ML model has g ≈ 0.40 and e = 1.0 while our model can yield g ≈ 0.73 and e ≈ 0.26 (at β ≈ 0.00583) [see Fig. 4(c)], which occurs when only 26% of vertices are given the higher capacity λ(l) = 2, and the other remaining 74% of vertices have the lower capacity λ(l) = 1. In other words, by assigning lower capacities to 74% of vertices, the network becomes much more robust. In reality, it is also interesting to observe the effect of noise on the dynamical process. In Ref. [20], when noise is introduced into the nonlinear dynamical system, it has been shown that noise changes the singularity at a special time to a statistical time distribution and shows various in- teresting behaviors. In the present work, we are interested in how the presence of noise influences the final cascading failure behavior within our scheme. Here, we introduce A High Robustness and Low Cost Model for Cascading Failures effects of noise as an erroneous assignment of the capac- ity function. In detail, at a given error probability ε, the vertex v is assigned the capacity c′v instead of its correct c′v = cv(1 + r), (5) where r is the uniform random variable with zero mean (r ∈ [−1, 1]). We believe that this erroneous behavior is plausible in reality, since the perfect knowledge for the true value of the load for each vertex may not be available, which may cause an erroneous assignment of the capacity on a vertex. In the limiting case of ε = 0, we recover our error-free results presented above. In Fig. 5, we report the results at α = 0.2 for the robustness g for the BA network as a function of β for different error probability ε [see Fig.2(a) for comparison]. It is seen that for small ε, the overall behavior is qualitatively the same as in Fig. 2(a), i.e., the existence of a well-developed robustness peak and gradual decrease as β is increased. The peak height of the robustness is found to decrease as ε is increased, indicating the negative effect of the noise. An interesting observation in Fig. 5 is that as ε becomes larger there exits a region of β in which the robustness is actually higher than the error-free case of ε = 0. In summary, we have suggested a new capacity model to cascading failures, by improving the existing ML capacity model in Ref. [8]. The main idea in our model is the same as in existing studies: In a highly heterogeneous network with a broad load distribution, vertices with large loads should be more protected by assigning large capacities. Different from other studies in which the capacity is as- signed in proportion to the load, i.e., c = λl, we generalize the model so that the proportionality constant λ is now changed to an increasing function λ(l) of l. In more detail, we use the Heaviside step function for λ(l) characterized by two parameters, the step height α, and the step posi- tion β. By applying this capacity model to the artificial BA network as well as the real network of the power grid, we have clearly shown that it is indeed possible to make the network more robust, while at the same time the cost to assign capacities is drastically reduced. We believe that our suggested model to assign capacities to vertices should be practically useful in designing infrastructure networks in an economic point of view. As a final remark, it needs to be pointed out that the model proposed in this work should be considered as only the first step to find the op- timal functional form λ(l) of the capacity as a function of the load. As a future work, we are planning to apply a sort of variational method to find the optimal functional form of λ(l). B.J.K. was supported by grant No. R01-2005-000- 10199-0 from the Basic Research Program of the Korea Science and Engineering Foundation. REFERENCES [1] Pastor-Satorras R. and Vespignani A., Evolution and Structure of the Internet: A Statistical Physics Ap- proach (Cambridge University Press, Cambridge, Eng- land, 2004); Albert R. and Barabási A.-L., Rev. Mod. Phys., 74 (2002) 47; Dorogovtsev S.N. and Mendes J.F.F., Adv. Phys., 51 (2002) 1079; Newman M.E.J., SIAM Rev., 45 (2003) 167. [2] Albert R., Jeong H. and Barabási A.-L., Nature, 406 (2000) 378. [3] Cohen R., Erez K., ben-Avraham D. and Havlin S., Phys. Rev. Lett., 85, (2000) 4626; ibid. 86 (2001) 3682. [4] Holme P., Kim B.J., Yoon C.N. and Han S.K., Phys. Rev. E, 65 (2002) 056109. [5] Wang B., Tang H.W., Guo C.H. and Xiu Z.L., Physica A, 363, (2006) 591; Wang B., Tang H.W., Guo C.H., Xiu Z.L. and Zhou T., ibid. 368 (2006) 607. [6] Goh K.I., Kahng B. and Kim D., Phys. Rev. Lett., 87 (2001) 278701. [7] U.S.-Canada Power System Outage Task Force, Final Report on the August 14th blackout in the United States and Canada: Causes and Recommendations (United States Department of Energy and National Resources Canada, April 2004.) [8] Motter A.E. and Lai Y.C., Phys. Rev. E, 66 (2002) 065102(R). [9] Motter A.E., Phys. Rev. Lett., 93 (2004) 098701. [10] Hayashi Y. and Miyazaki T., cond-mat/0503615. [11] Zhao L., Park K. and Lai Y.C., Phys. Rev. E, 70, (2004) 035101(R); Zhao L., Park K., Lai Y.C. and Ye N., ibid. 72 (2005) 025104. [12] Kinney R., Crucitti P., Albert R. and Latora V., Eur. Phys. J. B, 46 (2005) 101. [13] Crucitti P., Latora V. and Marchiori M., Phys. Rev. E, 69 (2004) 045104(R). [14] Holme P. and Kim B.J., Phys. Rev. E, 65 (2002) 066109. [15] Schäfer M., Scholz J. and Greiner M., Phys. Rev. Lett., 96 (2006) 108701. [16] Wu J. J, Gao Z.Y. and Sun H. J., Phys. Rev. E, 74, (2006) 066111. [17] Barabási A.-L. and Albert R., Science, 286 (1999) 509. [18] Newman M.E.J., Phys. Rev. E, 64, (2001) 016132; Pro. Natl. Acad. Sci., U.S.A. 98 (2001) 404. [19] Watts D.J. and Strogatz S.H., Nature, 393 (1998) 440. [20] Fogedby H. C., Poutkaradze V., Phys. Rev. E, 66, (2002) 021103. http://arxiv.org/abs/cond-mat/0503615

0704.0346

Diffuse X-ray Emission from the Carina Nebula Observed with Suzaku

Diffuse X-ray Emission from the Carina Nebula Observed with Suzaku Kenji Hamaguchi1,2, the Suzaku η Carinae team and the Carinae D-1 team 1CRESST and X-ray Astrophysics Laboratory NASA/GSFC, Greenbelt, MD 20771 2Universities Space Research Association, 10211 Wincopin Circle, Suite 500, Columbia, MD 21044 A number of giant HII regions are associated with soft diffuse X-ray emission. Among these, the Carina nebula possesses the brightest soft diffuse emission. The required plasma temperature and thermal energy can be produced by collisions or termination of fast winds from main-sequence or embedded young O stars, but the extended emission is often observed from regions apart from massive stellar clusters. The origin of the X-ray emission is unknown. The XIS CCD camera onboard Suzaku has the best spectral resolution for extended soft sources so far, and is therefore capable of measuring key emission lines in the soft band. Suzaku observed the core and the eastern side of the Carina nebula (Car-D1) in 2005 Aug and 2006 June, respectively. Spectra of the south part of the core and Car-D1 similarly showed strong L-shell lines of iron ions and K-shell lines of silicon ions, while in the north of the core these lines were much weaker. Fitting the spectra with an absorbed thin-thermal plasma model showed kT∼0.2, 0.6 keV and NH∼1−2×10 21 cm−2 with a factor of 2-3 abundance variation in oxygen, magnesium, silicon and iron. The plasma might originate from an old supernova, or a super shell of multiple supernovae. §1. Extended X-ray Emission from the Star Forming Region Soft X-ray emission nebulae with kT∼0.1–0.8 keV, log LX∼33-35 ergs s −1, and size of ∼1–103 pc accompany a number of giant HII regions (see Table 4 of Ref. 6). Chandra observations of extended emission in a few star forming clusters indicate that the emission may arise from the fast O star stellar winds thermalized either by wind-wind collisions or by a termination shock. However, the emission is often found outside of the massive stellar clusters, so that another origin, such as an otherwise unrecognized supernova remnant, cannot be ruled out. In principle, the origin of the diffuse emission can be determined by measuring its composition. For example, the plasma should be overabundant in nitrogen and neon if it originates from winds from nitrogen-rich Wolf-Rayet stars (WN), while it would be overabundant in oxygen if it arises from a Type II SNR. The temperature of the plasma, typically a few million degrees, makes soft X-ray band studies highly desirable, because of the presence in this band of strong lines from these elements, plus carbon, silicon and iron. The Carina Nebula, which contains several evolved and main-sequence massive stars such as η Car, WR 25 and massive stellar clusters such as Trumpler 14 (Tr 14), emits soft diffuse X-rays 10–100 times stronger than any other Galactic giant HII region (LX ∼10 35 ergs s−1).4) The high surface brightness made possible the discovery of the diffuse emission by the Einstein Observatory in the late 1970’s. The Einstein observations revealed that the diffuse emission tends to be associated typeset using PTPTEX.cls 〈Ver.0.9〉 http://arxiv.org/abs/0704.0346v1 2 K. Hamaguchi et al. with optically bright regions containing massive stars. Recent Chandra observations provided a point source free measurement of the diffuse flux,1) and suggested the presence of a north-south Fe and Ne abundance gradient.5) The X-ray CCD cameras (XISs: X-ray Imaging Spectrometer) onboard the Suzaku observatory have the best spectral resolution for extended soft X-ray emission and thus they provide good diagnostics of emission lines especially below ∼1 keV. §2. Suzaku and XMM-Newton Observations of the Carina Nebula Figure 1 shows a mosaic image of the Carina nebula between 0.4−7 keV created from 32 XMM-Newton observations. The image depicts several bright X-ray point sources: η Car (an LBV), WR25, WR22 (Wolf-Rayet stars), HD 93250, HD 93043 (O3 stars), and Tr 14, Tr 16 (massive stellar clusters). The image also clearly shows apparently extended emission toward the east-west direction. In a color image (e.g. Figure 1 of Ref. 2), XMM-Newton Image Gallery∗)) the emission is softer between Tr 14, WR 25 and η Car. We analyzed the Suzaku data of the core and the eastern side (named Car-D1) of the Carina nebula taken on 2005 Aug. 29 and 2006 June 5. The XIS FOVs of these observations are shown in Figure 1 with dotted lines. To investigate the color variation in detail, we divided the core region into two and thus extracted three spectra from two Suzaku observations (core-north, core-south and Car-D1). The background was reproduced with the night earth data. The spectra showed strong emission between 0.3 and 2 keV, which is probably dominated by soft diffuse emission associated with the Carina nebula, while the spectra above 2 keV may be explained with CXB, Galactic Ridge X-ray Emission, X-ray point sources resolved with Chandra and unresolved pre-main-sequence stars. Figure 2 shows an overlay of the BI spectra between 0.3–2 keV. The left panel compares spectra of the core-north region with the core-south region. A strong differ- ence is seen between 0.7 keV and 1.2 keV, which apparently is the source of the two colors of diffuse emission. The band in which the difference is found is dominated by emission lines from the iron L-shell complex. Additionally, the core-south spectrum shows a stronger Si line. The Car-D1 spectrum shows similar intensity in the Si and Fe lines to the core-south spectrum (right panel of Figure 2) while it shows relatively strong magnesium and oxygen lines. All these spectra look similar except for these emission lines. This suggests that the differences represent an elemental abundance variation, and not a temperature difference. This is supported by spectral fits of the individual spectra. All three spectra between 0.3−2 keV were reproduced by an absorbed 2T thin-thermal plasma models although the best-fit models are not formally acceptable. The plasma tempera- tures of all three regions are ∼0.2 and ∼0.6 keV, and their column densities are ∼3×1021 cm−2, which is consistent with extinction toward the Carina nebula.3) The abundances of some elements show a factor of 2−4 variations: the core-north region has a factor of 2 lower silicon abundance and a factor of 4 lower iron abundance ∗) http://xmm.esac.esa.int/external/xmm science/gallery/public Diffuse X-rays from the Carina nebula 3 Fig. 1. Mosaic image (∼90′×60′) of the Carina nebula between 0.4−7 keV created from 32 XMM- Newton observations. The image is created with the ESAS package, divided by the exposure map and smoothed with the adaptive smoothing technique. The dotted lines show the XIS FOVs of the Suzaku observations of η Car (right) and the Car-D1 field (left). The solid lines show source extraction regions for the spectral analysis. than the core-south region, while the Car-D1 region has a factor of 2 higher oxygen and magnesium abundances. On the other hand, spectral fits of the core region with higher sensitivity around 0.5 keV gave small upper-limits (.0.02 solar) of the nitrogen abundance. §3. Origin of the Diffuse Plasma The N/O abundance ratio inferred from the spectral fits is .0.4, over 20 times less than around η Car. The abundance distribution is totally contrary to that expected from stellar winds from evolved massive stars, unless the winds somehow heat the interstellar matter without enriching it, thus leaving the X-ray plasma with abundances typical of interstellar matter. At the same time, the X-ray luminosity of the Carina Nebula is about two orders of magnitude higher than that of other Galactic star forming regions, but the number of early O stars is only an order of magnitude higher (see Table 4 in Ref. 6). These results suggest an additional energy source is needed to power the X-ray emission in the Carina Nebula. An obvious possibility is one or more core-collapse supernovae (i.e. Type Ib,c or II), mentioned as a possibility by Ref. 6). The regions vary strongly in oxygen, magnesium, silicon, and iron abundances. These elements are products of core- collapse supernovae, and young SNRs such as Cas A and Vela show strong abundance 4 K. Hamaguchi et al. Fe L complex Fe L complex Fig. 2. Comparison of the XIS1 spectra between the fields – left: the core-north region (black) and the core-south region (grey), right: the Car-D1 field (black) and the core-south region (grey). The above labels demonstrate energies of emission lines detected (black) or concerned (grey) with this result. Emission lines with the solid lines showed variation in their line intensity. Low count rates of the Car-D1 spectrum below 1 keV is caused by degradation of soft response by progressive contamination on the XIS. variation from location to location. The total energy content in the hot gas of ∼2×1050 ergs is a modest fraction of the ∼1051 ergs of kinetic energy produced by a canonical supernova, while assuming an iron abundance of 0.30 solar, the total iron mass in the diffuse gas requires at least 3-5 supernovae. Acknowledgements K. H. is financially supported by a US Chandra grant No. GO3-4008A and US Suzaku grant. References 1) N. R. Evans, F. D. Seward, M. I. Krauss, T. Isobe, J. Nichols, E. M. Schlegel, and S. J. Wolk, Astrophysical Journal 2003 (589), 509 2) K. Hamaguchi, R. Petre, H. Matsumoto, M. Tsujimoto, S. S. Holt, Y. Ezoe, H. Ozawa, Y. Tsuboi, Y. Soong, S. Kitamoto, A. Sekiguchi, and M. Kokubun. Publication of Astro- nomical Society of Japan 2007 (59), 151 3) M. A. Leutenegger, S. M. Kahn, and G. Ramsay. Astrophysical Journal 2003 (585), 1015 4) F. D. Seward and T. Chlebowski. Astrophysical Journal 1982 (256), 530 5) L. K. Townsley. Proceeding of the STScI May Symposium, ”Massive Stars: From Pop III and GRBs to the Milky Way, 2006, (astro–ph/0608173) 6) L. K. Townsley, E. D. Feigelson, T. Montmerle, P. S. Broos, Y.-H. Chu, and G. P. Garmire. Astrophysical Journal 2003 (593), 874 http://arxiv.org/abs/astro--ph/0608173 Extended X-ray Emission from the Star Forming Region Suzaku and XMM-Newton Observations of the Carina Nebula Origin of the Diffuse Plasma

0704.0349

The Colin de Verdi\`ere number and graphs of polytopes

The Colin de Verdière number and graphs of polytopes Ivan Izmestiev ∗ Institut für Mathematik Technische Universität Berlin Str. des 17. Juni 136 10623 Berlin, Germany [email protected] July 25, 2008 Abstract The Colin de Verdière number µ(G) of a graph G is the maximum corank of a Colin de Verdière matrix for G (that is, of a Schrödinger operator on G with a single negative eigenvalue). In 2001, Lovász gave a construction that associated to every convex 3-polytope a Colin de Verdière matrix of corank 3 for its 1-skeleton. We generalize the Lovász construction to higher dimensions by in- terpreting it as minus the Hessian matrix of the volume of the polar dual. As a corollary, µ(G) ≥ d if G is the 1-skeleton of a convex d-polytope. Determination of the signature of the Hessian of the volume is based on the second Minkowski inequality for mixed volumes and on Bol’s condition for equality. 1 Introduction 1.1 The Colin de Verdière number At the end of 80’s, Yves Colin de Verdière introduced a graph parameter µ(G) based on spectral properties of certain matrices associated with the graph G. Definition 1.1 Let G be a graph with n vertices. A Colin de Verdière matrix for G is a symmetric n × n matrix M = (Mij) with the following properties. Research for this article was supported by the DFG Research Unit 565 “Polyhedral Surfaces”. http://arxiv.org/abs/0704.0349v3 (M1) M is a Schrödinger operator on G, that is < 0, if ij is an edge of G; = 0, if ij is not an edge of G and i 6= j. (M2) M has exactly one negative eigenvalue, and this eigenvalue is simple. (M3) If X is a symmetric n × n matrix such that MX = 0 and Xij = 0 whenever i = j or ij is an edge of G, then X = 0. The set of all Colin de Verdière matrices for graph G is denoted by MG. The Colin de Verdière number µ(G) is defined as the maximum corank of matrices from MG: µ(G) := max dimkerM. A Colin de Verdière matrix of maximum corank is called optimal. Basically, the Colin de Verdière number is the maximum multiplicity of the second least eigenvalue λ2 of a discrete Schrödinger operator M satis- fying a certain stability assumption (M3). By replacing M with M − λ2Id, we can make the second eigenvalue zero (M2), so that multiplicity be- comes corank. Definition 1.1 was motivated by the study of Schrödinger and Laplace operators associated with degenerating families of Riemannian metrics on surfaces. The parameter µ(G) turned out to be interesting on its own. In partic- ular, it posesses the minor monotonicity property: if a graph H is a minor of G, then µ(H) ≤ µ(G). By the Robertson-Seymour theorem this implies that graphs with µ(G) ≤ n can be characterized by a finite set of forbidden minors. For n up to four such characterizations are known and allow nice topological reformulations: e. g. µ(G) ≤ 3 iff G is planar (that is doesn’t have K5 or K3,3 as minors), and µ(G) ≤ 4 iff G is linklessly embeddable in 3 (that is doesn’t have any graph of the Petersen family as a minor). An overview of results and open problems on the Colin de Verdière number can be found in [4], [14], and [5]. The book [4] deals also with other spectral invariants arising from discrete Schrödinger and Laplace operators. 1.2 Nullspace representations and Steinitz representations Let M be a Colin de Verdière matrix for graph G with dimkerM = d. Choose a basis (u1, . . . , ud) for kerM ⊂ R n, fix a coordinate system in Rn, and read off the coordinates of (uα): (u1, . . . , ud) = (v1, . . . , vn) The map that associates to every vertex i of G the vector vi ∈ R d is called a nullspace representation of the graph G. Nullspace representations were studied in [11]. In a subsequent paper [10] Lovász showed that, for a 3-connected planar G, the nullspace repre- sentation with properly scaled vectors (vi) realizes G as the skeleton of a convex 3-polytope. Lovász provided also an inverse construction that as- sociated to every convex 3-polytope with 1-skeleton G a Colin de Verdère matrix of corank 3. The proof that the constructed matrix had an appro- priate signature was indirect, and a more geometric approach was desirable. 1.3 Hessian matrix of the volume as a Colin de Verdière matrix In this paper we relate the Lovász construction (that of a matrix from a poly- tope) to the mixed volumes. Our approach allows a straightforward gener- alization to higher dimensions. That is, we associate to every d-dimensional convex polytope with 1-skeleton G a Colin de Verdière matrix for G of corank d. As a consequence, the graph of a convex d-dimensional polytope has Colin de Verdière number at least d. This result is not really new, since it follows from the minor monotonicity of µ, from the fact that the graph of a d-polytope has Kd+1 as a minor [8], and from µ(Kd+1) = d. Our result is based on the following observation. Take a convex d- polytope P and deform it by shifting every facet parallelly to itself. Then the Hessian matrix of the volume of P , where partial derivatives are taken with respect to the distances of the shifts, has corank d and exactly one positive eigenvalue. Besides, the mixed partial derivative ∂2vol(P ) ∂xi∂xj is positive if the ith and the jth facets are adjacent, and vanishes otherwise. Thus the negative of the Hessian matrix satisfies conditions (M1) and (M2) from Definition 1.1. The condition (M3) follows quite easily, too. The signature of the Hessian of the volume is encoded in the second Minkowski inequality for mixed volumes together with Bol’s characteriza- tion of the case of equality. For simple polytopes, the determination of the signature of the Hessian is an essential part in the proof of the Alexandrov- Fenchel inequality. 1.4 Plan of the paper In Section 2.1 we recall the Lovász construction of a Colin de Verdière matrix for the skeleton of a convex 3-polytope Q. After inroducing some terminology and notation in Section 2.2, we show in Section 2.3 that the Lovász matrix is minus the Hessian matrix of the volume of the polar dual polytope Q∗. In Section 2.4, dealing with 3-polytopes, we point out an interesting identity (first found and used elsewhere [2]) between the Hessian matrix of vol(Q∗) and the Hessian matrix of another geometric quantity associated with Q. This gives another interpretation of the Lovász matrix M and relates the equality dimkerM = 3 with the infinitesimal rigidity of the polytope Q. In Section 3.1 we discuss the (im)possibility of inverting the construction, that is of finding a convex polytope whose Hessian matrix of the volume equals to a given Colin de Verdière matrix. In Section 3.2 we give an estimate of the negative eigenvalue (and thus of the spectral gap) for the Hessian matrices of the volume. Finally, in the Appendix we derive the signature of the Hessian from the second Minkowski inequality and Bol’s condition. Although this seems to be a folklore knowledge in narrow circles, we failed to find a written account on this subject. 1.5 Acknowledgements I am grateful to the organizers of the 2006 Oberwolfach conference “Discrete Differential Geometry”, where the idea of this paper was born. I also thank Ronald Wotzlaw for pointing me out a mistake in a preliminary version. 2 From a convex polytope to a Colin de Verdière matrix 2.1 Lovász construction Let us recall the Lovász construction of an optimal Colin de Verdière matrix associated with a polytopal representation of a graph in R3. Let Q ⊂ R3 be a convex polytope containing the coordinate origin in its interior. Let G be the 1-skeleton of Q. We denote the vertices of G by i, j, . . ., and the corresponding vertices of Q by vi, vj , . . .. Let Q ∗ be the polar dual of Q. The vertices of Q∗ are denoted by wf , wg, . . ., where f, g, . . . are faces of Q. For ij ∈ G, consider the edge vivj of Q and the dual edge wfwg of Q see Figure 1. It is easy to show that the vector wf − wg is orthogonal to both vectors vi and vj, hence parallel to their cross product vi × vj. Thus we have wf − wg = Mij(vi × vj), (1) with Mij < 0 (we agree to choose the labeling of wf and wg so that we get the correct sign). Further, consider the vector v′i = Mijvj, vi × vj Figure 1: To the definition of the matrix M . where the sum extends over all vertices of G adjacent to i. From (1) it is easy to see that vi × v i = 0. Thus there exists a real number Mii such that v′i = −Miivi. (2) Putting Mij = 0 for distinct non-adjacent vertices i and j of G, we complete the construction of the matrix M . Theorem 2.1 (Lovász, [10]) The matrix M is a Colin de Verdière matrix for the graph G. The equation (2) can be rewritten as Mijvj = 0. (3) Thus M has corank at least 3. Since µ(G) ≤ 3 for planar graphs, M is an optimal Colin de Verdière matrix for G. The proof of Theorem 2.1 goes through a deformation argument, using the fact that the space of convex 3-polytopes with a given graph is connected. 2.2 Polytopes with a given set of normals Here we fix some terminology and notation needed in the subsequent sec- tions. All polytopes in this paper are assumed to be convex. A facet of a d-dimensional polytope is a (d− 1)-dimensional face of it. We will study families of polytopes with fixed facet normals. Let v1, . . . , vn be vectors in Rd such that the coordinate origin lies in the interior of their convex hull. Consider a d× n matrix formed by row vectors v⊤i : V = (v1, . . . , vn) Definition 2.2 Denote by P(V ) the set of all convex polytopes with the outer facet normals v1, . . . , vn. Every polytope in P(V ) is the solution set of a system of linear inequal- ities: P (x) = {p ∈ Rd |V p ≤ x}, where x = (xi) i=1 ∈ R n. Denote by Fi(x) the facet of P (x) with the outer normal vi. We have Fi(x) = {p ∈ P (x) | v i p = xi}. The numbers xi are called the support parameters of the polytope P (x). The map P (x) 7→ x embeds P(V ) into Rn as an open convex subset. The support parameter xi is proportional to the signed distance from 0 to the affine hull of the facet Fi(x): xi = ‖vi‖ · hi. By vold we denote the volume of a d-dimensional polytope. We use the subscript because both vold(P ) and vold−1(Fi) will occur in our formulas. We omit the subscript at vol, when it seems reasonable to do so. 2.3 Interpreting and generalizing the Lovász construction By definition of the polar dual, we have Q∗ = {p ∈ R3 | v⊤i p ≤ 1 for all i}. Thus Q∗ can be viewed as an element of the set P(V ) of polytopes with facet normals (vi)i∈G. In terms of Section 2.2, Q ∗ = P (1, . . . , 1). Let’s vary the support parameters of Q∗ and look how does this change its volume. Lemma 2.3 Let M be the matrix constructed in Section 2.1. Then we have Mij = − ∂2vol(P (x)) ∂xi∂xj x=(1,...,1) where P (x) is as in Section 2.2. Proof . Let Fi(x) be the facet of P (x) with the normal vi. It is not hard to show that ∂vol3(P (x)) vol2(Fi(x)) Further, for i 6= j we have ∂vol2(Fi(x)) vol1(Fij(x)) ‖vj‖ sin θij ∆vol1(Fj) ∆vol2(P ) Figure 2: Partial derivatives of the volume with respect to the support parameters. if faces Fi(x) and Fj(x) are adjacent; otherwise this derivative is zero. Here Fij(x) is the common edge of Fi(x) and Fj(x), and θij is the angle between the vectors vi and vj (i. e. the outer dihedral angle at the edge Fij). The equations are illustrated in Figure 2 in one dimension lower and for ‖vi‖ = 1. Thus at x = (1, . . . , 1) we have ∂2vol(P (x)) ∂xi∂xj vol1(Fij(x)) ‖vi‖‖vj‖ sin θij ‖wf − wg‖ ‖vi × vj‖ = −Mij (4) for all i 6= j. To deal with the case i = j, differentiate the well-known identity vol2(Fj(x)) with respect to xi. This gives ∂2vol(P (x)) j 6=i ∂2vol(P (x)) ∂xi∂xj vj = 0. (5) In view of (3) and (4), we have ∂2vol(P (x)) |x=(1,...,1) = −Mii. � Lemma 2.3 suggests the following generalization of the Lovász construc- tion. Theorem 2.4 Let P (x0) = {p ∈ Rn | v⊤i p ≤ x i for all i} be a convex polytope with outer facet normals vi and support parameters x0i , i = 1, . . . , n. Let G be the dual 1-skeleton of P (x 0). Then the matrix M defined by Mij = − ∂2vol(P (x)) ∂xi∂xj is a Colin de Verdiére matrix for the graph G. The corank of M is equal to d. In particular, µ(G) ≥ d for every graph G that can be realized as the 1-skeleton of a d-dimensional polytope. Proof . Similarly to Lemma 2.3, for adjacent facets Fi and Fj we have ∂2vold(P (x)) ∂xi∂xj vold−2(Fij(x)) ‖vi‖‖vj‖ sin θij where Fij is their common (d− 2)-face, and θij is the angle between vi and vj. For non-adjacent Fi and Fj this derivative is zero. Therefore matrix M satisfies property (M1) from Definition 1.1. The proof of property (M2) is the most interesting part of the theo- rem. The signature of the Hessian of the volume is encoded in the second Minkowski inequality for mixed volumes enhanced by Bol’s condition for equality. Theorem A.10 in Section A states in particular that the matrix M has corank d. The kernel of M is easy to identify: due to the equation (5) it consists of the vectors ξ ∈ Rn such that ξi = v i p for some vector p ∈ R Assuming this description of kerM , let us prove that the matrix M satisfies property (M3). If MX = 0, then there are vectors p1, . . . , pn ∈ R such that Xij = v i pj for all i, j. Fix j. Then by assumption on X we have pj ⊥ vj and pj ⊥ vi for all ij ∈ G. But the normal vj to the face Fj and the normals to the neighboring faces span the space Rd. Thus we have pj = 0 for all j, which implies X = 0. As for the last sentence of the theorem, if G is the dual 1-skeleton of a convex polytope P , then G is the skeleton of the polar (P − p)∗, where p is any interior point of P . � 2.4 Case d = 3 and infinitesimal rigidity of convex polytopes In the case d = 3 there is another interpretation of the matrix M . As in Section 2.1, let Q be a convex polytope that has skeleton G and contains 0 in the interior. Triangulate the faces of Q by diagonals and cut Q into pyramids with apices at 0 and triangles of the triangulation as bases. De- note by ri the length of the edge that joins 0 to the vertex vi of Q. Now deform the pyramids by changing the lengths ri and leaving the lengths of boundary edges constant. During such deformation, the dihedral angles of the pyramids change, and the total angle ωi around the i-th edge can be- come different from 2π. By computing the derivatives of ωi explicitly, we obtain ([2], Theorem 3.11) vol1(Fij) sin θij = ‖vi‖‖vj‖ ·Mij, (7) where we use the notations from Section 2.3. If we change the variables xi to hi = ‖vi‖ · xi, so that hi is the distance of 0 from aff (Fi), then the equation (7) takes a particularly nice form ∂vol2(Fi) By (7), the matrix (∂ωi ) is obtained from the matrix M by multiplying the i-th row and the i-th column with ‖vi‖, for all i. This implies Corollary 2.5 The matrix (∂ωi ) is an optimal Colin de Verdière matrix for graph G. The fact that the matrix (∂ωi ) has corank 3 is equivalent to the infinitesi- mal rigidity of the polytope Q. Indeed, every infinitesimal deformation (dri) such that dωi = 0 for all i gives rise to an infinitesimal isometric deformation of Q. The resulting deformation is trivial iff it is produced by moving the apex 0 inside Q. Another interesting fact is that the matrix (∂ωi ) is the Hessian matrix of a geometric quantity related to the polytope Q (deformed by varying ri). Namely, put S(r) = riκi + ℓijθij, where κi = 2π − ωi is the “curvature” along the i-th radial edge, and ℓij = vol1(Fij) is the length of the edge vivj. Then the Schläfli formula implies = κi. Hence ∂2S(Q) ∂ri∂rj ∂2vol(Q∗) ∂hi∂hj and both matrices are equal to the negative of the Lovász matrix M , up to scaling the rows and columns by ‖vi‖. 3 Concluding remarks 3.1 What fails in the inverse construction Let M be a Colin de Verdère matrix for the graph G. Is there a convex polytope P such that M arises from P as a result of the construction de- scribed in Section 2.3? Of course, in general the answer is no, because G must be the dual skeleton of P , and P must have dimension d = dimkerM . In particular, all vertices of G must have degrees at least d. But, due to the minor monotonicity of µ, there exist trivalent graphs with µ(G) arbitrarily large. Nevertheless, it is worth looking at what fails when we try to reconstruct the polytope P from matrix M . Let u1, . . . , ud ∈ R n be a basis of kerM . Let v⊤i be the i-th row in the matrix (u1, . . . , ud). Then we have Mijvj = 0 (8) for all i. Therefore, the vectors v1, . . . , vn ∈ R d are good candidates for the outer normals to the faces of the polytope P . At this point we can already fail, if the following assumptions aren’t fulfilled: 1. vi 6= 0 for all i, and vi 6= vj for all i 6= j; 2. for every i, the projections vij of vj on v i for ij ∈ G satisfy the previous assumption and span v⊥i . We proceed assuming that these conditions hold. Codimension 2 faces Fij of P must be in 1-to-1 correspondence with the edges of G, and their volumes are determined by the matrix M : vold−2(Fij) = Aij := −Mij‖vi‖‖vj‖ sin θij, where θij is the angle between vi and vj . Lemma 3.1 For every i, there exists a convex (d−1)-dimensional polytope Fi ⊂ v i with outer facet normals vij and facet volumes Aij , ij ∈ G. Proof . By projecting the equation (8) on v⊥i , we obtain Mij · vij = 0. (9) Due to ‖vij‖ = ‖vj‖ sin θij, it follows that Aij · ‖vij‖ By Minkowski’s theorem [13, Section 7.1], this implies the existence of a polytope Fi as stated in the lemma. � The polytopes Fi in Lemma 3.1 should become facets of the polytope P . But here is the second point where the reconstruction can fail: the j-th facet Fij of Fi might be different from the i-th facet Fji of Fj ; the only thing we know is vold−2(Fij) = Aij = vold−2(Fji). In the case d = 3, however, this suffices: Fi are convex polygons and fit together along their edges to form a polytope P . Conditions 1. and 2. above hold if we assume that G is a 3-connected planar graph [11]. Thus for 3-connected planar graphs every Colin de Verdière matrix corresponds to a polytope. This is one of the results of [10]. The following example shows that even for highly connected graphs the number µ(G) can be bigger than the maximum dimension of a polytope with 1-skeleton G. Example Let Gn = K2,2,...,2 be the multipartite graph on 2n vertices (the graph of an n-dimensional cross-polytope). By [9], µ(Gn) = 2n − 3 for n ≥ 3. For n = 3, 4 the graph Gn can also be represented as the skeleton of a (2n − 3)-dimensional convex polytope: for n = 3 this is the octahedron, for n = 4 the join of two convex quadrilaterals in general position in R5. For n ≥ 5, however, there is no (2n − 3)-dimensional convex polytope with skeleton Gn. Indeed, by studying the Gale diagram [16, Lecture 6] of a d-polytope with d + 3 vertices, one can show that the complement to the graph of such polytope cannot have more than 4 edges. Note that the equation (9) is reminiscent of the definition of a (d − 2)- weight in [12]. 3.2 Negative eigenvalue Theorem 3.2 Let λ1 be the negative eigenvalue of the matrix (6). Then the following inequality holds: λ1 ≤ −d(d− 1) · vold(P (x ‖x0‖2 The equality takes place iff x0i = c · vold−1(Fi(x for all i and some constant c. Proof . By induction on d, it is easy to show that the function vold(P (x)) is a degree d homogeneous polynomial in x as long as the combinatorics of P (x) does not change. For different combinatorics, the polynomials have different coefficients. However, since vold(P (x)) is twice differentiable, we can apply Euler’s homogeneous function theorem twice at the point x0, independently on how generic the combinatorics of P (x0) is. This yields (x0)⊤Mx0 = −d(d− 1) · vold(P (x Since λ1 = min‖ξ‖=1 ξ ⊤Mξ, the inequality follows. Since λ1 is the unique negative eigenvalue of M , the inequality turns into equality iff Mx0 = λx0 for some λ. We have j = − ∂vold−1(Fi(x x0j = −(d− 1) · vold−1(Fi(x Thus Mx0 = λx0 is equivalent to x0i = c · vold−1(Fi(x , and the theorem is proved. � The number λ2 − λ1 is called the spectral gap. In our case λ2 = 0 by definition. Thus Theorem 3.2 provides an estimate on the spectral gap of the matrix M . Usually, one seeks to make the spectral gap as large as possible, but in order this to make sense for Colin de Verdère matices, one has to choose a matrix norm, [4, Chapter 5.7]. The norm of the matrix (6) is a function of its coefficients, which have a geometric meaning. Thus, as soon as the choice of a matrix norm is made, one can try to solve the problem of the spectral gap by geometric means (at least for 3-connected planar graphs, for which every optimal Colin de Verdière matrix can be realized through a polytope). A The second Minkowski inequality for mixed vol- umes and the signature of the matrix ∂2vol ∂xi∂xj The goal of this appendix is to prove Theorem A.10 that describes the sig- nature of the matrix (6). The theorem is derived from the second Minkowski inequality for mixed volumes and Bol’s condition for equality. The relation between the theory of mixed volumes and infinitesimal rigid- ity (as we know, the rank of matrix (6) accounts for the infinitesimal rigidity of the dual polytope, see Section 2.4) was noticed long ago [1, 15]. In the decades thereafter this phenomenon seemed to be forgotten. Quite recently, Carl Lee and Paul Filliman [7] discovered it again. A.1 The second Minkowski inequality and Bol’s condition Definition A.1 Let P,Q ⊂ Rd be convex bodies. A mixed volume of P and Q is a coefficient in the expansion vol(λP + µQ) = vol(Q, . . . , Q ︸ ︷︷ ︸ , P, . . . , P ︸ ︷︷ ︸ )λd−kµk (10) with λ, µ > 0, where A + B for A,B ⊂ Rd denotes the Minkowski sum. In particular, vol(P, . . . , P ) = vol(P ). In a similar way one defines the mixed volume of more than two convex bodies. It turns out that the mixed volume is polylinear with respect to the Minkowski addition and multiplication with positive scalars. A proof that the expansion (10) takes place and more information on mixed volumes can be found in [6, 13]. Theorem A.2 Let P,Q ⊂ Rd be convex bodies. Then the following holds: 1. (The second Minkowski inequality) vol(Q,P, . . . , P )2 ≥ vol(P ) · vol(Q,Q,P, . . . , P ). (11) 2. (Bol’s condition) Assume that dimQ = d. Then equality holds in (11) if and only if either dimP < d − 1 or P is homothetic to a (d − 2)- tangential body of Q. For a proof see [13, Theorem 6.2.1, Theorem 6.6.18]. Bol’s condition was conjectured by Minkowski but proved only decades later by Bol, [3]. Definition A.3 If P ⊂ Q ⊂ Rd are d-dimensional convex polytopes, then Q is called a p-tangential body of P iff P has a non-empty intersection with every face of Q of dimension at least p. A.2 Mixed volumes as derivatives of the volume By substituting in (10) λ = 1 and µ = t, we obtain vol(P + tQ) = vol(P ) + tdvol(Q,P, . . . , P ) d(d− 1) vol(Q,Q,P, . . . , P ) + · · · (12) for all t > 0, which can be seen as the Taylor expansion of vol. We will look at it in the case when P and Q are polytopes with the same sets of facet normals. The space P(V ) of all polytopes with outer facet normals v1, . . . , vn is defined in Section 2.2. We want to study the partial derivatives of the volume of P (x) ∈ P(V ) with respect to the support parameters x. For brevity, let’s use the notation vol(x) := vol(P (x)). Similarly, the mixed volume of polytopes from P(V ) will be written as a function of the support parameters: vol(x1, . . . , xd) := vol(P (x1), . . . , P (xd)). Now we would like to compute vol(x+ ty) with the help of (12). This is not as straightforward as it seems, because the support parameters behave not quite linearly under the Minkowski addition. We have P (ty) = tP (y) for t > 0. Also we have P (x) + P (y) ⊂ P (x + y), but the equality doesn’t always hold. To describe the cases in which we do have the equality, we need a new definition. Definition A.4 The normal cone N(F,P ) of the face F of a polytope P ⊂ d is the set of vectors w ∈ Rd such that (w⊤x) = max (w⊤x). The normal fan N(P ) is the decomposition of Rd into the normal cones of the faces of P . If the normal fan N(Q) subdivides the normal fan N(P ), then we write N(Q) > N(P ). Note that the normal fan of a polytope P ∈ P(V ) has the rays R+vi as 1-dimensional cones. The higher-dimensional cones of the normal fan deter- mine the combinatorics of P . Therefore polytopes with equal normal fans are sometimes called strongly isomorphic. We denote the normal fans of the polytopes from P(V ) by N(x) := N(P (x)). The following lemma is classical. Lemma A.5 If N(y) > N(x), then P (x) + P (y) = P (x+ y). Now we are ready to prove Lemma A.6 Let y ∈ P(V ) be such that N(y) > N(x). Then ∇yvol(x) = d · vol(y, x, . . . , x), ∇2yvol(x) = d(d− 1) · vol(y, y, x, . . . , x), where ∇y denotes the directional derivative along y. Proof . Due to Lemma A.5 we have P (x + ty) = P (x) + tP (y). By substi- tuting P = P (x) and Q = P (y) in (12), we obtain vol(x+ ty) = vol(x) + tdvol(y, x, . . . , x) d(d− 1) vol(y, y, x, . . . , x) + · · · , which implies the lemma. � Remark. For polytopes with the same normal fan (“strongly isomorphic polytopes”), there is the following description of mixed volumes. Denote P∆(V ) = {P (x) ∈ P(V ) |N(x) = ∆}. By induction on d, it is easy to show that there exists a homogeneous poly- nomial V∆ of degree d in n variables such that vol(P (x)) = V∆(x), for all x ∈ P∆(U). If we use the same symbol V∆ to denote the associated symmetric polylinear form, then we have vol(P (x(1)), . . . , P (x(d))) = V∆(x (1), . . . , x(d)) for all x(1), . . . , x(d) ∈ P∆(V ). A.3 From the second Minkowski inequality to the signature of the Hessian of the volume By geometric arguments similar to those in the proof of Lemma 2.3, the function vol is twice continuously differentiable on P(V ). Therefore the following definition makes sense. Definition A.7 Let x ∈ P(V ). Define a symmetric bilinear form Φ on Rn Φ(ξ, η) = ∇η∇ξvol(x). Let y ∈ P(V ) be such that N(y) > N(x). By combining Euler’s homo- geneous function theorem and Lemma A.6, we obtain Φ(x, x) = d(d− 1) vol(x, . . . , x), Φ(x, y) = d(d− 1) vol(y, x, . . . , x), Φ(y, y) = d(d− 1) vol(y, y, x, . . . , x). Lemma A.8 Let L ⊂ Rn be a 2-dimensional vector subspace such that x ∈ L. Then the restriction of the form Φ to L has signature (+,−) or (+, 0). Proof . Let y ∈ P(V ) be such that N(y) > N(x). The second Minkowski inequality (11) applied to P = P (x) and Q = P (y) can be rewritten as Φ(x, x) Φ(x, y) Φ(x, y) Φ(y, y) ≤ 0. (13) Since, moreover, Φ(x, x) = d(d−1) vol(P ) > 0, it follows that the restriction of Φ to span {x, y} has signature (+, 0) or (+,−). It remains to show that every 2-subspace L ∋ x can be represented as span {x, y} with N(y) > N(x). This is true since x is an interior point of the set {y ∈ P(V ) |N(y) > N(x)}. (When we perturb x, we can create new faces, but cannot destroy old ones.) � Lemma A.9 The form Φ has corank d. Proof . Let us exhibit a d-dimensional subspace of ker Φ. Associate with every point p ∈ Rd a vector p ∈ Rn with coordinates pi = 〈vi, p〉. The polytope P (x+p) is the translate of P (x) by p. Therefore the directional derivative ∇pvol(x) vanishes for all x, which implies Φ(p, η) = 0 for all η. Thus we have p ∈ kerΦ for all p ∈ Rd. Let ξ ∈ ker Φ. We need to show that ξ = p for some p ∈ Rd. Denote the span of x and ξ by L. Then, by Lemma A.8, the restriction Φ|L has signature (+, 0) and hence Rξ = L ∩ ker Φ. (14) Choose y ∈ L such that N(y) > N(x), and x and y are linearly independent. Then the degeneracy of Φ|L means that we have an equality in (13) and thus also in the Minkowski inequality for P = P (x) and Q = P (y). By Bol’s condition, see Theorem A.2, this happens if and only if the polytope P (x) is homothetic to a (d − 2)-tangential body of the polytope P (y). By studying Definition A.3, we see that in P(V ) it is equivalent to P (x) being homothetic to P (y). If P (x) is homothetic to P (y), then x = λy + p for some p ∈ Rd, thus p ∈ L. Since p ∈ kerΦ, it follows that Rp = L ∩ ker Φ. By comparing this to (14), we conclude that ξ = µp = µp for some µ ∈ R. Thus the kernel of Φ is confined to the vectors of the form p. � Theorem A.10 The form Φ has corank d and exactly one positive eigen- value, which is simple. Proof . The corank of Φ is computed in Lemma A.9. The form Φ has at least one positive eigenvector since Φ(x, x) > 0. Assume that it has more than one. Then there exists a 2-subspace of Rn on which Φ is positively definite. The subgroup of GL(Rn) that preserves Φ acts transitively on the cone of positive directions. Thus there is a positive 2-subspace L that passes through x. This contradicts Lemma A.8. Theorem is proved. � References [1] W. Blaschke. Ein Beweis für die Unverbiegbarkeit geschlossener kon- vexer Flächen. Gött. Nachr., pages 607–610, 1912. [2] A. Bobenko and I. Izmestiev. Alexandrov’s theorem, weighted Delau- nay triangulations, and mixed volumes. Ann. Inst. Fourier (Grenoble), 58(2):447–505, 2008. [3] G. Bol. Beweis einer Vermutung von H. Minkowski. Abh. Math. Sem. Hansischen Univ., 15:37–56, 1943. [4] Y. Colin de Verdière. Spectres de graphes, volume 4 of Cours Spécialisés. Société Mathématique de France, Paris, 1998. [5] Y. Colin de Verdière. Sur le spectre des opérateurs de type Schrödinger sur les graphes. In Graphes, pages 25–52. Ed. Éc. Polytech., Palaiseau, 2004. [6] G. Ewald. Combinatorial convexity and algebraic geometry, volume 168 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1996. [7] P. Filliman. Rigidity and the Alexandrov-Fenchel inequality. Monatsh. Math., 113(1):1–22, 1992. [8] B. Grünbaum. Convex polytopes, volume 221 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2003. Pre- pared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler. [9] A. Kotlov, L. Lovász, and S. Vempala. The Colin de Verdière number and sphere representations of a graph. Combinatorica, 17(4):483–521, 1997. [10] L. Lovász. Steinitz representations of polyhedra and the Colin de Verdière number. J. Combin. Theory Ser. B, 82(2):223–236, 2001. [11] L. Lovász and A. Schrijver. On the null space of a Colin de Verdière matrix. Ann. Inst. Fourier (Grenoble), 49(3):1017–1026, 1999. Sympo- sium à la Mémoire de François Jaeger (Grenoble, 1998). [12] P. McMullen. Weights on polytopes. Discrete Comput. Geom., 15(4):363–388, 1996. [13] R. Schneider. Convex bodies: the Brunn-Minkowski theory, volume 44 of Encyclopedia of Mathematics and its Applications. Cambridge Uni- versity Press, Cambridge, 1993. [14] H. van der Holst, L. Lovász, and A. Schrijver. The Colin de Verdière graph parameter. In Graph theory and combinatorial biology (Balaton- lelle, 1996), volume 7 of Bolyai Soc. Math. Stud., pages 29–85. János Bolyai Math. Soc., Budapest, 1999. [15] H. Weyl. Über die Bestimmung einer geschlossenen konvexen Fläche durch ihr Linienelement. Zürich. Naturf. Ges., 61:40–72, 1916. [16] G. M. Ziegler. Lectures on polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. Introduction The Colin de Verdière number Nullspace representations and Steinitz representations Hessian matrix of the volume as a Colin de Verdière matrix Plan of the paper Acknowledgements From a convex polytope to a Colin de Verdière matrix Lovász construction Polytopes with a given set of normals Interpreting and generalizing the Lovász construction Case d=3 and infinitesimal rigidity of convex polytopes Concluding remarks What fails in the inverse construction Negative eigenvalue The second Minkowski inequality for mixed volumes and the signature of the matrix (2 volxi xj) The second Minkowski inequality and Bol's condition Mixed volumes as derivatives of the volume From the second Minkowski inequality to the signature of the Hessian of the volume

0704.0350

Visible spectroscopic and photometric survey of Jupiter Trojans: final results on dynamical families

Visible spectroscopic and photometric survey of Jupiter Trojans: final results on dynamical families. ∗ Fornasier S.1,2, Dotto E.3, Hainaut O.4, Marzari F.5, Boehnhardt H.6, De Luise F.3, Barucci M.A.2 October 22, 2018 1 University of Paris 7, France 2 LESIA – Paris Observatory, France. 3 INAF – Osservatorio Astronomico di Roma, Italy; 4 European Southern Observatory, Chile; 5 Dipartimento di Fisica, Università di Padova, Italy; 6 Max-Planck Institute for Solar System Research, Katlenburg-Lindau, Germany Submitted to Icarus: December 2006 e-mail: [email protected] fax: +33145077144, phone: +33145077746 Running head: Investigation of Dynamical Families of Jupiter Trojans ∗Based on observations carried out at the European Southern Observatory (ESO), La Silla, Chile, ESO proposals 71.C-0650, 73.C-0622, 74.C-0577 http://arxiv.org/abs/0704.0350v1 Send correspondence to: Sonia Fornasier LESIA-Observatoire de Paris Batiment 17 5, Place Jules Janssen 92195 Meudon Cedex France e-mail: [email protected] fax: +33145077144 phone: +33145077746 Abstract We present the results of a visible spectroscopic and photometric survey of Jupiter Trojans belonging to different dynamical families. The survey was carried out at the 3.5m New Technology Telescope (NTT) of the European Southern Observatory (La Silla, Chile) in April 2003, May 2004 and January 2005. We obtained data on 47 objects, 23 belonging to the L5 swarm and 24 to the L4 one. These data together with those already published by Fornasier et al. (2004a) and Dotto et al. (2006), acquired since November 2002, constitute a total sample of visible spectra for 80 objects. The survey allows us to investigate six families (Aneas, Anchises, Mis- enus, Phereclos, Sarpedon, Panthoos) in the L5 cloud and four L4 fam- ilies (Eurybates, Menelaus, 1986 WD and 1986 TS6). The sample that we measured is dominated by D–type asteroids, with the exception of the Eurybates family in the L4 swarm, where there is a dominance of C– and P–type asteroids. All the spectra that we obtained are featureless with the exception of some Eurybates members, where a drop–off of the reflectance is detected shortward of 5200 Å. Similar features are seen in main belt C–type asteroids and commonly attributed to the intervalence charge transfer transition in oxidized iron. Our sample comprises fainter and smaller Trojans as compared to the literature’s data and allows us to investigate the properties of objects with estimated diameter smaller than 40–50 km. The analysis of the spectral slopes and colors versus the estimated diameters shows that the blue and red objects have indistinguishable size distribution, so any relationship between size and spectral slopes has been found. To fully investigate the Trojans population, we include in our anal- ysis 62 spectra of Trojans available in literature, resulting in a total sample of 142 objects. Although the mean spectral behavior of L4 and L5 Trojans is indistinguishable within the uncertainties, we find that the L4 population is more heterogeneous and that it has a higher abundance of bluish objects as compared to the L5 swarm. Finally, we perform a statistical investigation of the Trojans’s spectra property distributions as a function of their orbital and physical pa- rameters, and in comparison with other classes of minor bodies in the outer Solar System. Trojans at lower inclination appear significantly bluer than those at higher inclination, but this effect is strongly driven by the Eurybates family. The mean colors of the Trojans are similar to those of short period comets and neutral Centaurs, but their color distributions are different. Keywords: Trojan Asteroids – Photometry – Spectroscopy – – Asteroids families 1 Introduction Jupiter Trojans are small bodies of the Solar System located in the Jupiter Lagrangian points L4 and L5. Up to now more than 2000 Trojans have been discovered, ∼ 1150 belonging to the L4 cloud and ∼ 950 to the L5 one. The number of L4 Trojans with radius greater than 1 km is estimated to be around 1.6 ×105 (Jewitt et al., 2000), comparable with the estimated main belt population of similar size. The debate about the origin of Jupiter Trojans and how they were trapped in librating orbits around the Lagrangian points is still open to several possi- bilities. Considering that Trojans have orbits stable over the age of the Solar System (Levison et al, 1997, Marzari et al. 2003) their origin must date back to the early phase of the solar system formation. Some authors (Marzari & Scholl, 1998a,b; Marzari et al., 2002) suggested that they formed very close to their present location and were trapped during the growth of Jupiter. Morbidelli et al. (2005) suggested that Trojans formed in the Kuiper belt and were subsequently captured in the Jupiter L4 and L5 Lagrangian points during planetary migration, just after Jupiter and Saturn crossed their mu- tual 1:2 resonances. In this scenario, Jupiter Troians would give important clues on the composition and accretion of bodies in the outer regions of the solar nebula. Several theoretical studies conclude that Jupiter Trojan clouds are at least as collisionally evolved as main belt asteroids (Shoemaker et al., 1989; Binzel & Sauter, 1992; Marzari et al., 1997; Dell’Oro et al., 1998). This result is supported by the identification of several dynamical families, both in the L4 and L5 swarm (Shoemaker et al., 1989, Milani, 1993, Beaugé and Roig, 2001). Whatever the Trojan origin is, it is plausible to assume that they formed be- yond the frost line and that they are primitive bodies, are possibly composed of anhydrous silicates and organic compounds, and possibly still contain ices in their interior. Several observations of Trojans in the near infrared region (0.8-2.5 µm) have failed to clearly detect any absorption features indicative of water ice (Barucci et al, 1994; Dumas et al, 1998; Emery & Brown, 2003, 2004; Dotto et al., 2006). Also in the visible range Trojan spectra appear featureless (Jewitt & Luu, 1990; Fornasier et al., 2004a, Bendjoya et al., 2004; Dotto et al., 2006). Up to now only 2 objects (1988 BY1 and 1870 Glaukos) show the possible presence of faint bands (Jewitt & Luu, 1990). However, these bands are comparable to the peak to peak noise and are not yet confirmed. Recently, mineralogical features have been detected in emissivity spectra of three Trojan asteroids measured by the Spitzer Space Telescope. These fea- tures are interpreted as indicating the presence of fine-grained silicates on the surfaces (Emery et al. 2006). Several questions about Jupiter Trojans’ dynamical origin, physical prop- erties, composition and link with other groups of minor bodies such as outer main belt asteroids, cometary nuclei, Centaurs and KBOs are still open. In order to shed some light on these questions, we have carried out a spectro- scopic and photometric survey of Jupiter Trojans at the 3.5m New Technol- ogy Telescope (NTT) of the European Southern Observatory (La Silla, Chile) and at the 3.5m Telescopio Nazionale Galileo (TNG), La Palma, Spain. In this paper we present new visible spectroscopic and photometric data, ob- tained during 7 observing nights, carried out at ESO-NTT on April 2003, May 2004, and January 2005, for a total of 47 objects belonging to the L5 (23 objects) and L4 (24 objects) swarms. Considering also the results already published in Fornasier et al. (2004a) and Dotto et al. (2006), obtained in the framework of the same project, we collected a total sample of 80 Jupiter Trojan visible spectra, 47 belonging to the L5 clouds and 33 to the L4. This is the largest homogeneous data set available up to now on these primitive asteroids. The principal aim of our survey was the investigation of Jupiter Trojans belonging to different dynamical families. In fact, since dynamical families are supposed to be formed from the collisional disruption of parent bodies, the investigation of the surface properties of small and large family members can help in understanding the nature of these dynamical groups and might provide a glimpse of the interior structure of the larger primordial parent bodies. We also present an analysis of the visible spectral slopes for all the data in our survey along with those available in the literature, for a total sample of 142 Trojans. This enlarged sample allowed us to carry out a significant statistical investi- gation of the Trojans’ spectral property distributions, as a function of their orbital and physical parameters, and in comparison with other classes of mi- nor bodies in the outer Solar System. We also discuss the spectral slope distribution within the Trojan families. 2 Observations and data reduction [HERE TABLE 1 AND 2] The data were obtained in the visible range during 3 different observing runs at ESO-NTT: 10 and 11 April 2003 for the spectroscopic and pho- tometric investigation of 6 members of the 4035 1986 WD and 1 member of 1986 TS6 families; 25 and 26 May 2004 for a spectroscopic survey of L4 Eurybates family; 17, 18, and 19 January 2005 for the spectroscopic and pho- tometric investigation of 5 Anchises, 6 Misenus, 5 Panthoos, 2 Cloanthus, 2 Sarpedon and 3 Phereclos family members (L5 swarm). We selected our targets from the list of Jupiter Trojan families provided by Beaugé and Roig (2001 and P.E.Tr.A. Project at www.daf.on.br/ froig/petra/). The authors have used a cluster-detection algorithm called Hierarchical Clus- tering Method (HCM, e.g. Zappalà et al., 1990) to find asteroid families among Jupiter Trojans starting from a data–base of semi-analytical proper elements (Beaugé & Roig, 2001). The identification of families is performed by comparing the mutual distances with a suitable metric in the proper el- ements’ space. The clustering chain is halted when the mutual distance, measuring the incremental velocity needed for orbital change after the puta- tive parent body breakup, is larger than a fixed cut-off value. A lower cutoff implies a higher statistical significance of the family. Since families in L4 are on average more robust than those around L5 (Beaugé and Roig, 2001), we prefer to adopt a cutoff of 100 m/s for the L4 cloud and of 150 m/s for L5. For the very robust Eurybates family we decided to limit our survey to those family members defined with a cutoff of 70 m/s. All the data were acquired using the EMMI instrument, equipped with a 2x1 mosaic of 2048×4096 MIT/LL CCD with square 15µm pixels. For the spectroscopic investigation during May 2004 and January 2005 runs we used the grism #1 (150 gr/mm) in RILD mode to cover the wavelength range 4100–9400 Å with a dispersion of 3.1 Å/px (200 Å/mm) at the first order, while on April 2003 we used a different grism, the #7 (150 gr/mm), covering the spectral range 5200–9500 Å, with a dispersion of 3.6 Å/px at the first order. April 2003 and January 2005 spectra were taken through a 1 arcsec wide slit, while during May 2004 we used a larger slit (1.5 arcsec). The slit was oriented along the parallactic angle during all the observing runs in order to avoid flux loss due to the atmospheric differential refraction. For most objects, the total exposure time was divided into several (usually 2-4) shorter acquisitions. This allowed us to check the asteroid position in the slit before each acquisition, and correct the telescope pointing and/or tracking rates if necessary. During each night we also recorded bias, flat– field, calibration lamp (He-Ar) and several (6-7) spectra of solar analog stars measured at different airmasses, covering the airmass range of the science targets. During 17 January 2005, part of the night was lost due to some technical problems and only 2 solar analog stars were acquired. The ratio of these 2 stars show minimal variations (less than 1%) in the 5000–8400 Å range, but higher differences at the edges of this range. For this reason we omit the spectral region below 4800 Å for most of the asteroids acquired that night. The spectra were reduced using ordinary procedures of data reduction as described in Fornasier et al. (2004a). The reflectivity of each asteroid was obtained by dividing its spectrum by that of the solar analog star closest in time and airmass to the object. Spectra were finally smoothed with a median filter technique, using a box of 19 pixels in the spectral direction for each point of the spectrum. The threshold was set to 0.1, meaning that the original value was replaced by the median value if the median value differs by more than 10% from the original one. The obtained spectra are shown in Figs. 1–5. In Table 1 and Table 2 we report the circumstances of the observations and the solar analog stars used respectively for the L5 and L4 family members. [TABLE 3] The broadband color data were obtained during the April 2003 and Jan- uary 2005 runs just before the Trojans’ spectral observation. We used the RILD mode of EMMI for wide field imaging with the Bessell-type B, V, R, and I filters (centered respectively at 4139, 5426, 6410 and 7985Å). The ob- servations were carried out in a 2 × 2 binning mode, yielding a pixel scale of 0.33 arcsec/pixel. The exposure time varied with the object magnitude: typically it was about 12-90s in V, 30-180s in B, 12-70s in R and I filters. The CCD images were reduced and calibrated with a standard method (For- nasier et al., 2004a), and absolute calibration was obtained through the ob- servations of several Landolt fields (Landolt, 1992). The instrumental mag- nitudes were measured using aperture photometry with an integrating radius typically about three times the average seeing, and sky subtraction was per- formed using a 5-10 pixels wide annulus around each object. The results are reported in Table 3. From the visual inspection and the radial profiles analysis of the images, no coma was detected for any of the observed Trojans. On May 2004, as the sky conditions were clear but not photometric, we did not perform photometry of the Eurybates family targets. 3 Results [TABLE 4 AND 5] For each Trojan we computed the slope S of the spectral continuum using a standard least squared technique for a linear fit in the wavelength range between 5500 and 8000 Å. The choice of these wavelength limits has been driven by the spectral coverage of our data. We choose 5500 Å as the lower limit because of the different instrumental setup used during different ob- serving runs (with some spectra starting at wavelength ≥ 5200 Å), while beyond 8000 Å our spectra are generally noisier due to a combination of the CCD drop-off in sensitivity and the presence of the strong atmospheric water bands. The computed slopes and errors are listed in Table 4 and 5. The reported er- ror bars take into account the 1σ uncertainty of the linear fit plus 0.5%/103Å attributable to the use of different instruments and solar analog stars (esti- mated from the different efficiency of the grism used, and from flux losses due to different slit apertures). In Table 4 and 5 we also report the taxo- nomic class derived following the Dahlgren & Lagerkvist (1995) classification scheme. In the L5 cloud we find 27 D–, 3 DP–, 2 PD–, and 1 P–type objects. In the L4 cloud we find 10 C–type and 7 P–type objects inside the Eurybates family, while for the Menelaus, 1986 TS6 and 1986 WD families, including the data published in Dotto et al. (2006), we get 9 D–, 3 P–, 3C–, and 1 DP–type asteroids. The majority of the spectra are featureless, although some of the observed Eurybates’ members show weak spectral absorption features (Fig. 5). These features are discussed in the following section. We derived an estimated absolute magnitude H by scaling the measured V magnitude to r = ∆ = 1 AU and to zero phase assuming G=0.15 (Bowell et al., 1989). The estimated H magnitude of each Trojan might be skewed uncertain rotational phase, as the lightcurve amplitudes of Trojans might vary up to 1 magnitude. In order to investigate possible size dependence in- side each family, and considering that IRAS diameters are available for very few objects, we estimate the size using the following relationship: 1329× 10−H/5 where D is the asteroid diameter, p is the geometric albedo, and H is the abso- lute magnitude. We use H derived from our observations when available, and from the ASTORB.DAT file (Lowell observatory) for the Eurybates mem- bers, for which we did not carry out visible photometry. We evaluated the diameter for an albedo range of 0.03–0.07, assuming a mean albedo of 0.04 for these dark asteroids (Fernandez et al., 2003). The resulting D values are reported in Tables 4 and 5. 3.1 Dynamical families: L5 swarm 3.1.1 Anchises [FIGURE 1] We investigated 5 of the 15 members of the Anchises family (Fig. 1): 1173 Anchises, 23549 1994 ES6, 24452 2000 QU167, 47967 2000 SL298 and 124729 2001 SB173 on 17 January 2005. For 4 out of 5 observed objects we omit the spectral range below 4800Å due to low S/N ratio and problems with the solar analog stars. The spectral behavior is confirmed by photometric data (see Table 3). All the obtained spectra are featureless. The Anchises family survives at a cutoff corresponding to relative veloc- ities of 150 m/s. The biggest member, 1173 Anchises, has a diameter of 126 km (IRAS data) and has the lowest spectral slope (3.9 %/103Å) among the investigated family members. It is classified as P–type, while the other 4 members are all D–types. Anchises was previously observed in the 4000- 7400Å region by Jewitt & Luu (1990), who reported a spectral slope of 3.8 %/103Å, in perfect agreement with the value we found. The three 19-29 km sized objects have a steeper spectral slope (7.4-9.2 %/103Å), while the small- est object, 2001 SB173 (spectral slope = 14.78±0.99 %/103Å) is the reddest one (Table 4). Even with the uncertainties in the albedo and diameter, a slope–size rela- tionship is evident among the observed objects, with smaller–fainter members redder than larger ones (Fig. 7). 3.1.2 Misenus [FIGURE 2] For this family we investigated 6 members (11663 1997 GO24, 32794 1989 UE5, 56968 2000 SA92, 99328 2001 UY123, 105685 2000 SC51 and 120453 1988 RE12) out of the 12 grouped at a relative velocity of 150 m/s. The family survives with the same members also at a stringent cut-off velocity of 120 m/s. The spectra, together with magnitude color indices transformed into linear reflectance, are shown in Fig. 2, while the color indices are reported in Table 3. All the spectra are featureless with different spectral slope values covering the 4.6–15.9 %/103Å range (Table 4): 1988 RE12 has the lowest spectral slope and is classified as P–type, 3 objects (11663, 32794 and 2000 SC51) are in the transition region between P– and D– type, with very similar spectral behavior, while the two other observed members are D–types. Of these last, 56968 has the highest spectral slope not only inside the family (15.86 %/103Å) but also inside the whole L5 sample analyzed in this paper. All the investigated Misenus members are quite faint and have diameters of a few tens of kilometers. No clear size-slope relationship has been found inside this family (Fig. 7). No other data on the Misenus family members are available in the literature, so we do not know if the large gap between the spectral slope of 56968 and those of the other 5 investigated objects is real or it could be filled by other members not yet observed. If real, 56968 can be an interloper inside the family. 3.1.3 Panthoos [FIGURE 3] The Panthoos family has 59 members for a relative velocity cutoff of 150 m/s. We obtained new spectroscopic and photometric data of 5 members: 4829 Sergestus, 30698 Hippokoon, 31821 1999 RK225, 76804 2000 QE and 111113 2001 VK85 (Fig. 3). Three objects presented by Fornasier et al. (2004a) as belonging to the Astyanax family (23694 1997 KZ3, 32430 2000 RQ83, 30698 Hippokoon) and one to the background population (24444 2000 OP32) are now included among the members of the Panthoos family. Peri- odic updates of the proper elements can change the family membership. In particular the Astyanax group disappeared in the latest revision of dynami- cal families, and its members are now in the Panthoos family within a cutoff of 150m/s. The Panthos family survives also a cutoff of 120 m/s, with 7 members, and 90 m/s, with 6 members. We observed 30698 Hippokoon during two different runs (on 9 Nov. 2002 and on 18 Jan. 2005), and both spectral slopes and colors are in agreement inside the error bars (see Table 3, Table 4, and Fornasier et al., 2004a). No other data on the Panthoos family are available in the literature. The analysis of the 8 members (for 24444 only photometry is available) show featureless spectra with slopes that seem to slightly increase as the asteroid size decreases (Table 4 and Fig. 7). However, all the members have dimensions very similar within the uncertainties, making it difficult for any slope-size relationship to be studied. The largest member, 4829 Sergestus, is a PD–type with a slope of about 5 %/103Å, while all the other investigated members are D–types. 3.1.4 Cloantus [FIGURE 4] We observed only 2 out of 8 members of the Cloantus family (5511 Cloan- thus and 51359 2000 SC17, see Fig. 4) as grouped at a cutoff corresponding to relative velocities of 150 m/s. This family survives at a stringent cutoff and 3 members (including the two that we observed) also survive for relative velocities of 60 m/s. Both of the observed objects are D–types with very similar, featureless, reddish spectra (Table 4 and Fig. 7). 5511 Cloanthus was observed also by Bendjoya et al. (2004), who found a slope of 13.0±0.1 %/103Å in the 5000-7500 Å wavelength range, while we measure a value of 10.84±0.15 %/103Å. Our spectrum has a higher S/N ratio than the spectrum by Bendjoya et al. (2004), and it is perfectly matched by our measured color indices that confirm the spectral slope. This difference cannot be caused by the slightly different spectral ranges used to measure the slope, but could possibly be due to heterogeneous surface composition. 3.1.5 Phereclos The Phereclos family comprises 15 members at a cutoff of 150 m/s. The family survives with 8 members also at a cutoff of 120m/s. We obtained spectroscopic and photometric data of 3 members (9030 1989 UX5, 11488 1988 RM11 and 31820 1999 RT186, see Fig 4), that, together with the 4 spectra (2357 Phereclos, 6998 Tithonus, 9430 1996 HU10, 18940 2000QV49) already presented by Fornasier et al. (2004a), allow us to investigate about half of the Phereclos family population defined at a cutoff of 150m/s. The spectral slope of these objects, all classified as D–type except one PD–type (11488), varies from 5.3 to 11.3 %/103Å (Table 4). The size of the fam- ily members ranges from about 20 km in diameter for 31820 to 95 km for 2357, but we do not observe any clear slope-diameter relationship (Fig. 7 and Table 4). 3.1.6 Sarpedon We obtained new spectroscopic and photometric data of 2 members of the Sarpedon family (48252 2001 TL212 and 84709 2002 VW120), whose spectra and magnitude color indices are reported in Fig. 4 and Table 4. Including the previous observations (Fornasier et al., 2004a) of 4 other members (2223 Sarpedon, 5130 Ilioneus, 17416 1988 RR10, and 25347 1999 RQ116), we have measurements of 6 of the 21 members of this family dynamically defined at a cutoff of 150 m/s. All the 6 aforementioned objects, except 25347, constitute a robust clustering which survives up to 90 m/s with 9 members. The cluster which contains (2223) Sarpedon was also recognized as a family by Milani (1993). All the 6 investigated members have very similar colors (see Table 3) and spectral behavior. The spectral slope (Fig. 7) varies over a very restricted range, from 9.6 to 11.6 %/103Å (Table 4), despite a significant variation of the estimated size (from the 18 km of 17416 to the 105 km of 2223). Con- sequently, the surface composition of the Sarpedon family members appears to be very homogeneous. 3.2 Dynamical families: L4 swarm 3.2.1 Eurybates [FIGURE 5] Eurybates family members were observed in May 2004. The selection of the targets was made on the basis of a very stringent cutoff, corresponding to relative velocities of 70 m/s, that gives a family population of 28 objects. We observed 17 of these members (see Table 2) that constitute a very robust clustering in the space of the proper elements: all the members we studied, except 2002 CT22, survive at a cutoff of 40 m/s. The spectral behavior of these objects (Fig. 5) is quite homogeneous with 10 asteroids classified as C–type and 7 as P–type. The spectral slopes (Ta- ble 5) range from neutral to moderately red (from -0.5 to 4.6 %/103Å). The slopes of six members are close to zero (3 slightly negative) with solar-like col- ors. The asteroids 18060, 24380, 24420, and 39285, all classified as C–types, clearly show a drop off of reflectance for wavelength shorter than 5000–5200 Å. The presence of the same feature in the spectra of 2 other members (1996 RD29 and 28958) is less certain due to the lower S/N ratio. This absorp- tion is commonly seen on main belt C–type asteroids (Vilas 1994; Fornasier et al. 1999), where is due to the intervalence charge transfer transitions (IVCT) in oxidized iron, and is often coupled with other visible absorption features related to the presence of aqueous alteration products (e.g. phyl- losilicates, oxides, etc). These IVCTs comprise multiple absorptions that are not uniquely indicative of phyllosilicates, but are present in the spectrum of any object containing Fe2+ and Fe3+ in its surface material (Vilas 1994). Since no other phyllosilicate absorption features are present in the C-type spectra of the Eurybates family, there is no evidence that aqueous alteration processes occurred on the surface of these bodies. In Fig. 8 we show the spectral slopes versus the estimated diameters for the Eurybates family members. All the observed objects, except the largest member (3548) that has a diameter of about 70 km and exhibit a neutral (∼ solar-like) spectral slope, are smaller than ∼ 40 km and present both neutral and moderately red colors. The spectral slopes are strongly clustered around S = 2%/103Å, with higher S values restricted to smaller objects (D< 25 3.2.2 1986 WD [FIGURE 6] We investigated 6 out of 17 members of the 4035 1986 WD family that is dynamically defined at a cutoff of 130 m/s (Fig. 6 and Table 2). Three of our targets (4035, 6545 and 11351) were already observed by Dotto et al. (2006): for 6545 and 11351 there is a good consistency between our spectra and those already published. 4035 was observed also by Bendjoya et al. (2004): all the spectra are featureless, but Bendjoya et al. (2004) obtain a slope of 8.8 %/103Å, comparable to the one here presented, while Dotto et al. (2006) found a higher value (see Table 5). This could be interpreted as due to the different rotational phases seen in the three observations, and could indicate some inhomogeneities on the surface of 4035. The observed family members show heterogeneous behaviors (Fig. 8), with spectral slopes ranging from neutral values for the smaller members (24341 and 14707) to reddish ones for the 3 members with size bigger than 50 km (4035, 6545, and 11351). For this family, it seems that a size-slope relationship exists, with smaller members having solar colors and spectral slopes increasing with the object’ sizes. 3.2.3 1986 TS6 The 1986 TS6 family includes 20 objects at a cut-off of 100 m/s. We present new spectroscopy and photometry of a single member, 12921 1998 WZ5 (Fig. 6). The spectrum we present here is flat and featureless, with a spectral slope of 4.6±0.8%/103Å. Dotto et al. (2006) presented a spectrum obtained a month after our data (in May 2003) that has a very similar spectral slope 3.7± 0.8%/103Å. Previously, 12917 1998 TG16, 13463 Antiphos, 12921 1998 WZ5, 15535 2000 AT177, 20738 1999 XG191, and 24390 2000 AD177 were included in the Makhoan family. Refined proper elements now place all of these bodies in the 1986 TS6 family. In Fig. 8 we report the spectral slopes vs. estimated diameters of the 6 observed members. The family shows different spectral slopes with the presence of both P–type (12921 and 13463) and D–type asteroids (12917, 15535, 20738, and 24390). Due to the very similar diameters, a slope-size relationship is not found. [FIGURE 7 AND 8] 4 Discussion The spectra of Jupiter Trojan members of dynamical families show a range of spectral variation from C– to D–type asteroids. With the exception of the L4 Eurybates family, all the observed objects have featureless spectra, and we cannot find any spectral bands which could help in the identification of minerals present on their surfaces. The lack of detection of any mineralogy diagnostic feature might indicate the formation of a thick mantle on the Tro- jan surfaces. Such a mantle could be formed by a phase of cometary activity and/or by space weathering processes as demonstrated by laboratory exper- iments on originally icy surfaces (Moore et al., 1983; Thompson et al., 1987; Strazzulla et al., 1998; Hudson & Moore, 1999). A peculiar case is constituted by the Eurybates family, which shows a pre- ponderance of C–type objects and a total absence of D–types. Moreover, this is the only family in which some members exhibit spectral features at wavelengths shorter than 5000–5200 Å, most likely due to the intervalence charge transitions in materials containing oxidized iron (Vilas 1994). 4.1 Size vs spectral slope distribution: Individual families The plots of spectral slopes vs. diameters are shown in Fig. 7 and 8. A relationship between spectral slopes and diameters seems to exist for only three of the nine families we studied. In the Anchises and Panthoos families, smaller objects have redder spectra, while for the 1986 WD family larger objects have the redder spectra. Moroz et al. (2004) have shown that ion irradiation on natural complex hydrocarbons gradually neutralizes the spectral slopes of these red organic solids. If the process studied by Moroz et al. (2004) occurred on the surface of Jupiter Trojans, the objects having redder spectra have to be younger than those characterized by bluish-neutral spectra. In this scenario the largest and spectrally reddest objects of the 1986 WD family could come from the interior of the parent body and expose fresh material. In the case of the Anchises and Panthoos families the spectrally reddest members, being the smallest, could come from the interior of the parent body, or alternatively could be produced by more recent secondary fragmentations. In particular, small family members may be more easily resurfaced, as significant collisions (an impactor having a size greater than a few percent of the target), as well as seismic shaking and recoating by fresh dust, may occur frequently at small sizes. [FIGURE 9] 4.2 Size vs slope distribution: The Trojan population as a whole [TABLE 6] As compared to the data available in literature, our sample strongly con- tributed to the analysis of fainter and smaller Trojans, with estimated di- ameters smaller than 50 km. Jewitt & Luu (1990), analyzing a sample of 32 Trojans, found that the smaller objects were redder than the bigger ones. However, our data play against the existence of a possible color-dimension trend. In fact, the spectral slope’s range of the objects smaller than 50 km is similar to that of the larger Trojans, as shown in Fig. 9. The Eurybates family strongly contributes to the population of small spectrally neutral objects, filling the region of bodies with mean diameter D<40 km and with spectral slopes smaller than 3 %/103Å. In order to carry out a complete analysis of the spectroscopic and pho- tometric characteristics of the whole available data set on Jupiter Trojans, we considered all the visible spectra published in the literature: Jewitt & Luu (1990, 32 objects), Fitzimmons et al. (1994, 3 objects), Bendjoya et al. (2004, 34 objects), Fornasier et al. (2004a, 26 L5 objects), and Dotto et al. (2006, 24 L4 Trojans). We also add several Trojans spectra (11 L4 and 3 L5 Trojans) from the files available on line (Planetary data System archive, pdssbn.astro.umd.edu, and www.daf.on.br/∼lazzaro/S3OS2-Pub/s3os2.htm) from the SMASS I, SMASS II and S3OS2 surveys (Xu et al., 1995; Bus & Binzel, 2003; Lazzaro et al., 2004). Including all these data, we compile a sample of 142 different Trojans, 68 belonging to the L5 cloud and 74 belong- ing to the L4. We performed the taxonomic classification of this enlarged sample, on the basis of the Dahlgren and Lagerkvist (1995) scheme, by ana- lyzing spectral slopes computed in the range 5500-8000 Å. Different authors, of course, considered different spectral ranges for their own slope gradient evaluations: Jewitt & Luu (1990) and Fitzimmons et al. (1994) use the 4000-7400 Å and Bendjoya et al. (2004, Table 2) used a slightly different ranges around 5200-7500 Å. Since all the cited papers show spectra with lin- ear featureless trends, the different wavelength ranges used for the spectral gradient computation by Bendjoya et al. (2004) and Jewitt & Luu (1990) are not expected to influence the obtained slopes. In order to search for a dependency of the spectral slope distribution with the size of the objects, all observations (from this paper as well as from the literature) were combined. The objects were isolated in 5 size bins (smaller than 25 km, 25–50 km, 50–75 km, 75-100 km and larger than 100 km). Each bin contains between 20 and 50 objects. These subsamples are large enough to be compared using classical statistical tests: the t-test, which estimates if the mean values are compatible, the f-test, which checks if the widths of the distributions are compatible (even if they have different means), and the KS test, which compares directly the full distributions. A probability is computed for each test; a small probability indicates that the tested distri- butions are not compatible, i.e. the objects are not randomly extracted from the same population, while a large probability value has no meaning (i.e. it is not possible to assure that both samples come from the same population, we can just say in that case that they are not incompatible). In order to quantify the probability levels that we consider as significant, the same tests were run on randomized distributions (see Hainaut & Delsanti 2002 for the method). Since probability lower than 0.04-0.05 does not appear in these randomized distributions, we consider that values smaller than 0.05 indicate a significant incompatibility. Each sub-sample was compared with the four others – the results are sum- marized in Table 6. The average slope of the 5 bins are all compatible among each other. The only marginally significant result is that the width of the slope distribution among the larger objects (diam. > 100 km) is narrower than that of all the smaller objects. This narrower color distribution could be due to the aging processes affect- ing the surface of bigger objects, which are supposed to be older. The wider color distribution of small members is possibly related to the different ages of their surfaces: some of them could be quite old, while some other could have been recently refreshed. 4.3 Spectral slopes and L4/L5 Clouds [HERE FIGURES 10 AND 11] Considering only the Trojan observations reported in this paper, the aver- age slope is 8.84±3.03%/103Å for the L5 population, and 4.57±4.01%/103Å for the L4. Considering now all the spectra available in the literature, the 68 L5 Trojans have an average slope of 9.15±4.19%/103Å, and the 78 L4 objects, 6.10±4.48%/103Å. Performing the same statistical tests as above, it appears that these two populations are significantly different. In particular, the av- erage slopes are incompatible at the 10−5 level. Nevertheless, as described in Section 3.2.1, the Eurybates family members have quite different spectral characteristics than the other objects and con- stitute a large subset of the whole sample. Indeed, comparing their distribu- tion with the whole populations, they are found significantly different at the 10−10 level. In other words, the Eurybates family members do not constitute a random subset of the other Trojans. Once excluded the Eurybates family, the remaining 61 Trojans from the L4 swarm have an average slope of 7.33±4.24%/103Å. The very slight dif- ference of average slope between the L5 and remaining L4 objects is very marginally significant (probability of 1.6%), and the shape and width of the slope distributions are compatible with each other. The taxonomic classification we have performed shows that the majority (73.5%) of the observed L5 Trojans (Fig. 10) are D–type (slope > 7 %/103 Å) with featureless reddish spectra, 11.8% are DP/PD –type (slope between 5 and 7 %/103 Å), 10.3% are P–type, and only 3 objects are classified as C–type (4.4%). In the L4 swarm (Fig. 11), even though the D–type still dominate the population (48.6%), the spectral types are more heterogeneous as compared to the L5 cloud, with a higher percentage of neutral-bluish objects: 20.3% are P–type, 8.1% are DP/PD-type, 12.2% are C–type, and 10.8% of the bodies have negative spectral slope. The higher percentage of C– and P– type as compared to the L5 swarm is strongly associated with the presence of the very peculiar Eurybates family. Among 17 observed members 10 are classified as C–types (among which 3 have negative spectral slopes) and 7 are P–types. Considering the 57 asteroids that compose the L4 cloud without the Eurybates family, we find percentages of P, and PD/DP –types very similar to those of the L5 cloud (14.0% and 10.5% respectively), a smaller percentage of D–types (63.2%) and of the C–types (3.5%), and the presence of a 8.8% Trojans with negative spectral slopes. The visible spectra of the Eurybates members are very similar to those of C–type main belt asteroids, Chiron-like Centaurs, and cometary nuclei. This similarity is compatible with three different scenarios: the family could have been produced by the fragmentation of a parent body very different from all the other Jupiter Trojans (in which case the origin of such a peculiar parent must still be assessed); this could be a very old family where space weathering processes have covered any differences in composition among the family members and flattened all the spectra; this could be a young family where space weathering processes occurred within time scales smaller than the age of the family. In the last two cases the Eurybates family would give the first observational evidence of spectra flattened owing to space weathering processes. This would then imply, according to the results of Moroz et al. (2004), that its primordial composition was rich in complex hydrocarbons. The knowledge of the age of the Eurybates family is therefore a fundamental step to investigate the nature and the origin of the parent body, and to assess the effect of space weathering processes on the surfaces of its members. The present sample of Jupiter Trojans suggests a more heterogeneous composition of the L4 swarm as compared to the L5 one. As previously noted by Bendjoya et al. (2004), the L4 swarm contains a higher percentage of C– and P–type objects. This result is enhanced by members of the Eu- rybates family, but remains even when these family members are excluded. Moreover, the dynamical families belonging to the L4 cloud are more robust than those of the L5 one, surviving with densely populated clustering even at low relative velocity cut-off. We therefore could argue that the L4 cloud is more collisionally active than the L5 swarm. Nevertheless, we still cannot intepret this in terms of the composition of the two populations, since we cannot exclude that as yet unobserved C– and P–type families are present in the L5 cloud. 4.4 Orbital Elements [HERE FIGURE 12 and TABLES 7 and 8 ] We analyzed the spectral slope as a function of the Trojans’ orbital el- ements. As an illustration, Fig. 12 shows the B − R color distribution as a function of the orbital elements. In order to investigate variations with orbital parameters, the Trojan population is divided in 2 sub samples: those with the considered orbital element lower than the median value, and those with the orbital element higher than the median (by construction, the two subsamples have the same size). Taking a as an example, half the Trojans have a < 5.21AU, and half have a larger than this value. The mean color, the color dispersion, and the color distribution of the 2 subsamples are compared using the three statistical tests mentioned in Section 4.2. The method is discussed in details in Hainaut & Delsanti (2002). The tests are repeated for all color and spectral slope distributions. The results are the following. • q, perihelion distance: the color distribution of the Trojans with small q is marginally broader than that of Trojans with larger q. This result is not very strong (5%), and is dominated by the red-end of the visible wavelength. Removing the Eurybates from the sample maintains the result, at the same weak level. • e, eccentricity: the distribution shows a similar result, also at the weak 5% significance. The objects with larger e have broader color distribu- tion than those with lower e. This result is entirely dominated by the Eurybates’ contribution. • i, inclination: objects with smaller inclination are significantly bluer than those with larger i. This result is observed at all wavelengths. It is worth noting that this is contrary to what is usually observed on other Minor Bodies in the Outer Solar System survey (MBOSSes), where objects with high i, or more generally, high excitation E =√ e2 + sin2 i, are bluer (Hainaut & Delsanti, 2002; Doressoundiram et al., 2005). This can also be visually appreciated in Fig. 12. This result is also completely dominated by the Eurybates’ contribution. The non- Eurybates Trojans do not display this trend. • E = e2 + sin2 i, orbital excitation: the objects with small E are also significantly bluer than those with high E. This result is also com- pletely dominated by the Eurybates’ contribution. The non-Eurybates Trojans do not display this trend. In summary this analysis shows that the Eurybates sub-sample of the Trojans is well separated in orbital elements and in colors. For the other Minor Bodies in the outer Solar System, the relation be- tween color and inclination–orbital excitation (objects with a higher orbital excitation tend to be bluer) is interpreted as a relation between excitation and surface aging/rejuvenating processes (Doressoudiram et al., 2005). The Eurybates family has low excitation and neutral-blue colors, suggesting that the aging/rejuvenating processes affecting them are different from the other objects. This could be due to different surface compositions, different irradi- ation processes, or different collisional properties – which would be natural for a collisional family. 5 Comparison with other outer Solar System minor bodied 5.1 Introduction and methods [HERE FIGURES 13 AND 14] The statistical tests set described in section 4.2 has also been applied to compare the colors and the spectral slopes distribution of the Trojans with those of the other minor bodies in the outer Solar System taken from the updated, on-line version of the Hainaut & Delsanti (2002) database. Figure 13, as an example, displays the (R-I) vs (V-R) diagrams, while Fig. 14 shows the (B-V) and (V-R) color distributions, as well as the spectral slope distribution of the different classes of objects. The tests were performed on all the color indices derived from filters in the visible (UBVRI) and near infrared range (JHK) but in Table 7 and 8 we summarize the most significant results. In order to “calibrate” the significant probabilities, additional artificial classes are also compared: first, the objects which have an even internal number in the database with the odd ones. As this internal number is purely arbitrary, both classes are statistically indistinguishable. The other tested pair is the objects with a “1999” designation versus the others. Again, this selection criterion is arbitrary, so the pseudo-classes it generates are sub- sample of the total population, and should be indistinguishable. However, as many more objects have been discovered in all the other years than during that specific year, the size of these sub-samples are very different. This permits us to estimate the sensitivity of the tests on sample of very different sizes. Some of the tests found the arbitrary populations incompatible at the 5% level, so we use 0.5% as a conservative threshold for statistical significance of the distribution incompatibility 5.2 Results Table 7 and Fig. 14 clearly show that the Trojans’ colors distribution is different as compare to that of Centaurs, TNOs and comets. Trojans are at the same time bluer, and their distribution is narrower than all the other populations. Using the statistical tests (see Table 8), we can confirm the significance of these results. • The average colors of the Trojans are significantly different from those of all the other classes of objects (t-test), with the notable exception of the short period comet nuclei. Refining the test to the Eurybates/non- Eurybates, it appears that the Eurybates have marginally different mean colors, while the non-Eurybates average colors are indistinguish- able from those of the comets. • Considering the full shape of the distribution (KS test), we obtain the same results: the Trojans colors distributions are significantly differ- ent from those of all the other classes, with the exception of the SP comets, which are compatible. Again, this result becomes stronger separating the Eurybates: their distributions are different from those of the comets, while the non-Eurybates ones are indistinguishable. • The results when considering the widths of the color distributions (f- test) are slightly different. Classes of objects with different mean colors could still have the same distribution width. This could suggest that a similar process (causing the width of the distribution) is in action, but reached a different equilibrium point (resulting in different mean val- ues). This time, all classes are incompatible with the Trojans, including the comets, with strong statistical significance. In order to further explore possible similarities between Trojans and other classes, the comparisons were also performed with the neutral Centaurs. These were selected with S < 20%/103Å); this cut-off line falls in the gap between the ”neutral” and ”red” Centaurs (Peixinho et al., 2003, Fornasier et al., 2004b). The t-Test (mean color) only reveals a very moderate incompatibility be- tween the Trojans and neutral Centaurs, at the 5% level, i.e. only marginally significant. On the other hand, the f-Test gives some strong incompatibilities in various colors (moderate in B− V and H −K, very strong in R− I), but the two populations are compatible for most of the other colors. Similarly, only the R − I KS-test reveals a strong incompatibility. It should also be noted that only 18 neutral Centaurs are known in the database. In summary, while the Trojans and neutral Centaurs have fairly similar mean colors, their color distributions are also different. 6 Conclusions From 2002, we carried out a spectroscopic and photometric survey of Jupiter Trojans, with the aim of investigating the members of dynamical families. In this paper we present new data on 47 objects belonging to several dy- namical families: Anchises (5 members), Cloanthus (2 members), Misenus (6 members), Phereclos (3 members), Sarpedon (2 members) and Panthoos (5 members) from the L5 swarm; Eurybates (17 members), 1986 WD (6 mem- bers), and Menelaus (1 member) for the L4 swarm. Together with the data already published by Fornasier et al. (2004a) and Dotto et al. (2006), taken within the same observing program, we have a total sample of 80 Trojans, the largest homogeneous data set available to date on these primitive aster- oids. The main results coming from the observations presented here, and from the analysis including previously published visible spectra of Trojans, are the following: • Trojans’ visible spectra are mostly featureless. However, some mem- bers of the Eurybates family show a UV drop-off in reflectivity for wavelength shorter than 5000–5200 Å that is possibly due to interva- lence charge transfer transitions (IVCT) in oxidized iron. • The L4 Eurybates family strongly differs from all the other families in that it is dominated by C– and P–type asteroids. Also its spectral slope distribution is significantly different when compared to that of the other Trojans (at the 10−10 level). This family is very peculiar and is dynamically very strong, as it sur- vives also at a very stringent cutoff (40 m/s). Further observations in the near-infrared region are strongly encouraged to look for possible absorption features due to water ice or to material that experienced aqueous alteration. • The average spectral slope for the L5 Trojans is 9.15±4.19%/103Å, and 6.10±4.48%/103Å for the L4 objects. Excluding the Eurybates, the L4 average slope values becomes 7.33±4.24%/103Å. The slope distribu- tions of the L5 and of the non-Eurybates L4 are indistinguishable. • Both L4 and L5 clouds are dominated by D–type asteroids, but the L4 swarm has an higher presence of C– and P–type asteroids, even when the Eurybates family is excluded, and appears more heterogeneous in composition as compared to the L5 one. • We do not find any size versus spectral slope relationship inside the whole Trojans population. • The Trojans with higher orbital inclination are significantly redder than those with lower i. While this trend is the opposite of that observed for other distant minor bodies, this effect is entirely dominated by the Eurybates family. • Comparing the Trojans colors with those of other distant minor bod- ies, they are the bluest of all classes, and their colors distribution is the narrowest. This difference is mostly due to the Eurybates family. In fact, if we consider only the Trojan population without the Eurybates members, their average colors and overall distributions are not distin- guishable from that of the short period comets. However, the widths of their color distributions are not compatible. The similarity in the overall color distributions might be caused by the small size of the short period comet sample rather than by a physical analogy. The Trojans average colors are also fairly similar to those of the neutral Centaurs, but the overall distributions are not compatible. After this study, we have to conclude that Trojans have peculiar charac- teristics very different from those of all the other populations of the outer Solar System. Unfortunately, we still cannot assess if this is due to differences in the physi- cal nature, or in the aging/rejuvenating processes which modified the surface materials in different way at different solar distances. Further observations, mainly in V+NIR spectroscopy and polarimetry, are absolutely needed to better investigate the nature of Jupiter Trojans and to definitively assess if a genetical link might exist with Trans-Neptunian Objects, Centaurs and short period comets. Acknowledgments We thank Beaugé and Roig for kindly providing us with updated Trojan family list, and R.P. Binzel and J.P. Emery for their useful comments in the reviewing process. References Barucci, M. A., Lazzarin, M., Owen, T., Barbieri, C., Fulchignoni, M., 1994. Near–infrared spectroscopy of dark asteroids. Icarus 110, 287-291. Beaugé, C., Roig, F., 2001. A Semianalytical Model for the Motion of the Trojan Asteroids: Proper Elements and Families. Icarus 53, 391-415. Bendjoya, P., Cellino, A., Di Martino, M., Saba, L., 2004. Spectroscopic observations of Jupiter Trojans. Icarus 168, 374-384. Binzel, R. P., Sauter, L. M., 1992. Trojan, Hilda, and Cybele asteroids - New lightcurve observations and analysis. Icarus 95, 222-238. Bowell, E., Hapke, B., Domingue, D., Lumme, K., Peltoniemi, J., Harris, A.W., 2003. Application of photometric models to asteroids. In Asteroids II (R.P Binzel, T. Gehrels, M.S. Matthews, eds) Univ. of Arizona Press, Tucson, pp. 524–556. Bus, S. J., Binzel, R.P., 2003. Phase II of the Small Main-Belt Asteroid Spectroscopic Survey. The Observations. Icarus 158, 106–145. Dahlgren, M., Lagerkvist, C. I., 1995. A study of Hilda asteroids. I. CCD spectroscopy of Hilda asteroids. Astron. Astrophys. 302, 907-914. Dell’Oro, A., Marzari, P., Paolicchi F., Dotto, E., Vanzani, V., 1998. Trojan collision probability: a statistical approach. Astron. Astrophys. 339, 272-277. Doressoundiram, A., Peixinho, N., Doucet, C., Mousis, O., Barucci, M. A., Petit, J. M., Veillet, C., 2005. The Meudon Multicolor Survey (2MS) of Centaurs and trans-neptunian objects: extended dataset and status on the correlations reported. Icarus 174, 90–104. Dotto, E., Fornasier, S., Barucci, M. A., Licandro, J., Boehnhardt, H., Hainaut, O., Marzari, F., de Bergh, C., De Luise, F., 2006. The surface com- position of Jupiter Trojans: Visible and Near–Infrared Survey of Dynamical Families. Icarus 183, 420-434 Dumas, C., Owen, T., Barucci, M. A., 1998. Near-Infrared Spectroscopy of Low-Albedo Surfaces of the Solar System: Search for the Spectral Signa- ture of Dark Material. Icarus 133, 221-232. Emery, J. P., Brown, R. H., 2003. Constraints on the surface composition of Trojan asteroids from near-infrared (0.8-4.0 µm) spectroscopy. Icarus 164, 104-121. Emery, J. P., Brown, R. H., 2004. The surface composition of Trojan asteroids: constraints set by scattering theory. Icarus 170, 131-152. Emery, J. P., Cruikshank, D. P., Van Cleve, J., 2006. Thermal emission spectroscopy (5.2 38 µm of three Trojan asteroids with the Spitzer Space Telescope: Detection of fine-grained silicates. Icarus 182, 496-512. Fernandez Y. R., Sheppard, S. S., Jewitt, D. C., 2003. The Albedo Dis- tribution of Jovian Trojan Asteroids. Astron. J. 126, 1563-1574. Fitzsimmons, A., Dahlgren, M., Lagerkvist, C. I., Magnusson, P., Williams, I. P., 1994. A spectroscopic survey of D-type asteroids. Astron. Astrophys. 282, 634-642. Fornasier, S., Lazzarin, M., Barbieri, C., Barucci, M. A., 1999. Spec- troscopic comparison of aqueous altered asteroids with CM2 carbonaceous chondrite meteorites. Astron. Astrophys. 135, 65-73 Fornasier, S., Dotto, E., Marzari, F., Barucci, M.A., Boehnhardt, H., Hainaut, O., de Bergh, C., 2004a. Visible spectroscopic and photometric survey of L5 Trojans : investigation of dynamical families. Icarus, 172, 221– Fornasier, S., Doressoundiram, A., Tozzi, G. P., Barucci, M. A., Boehn- hardt, H., de Bergh, C., Delsanti A., Davies, J., Dotto, E., 2004b. ESO Large Program on Physical Studies of Trans-Neptunian Objects and Centaurs: fi- nal results of the visible spectroscopic observations. Astron. Astrophys. 421, 353-363. Hainaut, O. R., Delsanti, A. C., 2002. Colors of Minor Bodies in the Outer Solar System. A statistical analysis. Astron. Astroph. 389, 641-664. Hudson, R.L., Moore, M.H. 1999. Laboratory Studies of the Formation of Methanol and Other Organic Molecules by Water+Carbon Monoxide Ra- diolysis: Relevance to Comets, Icy Satellites, and Interstellar Ices. Icarus 140, 451-461. Jewitt, D. C., Luu, J. X., 1990. CCD spectra of asteroids. II - The Tro- jans as spectral analogs of cometary nuclei. Astron. J. 100, 933-944. Jewitt, D. C., Trujillo, C. A., Luu, J. X., 2000. Population and Size Dis- tribution of Small Jovian Trojan Asteroids. Astron. J. 120, 1140-1147 Landolt, A. U., 1992. UBVRI photometric standard stars in the magni- tude range 11.5–16.0 around the celestial equator. Astron. J . 104, 340-371, 436-491. Lazzaro, D., Angeli, C. A., Carvano, J. M., Mothé-Diniz, T., Duffard, R., Florczak, M., 2004. S3OS2: the visible spectroscopic survey of 820 asteroids. Icarus 172, 179–220. Levison, H., Shoemaker, E. M., Shoemaker, C. S., 1997. The dispersal of the Trojan asteroid swarm. Nature 385, 42-44. Marzari, F., Farinella, P., Davis, D. R., Scholl, H., Campo Bagatin, A., 1997. Collisional Evolution of Trojan Asteroids. Icarus 125, 39-49. Marzari, F., Scholl, H., 1998a. Capture of Trojans by a Growing Proto- Jupiter. Icarus 131, 41-51. Marzari, F., Scholl, H., 1998b. The growth of Jupiter and Saturn and the capture of Trojans. Astron. Astroph. 339, 278-285 Marzari, F., Scholl, H., Murray, C., Lagerkvist, C., 2002. Origin and Evo- lution of Trojan Asteroids. In Asteroids III, W. F. Bottke Jr., A. Cellino, P. Paolicchi, and R. P. Binzel (eds), University of Arizona Press, Tucson, 725-738. Marzari, F., Tricarico, P., Scholl, H., 2003. Stability of Jupiter Trojans investigated using frequency map analysis: the MATROS project. MNRAS 345, 1091-1100. Milani, A., 1993. The Trojan asteroid belt: Proper elements, stability, chaos and families. Celest. Mech. Dynam. Astron. 57, 59-94. Morbidelli, A., Levison, H. F., Tsiganis, K., Gomes, R., 2005. Chaotic capture of Jupiter’s Trojan asteroids in the early Solar System. Nature 435, 462-465. Moore, M.H., Donn, B., Khanna, R., A’Hearn, M.F., 1983. Studies of proton-irradiated cometary-type ice mixtures. Icarus 54, 388-405. Moroz L., Baratta G., Strazzulla G., Starukhina L., Dotto E., Barucci M.A., Arnold G., Distefano E. 2004. Optical alteration of complex organics induced by ion irradiation: 1. Laboratory experiments suggest unusual space weathering trend. Icarus 170, 214-228. Peixinho, N., Doressoundiram, A., Delsanti, A., Boehnhardt, H., Barucci, M. A., Belskaya, I., 2003. Reopening the TNOs color controversy: Centaurs bimodality and TNOs unimodality. Astron. Astrophys. 410, 29–32. Shoemaker, E. M., Shoemaker, C. S., Wolfe, R. F., 1989. Trojan aster- oids: populations, dynamical structure and origin of the L4 and L5 swarms. In Binzel, Gehrels, Matthews (Eds.), Asteroids II. Univ. of Arizona Press, Tucson, pp. 487-523. Strazzulla, G., 1998. Chemistry of Ice Induced by Bombardment with Energetic Charged Particles. In Solar System Ices (B. Schmitt, C. de Bergh, M. Festou, eds.), Kluwer Academic, Dordrecht, Astrophys. Space Sci Lib. Thompson, W.R., Murray, B.G.J.P.T., Khare, B.N., Sagan, C. 1987. Col- oration and darkening of methane clathrate and other ices by charged particle irradiation - Applications to the outer solar system. JGR 92, 14933-14947. Xu, S., Binzel, R. P., Burbine, T. H., Bus, S. J., 1995. Small main-belt asteroid spectroscopic survey: Initial results. Icarus 115, 1–35. Vilas, F. 1994. A quick look method of detecting water of hydration in small solar system bodies. LPI 25, 1439-1440. Zappala, V., Cellino, A., Farinella, P., Knez̆ević, Z., 1990. Asteroid fam- ilies. I - Identification by hierarchical clustering and reliability assessment. Astron. J. 100, 2030-2046. Tables Table 1: Observing conditions of the investigated L5 asteroids. For each object we report the observational date and universal time, total exposure time, number of acquisitions with exposure time of each acquisition, airmass, and the observed solar analogs with their airmass. Obj Date UT Texp (s) nexp air. Solar An. (air.) Anchises 1173 17 Jan 05 06:06 60 1×60s 1.42 HD76151 (1.48) 23549 17 Jan 05 07:20 480 2×240s 1.60 HD76151 (1.48) 24452 17 Jan 05 07:54 960 4×240s 1.44 HD76151 (1.48) 47967 17 Jan 05 05:34 800 2×400s 1.38 HD76151 (1.48) 2001 SB173 17 Jan 05 06:28 1200 2×600s 1.35 HD76151 (1.48) Cloanthus 5511 19 Jan 05 06:04 960 4×240s 1.26 HD76151 (1.12) 51359 19 Jan 05 04:13 660 1×660s 1.36 HD76151 (1.12) Misenus 11663 17 Jan 05 05:13 400 1×400s 1.21 HD44594 (1.12) 32794 18 Jan 05 03:13 1800 2×900s 1.39 HD28099 (1.44) 56968 17 Jan 05 04:31 400 2×400s 1.21 HD44594 (1.12) 1988 RE12 18 Jan 05 04:12 2000 2×1000s 1.31 HD28099 (1.44) 2000 SC51 18 Jan 05 06:09 1320 2×660s 1.16 HD44594 (1.17) 2001 UY123 18 Jan 05 06:46 1320 2×660s 1.32 HD44594 (1.17) Phereclos 9030 18 Jan 05 08:19 1000 1×1000s 1.37 HD44594 (1.17) 11488 19 Jan 05 03:31 1320 2×660s 1.99 HD76151 (1.12) 31820 19 Jan 05 07:02 1320 2×660s 1.35 HD76151 (1.11) Sarpedon 48252 18 Jan 05 02:32 1320 2×660s 1.30 HD28099 (1.44) 84709 19 Jan 05 05:35 1320 2×660s 1.34 HD76151 (1.12) Panthoos 4829 17 Jan 05 08:37 720 3×240s 1.45 HD76151 (1.48) 30698 18 Jan 05 01:54 1320 2×660s 1.73 HD28099 (1.44) 31821 18 Jan 05 05:27 1320 2×660s 1.35 HD28099 (1.44) 76804 17 Jan 05 03:35 1800 3×600s 1.38 HD44594 (1.12) 2001 VK85 18 Jan 05 07:31 2000 2×1000s 1.23 HD44594 (1.17) Table 2: Observing conditions of the investigated L4 asteroids. For each object we report the observational date and universal time, total exposure time, number of acquisitions with exposure time of each acquisition, airmass, and the observed solar analogs with their airmass. Obj Date UT Texp (s) nexp air. Solar An. (air.) Eurybates 3548 25 May 04 05:14 600 2×300s 1.02 SA107-684 (1.19) 9818 26 May 04 00:13 780 1×780s 1.19 SA102-1081(1.15) 13862 25 May 04 03:35 1200 2×600s 1.09 SA107-998 (1.15) 18060 25 May 04 02:47 1500 2×750s 1.07 SA107-998 (1.15) 24380 25 May 04 06:53 780 1×780s 1.18 SA107-684 (1.19) 24420 25 May 04 08:49 900 1×900s 1.59 SA112-1333 (1.17) 24426 26 May 04 00:13 1440 2×720s 1.13 SA107-684 (1.17) 28958 26 May 04 07:14 1800 2×900s 1.35 SA107-684 (1.17) 39285 25 May 04 05:40 2700 3×900s 1.09 SA107-684 (1.19) 43212 25 May 04 07:39 2340 3×780s 1.39 SA110-361 (1.15) 53469 25 May 04 02:05 1800 2×900s 1.04 SA107-998 (1.15) 65150 26 May 04 01:59 3600 4×900s 1.07 SA102-1081 (1.20) 65225 26 May 04 03:40 3600 4×900s 1.04 SA107-684 (1.17) 1996RD29 26 May 04 05:12 2700 3×900s 1.10 SA107-684 (1.17) 2000AT44 25 May 04 04:14 1800 2×900s 1.04 SA107-684 (1.19) 2002CT22 26 May 04 00:49 2400 4×600s 1.08 SA102-1081 (1.15) 2002EN68 26 May 04 08:10 1800 2×900s 1.62 SA107-684 (1.17) 1986 WD 4035 10 Apr 03 03:28 600 1×600s 1.09 SA107-684 (1.15) 6545 10 Apr 03 02:39 900 1×900s 1.16 SA107-684 (1.15) 11351 10 Apr 03 09:21 900 1×900s 1.28 SA107-684 (1.15) 14707 11 Apr 03 08:11 1200 1×1200s 1.15 SA107-684 (1.15) 24233 11 Apr 03 02:29 1200 1×1200s 1.39 SA107-684 (1.37) 24341 11 Apr 03 05:47 900 1×900s 1.16 SA107-684 (1.17) 1986 TS6 12921 10 Apr 03 07:33 900 1×900s 1.39 SA107-684 (1.40) Table 3: Visible photometric observations of L4 and L5 Trojans (ESO-NTT EMMI): for each object, date, computed V magnitude, B-V, V-R and V- I colors are reported. The given UT is for the V filter acquisition. The observing photometric sequence (V-R-B-I) took a few minutes. Object date UT V B-V V-R V-I 1986 WD 4035 10 Apr 03 03:11 16.892±0.031 0.752±0.040 0.473±0.042 0.926±0.055 4035 10 Apr 03 04:22 16.981±0.031 0.752±0.040 0.495±0.042 0.945±0.055 6545 10 Apr 03 02:22 17.558±0.031 0.734±0.041 0.499±0.042 0.935±0.055 11351 10 Apr 03 09:03 18.407±0.032 0.739±0.044 0.498±0.044 0.900±0.057 14707 11 Apr 03 06:46 18.666±0.031 0.751±0.041 0.401±0.033 0.804±0.055 14707 11 Apr 03 08:37 18.873±0.031 0.754±0.041 0.424±0.033 0.790±0.056 24233 11 Apr 03 01:33 18.894±0.034 0.704±0.051 0.481±0.037 0.899±0.058 24341 11 Apr 03 05:05 19.376±0.032 0.713±0.043 0.369±0.035 0.759±0.057 1986 TS6 12921 10 Apr 03 07:12 18.393±0.031 0.673±0.040 0.421±0.042 0.786±0.055 L5 cut off 150m/s Anchises 1173 17 Jan 05 05:54 16.595±0.024 0.811±0.034 0.402±0.035 0.805±0.038 23549 17 Jan 05 07:09 18.969±0.050 0.800±0.071 0.485±0.068 0.872±0.075 24452 17 Jan 05 07:48 18.757±0.043 0.872±0.056 0.441±0.056 0.847±0.066 47967 17 Jan 05 05:27 19.382±0.044 0.899±0.058 0.489±0.069 0.965±0.075 2001 SB173 17 Jan 05 06:20 19.882±0.043 0.992±0.060 0.503±0.064 0.927±0.078 Cloanthus 5511 19 Jan 05 05:52 17.968±0.020 0.906±0.027 0.442±0.027 0.968±0.032 51359 19 Jan 05 03:54 19.631±0.102 0.864±0.201 0.447±0.131 0.885±0.164 Misenus 11663 17 Jan 05 05:05 18.473±0.022 0.837±0.030 0.409±0.030 0.872±0.039 32794 18 Jan 05 03:07 19.685±0.038 0.923±0.065 0.393±0.056 0.879±0.057 56968 17 Jan 05 04:18 18.596±0.026 0.986±0.040 0.494±0.033 1.003±0.036 1988 RE12 18 Jan 05 04:00 20.892±0.081 0.826±0.132 0.388±0.108 0.871±0.106 2000 SC51 18 Jan 05 06:03 19.876±0.038 1.016±0.055 0.444±0.059 0.896±0.056 2001 UY123 18 Jan 05 06:41 19.869±0.047 0.890±0.058 0.537±0.056 0.971±0.063 Phereclos 9030 18 Jan 05 08:14 18.397±0.020 0.887±0.024 0.493±0.027 0.973±0.028 11488 19 Jan 05 02:57 18.931±0.066 0.868±0.101 0.430±0.079 0.848±0.084 31820 19 Jan 05 06:39 20.041±0.077 0.889±0.093 0.520±0.091 0.916±0.123 Sarpedon 48252 18 Jan 05 02:25 19.878±0.060 0.949±0.100 0.467±0.093 0.903±0.090 84709 19 Jan 05 05:10 19.862±0.068 0.855±0.087 0.462±0.090 1.010±0.094 Panthoos 4829 17 Jan 05 08:18 18.430±0.029 0.851±0.050 0.420±0.039 0.792±0.052 30698 18 Jan 05 01:45 19.353±0.036 – 0.472±0.042 0.865±0.047 31821 18 Jan 05 05:21 19.328±0.076 0.980±0.111 0.440±0.097 0.901±0.108 76804 17 Jan 05 03:21 19.471±0.065 0.803±0.082 0.446±0.070 0.889± 0.080 2001 VK85 18 Jan 05 07:23 20.179±0.038 0.822±0.063 0.462±0.048 1.020±0.050 Table 4: L5 families. We report for each target the absolute magnitude H and the estimated diameter (diameters marked by ∗ are taken from IRAS data), the spectral slope S computed between 5500 and 8000 Å and the taxonomic class (T) derived following Dahlgren & Lagerkvist (1995) classi- fication scheme. The asteroids marked with a were observed by Fornasier et al. (2004a), and their spectral slope values have been recomputed in the 5500-8000 Å wavelength range; asteroids 23694, 30698 and 32430, previously Astyanax members, have been reassigned to the Panthoos family due to re- fined proper elements. Obj H D (km) S (%/103Å) T Anchises 1173 8.99 ∗126+11 3.87±0.70 P 23549 12.04 26+4 8.49±0.88 D 24452 11.85 29+5 7.42±0.70 D 47967 12.15 25+4 9.21±0.78 D 2001 SB173 12.77 19+3 14.78±0.99 D Cloanthus 5511 10.43 55+8 10.84±0.65 D 51359 12.25 24+6 12.63±1.30 D Misenus 11663 10.95 44+7 6.91±0.70 DP 32794 12.77 19+3 6.59±0.88 DP 56968 11.72 30+5 15.86±0.71 D 1988 RE12 13.20 16+2 4.68±1.20 P 2000 SC51 12.69 20+3 6.54±0.98 DP 2001 UY123 12.75 19+3 8.28±0.88 D Phereclos a2357 8.86 ∗95+4 9.91±0.68 D a6998 11.43 34+5 11.30±0.75 D 9030 11.14 40+6 10.35±0.76 D a9430 11.47 35+5 10.02±0.90 D 11488 11.82 29+5 5.37±0.92 PD a18940 11.81 29+4 7.13±0.75 D 31820 12.63 20+3 7.53±0.80 D Sarpedon a2223 9.25 ∗95+4 10.20±0.65 D a5130 9.85 71+11 10.45±0.65 D a17416 12.83 18+3 10.80±0.90 D a25347 11.59 33+5 10.11±0.83 D 48252 12.84 18+3 9.62±0.82 D 84709 12.70 19+3 11.64±0.84 D Panthoos 4829 11.16 39+6 5.03±0.70 PD a23694 11.61 32+5 8.20±0.72 D 30698 12.14 25+4 8.23±1.00 D a30698 12.27 25+4 9.08±0.82 D a32430 12.23 25+4 8.12±1.00 D 31821 11.99 27+4 10.58±0.82 D 76804 12.16 25+4 7.29±0.71 D 2001 VK85 12.79 19+3 14.39±0.81 D Table 5: L4 Families. We report for each target the absolute magnitude H and the estimated diameter (diameters marked by ∗ are taken from IRAS data, while absolute magnitudes marked by ∗∗ are taken from the astorb.dat file of the Lowell Observatory), the spectral slope S computed between 5500 and 8000 Å, and the taxonomic class (T) derived following Dahlgren & Lagerkvist (1995) classification scheme. The asteroids marked with a were observed by Dotto et al. (2006), and their spectral slope values have been recomputed in the 5500-8000 Å wavelength range. Obj H D (km) S (%/103Å) T Eurybates 3548 9.50∗∗ ∗72+4 -0.18±0.57 C 9818 11.00∗∗ 42+6 2.12±0.72 P 13862 11.10∗∗ 40+6 1.59±0.70 C 18060 11.10∗∗ 40+6 2.86±0.60 P 24380 11.20∗∗ 38+6 0.34±0.65 C 24420 11.50∗∗ 33+5 1.65±0.70 C 24426 12.50∗∗ 21+3 4.64±0.80 P 28958 12.10∗∗ 25+4 -0.04±0.80 C 39285 12.90∗∗ 17+3 0.25±0.69 C 43212 12.30∗∗ 23+4 1.19±0.78 C 53469 11.80∗∗ 29+4 0.17±0.80 C 65150 12.90∗∗ 17+3 4.14±0.70 P 65225 12.80∗∗ 18+3 0.97±0.85 C 1996RD29 13.06∗∗ 16+3 2.76±0.89 P 2000AT44 12.16∗∗ 24+3 -0.53±0.83 C 2002CT22 12.04∗∗ 26+4 2.76±0.73 P 2002EN68 12.30∗∗ 23+3 3.60±0.98 P 1986 WD 4035 9.72 ∗68+5 9.78±0.61 D a4035 9.30∗∗ ∗68+5 15.19±0.61 D 6545 10.42 55+8 11.32±0.63 D a6545 10.00∗∗ 66+10 9.88±0.56 D 11351 10.88 44+7 10.26±0.67 D a11351 10.50∗∗ 53+8 10.44±0.61 D 14707 11.25 38+6 −9.4 -1.06±1.00 C 24233 11.58 33+5 −8.0 6.37±0.67 DP 24341 11.99 27+4 -0.26±0.71 C 1986 TS6 12917 11.61 32+5 10.98±0.68 D 12921 11.12 40+6 4.63±0.75 P a12921 10.70∗∗ 48+7 3.74±1.00 P 13463 11.27 37+6 4.37±0.65 P 15535 10.70 48+7 10.67±0.65 D 20738 11.67 31+5 8.84±0.70 D 24390 11.80 29+5 9.53±0.62 D Table 6: Results of the statistical analysis on the spectral slope distribution as a function of the diameters. For each test bin, the average slope and the dispersion are listed; the size of the sample is reported in parenthesis. For each pair of subsamples, the probability that both are randomly extracted from the same global sample is listed, as estimated by the t-, f- and ks-test, respectively. Low probability indicates significant differences between the subsamples. Diameter range 0–25 km 25–50 km 50–75 km 75–100 km >100 km S average±σ 7.17±4.79 (22) 6.92±4.69 (48) 8.91±4.68 (26) 6.74±5.85 (21) 7.87±2.88 (21) (%/103Å) 0–25 0.842 0.876 0.579 0.213 0.903 0.575 0.792 0.370 0.775 0.551 0.017 0.494 25–50 0.088 0.985 0.150 0.897 0.216 0.519 0.286 0.011 0.275 50–75 0.176 0.289 0.469 0.344 0.019 0.440 75–100 0.442 0.001 0.469 Table 7: Mean color indices and spectral slope of different classes of minor bodies of the outer Solar System. For each class the number of objects considered is also listed. Color Plutinos Cubewanos Centaurs Scattered Comets Trojans B-V 36 87 29 33 2 74 0.895± 0.190 0.973± 0.174 0.886± 0.213 0.875± 0.159 0.795± 0.035 0.777± 0.091 V-R 38 96 30 34 19 80 0.568± 0.106 0.622± 0.126 0.573± 0.127 0.553± 0.132 0.441± 0.122 0.445± 0.048 V-I 34 64 25 25 7 80 1.095± 0.201 1.181± 0.237 1.104± 0.245 1.070± 0.220 0.935± 0.141 0.861± 0.090 V-J 10 14 11 8 1 12 2.151± 0.302 1.750± 0.456 1.904± 0.480 2.041± 0.391 1.630± 0.000 1.551± 0.120 V-H 3 7 11 4 1 12 2.698± 0.083 2.173± 0.796 2.388± 0.439 2.605± 0.335 1.990± 0.000 1.986± 0.177 V-K 2 5 9 2 1 12 2.763± 0.000 2.204± 1.020 2.412± 0.396 2.730± 0.099 2.130± 0.000 2.125± 0.206 R-I 34 64 25 26 8 80 0.536± 0.135 0.586± 0.148 0.548± 0.150 0.517± 0.102 0.451± 0.059 0.416± 0.057 J-H 11 17 21 11 1 12 0.403± 0.292 0.370± 0.297 0.396± 0.112 0.348± 0.127 0.360± 0.000 0.434± 0.064 H-K 10 16 20 10 1 12 -0.034± 0.171 0.084± 0.231 0.090± 0.142 0.091± 0.136 0.140± 0.000 0.139± 0.041 Slope 38 91 30 34 8 80 (%/103Å) 19.852± 10.944 25.603± 13.234 20.601± 13.323 18.365± 12.141 10.722± 6.634 7.241± 3.909 Table 8: Statistical tests performed to compare the color and slope distributions of different classes of minor bodies (Plt= Plutinos, Resonant TNOs; QB1= Cubiwanos, Classical TNOs; Cent= Centaurs; Scat= scattered TNOs; Com= Short Period Comet nuclei) with those of Trojans. The first five columns consider all the Trojans, the second five only the Eurybates family, the third five only the non-Eurybates family Trojans. For each color, the first line shows the number of objects used for the comparison (2nd is the number of Trojans), and the second line reports the probability resulting from the test. A very low value indicates that the two compared distributions are not statistically compatible. Probabilities are in boldface when the size of the samples is large enough for the value to be meaningful. f-test Color All Trojans Only Eurybates Only NON-Eurybates Plt QB1 Cent Scat Com Plt QB1 Cent Scat Com Plt QB1 Cent Scat Com B-V 36 74 83 74 29 74 33 74 2 74 36 14 83 14 29 14 33 14 2 14 36 60 83 60 29 60 33 60 2 60 0.000 0.000 0.000 0.000 0.600 0.001 0.001 0.000 0.005 0.722 0.000 0.000 0.000 0.000 0.598 V-R 38 80 92 80 30 80 34 80 19 80 38 17 92 17 30 17 34 17 19 17 38 63 92 63 30 63 34 63 19 63 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 R-I 34 80 62 80 25 80 26 80 8 80 34 17 62 17 25 17 26 17 8 17 34 63 62 63 25 63 26 63 8 63 0.000 0.000 0.000 0.000 0.773 0.000 0.000 0.000 0.001 0.335 0.000 0.000 0.000 0.000 0.185 Slope 38 80 87 80 30 80 34 80 8 80 38 17 87 17 30 17 34 17 8 17 38 63 87 63 30 63 34 63 8 63 0.000 0.000 0.000 0.000 0.020 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 t-test Color All Trojans Only Eurybates Only NON-Eurybates Plt QB1 Cent Scat Com Plt QB1 Cent Scat Com Plt QB1 Cent Scat Com B-V 36 74 83 74 29 74 33 74 2 74 36 14 83 14 29 14 33 14 2 14 36 60 83 60 29 60 33 60 2 60 0.001 0.000 0.012 0.002 0.608 0.000 0.000 0.001 0.000 0.139 0.003 0.000 0.025 0.006 0.858 V-R 38 80 92 80 30 80 34 80 19 80 38 17 92 17 30 17 34 17 19 17 38 63 92 63 30 63 34 63 19 63 0.000 0.000 0.000 0.000 0.916 0.000 0.000 0.000 0.000 0.083 0.000 0.000 0.000 0.000 0.532 R-I 34 80 62 80 25 80 26 80 8 80 34 17 62 17 25 17 26 17 8 17 34 63 62 63 25 63 26 63 8 63 0.000 0.000 0.000 0.000 0.154 0.000 0.000 0.000 0.000 0.001 0.000 0.000 0.001 0.000 0.502 Slope 38 80 87 80 30 80 34 80 8 80 38 17 87 17 30 17 34 17 8 17 38 63 87 63 30 63 34 63 8 63 0.000 0.000 0.000 0.000 0.185 0.000 0.000 0.000 0.000 0.008 0.000 0.000 0.000 0.000 0.404 KS-test Color All Trojans Only Eurybates Only NON-Eurybates Plt QB1 Cent Scat Com Plt QB1 Cent Scat Com Plt QB1 Cent Scat Com B-V 36 74 83 74 29 74 33 74 2 74 36 14 83 14 29 14 33 14 2 14 36 60 83 60 29 60 33 60 2 60 0.001 0.000 0.001 0.004 0.330 0.002 0.000 0.035 0.000 0.065 0.003 0.000 0.002 0.047 0.468 V-R 38 80 92 80 30 80 34 80 19 80 38 17 92 17 30 17 34 17 19 17 38 63 92 63 30 63 34 63 19 63 0.000 0.000 0.000 0.000 0.040 0.000 0.000 0.000 0.000 0.008 0.000 0.000 0.000 0.000 0.056 R-I 34 80 62 80 25 80 26 80 8 80 34 17 62 17 25 17 26 17 8 17 34 63 62 63 25 63 26 63 8 63 0.000 0.000 0.000 0.000 0.201 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.587 Slope 38 80 87 80 30 80 34 80 8 80 38 17 87 17 30 17 34 17 8 17 38 63 87 63 30 63 34 63 8 63 0.000 0.000 0.000 0.000 0.088 0.000 0.000 0.000 0.000 0.002 0.000 0.000 0.000 0.000 0.211 Figure captions Fig. 1 - Reflectance spectra of 5 Anchises family members (L5 swarm). The photometric color indices are also converted to relative reflectance and overplotted on each spectrum. Spectra and photometry are shifted by 0.5 in reflectance for clarity. Fig. 2 - Reflectance spectra of 6 Misenus family members (L5 swarm). The photometric color indices are also converted to relative reflectance and overplotted on each spectrum. Spectra and photometry are shifted by 0.5 in reflectance for clarity. Fig. 3 - Reflectance spectra of 5 Panthoos family members (L5 swarm). The photometric color indices are also converted to relative reflectance and overplotted on each spectrum. Spectra and photometry are shifted by 0.5 in reflectance for clarity. For asteroid 30698, the B-V color is missing as a B filter measurement was not available. Fig. 4 - Reflectance spectra of 2 Cloantus, 3 Phereclos and 2 Sarpedon family members (L5 swarm). The photometric color indices are also con- verted to relative reflectance and overplotted on each spectrum. Spectra and photometry are shifted by 1.0 in reflectance for clarity. Fig. 5 - Reflectance spectra of the 17 Eurybates family members (L4 swarm). Spectra are shifted by 0.5 in reflectance for clarity. Fig. 6 - Reflectance spectra of the 6 1986 WD family members and 12921, which is a member of the 1986 TS6 family (all belonging to the L4 swarm). Spectra are shifted by 1.0 in reflectance for clarity. Fig. 7 - Plot of the spectral slope versus the estimated diameter for the families observed in the L5 swarm. Fig. 8 - Plot of the spectral slope versus the estimated diameter for the families observed in the L4 swarm. Fig. 9 - Plot of the observed spectral slopes versus the estimated diameter for the whole population of Jupiter Trojans investigated by us and available from the literature. The errors on slopes and diameters are not plotted to avoid confusion. Fig. 10 - Histogram of L5 Trojans taxonomic classes. Fig. 11 - Histogram of L4 Trojans taxonomic classes (Neg indicates ob- jects with negative spectral slope). Fig. 12 - Color distributions as functions of the absolute magnitude M(1, 1), the inclination i [degrees], the orbital semi-major axis a [AU], the perihelion distance q [AU], the eccentricity e, and the orbital energy E (see text for definition). We include all the available colors for distant minor bod- ies (TNOs, Centaurs, and cometary nuclei, see Hainaut & Delsanti 2002). The Plutinos (resonant TNOs) are red filled triangles, Cubiwanos (classical TNOs) are pink filled circles, Centaurs are green open triangles, Scattered TNOs are blue open circles, and Trojans are cyan filled triangles. Fig. 13 - V −R versus R−I color-color diagram for the observed Trojans and all distant minor bodies available in the updated Hainaut & Delsanti (2002) database. The solid symbols are for the Trojans (square for Eurby- bates, triangles for others). The open symbols are used as following: tri- angles for Plutinos, circles for Cubiwanos, squares for Centaurs, pentagons for Scattered, and starry square for Comets. The continuous line represents the ”reddening line”, that is the locus of objects with a linear reflectivity spectrum. The star symbol represents the Sun. Fig. 14 - Cumulative function and histograms of the B − V and V − R color distributions and of the spectral slope for all the considered classes of objects. The dotted line marks the solar colors. 4000 5000 6000 7000 8000 9000 Figure 1: 4000 5000 6000 7000 8000 9000 Figure 2: 4000 5000 6000 7000 8000 9000 Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: Figure 11: M(1,1) E a [AU] q [AU] Figure 12: Figure 13: Figure 14: Introduction Observations and data reduction Results Dynamical families: L5 swarm Anchises Misenus Panthoos Cloantus Phereclos Sarpedon Dynamical families: L4 swarm Eurybates 1986 WD 1986 TS6 Discussion Size vs spectral slope distribution:Individual families Size vs slope distribution: The Trojan population as a whole Spectral slopes and L4/L5 Clouds Orbital Elements Comparison with other outer Solar System minor bodied Introduction and methods Results Conclusions

0704.0351

FIRST-based survey of Compact Steep Spectrum sources, V. Milliarcsecond-scale morphology of CSS objects

Astronomy & Astrophysics manuscript no. 6364 c© ESO 2021 June 8, 2021 FIRST-based survey of compact steep spectrum sources V. Milliarcsecond-scale morphology of CSS objects M. Kunert-Bajraszewska1 and A. Marecki1 Toruń Centre for Astronomy, N. Copernicus University, 87-100 Toruń, Poland Received 8 September 2006; Accepted 7 March 2007 ABSTRACT Aims. Multifrequency VLBA observations of the final group of ten objects in a sample of FIRST-based compact steep spectrum (CSS) sources are presented. The sample was selected to investigate whether objects of this kind could be relics of radio−loud AGNs switched off at very early stages of their evolution or possibly to indicate intermittent activity. Methods. Initial observations were made using MERLIN at 5 GHz. The sources have now been observed with the VLBA at 1.7, 5 and 8.4 GHz in a snapshot mode with phase-referencing. The resulting maps are presented along with unpublished 8.4-GHz VLA images of five sources. Results. Some of the sources discussed here show a complex radio morphology and therefore a complicated past that, in some cases, might indicate intermittent activity. One of the sources studied – 1045+352 – is known as a powerful radio and infrared-luminous broad absorption line (BAL) quasar. It is a young CSS object whose asymmetric two-sided morphology on a scale of several hundred parsecs, extending in two different directions, may suggest intermittent activity. The young age and compact structure of 1045+352 is consistent with the evolution scenario of BAL quasars. It has also been confirmed that the submillimetre flux of 1045+352 can be seriously contaminated by synchrotron emission. Key words. galaxies: active, galaxies: evolution, quasars: absorption lines 1. Introduction Following early hypotheses (Phillips & Mutel, 1982; Carvalho, 1985) suggesting that the gigahertz-peaked spectrum (GPS) and compact steep spectrum (CSS) could be young objects, Readhead et al. (1996) proposed an evolutionary scheme uni- fying three classes of radio-loud AGNs (RLAGNs): symmet- ric GPS objects – CSOs (compact symmetric objects); sym- metric CSS objects – MSOs (medium-sized symmetric objects) and large symmetric objects (LSOs). In this scheme GPS/CSO sources with linear sizes less than 1 kpc1 would evolve into CSS/MSOs with subgalactic sizes (<20 kpc) and these in turn would eventually become LSOs during their lifetimes. Two pieces of evidence definitely point towards GPS/CSS sources being young objects: lobe proper motions (up to 0.3c) giving kinematic ages as low as ∼103 years for CSOs (Owsianik et al., 1998; Giroletti et al., 2003; Polatidis & Conway, 2003) and ra- diative ages typically ∼105 years for MSOs (Murgia et al., 1999). Although these AGNs are small-scale objects, in some cases CSO/GPS sources are associated with much larger ra- dio structures that extend out to many kiloparsecs. In these cases, it has been suggested that the CSO/GPS stage rep- resents a period of renewed activity in the life cycle of the AGN (Stanghellini et al., 2005, and references therein). Reynolds & Begelman (1997) have also proposed a model in which extragalactic radio sources are intermittent on timescales Send offprint requests to: M. Kunert-Bajraszewska e-mail: [email protected] 1 For consistency with earlier papers in this field, the following cosmological parameters have been adopted throughout this paper: H0=100 km s −1 Mpc−1 and q0=0.5. Throughout this paper, the spectral index is defined such that S ∝ να. of ∼104–105 years. Following the above scenarios and also an earlier suggestion by Readhead et al. (1994) and O’Dea & Baum (1997) that there exists a large population of compact, short- lived objects, Marecki et al. (2003, 2006) concluded that the evo- lutionary track proposed by Readhead et al. (1996) is only one of many possible tracks. A lack of stable fuelling from the black hole can inhibit the growth of a radio source, and consequently it will never reach the LSO stage, at least in a given phase of its activity. Observational support for the above ideas has been pro- vided by Gugliucci et al. (2005). They calculated the kinematic ages for a sample of CSOs with well-identified hotspots. It ap- pears that the kinematic age distribution drops sharply above ∼500 years, suggesting that in many CSOs activity may cease early. It is, therefore, possible that only some of them evolve any further. Our observations have shown that young, fading compact sources do indeed exist (Kunert-Bajraszewska et al., 2005; Marecki et al., 2006; Kunert-Bajraszewska et al., 2006, hereafter Papers II, III, and IV, respectively). A double source, 0809+404, described in Paper IV is our best example of a very compact – i.e. very young – fader. The VLBA multifrequency observations have shown it to have a diffuse, amorphous struc- ture, devoid of a dominant core and hotspots. Giroletti et al. (2005) have analysed the properties of a sample of small-size sources and found a very good example of a kiloparsec-scale fader (1855+37). It is to be noted that re-ignition of activity in compact radio sources is not ruled out. In this paper – the fifth and the last of the series – VLBA observations of 10 CSS and CSO sources that are potential candidates for compact faders or objects with intermittent activity are presented. One of these sources, 1045+352, is of particular interest not only because it http://arxiv.org/abs/0704.0351v2 2 M. Kunert-Bajraszewska and A. Marecki: FIRST-based survey of compact steep spectrum sources has a puzzling radio structure, but it also appears to be a broad absorption line (BAL) quasar. As their name somewhat suggests, BAL quasars have very broad, blue-shifted absorption lines arising from high-ionization transitions such as C IV, Si IV, N V, etc. (e.g C IV 1549Å). They constitute ∼10% of the optically selected radio-quiet quasars with the absorption arising from gas outflow at velocities up to ∼0.2 c (Hewett & Foltz, 2003). In fact, BAL quasars have been divided into two categories, as 10% of them also show absorp- tion troughs in low-ionization lines such as Mg II 2800Å. This group has been designated as LoBAL quasars and the others as HiBAL ones. The high ionization level and continuous absorp- tion over a wide velocity range is hard to reconcile with absorp- tion by individual clouds. Rather, they indicate that BAL regions exist in both BAL and non-BAL quasars and evidence, accumu- lated from optically selected BAL quasars, indicates an orienta- tion hypothesis to explain their nature. It would appear that BAL quasars are normal quasars seen along a particular line of sight, e.g. a line of sight skimming the edge of the accretion disk or torus (Weymann et al., 1991; Elvis, 2000). Murray et al. (1995) have proposed a model in which the line of sight to a BAL quasar intersects an outflow or wind that is not entirely radial, e.g. an outflow that initially emerges perpendicular to the accretion disk and is then accelerated radially. For quite a long time it was believed that BAL quasars were never radio-loud. This view was challenged by Becker et al. (1997), who discovered the first radio-loud BAL quasar when using the VLA FIRST survey to select quasar candidates. Five radio-loud BAL quasars were then identified in NVSS by Brotherton et al. (1998). Since then, the number of radio- loud BAL QSOs has increased considerably (Becker et al., 2000; Menou et al., 2001), following identification of new quasar can- didates selected from the FIRST survey. Most of the BAL quasars in the Becker et al. (2000) sample tended to be com- pact at radio frequencies with either a flat or steep radio spec- trum. Those with steep spectra could be related to GPS and CSS sources. A variety of their spectral indices also suggested a wide range of orientations, contrary to the interpretation favoured from optically selected quasars. Moreover, Becker et al. (2000) indicated that the frequency of BAL quasars in their sample was significantly greater (factor ∼2) than inferred from optically se- lected samples and that the frequency of BAL quasars appeared to show a complex dependence on radio loudness. The radio morphology of BAL quasars is important because it can indicate inclination in BALs, and therefore yields a di- rect test of the orientation model. However, information about the radio structure of BAL quasars is still very limited. Prior to 2006, only three BAL quasars, FIRST J101614.3+520916 (Gregg et al., 2000), PKS 1004+13 (Wills et al., 1999), and LBQS 1138−0126 (Brotherton et al., 2002) were known to have a double-lobed FR II radio morphology on kiloparsec scales, although this interpretation was doubtful for PKS 1004+13 (Gopal-Krishna & Wiita, 2000). Recently, the population of FR II-BAL quasars has increased to ten objects (excluding PKS 1004+13) following the discoveries of Gregg et al. (2006) and Zhou et al. (2006), although some of these still require confir- mation. Their symmetric structures indicate an “edge-on” ori- entation, which in turn supports an alternative hypothesis de- scribed as “unification by time”, with BAL quasars charac- terised as young or recently refuelled quasars (Becker et al., 2000; Gregg et al., 2000). There has been only one attempt (at 1.6 GHz with the EVN) to image radio structures of the smallest (and possibly the youngest) BAL quasars (Jiang & Wang, 2003) from the Becker et al. (2000) sample. This paper presents high frequency VLBA images of another very compact BAL quasar — 1045+352, which makes it the BAL quasar with the best known radio structure to date. 2. The observations and data reduction The five papers of this series are concerned with a sample of 60 candidates selected from the VLA FIRST catalogue (White et al., 1997)2 which could be weak CSS sources. The sample selection criteria have been given in Kunert et al. (2002) (hereafter Paper I). All the sources were initially observed with MERLIN at 5 GHz and the results of these observations led to the selection of several groups of objects for further study with MERLIN and the VLA (Paper II), as well as the VLBA and the EVN (Papers III and IV). The last of those groups contains 10 sources that, because of their structures (very faint “haloes” or possible core-jet structures), were not included in the other groups as they were less likely to be candidates for faders. However, to complete the investigation of the primary sample, 1.7, 5, and 8.4-GHz VLBA observations of 10 sources listed in Table 1 together with their basic properties, were carried out on 13 November 2004 in a snapshot mode with phase-referencing.3 Each target source scan was interleaved with a scan on a phase reference source and the total cycle time (target and phase- reference) was ∼9 minutes including telescope drive times, with ∼7 minutes actually on the target source per cycle. The cycles for a given target-calibrator pair were grouped and rotated round the three frequencies, although the source 1059+351 was only observed at 1.7 GHz with the VLBA because of its very low flux density as measured at 5 GHz by MERLIN (13 mJy). The whole data reduction process was carried out using standard AIPS procedures but, in addition to this, corrections for Earth orientation parameter (EOP) errors introduced by the VLBA correlator also had to be made. For each target source and at each frequency, the corresponding phase-reference source was mapped, and the phase errors so determined were applied to the target sources, which were then mapped using a few cycles of phase self-calibration and imaging. For some of the sources a final amplitude self-calibration was also applied. IMAGR was used to produce the final “naturally weighted”, total intensity im- ages shown in Figs. 1 to 10. Three of the ten sources (1056+316, 1302+356, 1627+289) were not detected in the 8.4-GHz VLBA observations, and 1425+287 has not been detected in any VLBA observations. Flux densities of the principal components of the sources were measured using the AIPS task JMFIT and are listed in Table 3. In addition to the observations described above, unpub- lished 8.4-GHz VLA observations of five sources – 1056+316, 1126+293, 1425+287, 1627+289, 1302+356 – made in A-conf. by Glen Langston (first four objects) and Patnaik et al. (1992) have been included (Figs. 3, 5, 9, 10, and 7, respectively). It was realised that because of poor u-v coverage at the higher frequencies, some flux density could be missing and the resul- tant spectral index maps were not considered to be reliable. Any calculation of spectral indices from the flux densities quoted in Table 3 should also be treated only as coarse approximations. For 1045+352, 30-GHz continuum observations using the Toruń 32-m radio telescope and a prototype (two-element 2 Official website: http://sundog.stsci.edu 3 Including this paper, the results of the observations of 46 sources out of 60 candidates from the primary sample have been published. The observations of 14 objects failed for different reasons. http://sundog.stsci.edu M. Kunert-Bajraszewska and A. Marecki: FIRST-based survey of compact steep spectrum sources 3 Table 1. Basic parameters of target sources Source RA Dec ID mR z S 1.4 GHz logP1.4GHz S 4.85 GHz α 4.85GHz 1.4GHz LAS LLS Name h m s ◦ ′ ′′ mJy W Hz−1 mJy ′′ h−1 kpc (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) 1045+352 10 48 34.247 34 57 24.99 Q 20.86 1.604 1051 27.65 439 −0.70 ∼0.50 2.1 1049+384 10 52 11.797 38 11 43.83 G 20.76 1.018 712 27.04 205 −1.00 0.14 0.6 1056+316 10 59 43.236 31 24 20.59 G 21.10 0.307∗ 459 25.72 209 −0.63 0.50 1.4 1059+351 11 02 08.686 34 55 10.74 G 19.50 0.594∗ 702 26.52 252 −0.82 3.03 11.5 1126+293 11 29 21.738 29 05 06.40 EF — — 729 — 213 −0.99 0.79 — 1132+374 11 35 05.927 37 08 40.80 G — 2.880 638 28.00 218 −0.86 ∼0.30 1.1 1302+356 13 04 34.477 35 23 33.93 EF — — 483 — 185 −0.77 ∼0.20 — 1407+369 14 09 09.528 36 42 08.06 q 21.51 0.996∗ 538 26.89 216 −0.73 ∼0.25 1.1 1425+287 14 27 40.281 28 33 25.78 EF — — 859 — 198 −1.18 0.75 — 1627+289 16 29 12.290 28 51 34.25 EF — — 526 — 162 −0.95 ∼0.65 — Description of the columns: (1) source name in the IAU format; (2) source right ascension (J2000) extracted from FIRST; (3) source declination (J2000) extracted from FIRST; (4) optical identification: G - galaxy, Q - quasar, EF - empty field, q - star-like object, i.e. unconfirmed QSO; (5) red magnitude extracted from SDSS/DR5; (6) redshift; (7) total flux density at 1.4 GHz extracted from FIRST; (8) log of the radio luminosity at 1.4 GHz; (9) total flux density at 4.85 GHz extracted from GB6; (10) spectral index between 1.4 and 4.85 GHz calculated using flux densities in columns (7) and (9); (11) largest angular size (LAS) measured in the 5-GHz MERLIN image – in most cases, as a separation between the outermost component peaks, otherwise “∼” means measured in the image contour plot; (12) largest linear size (LLS). ∗ photometric redshift extracted from SDSS/DR5 receiver) of the One-Centimeter Receiver Array (OCRA-p, Lowe et al., 2005) have also been made. The recorded output from the receiver was the difference between the signals from two closely-spaced horns effectively separated in azimuth so that atmospheric variations were mostly cancelled out. The ob- serving technique was such that the respective two beams were pointed at the source alternately with a switching cycle of ∼50 seconds for a period of ∼6 minutes, thus measuring the source flux density relative to the sky background on either side of the source. The telescope pointing was determined from azimuth and elevation scans across the point source Mrk 421. The pri- mary flux density calibrator that was used was the planetary neb- ula NGC 7027, which has an effective radio angular size of ∼8 arcseconds (Bryce et al., 1997) and for which a correction of the flux density scale had to be made. However, as NGC 7027 was at some distance from the target source, the point source 1144+402 was used as a secondary flux density calibrator. Corrections for the effects of the atmosphere were determined from system tem- perature measurements at zenith distances of 0◦ and 60◦. 3. Comments on individual sources 1045+352. The MERLIN and VLBA maps (Fig. 1) show this source to be extended in both the NE/SW and NW/SE directions. The central compact feature visible in all the maps is probably a radio core with a steep spectrum. The VLBA image at 1.7 GHz shows two symmetric protrusions – possibly jets – straddling the core in a NE/SW direction, the SW emission being weaker than in the NE. This structure is aligned with the NE/SW emission visible in the 5-GHz MERLIN image, but the more extended dif- fuse emission has been resolved out in the VLBA images. The 5-GHz VLBA image shows a core and a one-sided jet pointing to the East. Some compact features in a NE direction are also visible. The radio structure in the 8.4-GHz VLBA image is sim- ilar to that at 5 GHz: an extended radio core and a jet pointing in an easterly direction. The observed radio morphology of 1045+352 could indicate a restart of activity with the NE/SW radio emission being the first phase of activity, now fading away, and the extension in the NW/SE direction being a signature of the current active phase. However, the above is only one of a number of possible interpre- tations of the structure of 1045+352 – see further discussion in Sect. 4. According to Sloan Digital Sky Survey/Data Release 5 (SDSS/DR5), 1045+352 is a galaxy at RA= 10h48m34.s242, Dec=+34◦57′24.′′95, which is marked with a cross in the MERLIN map but the spectral observations carried out by Willott et al. (2002) have shown 1045+352 to be a quasar with a redshift of z = 1.604. It has been also classified as a HiBAL object based upon the observed very broad C IV absorption, and it is a very luminous submillimetre object with detections at both 850µm and 450µm (Willott et al., 2002). The total flux of 1045+352 at 30 GHz measured by us using OCRA-p is S 30GHz=69 mJy±7 mJy, which gives a steep spectral index α = −1.01 between 4.85 GHz and 30 GHz. 1049+384. The 5-GHz MERLIN image (Fig. 2) shows it as a triple core-jet structure with the brightest component re- solved into a double structure extended in a NW/SE direc- tion in the high resolution VLBA observations. The 1.7-GHz VLBA image shows four radio components (in agreement with Dallacasa et al., 2002), whereas the 5-GHz and 8.4-GHz VLBA maps show only three components. However, the 5-GHz VLBA image published by Orienti et al. (2004) shows all four compo- nents, and they suggest that the two western components and the two eastern ones are two independent radio sources. As pointed by Orienti et al. (2004), it is difficult to classify the object, al- though the idea that 1049+384 consists of two separate com- pact, double sources is not very plausible because of the very small separation, ∼ 0.09′′ (0.4 kpc), between these two poten- tial objects. Although the spectral index calculations are very uncertain, it is suggested that one of the eastern components at RA= 10h52m11.s797, Dec=+38◦11′44.′′027 is a radio core (in agreement with Orienti et al., 2004) from which jets emerge al- ternately in opposite directions. 1049+384 is a galaxy with a redshift z = 1.018 (Riley & Warner, 1994), but according to Allington-Smith et al. (1988) the optical spectrum of 1049+384 shows interme- diate properties between a galaxy and a quasar. The opti- 4 M. Kunert-Bajraszewska and A. Marecki: FIRST-based survey of compact steep spectrum sources 1045+352 4994.000 MHz peak flux density=230.21 mJy/beam, beam size=56 x 41 mas first contour level=0.12 mJy/beam RIGHT ASCENSION (J2000) 10 48 34.30 34.28 34.26 34.24 34.22 34.20 34 57 25.8 1045+352 1667.474 MHz peak flux density=118.71 mJy/beam, beam size=13.1 x 8.2 mas first contour level=0.80 mJy/beam RIGHT ASCENSION (J2000) 10 48 34.256 34.254 34.252 34.250 34.248 34.246 34.244 34.242 34.240 34 57 25.14 25.12 25.10 25.08 25.06 25.04 25.02 25.00 24.98 24.96 24.94 1045+352 4987.474 MHz peak flux density=13.64 mJy/beam, beam size=4.7 x 2.4 mas first contour level=0.14 mJy/beam RIGHT ASCENSION (J2000) 10 48 34.254 34.252 34.250 34.248 34.246 34.244 34 57 25.12 25.10 25.08 25.06 25.04 25.02 25.00 24.98 1045+352 8421.474 MHz peak flux density=4.03 mJy/beam, beam size=2.7 x 1.5 mas first contour level=0.14 mJy/beam RIGHT ASCENSION (J2000) 10 48 34.251 34.250 34.249 34.248 34.247 34.246 34.245 34 57 25.08 25.07 25.06 25.05 25.04 25.03 25.02 25.01 Fig. 1. The MERLIN 5-GHz (upper left) and VLBA 1.7, 5, and 8.4-GHz maps of 1045+352. Contours increase by a factor 2, and the first contour level corresponds to ≈ 3σ. A cross indicates the position of an optical object found using the SDSS/DR5. cal object was included in SDSS/DR5 (RA= 10h52m11.s802, Dec=+38◦11′44.′′00) and is marked in all maps with a cross. 1056+316. The 8.4-GHz VLA image (Fig. 3) shows this source to have a double structure that, in the 5-GHz MERLIN image, has been resolved into a radio core and probably a hotspot in a NW radio lobe. Both components are visible in the 1.7-GHz VLBA image, but neither has been detected in the higher fre- quency VLBA images. The two weak features on either side of the NW component in the 1.7-GHz VLBA image may be the remains of extended emission that has been resolved out. The optical counterpart of 1056+316 was included in SDSS/DR5 (RA= 10h59m43.s145, Dec=+31◦24′23.′′31), to- gether with a photometric redshift (Table 1). Its position is marked with a cross in 8.4-GHz VLA map. 1059+351. The 5-GHz MERLIN map (Fig. 4) shows a bright component that is probably a radio core, on almost opposite sides of which is emission from compact features (hotspots) within the two radio lobes. This structure agrees with the 1.4- GHz VLA observations presented by Gregorini et al. (1988) and Machalski & Condon (1983). Their images clearly show an S- shaped morphology of 1059+351 with two very diffuse compo- nents, the brighter one resolved into a double structure in 5-GHz VLA observations (Machalski, 1998). One of these two com- ponents is the NW hotspot visible in the 5-GHz MERLIN map, and the second is probably a radio core visible in both the 5-GHz MERLIN and 1.7-GHz VLBA images. The optical counterpart of 1059+351 was included in SDSS/DR5 (RA= 11h02m08.s727, Dec=+34◦55′08.′′79), to- gether with a photometric redshift (Table 1). The position of the optical object is marked with a cross in all maps and is well correlated with the position of the radio core. Machalski (1998) also measured a photometric redshift for 1059+351, which is z = 0.37 and which differs from that in SDSS/DR5. 1126+293. The VLA 8.4-GHz and MERLIN 5-GHz maps (Fig. 5) show three radio components, the brighter one proba- bly being the core that was resolved into a core-jet structure in M. Kunert-Bajraszewska and A. Marecki: FIRST-based survey of compact steep spectrum sources 5 1049+384 4994.000 MHz peak flux density=116.27 mJy/beam, beam size=62 x 38 mas first contour level=0.40 mJy/beam RIGHT ASCENSION (J2000) 10 52 11.84 11.83 11.82 11.81 11.80 11.79 11.78 11.77 11.76 38 11 44.6 1049+384 1667.474 MHz peak flux density=181.64 mJy/beam, beam size=11.6 x 8.2 mas first contour level=0.09 mJy/beam RIGHT ASCENSION (J2000) 10 52 11.810 11.805 11.800 11.795 11.790 11.785 38 11 44.14 44.12 44.10 44.08 44.06 44.04 44.02 44.00 43.98 43.96 43.94 1049+384 4987.474 MHz peak flux density=21.07 mJy/beam, beam size=4.2 x 2.3 mas first contour level=0.09 mJy/beam RIGHT ASCENSION (J2000) 10 52 11.805 11.800 11.795 11.790 38 11 44.08 44.07 44.06 44.05 44.04 44.03 44.02 44.01 44.00 43.99 43.98 1049+384 8421.474 MHz peak flux density=68.39 mJy/beam, beam size=2.6 x 1.2 mas first contour level=0.15 mJy/beam RIGHT ASCENSION (J2000) 10 52 11.800 11.798 11.796 11.794 11.792 11.790 11.788 11.786 38 11 44.06 44.05 44.04 44.03 44.02 44.01 44.00 43.99 Fig. 2. The MERLIN 5-GHz (upper left) map and VLBA 1.7, 5, and 8.4-GHz maps of 1049+384. Contours increase by a factor 2, and the first contour level corresponds to ≈ 3σ. Crosses indicate the position of an optical object found using the SDSS/DR5 . the 1.7-GHz VLBA image. The source was not detected in the 5 and 8.4-GHz VLBA observations. 1132+374. The 5-GHz MERLIN image shows (Fig. 6) a core-jet structure that was resolved into a triple CSO object in the 1.7- GHz VLBA image. The 5 and 8.4-GHz VLBA images show only two components: a hotspot in the NE lobe and a radio core. This source is identified with a very high redshift (z = 2.88) galaxy (Eales & Rawlings, 1996). 1302+356. This source was observed with the VLA at 8.4 GHz as a part of the JVAS survey (Patnaik et al., 1992). The result- ing map shows a slightly extended EW object (Fig. 7). The 5- GHz MERLIN image shows this to be a double source, and the weak (∼10 mJy) eastern component could be part of a jet. The bright component was resolved into a diffuse structure in the 1.7- GHz VLBA image. The 5-GHz VLBA image shows only a sin- gle component at the position of the maximum emission in the 1.7-GHz VLBA image, which is probably a radio core (Fig. 7). There is no trace of this source in the 8.4-GHz VLBA image. 1407+369. The 5-GHz MERLIN image shows a core-jet struc- ture in a NW direction that is resolved into a core and jet in 6 M. Kunert-Bajraszewska and A. Marecki: FIRST-based survey of compact steep spectrum sources 1056+316 8439.900 MHz peak flux density=118.41 mJy/beam, beam size=270 x 257 mas first contour level=0.06 mJy/beam RIGHT ASCENSION (J2000) 10 59 43.45 43.35 43.25 43.15 43.05 31 24 23 1056+316 4994.000 MHz peak flux density=80.83 mJy/beam, beam size=60 x 43 mas first contour level=0.16 mJy/beam RIGHT ASCENSION (J2000) 10 59 43.28 43.26 43.24 43.22 43.20 31 24 21.2 1056+316 1667.474 MHz peak flux density=9.10 mJy/beam, beam size=13.9 x 5.5 mas first contour level=0.30 mJy/beam RIGHT ASCENSION (J2000) 10 59 43.265 43.255 43.245 43.235 43.225 31 24 20.6 Fig. 3. The VLA 8.4-GHz map, MERLIN 5-GHz map, and VLBA 1.7-GHz map of 1056+316. Contours increase by a factor 2, and the first contour level corresponds to ≈ 3σ. A cross on the VLA map indicates the position of an optical object found using the SDSS/DR5. 1059+351 4994.000 MHz peak flux density=10.03 mJy/beam, beam size=89 x 69 mas first contour level=0.15 mJy/beam RIGHT ASCENSION (J2000) 11 02 08.85 08.80 08.75 08.70 08.65 08.60 34 55 10.0 1059+351 1667.474 MHz peak flux density=8.07 mJy/beam, beam size=10.9 x 7.9 mas first contour level=0.08 mJy/beam RIGHT ASCENSION (J2000) 11 02 08.735 08.730 08.725 08.720 34 55 08.80 08.75 08.70 08.65 08.60 Fig. 4. The MERLIN 5-GHz map and VLBA 1.7-GHz map of 1059+351. Contours increase by a factor 2, and the first contour level corresponds to ≈ 3σ. Crosses indicate the position of an optical object found using the SDSS/DR5. all the VLBA maps (Fig. 8). The optical object was included in SDSS/DR5 (RA= 14h09m09.s509, Dec=+36◦42′08.′′15) and is marked with a cross in all maps. The redshift quoted in Table 1 is photometric. 1425+287. Both the VLA 8.4-GHz and MERLIN 5-GHz images (Fig. 9) show a double structure for this source. The brighter component seems to be a radio core, although this cannot be confirmed because the source was not detected in the VLBA ob- servations (Fig. 9). 1627+289. Both the VLA 8.4-GHz and MERLIN 5-GHz images (Fig. 10) show this source to have a core-jet structure. The 1.7- GHz VLBA image shows only the central extended feature that was resolved into a core-jet structure in the 5-GHz VLBA image. The source was not detected in the 8.4-GHz VLBA image. 4. Discussion 4.1. 1045+352 — a BAL quasar 1045+352 is a HiBAL quasar with a very reddened spectrum showing a C IV broad absorption system (Willott et al., 2002). Its projected linear size is only 2.1 kpc, which is consistent with the observation of Becker et al. (2000) that, amongst radio loud quasars, broad absorption lines are more commonly observed in the smallest radio sources. It is a very luminous submillimetre object, which together with the template dust spectrum adopted by Willott et al. (2002), indicates this source to be a hyperluminous infrared quasar, with large amounts of dust in its host galaxy. Although 1045+352 is quite luminous at 151 MHz (2.88 Jy, Waldram et al., 1996), which suggests the presence of some extended emission and which, indeed, appears to be present in our MERLIN 5-GHz M. Kunert-Bajraszewska and A. Marecki: FIRST-based survey of compact steep spectrum sources 7 1126+293 8439.900 MHz peak flux density=65.97 mJy/beam, beam size=368 x 310 mas first contour level=0.08 mJy/beam RIGHT ASCENSION (J2000) 11 29 23.9 23.8 23.7 23.6 23.5 23.4 29 05 01 04 59 1126+293 4994.000 MHz peak flux density=60.53 mJy/beam, beam size=62 x43 mas first contour level=0.14 mJy/beam RIGHT ASCENSION (J2000) 11 29 21.80 21.75 21.70 21.65 29 05 07.5 1126+293 1667.474 MHz peak flux density=6.03 mJy/beam, beam size=14.0 x 4.2 mas first contour level=0.15 mJy/beam RIGHT ASCENSION (J2000) 11 29 21.762 21.760 21.758 21.756 21.754 21.752 21.750 21.748 21.746 29 05 06.50 06.48 06.46 06.44 06.42 06.40 06.38 06.36 06.34 06.32 06.30 Fig. 5. The VLA 8.4-GHz map, MERLIN 5-GHz map and VLBA 1.7-GHz map of 1126+293. Contours increase by a factor 2, and the first contour level corresponds to ≈ 3σ. image, the VLBA maps show the radio structure to be domi- nated by jets and a core. The 30-GHz flux density of 1045+352 is also high, as would be expected from the VLBA structure. Consequently, there could be synchrotron contamination of the submillimetre flux. As shown by Blundell et al. (1999), either the first-order or second-order polynomials can accurately pre- dict the shape of the radio spectrum. Both models have been ap- plied to the radio data of 1045+352 taken from the literature and from this paper (Fig. 11), and show that a non-thermal compo- nent could constitute at least ∼40% of the entire 850µm flux (the parabolic fit). The linear fit agrees with calculations based upon the 1.25 mm flux measured by Haas et al. (2006), who derived a value of 94% for the non-thermal component part of the detected 850µm flux. It has to be noted here that the linear fit should be treated as an upper limit for the synchrotron emission at submil- limetre wavelengths, since the spectrum may steepen in the inter- val between 30 GHz and the SCUBA wavebands. However, the above can indicate values of infrared emission and dust mass of 1045+352 lower than estimated (Willott et al., 2002). This also appears be consistent with the findings of Willott et al. (2003), who have shown that there is no difference between the submil- limetre luminosities of BAL and non-BAL quasars, which sug- gest that a large dust mass is not required for quasars to show BALs. The radio luminosity at 1.4 GHz is high (Table 1), making this source one of the most radio-luminous BAL quasars, with a value similar to that of the first known radio-loud BAL QSO with an FR II structure, FIRST J101614.3+520916 (Gregg et al., 2000). Following Stocke et al. (1992), a radio-loudness param- eter, R∗, defined as the K-corrected ratio of the 5-GHz radio flux to 2500Å optical flux (Table 2) was calculated. For this, a global radio spectral index, αradio = −0.8 and an optical spec- tral index, αopt = −1.0, were assumed, and the SDSS g ′ mag- nitude defined by Fukugita et al. (1996) was converted to the Johnson-Morgan-Cousins B magnitude using the formula given by Smith et al. (2002). Corrections were also made for intrin- 8 M. Kunert-Bajraszewska and A. Marecki: FIRST-based survey of compact steep spectrum sources 1132+374 4994.000 MHz peak flux density=122.40 mJy/beam, beam size=58 x 44 mas first contour level=0.18 mJy/beam RIGHT ASCENSION (J2000) 11 35 05.98 05.96 05.94 05.92 05.90 05.88 37 08 41.6 1132+374 1667.474 MHz peak flux density=38.64 mJy/beam, beam size=9.5 x 4.0 mas first contour level=0.40 mJy/beam RIGHT ASCENSION (J2000) 11 35 05.940 05.938 05.936 05.934 05.932 05.930 05.928 05.926 37 08 40.86 40.84 40.82 40.80 40.78 40.76 40.74 40.72 40.70 40.68 40.66 1132+374 4987.474 MHz peak flux density=12.57 mJy/beam, beam size=3.1 x 1.2 mas first contour level=0.14 mJy/beam RIGHT ASCENSION (J2000) 11 35 05.936 05.934 05.932 05.930 05.928 37 08 40.82 40.80 40.78 40.76 40.74 40.72 40.70 1132+374 8421.474 MHz peak flux density=8.74 mJy/beam, beam size=2.2 x 1.5 mas first contour level=0.10 mJy/beam RIGHT ASCENSION (J2000) 11 35 05.936 05.935 05.934 05.933 05.932 05.931 05.930 37 08 40.83 40.82 40.81 40.80 40.79 40.78 40.77 40.76 40.75 40.74 Fig. 6. The MERLIN 5-GHz (upper left) map and VLBA 1.7, 5, and 8.4-GHz maps of 1132+374. Contours increase by a factor 2, and the first contour level corresponds to ≈ 3σ. sic extinction (local to the quasar) calculated by Willott et al. (2002), who assumed a Milky-Way extinction curve. Even af- ter correction, log(R∗) > 1, which means that 1045+352 is still radio-loud object. The angle between the jet axis and the line of sight can be estimated using the core radio-to-optical lumi- nosity ratio defined by Wills & Brotherton (1995) as log(RV ) = log(Lcore) + 0.4MV − 13.69, where Lcore is a radio luminosity of the core at 5-GHz rest frequency (the core flux density at 5 GHz were taken from the VLBA image; see also Table 3), and MV is the K-corrected absolute magnitude calculated using transfor- mation equation V = g′−0.55(g′−r′)−0.03 (Smith et al., 2002). From this, a value of ∼3.2 has been obtained for 1045+352, im- plying an angle in the range θ ∼ 10◦ − 30◦ for the jet in the observed asymmetric MERLIN 5-GHz radio morphology, and can explain the high value of the radio-loudness parameter. An assumption of θ = 20◦ yields the deprojected linear size of the source of ∼ 6 kpc. As shown by White et al. (2006), BAL QSOs are systematically brighter than non-BAL objects, which indi- cates we are looking closer to the jet axis in quasars with BALs. Based upon the small inclination angles of their BAL quasars, Zhou et al. (2006) suggest that BAL features can be caused by polar disk winds. Also, Saikia et al. (2001) and Jeyakumar et al. (2005) found that the radio properties of CSS sources are con- sistent with the unified scheme in which the axes of the quasars are observed close to the line of sight. On the other hand, it has been shown (Saikia et al., 2001; Jeyakumar et al., 2005) that many CSS objects interact with an asymmetric medium in the central regions of their host galaxies, and this can cause the ob- served asymmetries. It is then likely that, also in the case of the CSS quasar 1045+352, the environmental asymmetries might play an important role. The jet power can be estimated from the relationship between the radio luminosity and the jet power given by Willott et al. (1999, Eq.(12)). However, because some of the flux density of the 1045+352 can be beamed, the calcu- lations have to be treated as an approximation. Assuming the 151-MHz flux density, which accounts for the extended emis- M. Kunert-Bajraszewska and A. Marecki: FIRST-based survey of compact steep spectrum sources 9 1302+356 8452.400 MHz peak flux density=109.96 mJy/beam, beam size=252 x 230 mas first contour level=0.09 mJy/beam RIGHT ASCENSION (J2000) 13 04 34.75 34.70 34.65 34.60 34.55 34.50 34.45 34.40 34.35 34.30 35 23 36 1302+356 4994.500 MHz peak flux density=129.54 mJy/beam, beam size=62 x 39 mas first contour level=0.18 mJy/beam RIGHT ASCENSION (B1950) 13 02 13.86 13.84 13.82 13.80 13.78 13.76 13.74 13.72 13.70 13.68 35 39 38.5 1302+356 1667.474 MHz peak flux density=19.62 mJy/beam, beam size=10.0 x 4.0 mas first contour level=0.14 mJy/beam RIGHT ASCENSION (J2000) 13 04 34.502 34.500 34.498 34.496 34.494 34.492 34.490 34.488 34.486 35 23 33.64 33.62 33.60 33.58 33.56 33.54 33.52 33.50 33.48 33.46 33.44 1302+356 4987.474 MHz peak flux density=4.23 mJy/beam, beam size=3.8 x 1.5 mas first contour level=0.07 mJy/beam RIGHT ASCENSION (J2000) 13 04 34.498 34.497 34.496 34.495 34.494 34.493 34.492 35 23 33.57 33.56 33.55 33.54 33.53 33.52 33.51 33.50 33.49 Fig. 7. The VLA 8.4-GHz map (upper left), MERLIN 5-GHz map (upper right) and VLBA 1.7 and 5-GHz maps of 1302+356. Contours increase by a factor 2, and the first contour level corresponds to ≈ 3σ. sion and the radio emission from the jets, the jet kinetic power is Q jet ∼ 10 44erg sec−1. The projected linear size D of a radio quasar or radio galaxy can be approximately related to the time, from the trig- gering of activity, as the relationship between these variables is only weakly dependent upon the radio luminosity. Using the model of radio source evolution from Willott et al. (1999), the age of 1045+352 was estimated to be ∼ 105 years (see also Willott et al., 2002; Rawlings et al., 2004). For the calcu- lations we assumed: θ = 20◦, β = 1.5, c1 = 2.3, n100 = 3000 e− m−3, a0 = 100 kpc (see Willott et al., 1999, for defini- tions). Both the MERLIN and VLBA high frequency images have revealed that two cycles of activity may have occurred dur- ing these ∼ 105 years. The extended NE/SW emission is prob- ably the remnant of the first phase of activity, which has been very recently replaced by a new phase of activity pointing in a NW/SE direction. It has been shown by Stanghellini et al. (2005) that the extended emission observed for small-scale objects can be the remnants of an earlier period of activity in these sources. In the case of 1045+352, renewal of activity has been accompa- nied by a reorientation of the jet axis. Several processes can be used to explain a jet reorientation in AGNs. There are strong observational and theoretical grounds for believing that accretion disks around black holes may be twisted or warped, and this can be caused by a number of pos- sible physical processes. In particular, if there is a misalignment between the axis of rotating black hole and the axis of its rotating accretion disk, then the Lense-Thirring precession produces a warp in the disk. This process is called the Bardeen-Peterson ef- fect (Bardeen & Petterson, 1975). According to Pringle (1997), disk warping can also be induced by internal instabilities in the accretion disk caused by radiation pressure from the central source. A reorientation of the jet axis may also result from a merger with another black hole. Merritt & Ekers (2002) have shown that a rapid change in jet orientation can be caused by even a mi- 10 M. Kunert-Bajraszewska and A. Marecki: FIRST-based survey of compact steep spectrum sources 1407+369 4994.500 MHz peak flux density=109.87 mJy/beam, beam size=52 x 46 mas first contour level=0.12 mJy/beam RIGHT ASCENSION (J2000) 14 09 09.56 09.54 09.52 09.50 09.48 09.46 36 42 08.8 1407+369 1667.474 MHz peak flux density=147.37 mJy/beam, beam size=10.2 x 5.1 mas first contour level=0.18 mJy/beam RIGHT ASCENSION (J2000) 14 09 09.516 09.512 09.508 09.504 09.500 36 42 08.30 08.25 08.20 08.15 08.10 08.05 1407+369 4987.474 MHz peak flux density=60.90 mJy/beam, beam size=3.6 x 2.0 mas first contour level=0.16 mJy/beam RIGHT ASCENSION (J2000) 14 09 09.512 09.510 09.508 09.506 09.504 36 42 08.22 08.20 08.18 08.16 08.14 08.12 08.10 1407+369 8421.474 MHz peak flux density=24.34 mJy/beam, beam size=2.1 x 1.0 mas first contour level=0.15 mJy/beam RIGHT ASCENSION (J2000) 14 09 09.511 09.510 09.509 09.508 09.507 09.506 36 42 08.18 08.17 08.16 08.15 08.14 08.13 08.12 Fig. 8. The MERLIN 5-GHz map (upper left) and VLBA 1.7, 5, and 8.4-GHz maps of 1407+369. Contours increase by a factor 2, and the first contour level corresponds to ≈ 3σ. Crosses indicate the position of an optical object found using the SDSS/DR5. nor merger because of a spin-flip of the central active black hole arising from the coalescence of inclined binary black holes. According to Liu (2004), the Bardeen-Peterson effect can also cause a realignment of a rotating SMBH and a misaligned ac- cretion disk, where the timescale of such a realignment t < 105 years. If it is assumed that the typical speed of advance of ra- dio lobes of young AGNs is υ ∼0.3c (Owsianik et al., 1998; Giroletti et al., 2003; Polatidis & Conway, 2003), then distorted jets of length, tυ <10 kpc for some CSS and GPS sources should be observed, although the character of these disturbances is not known. Liu (2004) shows that the interaction/realignment of a binary and its accretion disk leads to the development of X- shaped sources. 1045+352 is not a typical X-shaped source like 3C 223.1 or 3C 403 (Dennett-Thorpe et al., 2002; Capetti et al., 2002). However, according to Cohen et al. (2005) the realign- ment of a rotating SMBH followed by a repositioning of the ac- cretion disk and jets is a plausible interpretation for misaligned radio structures, even if they are not conspicuously X-shaped. It is likely that in young sources such as 1045+352, the gas has not yet settled into a regular disk following a merger event and that separate clouds of gas and dust reaching the very central regions of the source at different times disturb the sta- bility of the accretion disk and affect the jet formation. Later, these clouds could cause a renewal of activity. Numerical simu- lations of colliding galaxies show that these usually merge com- pletely after a few encounters in timescales up to ∼ 108 years (Barnes & Hernquist, 1996). According to Schoenmakers et al. (2000), multiple encounters between interacting galaxies can cause interruptions of activity and lead to the many types of sources that are observed in a restarted phase, such as double- double radio galaxies. Nevertheless, it is unclear whether such encounters can cause jet reorientation. On the other hand, the dense medium of a host galaxy can frustrate the jets, and their collisions with the dense surrounding medium can cause rapid bends through large angles. In the case of 1045+352, the VLBA images at the higher frequencies seem to show a jet emerging in M. Kunert-Bajraszewska and A. Marecki: FIRST-based survey of compact steep spectrum sources 11 1425+287 8439.900 MHz peak flux density=64.89 mJy/beam, beam size=305 x 267 mas first contour level=0.08 mJy/beam RIGHT ASCENSION (J2000) 14 27 38.7 38.6 38.5 38.4 38.3 38.2 28 33 17 1425+287 4994.500 MHz peak flux density=62.37 mJy/beam, beam size=74 x 40 mas first contour level=0.80 mJy/beam RIGHT ASCENSION (J2000) 14 27 40.40 40.35 40.30 40.25 40.20 28 33 27.5 Fig. 9. The VLA 8.4-GHz map and MERLIN 5-GHz map of 1425+287. Contours increase by a factor 2, and the first contour level corresponds to ≈ 3σ. Frequency (Hz) 1045+352 Fig. 11. Spectral Energy Distribution (SED) of 1045+352 from radio to submillimetre wavelengths. The errors are smaller than the size of the symbols; 1.25 mm point (Haas et al., 2006) is shown as a triangle, 850µm and 450 µm points (Willott et al., 2002) are shown as filled circles, radio observations are shown as asterisks. The solid curve is the parabolic fit f (x) = ax2+bx+c to all radio data (yi), with a = −0.14, b = 1.91, c = −5.68, and reduced χ2 = 12. The dashed curve is the linear fit f (x) = ax+ b to radio data with ν > 1GHz, with a = −0.86, b = 7.91, and reduced χ2 = 0.5. a S/SE direction, but being bent through ∼ 60◦ to a NE direction in the lower resolution 1.7-GHz image. The MERLIN lower res- olution 5-GHz image might indicate that the jet has been bent again and now emerges from the core in a NW direction. It is difficult to find a convincing argument in favour of one of the above-mentioned alternatives or to rule any of them out based upon the extensive multifrequency data on 1045+352 pre- sented here. However, if it is assumed that a merger is the most probable cause of the ignition and restart of activity in radio galaxies, this could mean that 1045+352 has undergone two merger events in a very short period of time (∼ 105), which is un- Table 2. 1045+352 properties Parameter Value u′ 22.12 g′ 21.38 r′ 20.81 i′ 20.14 z′ 20.08 AB 2.0 MB -22.05 (-24.05) AV 1.5 MV -22.83 (-24.33) log(R∗)(total) 4.9 (4.1) log(R∗)(core) 3.8 (3.0) Notes: Optical photometry from SDSS, corrected for Galactic extinc- tion. AV taken from Willott et al. (2002). Quantities in parentheses are corrected for intrinsic extinction. likely. More probable is that the ignition of activity in 1045+352 has occurred during a merger event that is, as yet, incomplete and that disturbed, misaligned radio jets result from the realignment of a rotating SMBH or intermittent gas injection that interrupts jet formation. 4.2. Other nine sources Three sources from our sample (1126+293, 1407+369, 1627+289) show one- or two-sided core-jet structures, indicat- ing that they are in an active phase of their evolution, although the core-jet structure of 1126+293 is controversial. Our images indicate that the western components are parts of the jet, which is possibly precessing or being bent by interactions with the inter- stellar medium. They could, however, also be hotspots of a radio lobe. Unfortunately, our high frequency VLBA observations are not sensitive enough to settle this problem. Three other sources (1056+316, 1132+374, 1425+287) have visible radio cores and parts of lobes or hotspots, indicating activity. 1132+374 is a CSO object. In the case of one source, 1059+351, the VLBA obser- 12 M. Kunert-Bajraszewska and A. Marecki: FIRST-based survey of compact steep spectrum sources 1627+289 8439.900 MHz peak flux density=75.42 mJy/beam, beam size=271 x 263 mas first contour level=0.07 mJy/beam RIGHT ASCENSION (J2000) 16 29 12.50 12.45 12.40 12.35 12.30 12.25 12.20 12.15 12.10 12.05 28 51 37 1627+289 4994.500 MHz peak flux density=77.77 mJy/beam, beam size=70 x 39 mas first contour level=0.15 mJy/beam RIGHT ASCENSION (J2000) 16 29 12.36 12.34 12.32 12.30 12.28 12.26 12.24 12.22 12.20 28 51 35.5 1627+289 1667.474 MHz peak flux density=40.35 mJy/beam, beam size=10.6 x 5.1 mas first contour level=0.30 mJy/beam RIGHT ASCENSION (J2000) 16 29 12.270 12.268 12.266 12.264 12.262 12.260 12.258 28 51 34.16 34.14 34.12 34.10 34.08 34.06 34.04 34.02 34.00 33.98 33.96 1627+289 4987.474 MHz peak flux density=5.57 mJy/beam, beam size=4.2 x 1.9 mas first contour level=0.15 mJy/beam RIGHT ASCENSION (J2000) 16 29 12.267 12.266 12.265 12.264 12.263 12.262 12.261 12.260 28 51 34.11 34.10 34.09 34.08 34.07 34.06 34.05 34.04 34.03 34.02 Fig. 10. The VLA 8.4-GHz map (upper left), MERLIN 5-GHz map (upper right), and VLBA 1.7 and 5-GHz maps of 1627+289. Contours increase by a factor 2, and the first contour level corresponds to ≈ 3σ. vations show only a radio core, although the 5-GHz MERLIN image of 1059+351 also shows remnants of the two radio lobes of its “S” shaped structure visible at the VLA resolutions (Machalski & Condon, 1983; Machalski, 1998). According to Taylor et al. (1996) and Readhead et al. (1996), “S” symmetry is observed in many compact sources and can be explained by precession of the central engine. 1059+351 is the largest source in our sample with a linear size of 45 kpc based upon its largest angular size measured from 1.46-GHz VLA image (Machalski & Condon, 1983). The compact 1049+384 and 1302+356 steep spectrum sources appeared to be low-frequency variables (LFV) at 151 MHz with very high (≥0.99) probabilities that their variabil- ity is real (Minns & Riley, 2000). According to them, LFV ob- jects are generally more compact than other CSS sources and tend to exhibit steeper spectra than typical CSS sources. This may be because of rapid spectral ageing, which might be ex- pected for frustrated sources, or it might simply be because the sources are at very high redshifts. 5. Conclusions VLBA, VLA, and MERLIN images of ten compact steep spectrum sources have been presented. One of these sources, 1045+352, is a very radio-luminous BAL quasar, whose com- plex structure suggests restarted activity. This may have resulted either from a merger event or from the infall of a cloud of gas, that had cooled in the halo of the galaxy into the core region of the source. The asymmetric radio jets of 1045+352 and the es- timated angle suggest that some of the emission can be boosted, although the intrinsic asymmetries cannot be ruled out. It has also been confirmed that the 850µm flux of 1045+352 can be severely contaminated by synchrotron emission, which may sug- gest less than previously estimated values of infrared emission and dust mass. Most of the radio-loud BAL quasars detected to M. Kunert-Bajraszewska and A. Marecki: FIRST-based survey of compact steep spectrum sources 13 Table 3. Flux densities of sources principal components from the VLBA observations Source RA DEC S1.7 GHz S5 GHz S8.4GHz θ1 θ2 PA Name h m s ◦ ′ ′′ mJy mJy mJy mas mas ◦ (1) (2) (3) (4) (5) (6) (7) (8) (9) 1045+352 10 48 34.248 34 57 25.044 303.2 − − 15.0 11.0 60 10 48 34.249 34 57 25.061 − 3.5 − 2.0 1.0 76 10 48 34.248 34 57 25.041 − 21.8 7.1 7.0 1.0 101 10 48 34.248 34 57 25.043 − 32.7 12.3 4.0 3.0 95 1049+384 10 52 11.803 38 11 44.018 13.6 − − 3.0 1.0 14 10 52 11.797 38 11 44.027 11.4 3.9 6.9 2.0 2.0 121 10 52 11.789 38 11 44.031 182.1 33.6 12.9 8.0 1.0 119 10 52 11.787 38 11 44.048 218.5 23.9 2.3 5.0 3.0 177 1056+316 10 59 43.254 31 24 20.106 8.8 − − 0.9 0.1 7 10 59 43.235 31 24 20.538 43.6 − − 33.0 8.0 6 1059+351 11 02 08.726 34 55 08.709 8.1 − − 0.7 0.3 124 1126+293 11 29 21.755 29 05 06.402 7.3 − − 3.0 1.0 84 11 29 21.753 29 05 06.401 10.4 − − 13.0 4.0 53 1132+374 11 35 05.934 37 08 40.810 124.1 6.6 1.6 18.0 2.0 57 11 35 05.932 37 08 40.775 36.3 13.8 9.4 2.0 0.4 8 11 35 05.931 37 08 40.715 14.5 − − 5.0 0.8 105 1302+356 13 04 34.495 35 23 33.534 46.8 5.9 − 11.0 6.0 97 13 04 34.494 35 23 33.538 60.5 − − 15.0 7.0 147 1407+369 14 09 09.504 36 42 08.195 81.0 1.9 − 17.0 3.0 138 14 09 09.508 36 42 08.164 192.8 76.7 42.0 8.0 1.5 141 14 09 09.508 36 42 08.152 − 9.5 4.8 0.7 0.2 140 1627+289 16 29 12.264 28 51 34.062 111.5 8.1 − 10.0 6.0 58 Description of the columns: (1) source name in the IAU format; (2) component right ascension (J2000) as measured at 1.7 GHz; (3) component declination (J2000) as measured at 1.7 GHz; (4) VLBA flux density in mJy at 1.7 GHz from the present paper; (5) VLBA flux density in mJy at 5 GHz from the present paper; (6) VLBA flux density in mJy at 8.4 GHz from the present paper; (7) deconvolved component major axis angular size at 1.7 GHz obtained using JMFIT; (8) deconvolved component minor axis angular size at 1.7 GHz obtained using JMFIT; (9) deconvolved major axis position angle at 1.7 GHz obtained using JMFIT. In the case the component is not visible in 1.7 GHz map the values for the last three columns are taken from the 5-GHz image. date have very compact radio structures similar to GPS and CSS sources which are thought to be young. Therefore, the compact structure and young age of 1045+352 fit well to the evolutionary interpretation of radio-loud BAL QSOs. According to the evolutionary model recently proposed by Lipari & Terlevich (2006), BAL quasars are young systems with composite outflows, and they are accompanied by absorption clouds. The radio-loud systems may be associated with the later stages of evolution, when jets have removed the clouds respon- sible for the generation of BALs. The effect of orientation could play a secondary role here. The above could explain the rarity of extended radio structures showing BAL features (Gregg et al., 2006). Acknowledgements. The VLBA is operated by the National Radio Astronomy Observatory (NRAO), a facility of the National Science Foundation (NSF) operated under cooperative agreement by Associated Universities, Inc. (AUI). This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Use has been made of the Sloan Digital Sky Survey (SDSS) Archive. The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions: The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, Princeton University, the United States Naval Observatory, and the University of Washington. We thank M. Gawroński for his help with the OCRA-p observations. The OCRA project was supported by the Polish Ministry of Science and Higher Education under grant 5 P03D 024 21 and the Royal Society Paul Instrument Fund. We thank P.J. Wiita for a discussion and P. Thomasson for reading of the paper and a number of suggestions. This work was supported by the Polish Ministry of Science and Higher Education under grant 1 P03D 008 30. References Allington-Smith, J., R., Spinrad, H., Djorgovski, S., & Liebert, J. 1988, MNRAS, 234, 1091 Bardeen, J. M., & Petterson, J. A. 1975, ApJ, 195, L65 Barnes, J. E., & Hernquist, L. 1996, ApJ, 471, 115 Becker, R. H., Gregg, M. D., Hook, I. M., et al. 1997, ApJ, 479, L93 Becker, R. H., White, R. L., Gregg, M. D., et al. 2000, ApJ, 538, 72 Blundell, K. M., Rawlings, S., & Willott, C. J. 1999, ApJ, 117, 677 Brotherton, M. S., van Breugel, W., Smith, R. J., et al. 1998, ApJ, 505, L7 Brotherton, M. S., Croom, S. M., De Breuck, C., Becker, R. H., & Gregg, M. D. 2002, AJ, 124, 2575 Bryce, M., Pedlar, A., Muxlow, T., Thomasson, P., & Mellema, G. 1997, MNRAS, 284, 815 Capetti, A., Zamfir, S., Rossi, P., et al. 2002, A&A, 394, 39 Carvalho, J. C. 1985, MNRAS, 215, 463 Cohen, A. S., Clarke, T. E., Ferretti, L., & Kassim, N. E. 2005, ApJ, 620, L5 Dallacasa D., Tinti, S., Fanti, C., et al. 2002, A&A, 389, 115 Dennett-Thorpe J., Scheuer, P. A. G., Laing, R. A., et al. 2002, MNRAS, 330, Eales, S., & Rawlings, S., 1996, ApJ, 460, 68 Elvis, M. 2000, ApJ, 545, 63 Fukugita, M., Ichikawa, T., Gunn, J. E., et al. 1996, AJ, 111, 1748 Giroletti, M., Giovannini, G., & Taylor, G. B. 2005, A&A, 441, 89 Giroletti, M., Giovannini, G., Taylor, G. B., et al. 2003, A&A, 399, 889 Gopal-Krishna, & Wiita, P. J. 2000, A&A, 363, 507 Gregg, M. D., Becker, R. H., Brotherton, M. S., et al. 2000, ApJ, 544, 142 Gregg, M. D., Becker, R. H., & de Vries, W. 2006, ApJ, 641, 210 Gregorini, L., Padrielli, L., Parma, P., & Gilmore, G. 1988, A&AS, 74, 107 Gugliucci, N. E., Taylor, G. B., Peck, A. B., & Giroletti, M. 2005, ApJ, 622, 136 14 M. Kunert-Bajraszewska and A. Marecki: FIRST-based survey of compact steep spectrum sources Haas, M., Chini, R., Muller, S. A. H., Bertoldi, F., & Albrecht, M. 2006, A&A, 445, 115 Hewett, P. C., & Foltz, C. B. 2003, AJ, 125, 1784 Jeyakumar, S., Wiita, P. J., Saikia, D. J., & Hooda, J. S., 2005, A&A, 432, 823 Jiang, D. R., & Wang, T. G. 2003, A&A, 397, L13 Kunert, M., Marecki, A., Spencer, R. E., Kus, A. J., & Niezgoda J. 2002, A&A, 391, 47 (Paper I) Kunert-Bajraszewska, M., Marecki, A., Thomasson, P., & Spencer, R. E. 2005, A&A, 440, 93 (Paper II) Kunert-Bajraszewska, M., Marecki, A., & Thomasson, P. 2006, A&A, 450, 945 (Paper IV) Lipari, S. L., & Terlevich, R. J. 2006, MNRAS, 368, 1001 Liu, F. K., 2004, MNRAS, 347, 1357 Lowe, S. R., 2005, PhD thesis, University of Manchester Machalski, J., & Condon, J. J. 1983, AJ, 88, 143 Machalski, J. 1998, A&AS, 128, 153 Marecki, A., Spencer, R. E., & Kunert, M. 2003, PASA, 20, 46 Marecki, A., Kunert-Bajraszewska, M., & Spencer, R. E. 2006, A&A, 449, 985 (Paper III) Menou, K., Vanden Berk, D. E., & Ivezić, Ž. 2001, ApJ, 561, 645 Merritt, D., & Ekers, R. D. 2002, Science, 297, 1310 Minns, A. R., & Riley, J. M. 2000, MNRAS, 318, 827 Murgia, M., Fanti, C., Fanti, R., et al. 1999, A&A, 345, 769 Murray, N., Chiang, J., Grossman, S. A., & Voit, G. M. 1995, ApJ, 451, 498 O’Dea, C. P., & Baum, S. A. 1997, AJ, 113, 148 Orienti, M., Dallacasa, D., Fanti C., et al. 2004, A&A, 426, 463 Owsianik, I., Conway, J. E., & Polatidis, A. G. 1998, A&A, 336, L37 Patnaik, A. R., Browne, I. W. A., Wilkinson, P. N., & Wrobel, J. M. 1992, MNRAS, 254, 655 Phillips, R. B., & Mutel, R. L. 1982, A&A, 106, 21 Polatidis, A. G., & Conway, J. E. 2003, PASA, 20, 69 Pringle, J. E. 1997, MNRAS, 292, 136 Rawlings, S., Willott, C. J., Hill, G. J., et al. 2004, MNRAS, 351, 676 Readhead, A. C. S., Xu, W., Pearson, T. J., Wilkinson, P. N., & Polatidis, A. G. 1994, in Compact Extragalactic Radio Sources, NRAO Workshop, ed. J. A. Zenzus, K. Kellermann, 17 Readhead, A. C. S., Taylor, G. B., Xu, W., et al. 1996, ApJ, 460, 612 Reynolds, C. S., & Begelman, M. C. 1997, ApJ, 487, L135 Riley, J. M., & Warner, P., J. 1994, MNRAS, 269, 166 Saikia, D. J., Jeyakumar, S., Salter, C. J., et al. 2001, MNRAS, 321, 37 Schoenmakers, A. P., de Bruyn, A. G., Röttgering, H. J. A., van der Laan, & Kaiser, C. R. 2000, MNRAS, 315, 371 Smith, J. A., Tucker, D. L., Kent, S., et al. 2002, AJ, 123, 2121 Stanghellini, C., O’Dea, C. P., Dallacasa, D., et al. 2005, A&A, 443, 891 Stocke, J. T., Morris, S. L., Weymann, J. T., & Foltz, C. B. 1992, ApJ, 396, 487 Taylor, G. B., Readhead, A. C. S., & Pearson, T. J. 1996, ApJ, 463, 95 Waldram, E. M., Yates, J. A., Riley, J. M., & Warner, P. J. 1996, MNRAS, 282, Weymann, R. J., Morris, S. L., Foltz, C. B., & Hewett, P. C. 1991, ApJ, 373, 23 White, R. L., Becker, R. H., Helfand, D. J., & Gregg, M. D. 1997, ApJ, 475, 479 White, R. L., Helfand, D. J., Becker, R. H., Glikman, E., & de Vries, W. 2007, ApJ, 654, 99 Willott, C. J., Rawlings, S., Blundell, K. M., & Lacy, M. 1999, MNRAS, 309, Willott, C. J., Rawlings, S., Archibald, E. N., & Dunlop, J. S. 2002, MNRAS, 331, 435 Willott, C. J., Rawlings, S., & Grimes, J. A. 2003, ApJ, 598, 909 Wills, B. J., & Brotherton, M. S. 1995, ApJ, 448, L81 Wills, B. J., Brandt, W. N., & Laor, A. 1999, ApJ, 520, L91 Zhou, H., Wang, T., Wang, H., et al. 2006, ApJ, 639, 716 List of Objects ‘1045+352’ on page 3 ‘1049+384’ on page 3 ‘1056+316’ on page 3 ‘1059+351’ on page 3 ‘1126+293’ on page 4 ‘1132+374’ on page 4 ‘1302+356’ on page 4 ‘1407+369’ on page 5 ‘1425+287’ on page 5 ‘1627+289’ on page 5 Introduction The observations and data reduction Comments on individual sources Discussion 1045+352 — a BAL quasar Other nine sources Conclusions

0704.0352

Investigation of relaxation phenomena in high-temperature superconductors HoBa2Cu3O7-d at the action of pulsed magnetic fields

Microsoft Word - article.doc Investigation of of of of relaxation phenomena in high-temperature superconductors HoBa2Cu3O7-δ at the action of pulsed magnetic fields J.G. Chigvinadze*, J.V. Acrivos**, S.M. Ashimov*, A.A. Iashvili*, T. V. Machaidze*, Th. Wolf*** * E. Andronikashvili Institute of Physics, 0177 Tbilisi, Georgia ** San Jose’ State University, San Jose’ CA 95192-0101, USA *** Forschungszentrum Karlsruhe, Institut für Festkörperphysik, 76021 Karlsruhe, Germany Summary It is used the mechanical method of Abrikosov vortex stimulated dynamics investigation in superconductors. With its help it was studied relaxation phenomena in vortex matter of high-temperature superconductors. It established that pulsed magnetic fields change the course of relaxation processes taking place in vortex matter. The study of the influence of magnetic pulses differing by their durations and amplitudes on vortex system of isotropic high-temperature superconductors system HoBa2Cu3O7-δ showed the presence of threshold phenomena. The small duration pulses doesn’t change the course of relaxation processes taking place in vortex matter. When the duration of pulses exceeds some critical value (threshold), then their influence change the course of relaxation process which is revealed by stepwise change of relaxing mechanical moment relτ . These investigations showed that the time for formatting of Abrikosov vortex lattice in HoBa2Cu3O7-δ is of the order of 20 �s which on the order of value exceeds the time necessary for formation of a single vortex observed in type II superconductors. 1. Introduction The present communication is devoted to the experimental investigation of relaxation phenomena in high-temperature superconductors of HoBa2Cu3O7-δ system. High-temperature superconductors are characterized by such high critical transition temperatures Тс in the superconducting state, they remain superconductors at temperatures when their thermal fluctuations energy becomes compared with the elastic energy, and also with the pinning energy [1]. It creates prerequisites for phase transitions. Due to the layered crystal structure and anisotropy, which is a characteristic high-temperature superconductors, they reveal conditions for the appearance of different phases on B-T diagram.( B is magnetic induction, T-is temperature)[2-13]. As example, Abrikosov vortex lattice begin melting near the critical Тс temperature what is followed by the essential change of vortex continuum flow dynamics along with sharp change of character (dynamics) of relaxation phenomena. In high-temperature superconductors it is observed such relaxation processes as a slow logarithmic decrease of captured flux with time at temperatures much below their superconductive critical transition temperature Тс [14-16]. The logarithmic character of relaxation is explained by the Anderson [17]. Near Тс, in the range of Abrikosov vortex lattice melting, the logarithmic character of relaxation is changed by the power one with 2/3 exponent [18]. Consequently, the study of relaxation processes in high-temperature superconductors is an important problem. 2. Experimental For Investigation it was used currentless mechanical method of Abrikosov vortex stimulated dynamics study by magnetic pulses revealing relaxation phenomena in vortex matter described in work [19]. This method is a development of currentless mechanical method of pinning investigations [20,21] and is based on pinning forces countermoments measurements and viscous friction, acting on a axially symmetrical superconducting sample in an outer (transverse) magnetic field. Countermoments of pinning forces and of viscous friction, acting on a superconductive sample from quantized vortex lines side (Abrikosov vortices) are defined the way as it was described in work [22,23]. The sensitivity of the method accordingly works [24], is equivalent to 10-8 V×cm-1 in the method of V-A characteristics. The high-temperature superconducting samples of HoBa2Cu3O7-δ system were prepared by the standard solid state reaction method. Samples were made cylindrical with height L=13mm and diameter d=6mm. Their critical temperature was Tc=92 K. The investigated samples were isotropic what was established by mechanical moment τ measurements appearing H > 1cH with the penetration of Abrikosov vortices into a freely suspended on a thin elastic thread superconducting sample. The appearance of such moment ατ sinMH= , characteristic for anisotropic superconductors, is related with penetrating Abrikosov vortices and the mean magnetic moment M of a sample which could deviate on angle α from the direction of outer magnetic field H . In superconducting anisotropic samples it is presented energetically favorable directions for the arrangement of emerging (penetrating) vortex lines which in their turn are fastened by pinning centers creating aforementioned moment τ . The lack of τ moment is characteristic for isotropic and investigated by us samples, no matter magnetic field value and its previous orientation in respect to H in the axial symmetry plane. Pulsed magnetic fields were created by Helmholtz coils. The value of pulsed magnetic fields was changed in Oeh 2002 ÷=∆ limits. In experiments it was used both single and continuous pulsed with repetition frequency ν from 2.5 s-1 to 500s-1 . The duration x of pulses was changed from 0,5 до 500 �s. Magnetic pulse could be directed both parallel h||H) and perpendicularly ( h⊥H) to applied steady magnetic field H , creating mixed state of superconducting sample. The standard pulsed generator and amplifier were used to feed Helmholtz coils. The current strength in coils reached up to 40÷50 A. Samples were high-temperature superconductors of HoBa2Cu3O7-δ system placed in the center between Helmholtz coils. The principal set-up of experiment is shown in fig.1 [19,20]. In experiments it is measured the rotation angle 2ϕ of sample depending on the angle of rotation of a torsion head 1ϕ , transmitting the rotation to a sample by means of suspension having the torsion stiffness K ≈4·10-1 [dyn•cm], which can be replaced when necessary by a less stiff or stiffer one. The measurements were carried out at a constant speed of rotation of the torsion head, making ω1=1,8·10 -2 rad/s . Angles of rotation поворота 2ϕ and 1ϕ were determined with an accuracy of ±4,6·10-3 and ±2,3·10-3 rad, respectively. The uniformity of the magnetic field’s strength along a sample was below H ∆ = 10-3. Fig. 1. The schematic diagram and the geometry of the experiment. 1-sample, 2-upper elastic filament, 3-lower filament, 4 - leading head, 5 - glass road. φ is angle between Mr and Hr To avoid effects, connected with the frozen magnetic fluxes, the lower part of the cryostat with the sample was put into a special cylindrical Permalloy screen, reducing the Earth magnetic field by the factor of 1200. After a sample was cooled by liquid nitrogen to the superconducting state, the screen was removed, a magnetic field of necessary intensity H was applied and the 2 1( )ϕ ϕ dependences were measured. To carry out measurements at different values of H , the sample was brought to the normal state by heating it to до T > cT at H =0, and only after returning sample and torsion head to the initial state 1 2 0ϕ ϕ= = , the experiment was repeated. 3. Results and discussions During rotation of the sample both of normal and superconducting states in the absence external magnetic field ( H =0) the 2ϕ dependence versus 1ϕ is linear and the condition is satisfied. tωϕϕ == 21 The character of the 2 1( )ϕ ϕ dependence is changed significantly, when the sample is in magnetic fields H > 1cH at T < cT . Typical 2 1( )ϕ ϕ dependences at T=77K and various magnetic fields for HoBa2Cu3O7-δ sample ( length of a cylindrical sample L=13mm and diameter d=6mm ) is shown in Fig.2. Fig.2. Dependence of the rotation angle of the sample HoBa2Cu3O7-δ 2ϕ on the rotation angle of the leading head 1ϕ in magnetic field H=1000 Oe at T=77K. Three distinct regions are observed in Fig.2. In the first (initial) region, the sample does not respond to the increase in 1ϕ , i.e. to the applied and increase with time torsion torque as 1ϕ ~ )( 21 ϕϕτ −= K or responds weakly. Such behavior of the sample can be explained by fact that Abrikosov vortices are not detached from pinning centers at small values of 1ϕ ~τ , but if the sample is still turned slightly, this can be caused by elastic deformation of magnetic force lines beyond it or, possibly, by separation of the most weakly fixed vortices. As it is seen from fig.2 , as soon as a certain critical value φсmin depending on H is reached , the first region under goes a transition to the second region in which the velocity of the sample increases gradually with 1ϕ increasing resulting from the progressive process of detachment of vortices from their corresponding pinning centers. One should expect that just in this region, in the rotating sample “the vortices fan” begins to unfold, in with the vortices are distributed according to the instantaneous angles of orientations with respect to the fixed external magnetic field. In this case the of orientation angles of separate vortex filaments are limited from frϕ to pinfr ϕϕ + , where frϕ is the angle on which the vortex filament can be turned with respect to H by forces of viscous friction with the matrix of superconductor, and pinϕ is the angle on with the vortex filament can be turned by the most strong pinning center, studied for the first time in [25]. The gradual transition (at high 1ϕ values) to the third region where the linear 2 1( )ϕ ϕ dependence was observed, allows one to define the countermoments of pinning forces pτ and frτ , independently. Just in this region, when 21 ωω = the torque τ , appeared to the uniformly rotating sample, is balanced by the countermoment pτ and frτ . In particular, in the case of continuously rotating sample with frequency 21 ωω = one could find similarly to [26,27] the expression for the total braking torque τ [19] . Indeed, if we consider in this case a vortex element moving with velocity ⊥υ perpendicular to sd then the average force acting on this elements is dsFdsfd l ⊥ += υ and the associated braking torque, exerted on the rotating specimen becomes: υτ fdrd where r is the vector pointing from the rotational axis to the vortex elements, lF is the pinning force per flux thread per unit length, and η is the viscosity coefficient. For a cylindrical specimen of radius R and height L integrating over the individual contribution of all vortex gives a total braking torque τ ωτττ 0+= p (1) with =τ , and B ηπτ = , Where B is the inductivity averaged over the sample, 0Φ is the flux quantum , L is the height and R is the radius of the sample. As it is shown in Fig.2, starting with the point (a), where 21 ωω = , to the superconducting sample uniformly rotating in the homogeneous stationary magnetic field H=1000 Oe, is applied stationary dynamic torsion moment fr p τττ += . If in this region the torsion head is stopped, then at the expense of relaxation processes connected with the presence of viscous forces acting on vortex filaments, the sample will continue the rotation in the same direction (with decreasing velocity) until it reaches a certain equilibrium position, depending on the H value. The Fig.3 shows curves of 2ϕ∆ time dependences at the stopped leading head for HoBa2Cu3O7-δ sample at T=77K and H=1000 Oe. Fig.3. Dependence of momentum relτ on time t after the stopping of rotating head for HoBa2Cu3O7-δ sample at T=77K and H=1000 Oe. If during the relaxation after rotation of sample one applies the pulsed magnetic field in parallel to the outer magnetic field H , then additional vortices, created as result of magnetic pulse, influence the structure already existing in the sample as “the vortex fan” what could result in the decrease of the angle of its unfolding or to its folding. The letter in its turn, would cause the additional change in the relaxation process taking place in the sample, and, correspondingly, results in the stepwise decrease of moment related with viscous forces frτ . But the change of relaxation process character and, correspondently, the stepwise decrease of moment could happen if the duration of magnetic pulse is larger as compared with the time necessary for creation of a new vortex structure, which will influence the superconducting sample relaxing in magnetic field. If it is the case, then at the small durations of magnetic pulses the relaxation curve, presented in Fig.3, doesn’t change, but when this duration becomes the order of a time for penetration of vortices into the sample and the creation of vortex structure, then the aforementioned change of relaxation processes could principally appear. Namely, this situation when the duration �x of magnetic pulses was larger then the time for Abrikosov vortex lattice creation �xс, have been described by us our previous work [19], when it was shown that the influence of one magnetic pulse �h≈400 Oe (�h||H) with duration 30�сек>�xс was stepwisely decreased the relτ moment and the relaxation process continued with the reduction relτ on a level as far as a new magnetic pulse similar the first one is not applied. In the presented work it was studied the influence of different duration and amplitude pulses on relaxation processes in vortex matter. The results shown in Fig.4 on action of single pulses of different durations on relaxation processes in vortex matter and, consequently, on mechanical moment relτ revealed that at small pulses durations up to 15 �s the relτ doesn’t change, but at duration of applied pulse >15 �s it is observed the stepwise change relτ , what speaks on the existence of the �xс threshold. Fig.4. Dependence of momentum � relτ on the duration �x of magnetic field single pulse � h=172 Oe applied in parallel to the main magnetic field H=1000 Oe at T=77K for HoBa2Cu3O7-δ sample. This way one could say that the Abrikosov vortex lattice creation time in high-temperature isotropic superconductor of HoBa2Cu3O7-δ makes value on the order of 20�s. This value approximately on the order of value higher then time for the single-vortex creation for the first time measured by G. Boato, G.Gallinaro and C. Rizzuto [28], who showed that this time is less than 10-5 sec. In work [19] it was also shown that continuous action of aforementioned pulses with the train frequency equal to 2,5 s-1 more sharply reveals their influence on relaxation processes in vortex matter and in these conditions the processes of penetration of vortices into superconductors bulk are made more sharply expressed. In fig.5 it is presented the clear picture of magnetic pulses continuous action with �h=172 Oe (�h||H) , and the duration 20�sec, what is larger than the �xс with the train frequency ν = 2,5 s-1. As it is seen from picture the pulses of 5, 10 and 15 �sec durations doesn’t change )(tfrel =τ which is observed at absence of magnetic pulses. The results presented in Fig.5 show that at durations of pulses in 20�sec, 30�sec and 40�sec the Abrikosov vortices penetrate into the superconductor. This way the threshold value on the magnetic pulses duration observed at the action of single pulses (Fig.4) coinside with the threshold observed when their repetition frequency is ν = 2.5 s-1. Fig.5. Dependence of momentum relτ on time t after the stopping of rotating head with the influence since t=5 min on the relaxation process of HoBa2Cu3O7-δ sample of the continuous magnetic field pulses h=172 Oe with ν=2,5 s-1 frequency and different durations x=5; 10; 15; 20; 30; and 40 �s. Pulsed magnetic field was parallel to the main magnetic field H=1000 Oe at T=77K. In Fig.6 it is presented the curve of relτ =f(t) dependence on time at the influence of magnetic pulses �h=172 (�h||H) the duration of which is below the time of Abrikosov vortex system creation �x=5�s<�xс (�xс≥15�s for the investigated HoBa2Cu3O7-δ). As it is seen from the picture when �x<�xс, the relaxation curve doesn’t change in spite the increase of the repetition frequency of magnetic pulses ν from 2.5 up to 500 s-1. As soon as the duration of pulses exceeds the critical value and becomes �x=30�s, the relaxation curve undergoes the essential (stepwise) change. For example in Fig.6 it is presented measurement for ν=5s-1 и ν= 500s-1. Fig.6. Dependence of momentum relτ on the time t after the stopping of rotating head with the influence since t=10 min on the HoBa2Cu3O7-δ sample relaxation process of the continuous pulses magnetic field with frequency ν=2,5 ÷500s-1 at �x=5�s< �xс , and also at �x= 30�s > �xс. The pulsed magnetic field �h=172 Oe was parallel to the main magnetic field H=1000 Oe at T=77K. And finally, we have observed the threshold on the value of applied pulses. In Fig.7 it is shown that in spite the fact that we applied magnetic pulses of the large duration 300�s>>�xс, much longer as compared with the time of Abrikosov vortex creation at small amplitudes of pulsed field �h ~7, 11, 14 Oe relτ =f(t) doesn’t change. The stepwise change of the relaxing moment relτ is revealed only at �h ~18 Oe and higher. Fig.6. Dependence of momentum relτ on the time t after the stopping of rotating head with the application after 5 minutes on the HoBa2Cu3O7-δ sample relaxation process of the single magnetic field pulses �h=(7÷36) Oe with duration �x= 300�s >>�xс. The pulsed magnetic field was parallel to the main magnetic field H=400 Oe at T=77K. The further investigations of relaxation phenomena are anticipated for anisotropic high-temperature superconductors among them in strongly anisotropic high-temperature superconductors of Bi-Pb-Sr-Ca-Cu-O system. 4. Conclusion The simple mechanical method of Abrikosov vortex stimulated dynamics investigations it was applied for the study of pulsed magnetic fields influence on relaxation phenomena in vortex matter of high- temperature superconductors. It was observed the change of relaxation processes in vortex matter as a result of pulsed magnetic field influence on it. The study of influence of different duration and amplitude pulsed magnetic fields influence was revealed the existence of threshold phenomena. A small duration pulse doesn’t change the course of relaxation processes in vortex matter of isotropic high- temperature superconductor HoBa2Cu3O7-δ. When the duration of pulses exceeds some critical value (threshold), then their influence change the course of relaxation processes. The latter is revealed in a stepwise decrease of relaxing mechanical momentum relτ , apparently, related with a sharp change of pinning and the rearrange of vortex system of superconducting sample as a result of penetration into its bulk of a new portion of vortices at application of pulsed field on the outer magnetic field creating the main vortex structure in the investigated HoBa2Cu3O7-δ sample. A new portion of vortices “shakes” the vortex lattice existing in a sample causing the detachment of vortices from a weak pinning centers what, apparently, is the reason for the stepwise decrease of mechanical momentum relτ . All these made it possible to define the Abrikosov vortex lattice creation time in HoBa2Cu3O7-δ which turned out to be on the order of value higher as compared with the time of single- vortex creation observed in type II superconductors. Acknowledgements The work was supported by the grants of International Science and Technology Center (ISTC) G-389 and G- 593. References: 1. V.M. Pan, A.V.Pan , Low Temperature Physics, v27, №9-10, pp. 991-1010. 2. Brandt E. H., Esquinazi P., Weiss W. C. Phys. Rev. Lett., 1991, v. 62. p.2330. 3. Xu Y., Suenaga M. Phys. Rev. 1991, v. 43. p. 5516 Kopelevich Y., Esquinazi P.arXiv: cond-mat/0002019. 4. E. Koshelev and V. M. Vinokur, Phys. Rev. Lett. 73, 3580– 3583 (1994). 5. E.W. Carlson, A.H. Castro Neto, and D.K.Campbell, Phys. Rev. Lett., 1991, v. 90, p.087001. 6. D. E. Farrell, J. P. Rice and D. M. Ginsberg, Phys. Rev. Lett., 1991, v. 67, pp.1165-1168. 7. S.M. Ashimov, J.G.Chigvinadze, Cond-mat/0306118. 8. V.M.Vinokur, P.S. Kes and A.E. Koshelev., Physica C 168, (1990), 29-39. 9. M.V. Feigelman, V.B. Geshkenbein , A.I. Larkin, Physica C 167 (1990) 177. 10. J.G. Chigvinadze, A.A. Iashvili , T.V. Machaidze, Phys.Lett.A. 300(2002) 524-528. 11. J.G. Chigvinadze , A.A. Iashvili , T.V. Machaidze, Phys.Lett.A.300(2002) 311-316. 12. C. J. Olson, G.T. Zimanyi, A.B. Kolton, N. Gronbech-Iensen, Phys.Rev. Lett. 85(2000)5416. C.J.Olson, C.Reichbardt, R.T.Scalettar, G.T.Zimanyi, cond-mat/0008350. 13. S.M. Ashimov, J.G. Chigvinadze, Physics Letters A 313 (2003) 238-242. 14. Muller K. A., Tokashige., Bednorz J. G.- Phys. Rev. Lett., 1987, v.58, p.1143. 15. Touminen M., Goldman A. M., McCartney M. L.- Phys. Rev. B, 1988, v.37, p.548. 16. Klimenko A.G., Blinov A.G., Vesin Yu.I., StarikovM.A.- Pis’ma Zh.Eksp. Teor. Fiz., 1987, v.46,Suppl., p.196. 17. Anderson P. W. - Phys. Rev. Lett., 1962, v.9, p.303. 18. A.A. Iashvili, T.V. Machaidze. L.T.Paniashvili, and J.G. Chigvinadze. Phys.,Chem., Techn., 1994, v. 7, N 2, pp. 297-300. 19. J.G. Chigvinadze, J.V. Acrivos S.M. Ashimov, A.A. Iashvili, T. V. Machaidze, Th. Wolf. Phys.Lett. A, 349, 264(2006). 20. E. L. Andronikashvili, J.G. Chigvinadze, R.M.Kerr, J. Lowell, K. Mendelsohn, J. S. Tsakadze. Cryogenics, v. 9, N2, pp.119-121 (1969). 21. J.G. Chigvinadze, Zh.Eksp.Teor.Fiz.,v. 65, N5, pp. 1923-1927 (1973). 22. S.M. Ashimov, I.A. Naskidashvili et al., Low Temp. Phys. 10, 479, (1984). 23. S.M. Ashimov and J.G. Chigvinadze. Physics Letters A, 313, pp. 238-242. (2003). 24. G. L. Dorofeev, E.F. Klimenko, Journ. Techn. Phys. 57, p.2291, (1987). 25. B.H. Heise Rev.Mod.Phys.36, 64 (1964). 26. M. Fuhrmans, C. Heiden, Proc. International Discussion (Sonnenberg, Germany, 1974), Göttingen, 1975, p. 223. 27. M. Fuhrmans, C. Heiden, Cryogenics, 125, 451 (1976). 28. G. Boato, G.Gallinaro and C. Rizzuto. Solid State Communications, vol. 3, pp.173-176(1965).

0704.0353

Spin and pseudospin symmetries and the equivalent spectra of relativistic spin-1/2 and spin-0 particles

Spin and pseudospin symmetries and the equivalent spectra of relativistic spin-1/2 and spin-0 particles P. Alberto Physics Department and Center for Computational Physics, University of Coimbra, P-3004-516 Coimbra, Portugal A. S. de Castro Departamento de F́ısica e Qúımica, Universidade Estadual Paulista, 12516-410 Guaratinguetá, SP, Brazil M. Malheiro Departamento de F́ısica, Instituto Tecnológico de Aeronáutica, CTA, 12228-900, São José dos Campos, SP, Brazil and Instituto de F́ısica, Universidade Federal Fluminense, 24210-340 Niterói, Brazil (Dated: November 4, 2018) We show that the conditions which originate the spin and pseudospin symmetries in the Dirac equation are the same that produce equivalent energy spectra of relativistic spin-1/2 and spin-0 particles in the presence of vector and scalar potentials. The conclusions do not depend on the particular shapes of the potentials and can be important in different fields of physics. When both scalar and vector potentials are spherical, these conditions for isospectrality imply that the spin- orbit and Darwin terms of either the upper component or the lower component of the Dirac spinor vanish, making it equivalent, as far as energy is concerned, to a spin-0 state. In this case, besides energy, a scalar particle will also have the same orbital angular momentum as the (conserved) orbital angular momentum of either the upper or lower component of the corresponding spin-1/2 particle. We point out a few possible applications of this result. PACS numbers: 11.30.-j,03.65.Pm When describing some strong interacting systems it is often useful, because of simplicity, to approximate the behavior of relativistic spin-1/2 particles by scalar spin-0 particles obeying the Klein-Gordon equation. An example is the case of relativistic quark models used for studying quark-hadron duality because of the added complexity of structure functions of Dirac particles as compared to scalar ones. It turns out that some results (e.g., the onset of scaling in some structure functions) almost do not depend on the spin structure of the particle [1]. In this work we will give another example of an observable, the energy, whose value may not depend on the spinor structure of the particle, i.e., whether one has a spin-1/2 or a spin-0 particle. We will show that when a Dirac particle is subjected to scalar and vector potentials of equal magnitude, it will have exactly the same energy spectrum as a scalar particle of the same mass under the same potentials. As we will see, this happens because the spin-orbit and Darwin terms in the second-order equation for either the upper or lower spinor component vanish when the scalar and vector potentials have equal magnitude. It is not uncommon to find physical systems in which strong interacting relativistic particles are subject to Lorentz scalar potentials (or position-dependent effective masses) that are of the same order of magnitude of potentials which couple to the energy (time components of Lorentz four-vectors). For instance, the scalar and vector (hereafter meaning time-component of a four-vector potential) nuclear mean-field potentials have opposite signs but similar magnitudes, whereas relativistic models of mesons with a heavy and a light quark, like D- or B-mesons, explain the observed small spin-orbit splitting by having vector and scalar potentials with the same sign and similar strengths [2]. It is well-known that all the components of the free Dirac spinor, i.e., the solution of the free Dirac equation, satisfy the free Klein-Gordon equation. Indeed, from the free Dirac equation (i~γµ∂µ −mc)Ψ = 0 (1) one gets (−i~γν∂ν −mc)(i~γ µ∂µ −mc)Ψ = (~ 2∂µ∂µ +m 2c2)Ψ = 0 , (2) where use has been made of the relation γµγν∂µ∂ν = ∂µ∂ µ. In a similar way, for the time-independent free Dirac equation we would have (cα · p+ βmc2)ψ = (−i~cα · ∇+ βmc2)ψ = Eψ , (3) http://arxiv.org/abs/0704.0353v1 where, as usual, ψ(r) = Ψ(r, t) exp (i E t/~), α = γ0γ and β = γ0. Then, by left multiplying Eq. (3) by cα ·p+βmc2, one gets the time-independent free Klein-Gordon equation (c2p2 +m2c4)ψ = (−~2c2∇2 +m2c2)ψ = E2ψ , (4) where the relation {β,α} = 0 was used. This all means that the free four-component Dirac spinor, and of course all of its components, satisfy the Klein-Gordon equation. This is not surprising, because, after all, both free spin-1/2 and spin-0 particles obey the same relativistic dispersion relation, E2 = p2c2 +m2c4, in spite of having different spinor structures and thus different wave functions. Since there is no spin-dependent interaction, one expects both to have the same energy spectrum. We consider now the case of a spin-1/2 particle subject to a Lorentz scalar potential Vs plus a vector potential Vv. The time-independent Dirac equation is given by [cα · p+ β(mc2 + Vs)]ψ = (E − Vv)ψ (5) It is convenient to define the four-spinors ψ± = P±ψ = [(I ± β)/2]ψ such that , (6) where φ and χ are respectively the upper and lower two-component spinors. Using the properties and anti- commutation relations of the matrices β and α we can apply the projectors P± to the Dirac equation (5) and decompose it into two coupled equations for ψ+ and ψ−: cα · pψ− + (mc 2 + Vs)ψ+ = (E − Vv)ψ+ (7) cα · pψ+ − (mc 2 + Vs)ψ− = (E − Vv)ψ− . (8) Applying the operator cα · p on the left of these equations and using them to write ψ+ and ψ− in terms of α · pψ− and α · pψ+ respectively, we finally get second-order equations for ψ+ and ψ−: c2p2 ψ+ + c [α · p∆]α · pψ+ E −∆+mc2 = (E −∆+mc2)(E − Σ−mc2)ψ+ (9) c2p2 ψ− + c [α · pΣ]α · pψ− E − Σ−mc2 = (E −∆+mc2)(E − Σ−mc2)ψ− (10) where the square brackets [ ] mean that the operator α · p only acts on the potential in front of it and we defined Σ = Vv + Vs and ∆ = Vv − Vs. The second term in these equations can be further elaborated noting that the Dirac αi matrices satisfy the relation αiαj = δij + iǫijkSk where Sk, k = 1, 2, 3, are the spin operator components. The second-order equations read now c2 p2 ψ+ + c [p∆] · pψ+ + [p∆]× p · S ψ+ E −∆+mc2 = (E −∆+mc2)(E − Σ−mc2)ψ+ (11) c2 p2 ψ− + c [pΣ] · pψ− + [pΣ]× p · S ψ− E − Σ−mc2 = (E −∆+mc2)(E − Σ−mc2)ψ−. (12) Now, if p∆ = 0, meaning that ∆ is constant or zero (if ∆ goes to zero at infinity, the two conditions are equivalent), then the second term in eq. (11) disappears and we have c2 p2ψ+ = (E −∆+mc 2)(E − Σ−mc2)ψ+ = [(E − Vv) 2 − (mc2 + Vs) 2]ψ+ , (13) which is precisely the time-independent Klein-Gordon equation for a scalar potential Vs plus a vector potential Vv[14]. Since the second-order equation determines the eigenvalues for the spin-1/2 particle, this means that when p∆ = 0, a spin-1/2 and a spin-0 particle with the same mass and subject to the same potentials Vs and Vv will have the same energy spectrum, including both bound and scattering states. This last sufficient condition for isospectrality can be relaxed to demand that just the combination mc2+Vs be the same for both particles, allowing them to have different masses. This is so because this weaker condition does not change the gradient of ∆ and Σ and therefore the condition p∆ = 0 will still hold. On the other hand, if the scalar and vector potentials are such that pΣ = 0, we would obtain a Klein-Gordon equation for ψ−, and again the spectrum for spin-0 and spin-1/2 particles would be the same, provided they are subjected to the same vector potential and mc2 + Vs is the same for both particles. If both Vs and Vv are central potentials, i.e., only depend on the radial coordinate, then the numerators of the second terms in equations (11) and (12) read [p∆] · pψ+ + [p∆]× p · S ψ+ = −~ ∆′L · S ψ+ (14) [pΣ] · pψ− + [pΣ]× p · S ψ− = −~ Σ′L · S ψ− , (15) where ∆′ and Σ′ are the derivatives with respect to r of the radial potentials ∆(r) and Σ(r), and L = r × p is the orbital angular momentum operator. From these equations ones sees that these terms, which set apart the Dirac second-order equations for the upper and lower components of the Dirac spinor from the Klein-Gordon equation and thus are the origin of the different spectra for spin-1/2 and spin-0 particles, are composed of a derivative term, related to the Darwin term which appears in the Foldy-Wouthuysen expansion, and a L · S spin-orbit term. If ∆′ = 0 (Σ′ = 0), then there is no spin-orbit term for the upper (lower) component of the Dirac spinor. In turn, since the second-order equation determines the energy eigenvalues, this means that the orbital angular momentum of the respective component is a good quantum number of the Dirac spinor. This can be a bit surprising, since one knows that in general the orbital quantum number is not a good quantum number for a Dirac particle, since L2 does not commute with a Dirac Hamiltonian with radial potentials. The reason why this does not happen in these cases was reported in Refs. [3, 4], and we now review it in a slight different fashion. Let us consider in more detail the case of spherical potentials such that ∆′ = 0. One knows that a spinor that is a solution of a Dirac equation with spherically symmetric potentials can be generally written as ψjm(r) = gj l(r) Yj lm(r̂) j l̃ m . (16) where Yj lm are the spinor spherical harmonics. These result from the coupling of spherical harmonics and two- dimensional Pauli spinors χms , Yj lm = 〈 l ml ; 1/2ms | j m 〉Ylmlχms , where 〈 l ml ; 1/2ms | j m 〉 is a Clebsch-Gordan coefficient and l̃ = l ± 1, the plus and minus signs being related to whether one has aligned or anti-aligned spin, i.e., j = l ± 1/2. The spinor spherical harmonics for the lower component satisfy the relation j l̃m = −σ · r̂Yj lm. The fact that the upper and lower components have different orbital angular momenta is related to the fact, mentioned before, that L2 does not commute with the Dirac Hamiltonian H = cα · p+ β(Vs +mc 2) + Vv = cα · p+ βmc 2 +ΣP+ +∆P− , (17) where P± are the projectors defined above. However, when ∆ ′ = 0, there is an extra SU(2) symmetry of H (so-called “spin symmetry”) as first shown by Bell and Ruegg [5]. When we have spherical potentials, Ginocchio showed that there is an additional SU(2) symmetry (for a recent review see [4]). The generators of this last symmetry are L = LP+ + α · pLα · pP− = 0 Up LUp , (18) where Up = σ · p/( p2) is the helicity operator. One can check that L commutes with the Dirac Hamiltonian, [H,L] = [cα · p,LP+ + α · pLα · pP−] + [∆, α · pLα · p] + [Σ,L] = [∆, α · pLα · p ] = 0 , (19) where the last equality comes from the fact that ∆′ = 0. The Casimir L2 operator is given by L2 = L2P+ + α · pP−. Applying this operator to the spinor ψjm (16), we get 2ψjm = L α · pL2 α · pψ− = ~2l(l+ 1)ψ+ α · p cL2 ψ+jm E −∆+mc2 = ~2l(l + 1)ψ+ + ~2l(l + 1)ψ− = ~2l(l + 1)ψjm , (20) where ψ±jm = P±ψjm and we used the relation, valid when ∆ ′ = 0, ψ+jm = (E −∆ +mc α · p ψ−jm. From (20) we see that ψjm is indeed an eigenstate of L 2. Thus the orbital quantum number of the upper component l is a good quantum number of the system when the spherical potentials Vs(r) and Vv(r) are such that Vv(r) = Vs(r)+C∆, where C∆ is an arbitrary constant. Also, according to we have said before, there is a state of a spin-0 particle subjected to these same spherical potentials (or, at least, with a scalar potential such that the sum Vs +mc 2 is the same) that has the same energy and the same orbital angular momentum as ψjm. In addition, the wave function of this scalar particle would be proportional to the spatial part of the wave function of the upper component. Note that the generator of the “spin symmetry” S is given by a similar expression as (18) just replacing L by ~/2σ [4, 5], meaning that S2 ≡ S2 = 3/4 ~2I so that spin is also a good quantum number, as would be expected. Actually, one can show that the total angular momentum operator J can be written as L + S, so that l, ml (eigenvalue of Lz), s = 1/2, ms (eigenvalue of Sz) are good quantum numbers. Then, of course, j and m = ml +ms are also good quantum numbers, but only in a trivial way, because there is no longer spin-orbit coupling. Therefore, in the spinor (16) one could just replace the spinor spherical harmonic Yj lm by Yl mlχms and Yj l̃m by −σ · r̂ Yl mlχms . Note that if ∆ is a nonrelativistic potential, ∆ ≪ mc2 and ∆′ ≪ m2c4/(~c), i.e., it is slowly varying over a Compton wavelength. In this case, the spin-orbit term will also get suppressed. In fact, the derivative of the ∆ potential is the origin of the well-known relativistic spin-orbit effect which appears as a relativistic correction term in atomic physics or in the v/c Foldy-Wouthuysen expansion (only the derivative of Vv appears because usually no Lorentz scalar potential Vs is considered, and therefore ∆ = Vv). When Σ′ = 0, or Vv(r) = −Vs(r) + CΣ, with CΣ an arbitrary constant, there is again a SU(2) symmetry, usually called pseudospin symmetry ([5, 6]) which is relevant for describing the single-particle level structure of several nuclei. This symmetry has a dynamical character and cannot be fully realized in nuclei because in Relativistic Mean-field Theories the Σ potential is the only binding potential for nucleons [7, 8]. For harmonic oscillator potentials this is no longer the case, since ∆, acting as an effective mass going to infinity, can bind Dirac particles [9, 10], even when Σ = 0. As before, in the special case of spherical potentials, there is another SU(2) symmetry whose generators are α · pLα · pP+ +LP− = Up LUp 0 . (21) In the same way as before, applying L̃ to ψjm, we would find that L̃ ψjm = ~ 2 l̃(l̃ + 1)ψjm, that is, this time it is the orbital quantum number of the lower component l̃ which is a good quantum number of the system and can be used to classify energy levels. Again, provided the vector and scalar potentials are adequately related, there would be a corresponding state of a spin-0 particle with the same energy and same orbital angular momentum l̃, and, furthermore, its wave function would be proportional to the spatial part of the wave function of the lower component. As before, the pseudospin symmetry generator S̃ can be obtained from L̃ by replacing L by ~/2σ. The good quantum numbers of the system would be, besides l̃, m , s̃ ≡ s = 1/2 and ms̃. Again, J = L̃ + S̃. It is interesting that, as has been noted by Ginocchio [9], the generators of spin and pseudospin symmetries are related through a γ5 transformation since S̃ = γ5Sγ5 and L̃ = γ5Lγ5. This property was used in a recent work to relate spin symmetric and pseudospin symmetric spectra of harmonic oscillator potentials [11]. There it was shown that for massless particles (or ultrarelativistic particles) the spin- and pseudo-spin spectra of Dirac particles are the same. In addition, this means that spin-symmetric massless eigenstates of γ5 would be also pseudo-spin symmetric and vice-versa. Since in this case ∆ = Σ = 0, or Vv = Vs = 0, this is, of course, just another way of stating the well-known fact that free massless Dirac particles have good chirality. Naturally, for free spin-1/2 particles described by spherical waves, both l and l̃ are good quantum numbers, which just reflects the fact that one can have free spherical waves with any orbital angular momentum for the upper or lower component and still have the same energy, as long as their linear momentum magnitude is the same, or, put in another way, the energy of a free spin-1/2 particle cannot depend on its direction of motion. In summary, we showed that when a relativistic spin-1/2 particle is subject to vector and scalar potentials such that Vv = ±Vs + C±, where C± are constants, its energy spectrum does not depend on their spinorial structure, being identical to the spectrum of a spin-0 particle which has no spinorial structure. This amounts to say that if the potentials have these configurations there is no spin-orbit coupling and Darwin term. If the scalar and vector potentials are spherical, one can classify the energy levels according to the orbital angular momentum quantum number of either the upper or the lower component of the Dirac spinor. This would then correspond to having a spin-0 particle with orbital angular momentum l or l̃, respectively. This spectral identity can of course happen only with potentials which do not involve the spinorial structure of the Dirac equation in an intrinsic way. For instance, a tensor potential of the form iβσµν (∂µAν − ∂νAµ) does not have an analog in the Klein-Gordon equation, so that one could not have a spin-0 particle with the same spectrum as a spin-1/2 particle with such a potential. This is the case of the so-called Dirac oscillator [12] (see [10] for a complete reference list), in which the Dirac equation contains a potential of the form iβσ0imωri = imωβα · r. Another important potential, the electromagnetic vector potential A, which is the spatial part of the electromagnetic four-vector potential, can be added via the minimal coupling scheme to both the Dirac and the Klein-Gordon equations. Since α · (p− eA)α · (p− eA) = (p− eA)2 + 2e~∇×A · S, the spectra of spin-0 and spin-1/2 particles cannot be identical as long as there is a magnetic field present, even though the condition Vv = ±Vs +C± is fulfilled. It is important also to remark that, since for an electromagnetic interaction Vv is the time-component of the electromagnetic four-vector potential, this last condition is gauge invariant in the present case, in which we are dealing with stationary states, i.e, time-independent potentials. So, in the absence of a external magnetic field (allowing, for instance, an electromagnetic vector potential A which is constant or a gradient of a scalar function), a spin-0 and spin-1/2 particle subject to the same electromagnetic potential Vv and a Lorentz scalar potential fulfilling the above relation would have the same spectrum. The remark made above about the similarity of spin-0 and spin-1/2 wave functions can be relevant for calculations in which the observables do not depend on the spin structure of the particle, like some structure functions. One such calculation was made by Paris [13] in a massless confined Dirac particle, in which Vv = Vs. It would be interesting to see how a Klein-Gordon particle would behave under the same potentials. More generally, this spectral identity can also have experimental implications in different fields of physics, since, should such an identity be found, it would signal the presence of a Lorentz scalar field having a similar magnitude as that of a time-component of a Lorentz vector field, or at least differing just by a constant. Acknowledgments We acknowledge financial support from CNPQ, FAPESP and FCT (POCTI) scientific program. [1] S. Jeschonnek and J. W. Van Orden, Phys. Rev. D 69, 054006 (2004). [2] P. R. Page, T. Goldman, and J. N. Ginocchio, Phys. Rev. Lett. 86, 204. [3] J. N. Ginocchio and A. Leviatan, Phys. Lett. B425, 1 (1998). [4] J. N. Ginocchio, Phys. Rep. 414 165 (2005). [5] J. S. Bell and H. Ruegg, Nucl. Phys. B98, 151 (1975). [6] J. N. Ginocchio, Phys. Rev. Lett. 78, 436 (1997). [7] P. Alberto, M. Fiolhais, M. Malheiro, A. Delfino, and M. Chiapparini, Phys. Rev. Lett. 86, 5015 (2001). [8] P. Alberto, M. Fiolhais, M. Malheiro, A. Delfino, and M. Chiapparini, Phys. Rev. C 65, 034307 (2002). [9] J. N. Ginocchio, Phys. Rev. Lett. 95, 252501 (2005). [10] R. Lisboa, M. Malheiro, A. S. de Castro, P. Alberto, and M. Fiolhais, Phys. Rev. C 69, 024319 (2004). [11] A. S. de Castro, P. Alberto, R. Lisboa, and M. Malheiro, Phys. Rev. C 73, 054309 (2006). [12] D. Itô, K. Mori, and E. Carriere, Nuovo Cimento A 51, 1119 (1967); M. Moshinsky and A. Szczepaniak, J. Phys. A 22, L817 (1989). [13] M. W. Paris, Phys. Rev. C 68, 025201 (2003). [14] There are some authors who introduce a scalar potential Vs in the Klein-Gordon equation by making the replacement m2c4 → m2c4 +V2 . Here we introduce it, as most authors do, as an effective mass m∗ 2 = (m+Vs/c 2)2, since it is the way that it is introduced in the Dirac equation. The two potentials are related by V2 = (mc2 + Vs) −m2c4. Acknowledgments References

0704.0354

General asymptotic solutions of the Einstein equations and phase transitions in quantum gravity

General asymptoti solutions of the Einstein equations and phase transitions in quantum gravity Dmitry Podolsky Helsinki Institute of Physi s, University of Helsinki, Gustaf Hällströmin katu 2, FIN00014, Helsinki, Finland Email: dmitry.podolsky�helsinki.� Abstra t We dis uss generi properties of lassi al and quantum theories of grav- ity with a s alar �eld whi h are revealed at the vi inity of the osmolog- i al singularity. When the potential of the s alar �eld is exponential and unbounded from below, the general solution of the Einstein equations has quasi-isotropi asymptoti s near the singularity instead of the usual anisotropi Belinskii - Khalatnikov - Lifshitz (BKL) asymptoti s. De- pending on the strength of s alar �eld potential, there exist two phases of quantum gravity with s alar �eld: one with essentially anisotropi be- havior of �eld orrelation fun tions near the osmologi al singularity, and another with quasi-isotropi behavior. The �phase transition� between the two phases is interpreted as the ondensation of gravitons. On leave from Landau Institute for Theoreti al Physi s, 119940, Mos ow, Russia. http://arxiv.org/abs/0704.0354v2 One pessimisti quotation from the golden era of �nding exa t solutions of the Einstein equations whi h re�e ted the relations between parti le theorists and experts in GR belongs to Ri hard Feynman. Taking part in the Interna- tional Conferen e on Relativisti Theories of Gravitation at Warsaw, he was writing to his wife [1℄: �I am not getting anything out of the meeting. I am learning nothing. ... I get into arguments outside the formal sessions (say, at laun h) whenever anyone asks me a question or starts to tell me about his �work�. The �work� is always: (1) ompletely un-understandable, (2) vague and inde�nite, (3) something orre t that is obvious and self-evident but worked out by a long and di� ult analysis, and presented as an important dis overy, or (4) a laim based on the stupidity of the author that some obvious and orre t fa t, a epted and he ked for years, is in fa t false ... (5) an attempt to do something probably impossible but ertainly of no utility whi h, it is �nally revealed in the end, fails ... or (6) just plan wrong ... Remind me not to ome to any more gravity onferen es!� Certainly, I am well aware of that the work presented in this essay ould belong to the lass (3) or (5) in the Feynman's lassi� ation (hopefully, not to the lass (6)!), but I will follow Feynman's own words [1℄: �We all do it for the fun of it� trying to �nd my fun in identifying some links whi h onne t the part of the ommon lore on general relativity named �Exa t solutions of the Einstein equations� to the problem of the GR quantization. Of ourse, Feynman's interest was in the quantization of GR by applying the path integral approa h working so well in QED. Solutions of the Einstein equa- tions de�ne saddle points of the a tion 2 S = Sgravity + Smatter of the quantum gravity with matter. However, the ontributions of these saddle points into the partition fun tion of the theory and �u tuations near them Dφmatter exp (Sgravity + Smatter) typi ally have zero measure. In other words, the probability for an almost any exa t solution to des ribe the observable features of the Universe or some parts of it, to appear somehow from the quantum foam realized near the singularity is in�nitely small, and the Feynman's anger is absolutely understandable. Well, almost absolutely... Of ourse, there are several lasses of solutions whi h will be important for the quantum part of the story, too, and one an without mu h thinking immediately identify some: 1. Attra tors : among them are Minkowski spa etime, de Sitter (at least in the sense of eternal in�ation [2℄) and anti de Sitter spa etimes (a set of AdS domains is mostly probably the global attra tor of GR realized as low-energy approximation of string theory [3℄); bla k holes (S hwarzs hild, Kerr, Reissner-Nordström, Kerr-Newman solutions), et . From now on, by the quantum theory of gravity we mean e�e tive QFT of spin 2 �elds [4℄ (plus matter �elds) � the one whi h parti les with energies E ≪ MP test. In this limit, the e�e ts of the non-renormalizability may be negle ted. Although we dis uss below the situation whi h is realized near the osmologi al singularity, we limit the dis ussion to time s ales t ≫ tP . 2. General solutions of the Einstein equations. As usual [5℄, a solution of the Einstein equations is regarded as general if it ontains su� ient number of arbitrary fun tions of oordinates. In the ase of Ri i-�at spa etimes, this number is 4, and is equal to 8 in the presen e of hydrodynami matter. While any non-attra tor type solution of the Einstein equations de�nes the saddle point for the path integral (1) whi h does have a vanishing ontribution into the overall partition fun tion, eventually it well settle down towards an attra tor solution due to the e�e t of lassi al perturbations and/or quantum �u tuations. The ontribution of attra tor type saddle points into the partition fun tion (1) is therefore signi� ant. However, the key word here is �eventually�. For any non-attra tor solution it takes a time tcoll before the solution rea hes its attra tor asymptoti s. Let us onstru t some initial state |Ψ(t = ti)〉 of quantum matter �elds in a urved spa etime and gravitons. The amplitude 〈Ψ(tf )|Ψ(ti)〉 is then de�ned by the path integral (1) al ulated on the losed S hwinger-Keldysh ontour from t = ti to t = tf and ba k. Then, if tf ≪ tcoll, the orresponding attra tor saddle point does not give any noti eable ontribution into the amplitude. it is ne essary to know the evolution of the quantum state |Ψ(t)〉 at time s ales t ≪ tcoll, we are for ed to pay mu h more attention to the type of saddle orresponding to general solutions of the Einstein equations. Certainly, the Einstein equations are hard to solve, and it is possible to �nd something like their general solution only in physi ally simpli�ed situations. As was �rst shown by Belinskii, Khalatnikov and Lifshitz [6℄, asymptoti ally, the general solutions of the Einstein equations near the osmologi al singularity have the very same form for an almost arbitrary hoi e of the matter ontent. This asymptoti s in the syn hronous frame is given by Kasner-like solution ds2 = dt2 − γαβ(t,x)dxαdxβ , (2) γαβ(t, x) = t lαlβ + t mαmβ + t nαnβ . (3) Both Kasner exponents p1, p2, p3 and Kasner axis ve tors lα, mα and nα are arbitrary fun tions of spa e oordinates. The Einstein equations provide two onstraints on the Kasner exponents p1 + p2 + p3 = 1, (4) p21 + p 2 + p 3 = 1, (5) as well as three other onstraints on arbitrary fun tions of spa e oordinates present in (3). Taking into a ount that the hoi e of syn hronous gauge g00 = 1, g0α = 0 (6) Of ourse, the time s ale tcoll itself is a fun tional of the initial state |Ψ(t = ti)〉. Often, it is impossible to hoose the globally syn hronous frame of referen e due to the limitations set by the asuality. However, everywhere in the text we dis uss the physi s in a given asual pat h. leaves the freedom to make three-dimensional spa e oordinate transformations, one an easily see that the total number of arbitrary oordinate fun tions in the Kasner-like solution (2),(3) is equal to 4 as it should be expe ted for a general solution of Einstein equations orresponding to an empty spa etime. In the presen e of the hydrodynami matter Kasner solution (2),(3) de- s ribes asymptoti behavior of metri s near the singularity, sin e omponents of energy-momentum tensor Tik grow slower at t → 0 then the omponents of the Ri i tensor. Higher order orre tions to the Kasner solution (2),(3), i.e., higher order terms in the expansion of γαβ(t, x) over powers of t play the role of perturbations whi h give rise to the time dependen e of Kasner exponents pi as well as Kasner axis ve tors lα, mα and nα and to well-known BKL haoti behavior. Therefore, the BKL solution is simultaneously a universal attra - tor for all solutions of the Einstein equations possessing a spa elike singularity. It means that no other saddle points ontribute into the amplitude (1) in the vi inity of the osmologi al singularity. In this essay, it will be �rst of all shown that in the presen e of a s alar �eld with potential V (φ) whi h is exponential and unbounded from below, the general asymptoti solution of the Einstein equations is di�erent from the BKL solution and is quasi-isotropi [8℄ (while the BKL solution is essentially anisotropi ). In parti ular, we will hoose potential of the form V (φ) = −|V0|ch (λφ) . (7) S alar �eld potentials of this form appear in problems related to gauged super- gravity models [10℄ and the ekpyroti s enario [11℄. The osmologi al singularity realized in su h theory is of the Anti de Sitter Big Crun h type. The physi s in its vi inity it is interesting by itself and even more so sin e this type of singularity seems to be realized quite often on the string theory lands ape [3℄. As in the ase dis ussed in [6℄, it is onvenient to perform all al ulations in the syn hronous frame of referen e where g00 = 1, g0α = 0, gαβ = −γαβ , α, β = 1 . . . 3, i.e., the spa etime interval has the form ds2 = dt2 − γαβ(t, x)dxαdxβ . (8) Near the hypersurfa e t = 0 whi h orresponds to the singularity, the spatial metri omponents behave as γαβ(t,x) = aαβ(x)t 2q + cαβ(x)t d + bαβ(x)t (i,j) (x)tfij . (9) With the same pre ision, one has in the vi inity of singularity φ(t,x) = ψ(x) + φ0(x)log(t) + φ1(x)t f1 + φ2(x)t f2 + · · · , (10) Whi h orresponds everywhere below to the spa elike hypersurfa e t = 0. If there is a s alar �eld in the matter ontent [7℄, BKL solution (3) remains general solution of the Einstein equations with hanged Kasner onstraints (4),(5). The quasi-isotropi solution for su h potentials was �rst found at the ba kground level in [9℄, where it was also shown that it is the attra tor. The goal we pursue in this essay is to prove that the quasi-isotropi solution is also general and to understand how its instability develops with the hange of the form of the potential. with dots orresponding to higher order terms of φ(t,x) expansion in powers of t. From the Einstein equations one �nds8 that the leading exponents in the expansions (9) and (10) are de�ned by the expressions , n = 2, d = 1− q, (11) ψ(x) = Const, φ0(x) = , f1 = 1− 3q, f2 = 2− q, (12) cαα(x) = 2λφ1(x), c α;β(x) = 1− 2q 1− 3q φ0φ1,α(x), (13) P̃ βα (x) + (1 − q)(qbγγ(x)δβα + (1 + q)bβα(x)) = e−ψλφ2(x), (14) − (1− q)bαα(x) = (1− q)φ0φ2(x)− −ψλφ2(x), (15) where P̃ βα (x) is the 3-dimensional Ri i tensor onstru ted from omponents of the tensor aβα(x) as omponents of metri tensor. Higher order terms in the expansions (9) and (10) an be self onsistently al ulated by using the Einstein equations and the orthogonality ondition β = δ α. (16) One an immediately �nd from Eq. (16) that the higher order exponents in the metri (9) are de�ned by fij = i+ 2j − (3i+ 2j − 2)q, (17) where i, j ∈ N. The n term in the metri expansion orresponds to i = 0, j = 1 and d term � to i = 1, j = 0. It is easy to see that there is no other exponents in the expansion (9). Let us examine the formulae (11)-(15) more losely and al ulate the number of arbitrary fun tions present in this solution. First of all, one an immediately see that the tensor aβα(x) is not onstrained by the Einstein equations. It has 6 omponents, and 3 of them an be made to be equal to 0 by a three-dimensional oordinate transformation (the remnant gauge freedom of the syn hronous gauge (6)). Sin e this tensor is used for lowering and rising the indi es and represents the leading term in the expansion (9), we will identify the term aαβt ba kground ontribution to γαβ(t, x). Furthermore, we see from Eqs. (14),(15) that bαβ an be re onstru ted from the known tensor aαβ . The tensor cαβ ontains three more arbitrary fun tions of oordinates. In- deed, it an be represented in the form cβα(x) = α + Y ;α + Y Y γγ δ α + c (TT)β α . (18) Due to the limitations of spa e we are unable to present the full derivation of the solution here. It will be given in the forth oming publi ation [12℄. The indi es of all matri es are lowered and raised by the tensor aαβ , for example, b From Eq. (13) one an see that its tra e part de�nes the value of φ2(x) on- tributing to Eq. (10) and therefore provides one arbitrary fun tion. Then, three omponents of the ve tor ontribution Yα(x) are �xed, and transverse tra eless part c (TT)β α (x) provides remaining two arbitrary fun tions. We also note that the cαβ term an be regarded as the leading term perturbation to the ba kground ontribution into γαβ . In parti ular, it ontains the ontribution of s alar per- turbations (related to the tra e of the tensor cβα) and tensor perturbations or gravitons (related to the transverse tra eless part of the tensor cβα). The total number of arbitrary fun tions in the solution (9),(10) is therefore 6, as one may expe t for the general solution of the Einstein equations with a s alar �eld. By analysis similar to [6℄, one may show [9, 12℄ that the ontri- butions of other matter �elds into the overall energy-momentum tensor grow slower at t → 0 than the ontribution of the s alar �eld. We on lude that the solution (9), (10) is the general asymptoti solution of Einstein equations (with arbitrary matter ontent) near the osmologi al singularity. Similarly to the BKL solution, the quasi-isotropi solution is the universal attra tor for all solutions of the Einstein equations with s alar �eld having the potential (7) and arbitrary additional matter ontent whi h possess the time-like singularity. Again, under onsidered onditions, no other saddle points ontribute into the amplitude (1) in the vi inity of the Big Crun h singularity. It is instru tive to understand how exa tly the transition from the quasi- isotropi regime (9),(10) near the singularity to the BKL anisotropi regime (3) happens. This transition an be a hieved by hanging the value of λwhile keeping V0 �xed (or vise versa). By onstru tion, 2q < d = 1− q, i.e., the exponent aαβt2q in the expansion (9) is leading. With the in rease of q, the value of d de reases and when q rea hes the riti al value qc = 1/3, the ontributions aαβ(x)t and cαβ(x)t into the expansion of the metri (9) be ome of the same order. Similarly, one an he k that the values of higher order exponents (17) de rease with the in rease of q. In parti ular, all exponents with di�erent i's and similar j's be ome of the same order of magnitude at qc = 1/3. At q > qc = 1/3 the general asymptoti solution of the Einstein equations near the singularity is given by Eq. (3) instead of Eq. In fa t, what we have just found is relevant for the quantum part of the story, too, and in a sense is analogous to the spontaneous symmetry breaking phenomenon in QFTs. Indeed, let us take the theory with a s alar �eld (Φ2 − v)2, (19) set Φ(x, 0) = 0 as an initial ondition and ontinuously hange the value of the parameter v. At v > 0 the solution Φ(t, x) = 0 of the lassi al equations of motion is perturbatively stable and orresponds to the true va uum of the theory at the quantum level. At ν < 0 the same solution be omes lassi ally unstable, and Φ(t, x) rea hes the �true� va uum value Φ = ± v during the time t ∼ 1 log 1 (with the VEV of the operator Φ̂ having similar behavior at the quantum level). Similar situation is realized in our ase. At q < qc = 1/3 the quasi-isotropi solution (9),(10) is the general solution of the Einstein equations; it is perturbatively stable by onstru tion (without any limitations on the weakness of the perturbations). At q > qc the quasi-isotropi solution be omes perturbatively unstable (perturbations de�ned by cαβ and higher order terms grow faster than the ba kground term aαβ at t→ 0). Vise versa, at q > qc = 1/3 the BKL anisotropi solution if the Einstein equations is general in the vi inity of the osmologi al singularity. It is stable by onstru tion with respe t to arbitrary perturbations and the stability is lost at q < qc. This analysis remains valid for the quantum situation sin e the anoni- al phase spa e is in one-to-one orresponden e with the spa e of solutions of lassi al �eld equations [13℄, and both quasi-isotropi and BKL solutions are (a) general and (b) universal attra tors for other solutions of the Einstein equations in the vi inity of the time-like singularity. The transition from the regime realized at q < 1/3 to the regime q > 1/3 probably orresponds in the quantum level to the ondensation of gravitational perturbations. Indeed, one an interpret the higher order ontributions in the expansion (9) as terms orresponding to the intera tion between gravitational degrees of freedom as well as higher order nonlinearities in the ba kground. Our on lusion is based on the fa t that at q = qc the spe trum of the exponents in the expansion (9) be omes in�nitely dense. It is also possible to show that the point of the �phase transition� qc = 1/3 orresponds at the lassi al level to the situation when the hoi e of globally syn hronous frame of referen e is impossible near the singularity [12℄. Let us summarize what have been found in the present essay. We have shown that in the presen e of the s alar �eld with exponential potential unbounded from below, the general asymptoti solution of the Einstein equations near the osmologi al singularity has quasi-isotropi behavior instead of anisotropi found by [6℄. We have argued that at the quantum level there should exist a phase transition between the quasi-isotropi and anisotropi phases, governed by the strength of the s alar �eld potential and interpreted this phase transition as the ondensation of gravitational perturbations. A knowledgements I am thankful to A.A. Starobinsky and D. Wesley for the dis ussions and to K. Enqvist for making helpful omments. While ondu ting this work, I was supported by Marie Curie Resear h training network HPRN-CT-2006-035863. One important omment regarding the quantization should be made. The quantum theory of the s alar �eld with the potential (7) is ta hyoni ally unstable and has neither well-de�ned asymptoti |out〉 states, nor 〈out|in〉 S-matrix. However, the S hwinger-Keldysh 〈in|in〉 S- matrix is de�ned, and it is possible to make sense of the orresponding time-dependent theory [12℄. Referen es [1℄ R. Feynman, as told R. Leighton, �What do you are what other people think?�, W.W. Norton, New-York, 1988. [2℄ A. Linde, �Parti le physi s and in�ationary osmology�, Harwood, Switzer- land, 1990 [hep-th/0503203℄; A. Linde, Phys. Lett. 175B, 395 (1983). [3℄ A. Ceresole, G. Dall'Agata, A. Giryavets, R. Kallosh and A. Linde, Phys. Rev. D 74 (2006) 086010 [hep-th/0605086℄; A. Linde, JCAP 0701 (2007) 022 [hep-th/0611043℄; T. Clifton, A. Linde, N. Sivanandam, JHEP 0702 (2007) 024 [hep-th/0701083℄. [4℄ C. Burgess, in �Towards quantum gravity�, ed. D. Oriti, Cambridge Uni- versity Press, 2006 [gr-q /0606108℄; C. Burgess, hep-th/0701053. [5℄ L.D. Landau and E.M. Lifshitz, �The lassi al theory of �elds�, Pergamon Press, 1979. [6℄ V.A. Belinskii, E.M. Lifshitz and I.M. Khalatnikov, Adv. Phys. 19, 525 (1970); V.A. Belinskii, E.M. Lifshitz and I.M. Khalatnikov, Adv. Phys. 31, 639 (1982); B. Berger, D. Gar�nkle, J. Isenberg, V. Mon rief, and M. Weaver, Mod. Phys. Lett. A 13, 1565 (1998). [7℄ V.A. Belinskii and I.M. Khalatnikov, Sov.Phys. JETP 36, 591 (1973); L. Andersson, A. Rendall, Commun. Math. Phys. 218, 479 (2001) [gr-q /0001047℄. [8℄ E.M. Lifshitz, I.M. Khalatnikov, ZhETF 39, 149 (1960) (in russian); E.M. Lifshitz, I.M. Khalatnikov, Sov. Phys. Uspekhi 6, 495 (1964). [9℄ J.K. Eri kson, D. Wesley, P. Steinhardt, N. Turok, Phys. Rev. D 69 (2004) 063514 [hep-th/0312009℄. [10℄ C.M. Hull, Class. Quant. Grav. 2, 343 (1985); R. Kallosh, A. Linde, S. Prokushkin, M. Shmakova, Phys. Rev. D 65, 105016 (2002) [hep-th/0110089℄; R. Kallosh, A. Linde, S. Prokushkin, M. Shmakova, Phys. Rev. D 66, 123503 (2002) [hep-th/0208156℄. [11℄ J. Khoury, B.A. Ovrut, P.J. Steinhardt, N. Turok, Phys. Rev. D 64, 123522 (2001) [hep-th/0103239℄; J. Khoury, B.A. Ovrut, N. Seiberg, P.J. Stein- hardt, N. Turok, Phys. Rev. D 65, 086007 (2002) [hep-th/0108187℄; E. Bu hbinder, J. Khoury, B. Ovrut, hep-th/0702154. [12℄ D. Podolsky, A. Starobinsky, in preparation. [13℄ Č. Crn ović and E. Witten, in Three hundred year of gravitation, eds. S.W. Hawking and W. Israel, Cambridge University Press, 1987, p. 676; G.J. Zu kerman, in Mathemati al aspe ts of string theory, San-Diego 1986, Ed. S.-T. Tau, Worlds S ienti� ,1987, p.259. http://arxiv.org/abs/hep-th/0503203 http://arxiv.org/abs/hep-th/0605086 http://arxiv.org/abs/hep-th/0611043 http://arxiv.org/abs/hep-th/0701083 http://arxiv.org/abs/gr-qc/0606108 http://arxiv.org/abs/hep-th/0701053 http://arxiv.org/abs/gr-qc/0001047 http://arxiv.org/abs/hep-th/0312009 http://arxiv.org/abs/hep-th/0110089 http://arxiv.org/abs/hep-th/0208156 http://arxiv.org/abs/hep-th/0103239 http://arxiv.org/abs/hep-th/0108187 http://arxiv.org/abs/hep-th/0702154

0704.0355

Trigonometric parallaxes of high velocity halo white dwarf candidates

Astronomy & Astrophysics manuscript no. white2v4 c© ESO 2018 November 4, 2018 Trigonometric parallaxes of high velocity halo white dwarf candidates ? C. Ducourant1, R. Teixeira2,1, N.C. Hambly3, B. R. Oppenheimer4, M.R.S. Hawkins3, M. Rapaport1, J. Modolo1, and J.F. Lecampion1 1 Observatoire Aquitain des Sciences de l’Univers, CNRS-UMR 5804, BP 89, 33270 Floirac, France. 2 Instituto de Astronomia, Geofı́sica e Ciências Atmosféricas, Universidade de São Paulo, Rua do Matão, 1226 - Cidade Universitária, 05508-900 São Paulo - SP, Brasil. 3 Scottish Universities Physics Alliance (SUPA), Institute for Astronomy, School of Physics, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK. 4 Department of Astrophysics, American Museum of Natural History, 79th Street at Central Park West, New York, NY 10024-5192, Received / Accepted ABSTRACT Context. The status of 38 halo white dwarf candidates identified by Oppenheimer et al. (2001) has been intensively discussed by various authors. In analyses undertaken to date, trigonometric parallaxes are crucial missing data. Distance measurements are mandatory to kinematically segregate halo object from disk objects and hence enable a more reliable estimate of the local density of halo dark matter residing in such objects. Aims. We present trigonometric parallax measurements for 15 candidate halo white dwarfs (WDs) selected from the Oppenheimer et al. (2001) list. Methods. We observed the stars using the ESO 1.56-m Danish Telescope and ESO 2.2-m telescope from August 2001 to July 2004. Results. Parallaxes with accuracies of 1–2 mas were determined yielding relative errors on distances of ∼ 5% for 6 objects, ∼ 12% for 3 objects, and ∼ 20% for two more objects. Four stars appear to be too distant (probably farther than 100 pc) to have measurable parallaxes in our observations. Conclusions. Distances, absolute magnitudes and revised space velocities were derived for the 15 halo WDs from the Oppenheimer et al. (2001) list. Halo membership is confirmed unambiguously for 6 objects while 5 objects may be thick disk members and 4 objects are too distant to draw any conclusion based solely on kinematics. Comparing our trigonometric parallaxes with photometric parallaxes used in previous work reveals an overestimation of distance as derived from photometric techniques. This new data set can be used to revise the halo white dwarf space density, and that analysis will be presented in a subsequent publication. Key words. Astrometry : trigonometric parallax – Dark matter – Galaxy : halo – Star : kinematics – white dwarfs. 1. Introduction In the last decade interest in the very cool, old white dwarf (WD) halo population has grown. This interest is motivated by the pos- sibility that these objects could account for a significant fraction of the baryonic dark matter of our Galaxy. This idea is in accord with discussions attempting to explain the microlensing events in the Large Magellanic Cloud in terms of a halo WD popula- tion – see, for example, Chabrier et al. 1996 and Hansen 1998. Alcock et al. 1999 suggested that massive compact halo objects (MACHOs) make up 20 to 100% of the dark matter in the halo, with MACHOs having typical mass m ∼ 0.5 M�; more recently, Calchi Novati et al. 2005 find a similar result from pixel lensing in the line of sight to M31. Hence, in this scenario the search for, and direct study of, halo WDs can provide constraints on the fraction of dark matter in the Milk Way that is attributable to these objects. Oppenheimer et al. (2001, hereafter OHDHS) identified 38 high proper motion WDs; from their kinematics, the authors Send offprint requests to: [email protected] ? Based on observations collected at the European Southern Observatory, Chile (067.D-0107, 069.D-0054, 070.D-0028, 071.D- 0005, 072.D-0153, 073.D-0028) concluded that they were members of a halo population. Since then an intense discussion concerning the status of these objects has taken place in the literature. A comprehensive review of this debate is presented in Hansen and Liebert 2003 where the conclusion is that the OHDHS interpretation is possibly over- stated, but that complete conclusions are not possible without further data. Other studies suggest that the disk and “thick disk” Galactic populations can be used to explain the great majority of the objects (Reid 2005, Kilic et al. 2005, Spagna et al. 2004, Crézé et al. 2004, Holopainen & Flynn 2004, Flynn et al. 2003, Silvestri et al. 2002). The importance of the high velocity WDs cannot be understated in other contexts (e.g. the star forma- tion history of the Galaxy, see also Davies, King & Ritter 2002, Hansen 2003, Montiero et al. 2006). Moreover, several stud- ies emphasise the importance of obtaining trigonometric par- allaxes for candidate halo WDs (Bergeron & Leggett 2002, Torres et al. 2002, Bergeron 2003). This is especially important for the coolest WDs, whose spectral energy distributions show remarkable departures from black–body distributions and which are proving to be difficult to model accurately (Kowalski 2006, Gates et al. 2004, Saumon & Jacobson 1999, Hansen 1998). In the presence of such radical changes to the WD spectrum, the as- sumption of a monotonic photometric parallax relation (e.g. as 2 C. Ducourant et al.: Parallaxes of halo white dwarf candidates used in OHDHS) could break down and estimates of intrinsic space velocities could be in error seriously. Furthermore, a recent paper (Bergeron et al. 2005) concludes that precise distances are mandatory to derive accurate kinematics and ages for the puta- tive halo WDs and in order to derive their evolutionary status. Aiming to clear up this question, in 2001 we started an ob- serving program with the ESO 1.56-m Danish and ESO 2.2-m telescopes to measure the trigonometric parallaxes of these stars. Trigonometric parallax measurements remain the only direct un- biased distance determination. They are of great importance in the debate about the status of cool halo white dwarfs because they are required to derive precise space velocities and ages which are used for distinguishing between halo and disk mem- bership. These trigonometric parallaxes lead to the re-calibration of photometric distances used until now in this debate and allow analysis of the cool halo white dwarf population with more con- fidence. Unfortunately, due to limited observing time, only 15 stars on the OHDHS list have been observed to date. However, this sub-sample provides important insight into the problem. 2. Observations Astrometric observations of 15 of the OHDHS list of 38 halo white dwarf candidates were performed at the ESO 2.2-m tele- scope equipped with the WFI wide–field mosaic camera (with 0.238 ′′/pixel, a field of view FOV = 34′ × 33′, 4 × 2 mosaic of 2k × 4k CCDs), through the ESO 845 I filter. To reduce astro- metric distortions and other instrumental effects, only data from chip 51 (with FOV = 8′× 16′) were used in this work; target stars were centered in the FOV of this chip. Four epochs of observation were acquired at maximum parallactic factor in Right Ascension in November 2002, July 2003, November 2003 and July 2004 with a total of 11 nights of observations. Two parallactic periods (four observations over 1.5 years) are required, at a minimum, for a unique determination of the parallax and proper motion. Two preliminary observing runs were performed at the ESO 1.56-m Danish telescope in July 2001 and July 2002 but the subsequent closure of the telescope forced the authors to move the program to the ESO 2.2-m telescope. Data acquired at the Danish telescope were not included in our final analysis to avoid systematic effects due to the use of two different telescopes. To minimize differential colour refraction effects (DCR), ob- servations were performed around the transit of targets with hour angles of less than 1 hour. Multiple exposures were taken at each observation epoch to reduce the astrometric errors and to estimate the precision of measurements. Exposure times varied from 100 to 600 seconds depending on the magnitude of the tar- get. Each field was observed from 20 to 35 times. 3. Astrometric Reduction 3.1. Measurements Frames were measured using the DAOPHOT II package (Stetson 1987), fitting a PSF. The significance level of a luminosity en- hancement over the local sky brightness which was regarded as real was set to 7σ. The PSF routine was used to define a stel- lar point spread function for each frame. Finally we obtained the (x, y) measured positions, the internal magnitudes and associated errors of all stars on each frame. There were typically 300 to 600 stars measured on each frame depending on the exposure time. From these, a selection on the error in magnitude (ERRMAG) as derived by the DAOPHOT II software was applied. Any ob- servation with ERRMAG ≥ 0.15m was rejected. Objects fainter than 1.5m brighter than a given image’s limiting magnitude were also rejected from the analysis. 3.2. Cross-Identification For each of the 15 different fields of view, we selected a “master” or fiducial image from the set of 20 to 35 images. This master frame for each object had the deepest limiting magnitude and highest image quality. For each of the other images for a given target object, the positions of all stars not rejected by the crite- ria above were then cross–identified to the master image’s star positions. Objects not detected on three or more frames were excluded, yielding 100 to 200 stars in common in each field. Frames containing less than Nmaster/3 stars in common with the master frame were removed from the solution (where Nmaster is the number of stars in the master frame). Note that the master frame is processed in an identical fashion to the other frames and is not assumed to be free of errors in the parallax solution. In other words, the fiducial frames are not taken as an error-free “truth”, but are simply used as a basis for coordinate transfor- mations and correlation of star positions that comprise the astro- metric grid used in the solution. 3.3. Differential Colour Refraction Atmospheric refraction changes the apparent positions of stars in ground–based observations and depends on the zenith distance of the observations. For precision astrometry this effect must be accounted for, because it can be many tens of milliarcsececonds at even relatively modest zenith distances. In our case, another effect becomes important as well, because the atmospheric re- fraction of our target stars will not be identical to that of the background stars used for our astrometric reference grid. Our target stars (WDs) and the background stars (typically main– sequence G or K stars) have different spectral energy distribu- tions. Therefore, atmospheric refraction will affect them differ- ently when observed through a given filter bandpass. This is called a differential colour refraction (DCR) and is known to cause spurious parallactic motion Monet et al. 1992. DCR can affect both the Right Ascension (RA) and Declination of the tar- get as derived with respect to the field stars. Observations in par- allax programs are planned to maximize the parallactic factor in RA so the parallax solution for the target will rely heavily on the RA measured. Therefore the parallax derived is mainly per- turbed by the DCR effects in RA which are critically dependent on the zenith distance of a given observation. We investigated the impact of such effects on the parallax of white dwarfs through simulations. Using the usual formula for atmospheric refraction, a blackbody approximation for white dwarf and background stellar spectra, the Besançon Galaxy model for background star characteristics (Robin et al. 1994) and ESO 845 filter limits, we computed the average differen- tial colour refraction effects between a white dwarf similar to those of our list with effective temperatures, Teff , in the range 4000 K to 11000 K (Bergeron et al. 2005, Table 2) and a typical background star (Teff ∼ 5000 K). We present in Fig. 1 the effects of DCR in RA for white dwarfs situated at δ = −30◦, covering the range of temperatures of our targets. Fig. 1 demonstrates that the impact of DCR effects were always less than 0.5 mas for observations taken with an hour angle of less than one hour. Therefore, our observations C. Ducourant et al.: Parallaxes of halo white dwarf candidates 3 were made specifically so that the hour angle never exceeded one hour, and DCR corrections were not applied in this work. Fig. 1. DCR effects in RA between a white dwarf of temperature Teff and a mean background stars (Teff=5000K) at a Declination of −30◦ (representative of our sample) for various hour angles of observation. The DCR effects appear to be always lower than 0.5 mas for observations performed at less than 1 hour from merid- ian which is the case of the present project. The DCR effects are then negligible compared with other sources of astrometric error and were not taken into account in this work. 3.4. Impact of Pixel Scale Errors on Parallax Proper motions (µx, µy) and trigonometric parallax (πxy) of tar- gets are determined by comparing the (x, y) measurements ex- pressed in pixels. A scaling factor S f , the image pixel scale, is applied to πxy to convert pixel measurements into physical units: π = S fπxy; d(pc) = π−1, with S f expressed in ′′/pixel. Derivation of the pixel scale can be achieved through a cross–correlation between the (x, y) positions of stars on a given master frame to corresponding values of (α, δ) for the subset of stars that are also in a reference catalogue. Here we used the 2MASS catalogue (Cutri et al. 2003) to determine the orienta- tion of the master frame on the sky and for the pixel scale deter- mination. We selected the 2MASS catalogue as a reference cat- alogue because of its accuracy and density although we note the absence of proper motion corrections. Nevertheless the epoch difference between our observations and the 2MASS catalogue (3 years) would result in negligible corrections to the catalogue positions with respect to the catalogue errors. Errors on the scale so determined, resulting from catalogue random errors, will produce errors in the distance determination of the target. It is therefore important to quantify the impact of the catalogue errors onto the distance of the target. To measure this impact in the present work, we assumed N reference stars equally spread over a square detector of side A. The classical equation relating the (x, y) measurements of a stars on the frame to its standard coordinates X(α, δ),Y(α, δ) in the tangent plane to the celestial sphere is (with a similar equation in the Y coordinate) X = (ax + by + c)1/F, (1) where (a,b,c) are the unknown “plate” constants and F the focal length of the telescope (typically the value indicated in the ref- erence manual). F is expressed in the same units as (x,y) and A (pixel, mm). It is then easy to show that a fair approximation of the variance of the estimation of parameter (a) is given by �cat, (2) where �cat is the catalogue precision (expressed in radians). Similar results can be found in Eichhorn & Williams 1963. We can express the parallax (in radians) as: πxy, (3) σ2π = π σ2a, σπ = �cat (4) with F ∼ 13m, A ∼ 0.03m, we evaluate here σπ ∼ 10−4π. The impact of the error of the catalogue on the parallax of the target is far below the measurement errors (typically a few milliarcsec- onds) and are therefore negligible. 3.5. Global Solution: Relative Parallax The astrometric reduction of the whole set of data of each field is performed iteratively through a global central overlap procedure (Hawkins et al. 1998, Eichhorn 1997) in order to determine simultaneously the position, the proper motion and the parallax of each object of the field. The following condition equations are written for each star on each of the N frames considered (including the master frame). These equations relate the measured coordinates to the stellar astrometric parameters: X0 + ∆X0 + µX(t − t0) + πFX(t) = a1x(t) + a2y(t) + a3 (5) Y0 + ∆Y0 + µY (t − t0) + πFY (t) = b1x(t) + b2y(t) + b3 (6) where (X0,Y0) are the known standard coordinate of the star at the epoch t0 of the master frame, and (x(t), y(t)) its measured coordinates on the frame (epoch t) to be transformed into the master frame system. ∆X0, ∆Y0, µX , µY and π are the unknown stellar astrometric parameters: (∆X0, ∆Y0) yield correction of the standard coordinates of the star on the master frame, (µX , µY ) are the projected proper motion in RA ∗cos(δ) and Dec, and π is the parallax. Coefficients (ai, bi) are the unknown frame parameters which describe the transformation to the master frame system. (FX , FY ) are the parallax factors in standard coordinates. The unknowns of this large over–determined system of equations are the stellar astrometric parameters of each object, and the trans- formation coefficients of each of the N frames considered. The system of equations is singular and therefore the derived solution is not unique; any solution will depend on the starting point of the iterations. The usual technique to obtain a particular solution is to introduce a set of constraints that the solution must satisfy. In this work we chose to set strictly to zero the mean parallax of the reference stars. We used a Gauss–Seidel type iterative method to solve the set of equations. At the first iteration all stellar parameters are assumed null, we then computed the plate constants which are 4 C. Ducourant et al.: Parallaxes of halo white dwarf candidates injected into the system of equations to derive the stellar param- eters. These results are then used as the starting point of the fol- lowing iteration. The iterative procedure converges usually at the second or third iteration. A test of elimination at 3σ is applied to remove poor observations either in the master frame fit or in the stellar parameters fit. The stellar parameters fit equations have been weighted by the mean residual of the master frame fit. This weighting represents the quality of the measurements. The stars used for the master frame fit are called here reference stars. We applied this global treatment to the various observations of the 15 fields observed and we derived for the targets a proper motion and parallax with associated variances. 3.6. Conversion from Relative to Absolute Parallax The parallaxes that we derived for our targets are relative to the reference stars (for which we used the constraint π = 0), sup- posed placed at infinite distance. In fact these reference stars are at a finite distance from Sun. We must therefore correct the rela- tive parallax of the target from an estimate of the mean distance of the reference stars to obtain the absolute parallax of the target. The choice we made to keep as many reference stars as possible in our calculation is interesting because statistically faint stars have smaller parallax and require smaller correction. There are several ways to estimate the mean distance of reference stars: statistical methods relying on a model of the Galaxy; spectroscopic parallax; and photometric parallax. For the corrections from relative parallax to absolute parallax we used a statistical method relying on simulations using the Besançon Galaxy model (Robin et al. 1994) to derive the theo- retical mean distance of reference stars. A simulation of each ob- served field was performed, providing catalogues of distance and apparent magnitude of simulated stars. We computed in these catalogues mean distances and associated dispersion in magni- tude bins of 0.2 mag, establishing a table of theoretical distances with respect to apparent magnitude. Then we considered our ob- served fields and we computed the weighted mean parallax and associated dispersion of our reference stars using the theoretical table. Finally we added this mean parallax of reference stars to the relative parallax of our target leading to the absolute parallax of the white dwarfs. We give in Table 1 the relative to absolute corrections in mil- liarcseconds as found from the Besançon Galaxy model in each of the field treated. 4. Results 4.1. Distances of Halo White Dwarf Candidates We present in Table 2 the proper motions and absolute parallaxes of the fifteen halo white dwarf candidates as derived from this work together with their absolute magnitude MV computed using CCD V magnitudes from Bergeron et al. (2005). One notices that WD2326–272, LP586–51, LP588–37, and WD2324–595 are too distant to have a measurable parallax. Eleven objects are at distances ranging from 19 pc to 90 pc from the Sun. The parallax errors are about 1–2 mas corresponding to relative precisions of 5 to 20%. WD2214–390, which is the closest and brightest object, has a σπ = 2.6 mas. This poor pre- cision is due to the short exposure time used to avoid saturation problems and corresponding lower signal–to–noise ratio. We present in Figs 8 and ?? the positions (empty circles), their weighted mean (filled circles) and associated error bars at Table 1. Relative to absolute corrections ∆π and associated RMS (σ) as found from the Besançon Galaxy model in the Galactic direction (l,b) together with number of reference stars (N*) in magnitude interval [Jmin,Jmax]. Target l b ∆π σ N* Jmin Jmax [◦] [◦] [mas] [mag] WD2214-390 2.79 -55.37 1.3 0.3 38 13.1 16.2 WD2242-197 40.01 -59.42 1.0 0.3 97 14.0 18.4 WD2259-465 344.30 -60.62 1.1 0.2 83 13.6 18.0 LHS542 72.40 -59.70 1.2 0.3 42 13.4 17.0 WD2324-595 321.83 -54.34 1.1 0.2 62 13.3 17.0 WD2326-272 27.66 -71.06 1.3 0.4 80 14.2 18.7 LHS4033 90.24 -61.96 1.3 0.2 39 14.2 16.5 LHS4041 351.44 -74.66 1.4 0.3 37 13.5 16.2 LHS4042 6.55 -76.61 1.5 0.4 38 13.3 16.6 WD0045-061 118.54 -68.96 1.5 0.3 54 13.5 17.7 F351-50 314.26 -83.50 0.3 0.2 53 14.1 18.1 LP586-51 128.88 -63.30 1.3 0.3 47 14.1 17.4 WD0135-039 149.30 -64.53 1.3 0.2 82 14.4 19.0 LP588-37 150.44 -61.52 1.4 0.2 57 13.6 17.7 LHS147 178.72 -73.56 1.5 0.3 43 13.4 16.8 each epoch of observation, together with the fitted path for the eleven most significant parallaxes, where π/σπ ≥ 4. 4.2. Comparison with Published Distances We have compared our results with available data from the lit- erature, employing both trigonometric and photometric paral- laxes measured previously. We give in Table 3 the comparison with published trigonometric parallaxes and in Figure 2 a com- parison of the parallaxes derived in this work with photomet- ric parallaxes (from OHDHS, where photometric parallax errors were 20%). Parameters of a weighted linear fit between pho- tometric and trigonometric parallaxes are: πtrig = a.πphot + b with a = 1.08+/-0.08 and b = 3.21+/-1.56 [mas] with a reduced χ2 =8.06. Table 3. Comparison of trigonometric parallaxes from this work (πThiswork) with published data (πext) for LHS 147 (Van Altena et al. 1995), LHS 4033 (Dahn et al. 2004) and LHS 542 (Bergeron et al. 2005). Target πThiswork πext ∆π [mas] [mas] [mas] LHS 542 29.6 +/- 1.8 32.2 +/- 3.7 2.6 LHS 147 14.8 +/- 1.8 14.0 +/- 9.2 –0.8 LHS 4033 30.1 +/-1.8 33.9 +/- 0.6 3.8 Our parallaxes are in excellent agreement with the 3 pre- viously published trigonometric parallaxes, within the errors (which are considerably smaller in two cases than published val- ues). In Fig. 2 one notices a clear systematic tendency of pho- tometric parallaxes to be underestimated. This overestimation of OHDHS distances is of importance in the calculation of WD kinematics and space density. C. Ducourant et al.: Parallaxes of halo white dwarf candidates 5 Table 2. Proper motion and absolute parallaxes of the fifteen halo white dwarf candidates, where µα∗ = µαcos(δ) and σµ=σµα∗=σµδ ; π and σπ are the parallax and its precision, Dist the derived distance in parsec and MV the absolute magnitude. No value is given for Dist and Mv when the parallax is not better than 3 σ. N* is the number of reference stars and Nf the Number of frames. Dphot is the photometric distance from OHDHS and V is extracted from Bergeron et al. (2005) when available, otherwise (cases marked by an asterix) it comes from Salim et al. (2004). Note that LHS 4041 is in the OHDHS sample, but is not listed in OHDHS Table 1 (see Table 4 of Salim et al. 2004) Name α δ Epoch V µα∗ µδ σµ π σπ Dist Mv N* Nf Dphot [J2000] [yr] [mag] [mas/yr] [mas] [pc] [mag] [pc] WD2214–390 22 14 34.727 –38 59 07.05 2003.5 15.92 1009 –350 2.9 53.5 2.6 19 14.78 38 28 24 WD2242–197 22 41 44.252 –19 40 41.41 2003.5 19.74 359 +48 3.1 11.1 2.3 90 14.89 97 27 117 WD2259–465 22 59 06.633 –46 27 58.86 2002.9 19.56 402 –153 1.8 22.7 1.3 44 16.49 83 32 71 LHS542 23 19 09.518 –06 12 49.92 2003.5 18.15 –615 –1576 1.8 29.6 1.8 34 15.58 42 33 42 WD2324–595 23 24 10.165 –59 28 07.95 2003.5 16.79 136 –562 1.8 (3.1) 1.5 —- —- 62 25 58 WD2326–272 23 26 10.718 –27 14 46.68 2002.9 ∗19.92 574 –85 2.7 (6.2) 2.4 —- —- 80 17 108 LHS4033 23 52 31.941 –02 53 11.76 2002.9 16.98 631 298 2.5 30.1 1.8 33 14.38 39 26 63 LHS4041 23 54 18.793 –36 33 54.60 2002.9 ∗15.46 21 –662 1.8 13.4 1.5 75 11.10 37 27 59 LHS4042 23 54 35.034 –32 21 19.44 2003.5 17.41 421 –37 2.2 13.9 1.8 72 13.13 38 25 85 WD0045–061 00 45 06.325 –06 08 19.65 2002.9 18.26 111 –668 1.9 30.1 1.9 33 15.59 54 27 44 F351–50 00 45 19.695 –33 29 29.46 2003.5 19.01 1820 –1476 2.1 28.3 1.4 35 16.63 53 34 37 LP586–51 01 02 07.181 –00 33 01.82 2002.9 18.18 350 –118 3.6 (2.4) 2.7 —- —- 47 24 120 WD0135–039 01 35 33.685 –03 57 17.90 2002.9 19.68 456 –180 3.4 13.3 2.9 75 15.26 82 21 146 LP588–37 01 42 20.770 –01 23 51.38 2002.9 ∗18.50 112 –328 3.4 (1.4) 4.5 —- —- 57 17 120 LHS147 01 48 09.120 –17 12 14.08 2002.9 17.62 –115 –1094 2.1 14.8 1.8 68 13.46 43 29 71 Fig. 2. Comparison of parallaxes derived in this work with pho- tometric parallaxes from OHDHS (errors are assumed 20% for πphot). Parameters of a weighted linear regression (diagonal line) between both types of parallaxes are π = 1.08πphot + 3.21 [mas] with a reduced χ2 = 8.06. The photometric distances are sys- tematically larger than the astrometric ones. 4.3. Proper Motions We have compared the proper motions derived here with the OHDHS proper motions in order to check wether some system- atic effects could affect our proper motions derived on a 1.5 yr time span and, as a result, our parallaxes. We present this com- parison in Fig. 3 and Fig. 4. Error bars are drawn in both co- ordinates but since the present work has much higher precision than the photographic astrometry, the error bars in x are not vis- ible. The slope of a linear regression between proper motions in α cos(δ) derived in this work with the OHDHS proper motions is 1.04 ± 0.02 with a reduced χ2 = 3.7. The equivalent linear fit in proper motions in Declination has a slope of 1.01 ± 0.02 with a reduced χ2 = 0.7. For F351-50 (the largest error bars in both figures), the accordance in RA and Dec proper motions is not good. This is due to a known problem of contamination by a background galaxy of the Schmidt plate measurements used in the OHDHS work. Nevertheless the accordance is within 2σ. These comparisons show excellent agreement between both sets of proper motions, and argue against any systematic effects from the present work. 4.4. Space Velocities We derived the Galactic space velocities U, V, W (Johnson and Soderblom 1987) for the white dwarfs using the distances and proper motions measured here together with radial velocities from Salim et al. 2004 (data available for 9 of the 15 white dwarfs treated here). Salim’s observed radial velocities were corrected for a mean gravitational redshift of +28km/s as suggested by the authors in their paper except in the case of the very massive white dwarf LHS4033 were the correction was taken from Dahn et al. 2004. U is radial toward the Galactic center, V is in the direction of rotation and W perpendicular to the Galactic disk. U,V and W were corrected for the Sun’s peculiar velocity (Mihalas and Binney (1981)). When no radial velocity was available from other studies, we assumed Vr = 0 km/s. This approximation is acceptable due to its minor impact on U,V velocities since the targets are located 6 C. Ducourant et al.: Parallaxes of halo white dwarf candidates Fig. 3. Comparison of proper motions in RA cos(δ) with the OHDHS proper motions. Error bars are drawn in both coordi- nates but since the present work has much higher precision than the photographic astrometry, error bars in abscissae are not visi- ble. The slope of a linear regression (dotted line) is 1.04 ± 0.02 indicating good accordance between both proper motion data sets with a reduced χ2 = 3.7. close to South Galactic Cap (the effect was investigated in OHDHS and shown to be negligible). We present in Figure 5 the distribution of velocities in the Galactic plane together with the velocity dispersion for the disk (right most)(1, 2 and 3 σ), thick disk (middle)(1, 2 and 3 σ)( Fuhrmann 2004) and halo (left) (1 and 2σ) (Chiba and Beers 2000) and in Figure 6 the component of mo- tion perpendicular to the Galactic plane. These two figures con- cern the 11 objects with parallax measured at the 4σ level or better. In Fig. 5 one notices that 4 of the 11 studied WDs have a velocity incompatible at the 3σ level with the kinematic of the disk and of the thick disk and that 6 of them are incompatible at a 2σ level. No star lies within the 1σ ellipse of the disk, primarily because of selection effects in the original proper motion survey that OHDHS based is based upon Hambly et al. 2005. Obviously the choice of the center and dispersions of halo, thick disk and disk ellipses is critical to classify objects as be- longing to a particular population. We adopted recent values which are in in the range of the values cited by Reid 2005 in his review: Disk (Fuhrmann 2004) : (U,V) = (7.7, −18.1) km/s, (σU , σV ) = (42.6, 22.6) km/s; thick disk (Fuhrmann 2004): (U, V) = (-18, −63) km/s, (σU , σV ) = (58, 41) km/s; halo (Chiba and Beers 2000): (U,V) = (0, −180) km/s, (σU , σV ) = (141, 106) km/s. 5. Discussion As discussed above, OHDHS sparked a lively debate about whether stellar remnants contribute to a significant fraction of the baryonic component of the putative dark matter halo of our Galaxy. The main criticisms have concerned interpretation, and we do not address those here. However, the photographic pho- Fig. 4. Comparison of proper motions in Declination derived in this work with the OHDHS proper motions. Error bars are drawn in both coordinates but since the present work has much higher precision than the photographic astrometry, error bars in abscissae are not visible. The slope of a linear regression (dotted line) is 1.01 ± 0.02 indicating a good accordance between both proper motion data sets with a reduced χ2 = 0.7. tometry and use of a single photometric parallax relation are also potential sources of systematic error. Both Salim et al. (2004) and Bergeron et al. (2005) have shown that the original photom- etry presented in OHDHS was as accurate as could be expected. Here, we address the question of the accuracy of photometric parallaxes directly via trigonometric determination of distances. In Fig. 2 we compare the trigonometric parallaxes derived here with the OHDHS photometric parallaxes. Parameters of a weighted linear regression between both types of parallaxes are π = 1.08 πphot + 3.21 with a reduced χ2 = 8.06. A clear under- estimation of photometric parallaxes is visible in this figure with only one point below the diagonal and three points more than 3σ above the relation. With the usual caveat of small number statistics, this indicates some level of non–Gaussian scatter, or at least a mean value for the relation that is not coincident with π = πphot. The photometric parallax overestimates the distance. This leads, of course, to an overestimation of tangential space velocities based on proper motion and distance (as an aside, we note that the quoted photometric parallax errors of 20% were conservatively overestimated by OHDHS). It is interesting to note that the mass distribution of hot (Teff > 12, 000 K) DA WDs is not Gaussian and has a broad tail on the high mass side (Należyty et al. 2005). Given that ra- dius r ∝ m−1/3 for WDs, we would expect photometric paral- laxes to tend to overestimate rather than underestimate distances since some of the sample may have higher than average mass, and correspondingly smaller radii, placing them nearer to the Sun than typical objects of the same colour. Adding in a sprin- kling of higher mass WDs with helium–dominated atmospheres will introduce further systematic overestimation of distances. It is almost certainly the case that the discrepant photometric par- allaxes for WD2259–465 and WD0135–039 are caused by these effects; indeed, this has been shown to be the case for LHS 4033 C. Ducourant et al.: Parallaxes of halo white dwarf candidates 7 Fig. 5. Distribution of velocities in the Galactic plane to- gether with the velocity dispersion for the disk (right most)(1, 2 and 3 σ), thick disk (middle)(1, 2 and 3 σ)( Fuhrmann 2004) and halo (left) (1 and 2σ) (Chiba and Beers 2000). Filled squares correspond to objects with a measured radial velocity (Salim et al. 2004) while open circles correspond to objects with no Vr measurement. Only objects with parallax measured at the 4σ level or better are plotted. which has a mass m ∼ 1.3 M� (Dahn et al. 2004). On the other hand, the low–mass side of the mass distribution is by no means perfectly Gaussian (e.g. due to low-mass, helium–core white dwarfs formed in close binaries). Moreover, any overestimation in distance leads to a corresponding underestimate of space den- sity using the 1/Vmax technique. So the interpretation of the re- sults from this relatively small sub–sample is rather complicated, and it is only through detailed simulations compared with much larger samples that significant progress is likely to be made con- cerning the question of the kinematic population of such objects. From the comparison of trigonometric and photometric parallaxes (Fig. 2) we recalibrated photometric distances of the original OHDHS sample and, using radial velocities from Salim et al. 2004, we derived their associated recalibrated space velocities. We present the recalibrated UV plane for the entire OHDHS sample in Fig. 7. When compared to Fig. 3 of OHDHS, the number of halo objects has diminished. From the 38 original OHDHS halo can- didates, 16 appear compatible with a halo status based on a 2σ cut with the disk and thick disk velocity distributions (a 3σ cut would reduce this number to 7), the remaining objects being now located within the disk and thick disk 2 sigma ellipses. In the lit- erature there is a large spread of the proposed values to charac- terise the thick disk and halo populations in terms of kinematics. For instance in Reid 2005 the velocity dispersions for thick disk vary from 50 to 69 km/s in the U direction and from 39 to 58 km/s in the V direction. Even the center of velocity ellipsoid varies from –30 to –63 km/s in the < V > coordinate from one author to another. All this makes it very difficult to separate ob- jects into halo and thick disk populations and requires a more detailed analysis which is beyond the scope of the present paper. Fig. 6. Component of motion perpendicular to the Galactic plane (W) as function of U2 + V2. Only objects with paral- lax measured at the 4σ level or better and with available radial velocity (Salim et al. 2004) arre plotted. The vertical line is the OHDHS U2 + V2 = 94 km/s cut. The conclusions of OHDHS about local halo WD density must be now reanalysed since the volume explored by their sur- vey has changed (re-calibrated distances) and the number of halo candidates has also changed. This will be the subject of a forth- coming paper. 6. Acknowledgements The authors wish to thank G. Daigne for helpful comments and CAPES/COFECUB, FAPESP organizations and INR for sup- porting the project. References Alcock, C. et al. 1999, in ASP Conf. Ser. 165, The third Stromlo Symposium: the Galactic Halo, ed. B.K. Gibson, T.S. Axelrod and M.E. Putman (San Francisco:ASP),362 Bergeron, P., Leggett, S. K. 2002, ApJ, 580, 1070 Bergeron, P. 2003, ApJ, 586, 201 Bergeron, P.; Ruiz, M.–T.; Hamuy, M.; Leggett, S. K.; Currie, M. J.; Lajoie, C.–P.; Dufour, P. 2005, ApJ, 625, 838 Calchi Novati, S. et al. 2005, A&A, 443, 911 Chabrier G., Segretain L. and Mra D., 1996, ApJ, 468, L21-L24 Chiba, M., and Beers, T.C., 2000, AJ, 119, 2843 Crézé, M., Mohan, V., Robin, A. C., Reylé, C., McCraken, H. J., Cuillandre, J.–C., Le Fèvre, O., Mellier, Y., 2004, A&A, 426, 65 Cutri R. M., Skrustskie M. F., Van Dyk S. et al., 2003 Dahn, C. C., Bergeron, P., Liebert, J., Harris, H. C., Canzian, B., Leggett, S. K., Boudreault, S. 2004, ApJ, 605, 400 Davies, M. B., King, A. R., Ritter, H. 2002, MNRAS, 333, 469 Eichhorn, H. and Williams, C.A. 1963, AJ, 68, 221 Eichhorn, H.1997, Astron. Astrophys., 327, 404 Flynn, C., Holopainen, J., Holmberg, J., 2003, MNRAS, 339, 817 Fuhrmann K. 2004, Astron. Nact. 325:3-80 Gates, E. et al. 2004, ApJ, 612, 129L Hansen, B. M. S., 1998, Nature, 394, 860 Hansen, B. M. S. 2003, ApJ, 582, 915 Hansen, B. M. S. and Liebert, J. 2003, ARA&A, 41,465 Hambly, N. C., Digby, A. P., Oppenheimer, B. R., 2005, ASPC, 334, 113 8 C. Ducourant et al.: Parallaxes of halo white dwarf candidates Fig. 7. Distribution of velocities of the original OHDHS sam- ple with recalibrated parallaxes in the Galactic plane together with the velocity dispersion for the disk (right most)(1, 2 and 3 σ), thick disk (middle) (1, 2 and 3 σ)( Fuhrmann 2004) and halo (left) (1 and 2σ) (Chiba and Beers 2000). Filled squares correspond to objects with a measured radial velocity (Salim et al. 2004) while open circles correspond to objects with no Vr measurement. Hawkins, M. R. S., Ducourant, C., Jones, H. R. A. and Rapaport, M., 1998, MNRAS, 294, 505 Holopainen, J., Flynn, C., 2004, MNRAS, 351, 721 Johnson, D. R. H., Soderblom, D. R. 1987, AJ, 93, 864 Kilic, M., Mendez, R. A., von Hippel, T., Winget, D. E., 2005, ApJ, 633, 1126 Kowalski, P. M. 2006, ApJ, 641, 488 Należyty, M., Madej, J., Althaus, L. G. 2005, ASPC 334, 107 Mihalas, D., Binney, J. 1981, ”Galactic astronomy”, second edition. Monet D.G., Dahn C.C., Vrba F.J., Harris H.C., Pier J.R., Luginbuhl C.B., Ables H.D., 1992, AJ, 103, 638 Montiero, H., Jao, W.–C., Henry, T., Subasavage, J., Beaulieu, T. 2006, ApJ, 638, Oppenheimer, B. R., Hambly, N. C., Digby, A. P., Hodgkin, S. T., Saumon, D. 2001, Science, 292, 698 (OHDHS) Reid, I. N., 2005, ARA&A, 43, 247 Robin, A., 1994, ApSS, 217, 163R Salim, S., Rich, R. M., Hansen, B. M., Koopmans, L. V. E., Oppenheimer, B. R., Blandford, R. D., 2004, ApJ, 601, 1075 Torres, S., Garcı́a–Berro, E., Burket, A., Isern, J. 2002, MNRAS, 336, 971 Saumon, D.; Jacobson, S. B. 1999, ApJ, 511, L107 Silvestri, N. M., Oswalt, T. D., Hawley, S. L. 2002, AJ, 124, 1118 Spagna, A., Carollo, D., Lattanzi, M. G., Bucciarelli, B. 2004, A&A, 428, 451 Stetson Peter B., 1987, PASP, 99, 191 Van Altena, W. F., Lee J. T., Hoffleit E. D. 1995, General Catalogue of Trigonometric Stellar Parallaxes, Fourth Edition, Yale University Observatory C. Ducourant et al.: Parallaxes of halo white dwarf candidates 9 Fig. 8. Observations along the fitted path expressed in mas. Introduction Observations Astrometric Reduction Measurements Cross-Identification Differential Colour Refraction Impact of Pixel Scale Errors on Parallax Global Solution: Relative Parallax Conversion from Relative to Absolute Parallax Results Distances of Halo White Dwarf Candidates Comparison with Published Distances Proper Motions Space Velocities Discussion Acknowledgements

0704.0356

AMR simulations of the low T/|W| bar-mode instability of neutron stars

AMR simulations of the low T/|W | bar-mode instability of neutron stars Pablo Cerdá-Durán, Vicent Quilis, and José A. Font Departamento de Astronomı́a y Astrof́ısica, Universidad de Valencia, Dr. Moliner 50, 46100 Burjassot (Valencia), Spain Abstract It has been recently argued through numerical work that rotating stars with a high degree of differential rotation are dynamically unstable against bar-mode de- formation, even for values of the ratio of rotational kinetic energy to gravitational potential energy as low as O(0.01). This may have implications for gravitational wave astronomy in high-frequency sources such as core collapse supernovae. In this paper we present high-resolution simulations, performed with an adaptive mesh re- finement hydrodynamics code, of such low T/|W | bar-mode instability. The complex morphological features involved in the nonlinear dynamics of the instability are re- vealed in our simulations, which show that the excitation of Kelvin-Helmholtz-like fluid modes outside the corotation radius of the star leads to the saturation of the bar-mode deformation. While the overall trends reported in an earlier investigation are confirmed by our work, we also find that numerical resolution plays an impor- tant role during the long-term, nonlinear behaviour of the instability, which has implications on the dynamics of rotating stars and on the attainable amplitudes of the associated gravitational wave signals. Key words: gravitational waves, hydrodynamics, instabilities, stars: neutron stars: rotation PACS: 97.60.Jd, 04.30.-w, 95.30.Lz 1 Introduction Neutron stars following a core collapse supernova are rotating at birth and can be subject to various nonaxisymmetric instabilities (see e.g. [1] for a re- view). Among those, if the rotation rate is high enough so that the ratio of rotational kinetic energy T to gravitational potential energy W , β ≡ T/|W |, exceeds the critical value βd ∼ 0.27, inferred from studies with incompressible Maclaurin spheroids, the star is subject to a dynamical bar-mode (l = m = 2 Preprint submitted to Elsevier 30 July 2021 http://arxiv.org/abs/0704.0356v1 f -mode) instability driven by hydrodynamics and gravity. Its study is highly motivated nowadays as such an instability bears important implications in the prospects of detection of gravitational radiation from newly-born rapidly rotating neutron stars. Simulations of the dynamical bar-mode instability are available in the litera- ture, both using simplified models based on equilibrium stellar configurations perturbed with suitable eigenfunctions [2,3,4,5], and more involved models for the core collapse scenario [6,7,8,9], and in either case both in Newtonian grav- ity and general relativity. Due to its superior simplicity the former approach has received much more attention, notwithstanding that the conclusions drawn from perturbed stellar models may not be straightforwardly extended to the collapse scenario. Newtonian simulations of triaxial instabilities following core collapse were first performed by [6]. These showed that the bar-mode instability sets in when β ≫ 0.27 and when the progenitor rotates rapidly and highly differentially. Such conditions are met when the (artificial) depletion of internal energy to trigger the collapse is large enough to produce a very compact core for which a significant spun-up can be achieved. More recently, three-dimensional simula- tions of the core collapse of rotating polytropes in general relativity have been performed by [7]. These authors studied the evolution of the bar-mode insta- bility starting with axisymmetric core collapse initial models which reached values of β ∼ 0.27 during the infall phase. These simulations showed that the maximum value of β achieved during collapse and bounce depends strongly on the velocity profile, the total mass of the initial core, and on the equa- tion of state. In agreement with the findings from the Newtonian simula- tions of [6], the bar-mode instability sets in if the progenitor rotates rapidly (0.01 ≤ β ≤ 0.02) and has a high degree of differential rotation. In addition, the artificial depletion of pressure and internal energy to trigger the collapse, leading to a compact core which subsequently spins up, also plays a key role in general relativity for a noticeable growth of the bar-mode instability. Whether the requirements inferred from numerical simulations are at all met by the collapse progenitors remains unclear. As shown by [10] magnetic torques can spin down the core of the progenitor, which leads to slowly rotating neu- tron stars at birth (∼ 10 − 15ms). The most recent, state-of-the-art compu- tations of the evolution of massive stars, which include angular momentum redistribution by magnetic torques and spin estimates of neutron stars at birth [11,12], lead to core collapse progenitors which do not seem to rotate fast enough to guarantee the unambiguous growth of the canonical bar-mode instability. Rapidly-rotating cores might be produced by an appropriate mix- ture of high progenitor mass (M > 25M⊙) and low metallicity (N. Stergioulas, private communication). In such case the progenitor could by-pass the Red Supergiant phase in which the differential rotation of the core produces a magnetic field by dynamo action which couples the core to the outer layers of the star, transporting angular momentum outwards and spinning down the core. According to [13] about 1% of all stars with M > 10M⊙ will produce rapidly-rotating cores. On the other hand, Newtonian simulations of the bar-mode instability from perturbed equilibrium models of rotating stars have shown that βd ∼ 0.27 independent of the stiffness of the equation of state provided the star is not strongly differentially rotating. The relativistic simulations of [5] yielded a value of β ∼ 0.24 − 0.25 for the onset of the instability, while the dynamics of the process closely resembles that found in Newtonian theory, i.e. unstable models with large enough β develop spiral arms following the formation of bars, ejecting mass and redistributing the angular momentum. As the degree of differential rotation becomes higher Newtonian simulations have also shown that βd can be as low as 0.14 [14]. More recently [15,16] have reported that rotating stars with an extreme degree of differential rotation are dynamically unstable against bar-mode deformation even for values of β of O(0.01). Given its recent discovery and its potential astrophysical implications for post- bounce core collapse dynamics and gravitational wave astronomy, we present in this paper high resolution simulations of such low T/|W | bar-mode in- stabilities. This work is further motivated in the light of the few numerical simulations available in the literature. Our main goal is to revisit the simula- tions by [15] on the low T/|W | bar-mode instability, and particularly to check how sensitive the onset and development of the instability is to numerical issues such as grid resolution. To this aim we perform Newtonian hydrody- namical simulations of a subset of models analyzed by [15] using an adaptive mesh refinement (AMR) code [17] which allows us to perform such three di- mensional simulations with the highest resolution ever used. Our simulations reveal the complex morphological features involved in the nonlinear dynamics of the instability, where the excitation of Kelvin-Helmholtz-like fluid modes influences the saturation of the bar-mode deformation. We advance that while the overall trends found by [15] are confirmed by our work, the resolution employed in the simulations does play a key role for the long-term behaviour of the instability and for the nonlinear dynamics of rotating stars, which has implications on the attainable amplitudes of the associated gravitational wave signals. We note that we plan to upgrade the existing AMR code to account for the effects of magnetic fields in order to attempt the current study in a more realistic setup. The present work is a step towards that goal. The paper is organized as follows: Section 2 gives a brief overview of the equations to solve. Their solution is outlined in Section 3 which also contains the bare details of the AMR code. The results of the simulations are discussed in Section 4. Finally Section 5 presents our conclusions. 2 Mathematical framework The evolution of a self-gravitating ideal fluid in the Newtonian limit is de- scribed by the hydrodynamics equations and Poisson’s equation: +∇ · (ρv) = 0 (1) + (v · ∇)v = −1 ∇p−∇φ (2) +∇ · [(E + p)v] = −ρv∇φ (3) ∇2φ = 4πGρ (4) where x, v = dx = (vx, vy, vz), and φ(t,x) are, respectively, the Eulerian coordinates, the velocity, and the Newtonian gravitational potential. The total energy density, E = ρǫ + 1 ρv2 , is defined as the sum of the thermal energy, ρǫ, where ρ is the mass density and ǫ is the specific internal energy, and the kinetic energy (where v2 = v2x + v y + v z). Pressure gradients and gravitational forces are the responsible for the evolution. An equation of state p = p(ρ, ǫ) closes the system. We use an ideal gas equation of state p = (Γ − 1)ρǫ with Γ = 2. The hydrodynamics equations, Eqs. (1–3), can be rewritten in flux-conservative form: ∂f(u) ∂g(u) ∂h(u) = s(u) (5) where u is the vector of unknowns (conserved variables): u = [ρ, ρvx, ρvy, ρvz, E] . (6) The three flux functions Fα ≡ {f , g,h} in the spatial directions x, y, z, respec- tively, are defined by f(u) = ρvx, ρv x + p, ρvxvy, ρvxvz, (E + p)vx g(u)= ρvy, ρvxvy, ρv y + p, ρvyvz, (E + p)vy h(u) = ρvz, ρvxvz, ρvyvz, ρv z + p, (E + p)vz and the source terms s are given by Table 1 Overview of the initial models and results of the simulations. The rows report the name of the model, the ratio of equatorial-to-polar radii (re/rp), the degree of differential rotation (Â), the ratio of kinetic to potential energy (T/|W |), the size of the computational grid (L) and the location of the corotation radius (rc) for the two resolutions used: high (AMR H) and low (AMR L). In models R1H and R2H the corotation radius lies outside the star. The real (frequency) and imaginary (growth rate) parts of the bar-mode σ2 are shown, for the low and high resolution simulation in comparison with the numerical results and linear analysis by [15]. Note that for model D3 no linear analysis results are available. Model D1 D2 D3 R1 R2 re/rp 0.805 0.605 0.305 0.305 0.255  0.3 0.3 0.3 1.0 1.0 T/|W | 0.039 0.085 0.149 0.253 0.275 L/re 4.06 3.73 3.21 4.25 4.03 rc/re AMR L 0.38 0.47 0.58 AMR H 0.36 0.48 0.56 - - Re(σ2)/Ω0 AMR L 0.76 0.58 0.41 AMR H 0.81 0.55 0.43 - 0.82 Shibata 0.80 0.60 0.45 0.92 0.75 linear 0.80 0.58 - 0.92 0.75 Im(σ2)/Ω0 AMR L 0.0042 0.0154 0.0200 AMR H 0.0089 0.0190 0.0240 0.0005 0.1960 Shibata 0.009-0.013 0.019-0.021 0.013 <0.002 0.23 linear 0.015 0.021 - <0.002 0.20 s(u) = , ρvx − ρvy − ρvz . (10) System (5) is a three-dimensional hyperbolic system of conservation laws with sources s(u). 3 Numerical approach For our study of the low T/|W | bar-mode instability we perform high-resolution simulations of rotating neutron stars using a Newtonian AMR hydrodynamics code called MASCLET [17]. The implementation of the AMR technique in the code follows the procedure developed by [18]. The hydrodynamics equations are solved using a high-resolution shock-capturing scheme based upon Roe’s Riemann solver and second-order cell reconstruction procedures, while Pois- son’s equation for the gravitational field is solved using multigrid techniques. The accuracy and performance of the MASCLET code has been assessed in a number of tests [17]. We note that the code was originally designed for cos- mological applications, and here it is applied to simulations of self-gravitating stellar objects for the first time. The simulations are performed with two different grid resolutions. The low resolution grid consists of a box of size L with 1283 zones, yielding a fixed resolution of L/128. We note that the effective resolution of our coarse grid is comparable to that used by [15]. Correspondingly, the high resolution grid consists of a base coarse grid of 1283 cells, and one level of refinement composed of patches with maximum size of 643 cells (323 coarse cells). This yields a grid resolution on the finest grid of L/256. This resolution is enough to resolve the structures simulated, and hence no deeper refinement levels are needed. The patches are dynamically allocated covering those regions of the star where the highest resolution is required (highest densities). Typically only one patch is needed for spheroidal models, and 4-8 in models with toroidal topology. The use of AMR techniques in our high resolution simulations, allows us to save about a factor 4 in CPU time and memory with respect to a unigrid simulation with 2563 cells. No symmetries are imposed in the simulations. To the best of our knowledge, in the investigations of the bar-mode instability performed by previous groups, grid resolutions as high as the ones we use here were never employed. As customary in grid-based codes [19,20] the vacuum surrounding the star is filled with a tenuous numerical atmosphere with density ρ/ρmax ≈ 10−12 and zero velocities, ρmax being the maximun density. Every grid cell with ρ/ρmax < 10 −6 is reset to the atmosphere values. A correct treatment of the atmosphere is essential for an accurate description of the stellar dynamics and correct computation of the growth rates of unstable modes. We have checked that values for the atmosphere higher than those we chose or a free evolution of the atmosphere altogether, lead to remarkable changes in the mode behaviour, growth rates, and frequencies. We have also checked that lower values for the atmosphere do not produce those changes, which ensures that our evolutions are not affected by the atmosphere values used in the simulations. 4 Results 4.1 Initial data Differentially rotating stellar models in equilibrium are built according to the method of [21], and used as initial data for the AMR evolution code. The stars obey a polytropic equation of state P = KρΓ with index Γ = 2. As [15] the profile of the angular velocity Ω is given by (̟/re)2 + Â2 , (11) where re is the equatorial radius of the star, Ω0 is the central angular ve- locity, ̟ is the distance to the rotation axis, and  parametrizes the degree of differential rotation, from  ≪ 1 for highly differentially rotating stars to  → ∞ for rigidly rotating stars. For comparison purposes these parameters are chosen as in some of the models of [15], and are summarized in Table 1. Models labelled D rotate with a high degree of differential rotation, as  = 0.3, and may therefore be subject to the low T/|W | bar-mode instability. We also consider models almost rigidly rotating, labelled R, prone to experience the “classical” bar-mode instability. Labels L and H in the models refer to low and high resolution respectively. Following [15] we perturb the initial density profile ρ(0) according to ρ = ρ(0) 1 + δ x2 − y2 , (12) the perturbation of the pressure given by the equation of state accordingly. A perturbation amplitude δ = 0.1 is used in all our simulations. As we show below this form of the perturbation excites the l = m = 2 bar-mode. In addition, grid discretization can leak small amounts of energy to all other possible modes, which could in principle grow provided they were unstable and the simulations were carried on for sufficiently long times. 4.2 Stability analysis To compare with [15] we calculate the distortion parameters η+ and η× (and η = (η2+ + η )1/2) defined as 0 0.5 1 1.5 2 Ω / Ω0 0 0.21 0.43 0.65 0.86 1.1 1.3 1.5 1.7 1.9 Fig. 1. Power spectra of Am from m = 1 to m = 8 for model D3H. Ixx − Iyy Ixx + Iyy , η× ≡ Ixx + Iyy , (13) where Iij(i, j = x, y, z) is the mass-quadrupole moment Iij = dx3ρ xixj . (14) For the study of the growth rate and interaction of the different angular modes within the star is useful to calculate the global quantity dx3ρ(x) e−imϕ, (15) and Am ≡ Am/A0. We follow the time evolution of modes with m ranging from 1 to 8. Since our initial equilibrium models are axisymmetric and have equatorial plane symmetry, all Am are zero initially, but once perturbed all 0 20 40 60 80 100 Fig. 2. Evolution of η for models R1H (upper panel) and R2H (lower panel). Expo- nential fits to the peaks in the growing phase are overplotted as solid lines. initial models exhibit a dominant m = 2 component. Assuming that the modes behave as e−i(σmt−mϕ), the real part of σm can be obtained by Fourier trans- forming Am. In particular Re(σ2), the bar-mode frequency, can be extracted from either A2 or η as both represent the same mode. This is the dominant mode in all our simulations and its frequency and growth rate are given in Table 1. The latter corresponds to the imaginary part of σ2, which is calcu- lated fitting an exponential to the peak values of η in the growing phase of the evolution until the modes saturate. Other modes are also identified in the simulations for values of Am with lower amplitudes. We have checked that these modes are harmonics of the l = m = 2 mode so that they follow to good accuracy the relation σm = mσp, σp being the pattern frequency, calculated as σp = σ2/2. This is shown for model D3H in Fig. 1 which displays the spec- trum of Am from m = 1 to m = 8 (in arbitrary units). The vertical dashed lines in this figure indicate the location of the integer multiples of the pattern frequency σp, their values indicated on the axis at the top of the figure. Each spectrum for each mode is normalized to its own maximum for plotting pur- poses. Note that the lower the mode amplitude the noisier the spectrum and the less accurate the relation σm = mσp. For the models of our sample subject to the “clasical” bar-mode deformation (R1H and R2H), our simulations yield a value of β between 0.253 and 0.275, in good agreement with the critical value for the onset of the dynamical bar- mode instability. Model R1H is stable and model R2H is unstable. The growth rates and frequencies reported in Table 1 agree with those of [15]. Note that 0 100 200 300 Fig. 3. Evolution of η for models D1 (upper panel), D2 (central panel) and D3 (lower panel). Dashed lines correspond to low resolution and solid lines to high resolution. Exponential fits to the peaks in the growing phase are overplotted as solid lines. for model R1H, which is stable, the frequency for the m = 2 mode cannot be computed. The time evolution of η for these two models is displayed in Fig. 2. For the unstable model R2H, our simulations show the formation of a bar which saturates for values of η+ and η× close to 1, i. e. in the full nonlinear regime. Fig. 3 shows the time evolution of η for models D in our sample, prone to suffer the low T/|W | bar-mode instability. Solid lines correspond to high resolution simulations and dashed lines to low resolution. For all three models the pattern frequencies σp are such that there exists a corotation radius inside the star, i.e. a radius at which the bar-mode rotates with the same angular velocity as the fluid. The location of the corotation radius for all models of our sample is reported in Table 1. As recently discussed by [22] the existence of such corotation radius is a potential requirement for the ocurrence of the instability. As becomes clear from Fig. 3, all models are unstable but grid resolution has an important effect on the saturation of the instability once the nonlinear phase has been reached, as well as in the long-term dynamics of the stars. In the linear phase of models D1H and D2H, the growth rates and frequencies agree with the results of [15] in both, the numerical simulations and the linear analysis (see Table 1). In the linear phase of model D3H, our frequencies are similar to the numerical results of [15], although our growth rates are about a factor two larger. We emphasize that no results are reported in the linear analysis for this model in the work of [15], and therefore this discrepancy can be an effect of the resolution used or of the characteristics of each numerical code. Increasing resolution leads to similar results in the frequencies but to higher growth rates. In the nonlinear phase, models D1 and D3 behave similarly for the two resolu- tions used (see Fig. 3), and also similarly to the results by [15] (compare with Fig. 3 of that paper). For model D2 we observe a radical change of behavior in the nonlinear phase of the mode evolution depending on the grid resolution. This has implications on the long-term dynamics of the star and, in particu- lar, on the attainable amplitudes of the gravitational radiation emitted, as we discuss below. It is worth mentioning the possibility that the unstable mode at the start of model D2H might excite some other mode in the corotation band, which could not otherwise be excited for lower grid resolution. As discussed by [23,24] in their study of differentially rotating shells, there are many zero-step modes in the band, so that the whole continuous spectrum could potentially be excited. In such case these modes would have very slow power-law growth. For all our models we have checked mass conservation along the evolution. The worst results are obtained for model D3H, for which mass is conserved within 2.5% error when the instability saturates. At the end of the simulation (after 48 orbital periods and 25000 iterations in the coarsest grid) the error has grown to only 6%. For all other models mass conservation is even more accurate. Note that these errors are within the round-off error of the code, and it is not related to the conservation properties of the numerical scheme itself. For a regular grid with 1283 cells and a simulation employing 25000 iterations, the accumulated round-off error (binomial distribution) using single-precision arithmetics, is about 1283 × 25000 × 10−8 = 0.0023 = 0.23%. Correspondingly, for a 2563 grid (with twice the number of iterations for the simulation) the error is about 0.9%. Taking into account that this error affects the nonlinear evolution of the system, it is not surprising to have an error at the level of a few percent by the end of our high resolution simulations, for all conserved quantities. Figure 4 shows the evolution of Am for model D3 and for m ranging from 1 to 8 for our two resolutions. According to this figure, the only two modes 1e-06 0.0001 0 100 200 1e-06 0.0001 m=1, 3, 5, 7 m=1, 3, 5, 6, 7, 8 Fig. 4. Evolution of |Am| for model D3 with low resolution (top) and high resolution (bottom). The m = 2 mode is represented with thick solid line, m = 4 with thin solid line, m = 6 with dashed line, m = 8 with dot-dashed line, and all other odd m with dotted lines. relevant for the dynamics of the star are m = 2 and m = 4. All other modes have smaller amplitudes and play no role in the dynamics. Note that for odd m modes, the value of the integrated quantity Am, if close to zero, is extremely sensitive to very small numerical asymmetries, which are induced by the patch creation scheme of our AMR code. This explains the resolution differences in the initial values for odd m modes in Fig. 4 (at t = 0 they start off at 10−8 level for the low resolution simulation), although they saturate at the same value irrespective of the resolution. An important diagnosis for the accuracy of the results is the location of the center of mass during an evolution. The round-off error of the numerical code imposes controlled errors in mass and linear momentum, which results in tiny displacements of the center of mass. However small (one numerical cell in our runs) this unphysical displacement may hinder the correct analysis of the 0 100 200 300 1e-06 0.0001 Fig. 5. Effects of the artificial displacement of the center of mass (of only one numerical cell) on the time evolution of |A1| for model D2H. The thin solid line shows a fictitiuos evolution resulting from the numerical artifact originated by the center of mass displacement. mode growth rates. For this reason all integrated quantities shown in Fig. 4 are computed after correcting for the displacement of the center of mass, xnew = xold − xCM, in a post-processing stage of the data analysis. Were this not done, a one-armed m = 1 mode would grow much faster than it should to bring up fictitious features in the plots. This is shown for model D2H in Fig. 5. The thick solid line in this figure corresponds to the evolution of the m = 1 mode taking into account the correction for the center of mass displacement, while the thin solid line is the corresponding evolution of this mode without the correction. 4.3 Gravitational waves The growth and saturation of the instability is also imprinted on the gravita- tional waves emitted. The gravitational waveforms h+ and h× for models D1, D2, and D3, computed using the standard quadrupole formula, are shown in Fig. 6. For a source of mass M located at a distance R those waveforms can be calculated from the dimensionless waveform amplitudes a+ and a× as h+,× = a+,× sin2 θ , (16) using G = c = 1 units. The resulting chirp-like signal in all the models, partic- ularly apparent for model D2L, indicates the presence of a bipolar distribution of mass within the star (see Sec. 4.4). -0.05 0 100 200 300 Fig. 6. Gravitational waves for models D1 to D3 extracted using the standard quadrupole formula. Thick (thin) solid lines correspond to low (high) resolution. Only the dimensionless waveform amplitude a+ is plotted. As mentioned before, the effects of grid resolution on the evolution of the nonlinear phase of the bar-mode are imprinted on the gravitational waveforms. Thick solid lines in Fig. 6 are the waveforms which correspond to the low- resolution models, and thin solid lines to the high-resolution counterparts. The evolution of η for model D3, displayed in Fig. 3, shows little deviations with grid resolution, and this translates into very similar gravitational wave patterns (bottom panel of Fig. 6), the differences becoming more noticeable in the nonlinear phase following saturation (Ω0t ≥ 75). For model D1 (top panel), the differences also become more apparent at later times during the evolution, in good agreement with the dissimilar behaviour of the matter dynamics in this model, as encoded in the evolution of η in Fig. 3. As happens for model D3 the first few cycles of the gravitational waveform, when the mode is still in the linear phase, are accurately captured for both resolutions. The major dependence of the waveform on the grid resolution is found for model D2. Again, the linear phase for the growth of the bar deformation is Ω 0 t= 70.40.0 Fig. 7. Snapshots of the density, vorticity, and specific angular momentum, for model D3H, at three representative instants of the evolution. All snapshots show slices of the stars in the equatorial plane. Quantities are normalized as follows: max, rew s , and l ϕ/(rev s ), where v s is the initial velocity at the surface of the star. accurately captured irrespective of the resolution (and agrees with the per- turbative results of [15]). This is signalled in the perfect overlapping of both gravitational waveforms during the first three cycles (see the middle panel of Fig. 6). However, the different nonlinear dynamics of the bar-mode deforma- tion for this model, shown in the middle panel of Fig. 3, is severely imprinted on the gravitational waveform. Model D2H emits gravitational waves which have roughly one order of magnitude smaller amplitude than those computed for the corresponding low resolution model. 4.4 Morphology We next describe the morphological features encountered during the evolution of some representative models. Fig. 7 shows three snaphsots of the evolution of model D3H for the density (top), the azimuthal component of the vorticity, ~wϕ = (∇ × ~v)ϕ (middle), and the specific angular momentum, ~l = ~r × ~v (bottom). From left to right the snapshots correspond to the initial time (Ω0t = 0), a time when the bar-mode instability is growing (Ω0t = 33.6), and the time when the instability saturates (Ω0t = 70.4). Only the equatorial plane of the stars is shown in all these plots. Animations of all simulations performed are available at www.uv.es/∼cerdupa/bars/. We note that our AMR code is able to dynamically place patches (e. g. between 4 and 8 in the D3H model) and evolves the system with continuous matching between patches, as exemplified in Fig. 7. The evolution of model D3H shows that as the m = 2 mode grows the star develops an ellipsoidal shape which remains spinning beyond saturation. Since the low β m = 2 mode saturates at lower values (η ∼ 0.1) than the classical bar-mode instability (η ∼ 1), no clear bars are visible in the density plot. At late times (Ω0t > 100) a “boxy” structure becomes apparent as the m = 4 mode has grown to almost similar amplitude as the m = 2 mode (see anima- tions and Fig. 4). No other global features can be seen, consistent with the fact that |Am| ≪ 1 for all modes other than m = 2 and 4. The vorticity plot shows that the m = 2 mode at Ω0t = 33.6 adopts the form of a two-armed spiral winding up around the central parts of the star. As the mode begins to saturate (Ω0t = 70.4) the spirals break apart into the outer layers in a turbu- lent flow reminiscent of the (shear) Kelvin-Helmholtz instability, and shock as they reach the atmosphere. These trends are also visible in the specific angular momentum plot. The presence of a corotation radius, at r/re = 0.56 for model D3H, seems to play a role in the growth and saturation of the instability, in agreement with the recent findings of [25]. As the bar-mode grows, pressure waves carry angular momentum outside the corotation radius, which is deposited in the outer layers of the star. This excites Kelvin-Helmholtz-like instabilities in the fluid that break the mode outside the corotation radius. When this happens the m = 2 instability stops growing and no more angular momentum is extracted. Figure 8 shows late-time snapshots of the equatorial plane distribution of the density perturbation, i.e. (ρ−ρ(0))/ρ(0)max, for models D2H and D3H. The times are chosen well inside the nonlinear and saturation phase of the instability. This figure helps to interpret the mode dynamics and its saturation along the lines mentioned before: During the evolution the density perturbations are shed in waves from the center towards the outer layers of the star. At late times, when the instability saturates, such shedding stops, and the density Fig. 8. Snapshots of the density perturbation at the equatorial plane for models D2H and D3H. The white solid curves indicate the location of the corotation radius. The white dashed boxes indicate the location of the patches for model D3H. perturbation reaches the largest values outside the corotation radius (depicted with white solid lines in Fig. 8), for either model. We note in passing that the corotation radius in all our high resolution models lies well inside the outer boundary of the finest box set up by the AMR refinement pattern. (see, e.g. the white dashed boxes depicted in the right panel of Fig. 8 indicating the location of the AMR patches for model D3H) This rules out the possibility of a numerical artifact resulting from the patch creation scheme of our AMR code being the cause for the different long-term evolution between low and high resolution models, particularly noticeable for model D2 in Fig. 3. Finally, Fig. 9 shows a comparison between models D2L and D2H at Ω0t = 101 (i.e. well within the nonlinear phase), to highlight the effects of the numer- ical resolution on the morphology. From top to bottom this panel shows a schlieren plot (|∇ log ρ|) , ~wϕ, and ~l. The resolution differences in the evolu- tion of model D2 become apparent from this figure. In particular, the “boxy” structure becomes much more clearly visible in the low resolution simulation (D2L), indicating an excessive growth rate of the m = 4 mode. The presence of pressure waves is emphasized in the schlieren plot, very accurately captured in model D2H. Those waves, once the flow is driven to turbulence past the corotation radius, redistribute the angular momentum in the outer layers of model D2L in a much more pronounced way than for model D2H. Ω 0 t= D2L D2H Fig. 9. Resolution comparison between models D2L and D2H once the instability has saturated. Only slices of the stars in the equatorial plane are shown. 5 Summary and outlook We have presented AMR high-resolution simulations of the low T/|W | bar- mode instability of extremely differentially rotating neutron stars. Our main motivation has been to revisit the simulations by [15] on such instability, assessing how sensitive the onset and development of the instability is to numerical issues such as grid resolution. We have addressed the importance of a correct treatment of delicate numerical aspects which may spoil three- dimensional simulations in (Cartesian) grid-based codes, always hampered by insufficient resolution, namely the handling of the low-density atmosphere sur- rounding the star, the correction for the center of mass displacement, and the mass and momentum conservation properties of the numerical scheme. Our simulations have revealed the complex morphological features involved in the nonlinear dynamics of the instability. We have found that in the nonlinear phase of the evolution, the excitation of Kelvin-Helmholtz-like fluid modes outside the corotation radii of the stellar models leads to the saturation of the bar-mode deformation. While the overall trends reported in the investigation of [15] are confirmed by our work, the resolution used to perform the simu- lations may play a key role on the long-term behaviour of the instability and on the nonlinear dynamics of rotating stars, which has only become apparent for some specific models of our sample (namely model D2). This, in turn, has implications on the attainable amplitudes of the associated gravitational wave signals. The work reported in this paper is a first step in our ongoing efforts of study- ing the dynamical bar-mode instability within the magnetized core collapse scenario. Acknowledgements The authors thank Harry Dimmelmeier, Nick Stergioulas, and Anna Wats for useful comments. Research supported by the Spanish Ministerio de Edu- cación y Ciencia (MEC; grants AYA2004-08067-C03-01, AYA2003-08739-C02- 02, AYA2006-02570). VQ is a Ramón y Cajal Fellow of the Spanish MEC. Computations performed at the Servei d’Informática de la Universitat de València (CERCA-CESAR). References [1] N. Stergioulas, Liv. Rev. Relativ. 6 (2003) 3 [2] J. E. Tohline, R. H. Durisen, & M. McCollough, ApJ 298 (1985) 220 [3] J. L. Houser, J. M. Centrella, & S. Smith, Phys. Rev. Lett. 72 (1994) 1314 [4] K. C. B. New, J. M. Centrella, & J. E. Tohline, Phys. Rev. D 62 (2000) 064019 [5] M. Shibata, T. W. Baumgarte, & S. L. Shapiro, ApJ 542 (2000) 453 [6] M. Rampp, E. Müller, & M. Ruffert, A&A 332 (1998) 969 [7] M. Shibata, & Y. Sekiguchi, Phys. Rev. D 71 (2005) 024014 [8] M. Saijo, Phys. Rev. D 71 (2005) 104038 [9] C. D. Ott, S. Ou, J. E. Tohline, & A. Burrows, ApJ 625 (2005) L119 [10] H. C. Spruit & E. S. Phinney, Nature 393 (1998) 139 [11] A. Heger, S. E. Woosley, & H. C. Spruit, ApJ 626 (2005) 350 [12] C. D. Ott, A. Burrows, T. A. Thompson, E. Livne, & R. Walder, ApJS 164 (2006) 130 [13] S. E. Woosley, & A. Heger, ApJ 637 (2006) 914 [14] J. M. Centrella, K. C. B. New, L. L. Lowe, & J. D. Brown, ApJ 550 (2001) [15] M. Shibata, S. Karino, & Y. Eriguchi, MNRAS 334 (2002) L27 [16] M. Shibata, S. Karino, & Y. Eriguchi, MNRAS 343 (2003) 619 [17] V. Quilis, MNRAS 352 (2004) 1426 [18] M. J. Berger, P. Colella, J. Comp. Phys. 82 (1989) 64 [19] J. A. Font, M. Miller, W.-M. Suen, & M. Tobias, Phys. Rev. D 61 (2000) 0044011 [20] M. D. Duez, P. Marronetti, S. L. Shapiro, & T. W. Baumgarte, Phys. Rev. D 67 (2003) 024004 [21] Y. Eriguchi, & E. Müller, A&A, 147 (1984) 161 [22] A. L. Watts, N. Andersson, & D. I. Jones, ApJ 618 (2005) L37 [23] A. L. Watts, N. Andersson, H. Beyer, & B. F. Schutz, MNRAS 342 (2003) 1156 [24] A. L. Watts, N. Andersson, & R. L. Williams, MNRAS 350 (2004) 927 [25] M. Saijo & S. Yoshida, MNRAS 368 (2006) 1429 Introduction Mathematical framework Numerical approach Results Initial data Stability analysis Gravitational waves Morphology Summary and outlook References

0704.0357

Evolutionary games on minimally structured populations

Evolutionary games on minimally structured populations Gergely J. Szöllősi∗ and Imre Derényi† Biological Physics Department Eötvös University, Budapest Abstract Population structure induced by both spatial embedding and more general networks of interaction, such as model social networks, have been shown to have a fundamental effect on the dynamics and outcome of evolutionary games. These effects have, however, proved to be sensitive to the details of the underlying topology and dynamics. Here we introduce a minimal population structure that is described by two distinct hierarchical levels of interaction, similar to the structured metapopulation concept of ecology and island models in population genetics. We believe this model is able to identify effects of spatial structure that do not depend on the details of the topology. While effects depending on such details clearly lie outside the scope of our approach, we expect that those we are able to reproduce should be generally applicable to a wide range of models. We derive the dynamics governing the evolution of a system starting from fundamental individual level stochastic processes through two successive meanfield approximations. In our model of population structure the topology of interactions is described by only two parameters: the effective population size at the local scale and the relative strength of local dynamics to global mixing. We demonstrate, for example, the existence of a continuous transition leading to the dominance of cooperation in populations with hierarchical levels of unstructured mixing as the benefit to cost ratio becomes smaller then the local population size. Applying our model of spatial structure to the repeated prisoner’s dilemma we uncover a novel and counterintuitive mechanism by which the constant influx of defectors sustains cooperation. Further exploring the phase space of the repeated prisoner’s dilemma and also of the “rock- paper-scissor” game we find indications of rich structure and are able to reproduce several effects observed in other models with explicit spatial embedding, such as the maintenance of biodiversity and the emergence of global oscillations. PACS numbers: 87.10.+e 87.23.-n ∗[email protected]; angel.elte.hu/˜ssolo †[email protected]; angel.elte.hu/˜derenyi http://arxiv.org/abs/0704.0357v3 mailto:[email protected] angel.elte.hu/~ssolo mailto:[email protected] angel.elte.hu/~derenyi I. INTRODUCTION The dynamics of Darwinian evolution is intrinsically frequency dependent, the fitness of in- dividuals is tightly coupled to the type and number of competitors. Evolutionary dynamics acts, however, on populations, not individuals and as a consequence depends on not only population composition, but also population size and structure. Evolutionary game theory came about as the result of the realization that frequency dependent fitness introduces strategic aspects to evolution [1, 2, 3]. More recently the investigation of the evolutionary dynamics of structured populations, where individuals only compete with some subset of the population, e.g. their neighbors in space or more generally in some graph [4, 5], has lead to the recognition that the success of different strategies can be greatly influenced by the topology of interactions within the population. Funda- mental differences were found – compared to well-mixed populations, where individuals interact with randomly chosen partners – in models that describe the evolution of cooperation (variants of the prisoner’s dilemma game [4, 6, 7, 8, 9]) or deal with the maintenance of biodiversity in the context of competitive cycles (variants of the rock-paper-scissors game [3, 10, 11, 12, 13, 14]). In order to investigate the coevolutionary dynamics of games on structured populations the full set of connections between a potentially very large number of individuals must be specified. This is only possible by reducing the number of degrees of freedom considered, either through postulating a highly symmetrical (such as lattices [4, 8, 16, 17, 18, 19, 20]) or fundamentally random connection structure (such as some random graph ensemble [21, 22]). The question of how one goes about the task of reducing the number of degrees of freedom – of choosing the relevant parameters to describe the population structure constrained to which individuals undergo evolution – is not trivial. Both the explicit spatial as well as the random graph ensemble approach have clear precedents in condensed matter physics and network theory, respectively. It is not, however, clear which – if either – approach best describes natural populations of cyclically competing species or societies composed of individuals playing the prisoner’s dilemma game. As an example let us consider colicin producing bacteria, that play the so called ”rock-paper- scissors” (RPS) game (for details see below). This system has recently been the subject of two experimental studies aimed at demonstrating the role of structured populations in the maintenance of diversity. In the first study [10, 11] bacteria were cultured in vitro in Petri dishes, effectively re- stricting competition between bacteria to neighbors on the (2D) Petri dish surface (Fig.1 top left), while in the second experiment [12] in vivo bacterial colonies were established in co-caged mice and their development was subsequently followed. In the case of the first experiment the analogy with explicit 2D spatial embedding (present by construction) is clear (Fig.1 bottom left). The pop- ulation structure of the second experiment is, however, clearly different. The bacteria in individual mice can be readily considered as locally well-mixed populations, the coevolutionary dynamics of which reduces in the standard meanfield limit to a system of non-linear differential equations (the adjusted replicator equations [24]). As the experiments show, however, migration of bacteria between mice may also occur – resulting in the observed cyclic presence of the three strains in individuals. There are two distinct scales of mixing present in the system. Bacteria within each mice compete with each other forming local populations – an unstructured neighborhood com- posed of individual bacteria, while also being exposed to migrants from mice with whom they share the cage, together forming a global population – an unstructured neighborhood composed of individual local populations (Fig.1 top and bottom right). This setup is referred to in the ecology literature – albeit in significantly different contexts – as a ”structured metapopulation” [26, 27] where structured here refers to the detailed consideration of the population dynamics of the indi- vidual populations (often called ”patches”) comprising the metapopulation and is also related to the finite island models of population genetics [28]. The above example of co-caged mice is not unique, we may readily think of other ecological or sociological examples where an approximation with hierarchical scales of mixing with no internal structure can be relevant (such as human societies with two distinct scales of mixing present, the first within individual nations the between them at an international level). We have, also, recently used a similar approach to construct a model of genetic exchange among bacteria of the same species (the bacterial equivalent of sex) with which we were able to take into account the effects of spatial and temporal fluctuations in a manner that can explain the benefit of such genetic exchange at the level of the individual [31]. In this paper we construct a hierarchical meanfield theory where the two distinct (i.e. local and global) scales of mixing are each taken into account in terms of two separate meanfield approx- imations and fluctuations resulting from finite population size on the local scale of mixing are also considered. We subsequently explore the similarities and differences between this and other models of structured populations in the case of the ”rock-paper-scissors” and prisoner’s dilemma games. Through these examples we suggest that our approach allows the separation of the effects of structured populations on coevolutionary dynamics into effects which are highly sensitive to and dependent on the details of the topology and those which only require the minimal structure Explicit spatial embeding Two distinct scales of mixing Petri dish experiment Co-caged mice experiment FIG. 1: (Color online) In the colicin version of the RPS game, strains that produce colicins (red/dark grey) kill sensitive (green/light grey) strains, that outcompete resistant (blue/black) strains, that outcom- pete colicin producing strains (toxin production involves bacterial suicide). Experiments [10] show that colicin-producing strains cannot coexist with sensitive or resistant strains in a well-mixed culture, yet all three phenotypes are recovered in natural populations. Two recent experiments have examined the role of population structure in the maintenance of diversity among colicin-producing bacteria. In the first [10] in vitro colonies were established on an agar substrate in Petri dishes, a setup which effectively limits compe- tition to neighbors on the petri dish in analogy with explicit spatial embedding in 2D. In the second [12] in vivo colonies were established in the intestines of co-caged mice, a setup which has two distinct scales of mixing, with no explicit structure on either scale. present in our approximation and can consequently (in terms of sensitivity to the details of the topology) be considered more robust. II. HIERARCHICAL MEANFIELD THEORY FOR TWO DISTINCT SCALES Let us consider an evolutionary game between d types (strategies) described by the d×d payoff matrix A with elements αkj . Assuming finite and constant population size, natural selection can be described at the level of the individual by the so called the Moran process [30], during which at each time step an individual is selected randomly from the population to be replaced (death) by the offspring of an individual that is chosen proportional to its fitness to reproduce (birth). This models a population in equilibrium, where the time scale of the population dynamics is set by the rate at which ”vacancies” become available in the population. The fitness of each individual depends on the payoff received from playing the game described by A with competitors (an individual of type k receiving a payoff αkj when playing with an individual of type j). In well-mixed populations, individuals can be considered to come into contact (compete) with equal probability with any member of the population excluding themselves – this allows one to calculate the fitness of an individual of type k in a meanfield manner, yielding πk = πbase + αkj(nj − δkj) N − 1 , (1) where nk is the number of individuals of type k in the population, k=1 nk = N is the size of the population, πbase is some baseline fitness and the Kronecker delta symbol δkj is equal to unity if k = j and is zero otherwise. From this we may calculate the transition probabilities of our stochastic process, i.e., the probability of an individual of type i being replaced by an offspring of an individual of type k is given by Tik = , (2) where π̄ = k=1 πknk/N . The state of any population is completely described by the frequency of the different strategies xk = nk/N . Due to the normalization k=1 xk = 1, the values of xk are restricted to the unit simplex Sd [3]. For d = 2 this is the interval [0, 1], S3 is the triangle with vertices {(1, 0, 0), (0, 1, 0), (0, 0, 1)} while S4 is a tetrahedron etc. As Traulsen et al. have recently shown [24, 25] for sufficiently large, but finite populations the above stochastic process can be well approximated by a set of stochastic differential equa- tions combining deterministic dynamics and diffusion (population drift) referred to as Langevin dynamics: ẋk = ak(x) + ckj(x)ξj(t), (3) where the effective deterministic terms ak(x) are given by ak(x) = (Tjk − Tkj) = xk πk(x)− π̄(x) π̄(x) , (4) ckj(x) are effective diffusion terms, that can also be expressed in terms of the transition probabil- ities as described in [25], and ξj are delta correlated 〈ξk(t)ξj(t′)〉 = δkjδ(t − t′) Gaussian white noise terms. As N → ∞ the diffusion term tends to zero as 1/ N and we are left with the modified replicator equation. In the context of our hierarchical mixing model the topology of connections can be described by two parameters, the populations size at the local scale of mixing N , and a second parameter µ, which tunes the strength of global mixing relative to the local dynamics. We take into account the second (global) scale of mixing – mixing among local populations – by introducing a modified version of the Moran process. In the modified process a random individual is replaced at each time step either with the offspring of an individual from the same population (local reproduction) or with an individual from the global population (global mixing). This is equivalent to considering the global population to be well-mixed at the scale of local populations. Let us consider a global population that is composed of M local populations of size N . In each local population vacancies become available that local reproduction and global mixing compete to fill. In any local population l the probability of an individual of some type k filling a new vacancy due to local reproduction must be proportional to the number of individuals of type k multiplied by their fitness i.e. πlkn k, where we consider π k to be determined only by interactions with individuals in the same local population according to equation (1). To describe the tendency of individuals of some type k in local population l to contribute to global mixing we introduce the parameters σlk. The choice of appropriate σlk depends on the details of the global mixing mechanism, for systems where only the offspring of individuals mix globally it is proportional to the fitness of a given type, while for mechanisms such as physical mixing, by e.g. wind or ocean currents, it may be identical for each type. Irrespective of the details, however, the probability of an individual of some type k filling in a new vacancy due to global mixing should be proportional to the global average of the number of individuals of type k multiplied by their mixing tendency, which we denoted as 〈σknk〉 = l=1 σ k/M , and the strength of global mixing µ. These consideration lead to the new transition probabilities: T̂ lik = k + µ〈σknk〉 k=1(π + µ〈σknk〉) k + µ〈σknk〉 N(π̄l + µ ¯〈σ〉) , (5) where π̄l = k/N and ¯〈σ〉 = 〈σknk〉/N . We have found that the results presented below are qualitatively the same for both the fitness dependent choice of σlk = π k and the fitness independent choice of σ k = 1. Therefore, in the fol- lowing we restrict ourselves to the somewhat simpler fitness independent choice of σlk = 1, which can be considered to correspond to some form of physical mixing mechanism. The transition probabilities (5) then reduce to: T̄ lik = π̄l + µ π̄l + µ . (6) We can see that after a vacancy appears either local reproduction occurs, with probability π̄l/(π̄l+ µ), or global mixing, with probability µ/(π̄l + µ). From (6) we may derive the Langevin equation describing the coevolutionary dynamics of population l from the ẋlk = âk(x l, 〈x〉) + ĉkj(x l, 〈x〉)ξj(t), (7) with the modified deterministic terms given by âk(x l, 〈x〉) = x k(πk(x l)− π̄(xl)) + µ(〈xk〉 − xlk) π̄(xl) + µ , (8) where the vector 〈x〉 = l=1 x l/M with components 〈xk〉 = l=1 x k/M describes the fre- quencies of the individual types in the global population and the diffusion terms ĉ(xl, 〈x〉) can be expressed in terms of the modified transition probabilities T̂ lik as above. Equations (7) describe the coevolutionary dynamics of the global population through the cou- pled evolution of the {x1, . . . ,xM} local populations. In the limit of a large number of local popu- lations (M → ∞) the distribution of the local populations over the space of population states (the simplex Sd) is described by a density function ρ(x) that is normalized over Sd, i.e., ρ(x) = 1. The time evolution of ρ(x) follows a d−1 dimensional advection-diffusion equation – the Fokker- Planck equation corresponding to eq. (7): ρ̇(x) = −∇{â(x, 〈x〉)ρ(x)}+ 1 b̂(x, 〈x〉)ρ(x) , (9) with the global averages 〈xk〉 = xkρ(x) coupled back in a self-consistent manner into the deterministic terms âk(x, 〈x〉) and the diffusion matrix b̂kj(x, 〈x〉) = i=1 ĉki(x, 〈x〉)ĉij(x, 〈x〉). For large local populations (N → ∞) the diffusion term vanishes as 1/N . The above advection-diffusion equation (9) presents an intuitive picture of the coevolutionary dynamics of the population at a global scale. We can see that local populations each attempt to follow the trajectories corresponding to the deterministic replicator dynamics, while under the influence of two additional opposing forces: (i) global mixing, which attempts to synchronize local dynamics and (ii) diffusion resulting from finite population size effects, which attempts to smear them out over the simplex. The strength of these forces are tuned by two parameters µ and N , respectively. If, further, the effects of synchronization are irrelevant, as for example in the case of populations where selection is externally driven by independent environmental fluctuations, we may replace the global population average with the time average of any single population. This is the approach we used in our study of genetic mixing in bacteria [31]. During our numerical investigations we found solving the advection-diffusion equation (9) nu- merically challenging, particularly in the N → ∞ limit. We resorted instead to solving the coupled Langevin equations (7) for large M = 104 − 105 to simulate the time evolution of ρ(x). III. COOPERATION IN POPULATIONS WITH HIERARCHICAL LEVELS OF MIXING The evolution of cooperation is a fundamental problem in biology, as natural selection under most conditions favors individuals who defect. Despite of this, cooperation is widespread in na- ture. A cooperator is an individual who pays a cost c to provide another individual with some benefit b. A defector pays no cost and does not distribute any benefits. This implies the payoff matrix b− c −c  , (10) where b is the benefit derived from playing with a cooperator while c is the cost for cooperation. From the perspective of evolutionary game theory, which equates payoff with fitness, the apparent dominance of defection is simply the expression of the fact that natural selection a priori selects for fitness of individuals and not the fitness of groups. Defection dominates cooperation in any well-mixed population [3]. Population structure in- duced by spatial structure [4, 18] and more general networks of interactions [21, 22, 23]) has, however, been found to facilitate the emergence and maintenance of cooperation. The mecha- nism responsible, termed spatial, or more generally, network reciprocity[32] depends strongly on biased infux movement of global average density of local populations drift deterministic dynamics x = 0 x = 1 ρ(x, t) ρ(x, t + ∆t)biased influx 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 b/c = 95, 97, 100, 111, 125 1000 10 100 1000 d b/c 60 70 80 90 100 110 FIG. 2: a In an infinitely large well-mixed population evolutionary dynamics is deterministic and leads to the extinction of cooperators as average fitness monotonically declines. The only stable fixed point corresponds to the point where the fraction of cooperators is zero (x = 0). To understand qualitatively the mechanism favoring cooperation in hierarchically mixed populations let us consider some density of local populations (ρ(x, t)) that is symmetric around its mean at time t. Due to global mixing all local populations are being driven toward the global average. Due to the influx bias, however, populations with a lower than average number of cooperators will be driven stronger (faster) than those on the other side of the average. Examining the density of local populations at some time t+∆t, this results in a net movement of the global average toward a larger fraction of cooperators. This is, of course, opposed by local reproduction that favors an increase in the number of defectors. For the global average to keep moving toward a higher number of cooperators and eventually to keep balance with local reproduction bias a density of local population with finite width is needed over which the effect of the influx bias can exert itself. It is drift caused by local population size that maintains this finite width, and this is the reason that the b/c threshold above which cooperation dominates depends on local population size. b Stationary density of local populations ρ(x) for different values of b/c with N = 100, µ = 0.1. c Transition toward a global dominance of cooperation for µ = 10. (triangles), µ = 1 (crosses), µ = 0.1 (squares), µ = 0.01 (circles) with N = 100. The critical value of b/c depends only weakly on µ changing by 20% over four orders of magnitude d Critical values of b/c as a function of N for different values of µ (notation as before). The dashed line corresponds to b/c = N . The critical b/c values were determined by numerically finding the inflection point of the transition curves. M = 103 was used throughout. the details of local topology. In particular, it seems that lattice like connectivity structures where three-site clique percolation occurs [17] and more general interaction graphs where the degree of nodes k does not exceed the ratio of benefit to cost (i.e. k < b/c) [22] are required for cooperation to be favored. Examining the effects of hierarchical mixing on the evolutionary dynamics of cooperation we found that a sharp, but continuous transition leads to the dominance of cooperation as the benefit to cost ratio becomes smaller then the local population size, i.e. b/c < N . If the benefit to cost ratio is larger then the local population size the global population is dominated by defectors. The mechanism leading to the dominance of cooperation arises due to the competition between local reproduction and global mixing. In local populations with lower average fitness – larger number of defectors – the influx of individuals from the global scale will be larger than in local populations with higher average fitness (cf. eq. (6) where the relative strength of the two terms on the left hand side depends on the sum of the average fitness of population l and µ). The crucial ingredient for cooperation to be successful is population drift introduced by finite local population size. It is bi- ased influx coupled with drift that can result in cooperation being favored in the global population (Fig 2.). IV. THE RPS GAME To explore the effects of hierarchical mixing in the context of games with three strategies we first turn to the case of the so called ”rock-paper-scissors” (RPS) game. In the original popular version of the game two players are afforded the chance to simultaneously display either rock (fist), paper (flat hand) or scissors (two fingers). If player one displays a flat hand while player two displays a fist, player one wins as paper wraps rock. Similarly scissors cut paper, and rocks smashes scissors. Several examples of this game have been found in nature (e.g. among lizards [33] ), but it is bacteria that have received the most experimental and theoretical attention. In ecology the often high diversity among microbial organisms in seemingly uniform envi- ronments, referred to as the ”paradox of the plankton”, has been difficult to understand. Several models based on spatially explicit game theoretical models have been proposed to explain this di- versity [10, 11, 13, 14]. These models are all variants of the RPS game played by colicin producing bacteria. Colicins are antibiotics produced by some strains of Echerichia coli. In experiments (see Fig.1) typically three strains are used: colicin producing (C), sensitive (S) and resistant (R). The (a) (b) ALLD µ = 0 µ = 0.01 µ = 0.2 µ = 0 µ = 0.1 FIG. 3: a Deterministic replicator dynamics (the N → ∞ limit) of the symmetric RPS game consists of neutrally stable orbits along which the product of the strategy frequencies xRxPxS is conserved. If global mixing is present (µ > 0) local populations deviate from these neutral orbits toward the global average 〈x〉. Considering the simplest system with global mixing, that consisting of M = 2 local populations we see that in the presence of global mixing population x1 and population x2 move toward each other, respec- tively moving closer and further from the barycentre of the triangle until they become synchronized and subsequently pursue a common orbit. For deterministic local dynamics (N → ∞) such synchronization invariably occurs for any M if µ > 0 and typically converges to the barycentre of the simplex for suffi- ciently homogeneous initial conditions. b The deterministic replicator dynamics of the repeated PD game is markedly different from that of the RPS game in that the internal fixed point is unstable and in the absence of global mixing only ALLD survives. Again turning to the simplest scenario with M = 2 we see that if µ = 0 any pair of populations x1 and x2 (gray and black lines) converge to the to the ALLD corner. As µ is increased above a critical value a second, stable configuration emerges: for a large subset of the possible initial conditions (all, but the left most x2) we see that one of the populations (x1) converges to ALLD , while the second (x1) approaches a limit cycle. If µ is increased further, the above configuration ceases to be stable, the population which initially converges to ALLD (x1) is subsequently ”pulled out” by global mixing, following which the two populations synchronize and are finally absorbed together in ALLD. Simulations, however, show that synchronization may be avoided for M > 2 if µ is not too large. coevolutionary dynamics of the three strains can be cast in terms of an RPS game, C strains kill S strains, but are outcompeted, by R strains, because toxin production involves the suicide of bacte- ria. The cycle is closed by S strains that outcompete R strains, because resistance requires mutant versions of certain membrane protein, which are less efficient than the wild type [10]. Despite the cyclic dynamics colicin-producing strains cannot coexist with sensitive or resistant strains in a well-mixed culture, yet all three phenotypes are recovered in natural populations. Local dispersal (modeled as explicit spatial embedding) has widely been credited with promoting the maintenance of diversity in this system [10, 11, 13, 14]. In its most symmetric form the RPS game is described by the payoff matrix 0 −ǫ ǫ ǫ 0 −ǫ −ǫ ǫ 0 , (11) and some πbase > ǫ. The dynamics of this game in an infinitely large well mixed population consists of neutral orbits along which the product xRxPxS is conserved. For any finite N , however, fluctuations lead to the inevitable extinction of all but one of the strategies [15]. Spatial population structure can avert this reduction in diversity [10, 13] through the emergence of a stable fixed point at the barycentre of the simplex . The effect of the gradual randomization of different lattice topologies (where a small number of edges are randomly rewired) on the dynamics of the game has also been investigated. A Hopf bifurcation leading to global oscillations was observed [34, 35] as the fraction of rewired links was increased above some critical value. Examining the dynamics of the symmetric RPS game in terms of our hierarchical meanfield approximation we observed that an internal fixed point emerged for N → ∞ (Fig.3a). More importantly, diversity was also maintained for finite local population sizes if global mixing was present. Simulations of the time evolution of ρ(x) also revealed a Hopf bifurcation leading to the oscillation of the global average as µ was increased above a critical value µc depending on N (Fig.4a). These results show that previous results obtained from simulations of populations constrained to different lattice topologies can be considered universal in the sense that not only lattices, but any population structure that can be approximated by two distinct internally unstruc- tured scales of mixing are sufficient for their existence. In the context of the ”paradox of the plankton” these results imply that aside of local dispersal (modeled as explicit spatial embedding) a minimal metapopulation structure (with local competition and global migration) can also facili- tate the maintenance of diversity in cyclic competition systems. 0.045 0.055 0.05 0.06 0.065 0.07 0.05 0.07 0.09 b d f µ = 0.05 1600 1400 1200 1000 800 600 400 µ = 0.05 0.16 0.12 0.08 0.04 0 0.08 0.06 0.08 0.06 0.04 0.02 N = 1000 N = 2000 N = 4000 N = 1200 small N cyclic TFTALLC N = 1000 µ = 0.05µ = 0.15 N = 4000 FIG. 4: (Color online) a In the case of the rock-paper-scissors game a Hopf bifurcation similar to that observed for populations evolving on gradually randomized lattices [34, 35] leads to the emergence of global oscillations (the red line indicates the trajectory of 〈x〉) if µ is larger than a critical value µc(N) (see video S1 [36]). The density ρ(x) is indicated with a blue color scale. b The ratio A of the area of the global limit cycle and the area of the simplex is plotted as a function of µ for three different values of N . For the repeated prisoner’s dilemma game the combination of finite local population size and global mixing µ > 0 can lead to a stationary solution (c) qualitatively similar to that observed for explicit spatial embedding e. This state is characterized by a stable global average (large dot), just as the lattice system (data not shown) and sustained local cycles of cooperation, defection and reciprocity, also similar to the lattice case where groups of ALLD (red, dark grey) individuals are chased by those playing TFT (blue, black), which are gradually outcompeted by ALLC (green, light grey). d As µ is decreased a discontinuous transition can be observed to the ALLD phase. The ratio I of populations on the internal cycle is plotted as a function of µ. The inset shows the transition for different values of N . f The same critical line in the µ-N plane can be approached by increasing N with µ fixed. A large hysteresis can be observed as N is decreased below the critical value indicating the discontinuous nature of the transition. We numerically simulated the time evolution of ρ(x) by integrating the stochastic differential equation system defined by eq. (7) for large M (104−105) throughout. For the RPS game we used πbase = 1 and ǫ = 0.5, while in the case of the repeated PD game we followed ref. [38], setting T = 5, R = 3, P = 1, S = 0.1,m = 10 and c = 0.8. Lattice simulations (e) where performed on 1000 × 1000 square lattice with an asynchronous local Moran process between neighbors and periodic boundary conditions. V. THE REPEATED PRISONER’S DILEMMA GAME In the general formulation of the prisoner’s dilemma (PD) game, two players have the choice to cooperate or to defect. Both obtain some payoff R for mutual cooperation and some lower payoff P for mutual defection. If only one of the players defects, while the other cooperates, the defector receives the highest payoff T and the cooperator receives the lowest payoff S. That is T > R > P > S and defection dominates cooperation in any well-mixed population. New strategies become possible, however if the game is repeated, and players are allowed to chose whether to defect or cooperate based on the previous actions of the opponent. In the following we consider, similar to refs. [37] and [38] that recently examined the role of finite population size and mutation and finite population size, respectively in terms of the repeated PD game with three strategies: always defect (ALLD), always cooperate (ALLC), and tit-for-tat (TFT). TFT cooperates in the first move and then does whatever the opponent did in the previous move. TFT has been a world champion in the repeated prisoner’s dilemma ever since Axelrod conducted his celebrated computer tournaments [7], although it does have weaknesses and may be defeated by other more complex strategies [39]. Previous results indicate that if only the two pure strategies are present (players who either always defect or ones who always cooperate) explicit spatial embedding [4] and some sufficiently sparse interaction graphs [22, 40] allow cooperation to survive and the behavior of populations is highly sensitive to the underlying topology of the embedding [17]. We have found that introducing global mixing into the PD game with only the two pure strategies present also allows cooperation to survive. The mechanism responsible for favoring cooperation in this case, however, depends on the details of the competition between local reproduction and global mixing. For more than two strategies these details are much less relevant and do not qualitatively influence the dynamics. We will, therefore, consider the delicate issues concerning the PD game with only the two pure strategies in a separate publication, and concentrate here on the repeated PD game with three strategies. To investigate the effect of global mixing on the repeated PD game with three possible strate- gies: ALLD, ALLC and TFT following Imhof et al. [38] we considered the payoff matrix: large N cyclic small N cyclic TFTALLC TFTALLC TFTALLC TFTALLC N = 10 N = 10 µ = 0.05 µ = 0.05 (iii) (iii) deterministic local populations (i) (iii) (i)(ii) effective single population N = 10 N = 10 µ = 0.15 µ = 0.15 (iii) FIG. 5: (Color online) Phase space for the repeated prisoner’s dilemma game on a population structure with two distinct scales (see video S2 [36]). Three different phases are possible depending on the values of µ and N : (i) only ALLD survives (ii) an internal limit cycles is maintained by global mixing due to a large density of local populations around the ALLD corner (iii) a globally oscillating self maintaining limit cycle is formed. For extreme values of µ the global dynamics reduces to that of some well-mixed population where only ALLD survives: As µ becomes negligible (µ ≪ πk for all k) we approach the limit of isolated local populations, while for µ ≫ πk we are left with a single synchronized population. Similarly for N = 2 – the smallest system with competition – the system can be described as a single well mixed population for any µ and ALLD again prevails. In the limit of deterministic local populations (N → ∞) all three phases can be found depending on the value of µ. The density ρ(x) is indicated with the color scale. A figure illustrating the phase space of the repeated prisoner’s dilemma game with fitness dependent global mixing is included in the supplementary material [36]. Rm Sm Rm Tm Pm T + P (m− 1) Rm− c S + P (m− 1)− c Rm− c , (12) where the strategies are considered in the order ALLC, ALLD, TFT, m corresponds to the number of rounds played and c to the complexity cost associated with conditional strategies (TFT). The dynamics of this game has a single unstable internal fixed point and the state where each member of the population plays ALLD is the only nontrivial stable equilibrium (Fig.3b). Introducing global mixing, between local well-mixed populations, however, causes new sta- tionary states to emerge . Three phases can be identified: (i) ALLD wins (ii) large fraction of local populations in the ALLD corner maintains local cycles of cooperation defection and reciprocity through providing an influx of defectors that prevent TFT players from being outcompeted by ALLC playing individuals (iii) a self maintaining internal globally oscillating cycle emerges. The simplest scenario of two (M = 2) deterministic (N → ∞) local populations coupled by global mixing (µ > 0) already leads to the emergence of phase (ii) as demonstrated in Fig.3b while phase (iii) only emerges for larger M . For larger M simulations show that in the limit of large local pop- ulations all global configurations with less than some maximum ratio of the populations I on the internal cycle are stable in phase (ii). A transition from phase (ii) to (i) happens as µ is decreased below a critical value µii→ic and I approaches zero as I = (1−µii→ic /µ) (data not shown). This can be understood if we considered that near the transition point a critical proportion C = µ(1− I) of ALLD individuals needs to arrive to stabilize local cycles of cooperation defection and reciprocity. At the critical point I = 0 and µ = µii→ic which implies C = µ c giving I = (1− µii→ic /µ) Exploring the N − µ phase space (Fig.5) we see that the transition from phase (i) to (ii) be- comes discontinuous for finite N (Fig.4d,e). Further, for any given value of N and µ the global configuration is described by a unique I due to the presence of diffusion. For appropriate values of the parameters the global average converges to a stationary value in phase (ii) similarly to case of explicit spatial embedding (Fig.4c). For very small (µ ≪ πk for all k) and very large (µ ≪ πk) values of µ the global dynamics can be reduced to that of some well-mixed population where only ALLD persists (Fig 5.). For small N we again have an effective well-mixed population – the only limit were defectors do not dominate is N → ∞. In comparison with previous results of Imhof et al. we can see that evolutionary cycles of cooperation defection and reciprocity can be maintained not only by mutation, but also by population structures with hierarchical levels of mixing. VI. DISCUSSION While it is, of course, clear that the reduction of any realistic population structure to a manage- able construction is always an approximation, it has not been clearly established what the relevant degrees of freedom are in terms of evolutionary dynamics. Meanfield approximations are a classic method of statistical and condensed matter physics and are routinely used to circumvent intractable combinatorial problems which arise in many-body systems. Cluster-meanfield approximations of sufficient precision [18, 19] have been developed that adequately describe the evolutionary dynam- ics of explicitly structured populations through systematically approximating the combinatorial complexity of the entire topology with that of small motif of appropriate symmetry. The effects of more minimal effective topologies have, however, not been investigated previously. In the above we have shown that straightforward hierarchical application of the meanfield approximation (the assumption of a well-mixed system) surprisingly unveils a new level of complexity. In the broader context of ecological and population genetics research on structured populations our model can be described as a metapopulation model. The term ’metapopulation’ is, however, often used for any spatially structured population [27], and models thereof. More restrictive defi- nitions of the term are often implied in the context of ecology and population genetics literature. The foundations of the classic metapopulation concept where laid down by Levin’s vision of a ”metapopulation” as a population of ephemeral local populations prone to extinction. A classic metapopulation persists, like an ordinary population of mortal individuals, in a balance between ’deaths’ (local extinctions) and ’births’ (establishment of new populations at unoccupied sites) [27]. This classic framework is most wide spread in the ecology literature, a less often employed extension is the concept of a structured metapopulation where the state of the individual popula- tions is considered in more detail, this is more similar to our concept of hierarchical mixing, but differs in considering the possibility of local extinctions. The effects of finite population size and migration, which our model considers, has been of more central concern in the population genetics literature. The analog of Levin’s classic metapop- ulation concept is often referred to as the ’finite-island’ model [28] the effective population genetic parameters describing which, have been explored in detail[29]. The study of the population ge- netics of spatially subdivided populations in fact predates Levin, Wright having emphasised the capacity of drift in small populations to bring about genetic differentiation in the face of selection and/or migration several decades prior[28]. Our hierarchical mixing model treats the coevolutionary dynamics of evolutionary games on structured populations in a manner similar to the most simple population genetic models of spa- tially subdivided populations, focusing on the parallel effects of selection, drift and migration. It goes beyond these models both in considering the effects of frequency dependent selection (and the strategic aspects of the evolutionary dynamics this implies) and in using a self-consistent ap- proach to describe the global state of the subdivided population. Also, in order to maintain a connection with previous work on the effects of spatial structure on evolutionary games, which rely on Nowak’s concept of spatial games [4], with individuals restricted to interact, and hence compete, only with neighbours as defined by some topology of interaction, we develop our model from the level of the individual by introducing a modified version of the Moran process – and not by extending the Wright-Fisher process (which considers discrete generations and binomial sam- pling to account for finite population size). The effective population structure described by our hierarchical mixing model can be thought of as a population of individuals, interactions among which are specified by the edges of a hierarchically organized random graph. The fundamental difference in our picture is that the edges of this graph of interactions are not considered to be fixed, but are instead in a constant state of change, being present with a different probability be- tween pairs of individuals who share the same local population and between pairs of individuals who do not (Fig.1.). We consider annealed randomness, which in contrast to the usual quenched picture of fixed edges is insensitive to the details of topology. Our approach we believe best facil- itates the exploration of the effects of changing the relative strengths of drift and migration in the context of evolutionary games on structured populations. Examining the effects of hierarchical mixing in the context of the evolution of robustness we demonstrated that biased influx coupled with drift can result in cooperation being favored, pro- vided the ratio of benefit to cost exceeds the local population size. This result bears striking resemblance to that of Ohtsuki et al. [22], who were able to calculate the fixation probability of a randomly placed mutant for any two-person, two-strategy game on a regular graph and found that cooperation is favored provided the ratio of benefit to cost exceeds the degree of the graph. Our results demonstrate that this rule extends to the minimal spatial structure induced by hierarchical levels of mixing. Applying our model of spatial structure to the repeated prisoners dilemma revealed that a con- stant influx of defectors can help to stabilize cycles of cooperation, defection, and reciprocity through preventing the emergence of an intermittent period of ALLC domination in the popula- tion, which would present a situation that ”leaves the door wide open” to domination by defectors. While previous work has been done on the effects of ”forcing” cooperation [41] the idea that an influx of defectors can in fact stabilize the role of reciprocity in promoting cooperation has not been proposed previously. It seems highly unlikely that this mechanism can be explained in terms of kin or multilevel (group) selection, the similarities between which in structured populations have recently been the subject of intensive debate (see e.g. [42] and [43] or [44] and [45]). Kin selection can operate whenever interactions occurring among individuals who share a more recent common ancestor than individuals sampled randomly from the whole population [45] are relevant. In our case it is the interaction between defectors, arriving from the global scale, and TFT players present at the local scale that is important, and not the interaction between individuals in the local population, who may be thought of as sharing a recent common ancestor due to local dispersal. Also, while the concept of multilevel selection presents a promising framework for the study of evolution of cooperation, it must nonetheless be possible to derive it from ”first principles” – just as kin selection can be cast as an emergent effect of local dispersal. While there has been considerable work on studying the evolutionary games on graphs and highly symmetric spatial structures very little attention has been paid to the effects of more min- imal effective population structures, despite their widespread application in ecology and popula- tion genetics, fields from which evolutionary game theory was born and must ultimately reconnect with. We believe that the minimal population structure that such a hierarchical meanfield theory describes is potentially more relevant in a wide range of natural systems, than more subtle setups with a delicate dependence on the details and symmetries of the topology. We showed through two examples that such structure is sufficient for the emergence of some phenomena previously only observed for explicit spatial embedding, demonstrating the potential of our model to identify robust effects of population structure on the dynamics of evolutionary games that do not depend on the details of the underlying topology. The practical advantage of our approach, lies in its ability to readily determine whether or not some feature of a structured population depends on the topological details of local interactions. Recent simulation result concerning the dynamics of public goods games on different popula- tion structures [9, 46] and experiments where global mixing in an RPS like bacteria-phage system lead to the emergence of a ”Tragedy of the commons” scenario [47] should all be amicable to analysis in terms of our method. VII. ACKNOWLEDGMENTS This work was partially supported by the Hungarian Scientific Research Fund under grant No: OTKA 60665. VIII. APPENDIX Our approach readily generalizes for an arbitrary number of hierarchical mixing levels. For three levels of mixing we may consider the global population to be comprised of M subpopula- tions each of which is in turn subdivided into M local populations. With m ∈ {1, · · · ,M} running over subpopulations and l ∈ {1, · · · ,M} over local populations the transition probabilities can be written as: T̂mlik = πmlk n k + µ (1)〈(1)σml′k 〉l′ + µ(2)〈〈(2)σm k 〉l′〉m′ k=1(π + µ(1)〈(1)σml′ 〉l′ + µ(2)〈〈(2)σm 〉l′〉m′) , (13) where primed indices indicate the scale of mixing over which the average is taken, µ(1) describes the strength of mixing, and the (1)σmlk the tendencies of mixing among local populations within a subpopulation, while µ(2) describes the strength of mixing, and the (2)σmlk the tendencies of mixing among subpopulations in the global population. [1] Fisher R.A. The Genetical Theory of Natural Selection (Oxford University Press, 1930). [2] Maynard Smith, J. & Price, G. Nature (London) 246, 15-18 (1930). (doi:10.1038/246015a0) [3] Hofbauer, J. & Sigmund, K. Evolutionary Game Theory and Population Dynamics (Cambridge Uni- versity Press, 1998). [4] Nowak, M.A. & May, R.M. Nature (London) 359, 826-829 (1992). (doi:10.1038/359826a0) [5] Lieberman, E., Hauert, C. & Nowak, M.A. Nature (London) 433, 312-316 (2005). (doi:10.1038/nature03204) [6] Axelrod, R. & Hamilton, W.D. Science 211, 1390-1396 (1981). (doi:10.1126/science.7466396) [7] Axelrod, R. The Evolution of Cooperation (Basic Books, New York, 1984). [8] Nowak, M.A., Bonhoeffer S. & May, R.M. Proc. Natl. Acad. Sci. USA 91, 4877-4881 (1994). (doi:10.1073/pnas.91.11.4877) [9] Szabó, Gy. & Hauert, C. Phys. Rev. Lett. 89, 118101 (2002). (doi:10.1103/PhysRevLett.89.118101) [10] Kerr, B., Riley, M.A., Feldman, M.W. & Bohannan, B.J. Nature (London) 418, 171-174 (2002). (doi:10.1038/nature00823) [11] Nowak, M.A. & Sigmund, K. Nature (London) 418, 138-139 (2002). (doi:10.1038/418138a) [12] Kirkup, B.C. & Riley, M.A. Nature (London) 428, 412-414 (2004). (doi:10.1038/nature02429) [13] Czárán, T.L., Hoekstra, R.F. & Pagie, L. Proc. Natl. Acad. Sci. USA 99, 786-790 (2002). (doi:10.1073/pnas.012399899) [14] Lenski, R.E. & Riley, M.A. 99, 556-558 (2002). (doi:10.1073/pnas.02264199) [15] Reichenbach, T., Mobilia, M. & Frey, E. Phys. Rev. E 74, 051907 (2006). (doi:10.1103/PhysRevE.74.051907) [16] Szabó, Gy. & Tőke, Cs. Phys. Rev. E 58, 69 (1998). (doi:10.1103/PhysRevE.58.69) [17] Szabó, Gy., Vukov, J. & Szolnoki, A. Phys. Rev. E 72, 047107 (2005). (doi:10.1103/PhysRevE.72.047107) [18] Hauert, C. & Szabo, Gy. Am. J. Phys. 73, 405-414 (2005). (doi:10.1119/1.1848514) [19] Hui, C. & McGeochb, M.A. Bull. Math. Biol. 69, 659-676 (2007). (doi:10.1007/s11538-006-9145-1) [20] Hui, C., Zhang, F., Han, X. & Lid, Z. Ecol. Mod. 184, 397-412 (2005). (doi:10.1016/j.ecolmodel.2004.11.004) [21] Santos, F.C. & Pacheco, J.M. Phys. Rev. Lett. 95, 098104 (2005). (doi:10.1103/PhysRevLett.95.098104) [22] Ohtsuki, H., Hauert, C., Lieberman, E. & Nowak, M.A. Nature (London) 441, 502-505 (2006). (doi:10.1038/nature04605) [23] Wang, S., Szalay, M.S., Zhang, C. & Csermely, P. PLoS ONE 3 e1917 (2008). (doi:10.1371/journal.pone.0001917) [24] Traulsen, A., Claussen, J.C. & Hauert, C. Phys. Rev. Lett. 95, 238701 (2005). (doi:10.1103/PhysRevLett.95.238701) [25] Traulsen, A., Claussen, J.C. & Hauert, C. Phys. Rev. E 74, 011901 (2006). (doi:10.1103/PhysRevE.74.011901) [26] Hanski, I.A. & Gilpin, M.E. in Metapopultaion Biology (Academic Press, 1997), pp. 93-108. [27] Hanski, I.A. Nature 396, 41 (1998). (doi:10.1038/23876) [28] Pannell, R.P. & Charlesworth B. Philosophical Transactions: Biological Sciences 355, 1851-1864 (2000). (doi:10.1098/rstb.2000.0740) [29] Whitlock M.C. & Barton N.H. Genetics 146, 427-41 (1997). [30] Moran, P.A.P. The Statistical Processes of Evolutionary Theory (Clarendon, 1962) [31] Szöllősi, G.J., Derényi, I. & Vellai, T. Genetics 174, 2173-2180 (2006). (doi:10.1534/genetics.106.063412) [32] Nowak, M. Science 314, 1560-1563 (2006). (doi:10.1126/science.1133755) [33] Sinervo, B. & Lively, C.M. Nature (London) 380, 240-243 (1996). (doi:10.1038/380240a0) [34] Szolnoki, A. & Szabó, Gy. Phys. Rev. E 70, 037102 (2004). (doi:10.1103/PhysRevE.70.037102) [35] Szabó, Gy., Szolnoki, A. & Izsák, R. J. Phys. A 37, 2599-2606 (2004). (doi:10.1088/0305-4470/37/7/006) [36] See EPAPS Document No. [number will be inserted by publisher] for videos S1,S2 and S3. [37] Nowak M.A., Sasaki, A., Taylor, C. & Fudenberg, D. Nature (London) 428, 646-650 (2004). (doi:10.1038/nature02414) [38] Imhof, L.A., Fudenberg, D. & Nowak, M.A. Proc. Natl. Acad. Sci. USA 102, 10797-10800 (2005). (doi:10.1073/pnas.0502589102) [39] Molander, P. J. Conflict Resolut. 29, 611-618 (1985). (doi:10.1177/0022002785029004004) [40] Taylor, P.D., Day T. & Wild G. Nature (London) 447, 469-472 (2007). (doi:10.1038/nature05784) [41] Szabó, Gy., Antal, T., Szabó, P. and Droz, M. Phys. Rev. E 62, 1095-1103 (2000). (doi:10.1103/PhysRevE.62.1095) [42] Killingback, T., Bieri, J., & Flatt, T. Proc. R. Soc. B. 273 1477-1481 (2006). (doi:10.1098/rspb.2006.3476) [43] Grafen, A. Proc. R. Soc. B 274, 713-719 (2007). (doi:10.1098/rspb.2006.0140) [44] Traulsen, A. & Nowak, M.A. Proc. Natl. Acad. Sci. USA 103, 10952-10955 (2007). (doi:10.1073/pnas.0602530103) [45] Lehmann L., Keller, L., West, S. & Roze, D. Proc. Natl. Acad. Sci. USA 104, 6736-6739 (2007). (doi:10.1073/pnas.0700662104) [46] Hauert, C., Silvia De Monte, S. , Hofbauer, J. & Sigmund K. Science 296, 1129-1132 (2002). (doi:10.1126/science.1070582) [47] Kerr, B., Neuhauser, C., Bohannan, B.J.M. & Dean A.M. Nature (London) 442, 75-78 (2006). (doi:10.1038/nature04864) Introduction Hierarchical Meanfield Theory for Two Distinct Scales Cooperation in populations with hierarchical levels of mixing The RPS Game The Repeated Prisoner's Dilemma Game Discussion Acknowledgments Appendix References

0704.0358

Flavor Physics in SUSY at large tan(beta)

Flavor Physics in SUSY at large tan β Paride Paradisi Departament de F́ısica Teòrica and IFIC, Universitat de València-CSIC, E-46100, Burjassot, Spain. We discuss the phenomenological impact of a particularly interesting corner of the MSSM: the large tanβ regime. The capabilities of leptonic and hadronic Flavor Violating processes in shedding light on physics beyond the Standard Model are reviewed. Moreover, we show that tests of Lepton Universality in charged current processes can represent an interesting handle to obtain relevant information on New Physics scenarios. I. INTRODUCTION Despite the great phenomenological success of the Standard Model (SM), it is natural to consider this the- ory only as the low-energy limit of a more general model. The direct exploration of New Physics (NP) particles at the TeV scale will be performed at the upcoming LHC. A complementary strategy in looking for NP is provided by high-precision low-energy experiments where NP could be detected through the virtual effects of NP particles. In particular, flavor-changing neutral-current (FCNC) transitions may exhibit a sensitivity reach even beyond that achievable by the direct searches at the LHC while representing, at the same time, the best (or even the only) tool to extract information about the flavor structures of NP theories. In view of the above considerations, it is clear that flavor physics provides necessary and complementary in- formation to those obtainable by the LHC. Besides FCNC decays, also the Lepton Flavor Univer- sality (LFU) tests (Kℓ2 and πℓ2) offer a unique opportu- nity to probe the SM and thus, to shed light on NP: the smallness of NP effects is more than compensated by the excellent experimental resolution and the good theoreti- cal control. II. LFV IN SUSY The discovery of neutrino masses and oscillations has unambiguously pointed out the existence of the Lepton Flavor Violation (LFV) thus, we expect this phenomenon to occur also in the charged-lepton sector. Within a SM framework with massive neutrinos, FCNC transitions in the lepton sector like ℓi → ℓjγ are strongly suppressed by the GIM mechanism at the level of B(ℓi → ℓjγ) ∼ (mν/mW ) 4 ∼ 10−50 well beyond any realistic experimental resolution [1]. In this sense, the search for FCNC transitions of charged leptons is one of the most promising directions where to look for physics beyond the SM. Within a SUSY framework, LFV effects originate from any misalignment between fermion and sfermion mass eigenstates. In particular, if the light neutrino masses are obtained via a see-saw mechanism, the radiatively induced LFV entries in the slepton mass matrix (m2 are given by [2]: )i6=j ≈ − ν )i6=j ln , (1) where MX denote the scale of SUSY-breaking media- tion andm0 the universal supersymmetry breaking scalar mass. Since the see–saw equation 1 allows large (YνY entries, sizable effects can stem from this running [2]. The determination of (m2 )i6=j would imply a complete knowledge of the neutrino Yukawa matrix (Yν)ij , which is not possible even if all the low-energy observables from the neutrino sector were known. As a result, the predic- tions of leptonic FCNC effects will remain undetermined even in the very optimistic situation where all the rele- vant NP masses were measured at the LHC. This is in contrast with the quark sector, where similar RGE contributions are completely determined in terms of quark masses and CKM-matrix elements. More stable predictions can be obtained embedding the SUSY model within a Grand Unified Theory (GUT) where the see-saw mechanism can naturally arise (such as SO(10)). In this case the GUT symmetry allows us to obtain some hints about the unknown neutrino Yukawa matrix Yν . Moreover, in GUT scenarios there are other contributions stemming from the quark sector [3]. These effects are completely independent from the structure of Yν and can be regarded as new irreducible LFV contribu- tions within SUSY GUTs. For instance, within SU(5), as both Q and ec are hosted in the 10 representation, the CKM matrix mixing the left handed quarks will give rise to off diagonal entries in the running of the right-handed slepton soft masses [3]. There exist to different classes of LFV contributions to rare decays: i) Gauge-mediated LFV effects through the exchange of gauginos and sleptons, ii) Higgs-mediated LFV effects through effective non- holomorphic Yukawa interactions [4] . 1 The effective light-neutrino mass matrix obtained from a see- saw mechanism is mν = −YνM̂ ν 〈Hu〉 2, where M̂R is the 3 × 3 right-handed neutrino mass matrix and Yν are the 3 × 3 Yukawa couplings between left- and right-handed neutrinos (the potentially large sources of LFV), and 〈Hu〉 is the vacuum expectation value of the up-type Higgs. http://arxiv.org/abs/0704.0358v1 The above contributions decouple with the heaviest mass in the slepton/gaugino loops mSUSY (case i)) or with the heavy Higgs mass mH (case ii)). In principle, mH and mSUSY refers to different mass scales. Higgs mediated effects start being competitive with the gaugino mediated ones when mSUSY is roughly one order of magnitude heavier then mH and for tanβ ∼ O(50) [5]. While the appearance of LFV transitions would un- ambiguously signal the presence of NP, the underlying theory generating LFV phenomena will remain undeter- mined, in general. A powerful tool to disentangle among NP theories is the study of the correlations of LFV transitions among same families [5, 6, 7]. Interestingly enough, the predictions for the correla- tions among LFV processes are very different in the gauge- and Higgs-mediated cases [5]. In this way, if sev- eral LFV transitions are observed, their correlated anal- ysis could shed light on the underlying mechanism of LFV. In the case of gauge-mediated LFV amplitudes the ℓi → ℓjℓkℓk decays are dominated by the ℓi → ℓjγ ∗ dipole transition, which leads to the unambiguous prediction: B(ℓi → ℓjℓkℓk) B(ℓi → ℓjγ) B(µ− e in Ti) B(µ→eγ) ≃αel . (3) If some ratios different from the above were discovered, then this would be clear evidence that some new process is generating the ℓi → ℓj transition, with Higgs mediation being a potential candidate 2. As regards the Higgs mediated case, Br(τ → ljγ) still gets generally the largest contribution among all the pos- sible LFV decay modes [5]. The following approximate relations hold [5]: Br(τ → ljγ) Br(τ → ljη) & 1 , Br(τ → ljη) Br(τ → ljµµ) 3+5δjµ . (4) Br(τ → ljee) Br(τ → ljµµ) 3+5δjµ . (5) Br(µ → eγ) Br(µAl → eAl) ∼ 10 , Br(µ → eee) Br(µ → eγ) ∼ αel . (6) On the other hand, a correlated study of processes of the same type but relative to different family transitions, like 2 As recently shown in [7], a powerful tool to disentangle between Little Higgs models with T parity (LHT) and SUSY theories is a correlated analysis of LFV processes. In fact, LHT and SUSY theories predict very different correlations among LFV transitions [7]. Br(µ → eγ)/Br(τ → µγ) ∼ [(m2 )21/(m 2, provides important information about the unknown structure of the LFV source, i.e. (m2 )i6=j . III. LFU IN SUSY High precision electroweak tests, such as deviations from the SM expectations of the LFU breaking, represent a powerful tool to probe the SM and, hence, to constrain or obtain indirect hints of new physics beyond it. Kaon and pion physics are obvious grounds where to perform such tests, for instance in the π → ℓνℓ and K → ℓνℓ decays, where l = e or µ. In particular, the ratios B(P → µν) B(P → eν) can be predicted with excellent accuracies in the SM, both for P = π (0.02% accuracy [8]) and P = K (0.04% accuracy [8]), allowing for some of the most significant tests of LFU. As recently pointed out in Ref. [9], large departures from the SM expectations can be generated within a SUSY framework with R-parity only once we assume i) LFV effects, ii) large tanβ values. Denoting by ∆r NP the deviation from µ−e universal- ity in RK due to NP, i.e.: R K = (R K )SM 1 + ∆r it turns out that [9]: |∆31R | 2 tan6β. (8) The deviations from the SM could reach ∼ 1% in the K case [9] (not far from the present experimental res- olution [10]) and ∼ few × 10−4 in the R π case while maintaining LFV effects in τ decays at the 10−10 level. In the pion case the effect is quite below the present experi- mental resolution [11], but could well be within the reach of the new generation of high-precision πℓ2 experiments planned at TRIUMPH and at PSI. Larger violations of LFU are expected in B → ℓν decays, with O(10%) devi- ations from the SM in R B and even order-of-magnitude enhancements in R B [12]. IV. FLAVOR PHYSICS AT LARGE tanβ AND DARK MATTER Within the MSSM, the scenario with large tanβ and heavy squarks is particularly interesting. On the one hand, values of tanβ ∼ 30–50 can allow the unification of top and bottom Yukawa couplings, as predicted in well-motivated grand-unified models [13]. On the other hand, a Minimal Flavor Violating (MFV) structure [14] with heavy (∼ TeV ) soft-breaking terms in the quark sector and large tanβ ∼ 30 − 50 values leads to in- teresting phenomenological virtues [12, 15]: the present (g − 2)µ anomaly and the upper bound on the Higgs boson mass can be easily accommodated, while satisfy- ing all the present tight constraints in the electroweak and flavor sectors. Additional low-energy signatures of this scenario could possibly show up in the near future in B(Bu → τν), B(Bs,d → ℓ +ℓ−) and B(B → Xsγ). In the following, as discussed in [16], we analyze the above scenario under the additional assumption that the relic density of a Bino-like lightest SUSY particle (LSP) ac- commodates the observed dark matter distribution 0.094 ≤ ΩCDMh 2 ≤ 0.129 at 2σ C.L. . (9) In the regime with large tanβ and heavy squarks, the relic-density constraints can be easily satisfied mainly in the so called A-funnel region [17] where MB̃ ≈ MA/2. The combined constraints from low-energy observables and dark matter in the tanβ–MH plane are illustrated in Figure 1 (left). The light-blue areas are excluded since the stau turns out to be the LSP, while the yellow band denotes the allowed region where the stau coannihilation mechanism is also active. The remaining bands corre- spond to the following constraints/reference-ranges from low-energy observables: • B → Xsγ [1.01 < RBsγ < 1.24]: allowed region between the two blue lines. • aµ [2 < 10 −9(aexpµ − a µ ) < 4 [18]]: allowed region between the two purple lines. • B → µ+µ− [Bexp < 8.0×10−8 [19]]: allowed region below the dark-green line. • ∆MBs [∆MBs = 17.35 ± 0.25 ps −1 [20]]: allowed region below the gray line. • B → τν [0.8 < RBτν < 0.9]: allowed region be- tween the two black lines [ red (green) area if all the other conditions (but for aµ) are satisfied]. From Figure 1 (right), we deduce that there is a quite strong correlation between ∆aµ and B(Bu → τν) thanks to the A-funnel region condition MH ≈ 2M1. A SUSY contribution to aµ of O(10 −9) generally implies a sizable effect in 0.7 < B(Bu → τν) < 0.9. A more precise de- termination of B(Bu → τν) is therefore a key element to test this scenario. The interplay of B physics observables, dark-matter constraints, ∆aµ of O(10 −9), and LFV rates is shown in Figure 2. For a natural choice of |δ12LL| = 10 B(µ → eγ) is in the 10−12 range, i.e. well within the reach of MEG [21] experiment. On the other hand, B(τ → µγ) lies within the 10−9 range for a |δ23LL| = 10 that is a natural size expected in many models. Acknowledgments I wish to thank the conveners of WG3 for the kind invitation and G. Isidori, F. Mescia and D. Temes for collaborations on which this talk is partly based. I also acknowledge support from the EU contract No. MRTN- CT-2006-035482, ”FLAVIANET” and from the Spanish MEC and FEDER FPA2005-01678. [1] W. M. Yao et al. [Particle Data Group], J. Phys. G 33 (2006) 1 [hppt://pdg.lbl.gov]. [2] F. Borzumati and A. Masiero, Phys. Rev. Lett. 57 (1986) [3] R. Barbieri and L. J. Hall, Phys. Lett. B 338 (1994) 212 [hep-ph/9408406]; R. Barbieri, L. J. Hall and A. Strumia, Nucl. Phys. B 445 (1995) 219 [hep-ph/9501334]; L. Cal- ibbi, A. Faccia, A. Masiero and S. K. Vempati, Phys. Rev. D 74, 116002 (2006) [hep-ph/0605139]. [4] K. S. Babu and C. Kolda, Phys. Rev. Lett. 89 (2002) 241802 [hep-ph/0206310]. [5] P. Paradisi, JHEP 0602, 050 (2006) [hep-ph/0508054]; P. Paradisi, JHEP 0608, 047 (2006) [hep-ph/0601100]. [6] A. Brignole and A. Rossi, Nucl. Phys. B 701, 3 (2004) [hep-ph/0401100]. [7] M. Blanke, A. J. Buras, B. Duling, A. Poschenrieder and C. Tarantino, hep-ph/0702136. [8] W.J. Marciano and A. Sirlin, Phys.Rev.Lett. 71 3629 (1993); M.Finkemeier, Phys.Lett. B 387 391 (1996). [9] A. Masiero, P. Paradisi and R. Petronzio, Phys. Rev. D 74, 011701 (2006) [hep-ph/0511289]. [10] L. Fiorini [NA48/2 Collaboration], talk presented at EPS 2005 July 21st-27th 2005 (Lisboa, Portugal). [11] G. Czapek et al., Phys. Rev. Lett. 70 (1993) 17; D. I. Britton et al., Phys. Rev. Lett. 68 (1992) 3000. [12] G. Isidori and P. Paradisi, Phys. Lett. B 639 (2006) 499 [hep-ph/0605012]. [13] G. Anderson, S. Raby, S. Dimopoulos, L. J. Hall and G. D. Starkman, Phys. Rev. D 49 (1994) 3660 [hep-ph/9308333]. [14] G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, Nucl. Phys. B645 (2002) 155. [15] E. Lunghi, W. Porod and O. Vives, Phys. Rev. D 74 (2006) 075003 [hep-ph/0605177]. [16] G. Isidori, F. Mescia, P. Paradisi and D. Temes, hep-ph/0703035. [17] J. R. Ellis, L. Roszkowski and Z. Lalak, Phys. Lett. B 245 (1990) 545. [18] K. Hagiwara, A. D. Martin, D. Nomura and T. Teubner, hep-ph/0611102; M. Passera, Nucl. Phys. Proc. Suppl. 155 (2006) 365 [hep-ph/0509372]. [19] R. Bernhard et al. [CDF Collab.], hep-ex/0508058. [20] A. Abulencia et al. [CDF - Run II Collab.], Phys. Rev. Lett. 97 (2006) 062003 [AIP Conf. Proc. 870 (2006) 116] [hep-ex/0606027]. [21] M. Grassi [MEG Collaboration], Nucl. Phys. Proc. Suppl. 149 (2005) 369. http://arxiv.org/abs/hep-ph/9408406 http://arxiv.org/abs/hep-ph/9501334 http://arxiv.org/abs/hep-ph/0605139 http://arxiv.org/abs/hep-ph/0206310 http://arxiv.org/abs/hep-ph/0508054 http://arxiv.org/abs/hep-ph/0601100 http://arxiv.org/abs/hep-ph/0401100 http://arxiv.org/abs/hep-ph/0702136 http://arxiv.org/abs/hep-ph/0511289 http://arxiv.org/abs/hep-ph/0605012 http://arxiv.org/abs/hep-ph/9308333 http://arxiv.org/abs/hep-ph/0605177 http://arxiv.org/abs/hep-ph/0703035 http://arxiv.org/abs/hep-ph/0611102 http://arxiv.org/abs/hep-ph/0509372 http://arxiv.org/abs/hep-ex/0508058 http://arxiv.org/abs/hep-ex/0606027 FIG. 1: Left plot: Combined constraints from low-energy observables and dark matter in the tan β–MH plane setting [µ,Mℓ̃] = [0.5, 0.4] TeV. The light-blue area is excluded by the dark-matter conditions [16]. Within the red (green) area all the reference values of the low-energy observables (but for aµ) are satisfied. The yellow band denote the area where the stau coannihilation mechanism is active (1 < Mτ̃R/MB̃ < 1.1); in this area the A-funnel region (where MH ≈ 2M1) and the stau coannihilation region overlap. Right plot: ∆aµ = (gµ−g µ )/2 vs. the slepton mass within the funnel region taking into account the B → Xsγ constraint and setting RBτν > 0.7 (blue), RBτν > 0.8 (red), RBτν > 0.9 (green) [16]. The supersymmetric parameters have been varied in the following ranges: 200 GeV ≤ M2 ≤ 1000 GeV, 500 GeV ≤ µ ≤ 1000 GeV, 10 ≤ tan β ≤ 50. In both plots, we have set AU = −1 TeV, Mq̃ = 1.5 TeV, and imposed the GUT relation M1 ≈ M2/2 ≈ M3/6. FIG. 2: Isolevel curves for B(µ → eγ) and B(τ → µγ) assuming |δ12LL| = 10 −4 and |δ23LL| = 10 −2 in the tan β–MH plane [16]. The green/red areas correspond to the allowed regions for the low-energy observables illustrated in Figure 1 for [µ,M [0.5, 0.4] TeV.

0704.0359

Some properties of the complex Monge-Ampere operator in Cegrell's classes and applications

arXiv:0704.0359v1 [math.CV] 3 Apr 2007 Some properties of the complex Monge-Ampère operator in Cegrell’s classes and applications NGUYEN VAN KHUE and PHAM HOANG HIEP Abstract. In this article we will first prove a result about convergence in capacity. Using the achieved result we will obtain a general decompositon theorem for complex Monge- Ampère measues which will be used to prove a comparison principle for the complex Monge-Ampère operator. 2000 Mathematics Subject Classification: Primary 32W20, Secondary 32U15. Key words and phrases: complex Monge-Ampère operator, plurisubharmonic function. This work was supported by the National Research Program for Natural Sciences, Vietnam. 1. Introduction Let Ω be a bounded hyperconvex domain in Cn. By PSH(Ω) we denote the set of plurisub- harmonic (psh) functions on Ω. In [BT 1,2] the authors established and used the compari- son principle to study the Dirichlet problem in PSH∩L∞loc(Ω). Recently, Cegrell introduced a general class E of psh functions on which the complex Monge-Ampère operator (ddc.)n can be defined. He obtained many important results of pluripotential theory in the class E . For example, the ones on the comparison principle and solvability of the Dirichlet problem (see [Ce 1-3]). The main result of our paper are Theorem 4.1 and some Xing type comparision principles. Theorem 4.1 is generalize Lemma 5.4 in [Ce1], Lemma 7.2 in [Åh] and Lemma 3.4 in [Ce3]. For definitions of Cegrell’s classes see Section 2. After giving some preliminaries, we start in Proposition 3.1 with a comparison principle, which is analogous to a comparison principle due to Xing (Lemma 1 in [Xi1]). It should be observed that our proof is quite different from Xing’s proof, and the inequality we obtain is slightly stronger than Xing’s inequality, even in the case of bounded psh functions. Using Proposition 3.1, we give in Theorem 3.5 a sufficient condition for Cn-capacity convergence of a sequence of psh functions in the class F . This result should be compared to Theorem 3 of [Xi1] where the situation of bounded psh functions was studied. Applying Theorem 3.5 we give generalizations of recent results in [Cz] and [CLP] about convergences of multipole Green functions and a criterion for pluripolarity, respectively. Section 4 focuses on Theorem 4.1 and Theorem 4.9. By applying Theorem 4.1 we give some results on class Cegrell’s classes. We prove in Proposition 4.4 a local estimate for the Monge-Ampère measure in terms of the Beford- Taylor relative capacity. As an application, we give in Theorem 4.5 a decomposition result for Monge-Ampère measure, which is similar in spirit to Theorem 6.3 in [Ce1]. From Proposition 3.1 and Theorem 4.1 we obtain easily a Xing type comparison principle for functions in classes F and E . Acknowledgment. We are grateful to Professor Urban Cegrell for useful discussions that helped to improve the paper. We are grateful to Per Åhag for fruitful comments. This work is supported by the National Research Program for Natural Sciences, Vietnam. http://arxiv.org/abs/0704.0359v1 2. Preliminaries First we recall some elements of pluripotential theory that will be used throughout the paper. All this can be found in [BT2], [Ce1], [Ce2], [Le]. 2.1. We will always denote by Ω a bounded hyperconvex domain in Cn unless other wise stated. The Cn-capacity in the sense of Bedford and Taylor on Ω is the set function given Cn(E) = Cn(E,Ω) = sup{ (ddcu)n : u ∈ PSH(Ω), −1 ≤ u ≤ 0} for every Borel set E in Ω. It is proved in [BT2] that Cn(E) = (ddch∗E,Ω) where h∗E,Ω is the upper regularization of the relative extremal function hE,Ω for E (relative to Ω) i.e., hE,Ω(z) = sup{u(z) : u ∈ PSH −(Ω), u ≤ −1 on E}. The following concepts are taken from [Xi1] and [Xi2] ∗A sequence of functions uj on Ω is said to converge to a function u in Cn-capacity on a set E ⊂ Ω if for every δ > 0 we have Cn({z ∈ E : |uj(z) − u(z)| > δ}) → 0 as j → ∞. ∗A family of positive measures {µα} on Ω is called uniformly absolutely continuous with respect to Cn-capacity in a set E ⊂ Ω if for every ǫ > 0 there exists δ > 0 such that for each Borel subset F ⊂ E with Cn(F)< δ the inequality µα(F)< ǫ holds for all α. We write µα ≪ Cn in E uniformly for α. 2.2. The following classes of psh functions were introduced by Cegrell in [Ce1] and [Ce2] E0 = E0(Ω) = {ϕ ∈ PSH −(Ω) ∩ L∞(Ω) : lim ϕ(z) = 0, (ddcϕ)n < +∞}, F = F(Ω) = {ϕ ∈ PSH−(Ω) : ∃ E0(Ω) ∋ ϕj ց ϕ, sup (ddcϕj) n < +∞}, E = E(Ω) = {ϕ ∈ PSH−(Ω) : ∃ ϕK ∈ F(Ω) such that ϕK = ϕ on K, ∀K ⊂⊂ Ω}, Ea = Ea(Ω) = {u ∈ E(Ω) : (ddcu)n(E) = 0 ∀ E is pluripolar in Ω}. For each u ∈ F(Ω), we set e0(u) = (ddcu)n. 2.3. Let A = {(wj , νj)}j=1,...,p be a finite subset of Ω × R +. According to Lelong (see [Le]), the pluricomplex Green function with poles in A is defined by g(A)(z) = sup{u(z) : u ∈ LA} where LA = {u ∈ PSH −(Ω) : u(z) − νj log |z − wj | ≤ O(1) as z → wj , j = 1, ..., p} ν(A) = νnj , Â = {wj}j=1,...,p. 2.4. We write lim [u(z) − v(z)] ≥ a if for every ǫ > 0 there exists a compact set K in Ω such that u(z) − v(z) ≥ a− ǫ for z ∈ (Ω\K) ∩ {u > −∞} v(z) = −∞ for z ∈ (Ω\K) ∩ {u = −∞}. 2.5. Xing’s comparison principle (see Lemma 1 in [Xi1]). Let Ω be a bounded open subset in Cn and u, v ∈ PSH∩L∞(Ω) satisfy lim [u(z)− v(z)] ≥ 0. Then for any constant r ≥ 1 and all wj ∈ PSH(Ω) with 0 ≤ wj ≤ 1, j = 1, 2, ..., n we have (n!)2 {u<v} (v − u)nddcw1 ∧ ... ∧ dd {u<v} (r − w1)(dd v)n ≤ {u<v} (r − w1)(dd 3. Some convergence theorems In order to study the convergence of a sequence of psh functions in Cn-capacity, we start with the following. 3.1. Proposition. a) Let u, v ∈ Fsuch that u ≤ v on Ω. Then for 1 ≤ k ≤ n (v − u)kddcw1 ∧ ... ∧ dd cwn + (r − w1)(dd cv)k ∧ ddcwk+1 ∧ ... ∧ dd (r − w1)(dd cu)k ∧ ddcwk+1 ∧ ... ∧ dd for all wj ∈ PSH(Ω), 0 ≤ wj ≤ 1, j = 1, ..., k, wk+1, ..., wn ∈ F and all r ≥ 1. b) Let u, v ∈ E such that u ≤ v on Ω and u = v on Ω\K for some K ⊂⊂ Ω. Then for 1 ≤ k ≤ n (v − u)kddcw1 ∧ ... ∧ dd cwn + (r − w1)(dd cv)k ∧ ddcwk+1 ∧ ... ∧ dd (r − w1)(dd cu)k ∧ ddcwk+1 ∧ ... ∧ dd for all wj ∈ PSH(Ω), 0 ≤ wj ≤ 1, j = 1, ..., k, wk+1, ..., wn ∈ E and all r ≥ 1. We proceed through some lemmas. 3.2. Lemma. Let u, v ∈ PSH ∩ L∞(Ω) such that u ≤ v on Ω and lim [u(z) − v(z)] = 0. (v − u)kddcw ∧ T ≤ k (1 − w)(v − u)k−1ddcu ∧ T for all w ∈ PSH(Ω), 0 ≤ w ≤ 1 and all positive closed currents T . Proof. First, assume u, v ∈ PSH∩L∞(Ω), u ≤ v on Ω and u = v on Ω\K, K ⊂⊂ Ω. Then, using the Stokes formula we obtain (v − u)kddcw ∧ T = (v − u)kddc(w − 1) ∧ T (w − 1)ddc(v − u)k ∧ T = −k(k − 1) (1 − w)d(v − u) ∧ dc(v − u) ∧ T (1 − w)(v − u)k−1ddc(u− v) ∧ T (1 − w)(v − u)k−1ddc(u− v) ∧ T (1 − w)(v − u)k−1ddcu ∧ T. General case, for each ǫ > 0 we set vǫ = max(u, v − ǫ). Then vǫ ր v on Ω, vǫ ≥ u on Ω and vǫ = u on Ω\K for some K ⊂⊂ Ω. Hence (vǫ − u) kddcw ∧ T ≤ k (1 − w)(vǫ − u) k−1ddcu ∧ T. Since 0 ≤ vǫ − u ր v − u as ǫ ց 0, letting ǫ ց 0 we get (v − u)kddcw ∧ T ≤ k (1 − w)(v − u)k−1ddcu ∧ T. 3.3. Lemma. Let u, v ∈ PSH ∩ L∞(Ω) such that u ≤ v on Ω and lim [u(z) − v(z)] = 0. Then for 1 ≤ k ≤ n (v − u)kddcw1 ∧ ... ∧ dd cwn + (r − w1)(dd cv)k ∧ T (r − w1)(dd cu)k ∧ T. for all w1, ..., wk ∈ PSH(Ω), 0 ≤ wj ≤ 1 ∀ j = 1, ..., k, wk+1, ..., wn ∈ E and all r ≥ 1. Proof. To simplify the notation we set T = ddcwk+1 ∧ ... ∧ dd First, assume that u, v ∈ PSH ∩ L∞(Ω), u ≤ v on Ω, and u = v on Ω\K, K ⊂⊂ Ω. Using Lemma 3.2 we get (v − u)kddcw1 ∧ ... ∧ dd cwn ≤ k (v − u)k−1ddcw1 ∧ ... ∧ dd cwk−1 ∧ dd cu ∧ T ≤ ... (v − u)ddcw1 ∧ (dd cu)k−1 ∧ T (v − u)ddcw1 ∧ [ (ddcu)i ∧ (ddcv)k−i−1] ∧ T (w1 − r)dd c(v − u) ∧ [ (ddcu)i ∧ (ddcv)k−i−1] ∧ T (r − w1)dd c(u− v) ∧ [ (ddcu)i ∧ (ddcv)k−i−1] ∧ T (r − w1)[(dd cu)k − (ddcv)k] ∧ T. General case, for each ǫ > 0 we put vǫ = max(u, v − ǫ). Then vǫ ր v on Ω, vǫ ≥ u on Ω and vǫ = u on Ω\K for some K ⊂⊂ Ω. Hence (vǫ − u) kddcw1 ∧ ... ∧ dd cwn + (r − w1)(dd k ∧ T (r − w1)(dd u)k ∧ T. Observe that 0 ≤ vǫ − u ր v− u and (dd k ∧ T → (ddcv)k ∧ T weakly as ǫ ց 0, r−w1 is lower semicontinuous, by letting ǫ ց 0 we have (v − u)kddcw1 ∧ ... ∧ dd cwn + (r − w1)(dd cv)k ∧ T (r − w1)(dd cu)k ∧ T. The proof is finished. Proof of Proposition 3.1. a) Let E0 ∋ uj ց u and E0 ∋ vj ց v as in the definition of F . Replace vj by max(uj , vj) we may assume that uj ≤ vj for j ≥ 1. By Lemma 3.3 we have (vj − ut) kddcw1 ∧ ... ∧ dd cwn + (r − w1)(dd k ∧ ddcwk+1 ∧ ... ∧ dd (r − w1)(dd k ∧ ddcwk+1 ∧ ... ∧ dd for t ≥ j ≥ 1. By Proposition 5.1 in [Ce2] letting t → ∞ in the above inequality we have (vj − u) w1 ∧ ... ∧ dd (r − w1)(dd k ∧ T (r − w1)(dd cu)k ∧ T for j ≥ 1. Next letting j → ∞ again by Proposition 5.1 in [Ce2] we get the desired conclusion. b) Let G,W be open sets such that K ⊂⊂ G ⊂⊂ W ⊂⊂ Ω. According to the remark following Definition 4.6 in [Ce2] we can choose a function ṽ ∈ F such that ṽ ≥ v and ṽ = v on W . Set u on G ṽ on Ω\G Since u = v = ṽ on W\K we have ũ ∈ PSH−(Ω). It is easy to see that ũ ∈ F , ũ ≤ ṽ and ũ = u on W . By a) we have (ṽ − ũ)kddcw1 ∧ ... ∧ dd cwn + (r − w1)(dd cṽ)k ∧ ddcwk+1 ∧ ... ∧ dd (r − w1)(dd ũ)k ∧ ddcwk+1 ∧ ... ∧ dd Since ũ = ṽ on Ω\G we have (ṽ − ũ)kddcw1 ∧ ... ∧ dd cwn + (r − w1)(dd cṽ)k ∧ ddcwk+1 ∧ ... ∧ dd (r − w1)(dd cũ)k ∧ ddcwk+1 ∧ ... ∧ dd Since ũ = u, ṽ = v on W and u = v on Ω\K we obtain (v − u)kddcw1 ∧ ... ∧ dd cwn + (r − w1)(dd cv)k ∧ ddcwk+1 ∧ ... ∧ dd (r − w1)(dd cu)k ∧ ddcwk+1 ∧ ... ∧ dd 3.4. Proposition. Let u, v ∈ F and u ≤ v on Ω. Then (v − u)nddcw1 ∧ ... ∧ dd (−w1)[(dd u)n − (ddcv)n] for all wj ∈ PSH(Ω), −1 ≤ wj ≤ 0, j = 1, ..., n. Proof. The proposition follows from Proposition 3.1 with k = n, r = 1 and wj are replaced by wj + 1. 3.5. Theorem. Let u, uj ∈ F and uj ≤ u for j ≥ 1. Assume that sup (ddcuj) n < +∞ and ||(ddcuj) n − (ddcu)n||E → 0 as j → ∞ for all E ⊂⊂ Ω. Then uj → u in Cn-capacity on every E ⊂⊂ Ω as j → ∞. Proof. Let Ω′ ⊂⊂ Ω and δ > 0. Put Aj = {z ∈ Ω′ : |uj − u| ≥ δ} = {z ∈ Ω′ : u− uj ≥ δ}. We prove that Cn(Aj) → 0 as j → ∞. Given ǫ > 0. By quasicontinuity of u and uj , there is an open set G in Ω such that Cn(G) < ǫ, and uj |Ω\G, u|Ω\G are continuous. We have Aj = Bj ∪ {z ∈ G : u− uj ≥ δ}. where Bj = {z ∈ Ω′\G : u− uj ≥ δ} are compact sets in Ω and Cn(Aj) ≤ lim Cn(Bj) + ǫ We claim that lim Cn(Bj) = 0. By Proposition 3.4 we have Cn(Bj) = (ddch∗Bj ) (u− uj) n(ddch∗Bj ) (−h∗Bj )[(dd n − (ddcu)n] {||(ddcuj) n − (ddcu)n||K + (−hΩ′)[(dd n + (ddcu)n]} {||(ddcuj) n − (ddcu)n||K + sup |hΩ′ |[sup (ddcuj) (ddcu)n]}. As lim hΩ′(z) = 0 there exists K ⊂⊂ Ω such that |hΩ′ |[sup (ddcuj) (ddcu)n] < ǫ. By the hypothesis ||(ddcuj) n − (ddcu)n||K < ǫ for j > j0. Cn(Bj) < 2ǫ for j > j0. This proves the claim and hence the theorem. As an application of Theorem 3.5 we have the following 3.6. Proposition. Let g(Aj) be multipolar Green functions on Ω such that Âj = {w 1, ..., w } → ∂Ω and sup ν(Aj) = sup )n < +∞ Then g(Aj) → 0 as j → ∞ in Cn-capacity. Proof. By the hypothesis we have (ddcg(Aj)) n(Ω) = sup ν(Aj) < +∞ ||(ddcg(Aj)) n||K → 0 as j → ∞ for all K ⊂⊂ Ω. Theorem 3.5 implies that g(Aj) → 0 as j → ∞ in Cn-capacity. This section ends up with a criterion for pluripolarity 3.7. Theorem. Let uj ∈ F such that sup (ddcuj) n < +∞. Then there is a constant A > 0 such that i)( lim ∗ ∈ F . ii)Cn({z ∈ Ω : ( lim ∗(z) < −t}) ≤ A iii){z ∈ Ω : lim uj(z) = −∞} is pluripolar. Proof. i) For each j ≥ 1 put vj = sup{uj, uj+1, ...}. By [Ce2] v j ∈ F and (ddcv∗j ) n ≤ sup (ddcuj) n < +∞. By [Ce2] we have v∗j ց v ∈ F . ii) By Proposition 3.1 in [CKZ] we have Cn{z ∈ Ω : ( lim ∗(z) < −t} = Cn{z ∈ Ω : v(z) < −t} ≤ 2ne0(v) where A = 2ne0(v). iii) According to [BT2] we have Cn{z ∈ Ω : lim uj(z) = −∞} = Cn{z ∈ Ω : v(z) = −∞} = 0. Remark. Theorem 3.7 in the case where uj are multipole Green functions was proved by D.Coman, N.Levenberg and A.Poletsky in Theorem 4.1 of [CLP]. 4. Some properties of the Cegrell’s classes and applications In this section, first we prove the following 4.1. Theorem. Let u, u1, ..., un−1 ∈ E , v ∈ PSH −(Ω) and T = ddcu1 ∧ ... ∧ dd cun−1. ddc max(u, v) ∧ T |{u>v} = dd cu ∧ T |{u>v}. We need the following well-known fact. 4.2. Lemma. Let µ be a measure on Ω and f : Ω → R a measurable function on Ω. The following are equivalent i)µ(E) = 0 for all Borell sets E ⊂ {f 6= 0}. fdµ = 0 for every measurable set E in Ω. Proof. i)⇒ii) follows from: fdµ = E\{f=0} fdµ + E∩{f=0} fdµ = 0 ii)⇒i). It suffices to show that µ = 0 on every Xδ = {f > δ > 0}. By the Hahn decomposition theorem, there exist measurable subsets X+ and X− of Xδ such that Xδ = = ∅ and µ ≥ 0 on X+ , µ ≤ 0 on X− . We have δµ(X+ fdµ = 0 δµ(X− fdµ = 0 Hence, µ(X+ ) = µ(X− ) = 0. Therefore, we have µ = 0 on Xδ. Proof of Theorem 4.1. a) First we prove the proposition for v ≡ a < 0. According to the remark following Definition 4.6 in [Ce2], without loss of generality we may assume that u, u1, ..., un−1 ∈ F . Using Theorem 2.1 in [Ce2] we can find E0 ∩ C(Ω̄) ∋ u j ց u, E0 ∩ C(Ω̄) ∋ u ց uk, k = 1, ..., n− 1. Since {uj > a} is open we have ddc max(uj, a) ∧ Tj |{uj>a} = dd cuj ∧ Tj |{uj>a}. Thus from the inclusion {u > a} ⊂ {uj > a} we obtain ddc max(uj , a) ∧ Tj |{u>a} = dd cuj ∧ Tj |{u>a}. where Tj = dd 1 ∧ ... ∧ dd n−1. By Corollary 5.2 in [Ce2], it follows that max(u− a, 0)ddc max(uj , a) ∧ Tj → max(u− a, 0)dd c max(u, a) ∧ T. max(u− a, 0)ddcuj ∧ Tj → max(u− a, 0)dd cu ∧ T. Hence max(u− a, 0)[ddc max(u, a) ∧ T − ddcu ∧ T ] = 0. Using Lemma 4.2 we have ddc max(u, a) ∧ T = ddcu ∧ T on {u > a}. b) Assume that v ∈ PSH−(Ω). Since {u > v} = {u > a > v}, it suffices to show ddc max(u, v) ∧ T = ddcu ∧ T on {u > a > v} for all a ∈ Q−. Since max(u, v) ∈ E , by a) we have ddc max(u, v) ∧ T |{max(u,v)>a} = dd c max(max(u, v), a) ∧ T |{max(u,v)>a} = ddc max(u, v, a) ∧ T |{max(u,v)>a}. (2) ddcu ∧ T |{u>a} = dd c max(u, a) ∧ T |{u>a}. Since max(u, v, a) = max(u, a) on set open {a > v} , we have (3) ddc max(u, v, a) ∧ T |{a>v} = dd c max(u, a) ∧ T |{a>v}. Since {u > a > v} ⊂ {u > a}, {a > v}, {max(u, v) > a} and (1), (2), (3) we have ddc max(u, v) ∧ T |{u>a>v} = dd cu ∧ T |{u>a>v}. The next result is an analogue of an inequality due to Demaily in [De2] 4.3. Proposition. a) u, v ∈ E such that (ddcu)n({u = v = −∞}) = 0. Then (ddc max(u, v))n ≥ 1{u≥v}(dd cu)n + 1{u<v}(dd where 1E denotes the characteristic function of E. b) Let µ be a positive measure which vanishes on all pluripolar subsets of Ω. Suppose u, v ∈ E such that (ddcu)n ≥ µ, (ddcv)n ≥ µ. Then (ddc max(u, v))n ≥ µ. Proof. a) For each ǫ > 0 put Aǫ = {u = v − ǫ}\{u = v = −∞}. Since Aǫ ∩ Aδ = ∅ for ǫ 6= δ there exists ǫj ց 0 such that (dd cu)n(Aǫj ) = 0 for j ≥ 1. On the other hand, since (ddcu)n({u = v = −∞}) = 0 we have (ddcu)n({u = v− ǫj}) = 0 for j ≥ 1. Since Theorem 4.1 it follows that (ddc max(u, v − ǫj)) n ≥ (ddc max(u, v − ǫj)) n|{u>v−ǫj} + (dd c max(u, v − ǫj)) n|{u<v−ǫj} = (ddcu)n|{u≥v−ǫj} + (dd cv)n|{u<v−ǫj} = 1{u≥v−ǫj}(dd cu)n + 1{u<v−ǫj}(dd ≥ 1{u≥v}(dd cu)n + 1{u<v−ǫj}(dd cv)n. Letting j → ∞ and by Remark under Theorem 5.15 in [Ce2] we get (ddc max(u, v))n ≥ 1{u≥v}(dd cu)n + 1{u<v}(dd because max(u, v − ǫj) ր max(u, v) and 1{u<v−ǫj} ր 1{u<v} as j → ∞. b) Argument as a) 4.4. Proposition. Let u1, ..., uk ∈ PSH(Ω) ∩ L ∞(Ω) and uk+1, ..., un ∈ E . Then ddcu1 ∧ ... ∧ dd cun = O((Cn(B)) n ) for all Borel sets B ⊂ Ω′ ⊂⊂ Ω. B(a,r) ddcu1 ∧ ... ∧ dd cun = o((Cn(B(a, r))) n ) as r → 0 for all a ∈ Ω. where B(a, r) = {z ⊂ Cn : |z − a| < r} Proof. We may assume that 0 ≤ uj ≤ 1 for j = 1, ..., k. On the other hand, by the remark following Defintion 4.6 in [Ce2] we again may assume that uk+1, ..., un ∈ F . i) For each open set B ⊂⊂ Ω, applying Proposition 3.1 we get ddcu1 ∧ ... ∧ dd cun = (−h∗B) kddcu1 ∧ ... ∧ dd (−h∗B) kddcu1 ∧ ... ∧ dd (1 − u1)(dd ch∗B) k ∧ ddcuk+1 ∧ ... ∧ dd (ddch∗B) k ∧ ddcuk+1 ∧ ... ∧ dd ≤ k![ (ddch∗B) n ∧ [ (ddcuk+1) n ∧ ... ∧ [ (ddcun) (by Corollary 5.6 in [Ce2]) ≤ k!(e0(uk+1)) n ...(e0(un)) n .[Cn(B)] ≤ constants.[Cn(B)] Hence ddcu1 ∧ ... ∧ dd cun ≤ constants.[Cn(B)] for all Borel set B ⊂ Ω. ii) By Proposition 3.1 we have (−ϕ)kddcu1 ∧ ... ∧ dd un ≤ k! (1 − u1)(dd ϕ)k ∧ ddcuk+1 ∧ ... ∧ dd (ddcϕ)k ∧ ddcuk+1 ∧ ... ∧ dd cun < +∞. Hence (−ϕ)k ∈ L1(dd cu1 ∧ ... ∧ dd cun) for all ϕ ∈ F(Ω). Given a ∈ Ω let r0, R0 such that B(a, r0) ⊂⊂ Ω ⊂⊂ B(a, R0). Then |z − a| ≤ ga(z) ≤ log |z − a| for all z ∈ Ω, where ga denotes the Green function of Ω with pole at a. Since (−ga) L1(dd cu1 ∧ ... ∧ dd cun), it follows that B(a,r) (−ga) kddcu1 ∧ ... ∧ dd cun → 0 as r → 0 Hence (log r0 − log r) B(a,r) ddcu1 ∧ ... ∧ dd cun ≤ B(a,r) (−ga) kddcu1 ∧ ... ∧ dd cun → 0 as r → 0. This means that B(a,r) ddcu1 ∧ ... ∧ dd cun = o(( log r0 − log r )k) as r → 0 Combining this with the inequality Cn(B(a, r),Ω) ≥ Cn(B(a, r), B(a, R0)) = ( logR0 − log r )n = O(( log r0 − log r)n we get B(a,r) ddcu1 ∧ ... ∧ dd cun = o((Cn(B(a, r))) The next result should be compared with Theorem 6.3 in [Ce1] 4.5. Theorem. Let u1, ..., un ∈ E . Then there exists ũ ∈ E a such that ddcu1 ∧ ... ∧ dd cun = (dd cũ)n + ddcu1 ∧ ... ∧ dd cun|{u1=...=un=−∞}. Proof. First, we write ddcu1 ∧ ... ∧ dd cun = µ + dd cu1 ∧ ... ∧ dd cun|{u1=...=un=−∞}. where µ = ddcu1 ∧ ... ∧ dd cun|{u1>−∞}∪...∪{un>−∞}. It is easy to see that µ ≪ Cn in every E ⊂⊂ Ω. Indeed, by Theorem 4.1 we have ddcu1 ∧ ... ∧ dd cun|{u1>−j} = dd c max(u1,−j) ∧ ... ∧ dd cun|{u1>−j}. Hence, by Proposition 4.4 (i) it follows that ddcu1 ∧ ... ∧ dd cun|{u1>−j} ≪ Cn in every E ⊂⊂ Ω. Next, it remains to show that there exists ũ ∈ Ea such that µ = (ddcũ)n. Let {Ωj} be an increasing exhaustion sequence of Ω. For each j ≥ 1 put µj = µ|Ωj . By [Åh] there exists ũj ∈ F such that (dd cũj) n = µj . Notice that µj ր µ and (ddcũj) n ≤ µ ≤ (ddc(u1 + ... + un)) Applying the comparison principle we obtain ũj ց ũ ≥ u1 + ... + un ∈ E . Hence, ũ ∈ Ea and (ddcũ)n = lim (ddcũj) n = µ. The proof is thereby completed. 4.6. Corollary. u1, ..., un ∈ E . Then the following are equivalent i) ddcu1 ∧ ... ∧ dd cun ≪ Cn in every E ⊂⊂ Ω. {u1=...=un=−∞} ddcu1 ∧ ... ∧ dd cun = 0. {u1<−s,...,un<−s}∩E ddcu1 ∧ ... ∧ dd cun → 0 as s → +∞ for all E ⊂⊂ Ω. Proof. Direct application of Theorem 4.5. The comparison principle for class F was studied in [Ce3] and [H1]. By using Proposition 3.1 and Theorem 4.1 we prove a Xing type comparison principle for F 4.7. Theorem. Let u ∈ F , v ∈ E and 1 ≤ k ≤ n. Then {u<v} (v − u)kddcw1 ∧ ... ∧ dd cwn + {u<v} (r − w1)(dd cv)k ∧ ddcwk+1 ∧ ... ∧ dd {u<v}∪{u=v=−∞} (r − w1)(dd cu)k ∧ ddcwk+1 ∧ ... ∧ dd for all wj ∈ PSH(Ω), 0 ≤ wj ≤ 1, j = 1, ..., k, wk+1, ..., wn ∈ F and all r ≥ 1. Proof. Let ǫ > 0. We set ṽ = max(u, v − ǫ). By a) in Proposition 3.1 we have (ṽ − u)kddcw1 ∧ ... ∧ dd cwn + (r − w1)(dd cṽ)k ∧ ddcwk+1 ∧ ... ∧ dd (r − w1)(dd cu)k ∧ ddcwk+1 ∧ ... ∧ dd Since {u < ṽ} = {u < v − ǫ} and Theorem 4.1 we have {u<v−ǫ} (v− ǫ−u)kddcw1 ∧ ...∧ dd cwn + {u≤v−ǫ} (r−w1)(dd cv)k ∧ ddcwk+1 ∧ ...∧ dd {u≤v−ǫ} (r − w1)(dd cu)k ∧ ddcwk+1 ∧ ... ∧ dd {u<v}∪{u=v=−∞} (r − w1)(dd cu)k ∧ ddcwk+1 ∧ ... ∧ dd Letting ǫ ց 0 we obtain {u<v} (v − u)kddcw1 ∧ ... ∧ dd cwn + {u<v} (r − w1)(dd cv)k ∧ ddcwk+1 ∧ ... ∧ dd {u<v}∪{u=v=−∞} (r − w1)(dd cu)k ∧ ddcwk+1 ∧ ... ∧ dd 4.8. Corollary. Let u ∈ Ea such that u ≥ v for all functions v ∈ E satisfying (ddcu)n ≤ (ddcv)n. Then {u<v} (v − u)nddcw1 ∧ ... ∧ dd {u<v} (r − w1)(dd {u<v} (r − w1)(dd for all v ∈ E , r ≥ 1 and all w1, ..., wn ∈ PSH(Ω), 0 ≤ w1, ..., wn ≤ 1. Proof. Let {Ωj} be an increasing exhaustion sequence of relatively compact subdomains of Ω. Set µj = 1Ωj1{u>−j}(dd cu)n, where 1E denotes the characteristic function of E ⊂ Ω. Applying Theorem 4.1 we have µj = 1Ωj1{u>−j}(dd c max(u,−j))n ≤ 1Ωj (dd c max(u,−j))n. Take φ ∈ E0(Ω) ∩ C(Ω̄). Put φj = max(u,−j, ajφ) where aj = . Then φj = max(u,−j) on Ωj+1, φj ∈ E0 and µj ≤ 1Ωj (dd c max(u,−j))n = 1Ωj (dd n ≤ (ddcφj) By Ko lodziej’s theorem (see [Ko]) there exists uj ∈ E0 such that (ddcuj) n = µj = 1Ωj1{u>−j}(dd cu)n, ∀ j ≥ 1. for all j ≥ 1. By the comparison principle we have uj ց ũ ≥ u. On the other hand, since (ddcu)n({u = −∞}) = 0, it follows that (ddcuj) n = 1Ωj1{u>−j}(dd cu)n → (ddcu)n weakly as j → ∞. Thus (ddcũ)n = lim (ddcuj) n = (ddcu)n. By the hypothesis we have ũ = u. Applying Theorem 4.7 we get {uj<v} (v − uj) nddcw1 ∧ ... ∧ dd cwn + {uj<v} (r − w1)(dd {uj<v} (r − w1)(dd {uj<v} (r − w1)(dd cu)n. Letting j → ∞ we obtain {u<v} (v − u)nddcw1 ∧ ... ∧ dd cwn + {u<v} (r − w1)(dd Arguing as in Theorem 4.7 we prove a Xing type comparison principle for E . 4.9. Theorem. Let u, v ∈ E and 1 ≤ k ≤ n such that lim [u(z) − v(z)] ≥ 0. Then {u<v} (v − u)kddcw1 ∧ ... ∧ dd cwn + {u<v} (r − w1)(dd cv)k ∧ ddcwk+1 ∧ ... ∧ dd {u<v}∪{u=v=−∞} (r − w1)(dd cu)k ∧ ddcwk+1 ∧ ... ∧ dd for all wj ∈ PSH(Ω), 0 ≤ wj ≤ 1, j = 1, ..., k, wk+1, ..., wn ∈ E and all r ≥ 1. Proof. Let ǫ > 0. We set ṽ = max(u, v − ǫ). By b) in Proposition 3.1 we have (ṽ − u)kddcw1 ∧ ... ∧ dd cwn + (r − w1)(dd cṽ)k ∧ ddcwk+1 ∧ ... ∧ dd (r − w1)(dd u)k ∧ ddcwk+1 ∧ ... ∧ dd Since {u < ṽ} = {u < v − ǫ} and Theorem 4.1 we have {u<v−ǫ} (v− ǫ−u)kddcw1 ∧ ...∧ dd cwn + {u≤v−ǫ} (r−w1)(dd cv)k ∧ ddcwk+1 ∧ ...∧ dd {u≤v−ǫ} (r − w1)(dd cu)k ∧ ddcwk+1 ∧ ... ∧ dd {u<v}∪{u=v=−∞} (r − w1)(dd cu)k ∧ ddcwk+1 ∧ ... ∧ dd Letting ǫ ց 0 we obtain {u<v} (v − u)kddcw1 ∧ ... ∧ dd cwn + {u<v} (r − w1)(dd cv)k ∧ ddcwk+1 ∧ ... ∧ dd {u<v}∪{u=v=−∞} (r − w1)(dd u)k ∧ ddcwk+1 ∧ ... ∧ dd References [Åh] P. Åhag, The complex Monge-Ampère operator on bounded hyperconvex domains, Ph. D. Thesis, Ume̊a University, (2002). [Bl1] Z. Blocki, On the definition of the Monge-Ampère operator in C2, Math. Ann., 328 (2004), 415-423. [Bl2] Z. Blocki, Weak solutions to the complex Hessian equation, Ann. Inst. Fourier 55 (2005), 1735-1756. [BT1] E. Bedford and B.A.Taylor, The Dirichlet problem for the complex Monge-Ampère operator. Invent. Math.37 (1976), 1-44. [BT2] E. Bedford and B.A.Taylor, A new capacity for plurisubharmonic functions. Acta Math., 149 (1982), 1-40. [BT3] E. Bedford and B.A.Taylor, Fine topology, Silov boundary, and (ddc)n. J. Funct. Anal. 72 (1987), 225-251. [Ce1] U. Cegrell, Pluricomplex energy. Acta Math., 180 (1998), 187-217. [Ce2] U. Cegrell, The general definition of the complex Monge-Ampère operator. Ann. Inst. Fourier (Grenoble) 54 (2004), 159-179. [Ce3] U. Cegrell, A general Dirichlet problem for the complex Monge-Ampère operator, preprint (2006). [CKZ] U. Cegrell, S. Ko lodziej and A. Zeriahi, Subextention of plurisubharmonic functions with weak singularities. Math. Zeit., 250 (2005), 7-22. [Cz] R. Czyz, Convergence in capacity of the Perron-Bremermann envelope, Michigan Math. J., 53 (2005), 497-509. [CLP] D. Coman, N. Levenberg and E.A. Poletsky,Quasianalyticity and pluripolarity, J. Amer. Math. Soc., 18 (2005), 239-252. [De1] J-P. Demailly, Monge-Ampère operators, Lelong Numbers and Intersection theory, Complex Analysis and Geometry, Univ. Ser. Math., Plenum, New York, 1993, 115-193. [De2] J-P. Demailly, Potential theory in several variables, preprint (1989). [Ko] S. Ko lodziej, The range of the complex Monge-Ampère operator, II, Indiana Univ. Math. J., 44 (1995), 765-782. [H1] P. Hiep, A characterization of bounded plurisubharmonic functions, Ann. Polon. Math., 85 (2004), 233-238. [H2] P. Hiep, The comparison principle and Dirichlet problem in the class Ep(f), p > 0, Ann. Polon. Math., 88 (2006), 247-261. [Le] P .Lelong, Notions capacitaires et fonctions de Green pluricomplexes dans les espaces de Banach. C.R. Acad. Sci. Paris Ser. Imath., 305:71-76, 1987. [Xi1] Y. Xing, Continuity of the complex Monge-Ampère operator. Proc. of Amer. Math. Soc., 124 (1996), 457-467. [Xi2] Y. Xing, Complex Monge-Ampère measures of pluriharmonic functions with bounded values near the boundary. Cand. J. Math., 52, (2000),1085-1100. Department of Mathematics Hanoi University of Education (Dai hoc Su Pham Hanoi). Cau giay, Ha Noi, VietNam E-mail: phhiep−[email protected]

0704.0360

Torsional oscillations of longitudinally inhomogeneous coronal loops

Astronomy & Astrophysics manuscript no. 7246 c© ESO 2019 August 20, 2019 Torsional oscillations of longitudinally inhomogeneous coronal loops T.V. Zaqarashvili1 & K. Murawski2 1 Georgian National Astrophysical Observatory (Abastumani Astrophysical Observatory), Kazbegi Ave. 2a, Tbilisi 0160, Georgia 2 Group of Astrophysics and Gravity Theory, Institute of Physics, UMCS, ul. Radziszewskiego 10, 20-031 Lublin, Poland received / accepted ABSTRACT Aims. We explore the effect of an inhomogeneous mass density field on frequencies and wave profiles of torsional Alfvén oscillations in solar coronal loops. Methods. Dispersion relations for torsional oscillations are derived analytically in limits of weak and strong inhomogeneities. These analytical results are verified by numerical solutions, which are valid for a wide range of inhomogeneity strength. Results. It is shown that the inhomogeneous mass density field leads to the reduction of a wave frequency of torsional oscillations, in comparison to that of estimated from mass density at the loop apex. This frequency reduction results from the decrease of an average Alfvén speed as far as the inhomogeneous loop is denser at its footpoints. The derived dispersion relations and wave profiles are important for potential observations of torsional oscillations which result in periodic variations of spectral line widths. Conclusions. Torsional oscillations offer an additional powerful tool for a development of coronal seismology. Key words. Magnetohydrodynamics (MHD) – Sun: corona – Sun: oscillations 1. Introduction Recent space-based observations revealed a presence of vari- ous kinds of magnetohydrodynamic (MHD) waves and oscilla- tions in the solar corona. These observations as well as mod- eling of MHD waves are important as these waves contribute to the coronal heating problem (Roberts 2000) and they may consist unique tool of a coronal seismology (Edwin & Roberts 1983, Nakariakov & Ofman 2001). Fast kink (Aschwanden et al. 1999, Nakariakov et al. 1999, Wang & Solanki 2004) and sausage (Nakariakov 2003, Pascoe et al. 2007) as well as slow (de Moortel et al. 2002, Wang et al. 2003) magnetosonic oscil- lations were observed to be associated either with or without a solar flare. Analytical studies of these oscillations in coronal loops were carried on over the last few decades, amongst others, by Edwin & Roberts (1982, 1983), Poedts & Boynton (1996), Nakariakov (2003), Van Doorsselaere et al. (2004a,b), Ofman (2005), Verwichte et al. (2006) and Diáz et al. (2006). Coronal loops act as natural wave guides for magnetosonic and torsional Alfvén waves. The later are purely azimuthal os- cillations in cylindrical geometry. In the linear regime, Alfvén oscillations do not lead to mass density perturbations. As a re- sult, contrary to magnetosonic waves, torsional Alfvén waves can be observed only spectroscopically. While propagating from the base of the solar corona along open magnetic field lines, these waves may lead to an increase of a spectral line width with height (Hassler et al. 1990, Banerjee et al. 1998, Doyle et al. 1998). In closed magnetic field structures, such as coronal loops, these waves can be observed indirectly as periodic variations of non-thermal broadening of spectral lines (Zaqarashvili 2003). Alongside magnetosonic waves, torsional oscillations can be used to infer, in the framework of coronal seismology, plasma properties inside oscillating loops. These oscillations are an ideal Send offprint requests to: T. Zaqarashvili e-mail: [email protected] tool of coronal seismology as their phase speed depends alone on plasma quantities within the loop, while wave speeds of magne- tosonic oscillations are influenced by plasma conditions in the ambient medium. Having known mass density within a loop, coronal seismology, that is based on torsional oscillations, en- ables to estimate a magnetic field strength. Torsional oscilla- tions are potentially important in the context of rapid attenua- tion of coronal loop kink oscillations (Aschwanden et al. 1999, Nakariakov et al. 1999). One of a few suggested mechanisms of the attenuation is a resonant absorption of fast magnetosonic kink waves by azimuthal Alfvén waves (Ruderman & Roberts 2002). This process may lead to a formation of torsional oscil- lations in the outer part of a loop. As a result, spotting torsional oscillations after the kink mode was attenuated would serve as an evidence of this attenuation mechanism. A theoretical study of Alfvén oscillations in a coronal loop was carried on recently by Gruszecki et al. (2007) who con- sidered impulsively generated oscillations in two-dimensional straight and curved magnetic field topologies. They found that lateral leakage of Alfvén waves into the ambient corona is neg- ligibly small. However, mass density profiles were adopted ho- mogeneous within the loop, while the real conditions there are much more complex. Despite of significant achievements in a development of re- alistic models there is still much more effort required to develop our knowledge of wave phenomena in coronal loops. A goal of this paper is to study the influence of inhomogeneous mass den- sity fields on spectrum of torsional oscillations. The paper is or- ganized as follows. Analytical solutions for torsional oscillations in a longitudinally inhomogeneous coronal loop are presented in Sect. 2. The numerical results are showed in Sect. 3. Guidelines for potential observations of these oscillations are presented in Sect. 4. This paper is concluded by a discussion and a short sum- mary of the main results in Sect. 5. http://arxiv.org/abs/0704.0360v1 2 T. Zaqarashvili & K. Murawski: Torsional oscillations of a coronal loop 2. Analytical model of torsional oscillations We consider a coronal loop of its inhomogeneous mass density ̺0(z) and length 2L, that is embedded in a uniform magnetic field B = B0ẑ. Small amplitude torsional Alfvén waves in a cylindri- cal coordinate system (r, φ, z), in which plasma profiles depend on a longitudinal coordinate z only, can be described by the fol- lowing linear equations: 4π̺0(z) , (1) , (2) where uφ and bφ are the velocity and magnetic field components of Alfvén waves. These equations can be easily cast into a single wave equa- V2A(z) = 0 , (3) where VA(z) = B0/ 4π̺0(z) is the Alfvén speed. Assuming that uφ ∼ exp(iωt), where ω is a wave frequency, we get the equation uφ = 0 . (4) For a trapped solution uφ must satisfy line-tying boundary con- ditions which are implemented by setting uφ(z = ±L) = 0 . (5) Equation (4) with condition (5) consists the well-known Sturm- Liuville problem which solution depends on the profile of VA(z). We model the coronal loop by a rarefied plasma at the loop apex (at z = 0) and by a compressed plasma at the loop footpoints (z = ±L). Specifically, we adopt ̺0(z) = ̺00 1 + α2 , (6) where ̺00 is the mass density at the loop apex and α 2 is a param- eter which defines a strength of the inhomogeneity. For α2 = 0 the above mass density profile corresponds to a homogeneous loop, while for a larger value of α2 the medium is more inhomo- geneous. Figure 1 illustrates ̺0(z) for α 2 = 50. The mass density is described by Eq. (6) with ̺00 = 10 −12 kg m−3 and L = 25 Mm. Note that plasma is compressed at z = ±L. Substituting Eq. (6) into Eq. (4), we obtain 1 + α2 uφ = 0 , (7) where VA0 = B0/ 4π̺00. With a use of the notation y ≡ uφ, x ≡ z, a ≡ − Eq. (7) can be rewritten in the form of Weber (parabolic cylinder) equation (Abramowitz & Stegun 1964) y = 0 . (9) Fig. 1. Spatial profile of the background mass density, ̺0(z), given by Eq. (6) with α2 = 50. The mass density and length are expressed in units of 10−12 kg m−3 and 1 Mm, respectively. Standard solutions to this equation are called Weber (parabolic cylinder) functions (Abramowitz & Stegun 1964) W(a,±x) = (coshπa)1/4 G1y1(x) ∓ 2G3y2(x) , (10) where , G3 = and y1(x), y2(x) are respectively even and odd solutions to Eq. (9) y1(x) = 1 + a + · · · , y2(x) = x + a + · · · . 2.1. Two limiting solutions Periodic solutions to Eq. (9) can be written analytically in the limiting cases: (a) for a large value of a but a moderate value of x; (b) for a large x but a moderate a. The first (second) case corresponds to α2 ≪ 1 (α2 ≫ 1). 2.1.1. Weakly inhomogeneous plasma We consider first the case of a weakly inhomogeneous mass den- sity field, i.e. α2 ≪ 1. In this case we have a < 0, −a≫ x2, p ≡ −a . (12) We adopt the following expansion (Abramowitz & Stegun 1964): W(a, x) + iW(a,−x) = 2W(a, 0) exp [vr + i(px + π/4 + vi)] , (13) where W(a, 0) = , (14) vr = − (x/2)2 (2p)2 2(x/2)4 (2p)4 + · · · , vi = 2/3(x/2)3 + · · · . (15) T. Zaqarashvili & K. Murawski: Torsional oscillations of a coronal loop 3 As a result of relation −a≫ x2 we have from Eq. (13) W(a, x) = 2W(a, 0) exp cos ζ , (16) W(a,−x) = 2W(a, 0) exp sin ζ , (17) ζ ≡ px + π/4 + . (18) The general solution to Eq. (9) is uφ = c1W(a, x) + c2W(a,−x) , (19) where c1 and c2 are constants. For a homogeneous loop, i.e. α2 = 0, we recognize the well known solution uφ ∼ c1 cos (kz + π/4) + c2 sin (kz + π/4). (20) Here wave number k satisfies the following homogeneous dis- persion relation: . (21) Line-tying boundary conditions of Eq. (5) lead then to discrete values of the wave frequency, viz. 1 + α2/6 , n = 1, 2, 3, . . . . (22) From this dispersion relation we infer that in a comparison to the loop with a homogeneous mass density distribution, ̺00, the weakly inhomogeneous mass density field results in a decrease of a wave frequency. This reduction is a consequence of the fact that the inhomogeneous loop is denser at its footpoints, so the average Alfvén speed is decreased. To show this, we compare the results for the inhomogeneous loop with the homogeneous loop with the same average density, so that both loops contain exactly the same mass (Andries et al. 2005). We introduce a frequency difference ∆ωn = ωn − ω̄n , (23) where ω̄n = V̄A0 = 4π ¯̺0 corresponds to the average mass density ¯̺0 = ̺0(z)dz = ̺00 . (25) Substituting Eq. (25) into Eq. (24), we obtain ω̄n = 1 + α2/3 . (26) From Eqs. (23) and (26) we find that ∆ωn ≤ 0. Here we infer that in comparison to the average mass density case the wave frequency is reduced, but as a result of α2 ≪ 1 the frequency reduction is small. This is in a disagreement with Fermat’s law and with the results of Murawski et al. (2004) who showed that sound waves experience frequency increase in a case of a space- dependent random mass density field. 2.1.2. Strongly inhomogeneous plasma We discuss now a strongly inhomogeneous mass density case, i.e. α2 ≫ 1. This case corresponds to x ≫ |a|. In this limit we get (Abramowitz & Stegun 1964) W(a, x) = 2k/x(s1(a, x) cos(ξ) − s2(a, x) sin(ξ)) , (27) W(a,−x) = 2/kx(s1(a, x) sin(ξ) − s2(a, x) cos(ξ)) , (28) where − a ln x + argΓ(1/2 + ia) , (29) 1 + e2πa − eπa, (30) s1(a, x) ∼ 1 + 1!2x2 2!22x4 − · · · , (31) s2(a, x) ∼ − 1!2x2 2!22x4 + · · · (32) ur + ivr = Γ(r + 1/2 + ia)/Γ(1/2 + ia), r = 2, 4, . . . . (33) The boundary conditions of Eq. (5) lead to the discrete fre- quency spectrum . (34) Here we infer that the strongly inhomogeneous mass den- sity field results in a significant decrease of a wave frequency in comparison to the case of the loop with the constant density, ̺00. This wave frequency decrease is a consequence of the fact that the inhomogeneous loop is denser at its footpoints. Substituting Eq. (34) into Eq. (23) we find that ∆ωn > 0. This wave frequency decrease, in a comparison to the case of an average mass density is now in an agreement with Fermat’s law and with the results of Murawski et al. (2004). 3. Numerical results Numerical simulations are performed for Eqs. (1), (2) with an adaptation of CLAWPACK which is a software package de- signed to compute numerical solutions to hyperbolic partial dif- ferential equations using a wave propagation approach (LeVeque 2002). The simulation region (−L, L) is covered by an uniform grid of 600 numerical cells. We verified by convergence studies that this grid does not introduce much numerical diffusion and as a result it represents well the simulation region. We set reflect- ing boundary conditions at the left and right boundaries of the simulation region. Figure 2 shows a spatial profile of velocity uφ(z) for α 2 = 50, drawn at t = 1000 s (solid line). This spatial profile results from the initial Gaussian pulse that was launched at t = 0 in the center of the simulation region, at z = 0. It is noteworthy that the sine- wave profile of Eq. (20), which is valid for α2 = 0 (dashed line), is distorted by the strong inhomogeneity which takes place for the case of α2 = 50. As a consequence of the inhomogeneity wave period is al- tered. Figure 3 displays wave period P vs. inhomogeneity param- eter α2. Diamonds represent the numerical solutions while the solid lines correspond to the analytical solution to Eqs. (22) (top panel) and (34) (bottom panel). Wave periods were obtained by Fourier analysis of the wave signals that were collected in time at the fixed spatial location, z = 0. It is discernible that the nu- merical data fits quite well to the analytical curves. A growth of 4 T. Zaqarashvili & K. Murawski: Torsional oscillations of a coronal loop Fig. 2. Numerically evaluated velocity profile uφ at t = 1000 s for α2 = 50 (solid line). This profile corresponds to the mode number n = 1. Note that as a result of strong inhomogeneity, uφ departs from the sine-wave which corresponds to α2 = 0. The dashed line corresponds to Eq. (20) with c1 = c2 = 0.5. Fig. 3. Wave period P = ω/2π vs. α2 for the mode number n = 1. Diamonds correspond to the numerical solutions to Eqs. (1), (2). Solid lines are drawn with the use of the analytical solution to Eqs. (22) and (34). The wave period is expressed in seconds. wave period P with α2 results from wave scattering on centers of the inhomogeneity and it can be explained on simple physical grounds. In an inhomogeneous field wave frequency ωn of the torsional oscillations can be estimated from the following for- mula: V̄A0 , (35) where V̄A0 is the averaged Alfvén speed that is expressed by Eq. (24). Using P = 2π/ωn we obtain 4π ¯̺0 . (36) As ¯̺0 grows with α, the growth of P with α results in. 4. Potential observations of torsional oscillations Torsional oscillations of a coronal loop may result in periodic variations of spectral line non-thermal broadening (expressed by a half line width, ∆λB, hereafter HW) (Zaqarashvili 2003). For a homogeneous loop, HW can be expressed as ∆λB = uVA0λ |sin(ωnt)sin(knz)| , (37) where u is an amplitude of oscillations, λ is a wave length of the spectral line and c is the light speed. Periodic variations of spectral line width depend on a height above the solar surface: a strongest variation corresponds to the wave antinode and the place of a lack of line width variation corresponds to the nodes (loop footpoints). Therefore, time series of spectroscopic ob- servations may allow to determine a wave period. Knowing a length of the loop, we may estimate the Alfvén speed, which in turn gives a possibility to infer the magnetic field strength in the corona. We estimate the expected value of line width variations which result from torsional oscillations. For a typical coronal Alfvén speed of ∼ 800 km/s, an amplitude of linear torsional oscillation can be ∼ 40 km/s, which consists 5% of the Alfvén speed. For the ”green” coronal line Fe XIV (5303 Å) from Eq. (37) we obtain ∆λB ≈ 0.7 Å. (38) This value is about twice larger than the original thermal broad- ening of Fe XIV line. As a consequence, torsional oscillations can be detected in time series of the green coronal line spectra. For a weakly inhomogeneous distribution of mass density along a loop, Eq. (22) enables to estimate the Alfvén speed at the loop apex with the help of the observed period of HW vari- ation and a loop length. For a strongly inhomogeneous density profile along a loop, Eq. (34) shows that a wave period of tor- sional oscillations is not just the ratio of the loop length to the Alfvén speed, but it strongly depends on the rate of inhomo- geneity, α2. Therefore, an additional effort is required in order to apply the method of coronal seismology for torsional oscil- lations. A spatial variation of mass density along the loop can be estimated by a direct measurement of spectral line intensity variation along the loop. Then, the estimated variation can be fit- ted to Eq. (6), and hence a value of α2 can be inferred. Eq. (34) provides a value of VA0 at the loop summit. Another possibility is to collect time series of spectroscopic observations at differ- ent positions of the loop. A spatial variation of line width along the loop may be compared to the theoretical plot of uφ (Fig. 2), which enables to estimate α2 and consequently Alfvén speed at the loop apex (with a use of Eqs. (22) or (34)). T. Zaqarashvili & K. Murawski: Torsional oscillations of a coronal loop 5 5. Discussion and summary It is commonly believed that Alfvén waves are generated in the solar interior either by convection (granulation, supergranula- tion) or by any other kinds of plasma flow (differential rotation, solar global oscillations). Due to their incompressible nature, these waves may carry energy from the solar surface to the solar corona and therefore they may significantly contribute to coro- nal heating and solar wind acceleration. In closed magnetic loops the Alfvén waves may set up the standing torsional oscillations, while in opened magnetic structures these waves may propagate up to the solar wind. As a result, observations of Alfvén waves can be of vital importance to the problems of plasma heating and particle acceleration. The Alfvén waves that propagate along open magnetic field lines may lead to a growth of a spectral line width with height (Hassler et al. 1990, Banerjee et al. 1998; Doyle et al. 1998). However, at some altitudes the spectral line width reveals a sud- den fall off (Harrison et al. 2002; O’Shea et al. 2003, 2005). This phenomenon was recently explained by resonant energy transfer into acoustic waves (Zaqarashvili et al. 2006). On the other hand, the photospheric motions may set up tor- sional oscillations in closed magnetic loop systems, which can be observed spectroscopically as periodic variations of spectral line width (Zaqarashvili 2003). As a result, the observation of Alfvén waves can be used as an additional powerful tool of coro- nal seismology; the observed period and loop mean length en- ables to estimate the Alfvén speed within a loop, which in turn makes it possible to infer a mean magnetic field strength. Besides their photospheric origin, torsional Alfvén waves can be generated in the solar corona in a process of resonant ab- sorption of the global oscillations (Ruderman & Roberts 2002, Goossens et al. 2002, Andries et al. 2005, Terradas et al. 2006). These oscillations may excite Alfvén waves in the outer inho- mogeneous part of a loop, leading to attenuation of global oscil- lations and amplification of torsional oscillations. These Alfvén oscillations can be detected as periodic variations of spectral line width. As a consequence, observations of Alfvén waves can be a key for a determination of a damping mechanism of the loop global oscillations. Dynamics of torsional Alfvén waves in a homogeneous loop can be easily solved. However, real coronal loops are longitudi- nally inhomogeneous, which leads to alteration of wave dynam- ics (Arregui et al. 2005, 2007, Van Doorsselaere et al. 2004a,b, Donnelly et al. 2006, Dymova & Ruderman 2006, McEwan et al. 2006). Therefore, the dynamics of Alfvén waves in longitudi- nally inhomogeneous coronal loops must be understood in order to provide analytical basis for potential observations of torsional oscillations. In this paper we discussed by analytical and numerical means evolution of torsional Alfvén waves in an inhomogeneous mass density field. The analytical efforts resulted in dispersion relations which were obtained for a specific choice of an equilib- rium mass density profile. These dispersion relations were writ- ten explicitly for two limiting cases: (a) weekly inhomogeneous and (b) strongly inhomogeneous mass density fields. From these dispersion relations we inferred that the inhomogeneity results in a wave frequency reduction in comparison to that of estimated at the loop summit. This analytical finding is supported by the nu- merical data which reveals that frequency reduction takes place outside the region of validity of the analytical approach. As a result of that we claim that a reduction of wave frequency is ubiquitous for the inhomogeneous mass density field we consid- ered. This reduction is a consequence of wave scattering on in- homogeneity centers and it results from reduction of the average Alfvén speed within a coronal loop. This frequency reduction has important implications as far as wave observations are con- cerned. The analytical formulae can be used for estimation of coronal plasma parameters and therefore torsional Alfvén waves consist an additional powerful tool of coronal seismology. Acknowledgments: The authors express their thanks to the referee, Prof. S. Poedts, for his stimulating comments. The work of T.Z. is supported by the grant of Georgian National Science Foundation GNSF/ST06/4-098. A part of this paper is sup- ported by the ISSI International Programme ”Waves in the Solar Corona”. References Abramowitz, M., & Stegun, I.A. 1964, Handbook of Mathematical Functions (Washington, D.C.: National Bureau of Standards) Andries, J., Goossens, M., Hollweg, J. V., Arregui, I., & Van Doorsselaere, T. 2005, A&A, 430, 1109 Arregui, I., Van Doorsselaere, T., Andries, J., Goossens, M., & Kimpe, D. 2005, A&A, 441, 361 Arregui, I., Andries, J., Van Doorsselaere, T., Goossens, M., & Poedts, S. 2007, A&A, 463, 333 Aschwanden, M.J., Fletcher, L., Schrijver, C.J., & Alexander, D. 1999, ApJ, 520, Banerjee, D., Teriaca, L., Doyle, J., & Wilhelm, K. 1998, A&A, 339, 208 De Moortel I., Ireland J., Walsh R. W. & Hood A. W. 2002, Sol. Phys., 209, 61 Diáz, A., Zaqarashvili, T.V., & Roberts, B. 2006, A&A, 455, 709 Donnelly, G.R., Diáz, A., & Roberts, B. 2006, A&A, 457, 707 Doyle, J., Banerjee, D. & Perez, M. 1998, Sol. Phys., 181, 91 Dymova, M. V., & Ruderman, M. S. 2006, A&A, 457, 1059 Edwin, P.M., & Roberts, B. 1982, Sol. Phys., 76, 239 Edwin, P.M., & Roberts, B. 1983, Sol. Phys., 88, 179 Goossens, M., Andries, J., & Aschwanden, M.J. 2002, A&A, 394, L39 Gruszecki, M., Murawski, K., Solanki, S., & Ofman, L. 2007, A&A, (in press) Harrison, R.A., Hood, A.W., & Pike, C.D. 2002, A&A, 392, 319 Hassler, D.M., Rottman, G.J., Shoub, E.C., & Holzer, T.E. 1990, ApJ, 348, L77 LeVeque, R.J. 2002, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press McEwan, M. P., Donnelly, G. R., Diaz, A. J. & Roberts, B. 2006, A&A, 460, Murawski, K., Nocera, L. and Pelinovsky, E. N. 2004, Waves in Random Media, 14, 109 Nakariakov, V.M., Ofman, L., Deluca, E.E., Roberts, B., & Davila, J.M. 1999, Science, 285, 862 Nakariakov, V.M., & Ofman, L. 2001, A&A, 372, L53 Nakariakov, V.M. 2003, in The Dynamic Sun (Ed. B. Dwivedi), CUP O’Shea, E., Banerjee, D., & Poedts, S. 2003, A&A, 400, 1065 O’Shea, E., Banerjee, D., & Doyle, J.G. 2005, A&A, 436, L35 Ofman, L. 2005, Adv. Space Res., 36, 1772 Pascoe, D.J., Nakariakov, V.M. & Arber, T.D. 2007, A&A, 461, 1149 Poedts, S. & Boynton, G.C. 1996, A&A, 306, 610 Roberts, B. 2000, Sol. Phys., 193, 139 Ruderman, M.S., & Roberts, B. 2002, ApJ, 577, 475 Terradas, J., Oliver, R., & Ballester, J.L. 2006, ApJ, 642, 533 Verwichte, E., Foullon, C., & Nakariakov, V.M. 2006, A&A, 449, 769 Wang, T., Solanki, S.K., Innes, D.E., Curdt, W., & Marsch, E. 2003, A&A, 402, Wang, T.J., & Solanki S.K. 2004, A&A, 421, L33 Zaqarashvili, T.V., 2003, A&A, 399, L15 Zaqarashvili, T.V., Oliver, R., & Ballester, J.L. 2006, A&A, 456, L13 Van Doorsselaere, T., Andries, J., Poedts, S. & Goossens, M., 2004a, ApJ, 606, Van Doorsselaere, T., Debosscher, A., Andries, J. & Poedts, S., 2004b, A&A, 424, 1065 Introduction Analytical model of torsional oscillations Two limiting solutions Weakly inhomogeneous plasma Strongly inhomogeneous plasma Numerical results Potential observations of torsional oscillations Discussion and summary

0704.0361

Pseudo-random Puncturing: A Technique to Lower the Error Floor of Turbo Codes

Pseudo-random Puncturing: A Technique to Lower the Error Floor of Turbo Codes Ioannis Chatzigeorgiou, Miguel R. D. Rodrigues, Ian J. Wassell Digital Technology Group, Computer Laboratory University of Cambridge, United Kingdom Email: {ic231, mrdr3, ijw24}@cam.ac.uk Rolando Carrasco School of EE&C Engineering University of Newcastle, United Kingdom Email: [email protected] Abstract— It has been observed that particular rate-1/2 partially systematic parallel concatenated convolutional codes (PCCCs) can achieve a lower error floor than that of their rate-1/3 parent codes. Nevertheless, good puncturing patterns can only be identified by means of an exhaustive search, whilst convergence towards low bit error probabilities can be problematic when the systematic output of a rate-1/2 partially systematic PCCC is heavily punctured. In this paper, we present and study a family of rate-1/2 partially systematic PCCCs, which we call pseudo-randomly punctured codes. We evaluate their bit error rate performance and we show that they always yield a lower error floor than that of their rate-1/3 parent codes. Furthermore, we compare analytic results to simulations and we demonstrate that their performance converges towards the error floor region, owning to the moderate puncturing of their systematic output. Consequently, we propose pseudo-random puncturing as a means of improving the bandwidth efficiency of a PCCC and simultaneously lowering its error floor. I. INTRODUCTION Although in certain applications, such as satellite communications, link reliability is of essence and low rate codes are used to support it, bandwidth occupancy is more important in wireless communications and hence high rate codes are preferred. A high rate convolutional code can be obtained by periodic elimination, known as puncturing, of particular codeword bits from the output of a parent low rate convolutional encoder. Extensive analyses on punctured convolutional codes have shown that their performance is always inferior to the performance of their low rate parent codes (e.g. see [1], [2]). The performance of punctured parallel concatenated convolutional codes (PCCCs), also known as punctured turbo codes, has also been investigated. Design considerations have been derived by analytical [3]–[5] as well as simulation-based approaches [6]–[8], while upper bounds on the bit error probability (BEP) were evaluated in [5], [9]. Punctured turbo codes are usually classified as systematic, partially systematic or non-systematic depending on whether all, some or none of their systematic bits are transmitted [7]. Recent papers [7]–[9] have demonstrated that partially systematic PCCCs yield lower error floors than systematic PCCCs of the same rate. In [10] we showed that rate-1/2 non-systematic PCCCs can achieve error floors, which are lower even than those of their rate-1/3 parent PCCCs. This interesting outcome is valid when maximum-likelihood (ML) decoding is employed. When suboptimal iterative decoding is used, the absence of received systematic bits causes erroneous decisions, which prohibit the iterative decoder from converging to the error floor. Nevertheless, we demonstrated that rate-1/2 child codes, whose BEP performance converges towards an error floor which is lower than that of their rate-1/3 parent PCCC, can still be found by means of an exhaustive search. During this process, the union bound on the BEP of each rate-1/2 punctured PCCC is computed and compared to the union bound of the rate-1/3 parent PCCC. Note that the union bound coincides with the error floor of the code for high values of Eb/N0 [11]. Punctured PCCCs that achieve a bound lower than that of their rate-1/3 parent PCCC are selected. Computation of the exact union bound on the BEP of a punctured PCCC becomes intensive as the interleaver size increases. In [12] we presented a simple technique to approximate the union bound of a turbo code and we demonstrated that this approximation is very accurate when a large interleaver size is used. We used our technique to identify a family of rate-1/2 partially systematic PCCCs, which we called pseudo-randomly punctured PCCCs (PRP-PCCCs). Although we did not explore their BEP performance in detail, we observed that particular PRP-PCCC configurations could achieve a lower error floor than that of their parent codes. This paper builds upon the work carried out in [10] and [12]. Initially, we provide analytical expressions for the parameters that influence the bit error performance of PCCCs. We then evaluate those parameters and compute the union bound approximations for both rate-1/3 parent PCCCs and rate-1/2 PRP-PCCCs. We demonstrate that the latter always exhibit a lower error floor than the former, when large interleaver sizes are considered. In order to verify our theoretical analysis, we compare analytic results to simulations for specific PCCC configurations. The paper concludes with a summary of the main contributions. II. PERFORMANCE EVALUATION OF PCCCS Turbo codes, in the form of symmetric rate-1/3 PCCCs, consist of two identical rate-1/2 recursive systematic convolutional encoders separated by an interleaver of size N [13]. The information bits are input to the first constituent convolutional encoder, while an interleaved version of the information bits are input to the second convolutional encoder. http://arxiv.org/abs/0704.0361v1 The output of the turbo encoder consists of the systematic bits of the first encoder, which are identical to the information bits, the parity check bits of the first encoder and the parity check bits of the second encoder. The bit error probability Pb of a PCCC employing ML soft decoding, on an additive white Gaussian noise (AWGN) channel, is upper bounded as follows Pb ≤ P b (1) where the union bound P u is defined as P ub = P (w). (2) Here, the sum runs over all possible values of input information weight w, with P (w) being the contribution to the union bound P u of only those codeword sequences which were generated by input sequences of a specific information weight w. An individual contribution P (w) is given by [11], P (w) = Bw,dQ 2R · Eb , (3) where N is the interleaver size, R is the code rate of the turbo encoder and Bw,d denotes the number of codeword sequences having overall output weight d, which were generated by input information sequences of weight w. In [11] it was shown that the union bound on the BEP of a PCCC using a uniform interleaver of size N coincides with the average of the union bounds obtainable from the whole class of deterministic interleavers of size N . For small values of N , the union bound can be very loose compared with the actual performance of turbo codes using specific deterministic interleavers. However, for N≥1000, it has been observed that randomly generated interleavers generally perform better than deterministic interleaver designs [15]. Consequently, the union bound provides a good indication of the actual bit error rate performance of a PCCC operating in the error floor region, when long interleavers are considered. Derivation of all coefficients Bw,d becomes a computationally intensive process as the interleaver size increases, especially when punctured PCCCs are considered [12]. However, the union bound can be approximated as follows P ub ≈ P (w=2), (4) when long interleavers are used. This approximation is based on a number of observations: 1) Codeword sequences, which were generated by input sequences having the minimum possible information weight, become the main contributors to the bit error rate performance, as the size N of the interleaver increases [12], [16]. 2) Owning to the structure of the constituent encoders, the minimum information weight of an input sequence is always equal to two [16]. Therefore, P (w=2) is the dominant contribution to the union bound over a broad range of bit error probabilities [12], [16] and can be used to predict the error floor of turbo codes. Throughout this paper, we use the union bound approximation as the basis for the analytic performance comparison of turbo codes. In particular, if P and P ′ are two PCCCs using long interleavers of identical size, we say that P yields a lower error floor than that of P ′ when their bound approximations, PP(2) and PP (2) respectively, satisfy PP(2) < PP (2). (5) The above condition can be expanded using (3) as follows 2RP Eb A. (6) It was demonstrated in [16] that the free effective distance, df, which conveys the minimum weight of a codeword sequence for a weight-2 input information sequence, has a major impact on the performance of a turbo code. Consequently, if dPf and dP f denote the free effective distances of P and P respectively, condition (6) collapses to 2RP Eb A, (7) which only considers the first non-zero, that is the most significant, term of each sum. Function Q(ξ) is a monotonically decreasing function of ξ, where ξ is a real number. Therefore, if ξ1 and ξ2 are real numbers, with ξ1 > ξ2, we deduce that Q(ξ1) < Q(ξ2), and vice versa, i.e, Q(ξ1) < Q(ξ2) ⇔ ξ1 > ξ2. (8) Consequently, inequality (7) reduces to RPdPf > R f , (9) BP2,df ≤ B . (10) When the code rates are equal, the free effective distance of turbo codes plays a role similar to that of the free distance of convolutional codes, since the performance criterion (9) is simplified to dPf > d f . (11) Expressions (9) and (10) will be the basis for the comparison of the BEP performance in the error floor region of two PCCCs. III. DETERMINATION OF PARAMETERS THAT INFLUENCE THE PERFORMANCE OF TURBO CODES We will now determine the various parameters that affect performance for two classes of turbo codes: conventional rate-1/3 PCCCs and pseudo-randomly punctured rate-1/2 PCCCs. The turbo codes considered throughout this paper are symmetric, i.e., the two constituent encoders are identical. A. Rate-1/3 PCCCs Criteria (9) and (10) require knowledge of the free effective distance df and the coefficient B2,df of each PCCC. In the remainder of the paper, we use the abbreviation “Par” to denote a rate-1/3 parent PCCC. Its free effective distance dParf can be expressed as the sum of the minimum weight dmin of the codeword sequence generated by the first constituent encoder, and the minimum weight zmin of the parity check sequence generated by the second constituent encoder, when a sequence of information weight w=2 in input to the PCCC dParf = dmin + zmin. (12) Taking into account that the turbo codes are symmetric and the weight umin of the systematic output sequence is always 2 since w=2, we can write dParf = (umin + zmin) + zmin = 2 + 2zmin. (13) The number BPar of codeword sequences, generated by a turbo encoder using a uniform interleaver of size N , can be associated with the number B2,dmin of codeword sequences having weight dmin, generated by the first constituent encoder, and the number B2,zmin of parity check sequences having weight zmin, generated by the second constituent encoder, if we elaborate on the expressions described in [11]. In particular, we obtain B2,dmin · B2,zmin ) , (14) where B2,dmin and B2,zmin return the same value, since they both consider the same trellis paths. Note that the first index in the above notations refers to the input information weight, which is two. It was shown in [16] that good rate-1/3 PCCCs are obtained when their feedback generator polynomial GR is chosen to be primitive, whilst their feedforward generator polynomial GF is different than GR. The period L of a primitive polynomial is given by [17] L = 2ν − 1, (15) where ν is the order of the polynomial, or equivalently, the memory size of each constituent code. We demonstrated in [12] that when a primitive feedback generator polynomial is used, the minimum weight zmin and the coefficient B2,2,zmin can be expressed as zmin = 2 ν−1 + 2, B2,dmin = B2,zmin = N − L, respectively. Consequently, expression (13) assumes the form dParf = 6 + 2 ν , (17) whilst, if we combine (14) and (16), the coefficient BPar be expressed as a function of the intlerleaver size N and the period L, as follows 2(N − L)2 N(N − 1) . (18) In the special case when the size N of the interleaver is an integer multiple of the period L of the feedback generator polynomial, i.e., N=µL, we can rewrite (18) as 2L(µ− 1)2 µ(µL− 1) . (19) B. Rate-1/2 Pseudo-randomly Punctured PCCCs A high rate PCCC can be obtained by periodic elimination of specific codeword bits from the output of a rate-1/3 parent PCCC. A puncturing pattern P can be represented by a 3×M matrix as follows: p1,1 p1,2 . . . p1,M p2,1 p2,2 . . . p2,M p3,1 p3,2 . . . p3,M  , (20) where M is the puncturing period and pi,m ∈ {0, 1}, with i=1, 2, 3 and m=1, . . . ,M . For pi,m=0 the corresponding output bit is punctured, otherwise it is transmitted. The first and second rows of the pattern are used to puncture the systematic and parity check outputs, respectively, of the first constituent encoder. The third row determines which parity check bits from the output of the second constituent encoder will be punctured. Pseudo-random puncturing has been described in [12], in detail. It is applied to rate-1/3 PCCCs, which use primitive feedback generator polynomials, hence the polynomial period L is also given by (15). The puncturing pattern can be constructed once the parity check sequence y = (y0, y1, . . . , yL) for an input sequence x = (1, 0, . . . , 0) of length L+1, has been obtained at the output of the first constituent encoder. As long as a trail of zeros follows the first non-zero input bit, the component encoder behaves like a pseudo-random generator, hence the parity check bits from y1 to yL form a pseudo-random sequence. We set the elements of the second row of the puncturing pattern to be equal to the bits of this pseudo-random sequence, but circularly shifted rightwards by one, i.e., p2,m+1 = ym for m=1, . . . , L. Note that in pseudo-random puncturing, the puncturing period M is equal to the period L of the feedback polynomial, i.e., M=L. The first row of the pattern is set to be the complement of the second row, thus p1,m=1− p2,m. In order to achieve a code rate of 1/2, we do not puncture the parity check output of the second constituent encoder, hence all the elements of the third row are set to one, i.e., p3,m=1. As an example, let us consider a rate-1/3 PCCC with generator polynomials (GF , GR)=(5, 7)8 in octal form. The memory size of each constituent encoder is ν = 2, thus the period of GR is found to be L = 2 2−1 = 3. Consequently, we set the input sequence to (1, 0, 0, 0) and we obtain the parity check sequence (1, 1, 1, 0) at the output of the first constituent encoder. The block of the last L=3 parity check bits, i.e., (1, 1, 0), forms a pseudo-random sequence. If we circularly shift the bits of this pseudo-random sequence to the right by one and map them to the elements of the second row of the puncturing pattern, we obtain [0 1 1]. Eventually the puncturing pattern, based on which the rate-1/2 PRP-PCCC is generated from the rate-1/3 parent PCCC, assumes the form 1 0 0 0 1 1 1 1 1  . (21) We emphasize that the puncturing pattern depends on the generator polynomials of the rate-1/3 parent PCCC, hence different polynomials yield different puncturing patterns. Furthermore, a rate-1/2 PRP-PCCC can be obtained only if the parent PCCC uses primitive feedback generator polynomials. We have previously determined [12] the minimum weight d′min of the codeword sequence generated by the first constituent encoder, when a sequence of information weight w = 2 in input to the rate-1/2 PRP-PCCC. In particular, we found that d′min = 2 ν−2 + 2. (22) The parity check sequence generated by the second constituent encoder is not punctured, thus its minimum weight is also given by (16). Therefore, we can compute the free effective distance dPRPf of a rate-1/2 PRP-PCCC as follows dPRPf = d min + zmin = (2ν−2 + 2) + 2ν−1 + 2 = 4 + 3(2ν−2). Every time a particular column m of the puncturing pattern is active during the N time steps of the coding process, codeword sequences having minimum weight d′min are generated. Their exact number, Am, can be computed using the expressions in [12]. In particular, we find that for M =L the number of minimum-weight codeword sequences Am, generated when column m is active, is given by ⌊N/M⌋ − 1, if (N mod M)<m ⌊N/M⌋ , otherwise, where (ξ1 mod ξ2) denotes the remainder of division of ξ1 by ξ2, and ⌊ξ⌋ denotes the integer part of ξ. In order to facilitate our analysis, we assume that the interleaver size N is an integer multiple of the puncturing period M , i.e., N=µM , where µ is a positive integer. Hence, (24) collapses to Am = µ− 1, (25) since (N mod M) is always zero and m>0. It has been demonstrated in [12] that minimum-weight codeword sequences can be obtained only when the active column m is in the range 2 ≤ m ≤ M ; every time one of these M−1 columns of the puncturing pattern is active, Am minimum-weight codeword sequences are generated. Consequently, the total number of codeword sequences having weight d′min assumes the value B2,d′ = (M − 1)Am, (26) or, equivalently B2,d′min = (L− 1)(µ− 1), (27) where M has been replaced by L, since they are equal quantities and they can be used interchangeably. Similarly to the second constituent encoder of the rate-1/3 parent PCCC, the second constituent encoder of the rate-1/2 PRP-PCCC also generates a total of B2,zmin sequences having weight zmin, since its parity check output is not punctured. Consequently, the coefficient BPRP or a rate-1/2 PRP-PCCC can be expressed as BPRP2,df = B2,d′min · B2,zmin [(L− 1)(µ− 1)] · (N − L) 2(L− 1)(µ− 1)2 µ(µL− 1) invoking (14), which can be used when PCCCs employing uniform interleavers of size N are considered. IV. PERFORMANCE COMPARISON OF ANALYTIC TO SIMULATION RESULTS Having evaluated the parameters that influence the performance of the PCCCs under investigation, we are now in the position to explore whether a rate-1/2 PRP-PCCC exhibits a lower bound approximation than that of its rate-1/3 parent PCCC. We observe that dPRPf can be expressed in terms of dParf , if we subtract (17) from (23) dPRPf = d f − (2 + 2 ν−2). (29) Coefficient BPRP can also be represented in terms of BPar we divide (28) by (19) BPRP2,df = BPar2,df . (30) According to (9) and (10), if both conditions dPRPf > dParf (31) < BPar are satisfied, a rate-1/2 PRP-PCCC yields a lower bound approximation than that of its rate-1/3 parent code. We deduce from (30) that BPRP is always less than BPar , thus the second condition holds true. The first condition assumes the following form, if we substitute dPRPf with its equivalent, based on (29), dParf > 6 + 3(2 ν−2). (33) Nevertheless, we have shown in (17) that the free effective distance of the parent PCCC is given by dParf =6+ 2 ν , which can be rewritten as dParf = 6 + 4(2 ν−2). Therefore, dParf is always greater than 6 + 3(2ν−2), and hence, both conditions are satisfied. The outcome of this investigation reveals that rate-1/2 PRP-PCCCs using long interleavers are always expected to 0 1 2 3 4 5 6 (dB) Simulation, Rate−1/3 PCCC (ν=2) Bound Approx., Rate−1/3 PCCC (ν=2) Simulation, Rate−1/2 PRP−PCCC (ν=2) Bound Approx., Rate−1/2 PRP−PCCC (ν=2) Simulation, Rate−1/3 PCCC (ν=3) Bound Approx., Rate−1/3 PCCC (ν=3) Simulation, Rate−1/2 PRP−PCCC (ν=3) Bound Approx., Rate−1/2 PRP−PCCC (ν=3) Fig. 1. Comparison of bound approximations to simulation results. The exact log-MAP algorithm is applied over 8 iterations and an interleaver size of 1, 000 bits is used. yield a lower bound approximation, or equivalently a lower error floor, than that of their rate-1/3 parent codes. Fig.1 compares bound approximations to simulation results for rate-1/3 parent PCCCs and rate-1/2 PRP-PCCCs of memory size ν =2 and ν =3, over the AWGN channel. For ν=2, the generator polynomials of the PCCCs are taken to be (GF , GR)=(5, 7)8, whilst for ν=3, the PCCCs are described by (GF , GR) = (17, 15)8. The component decoders employ the conventional exact log-MAP algorithm [18]. A moderate interleaver size of 1, 000 bits has been chosen, so as to allow the bit error rate performance of the PCCCs to approach the corresponding bound approximations at BEPs in the region of 10−6 to 10−7. As expected, Fig.1 confirms that for high values of Eb/N0, the BEP of each rate-1/2 PRP-PCCC is indeed lower than that of the corresponding rate-1/3 parent code, whilst after 8 iterations the performance curves of all turbo codes approach the respective bound approximation curves. V. CONCLUSION In previous work [9], [10], [12] we introduced techniques to evaluate the performance of punctured PCCCs and we observed that, in some cases, the error floor could be lowered by reducing the rate of a PCCC from 1/3 to 1/2. Nevertheless, good puncturing patterns were identified by means of an exhaustive search, whilst convergence towards low bit error probabilities of those rate-1/2 PCCCs whose systematic output was heavily punctured, had to be investigated. In this paper, we established that rate-1/2 pseudo-randomly punctured PCCCs, which form a subset of rate-1/2 partially systematic PCCCs, not only approach the error floor region for an increasing number of iterations but always yield a lower error floor than that of their rate-1/3 parent codes. Consequently, pseudo-random puncturing can be used to reduce the rate of a PCCC from 1/3 to 1/2 and at the same time achieve a coding gain at low bit error probabilities. REFERENCES [1] J. Hagenauer, “Rate compatible punctured convolutional codes and their applications,” IEEE Trans. Commun., vol. 36, pp. 389–400, Apr. 1988. [2] D. Haccoun and G. Bégin, “High-rate punctured convolutional codes for Viterbi and sequential decoding,” IEEE Trans. Commun., vol. 37, pp. 1113–1125, Nov. 1989. [3] Ö. Açikel and W. E. Ryan, “Punctured turbo-codes for BPSK/QPSK channels,” IEEE Trans. Commun., vol. 47, pp. 1315–1323, Sept. 1999. [4] F. Babich, G. Montorsi, and F. Vatta, “Design of rate-compatible punctured turbo (RCPT) codes,” in Proc. Int. Conf. Comm. (ICC’02), New York, USA, Apr. 2002, pp. 1701–1705. [5] M. A. Kousa and A. H. Mugaibel, “Puncturing effects on turbo codes,” Proc. IEE Comm., vol. 149, pp. 132–138, June 2002. [6] M. Fan, S. C. Kwatra, and K. Junghwan, “Analysis of puncturing pattern for high rate turbo codes,” in Proc. Military Comm. Conf. (MILCOM’99), New Jersey, USA, Oct. 1999, pp. 547–550. [7] I. Land and P. Hoeher, “Partially systematic rate 1/2 turbo codes,” in Proc. Int. Symp. Turbo Codes, Brest, France, Sept. 2000, pp. 287–290. [8] Z. Blazek, V. K. Bhargava, and T. A. Gulliver, “Some results on partially systematic turbo codes,” in Proc. Vehicular Tech. Conf. (VTC-Fall’02), Vancouver, Canada, Sept. 2002, pp. 981–984. [9] I. Chatzigeorgiou, M. R. D. Rodrigues, I. J. Wassell, and R. Carrasco, “A novel technique for the evaluation of the transfer function of punctured turbo codes,” in Proc. IEEE Intl. Conf. Comm. (ICC’06), Istanbul, Turkey, July 2006. [10] ——, “Can punctured rate-1/2 turbo codes achieve a lower error floor than their rate-1/3 parent codes?” in Proc. IEEE Information Theory Workshop (ITW’06), Chengdu, China, Oct. 2006. [11] S. Benedetto and G. Montorsi, “Unveiling turbo codes: Some results on parallel concatenated coding schemes,” IEEE Trans. Inform. Theory, vol. 42, pp. 409–429, Mar. 1996. [12] I. Chatzigeorgiou, M. R. D. Rodrigues, I. J. Wassell, and R. Carrasco, “A union bound approximation for rapid performance evaluation of punctured turbo codes,” in Proc. Conference on Information Sciences and Systems (CISS’07), Baltimore, USA, Mar. 2007. [13] C. Berrou and A. Glavieux, “Near optimum error correcting coding and decoding: Turbo codes,” IEEE Trans. Commun., vol. 44, pp. 1261–1271, Oct. 1996. [14] W. E. Ryan, “Concatenated convolutional codes and iterative decoding,” in Wiley Encyclopedia on Telecommunications, J. G. Proakis, Ed. Hoboken, New Jersey: Wiley-Interscience, 2003, pp. 556–570. [15] E. K. Hall and S. G. Wilson, “Design and analysis of turbo codes on rayleigh fading channels,” IEEE J. Select. Areas Commun., vol. 16, pp. 160–174, Feb. 1998. [16] S. Benedetto and G. Montorsi, “Design of parallel concatenated convolutional codes,” IEEE Trans. Commun., vol. 44, pp. 591–600, May 1996. [17] F. J. MacWilliams and N. J. A. Sloane, “Pseudo-random sequences and arrays,” Proc. IEEE, vol. 64, pp. 1715–1729, Dec. 1976. [18] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimising symbol error rate,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 284–287, Mar. 1974. Introduction Performance Evaluation of PCCCs Determination of Parameters that Influence the Performance of Turbo Codes Rate-1/3 PCCCs Rate-1/2 Pseudo-randomly Punctured PCCCs Performance Comparison of Analytic to Simulation Results Conclusion References

0704.0362

The Arctic Circle Revisited

arXiv:0704.0362v1 [math-ph] 3 Apr 2007 The Arctic Circle Revisited F. Colomo and A.G. Pronko Abstract. The problem of limit shapes in the six-vertex model with domain wall boundary conditions is addressed by considering a specially tailored bulk correlation function, the emptiness formation probability. A closed expression of this correlation function is given, both in terms of certain determinant and multiple integral, which allows for a systematic treatment of the limit shapes of the model for full range of values of vertex weights. Specifically, we show that for vertex weights corresponding to the free-fermion line on the phase diagram, the emptiness formation probability is related to a one-matrix model with a triple logarithmic singularity, or Triple Penner model. The saddle-point analysis of this model leads to the Arctic Circle Theorem, and its generalization to the Arctic Ellipses, known previously from domino tilings. 1. Introduction The Arctic Circle has first appeared in the study of domino tilings of large Aztec diamonds [EKLP, JPS]. The name originates from the fact that in most configurations the dominoes are ‘frozen’ outside the circle inscribed into the dia- mond, while the interior of the circle is a disordered, or ‘temperate’, zone. Further investigations of the domino tilings of Aztec diamonds, such as details of statistics near the circle, can be found in [CEP, J1, J2]. Here we mention that the Arctic Circle is a particular example of a limit shape in dimer models, in the sense that it describes the shape of a spatial phase separation of order and disorder. Apart from domino tilings, many more examples have been discussed recently, see, among others, papers [CKP, CLP, KO, KOS, OR]. As long as only dimer models are considered, this amounts to restrict to dis- crete free-fermionic models, although with nontrivial boundary conditions. Indeed, many of them can be viewed as a six-vertex model at its Free Fermion point (the correspondence being however usually not bijective), with suitably chosen fixed boundary conditions. In particular, this is the case of domino tilings of Aztec dia- monds [EKLP], and the corresponding boundary conditions of the six-vertex model are the so-called Domain Wall Boundary Conditions (DWBC). Hence the problem of limit shapes extends to the six-vertex model with generic weights, and with fixed boundary conditions, among which the case of DWBC is the most interesting. Historically, the six-vertex model with DWBC was first considered in paper [K] within the framework of Quantum Inverse Scattering Method [KBI] to prove the Gaudin hypothesis for norms of Bethe states. The model was subsequently solved 2000 Mathematics Subject Classification. 15A52, 82B05, 82B20, 82B23. http://arxiv.org/abs/0704.0362v1 2 F. COLOMO AND A.G. PRONKO in paper [I] where a determinant formula for the partition function was given; see also [ICK] for a detailed exposition. Quite independently, the model was later found, under certain restrictions on the vertex weights, to be deeply related with enumerations of alternating sign matrices (see, e.g., [Br] for a review) and, as already mentioned, to domino tilings of Aztec diamonds [EKLP]. Concerning the problem of limit shapes for the six-vertex model with DWBC, as far as the Free Fermion point is considered, the relation with domino tilings provided apparently an indirect proof of the corresponding Arctic Circle. The non- bijective nature of the correspondence between the two models asked for more direct results, purposely for the free-fermion six-vertex model, see [Zi1, FS, KP]. Out of the Free Fermion point, however, only very few analytical results are available, such as exact expressions for boundary one-point [BPZ] and two-point [FP, CP1] correlation functions. The present knowledge on the subject is based mainly on numerics [E, SZ, AR]; some steps towards finding the limit shapes of the model have been done recently in [PR]. In the present note we propose a rather direct strategy to address the problem: after briefly reviewing the six-vertex model with DWBC, we define a bulk corre- lation function, the Emptiness Formation Probability (EFP), which discriminates the ordered and disordered phase regions. We give for this correlation function two equivalent representations, in terms of a determinant and of a multiple integral. The core derivation of EFP is heavily based on the Quantum Inverse Scattering Method [KBI], along the lines of papers [BPZ, CP1]; it is out of the scope of the present paper, corresponding details being given in a separate publication [CP4]. Here our aim is to demonstrate how the limit shapes for the considered model can be extracted from EFP in a suitable scaling limit, by making use of ideas and techniques of Random Matrix Models. To be more specific, and to establish a contact with previous results, we spe- cialize here our further discussion to the case of free-fermion six-vertex model. We show that the asymptotic analysis of multiple integral formula for EFP in the scal- ing limit reduces to a saddle-point problem for a one-matrix model with a triple logarithmic singularity, or triple Penner model. We argue that the limit shape cor- responds to condensation of all saddle-point solutions to a single point. This allows us to recover the known Arctic Circle and Ellipses. As a comment to our approach, it is to be stressed that it is directly tailored on the six-vertex model, rather than domino tilings. For this reason it is not restricted to the free-fermion models, even if, of course, further significant efforts might be necessary, essentially from the point of view of Random Matrix Model reformu- lation, for application to more general situations. On the basis of our previous results in [CP2], however, the application of the method to the particular case of the so-called Ice Point of the model should be straightforward. This would provide the limit shape of alternating sign matrices. 2. The model 2.1. The six-vertex model. The six-vertex model (for reviews, see [LW, Ba]) is formulated on a square lattice with arrows lying on edges, and obeying the so-called ‘ice-rule’, namely, the only admitted configurations are such that there are always two arrows pointing away from, and two arrows pointing into, each lattice vertex. An equivalent and graphically simpler description of the configurations of THE ARCTIC CIRCLE REVISITED 3 w1 w2 w3 w4 w5 w6 Figure 1. The six allowed types of vertices in terms of arrows and lines, and their Boltzmann weights. Figure 2. A possible configuration of the six-vertex model with DWBC at N = 4, in terms of arrows and lines. the model can be given in terms of lines flowing through the vertices: for each arrow pointing downward or to the left, draw a thick line on the corresponding edge. This line picture implements the ‘ice-rule’ in an automated way. The six possible vertex states and the Boltzmann weights w1, w2, . . . , w6 assigned to each vertex according to its state are shown in Figure 1. 2.2. Domain Wall Boundary Conditions. The Domain Wall Boundary Conditions (DWBC) are imposed on the N×N square lattice by fixing the direction of all arrows on the boundaries in a specific way. Namely, the vertical arrows on the top and bottom of the lattice point inward, while the horizontal arrows on the left and right sides point outward. Equivalently, a generic configuration of the model with DWBC can be depicted by N lines flowing from the upper boundary to the left one. A possible state of the model both in terms of arrows and of lines is shown in Figure 2. 2.3. Partition function. The partition function is defined, as usual, as a sum over all possible arrow configurations, compatible with the imposed DWBC, each configuration being assigned its Boltzmann weight, given as the product of all the corresponding vertex weights, arrow configurations with DWBC wn11 w 2 . . . w Here n1, n2, . . . , n6 denote the numbers of vertices with weights w1, w2, . . . , w6, respectively, in each arrow configuration (n1 + n2 + · · ·+ n6 = N 2.4. Anisotropy parameter and phases of the model. The six-vertex model with DWBC can be considered, with no loss of generality, with its weights invariant under the simultaneous reversal of all arrows, w1 = w2 =: a , w3 = w4 =: b , w5 = w6 =: c . 4 F. COLOMO AND A.G. PRONKO Under different choices of Boltzmann weights the six-vertex model exhibits different behaviours, according to the value of the parameter ∆, defined as a2 + b2 − c2 It is well known that there are three physical regions or phases for the six-vertex model: the ferroelectric phase, ∆ > 1; the anti-ferroelectric phase, ∆ < −1; the disordered phase, −1 < ∆ < 1. Here we restrict ourselves to the disordered phase, where the Boltzmann weights are conveniently parameterized as a = sin(λ+ η) , b = sin(λ− η) , c = sin 2η . (2.1) With this choice one has ∆ = cos 2η. The parameter λ is the so-called spectral parameter and η is the crossing parameter. The physical requirement of positive Boltzmann weights, in the disordered regime, restricts the values of the crossing and spectral parameters to 0 < η < π/2 and η < λ < π − η. The special case η = π/4 (or ∆ = 0) is related to free fermions on a lattice, and there is a well-known correspondence with dimers and domino tilings. In particular, at λ = π/2, the ∆ = 0 six-vertex model with DWBC is related to the domino tilings of Aztec diamond. For arbitrary λ ∈ [π/4, 3π/4], we shall refer to the ∆ = 0 case as the Free Fermion line. The case η = π/6 (i.e. ∆ = 1/2) and λ = π/2, where all weights are equal, a = b = c, is known as the Ice Point; all configurations are given the same weight. In this case there is a one to one correspondence between configurations of the model with DWBC and N ×N alternating sign matrices. 2.5. Phase separation and limit shapes. The six-vertex model exhibits spatial separation of phases for a wide choice of fixed boundary conditions, and, in particular, in the case of DWBC. Roughly speaking, the effect is related to the fact that ordered configurations on the boundary can induce, through the ice-rule, a macroscopic order inside the lattice. The notion of phase separation acquires a precise meaning in the scaling limit, that is the thermodynamic/continuum limit, performed by sending the number of sites N to infinity and the lattice spacing to zero, while keeping the total size of the lattice fixed, e.g., to 1. On a finite lattice, several macroscopic regions may appear, which in the scaling limit are expected to be sharply separated by some curves, the so-called Arctic curves. For the six-vertex model with DWBC the shape of the Artic curve, or limit shape, has been found rigorously only on the Free Fermion line, and for the closely related domino tilings of Aztec diamond [JPS, CEP, Zi1, FS, KP]. For generic values of weights the limit shapes are not known, but the whole picture is strongly supported both numerically [E, SZ, AR] and analytically [KZ, Zi2, BF, PR]. 3. Emptiness Formation Probability 3.1. Definition. We shall use the following coordinates on the lattice: r = 1, . . . , N labels the vertical lines from right to left; s = 1, . . . , N labels the horizontal lines from top to bottom. We may now introduce the correlation function FN (r, s), measuring the probability for the first s horizontal edges between the r-th and r+1-th line to be all ‘full’ (i.e. thick in the line picture, or with a left arrow in the THE ARCTIC CIRCLE REVISITED 5 Figure 3. Emptiness Formation Probability. The sum in (3.1) is performed over all configurations compatible with the drawn ar- rows. standard picture of the six-vertex model): FN (r, s) = ‘constrained’ arrow configurations with DWBC wn11 w 2 . . . w 6 . (3.1) Here the sum is performed over all arrow configurations on the N × N lattice, subjected to the restriction of DWBC, and to the condition that all arrows on the first s edges between the r-th and r + 1-th line should point left, see Figure 3. Although this correlation function may appear rather sophisticated, it is com- putable in some closed form by means of the Quantum Inverse Scattering Method, on which DWBC are indeed tailored. It is the natural adaptation of the Empti- ness Formation Probability of quantum spin chains to the present model. For this reason, and to link to the common practice in the quantum integrable models com- munity, even if FN (r, s) actually describes ‘fullness’ formation probability, we shall call it Emptiness Formation Probability (EFP). 3.2. Qualitative discussion of FN (r, s). Let us restrict ourselves to the dis- ordered regime, −1 < ∆ < 1, for definiteness. From previous analytical and nu- merical work, in the large N limit the emergence of a limit shape, in the form of a continuous closed curve touching once each of the four sides of the lattice, is ex- pected. It follows that five regions emerge in the lattice: a central region, enclosed by the curve, and four corner regions, lying outside the closed curve and delimited by the sides of the lattice. The central region is disordered, while the four corners are frozen, with mainly vertices of type 1, 3, 2, 4 (see Figure 1) appearing in the top-left, top-right, bottom-right and bottom-left corner, respectively. By construction, EFP is expected to be almost one in frozen regions of type 1, or 3, bordering the top side of the lattice, and to be rather small otherwise. DWBC exclude a region of type 3 to emerge in the upper part of the lattice. Hence FN (r, s) describes, at a given value of r, as s increases, a transition from a frozen region of vertices of type 1, where FN (r, s) ∼ 1, to a generic region where FN (r, s) ∼ 0. It follows that FN (r, s) can describe only the upper left portion of the closed curve, between its top and left contact points. Nevertheless, it should be mentioned that the full curve can be built from the knowledge of its top left portion, just 6 F. COLOMO AND A.G. PRONKO exploiting the crossing symmetry of the six-vertex model. Hence EFP, FN (r, s), is well suited to describe limit shapes. 3.3. Some notations. For a given choice of parameters λ, η we define sin 2η sin(λ+ η) sin(λ− η) and the integration measure on the real line µ(x) := ex(λ−π/2) sinh(ηx) sinh(πx/2) related to ϕ as follows: µ(x) dx . Let us introduce the complete set of monic orthogonal polynomial {Pn(x)}n=0,1,... associated to the integration measure µ(x), with the orthogonality relation Pn(x)Pm(x)µ(x) dx = hnδnm . The square norms hn are completely determined by the measure µ(x), and may be expressed, in principle, in terms of its moments. In the following we shall be interested in the complete set of orthogonal polynomials {Kn(x)}n=0,1,... defined as Kn(x) = n!ϕ n+1 1 Pn(x) . We moreover define ω(ǫ) := sin(ǫ) sin(ǫ− 2η) , ω̃(ǫ) := sin(ǫ) sin(ǫ + 2η) Note that the following relation holds a2 ω̃ − 2∆ab ω̃ω + b2 ω = 0 , (3.2) allowing to express ω̃ in terms of ω. 3.4. Determinant representation. For EFP in the six-vertex model with DWBC, the following representation holds: FN (r, s) = (−1) s det 1≤j,k≤s KN−k(∂ǫj ) [ω(ǫj)] [ω(ǫj)− 1] 1≤j<k≤s [ω̃(ǫj)− 1] [ω(ǫk)− 1] ω̃(ǫj)ω(ǫk)− 1 ǫ1=0,...,ǫs=0 . (3.3) This representation has been obtained in the framework of the Quantum Inverse Scattering Method [KBI], along the lines of analogous derivations worked out for one-point and two-point boundary correlation functions of the model [BPZ, CP1]. The details of the derivation can be found in [CP4]. THE ARCTIC CIRCLE REVISITED 7 3.5. The boundary correlation function. If we consider expression (3.3) when s = 1, we recover the boundary polarization, introduced and computed in [BPZ]. It is convenient to consider the closely related boundary correlation function HN (r) := FN (r, 1)− FN (r − 1, 1) . As shown in [BPZ, CP1], the following representation holds: HN (r) = KN−1(∂ǫ) [ω(ǫ)]N−r [ω(ǫ)− 1]N−1 We define the corresponding generating function hN (z) := HN (r) z r−1 . (3.4) Noticing that ω(ǫ) → 0 as ǫ → 0, it can be shown that, given any arbitrary function f(z) regular in a neighbourhood of the origin, the following inverse representation holds KN−1(∂ǫ)f(ω(ǫ)) (z − 1)N−1 hN(z)f(z) dz . (3.5) Here C0 is a closed counterclockwise contour in the complex plane, enclosing the origin, and no other singularity of the integrand. 3.6. Multiple integral representation. Plugging (3.5) into representation (3.3), we readily obtain the following multiple integral representation for EFP: FN (r, s) = · · · dsω det 1≤j,k≤s hN−k+1(ωj) ωj − 1 ωN−r−1j (ωj − 1)N 1≤j<k≤s (ω̃j − 1)(ωk − 1) ω̃jωk − 1 . (3.6) Here ω̃j ’s should be expressed in terms of ωj ’s through (3.2). Indeed, due to (3.5), relation (3.2) for functions ω(ǫ), ω̃(ǫ), translates directly into the same relation between ωj and ω̃j , j = 1, . . . , s. Representation (3.6), and all results in this Section hold for any choice of param- eters λ and η within the disordered regime. Moreover, by analytical continuation in parameters λ and η, these results can be easily extended to all other regimes. The determinant in expression (3.6) is a particular representation of the parti- tion function of the six-vertex model with DWBC, when the homogeneous limit is performed only on a subset of the spectral parameters [CP3]. The structure of the previous multiple integral representation therefore closely recalls analogous ones for the Heisenberg XXZ quantum spin chain correlation functions [JM, KMT]. For generic values of λ and η, the orthogonal polynomialsKn(x), or the generat- ing function hN (z), are known only in terms of rather implicit representations. For- tunately, there are three notable exceptions [CP2]: the Free Fermion line (η = π/4, −π/4 < λ < π/4, ∆ = 0), the Ice Point (η = π/6, λ = π/2, ∆ = 1/2), and the Dual Ice Point (η = π/3, λ = π/2, ∆ = −1/2). In these three cases, the Kn(x) turn out to be classical orthogonal polynomials, namely Meixner-Pollaczek, Continuous Hahn and Continuous Dual Hahn polynomial, respectively. Correspondingly, the 8 F. COLOMO AND A.G. PRONKO generating function can be represented explicitly in terms of terminating hyperge- ometric functions that may simplify considerably further evaluation of EFP. In the next Section we shall focus on the case of Free Fermion line. 4. Multiple integral representation at ∆ = 0 4.1. Specialization to η = π/4. We shall now restrict ourselves to the case η = π/4. We have ∆ = 0, and the six-vertex model reduces to a model of free fermions on the lattice. The parameter λ can still assume any value in the interval (−π/4, π/4). It is convenient to trade λ for the new parameter τ = tan2(λ− π/4) , 0 < τ < ∞ . The symmetric point (related to the domino tiling of Aztec Diamond) corresponds now to τ = 1. For generic values of τ we have: ω̃ = −τω . The generating function (3.4) is known explicitely (see [CP2] for details): hN (z) = 1 + τz 1 + τ Plugging this expression into (3.6), we get FN (r, s) = · · · dsω det 1≤j,k≤s (1 + τωj)(ωj − 1) (1 + τ)ωj ωN−r−1j (ωj − 1)N 1≤j<k≤s (1 + τωj)(ωk − 1) 1 + τωjωk . (4.1) 4.2. Symmetrization. After extracting a common factor (1 + τωj)(ωj − 1) (1 + τ)ωj from the determinant in (4.1), we recognize it to be of Vandermonde type. We can therefore collect from the integrand of (4.1) the double product 1≤j<k≤s (1 + τωj)(ωj − 1) (1 + τ)ωj (1 + τωk)(ωk − 1) (1 + τ)ωk (1 + τωj)(ωk − 1) 1 + τωjωk Noticing that the integration and the remaining of integrand are fully symmetric under permutation of variables ω1, . . . , ωj , we can perform total symmetrization of the previous double product over all its variables, with the result (−1)s(s−1)/2 ωs−1j 1≤j<k≤s (ωj − ωk) THE ARCTIC CIRCLE REVISITED 9 Hence, we finally obtain the following representation for EFP on the Free Fermion line: FN (r, s) = (−1)s(s+1)/2 s!(1 + τ)s(N−s)(2πi)s · · · 1≤j<k≤s (ωj − ωk) (1 + τωj) (ωj − 1)s ω . (4.2) The appearance of a squared Vandermonde determinant in this expression naturally recalls the partition functions of s× s Random Matrix Models. 5. Triple Penner model and Arctic Ellipses 5.1. Scaling limit. We shall now address the asymptotic behaviour of expres- sion (4.2) for EFP in the ∆ = 0 case. We are interested in the limit N, r, s → ∞, while keeping the ratios r/N = x , s/N = y , fixed. In this limit, x, y ∈ [0, 1] will parameterize the unit square to which the lattice is rescaled. Correspondingly EFP is expected to approach a limit function F (x, y) := lim FN (xN, yN) , x, y ∈ [0, 1] . We shall exploit the standard approach developed for instance in the investigation of asymptotic behaviour for Random Matrix Models. Before this let us however point out some facts which holds already for any finite value of s. 5.2. A useful identity. Consider the quantity IN (r, s) := (−1)s(s+1)/2 s!(1 + τ)s(N−s)(2πi)s · · · 1≤j<k≤s (ωj − ωk) (1 + τωj) (ωj − 1)s ω which differs from (4.2) only in the integration contours. Here C1 is a closed, clockwise oriented contour (note the change in orientation) in the complex plane enclosing point ω = 1, and no other singularity of the integrand. We have the identity IN (r, s) = 1 (5.1) for any integer r, s = 1, . . . , N . The simplest way to prove the previous identity is by shifting ωj → ωj + 1, and rewriting IN (r, s) as an Hankel determinant; indeed we have IN (r, s) = (−1)s(s−1)/2 (1 + τ)s(N−s) 1≤j,k≤s ωj+k−2−s(1 + τ + τω)N−s (1 + ω)r The entries of the Hankel matrix vanish for j+k > s+1, and hence the determinant is simply given by the product of the antidiagonal entries, j + k = s + 1 (modulo a sign (−1)s(s−1)/2 emerging from the permutation of all columns). Identity (5.1) follows immediately. 10 F. COLOMO AND A.G. PRONKO 5.3. Saddle-point evaluation for large N and finite s. When using the saddle-point method in variables ω1, . . . , ωs to evaluate the behaviour of FN (r, s) for large N and r, and finite s, it is rather easy to see that the saddle-point equations decouple at leading order, and that each saddle-point will be on the real axis, contributing with a factor e−NSj with Sj positive. If a given saddle-point is smaller than 1, the contour C0 can be deformed through the saddle-point without encountering any pole, and its contribution will vanish as e−NSj in the large N limit. If however the saddle-point, still on the real axis, happens to be larger than 1, the deformation of the contour C0 through the saddle-point will pick up the contribution of the pole at ω = 1 (with a reversed orientation of the contour), and the j-th integral will behave as 1 + e−NSj . Hence, in the large N limit (at fixed s) the quantity FN (r, s) will vanish unless all the saddle-points are greater than 1, in which case FN (r, s) ∼ IN (r, s) = 1. Note that in the present situation the s saddle-points coincide. A detailed analysis shows that in this case the position of the s saddle-points depends on the value x = r/N as τ(1−x) . In correspondence to the value x0 = , for which these saddle- points are exactly 1, the function F (x, 0) has a step discontinuity. More precisely, it is easy to show that for x ∈ [0, 1], F (x, 0) = θ(x − x0), where θ(x) is Heaviside step function. From a physical point of view x0 is the contact point between the limit shape and the boundary. What have been discussed here can easily be verified in the case s = 1. The extension to finite s > 1 is rather direct as well. 5.4. Saddle-point equation. Having in mind the analogy with s × s Ran- dom Matrix Models, and the scaling limit specified in Section 5.1, we rewrite our expression for FN (r, s) at ∆ = 0 as follows: FN (r, s) = (−1)s(s+1)/2 s!(1 + τ)s 2(1/y−1)(2πi)s · · · dsω exp j,k=1 j 6=k ln |ωj − ωk| ln(τωj + 1)− ln(ωj − 1)− ln(ωj) . (5.2) Both sums in the exponent are O(s2). The corresponding (coupled) saddle-point equations read ωj − 1 (1/y − 1)τ τωj + 1 k 6=j ωj − ωk . (5.3) A standard physical picture reinterprets the saddle-point equations as the equi- librium condition for the positions of s charged particle confined to the real axis, with logarithmic electrostatic repulsion, in an external potential. In the present case the latter can be seen as generated by three external charges, 1, x/y, and −(1/y − 1) at positions 1, 0, and −1/τ , respectively. It is natural to refer to this model as the triple Penner model. Although the simple Penner [P] matrix model has been widely investigated, not so much is known about the much more compli- cate double Penner model [M, PW]. We have not been able to trace any previous study concerning the triple Penner model. THE ARCTIC CIRCLE REVISITED 11 5.5. The exact Green function at finite s. To investigate the structure of solutions of the saddle-point equations (5.3) for large s we first introduce the Green function Gs(z) = z − ωj which, if the ωj ’s solves (5.3), has to satisfy the differential equation: z(z − 1)(τz + 1) sG′s(z) + s 2G2s(z) − s(αz2 + βz + γ)sGs(z) τs(s− 1)− αs2 z + (1 − τ)s(s− 1)− βs2 +Ω 2τs(s− 1)− αs2 . (5.4) The coefficients α, β and γ are readily obtained as the coefficients of the second order polynomial appearing in the numerator, when setting to common denominator the left hand side of (5.3). We give them explicitly for later convenience: α = τ , β = + (1− τ) , γ = − The derivation of the differential equation is very standard (see, e.g., [SD]). The left hand side is built by suitably combining the explicit definition of the Green function and its derivative. The result has to be a polynomial of the first degree in z, whose coefficients are constructed by matching the leading and first subleading behaviour of the left hand side as |z| → ∞. 5.6. The first moment Ω. The quantity Ω appearing in (5.4) is defined as the first moment of the solutions of the saddle-point equations: It is related in a obvious way to the first subleading coefficient of Gs(z); indeed, from the definition of the Green function, it is evident that Gs(z) = +O(z−3) , |z| → ∞ . It is worth to emphasize that Ω is still unknown, and that in principle its value should be determined self consistently by first working out the explicit solution of Gs(z) (which will depend implicitly on Ω), from (5.4) and then demanding that j=1 ωj evaluated from this solution coincides with Ω. The appearance of the un- determined parameter Ω is a manifestation of the ‘two-cuts’ nature of the Random Matrix Model related to (5.2), see, e.g., par. 6.7 of [D1]. 5.7. The asymptotic Green function. We are now in condition to perform the large s (and large N , r) limit at fixed x, y. In the limit, we can neglect terms of order O(s) in the differential equation (5.4), which therefore reduces to an algebraic equation for the limiting Green function G(z): z(z−1)(τz+1)[G(z)]2−(αz2+βz+γ)G(z) = (τ−α)z+(1−τ−β)+Ω(2τ−α) . (5.5) The previous algebraic equation has to be supplemented by the normalization con- dition G(z) ∼ , |z| → ∞ . (5.6) 12 F. COLOMO AND A.G. PRONKO Hence the Green function describing the large s asymptotic distribution of solutions for the saddle equation (5.3) reads: G(z) = 2z(z − 1)(τz + 1) (αz2 + βz + γ) (αz2 + βz + γ)2 + 4z(z − 1)(τz + 1)[(τ − α)z +Ω(2τ − α) + 1− τ − β] (5.7) We have selected the positive branch of the square root, to satisfy the normalization condition (note that the coefficient of z4 under the square root is (α − 2τ)2, and α − 2τ is negative for any x, y ∈ [0, 1]). However, the expression for G(z) is not completely specified yet, because Ω is still undetermined. 5.8. Limit shape and condensation of roots. The polynomial under the square root is of fourth order, hence G(z) will have in general two cuts in the complex plane. The emergence of a two-cut problem was already expected from the appearance of the undetermined first moment Ω in (5.4). The discontinuity of G(z) across these cuts defines, when positive, the density of solutions of the saddle- point equations (5.3) when s → ∞. The problem of explicitly finding this density, for arbitrary α, β, γ (or x, y), is a formidable one, not to mention the evaluation of the corresponding ‘free energy’, and of the saddle-point contribution to the integral in (5.2). But our aim is much more modest, since we are presently interested only in the expression of the limit shape, i.e. in the curve in the square x, y ∈ [0, 1], delimiting regions where F (x, y) = 0 from regions where F (x, y) = 1. Of course we are here somehow assuming that the transition of F (x, y) from 0 to 1 is stepwise in the scaling limit, but this is supported both by the physical interpretation of EFP (in the disordered region, by definition, the number of ‘thin’ lines is macroscopic, and the probability of finding no ‘thin’ horizontal edges immediately vanishes in the scaling limit) and by the discussion of Section 5.3. As explained in the discussion of the double Penner model in paper [PW], the logarithmic wells in the potential can behave as condensation germs for the saddle- point solutions. In our case, this can role can be played only by the ‘charge’ at ω = 1 in the Penner potential since the charge at ω = −1/τ is always repulsive, while the one at ω = 0 is larger than 1, at least in the region of interest. [PW] have shown that condensation can occur only for charges less than or equal to 1, since this will be the fraction of condensed solutions. This consideration, together with the expected stepwise behaviour and the discussion in Section 5.3, suggest the following picture for the evolution of saddle-point solution density from the disordered region, F (x, y) ∼ 0, to the upper left frozen region, F (x, y) ∼ 1: in the disordered region there is a macroscopic fraction of solutions which are real and smaller than 1, while in the upper left frozen region this fraction vanish. On the basis of the discussion here and in Sections 3.2 and 5.3, we shall assume that at the transition between the two regions all saddle-point solutions have condensed at ω = 1. 5.9. Main assumption. We claim that the Arctic curve in the square x, y ∈ [0, 1] separating the disordered phase from the upper left frozen phase is defined by the condition that all solutions of the saddle-point equation lies at ω = 1. In the derivation of the limit shape, this is indeed the only assumption to which we are unable to provide a proof. There is in fact no guarantee, at this level, for THE ARCTIC CIRCLE REVISITED 13 this possibility to occur, and limit shapes could in principle emerge from a different condition. But if for some values of x, y ∈ [0, 1] we have all solutions of the saddle- point equation condensing at ω = 1, then this provides a transition mechanism from 0 to 1 for F (x, y), and this might correspondingly define some limit shape. If all saddle-point solutions condensate at ω = 1, then we obviously have: Ω = 1 . Moreover, the complicate expression (5.7) for G(z) should simply reduce to G(z) = z − 1 , (5.8) since we expect to have no cuts, and only one pole at z = 1 with unit residue. 5.10. Arctic Ellipses. Consider the quartic polynomial under the square root in (5.7). It is convenient to rewrite it in terms of α̃ := 2τ − α = τ β̃ := 2− β = τ x+ y − 1 y − x γ̃ := −γ = (5.9) Note that α̃ and γ̃ are always positive for x, y ∈ [0, 1]. When Ω = 1, our quartic polynomial reads α̃2z4 + 2α̃β̃z3 + (β̃2 + 2α̃γ̃)z2 + 2β̃γ̃z + γ̃2 , which may be equivalently rewritten as (α̃z2 + β̃z + γ̃)2 . We see that the quartic polynomial reduces to a perfect square, and hence, when Ω = 1, the two cuts of G(z) disappear, as expected. Now, when Ω = 1, in our new notations, the Green function reads: G(z) = [(2τ − α̃)z2 + (2− β̃)z − γ̃] + (α̃z2 + β̃z + γ̃)2 2z(τz + 1)(z − 1) . (5.10) We now require the coefficients α̃, β̃, γ̃ to be such that the polynomial under the square root combines with the first part of the numerator in (5.10) to give 2z(τz+1) and simplify the Green function according to (5.8). Once we have chosen a given branch of the square root (the positive one, in order to satisfy normalization condition (5.6)), it is obvious that the required simplification can occur for any z in the complex plane only if the second order polynomial α̃z2 + β̃z + γ̃ does not change its sign, i.e. only if its two roots coincide, implying: β̃2 − 4α̃γ̃ = 0 . Rewriting the last relation in terms of x, y, through (5.9), we readily get (1 + τ)2x2 + (1 + τ)2y2 − 2(1− τ2)xy − 2τ(1 + τ)x − 2τ(1 + τ)y + τ2 = 0 . We have therefore recovered the limit shape, which in this Free Fermion case is the well-known Arctic Ellipse (Arctic Circle for τ = 1) [JPS, CEP]. We recall that, as discussed in Section 3.2, F (x, y) is non-vanishing only in the upper left region 14 F. COLOMO AND A.G. PRONKO of the unit square. Therefore, concerning EFP, only the upper left portion of the Arctic curve, between the two contact points at ( τ , 0) and (1, 1 ), is relevant. 6. Concluding remarks Our starting point has been the definition of a relatively simple but relevant correlation function for the six-vertex model with DWBC, the Emptiness Formation Probability. We have provided both a determinant representation and a multiple integral representation for the proposed correlation function. This is the first ex- ample in literature of a bulk (as opposed to boundary) correlation function for the considered model, for generic weights. The multiple integral representation, specialized to the Free Fermion case, has been studied in the scaling limit. In the standard picture of Random Matrix Mod- els, we recognize the emergence of a triple Penner model. Assuming condensation of the roots of saddle point equations in correspondence to a limit shape, we recover the well-known Arctic Circle and Ellipse. It would be interesting to investigate whether universality considerations of Random Matrix Models (see, e.g., [D2]) can be extended to the Penner model in the neighbourhood of its logarithmic singular- ities. This would imply directly the results of [CEP, J1, J2] on the Tracy-Widom distribution and the Airy process, emerging in a suitably rescaled neighbourhood of the Arctic Ellipse. It is worth to stress that the multiple integral representation for EFP presented in Section 3 can be studied beyond the usual Free Fermion situation. We expect that condensation of roots of the saddle point equation in correspondence of the limit shape is a general phenomenon. We believe that this assumption could be of importance in addressing the problem of limit shapes in the six-vertex model with DWBC. Our derivation of the limit shape in the Free Fermion case uses the explicit knowledge of function hN (z), standing in the multiple integral representation (3.6). It is worth mentioning that function hN(z) is also known explicitly at Ice Point, (∆ = 1/2), and Dual Ice Point, (∆ = −1/2), being expressible in terms of (poly- nomial) Gauss hypergeometric function [Ze, CP2]. For instance, at Ice Point the triple Penner model discussed above generalizes to a two-matrix Penner model. This model can be studied along the lines presented here, thus providing a solution to the longstanding problem of limit shape for Alternating Sign Matrices. Acknowledgements We thank Nicolai Reshetikhin for useful discussion, and for giving us a draft of [PR] before completion. FC is grateful to Percy Deift, and Courant Institute of Mathematical Science, for warm hospitality. AGP thanks INFN, Sezione di Firenze, where part of this work was done. We acknowledge financial support from MIUR PRIN program (SINTESI 2004). One of us (AGP) is also supported in part by Civilian Research and Development Foundation (grant RUM1-2622-ST- 04), by Russian Foundation for Basic Research (grant 04-01-00825), and by the program Mathematical Methods in Nonlinear Dynamics of Russian Academy of Sciences. This work is partially done within the European Community network EUCLID (HPRN-CT-2002-00325), and the European Science Foundation program INSTANS. THE ARCTIC CIRCLE REVISITED 15 References [AR] D. Allison and N. Reshetikhin, Numerical study of the 6-vertex model with domain wall boundary conditions, Ann. Inst. Fourier (Grenoble) 55 (2005) 1847–1869. [Ba] R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic press, San Diego, 1982. [Br] D. M. Bressoud, Proofs and Confirmations: The Story of the Alternating Sign Matrix Con- jecture, Cambridge University Press, Cambridge, 1999. [BF] P. Bleher and V. Fokin, Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions. Disordered Phase, preprint (2005) arXiv:math-ph/0510033. [BPZ] N.M. Bogoliubov, A.G. Pronko, and M.B. Zvonarev, Boundary correlation functions of the six-vertex model, J. Phys. A: Math. Gen. 35 (2002) 5525–5541. [CEP] H. Cohn, N. Elkies and J. Propp, Local statistics for random domino tilings of the Aztec diamond, Duke Math. J. 85 (1996) 117–166. [CKP] H. Cohn, R. Kenyon and J. Propp, A variational principle for domino tilings, J. Amer. Math. Soc. 14 (2001) 297–346 [CLP] H. Cohn, M. Larsen and J. Propp, The shape of a typical boxed plane partition, New York J. Math. 4 (1998) 137–165. [CP1] F. Colomo and A.G. Pronko, On two-point boundary correlations in the six-vertex model with DWBC, J. Stat. Mech.: Theor. Exp. JSTAT(2005)P05010, arXiv:math-ph/0503049. [CP2] F. Colomo and A.G. Pronko, Square ice, alternating sign matrices, and classical orthogonal polynomials, J. Stat. Mech.: Theor. Exp. JSTAT(2005)P01005, arXiv:math-ph/0411076. [CP3] F. Colomo and A.G. Pronko, The role of orthogonal polynomials in the six-vertex model and its combinatorial applications, J. Phys. A: Math. Gen. 39 (2006) 9015–9033. [CP4] F. Colomo and A.G. Pronko, Emptiness Formation Probability in the Domain Wall six- vertex marodel, in preparation. [D1] P. Deift, Orthogonal Polynomials and Random Matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, Amer. Math. Soc., Providence, RI, 2000. [D2] P. Deift, Universality for mathematical and physical systems, preprint (2006) arXiv: math-ph/0603038. [E] K. Eloranta, Diamond Ice, J. Statist. Phys. 96 (1999) 1091–1109. [EKLP] N. Elkies, G. Kuperberg, M. Larsen and J. Propp, Alternating sign matrices and domino tilings , J. Algebraic Combin. 1 (1992) 111–132; 219–234. [FP] O. Foda and I. Preston, On the correlation functions of the domain wall six-vertex model, J. Stat. Mech.: Theor. Exp. JSTAT(2004)P11001. [FS] P.L. Ferrari and H. Spohn, Domino tilings and the six-vertex model at its free fermion point, J. Phys. A: Math. Gen. 39 (2006) 10297–10306. [I] A.G. Izergin, Partition function of the six-vertex model in the finite volume, Sov. Phys. Dokl. 32 (1987) 878. [ICK] A.G. Izergin, D.A. Coker and V.E. Korepin, Determinant formula for the six-vertex model, J. Phys. A: Math. Gen. 25 (1992) 4315–4334. [J1] K. Johansson, Non-intersecting paths, random tilings and random matrices, Probab. Theory Related Fields 123 (2002) 225–280. [J2] K. Johansson, The arctic circle boundary and the Airy process, Annals of Probability 33 (2005) 1–30. [JM] M. Jimbo and T. Miwa, Algebraic analysis of solvable lattice models, CBMS Lecture Notes Series, vol. 85, Amer. Math. Soc., Providence, RI (1995). [JPS] W. Jockush, J. Propp and P. Shor, Random domino tilings and the arctic circle theorem, preprint (1995) arXiv:math.CO/9801068. [K] V.E. Korepin, Calculation of norms of Bethe wave functions, Comm. Math. Phys. 86 (1982) 391–418. [KBI] V.E. Korepin, N.M. Bogoliubov, and A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, Cambridge, 1993. [KMT] N. Kitanine, J. M. Maillet and V. Terras, Correlation functions of the XXZ Heisenberg spin-1/2 chain in a magnetic field, Nucl. Phys. B 567 (2000) 554–582. [KO] R. Kenyon and A. Okounkov, Limit shapes and the complex Burgers equation, preprint (2005) arXiv:math-ph/0507007. 16 F. COLOMO AND A.G. PRONKO [KOS] R. Kenyon, A. Okounkov and S. Sheffield, Dimers and Amoebae, Ann. of Math. (2) 163 (2006) 1019–1056. [KP] V. Kapitonov and A. Pronko, On the arctic ellipse phenomenon in the six-vertex model, in preparation. [KZ] V. Korepin, P. Zinn-Justin, Thermodynamic limit of the Six-Vertex Model with Domain Wall Boundary Conditions, J. Phys. A 33 (2000) 7053–7066. [LW] E.H. Lieb and F.Y. Wu, Two-dimensional ferroelectric models, in Phase Transitions and Critical Phenomena, Vol. 1, edited by C. Domb and M.S. Green, Academic Press, London, 1972, pp. 321–490. [M] Yu. Makeenko, Critical Scaling and Continuum Limits in the D > 1 Kazakov-Migdal Model, Int.J.Mod.Phys. A10 (1995) 2615–2660. [OR] A. Okounkov and N. Reshetikhin, Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Amer. Math. Soc. 16 (2003) 581–603, [P] R.C. Penner, Perturbative series and the moduli space of Riemann surfaces, J. Diff. Geom. 27 (1988) 35–53. [PR] K. Palamarchuk and N. Reshetikhin, The six-vertex model with fixed boundary conditions, in preparation. [PW] L. Paniak and N. Weiss, Kazakov-Migdal Model with Logarithmic potential and the Double Penner Matrix Model, J. Math. Phys. 36 (1995) 2512–2530. [SD] B. Sriram Shastry and A. Dhar, Solution of a generalized Stieltjes problem J. Phys. A: Math. Gen. 34 6197-6208. [SZ] O.F. Syljuasen and M.B. Zvonarev, Directed-loop Monte Carlo simulations of Vertex models, Phys. Rev. E 70 (2004) 016118. [Ze] D. Zeilberger, Proof of the refined alternating sign matrix conjecture, New York J. Math. 2 (1996) 59–68. [Zi1] P. Zinn-Justin, The influence of boundary conditions in the six-vertex model, preprint (2002) arXiv:cond-mat/0205192. [Zi2] P. Zinn-Justin, Six-Vertex Model with Domain Wall Boundary Conditions and One-Matrix Model, Phys. Rev. E 62 (2000), 3411–3418. I.N.F.N., Sezione di Firenze and Dipartimento di Fisica, Università di Firenze, Via G. Sansone 1, 50019 Sesto Fiorentino (FI), Italy E-mail address: [email protected] Saint Petersburg Department of Steklov Mathematical Institute of Russian Acad- emy of Sciences, Fontanka 27, 191023 Saint Petersburg, Russia E-mail address: [email protected]

0704.0363

Time and motion in physics: the Reciprocity Principle, relativistic invariance of the lengths of rulers and time dilatation

arXiv:0704.0363v2 [physics.gen-ph] 10 Feb 2009 Time and motion in physics: the Reciprocity Principle, relativistic invariance of the lengths of rulers and time dilatation J.H.Field Département de Physique Nucléaire et Corpusculaire Université de Genève . 24, quai Ernest-Ansermet CH-1211 Genève 4. E-mail: [email protected] Abstract Ponderable objects moving in free space according to Newton’s First Law con- stitute both rulers and clocks when one such object is viewed from the rest frame of another. Together with the Reciprocity Principle this is used to demonstrate, in both Galilean and special relativity, the invariance of the measured length of a ruler in motion. The different times: ‘proper’, ‘improper’ and ‘apparent’ appearing in different formulations of the relativistic time dilatation relation are discussed and exemplified by experimental applications. A non-intuitive ‘length expansion’ effect predicted by the Reciprocity Principle as a necessary consequence of time dilatation is pointed out. PACS 03.30.+p http://arxiv.org/abs/0704.0363v2 1 Introduction The standard text-book presentation of special relativity follows closely that of Ein- stein’s seminal paper of 1905 [1] in basing the theory on the Special Relativity Principle, classical electromagnetism and the postulate of constant light speed. However an alterna- tive and conceptually simpler approach to the physics of space and time, in the absence of gravitational fields, is possible in which it is not necessary to consider light signals, classical electromagnetism, or indeed, any dynamical theory whatsoever. The Lorentz transformation (LT) was first derived in this way by Ignatowsky [2] in 1910. Purely mathematical considerations imply, in such a derivation of the LT, the existence of a maximum relative velocity, V , of two inertial frames. Use of relativistic kinematics then shows that V is equal to the speed of light, c, when light in identified as a manifestation of the propgation in space-time of massless particles –photons [3]. In this way Einstein’s mysterious second postulate is derived from first principles. The fundamental axiom un- derlying such an approach is the Reciprocity Principle (RP) [4, 3], discussed in Section 3 below, relating the the relative velocities of two inertial frames. Derivations of the LT and the parallel velocity addition formula based on the RP and other simple axioms are given in Ref. [3]. In the present paper the space-time properties of ponderable1 physical bodies in free space, as described by Newton’s First Law of mechanics, are used together with the RP, to demonstrate the invariance of the measured length of a ruler in uniform motion. The proof given is valid in both Galilean and special relativity, since Newton’s First Law and the RP hold in both theories. The analysis presented is based on a careful definition of physical time concepts. In particular, the ‘frame time’ or ‘proper times’ that appear in in the RP, are distinguished from the ‘improper time’ or ‘apparent time’ (of a moving clock) that appear in the Time Dilatation (TD) relation of special relativity. The paper is organised as follows: The following section contains an elementary discus- sion of the concepts of ‘space’, ‘time’ and ‘motion’ in physics, in connection with Newton’s First Law. In Section 3, the RP is introduced and discussed in relation to Newton’s First Law. It is pointed out that, because of the RP, ‘rulers are clocks’ and ‘clocks are rulers’ when the motion of ponderable bodies in free space is considered. In Section 4 the RP is used to demonstrate the invariance of the measured length of a uniformly moving ruler. In Section 5 the operational meanings of the time symbols appearing in the TD formula of special relativity are discussed. This may be done in a ‘clock oriented’ manner in terms of ’proper’ and ’improper’ times of the observed clock, or in an ‘observer oriented’ manner in terms of the proper time of the observer’s local clock and the ‘apparent time’, as seen by the observer, of the moving clock. Two specific experiments are described to exemplify the operational meanings of the time symbols in the TD formula. A non-intuitive ‘length expansion’ effect is found to relate similarly defined spatial intervals corresponding to the observation of an event either in the rest frame of the clock, or in a frame in which it is in uniform motion. 1That is bodies, with a non-vanishing Newtonian mass, which may be associated with an inertial frame in which the body is at rest. No such frame may be associated with a massless object. The results of the present paper show that the ‘length contraction’ effect and the correlated ‘relativity of simultaneity’ effect of conventional special relativity do not exist. A detailed discussion of the reason for the spurious nature of these effects of conventional special relativity theory may be found in Refs. [5, 6, 7, 8, 9, 10] . However, a genuine ‘relativistic length contraction’ effect does occur when distances be- tween spatial coincidences of moving objects are observed from different inertial frames [11]. Also a genuine ‘relativity of simultaneity’ effect occurs when clocks at rest in two different inertial frames are viewed from a third one [12, 13]. An alternative derivation, directly from the RP, of the invariance of the measured spatial separation of two objects at rest in the same inertial frame as well as the absence of the conventional ‘relativity of simul- taneity’ effect is given in Ref. [9]. 2 Physical time and Newton’s First Law of Mechan- In physics the concepts of ‘time’ and ‘motion’ are inseparable. In a world in which motion did not exist the physical concept of time would be meaningless. Similarly the physical concepts of ‘space’ and ‘motion’ are inseperable. Without the concept of space, no operational definition of motion is possible. The concept of historical time –the time of the everyday world of human existence– requires the introduction of the further, and equivalent, concepts of ‘uniform motion’ and ‘cyclic motion with constant period’. For example, the unit of time the ‘year’ is identified with the (assumed constant) period of rotation of the Earth about the Sun. The idea of uniform motion entered into physics in a quantitative way with the for- mulation of Newton’s First Law [14] Every body continues in its state of rest, or uniform motion in a right line unless it is compelled to change that state by forces impressed upon This law gives an operational meaning to the physical concept of ‘uniform motion’. It is defined by observations of the position of any ponderable object in ‘free space’ i.e. in the absence of any mechanical interaction of the object with other objects. There is a one-to-one correspondence between such a ponderable object and an ‘inertial frame’ of relativity theory. As will be discussed in the following section, one such ponderable object, O, constitutes both a ruler and a clock for an observer in the rest frame of another such object, O’, and vice versa. When time is measured by using a cyclic physical phenomenon, e.g. an analogue clock, time measurement reduces to recording the result of a spatial (or angular) measurement. There is a one-to-one correspondence between the spatial coincidence of a stationary ‘mark’ on the face of the clock and a moving ‘pointer’, constituted by the hand of the clock, and the time measurement [6]. A ‘time interval’ is measured by the angular separation of two such ‘pointer-mark coincidences’. The implicit assumption is that the motion of the pointer is ‘uniform’. There is an evident logical circularity here since ‘equal’ time intervals measured by such an analogue clock assume that the angular velocity of the hand is constant, whereas constant angular velocity is established by observation of equal angular increments for equal time time intervals (i.e. also equal angular increments) recorded by a second clock of supposedly known uniform rate. In practice, this conundrum is resolved by an appeal to physics. For example, an undamped pendulum in a uniform gravitational field is predicted, by the laws of mechanics, to have a constant period of oscillation. Quantum mechanics predicts the same transition frequency and mean lifetimes for two identical atoms in the same excited state, in the same physical environment, etc. Measurements of ‘time’ are then ultimately observations of spatial phenomena, e.g. the time measurement corresponding to observation of the number displayed by a digital clock is a spatial perception. This will also be the case for time measurements related to observation of two ponderable objects O and O’ in motion in free space that will now be discussed. 3 The Reciprocity Principle: rulers are clocks, and clocks are rulers Consider two non-interacting ponderable objects O and O’, with arbitary motions in free space. They are placed at the origins of inertial coordinate systems S and S’ with axes orientated so that the x and x′ axes are parallel to the relative velocity vector of O and O’. Without any loss of generality for the following discussion, it may be assumed that O and O’ lie on the common x-x′ axis. The Reciprocity Principle (RP) [4, 3, 9] is defined by the equation: v = vO′O = ∂xO′O = −vOO′ (3.1) where xO′O ≡ xO′−xO and x , or in words: ‘If the velocity of O’ relative to O is ~v, the velocity of O relative to O’ is - ~v’. In many discussions of special relativity, the RP is taken as ‘obvious’ and is often not even declared as a separate axiom. This is the case, for example, in Einstein’s 1905 special relativity paper [1]. However, as first demonstrated by Ignatowsky in 1910 [2], it is sufficient, together with some other weaker axioms such as the homogeneity of space or single-valuedness of the transfomation equations, to derive [3] the space-time Lorentz transformation and hence the whole of special relativity theory. Eqn(3.1) looks very similar to the equation defining the relative velocity of two objects A and B as observed in a single inertial reference frame (say S): vAB ≡ vA − vB = d(xA − xB) = −vBA (3.2) The crucial difference is the appearence in the RP, (3.1), of two different times t and t′. The time t is the ‘frame time’ of S. i.e. the time registered by a synchronised clock at rest, at any position in S, according to an observer also a rest in S. The frame time t′ is similarly defined by an array of synchronised clocks at rest in S’. Eqn(3.1) (and its integral) gives a relation between the times t and t′ Both t and t′ correspond to ‘proper times’ of clocks at rest, whereas, as explained in Section 4 below, the Lorentz transformation relates instead a proper time to an ‘improper time’ –the observed time of a clock in uniform motion. Suppose now that O and O’ are equipped with local clocks that are observed to run at exactly the same rate when they are both at rest in the same inertial frame. The direction of the relative velocity vector ~v of O’ relative to O is such that they are approaching each other at the frame times t and t′. The spatial separations of O and O’ in S and S’ are ℓ(t) and ℓ′(t′) respectively, at times t and t′. Using the RP, a spatial coincidence of O and O’ will be observed at the time tOO′ = t+ (3.3) in S, and = t′ + ℓ′(t′) (3.4) in S’. The OO’ coincidence event will be mutually simultaneous in the frames S and S’. Note that the OO’ spatial coincidence that is mutually simultaneous in S and S’ constitutes a pair of reciprocal pointer mark coincidences. In S the mark is at the position of O and the moving pointer at the position of O’, whereas in S’ the position of O’ constitutes the mark and the position of O the pointer. A corollary is that all such pairs of reciprocal pointer mark coincidences are mutually simultaneous. This is the basis of the ‘system external synchronistation’ [15] as introduced in Einstein’s first special relativity paper [1] to synchronise clocks at rest in different inertial frames when they are in spatial coincidence. The observation of the OO’ coincidence event in both frames can be used to give a condition that any other pair of events, one observed in S, the other observed in S’ are mutually simultaneous. If the time of an event in S is t̃ and another event in S’ is t̃′ they will be ‘mutually simultaneous’ providing that: t̃′ − t̃ = t′ − tOO′ (3.5) Combining (3.3)-(3.5) gives: t̃′ − t̃ = t′ − tOO′ = t ′ − t+ ℓ′(t′)− ℓ(t) (3.6) If now events occuring at times t in S and t′ in S’ are mutually simultaneous, it follows from (3.5) and (3.6) that ℓ(t) = ℓ′(t′), so that events which occur when O and O’ have the same spatial separation in S and S’ are mutually simultaneous. A special case occurs if the clock arrays in S and S’ are mutually synchronised so that ℓ(t) = ℓ′(t′ = t). There is then a direct correlation between either t or t′ and the spatial separation of O and O’: When mutually synchronised clocks in the frames S and S’ have the same reading, O and O’ have the equal spatial separations in S and S’, and conversely, When O and O’ have equal spatial separations in the frames S and S’, mutually synchronised clocks in S and S’ have the same reading. The dependence of ℓ on t in Eqn(3.3) and ℓ′ on t′ in Eqn(3.4) means that each of the objects may be considered to be an ‘inertial clock’ by an observer in the rest frame of the other one. That is, t is measured by the spatial separation of O’ from O in S and t′ is measured by the spatial separation of O from O’ in S’. Conversely, after mutual synchronisation of the clock arrays in S and S’ at the instant when O and O’ are in spatial coincidence, t measures the spatial separation of O’ and O in S (and so is effectively a ruler in this frame) while t′ measures the spatial separation of O’ and O in S’, constituting a ruler in this frame. Matching of these measurements of the separation of O and O’ with the lengths of physical rulers at rest in S and S’ is now used to demonstrate the invariance of the measured length of the length of a ruler in uniform motion –that is, the absence of any relativistic length contraction effect– in this case. 4 Invariance of the measured length of a ruler in uni- form motion Figure 1: Rulers attached to objects O and O’ are viewed from the frame S (left) and S’ (right). The equality of the separations of O and O’ in S and S’ at time t = t′ = L/v, predicted by the RP, is used to establish the invariance of the measured length of the moving ruler R’ in S, or of the moving ruler R in S’ (see text). Suppose that O and O’ are equipped with rulers R and R’, parallel to the x-x′ axis as shown in Fig.1. O coincides with the mark MR(0) of the ruler R and O’ with the mark MR′(10) of the ruler R’. A t = t′ = 0 (Fig.1a) O and O’ are in spatial coincidence. The clock arrays in S and S’ are mutually synchronised at this time. The length of each ruler in its rest frame is L. The object O’ now moves along the ruler R, being in spatial coincidence with different marks of the ruler at different times. The object O moves in a similar manner along the ruler R’. At any given time t the separation of O and O’ in S is given by the corresponding ‘Pointer Mark Coincidence’ (PMC): PMC(O′, t) ≡ O′(t)@MR(J) (4.1) where the symbol before the ampersand denotes the moving ‘pointer’, and the symbol after it the stationary ‘mark’ with which it is spatial coincidence2. Since PMC(O, t) ≡ O(t)@MR(0) for all t (4.2) and x[MR(0)] = 0 it follows that the separation of O and O’ in the frame S at time t is given by: dO′O(t) = x[MR(J)]− x[MR(0)] = x[MR(J)] (4.3) where x[MR(J)] = and where, in Fig.1, Jmax = 10, is the ordinal number of the mark at the end of the ruler. Thus the x-coordinate origin is at MR(0). Defining in a similar manner a PMC in the frame S’: PMC(O, t′) ≡ O(t′)@MR′(K) (4.4) and since PMC(O′, t′) ≡ O′(t′)@MR′(10) for all t′ (4.5) the separation of O and O’ in S’ at the time t′ is (t′) = x′[MR′(10)]− x′[MR′(K)] (4.6) where x′[MR′(K)] = and where, in Fig.1, Kmax = 10. The spatial configurations in S and S’ at the times t = t′ = L/v are shown in Fig.1b. The corresponding PMC are: PMC(O′, L/v) ≡ O′(L/v)@MR(10) (4.7) PMC(O, L/v) ≡ O(L/v)@MR′(0) (4.8) It follows from (4.3) and (4.6) that dO′O(L/v) = x[MR(10)]−x[MR(0)] = L = x ′[MR′(10)]−x′[MR′(0)] = d′ (L/v) (4.9) Since O’ coincides with MR′(10) at all times it follows that, at t = L/v x[MR′(10)] = x[O′] = x[MR(10)] (4.10) Also, since O is in spatial coincidence with MR′(0) at t = t′ = L/v it follows that at t = L/v, x[MR′(0)] = x[O] = x[MR(0)] = 0 (4.11) 2This notation was introduced in Ref. [6]. Note the similarity with an e-mail address Eqns(4.9)-(4.11) then give at t = L/v: x[MR′(10)]− x[MR′(0)] = x[MR(10)]− x[MR(0)] = L (4.12) That is, the measured length of the moving ruler R’ in the frame S, at t = L/v, is the same as the length of the same ruler at rest –there is no ‘length contraction’ effect. A similar calculation for the length of the ruler R as measured in the frame S’ gives, at t′ = L/v: x′[MR(10)]− x′[MR(0)] = x′[MR′(10)]− x′[MR′(0)] = L (4.13) The length of the moving ruler R as measured in S’, at t′ = L/v, is the same as the length of the same ruler at rest. The above calculations have used the equality of the spatial separations of O and O’ in S and S’ at equal times of mutually synchronised clocks in these frames, that follows from the RP, to establish, via corresponding PMCs, the equality of the measured lengths of a ruler at rest, or in motion. Note that nowhere in any of the calculations was the Lorentz transformation invoked. In fact the calculations are the same in Galilean and special relativity, since the RP is equally valid for both. 5 The time dilatation effect; proper, improper and apparent time intervals All the times considered above were ‘frame times’ i.e. t and t′ are the times recorded by a synchronised clock at rest at any position in S and S’ as viewed by an observer at rest in these respective frames. In order to discuss the time dilatation effect it will be found convenient to use the notation t(S), t′(S ′) for the frame times where the arguments S, S’ specify the reference frame of the observer of the clock. Such times are proper times of such a clock. The Lorentz transformation relates the space-time coordinates (x′,t′(S ′)) of an event specified in the frame S’ to those of the same event, (x,t′(S)) as observed in S, or vice versa. The times t(S ′)[ t′(S)] which are those of clocks at rest in S[S’], as viewed from S’[S] are called improper times. The space-time LT gives the following invariant interval relation between corresponding space and time intervals in the frames S and S’: c2(∆τ ′)2 = c2(∆t′(S))2 − (∆x)2 = c2(∆t′(S ′))2 − (∆x′)2 (5.1) where ∆x ≡ x2 − x1 etc, while the inverse LT gives: c2(∆τ)2 = c2(∆t(S ′))2 − (∆x′)2 = c2(∆t(S))2 − (∆x)2 (5.2) In order to use the general interval relation (5.1) to derive the time dilatation effect it is necessary to identify the time interval ∆t′(S ′) with the proper time interval of a clock at rest in S’ (∆x′ = 0), and with equation of motion in S: ∆x = v∆t′(S). Using the latter equation to eliminate ∆x from (5.1) and setting ∆x′ = 0 yields the time dilatation (TD) relation: ∆t′(S) = γ∆t′(S ′) (5.3) Figure 2: An experiment to illustrate the TD effcet viewed from S (left) and S’ (right). a) The pulsed lamp PL at rest in S flashes at time t(S) = L/v and PL’ at rest in S’ flashes at time t′(S ′) = L/v. b) The light signal from PL is observed at time t(S ′) = γL/v in the frame S’, that from PL’ at time t′(S) = γL/v in the frame S. The PMCs corresponding to the positions of observation of the signals in the different frames are indicated. See text for discussion. Figure 3: Spatial configurations in the frame S (left) and the frame S’ (right) are viewed at different times. a) t(S) = t′(S ′) = 0; the Λ is created and moves to the right in the plane of the figure with speed v = 3c/2. b) t(S) = t′(S ′) = T ′; the Λ is observed to decay in the frame S’. The decay products move in the plane of the figure perpendicular to the direction of motion of the Λ. c) t(S) = t′(S ′) = γT ′; the Λ is observed to decay in the frame S. See text for discussion. The momentum vectors of the p and π− are drawn to scale in the different reference frames. The spatial position of each particle is at the tail of the corresponding momentum vector. where γ ≡ 1/ 1− (v/c)2, relating the improper to the proper time of a clock at rest in S’. In a similar manner the interval relation (5.2) gives the TD relation for a clock at rest in S and observed from S’: ∆t(S ′) = γ∆t(S) (5.4) It is important to note the existence of four different time symbols, with different opera- tional meanings in Eqns(5.3) and (5.4). The proper times t(S) and t′(S ′) (corresponding to the ‘frame times’ t and t′ of the previous sections) and the improper times t(S ′) and t(S ′). The notation for these times just introduced may be called ‘clock oriented’ since only the readings of a single clock (observed either at rest, or in motion) appear in the TD relations. In any actual experiment where the TD effect in measured, two clocks are nec- essary, the observed moving clock, and another one at rest to measure the corresponding time interval in the observer’s proper frame. If a clock at rest in S’ is observed from S as in Eqn(5.3), the time interval ∆t′(S) is actually that, ∆τ , recorded by a similar clock, at rest in S while ∆t′(S ′) is the corresponding time interval recorded by the (slowed-down) moving clock. Since the observed rate of the moving clock depends on its motion, ∆t′(S ′) is not a proper time interval for the observer in S. From the view-point of the latter this is an ‘apparent’ (velocity-dependent) time interval that may be denoted simply as ∆t′, to distinguish it from the observer’s proper time interval ∆τ . This gives an alternative ‘observer oriented’ time notation for the TD relations (5.3) and (5.4) above: ∆τ = γ∆t′ (5.5) ∆τ ′ = γ∆t (5.6) This alternative notation has beeen employed in several previous papers by the present author [6, 8, 11, 12, 13, 16]. In order to apply the TD relations (5.3) and (5.4), or (5.5) and (5.6), to any actual or imagined experiment an operational definition must be given to the improper time intervals of Eqns(5.3) and (5.4) or the apparent time intervals of (5.5) and (5.6). Two examples of such definitions will be given, the first in a thought experiment to illustrate the physical meaning of the TD effect, the second in an actual experiment typical of many performed in particle physics, where the TD effect is used to measure the proper decay time of an unstable particle. However as will be seen, the thought experiment and actually realisable (and many times realised) one are similar in all essential features. What notation is most convenient depends on the experiment considered. In the observation of the TD effect in the last CERN muon g-2 experiment [17] where the time interval ∆τ was directly measured by clocks in the laboratory frame, and ∆t′ was the known muon rest-frame lifetime, it was natural to use Eqn(5.5). For the second of the two experiments considered below where ∆τ is not directly measured but inferred from spatial measurements in the frame S, the relation (5.3) relating connecting a proper time in the frame S’ to an improper time in the frame S, is used. In the thought experiment it is imagined that the objects O, O’ are each equipped with local pulsed lamps PL, PL’. The objects O, O’ are in spatial coincidence at times t(S) = t′(S ′) = 0 and are attached to rulers of length 2L in similar spatial configurations to that shown in Fig.1a. The objects move apart with relative velocity v = 3c/2. As shown in Fig.2a, at the times t(S) = t′(S ′) = L/v, PL and PL’ both flash, producing an isotropic pulse of photons. The observation times in S of the photon signal produced by PL’, and in S’ of the photon signal produced by PL, are given by Eqns(5.3) and (5.4) respectively. Since γ = 2, these observations occur at the times t(S) = t′(S ′) = γL/v = 2L/v. The corresponding spatial configurations of O and O’ at these times shown in Fig.2b. It can be seen that the observation times of the light flashes in S and S’ correspond to different PMCs of the objects O and O’ and to different spatial separations of the objects: In S PL : PMC(MR′(10), L/v) ≡ MR′(10)@0 = MR′(10)@MR(0) (5.7) PL′ : PMC(O′, γL/v) ≡ O′@MR(20) = MR′(20)@MR(20) (5.8) In S′ PL′ : PMC(MR(10), L/v) ≡ MR(10)@O′ = MR(10)@MR′(20) (5.9) PL : PMC(O, γL/v) ≡ O@MR′(0) = MR(0)@MR′(0) (5.10) ℓ(γL/v) ℓ(L/v) ℓ′(γL/v) ℓ′(L/v) vγL/v vL/v) = γ (5.11) The relations in (5.11) follow directly from the RP, while the PMC in (5.7)-(5.10) are obtained from the geometry of Fig.2 and the invariance of the lengths of the moving rulers derived in Section 3 above. The different PMC corresponding to observations of the light flashes emitted by PL and PL’ in different frames in (5.7)-(5.10) is deeply perplexing for common-sense con- cepts of space and time. For example the photon bunches emitted by PL’ correspond to MR(10)@MR′(20) in S’ and to MR′(20)@MR(20) in S. In some discussions of time dilatation this apparent paradox is avoided by invoking a hypothetical contraction of a moving ruler by a factor 1/γ [18]. This has the effect of shortening the moving ruler R by a factor 1/2 in the right hand figure in Fig.2a, so that the PMC corresponding to the flashing of PL’ becomes MR(20)@MR′(20), the same as in S with inversion of pointer and mark. However, as demonstrated above, there is no such length contraction effect, which, as pointed out elsewhere [5, 6, 7, 8, 9] is a spurious consequence of misin- terpreting the space-time Lorentz transformation. Indeed the possibility of such a length contraction effect is already excluded by inspection of Fig.2a. In the right hand figure, the PMC correponding to the moving object O considered as a pointer is MR(0)@MR′(10). Since O is in motion and R’ at rest no hypothetical length contraction effect operates here. In the left hand figure the mutually simultaneous PMC in S is MR′(10)@MR(0) so that at t(S) = t′(S ′) = L/v observers in S and S’ see reciprocal PMCs, i.e. ones related by exchange of the pointer and mark symbols. If however the length contrac- tion effect exists, the observer in S will see instead that the PMC corresponding to O is MR′(0)@MR(0) at time t(S) = L/v. But from the RP this PMC must correspond to the times t(S) = t′(S ′) = 2L/v (see Fig.2b) contrary to the assumption that t(S) = L/v. The length contraction hypothesis therefore contradicts the corollory of the RP that states that mutually simultaneous events in two frames have reciprocal PMCs, since it implies that the reciprocal PMCs MR′(0)@MR(0) and MR(0)@MR′(0) are not mutually simul- taneous. The second example of a TD experiment illustrates a typical application of the effect in particle physics (see Fig.3). A π− meson interacts with a proton in a thin plastic target T to produce a Λ hyperon via the reaction3 π−p → ΛK0 The hyperon moves with 3The results of an actual such experiment constructed to test the ∆S = ∆Q rule in semileptonic neutral kaon decays are described in Ref. [19]. velocity v = 3c/2 perpendicular to the plane of the target in the laboratory frame S. After the time t′(S ′) = T ′ in its rest frame S’, it decays to a proton and a negative pion: Λ → pπ−. These decay products are observed in the laboratory system. The experiment is in every way similar to that shown in Fig.2. The object O is replaced by the target T, the object O’ by the undecayed Λ or the kinematical system constructed from its decay products. The photon pulse emitted by PL’ is replaced by the decay products of the Λ. By reconstructing the trajectories of the decay p and π− in a particle detector the position of the decay event and hence the decay length lD –the distance between the point of production and decay of the Λ– in the frame S can be measured. Identification of the p and π− and measurement of their momenta (typically by measurement of the curvature of their trajectories in a known magnetic field ) enables the momentum P and the energy E of the Λ to be determined. Since v = Pc2/E and γ = E/(mΛc 2) where mΛ is the mass of the Λ, the proper decay time of the Λ is given by Eqn(5.3) as: T ′ = ∆t′(S ′) = ∆t′(S) (5.12) The spatial configurations of T and the Λ at different times in the frames S and S’ are shown in Fig.3. The spatial separations of T and the Λ at the observed instant of decay in S and S’ obey the relation (5.11). This implies that this separation, in changing the frame of observation from the rest frame of the Λ to the laboratory system in which it is motion, undergoes a ‘length expansion’ by the factor γ. In accordance with Eqn(5.11), it can be seen that this is a necessary consequence of the RP, given the existence of the TD effect. The mutally simultaneous events in S and S’ shown in Fig.3c, correspond, as they must, to equal spatial separations of T and the physical object constituted by the decay products, p and π−, of the Λ. However, in the frame S, these particles have just been created and have vanishing spatial separation, whereas in S’ they are spatially separated by a distance corresponding to a time-of-flight (γ− 1)T ′. This also seems highly paradoxical when interpreted by commonsense classical concepts of space and time. Acknowledgement I thank the referee of the journal that rejected Ref. [11] for publication for correspon- dence that was important for the clarificatiion of the ideas expressed in both the latest version of Ref. [11] and the present paper. Added Note The calculations presented in the present paper are flawed by a major conceptual misunderstanding which is rectified in later papers [20, 21] treating similar subjects. At the time of writing the present paper, the author had correctly understood the spurious nature of the ‘relativity of simultaneity’and ‘length contraction’ effects of con- ventional special relativity theory [5, 7, 8, 10] but had not yet drawn the simple conclusion that the existence of the genuine and experimentally-confirmed time dilatation effect then necessarily implies that the Reciprocity Principle, as generally understood, also breaks down in special relativity. This point is easily understood by considering the first member of Eqn(3.1), written in a simplified notation as: dxO′O Transforming into the frame S’, the invariance of length intervals implies that dxO′O = −dx Since the time dilatation relation gives dt = γdt′, the Reciprocity Principle of (3.1) is replaced by: dxO′O so that = −γv to be compared with v′ = −v given by (3.1). The detailed calculations presented in Section 4 are correct and logically coherent given the initial assumptions, but the configurations shown in the frame S’ in Fig.1 do not correspond to observations in this frame of the coincidence events specified in the frame S in the same space-time experiment. If this were the case, in the S’ frame configurations in Fig.1 v should be replaced by γv and t and t′ should be related by time dilatation relation t = γt′. In fact, what are shown in Fig.1 and considered in Section 4 are the configurations in S of a primary experiment and in S’ of the corresponding but physically independent reciprocal experiment [20, 22]. Nevertheless, the invariance of corresponding length intervals can be derived [21] by considering the configurations in S and S’ in Fig.1b in the case that they are corresponding ones, at the same epoch, in the same space-time experiment. In this case, as explained above, the speed of O in S’ should be γv, not v. Consider, however, an object Õ with the same x′ coordinate as O that does have the velocity v. The separation L′ of O and O’ in S’ is then equal to that between O’ and Õ. at the epoch of Fig.1b. Compare now the configuration of O and O’ in S, with separation L with the corresponding one of Õ. and O’ in S’ with separation L′. From the symmetry of the configurations it can be seen that both L and L′ can depend only on v: L = L(v), L′ = L′(v). The reciprocity of the two configurations is now invoked to give the condition, as stated by Pauli [23]: The contraction of length at rest in S’ and observed from S is equal to the length at rest in S as observed from S’. The ‘length at rest in S’ ’ is L′ which ‘as observed from S’ is L, whereas the ‘length at rest in S ’ is L which ‘as observed fron S’ ’ is L′. Denoting the contraction factor by α(v), the above condition states that L = α(v)L′, L′ = α(v)L which implies that L = α(v)2L or α(v)2 = 1 so that L = L′ and the spatial separation between O and O’ is the same in S and S’ at corresponding epochs. The same conclusion is more simply reached by noting the symmetry of the configurations of O,O’ in S and Õ,O’ in S’. and applying Leibnitz’ Principle of Sufficient Reason [21]. If, therefore, in the primary experiment, shown in S in Fig.2b and S’ in Fig.2a, the configuration in S’ in Fig.2a is to correctly represent that corresponding to the configura- tion in S in Fig.2b, the velocity v in S’ should be replaced by γv, so that when PL’ flashes O’ is aligned with MR(20) in both S and S’. In the reciprocal experiment, shown in S in Fig.2a and S’ in Fig.2b, v in S in Fig.2a should be replaced by γv so that O is aligned with MR’(0) in both S and S’ when PL flashes. Similarly, in the thought experiment of Fig.5, if the S’ frame configurations on the right side of the figure are to represent observations in this frame of events shown in S by the configurations on the left side, instead of what are actually shown which are configurations of the physically independent reciprocal experiment, v should be replaced by γv in all the S’ frame configurations. In this case, there is no mismatch between the spatial position of the decay event in the two frames and the claimed ‘length expansion’ effect does not occur. Indeed the claimed ‘... different PMC corresponding to observations of the light flashes emitted by PL and PL’ in different frames in (5.7)-(5.10)’ is not only ‘...deeply perplex- ing for common-sense concepts of space and time.’ it is the absurd (self-contradictory) consequence of assuming, at the same time, that length intervals are invariant, time di- latation occurs and the conventional interpretation of the Reciprocity Principle holds. In conventional special relativity theory time dilatation and the Reciprocity Principle are reconciled by invoking the spurious ‘length contraction’ effect dxO′O = −γx [18]. so that v′ = −v. The correct physical interpretation of the Reciprocity Principle is actually the definition of the configuration in S’ of the physically-independent experiment that is reciprocal to the primary one specified by the standard configuration of the frames S and S’ [20, 22]. References [1] A.Einstein,‘Zur Elektrodynamik bewegter Korper’, Annalen der Physik 17, 891 (1905). English translation by W.Perrett and G.B.Jeffery in ‘The Principle of Relativ- ity’ (Dover, New York, 1952) P37-P65, or in ‘Einstein’s Miraculous Year’ (Princeton University Press, Princeton, New Jersey, 1998) P123-P161. [2] W.Ignatowsky, ‘Einige allgemine Bermerkungun Zum Relatitivitäsprinzip’, Phys. Zeitschr. 11 972 (1910). [3] J.H.Field, ‘A New Kinematical Derivation of the Lorentz Transformation and the Particle Description of Light’, Helv. Phys. Acta. 70 542-564 (1997); arXiv pre-print: http://xxx.lanl.gov/abs/physics/0410262. Cited 27 Oct 2004. [4] V.Berzi and V.Gorini, ‘Reciprocity Principle and the Lorentz Transformation’, Journ. Math. Phys. 10 1518-1524 (1969). [5] J.H.Field, ‘The Local Space-Time Lorentz Transformation: a New Formulation of Special Relativity Compatible with Translational Invariance’, arXiv pre-print: http://xxx.lanl.gov/abs/physics/0501043. Cited 30 Nov 2007. [6] J.H.Field, ‘The physics of space and time I: The description of rulers and clocks in uniform translational motion by Galilean or Lorentz transformations’, arXiv pre- print: http://xxx.lanl.gov/abs/physics/0612039. Cited 28 Mar 2008. [7] J.H.Field, ‘Uniformly moving clocks in special relativity: Time dilatation, but but no relativity of simultaneity or length contraction’, arXiv pre-print: http://xxx.lanl.gov/abs/physics/0603135. Cited 4 Dec 2008. [8] J.H.Field, ‘Clock rates, clock settings and the physics of the space-time Lorentz transformation’, arXiv pre-print: http://xxx.lanl.gov/abs/physics/0606101. Cited 4 Dec 2007. [9] J.H.Field, ‘Absolute simultaneity: Special relativity without light signals or synchro- nised clocks’, arXiv pre-print: http://xxx.lanl.gov/abs/physics/0604010. Cited 6 Nov 2008. [10] J.H.Field, ‘Translational invariance and the space-time Lorentz transformation with arbitary spa- tial coordinates’, arXiv pre-print: http://xxx.lanl.gov/abs/physics/0703185. Cited 15 Feb 2008. [11] J.H.Field, ‘Relativistic velocity addition and the relativity of space and time inter- vals’, arXiv pre-print: http://xxx.lanl.gov/abs/physics/0610065. Cited 6 Feb 2009. [12] J.H.Field, ‘The train/embankment thought experiment, Einstein’s second pos- tulate of special relativity and relativity of simultaneity’, arXiv pre-print: http://xxx.lanl.gov/abs/physics/0606135. Cited 9 Jan 2009. [13] J.H.Field, ‘Muon decays in the Earth’s atmosphere, time dilatation and relativity of simultaneity’, arXiv pre-print: http://xxx.lanl.gov/abs/physics/0606188. Cited 22 Jan 2009. [14] I.B.Cohen and R.S.Westfall ‘Newton’ (W.W.Norton Company, New York, 1995) P233. [15] R.Mansouri and R.U.Sexl, ‘A Test Theory of Special Relativity: I Simultaneity and Clock Synchronisation’, Gen. Rel. Grav. 8, 497-513 (1977). [16] J.H.Field, ‘The physics of space and time II: A reassessment of Einstein’s 1905 special relativity paper’, arXiv pre-print: http://xxx.lanl.gov/abs/physics/0612041. Cited 14 Apr 2008. [17] J.Bailey et al., ‘Measurements of relativistic time dilation for positive and negative muons in circular orbit’, Nature 268, 301 (1979). [18] N.D.Mermin ‘Its About Time’ (Princeton University Press, Princeton 2005) Figure 6.3. P67. [19] J.C.Hart et. al ,‘ A test of the ∆S = ∆Q rule in Ke3 decay’, Nucl. Phys.B 66 317 (1973). [20] J.H.Field, ‘Primary and reciprocal space-time experiments, relativistic reciprocity re- lations and Einstein’s train-embankment thought experiment’, arXiv pre-print:http: //xxx.lanl.gov/abs/0807.0158. Cited 1 Jul 2008. [21] J.H.Field, ‘Space-time attributes of physical objects and the laws of space-time physics’, arXiv pre-print: http://xxx.lanl.gov/abs/0809.4121. Cited 24 Sep 2008. [22] J.H.Field, ‘The physics of space and time III: Classification of space-time experiments and the twin paradox’, arXiv pre-print: http://xxx.lanl.gov/abs/0806.3671. Cited 23 Jun 2008. [23] W.Pauli, ‘Relativitätstheorie’ (Springer, Berlin 2000). English translation,‘Theory of Relativity’ (Pergamon Press, Oxford, 1958) Section 4, P11.

0704.0364

B --> rho K* decays and other rare vector-vector modes

B → ρK∗ decays and other rare vector-vector modes ∗ G. Vasseur† DSM/DAPNIA/SPP, CEA/Saclay, F-91191 Gif-sur-Yvette, France The recent analyses of the following rare vector-vector decays of the B meson are presented: ∗, ωK∗, ωρ, ωω, and ωφ charmless final states. The latest results indicate that the fraction of longitudinal polarization is about 0.5 in penguin-dominated modes and close to 1 for tree-dominated modes. I. MOTIVATION The search for rare charmless hadronic decays of the B meson to vector-vector final states has become a quite active field in the experiments at the B factories, Belle at KEK and BABAR at SLAC. As a lot of these decays have not yet been seen, the first goal of these studies is to observe such modes and measure their branching fraction. The measurements can then be compared to theoretical predictions. The direct CP-violation asymmetry in these modes can also be measured. It is defined as ACP = (Γ Γ+)/(Γ−+Γ+), where the superscript on the total width Γ indicates the sign of the b-quark charge in the B me- son. Some modes can be used for further CP studies. In fact, the result on B+ → ρ+K∗0 has already been used to constrain the effect of the penguin amplitude on the measurement of the angle α of the unitarity triangle from B0 → ρ+ρ− using SU(3) flavor symmetry [1]. A hot topic is the measurement of the fraction of lon- gitudinal polarization. The helicity angles θ1 and θ2 of the two vector mesons are defined as the angles between the vector meson direction in the B meson rest frame and the direction of one of its decay product in the vec- tor meson rest frame, as illustrated on one example in Fig. 1. Integrating over the φ angle between the decay planes of the two vector mesons, the fraction of longitu- dinal polarization fL can be extracted from the angular θK*0 θρ +K*0 FIG. 1: Definition of the helicity angles in the case of the vector-vector B+ → ρ+K∗ decay. ∗Presented at the 4th International Workshop on the CKM Uni- tariry Triangle, Nagoya, Japan, December 12-16, 2006. Preprint DAPNIA-06-601. †Electronic address: [email protected] dependence of the decay rate, which is proportional to (1− fL) sin 2 θ1 sin 2 θ2 + fL cos 2 θ1 cos 2 θ2. A value of fL close to unity of order (1−O( 2 )) is ex- pected for light vector mesons from helicity conservation. This is expected to be true for both tree and penguin dia- grams. However the experimental situation is more com- plex. If fL has indeed been measured close to 1 in the tree-dominated B → ρρ modes [2], it is surprisingly close to 0.5 in the penguin-dominated B → φK∗ modes [3]. This effect is not yet understood. There are several possi- ble explanations, either within the Standard Model, such as rescattering in the final state, contribution from anni- hilation or electroweak penguin diagrams, and transverse gluon [4], or in new physics outside the Standard Model. To have a better picture, it is important to measure other vector-vector modes, both tree-dominated, like B → ωρ and B0 → ωω, and penguin-dominated like B → ρK∗ and B → ωK∗. The recent studies of the B → ρK∗ modes are reviewed in section II, the ones involving an ω meson in section III. Charge-conjugate modes are implied throughout. II. B → ρK MODES A. Introduction The B → ρK∗ charmless decays proceed through dominant gluonic penguin loops and doubly Cabibbo- suppressed tree processes, as shown in Fig. 2. The ex- ternal tree diagram is only possible with a K∗+, and the color-suppressed internal tree diagram with a ρ0. Hence B+ → ρ+K∗0 is pure penguin. According to isospin symmetry, the two modes with a charged ρ are expected to have a branching fraction twice as large as the two modes with a neutral ρ. FIG. 2: Feynmann diagrams for the B → ρK∗ decay: gluonic penguin, external tree and internal tree diagrams. http://arxiv.org/abs/0704.0364v1 mailto:[email protected] B. Results from Belle Mbc (GeV/c ∆E (GeV) FIG. 3: Projections of Mbc for events in the ∆E signal region (left) and of ∆E in the Mbc signal region (right). The solid curves show the results of the fit. The dashed curve is the signal contribution. The hatched histograms represent the continuum background. The sum of the b → c and continuum background component is shown as dot-dashed lines. 0.4 0.8 1.2 1.6 2.0 M(π+π0) (GeV/c2) 0.64 0.84 1.04 1.24 1.44 M(K+π-) (GeV/c2) FIG. 4: Signal yields obtained from the Mbc-∆E distribution in bins of M(π+π0) (left) for events in the K∗0 region and in bins of M(K+π−) (right) for events in the ρ+ region. The points with error bars show the data. Solid curves show the results of the fit. Hatched histograms are for the nonresonant component. Belle was the first experiment in 2005 to publish a re- sult on the observation of the B+ → ρ+K∗0 mode [5], on a sample of 275 millions of BB̄ pairs. A signal of B+ → π+π0K+π− is extracted from the e+e− → qq̄ continuum and BB̄ backgrounds in an extended un- binned maximum-likelihood fit using the B meson beam- constrained mass Mbc and energy difference ∆E, as shown in Fig 3. The B+ → ρ+K∗0 signal is extracted by fits to Mbc and ∆E in bins of the vector meson masses M(π+π0) and M(K+π−), as shown in Fig 4. This is necessary because there is a large nonresonant ρKπ background, which gives a continuum in the distribution of M(K+π−). Nethertheless there is a clear B+ → ρ+K∗0 signal of 85± 16 events with a significance of 5.2 σ. As for fL, it is obtained by fitting simultaneously the signal yields obtained from Mbc-∆E fits in bins of the two helicity angles, assuming an S-wave Kπ system in the ρKπ background. The results for the branching fraction and fL in B → ρ+K∗0 are: B = (8.9± 1.7± 1.2) 10−6, fL = 0.43± 0.11 +0.05 −0.02. The value found for fL is similar to the one found in φK and its error is about twice as large as in φK∗. C. Results from BABAR (GeV)0π+πm 0.5 1 1.5E (GeV)0π+πm 0.5 1 1.5E (GeV)-π+πm 0.6 0.8 1 1.2 1.4E (GeV)-π+πm 0.6 0.8 1 1.2 1.4E (GeV)-π+Km 0.8 1 1.2 1.4E (GeV)-π+Km 0.8 1 1.2 1.4E (GeV)-π+Km 0.8 1 1.2 1.4E (GeV)-π+Km 0.8 1 1.2 1.4E FIG. 5: sPlots for the ππ (top) and Kπ (bottom) invariant masses in the B+ → ρ+K∗ (left) and B0 → ρ0K∗ /B0 → f0(980)K ∗0 (right) analyses. The points with error bars show the data. The solid curve shows the signal and nonresonant background contribution, the dashed curve is the nonreso- nant background contribution (ρKπ except for the top right plot where it represents the sum of f0(1370)K ∗, ππK∗, and ππKπ). The arrows show the standard mass windows used in the final fit. More recently BABAR published an anlysis of all four B → ρK∗ modes [6], performed on a sample of 232 mil- lions of BB̄ pairs. It is based on an unbinned maximum- likelihood fit, using seven variables: the B meson energy- substituted mass mES and energy difference ∆E, a neural network output or a Fischer discriminant combining sev- eral event shape variables, the two vector meson masses, and the two helicity angle cosines. The fit allows the simultaneous extraction of the branching ratio and the fraction of longitudinal polarization. The major challenge in the analysis comes from the nonresonant backgrounds, which share the same final state as the signal. They are studied by enlarging the vector meson mass windows, as illustrated in Fig. 5. As in Belle, a large ρKπ background is seen in the mKπ dis- tribution in the B+ → ρ+K∗0 mode. The Kπ system in this background is measured to be mostly S-wave. The situation is even more complex in the B0 → ρ0K∗0 mode, since in addition to the ρKπ background there are sev- eral contributions seen in the mππ distribution for a ρ in contrast to the one for a ρ+. The f0(980) can be seen clearly. In fact B → f0(980)K ∗, which is a scalar-vector TABLE I: Results from BABAR on the B → ρK∗ modes: signal yield with its statistical uncertainty, significance (systematic un- certainties included), branching fraction (90% confidence level upper limit in parentheses), fraction of longitudinal polarization and direct CP asymmetry. (The numbers in brackets are not quoted as measurements.) Mode Signal yield Significance (σ) B(×10−6) fL ACP ∗+ 51± 24 2.5 < 6.1 (3.6± 1.7± 0.8) [0.9 ± 0.2] ∗+ 60± 24 1.6 < 12.0 (5.4± 3.6± 1.6) ∗0 194± 29 7.1 9.6± 1.7± 1.5 0.52 ± 0.10± 0.04 −0.01 ± 0.16± 0.02 ∗0 185± 30 5.3 5.6± 0.9± 1.3 0.57 ± 0.09± 0.08 0.09 ± 0.19± 0.02 a) b) c) d) (GeV)ESm (GeV)ESm 5.25 5.26 5.27 5.28 )2 (GeV/cESm )2 (GeV/cESm 5.26 5.27 5.28 5.29 FIG. 6: Projections of mES of events passing a signal likeli- hood threshold for (a) B+ → ρ0K∗ , (b) B+ → ρ+K∗ , (c) ∗+, (d) B0 → ρ0K∗ , (e) B+ → f0(980)K ∗+, and (f) B0 → f0(980)K ∗0. The points with error bars show the data. The solid curve is the fit function, the dashed curve is the total background contribution, and the dotted curve is the continuum background contribution. mode, is considered as another signal to be measured in the same maximum-likelihood fit. Also present are con- tributions from the f0(1370) and nonresonant ππ. The yields of the nonresonant backgrounds are fitted in the enlarged mass windows, then extrapolated to the stan- dard ones and fixed in the final fit with the standard mass windows. The projection plots in the B mass shown in Fig. 6 il- lustrate the extraction of the signal from the continuum and BB̄ backgrounds in the four B → ρK∗ channels and the two B → f0(980)K ∗ modes. Table I summarizes the results. No significant enough signals are observed for B0 → ρ−K∗+ and B+ → ρ0K∗+, where upper limits at the 90 % confidence level are set on the branching ratios. For the latter a related signal B+ → f0(980)K ∗+ is ob- served with a significance of 5.0 σ and a measured branch- ing fraction of (5.2 ± 1.2 ± 0.5) 10−6. In B+ → ρ+K∗0, the result is in very good agreement with the result from Belle, with a similar precision. The B0 → ρ0K∗0 mode is observed for the first time. The ratio between the branch- ing fractions in these two modes is compatible with the factor 2 expected from isospin symmetry. The value of ACP is measured in the two significant modes to be compatible with 0, as expected since there is one dominant diagram. Finally fL is found close to 0.5 in these two modes. It is compatible with the measurement from Belle and has about the same precision. It is again similar to the value found for φK∗. III. MODES WITH ω 40 +ρω 10 ωω 20 *0Kω 20 0ρω 40 +ρω 20 ωω E (GeV) ∆ -0.2 -0.1 0 0.1 0.2 (GeV/cESM 5.25 5.26 5.27 5.28 5.29 20 0fω FIG. 7: Projections of ∆E (left) and mES (right) of events passing a signal likelihood threshold for, from top to bottom, ∗0, B+ → ωK∗ , B0 → ωρ0, B+ → ωρ+, B0 → ωω, B0 → ωφ, and B0 → ωf0(980). The points with error bars show the data. The solid curve is the fit function, the dashed curve is the signal contribution, and the dot-dashed curve is the background contribution. TABLE II: Results from BABAR on modes involving an ω meson: signal yield with its statistical uncertainty, significance (systematic uncertainties included), branching fraction (90% confidence level upper limit in parentheses), fraction of longitudinal polarization and direct CP asymmetry. (The numbers in brackets are not quoted as measurements.) Mode Signal yield Significance (σ) B(×10−6) fL ACP ∗0 55± 20 2.4 < 4.2 (2.4± 1.1± 0.7) [0.71 ± 0.25] ∗+ 8± 16 0.4 < 3.4 (0.6± 1.3± 1.0) −18± 16 0.6 < 1.5 (−0.6± 0.7+0.8−0.3) + 156± 32 5.7 10.6± 2.1+1.6−1.0 0.82 ± 0.11± 0.02 0.04 ± 0.18± 0.02 → ωω 48+24−19 2.1 < 4.0 (1.8 −0.9 ± 0.4) [0.71 ± 0.25] → ωφ 3.1±+4.4−8.5 0.3 < 1.2 (0.1± 0.5± 0.1) ρθcos -0.5 0 0.5 | ωθ|cos 0 0.5 1 FIG. 8: Projections of the helicity-angle cosines for ω (left) and ρ+ (right) of events passing a signal likelihood threshold from the fit for B+ → ωρ+ decays. The points with error bars show the data. The solid curve is the fit function, the dashed curve is the signal contribution, and the dot-dashed curve is the background contribution. On the same sample of 232 millions of BB̄ pairs, BABAR has also recently published a search for several vector-vector modes involving an ω meson [7]: B0 → ωK∗0, B+ → ωK∗+, B0 → ωρ0, B+ → ωρ+, B0 → ωω, and B0 → ωφ. The related vector-scalar mode B0 → ωf0(980) was also searched for. An earlier search for B → ωK∗ and B → ωρ on 89 millions of BB̄ pairs resulted in the first observation of the B+ → ωρ+ chan- nel [8]. The analysis is also based on an extended unbinned maximim-likelihood fit using the same seven variables as in the previous section. Nonresonant ππ and Kπ back- grounds are fixed in the fit as determined from extrap- olations from higher-mass regions. The projection plots of ∆E and mES of Fig. 7 illustrate the extraction of the signal from the continuum and BB̄ backgrounds in all these modes. In most of them, no significant enough sig- nal is seen. The only channel where a significant signal is observed is B+ → ωρ+. Its measured branching fraction is about 2 standard deviations smaller than the one of B+ → ρ+ρ0 [2], while these two branching fractions are naively expected to be equal. Table II summarizes the results in all the modes. To calculate the branching frac- tion, fL is left free in the fit for the three modes with a signal significance greater than 2σ and is fixed otherwise. Upper limits at the 90 % confidence level are set on the branching fractions for the modes other than B+ → ωρ+. The maximum-likelihood fit also provides the value of fL in B → ωρ+, which is found to be 0.82 ± 0.11, a high value expected for this tree-dominated mode. This is illustrared in the projection plots of the helicity angle cosines shown in Fig. 8. The direct CP asymmetry is also measured and found to be compatible with 0. IV. CONCLUSION In summary, improved analyses with explicit consider- ation of nonresonant backgrounds have been performed on several charmless hadronic vector-vector decays of the B meson. The B+ → ωρ+, B+ → ρ+K∗0, and B0 → ρ0K∗0 modes have been observed and measured in the past few years. Improved upper limits have been set on the branching fraction of other vector-vector modes. The recent results on vector-vector modes have also brought more pieces to the polarization puzzle. The penguin-dominated B+ → ρ+K∗0 and B0 → ρ0K∗0 modes have a fraction of longitudinal polarization of about 0.5 like φK∗, while the tree-dominated B+ → ωρ+ mode has one closer to 1 like ρρ. As a lot of charmless vector-vector modes have not yet been observed, new re- sults can be expected with more data. [1] M. Beneke et al., Phys. Lett. B 638, 68 (2006). [2] A. Somov, contribution to this conference. [3] K.F. Chen, contribution to this conference. [4] G.W.S. Hou, contribution to this conference. [5] J. Zhang et al., Phys. Rev. Lett. 95, 141801 (2005). [6] B. Aubert et al., Phys. Rev. Lett. 97, 201801 (2006). [7] B. Aubert et al., Phys. Rev. D 74, 051102 (2006). [8] B. Aubert et al., Phys. Rev. D 71, 031103 (2005).

0704.0365

Extending the theory of phonon-mediated superconductivity in quasi-2D

arXiv:0704.0365v1 [cond-mat.supr-con] 3 Apr 2007 7 Extending the theory of phonon-mediated superconductivity in quasi-2D J.P.Hague Department of Physics, Loughborough University, Loughborough, LE11 3TU Abstract. I present results from an extended Migdal–Eliashberg theory of electron-phonon inter- actions and superconductivity. The history of the electron-phonon problem is introduced, and then study of the intermediate parameter regime is justified from the energy scales in the cuprate su- perconductors. The Holstein model is detailed, and limiting cases are examined to demonstrate the need for an extended theory of superconductivity. Results of the extended approximation are shown, including spectral functions and phase diagrams. These are discussed with reference to Hohenberg’s theorem, the Bardeen–Cooper–Schrieffer theory and Coulomb repulsion. [Published in: Lectures on the physics of highly correlated electron systems X, p255-264, AIP Conference Proceedings vol. 846 (2006)] INTRODUCTION Over the past half-century, the study of the role of electron-phonon interactions in condensed matter physics has been an active and controversial field. Initially of interest from the point of view of thermal properties, early models of the interactions between lattice vibrations and electrons included the continuum Fröhlich model [1]. Interest in electron-phonon interactions increased dramatically when in 1957, Bardeen, Cooper and Schrieffer (BCS) published their famous theory of superconductivity [2], which directly implicated phonons as the microscopic mechanism for the low temperature absence of resistivity in a variety of metals. Until the discovery of the cuprate superconductors by Bednorz and Müller in 1986 [3], the BCS picture was found to account well for all superconducting materials - a remarkable success for a simple mean-field theory which is only applicable at weak coupling! Soon after the realisation that phonons were responsible for superconductivity, Eliash- berg extended the theoretical description beyond the absolute weak coupling theory with the famous Eliashberg equations [4]. In doing this, he built on the earlier work of Migdal, who argued that a simple resummation of a certain class of Feynmann diagrams should be sufficient to describe the limit of low phonon frequency [5]. Eliashberg’s theory can be argued to be one of the first applications of the dynamical mean-field theory (DMFT) [6], since (in its original sense) it ignores spatial fluctuations (momentum dependence) in the self-energy, while keeping frequency dependent (dynamical) effects. The purpose of this paper is to describe an extension to the theory of superconductivity from electron-phonon interactions. The approach goes beyond the Eliashberg theory by introducing the effects of spatial fluctuations and higher order terms in the perturbation theory. The aim is to develop a theory which can be used for systems with stronger coupling, larger phonon frequencies and reduced dimensionality. I begin by motivating http://arxiv.org/abs/0704.0365v1 the need for a more sophisticated theory from the experimental viewpoint. I also discuss limiting cases of the Holstein model, and how the large phonon frequency limit of that model implies that the conventional theories of superconductivity are incomplete. I then introduce the approximations needed to develop a more sophisticated theory. Finally I present some results from the new approximation, and discuss them in relation to Cuprate superconductors, and also with regard to conventional theories and the exact Hohenberg theorem [7]. MOTIVATION When the high-temperature cuprate superconductors were discovered in 1986 [3], the possibility that phonons could be attributed to the microscopic mechanism was quickly discounted by many people. In part, this was due to the absence of an isotope effect at optimal doping, and also an assumption that phonon-mediated superconductivity could not occur above 30K. The mechanism for high-TC superconductors remains highly controversial, and many different hypotheses are suggested (some examples are spin fluctuations [8] and exotic phonon mechanisms such as bipolarons [9]). An increasing body of evidence shows that phonons as well as Coulomb repulsion have an effect on the physics of the cuprate materials. I shall give a brief review of the current experimental situation in this section, and argue that (1) Electron-phonon interactions need to be treated on an equal footing to Coulomb repulsion if the Cuprates are to be understood, and (2) In order to treat the phonons in the Cuprates, extensions to the current theories of electron-phonon interactions and phonon-mediated superconductivity are required. There are several experiments demonstrating strong electron-phonon coupling in the cuprates. The most compelling is the existence of a strong isotope effect on exchanging O16 for O18 [10]. There are also some more recent experiments which demonstrate the effects of electron-phonon interactions in a transparent manner. Figure 1 shows schematic representations of electron and phonon dispersions in the cuprates. Panel (a) details the main features of the electronic dispersion measured by Angle-Resolved Photo-Emmission Spectroscopy (ARPES) in the [11] direction [11]. At energies close to the Fermi-surface, there are coherent excitations with a long lifetime. As εk = |ω0 − εF | is approached, the gradient of the dispersion changes at a sharp kink. The phonon is of the transverse optic variety, and its frequency (ω0) is of the order of 100meV. It suffices here to mention that this is very large. The ratio of the gradients above and below the kink is related to the dimensionless coupling constant (λ = g2/tω0), and it is found that λ can take values of up to 2 [11]. Panel (b) shows a schematic representation of some neutron scattering results measuring the phonon dispersion [12, 13]. Above the transition temperature, this looks like the solid line, but as the system moves from normal to superconducting state, the spectral weight in the circled area vanishes. This indicates that the superconductivity (bound pairs of electrons) affects the phonons, and is additional evidence for a strong electron-phonon coupling. A frequent misconception about the cuprates is that electron-phonon terms in the Hamiltonian can be neglected on the basis that they are small. To demonstrate that this is not the case, figure 2 shows approximate energy scales in the cuprates. The largest energy by far is the Coulomb repulsion (or Hubbard U ) which weighs in at some 10eV. (a) (b) INCOHERENT COHERENT WEIGHT LOST ON TRANSITION kk F 0 PHONON DISPERSIONELECTRON DISPERSION FIGURE 1. Schematics showing the effect of electron-phonon interactions on the electron and phonon dispersions in the cuprates. Both panels describe measurements along the [11] direction. Panel (a) shows a schematic representation of the electronic dispersion measured by Angle-Resolved Photo-Emmission Spectroscopy (ARPES) [11]. At energies close to the Fermi-surface, there are coherent excitations with a long lifetime. As εk = |ω0 − εF | is approached, the gradient of the dispersion changes and a kink is introduced. The phonon is of the transverse optic variety, and its frequency (ω0) is ∼ 75meV. The ratio of the gradients above and below the kink is related to the coupling constant [11]. Panel (b) shows a schematic representation of some neutron scattering results measuring the phonon dispersion [12, 13]. Above the transition temperature, this looks like the solid line, but as the system moves from the normal to the superconducting state, the spectral weight in the shaded area vanishes. This indicates that the superconducting state affects the phonons, and is further evidence for strong electron-phonon coupling. Next is the intersite hopping integral t, which is of the order of 1eV. Using a simple 2nd order perturbation theory at strong coupling, an effective exchange interaction is generated [14], with J = t2/U of the order of 100meV. This J often used to argue for a spin-fluctuation theory of high-TC superconductivity that neglects phonons. The problem with this viewpoint is immediately clear if one reviews the experimental data. First, the energies of the phonons are also approximately 100meV, so they cannot be treated as a small energy scale. Second, a dimensionless coupling constant of order unity implies dimensionfull coupling g with similar magnitude. Thus with three very close energy scales, it is important that the contributions from both phonon and Coulomb mechanisms are treated on equal footing in a theory for the cuprates. Unfortunately, as I discuss in the next section, current theories of electron-phonon interactions are not capable of handling the large phonon energies and coupling constants in the cuprates. The remainder of this paper focuses on how the theory can be extended to describe this regime. MODEL AND LIMITS A generic model of electron-phonon interactions includes the motion of the electrons Hel, the motion of the ions (or phonons) Hph and the interaction between the electrons and the phonons (which may be absorbed or emitted) which is denoted Hel−ph. In this Energy: 10meV 100meV 1eV 10eV T J t U FIGURE 2. Schematic showing the energy scales in the cuprates. The largest energy by far is the Coulomb repulsion (or Hubbard U) of order 10eV. The intersite hopping integral t, is ∼1eV. Using a simple 2nd order perturbation theory, an effective exchange interaction is generated, with J = t2/U of the order of 100meV. This J is then used to argue for the spin-fluctuation theory of high TC. However, the energies of the phonons are also approximately 100meV and the dimensionful coupling g has around the same value. Thus with 3 similar energy scales, it is important that the contributions from both spin- fluctuations and phonon mechanisms are treated on equal footing. way, H = Hel +Hel−ph +Hph is the total Hamiltonian. Hel = ∑ kck ≈− ∑ <i j>σ tc†iσ c jσ (1) Hel−ph =−∑ k−qck(b q +b−q)≈−∑ niσ gri (2) Hph = ∑ b†kbk + Mω20 r The first term in the Hamiltonian is the general form for free electrons, i.e. the total energy is the sum of the kinetic energies of all occupied states. In a special case, which is known as the Holstein Hamiltonian, the electrons in a tight binding model may hop between nearest-neighbour sites only, and εk = −2t ∑Di=1 cos(ki), where t is the overlap integral. In the generic form of the electron-phonon interaction, an electron may be scattered by absorbing a phonon with momentum −q or emitting a phonon with momentum q. An additional approximation uses a momentum independent electron- phonon coupling, g, and in that case the Fourier transform shows that the second term connects the local ion displacement, ri to the local electron density. Finally, the free phonon term may be simplified by using the Einstein approximation ωk ≈ ω0 and Fourier transforming, the bare phonon Hamiltonian is shown to be a series of independent simple harmonic oscillators at each site index. The creation of electrons and phonons is represented by c† and b† respectively, pi is the ion momentum and M the ion mass. By choosing t = 0.25, a bandwidth of W = 2 is chosen. A small interplanar hopping of t⊥ = 0.01 is included to remove the logarithmic singularity in the 2D density of states at ε = 0. Figure 3 shows the parameter space of the Holstein model. For very large phonon frequency, the effective interaction is instantaneous, and a Lang–Firsov transformation [15] results in an attractive Hubbard model (which is one of the standard models for correlated electron systems) [16]. Alternatively, taking the limit of very small phonon frequency, a fast moving electron cannot ‘see’ the nuclei move in the time it takes to EXTENDED THEORY CUPRATE? FIGURE 3. Parameter space of the Holstein model. For very large phonon frequency, the effective interaction is instantaneous, and a Lang–Firsov transformation results in an attractive Hubbard model. Alternatively, taking the limit of very small phonon frequency, a fast moving electron cannot ‘see’ the phonons move, and the problem maps to a static disorder problem (similar to the Falikov–Kimball model [19]). This makes the phonon problem extremely hard, and little is known about the middle of the parameter space. The range of the Eliashberg theory is shown in the bottom left corner. The expected position of the cuprates is shown as the single diamond. The expected validity of an extended theory including all 2nd order Feynman diagrams is also shown. traverse many sites, so the problem maps to a static disorder problem (which is essen- tially uncorrelated). One may therefore think of the phonon frequency as possessing the ability to “tune” the effect of correlations, and one therefore obtains a second motivation for the study of electron-phonon systems of trying to understand electronic correlations [17]. The correlation tuning makes the phonon problem extremely hard, and little is known about the intermediate regime of the parameter space. The range of the Eliash- berg theory is shown in the bottom left corner. Contrary to Migdal’s assumption, the theory cannot extend beyond intermediate coupling since renormalisation of the effec- tive mass reduces εF invalidating the condition (Migdal’s theorem) ω0 ≪ εF [18, 9]. The approximate position of the phonon parameters in the cuprates is shown as the single di- amond. It is essential to correct the theory for weak to intermediate coupling at larger phonon frequencies. The extension is clear by looking at the large phonon frequency limit. The Hubbard limit requires that all 2nd order processes in U are included in the self-energy, or the incorrect weak coupling limit is found. An extended theory including all 2nd order Feynman diagrams is required to understand the weak coupling limit, from small to large phonon frequency. FIGURE 4. Series of Feynman diagrams used in the current approximation. Σ is the electron and Π the phonon self-energy. Series (a) is the Migdal-Eliashberg approximation and (b) the vertex corrected series. EXTENDING THE ELIASHBERG THEORY Extending the Eliashberg theory involves inserting the lowest order vertex corrections into the electron and phonon self energies. In the Eliashberg theory, emitted phonons are reabsorbed in a last-out-first-in order. Vertex corrections essentially allow this order to be changed once. Such contributions are shown diagrammatically in figure 4. All the diagrams must be included in the calculation, or electron number would not be conserved. Momentum dependence is included in the approximation, which is essential in low-dimensions. The inclusion of vertex corrections leads to double 2-fold integration over the Brillouin zone in combination with a double sum over matsubara frequencies, which is time consuming for the numerics. In order to reduce the number of points in k- space while maintaining the thermodynamic limit, the dynamical cluster approximation is applied [20]. Additionally, superconducting states can be considered by using the Nambu formalism. The full details of the implementation of the extended approximation can be found in references [21] and [22]. Using a maximum entropy technique, it is possible to compute the spectral function from the Matsubara axis Green function. Figure 5 shows the spectral function of the Holstein model calculated using the extended Migdal–Eliashberg theory. The results are qualitatively similar to ARPES measurements of the cuprates. In particular the change between incoherent and coherent particles occurs at the phonon frequency (shown as the dashed line), associated with a kink in the [11] direction. It is noted here that the effect of the phonon self-energy is a softening of the phonon mode. In the standard ME theory in 2D, the mode at the (π ,π) point is completely softened, leading to a fatal instability of the theory. However, the vertex corrections act against this softening, and relieve the instability. In such a way, it is clear that a vertex corrected Eliashberg theory is essential for the study of quasi-2D materials [21]. One can also compute properties in the superconducting state. One such property is the momentum-dependent pairing density, ns(k) = T ∑n F(iωn,k), where F(iωn,k) is the anomalous Green function associated with the pairing of electrons with momentum k and −k. It is possible to transform the momentum dependent order parameter to determine the magnitude of individual spherical harmonics. Figure 6 shows such a decomposition. A cluster size of NC = 64 is used, with U = 0.6 and ω0 = 0.4. Note ω0=0.2, U=0.3, DCA(VC) -2 -1 A(k,ω) FIGURE 5. Spectral function of the Holstein model in the extended Migdal–Eliashberg theory. The results are qualitatively similar to ARPES measurements of the cuprates. In particular the change between incoherent and coherent particles occurs at the phonon frequency, associated with a kink in the [11] direction. ©Institute of physics publishing 2003 [21]. how higher order harmonics develop as the filling is increased. In particular, it can be seen that no single harmonic (such as the s-wave symmetry) is sufficient to describe the order parameter. Some of the higher order terms come about due to increased pairing at momentum k = (π/2,π/2), in particular, pairs with angular momentum. Finally, by varying the temperature and chemical potential, the phase diagram can be computed. Figure 7 shows phase diagrams of the Holstein model for the different approximations. U = 0.6 and ω0 = 0.4.The top diagram shows the result from the Eliashberg approximation (dynamical mean-field theory NC = 1). On the bottom the results from the current approximation with NC = 4 are shown. The superconducting order is suppressed close to half filling. Assuming a form for the density of states in 2D (with small interplane hopping) of D(ε) = (1− t log((ε2+ t2⊥)/16t 2))/tπ2 (for |ε|< 4t) [23], which matches the full density of states with reasonable accuracy. From this the BCS result may be calculated using the expression TC(n) = 2ω0 exp(−1/|U |D(µ(n)))/π , (4) with the chemical potential taken from the self-consistent solution for a given n. This result also drops off monotonically. Results in the dilute limit are in good agreement with the BCS result (line with points). Close to half-filling, the DMFT result is significantly smaller than the BCS result (which predicts TC(n = 1)> 0.07). The difference in results -0.025 -0.02 -0.015 -0.01 -0.005 0.005 0.01 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 s, m=0 d, m=0 g, m=0 g, m=4,-4 FIGURE 6. Decomposition of the order parameter into spherical harmonics. A cluster size, NC = 64 is used, with U = 0.6 and ω0 = 0.4. Note how higher order harmonics develop as the filling is increased. In particular, the g harmonics can be almost as strong as the s harmonics at n = 1.45. ©Institute of physics publishing 2005 [22]. between the two mean-field theories at half-filling is due to the self-consistency in the DMFT. When vertex corrections and spatial fluctuations are included, the dilute limit is relatively unchanged. However at half-filling, there is a huge drop in the transition temperature. The suppression at half-filling is a manifestation of Hohenberg’s theorem, which implies that there may be no superconducting order in 2D. Here I have computed for quasi-2D, so it is interesting that in real materials with low dimensional character the maximum in superconductivity is shifted away from half-filling. CONCLUDING REMARKS I end the paper with a warning for constructing theories of high-temperature supercon- ductivity using electron-phonon interactions alone, while neglecting the Coulomb re- pulsion. If one takes the phase diagrams from the previous section, and assigns similar energy scales to those in the cuprates, it is possible to obtain a temperature in Kelvins for the maximum in the phase diagram at n = 1.2. This comes out as around 172K - one could say approximately the TC in the cuprates. So why isn’t this the solution for the cuprates? Cuprates are very tightly bound ma- terials, which is why the “Fermi energy” is low, and the ratio ω0/εF is large enough to justify extending Eliashberg theory. The problem is that a small Fermi energy also means the the Hubbard U is a comparatively large quantity. On a simple mean-field level, one can include the Coulomb repulsion in the theory of superconductivity. For ex- ample, the Eliashberg equations can be extended to include an effective electron-electron interaction (otherwise known as the Coulomb pseudopotential µC). The effect of this is to modify λ → λ − µC. Substitution into equation 4 means that the transition temper- 0.02 0.04 0.06 0.08 U=0.6, ω0=0.4, Nc=4, VC 0.01 0.02 0.03 0.04 1 1.2 1.4 1.6 1.8 0.02 0.04 0.06 0.08 0.02 0.04 0.06 0.08 0.01 0.02 0.03 0.04 1 1.2 1.4 1.6 1.8 0.02 0.04 0.06 0.08 U=0.6, ω0=0.4, Nc=1 FIGURE 7. Phase diagrams of the Holstein model. U = 0.6 and ω0 = 0.4. The top diagram shows the result from the Eliashberg approximation (dynamical mean-field theory NC = 1). Also shown is the BCS result (line with points). On the bottom the results from the current approximation with NC = 4 are shown. The superconducting order is suppressed close to half filling in the vertex corrected theory. ©Institute of physics publishing 2005 [22]. ature is considerably reduced, or that superconductivity of the BCS type is completely destroyed. Any phonon-based mechanism for the cuprates must address this point and be compatible with the electron-electron interaction. Alternatively (and this is a warn- ing against the other extreme) on the basis of the similarity of energy scales, any spin- fluctuation mechanism (which is essentially Coulombic) must also treat the phonons (or at least be compatible with them) to be plausible. ACKNOWLEDGMENTS I sincerely thank the organising committee of the course for their generous financial support. Aspects of this research were carried out under the MPIPKS guest scien- tist program, and as a visitor at the University of Leicester. I thank A.S.Alexandrov, J.L.Beeby, E.M.L.Chung, N.d’Ambrumenil, J.K.Freericks, M.Jarrell, P.E.Kornilovitch, J.H.Samson and M.Yethiraj for stimulating discussions, both about this work and the problems of electron-phonon interactions and superconductivity in general. I acknowl- edge support at Loughborough University under EPSRC grant no. EP/C518365/1. REFERENCES 1. H.Fröhlich. Phys. Rev., 79:845, 1950. 2. J.Bardeen, L.N.Cooper, and J.R.Schrieffer. Phys. Rev., 108:1175, 1957. 3. J.G.Bednorz and K.A.Müller. Z. Phys. B, 64:189, 1986. 4. G.M.Eliashberg. JETP letters, 11:696, 1960. 5. A.B.Migdal. JETP letters, 7:996, 1958. 6. W.Metzner and D.Vollhardt. Phys. Rev. Lett., 62:324, 1989. 7. P.C.Hohenberg. Phys. Rev., 158:383, 1967. 8. P.W.Anderson. The theory of superconductivity in the high-TC cuprates. Princeton University Press, 1997. 9. E.K.H.Salje, A.S.Alexandrov, and W.Y.Liang, editors. Polarons and Bipolarons in high-TC super- conductors and related materials. Cambridge University Press, 1995. 10. G.M.Zhao, M.B.Hunt, H.Keller, and K.A.Müller. Nature, 385:236, 1997. 11. A.Lanzara, P.V.Bogdanov, X.J.Zhou, S.A.Kellar, D.L.Feng, E.D.Lu, T.Yoshida, H.Eisaki, A.Fujimori, K.Kishio, J.-I.Shimoyama, T.Noda, S.Uchida, Z.Hussa, and Z.-X.Shen. Nature, 412:6846, 2001. 12. R.J.McQueeney, Y.Petrov, T.Egami, M.Yethiraj, G.Shirane, and Y.Endoh. Phys. Rev. Lett., 82:628, 1999. 13. J-.H.Chung et al. Phys. Rev. B, 67:014517, 2003. 14. F.Gebhard. The Mott Metal-Insulator Transition - Models and Methods, volume 137 of Springer tracts in modern physics. Springer, Heidelberg, 1997. 15. I.G.Lang and Yu.A.Firsov. Sov. Phys. JETP, 16:1301, 1963. 16. J.Hubbard. Proc. R. Soc. London Ser. A, 276:238, 1963. 17. J.P.Hague and N.d’Ambrumenil. J. Low Temp. Phys., 140:77–89, 2005. 18. J.P.Hague and N.d’Ambrumenil. cond-mat/0106355. 19. A.J.Millis, R.Mueller, and B.I.Shraiman. Phys. Rev. B, 54:5389, 1996. 20. M.Hettler, A.N.Tahvildar-Zadeh, M.Jarrell, T.Pruschke, and H.R.Krishnamurthy. Phys. Rev. B, 58:7475, 1998. 21. J.P.Hague. Electron and phonon dispersions of the two dimensional Holstein model: Effects of vertex and non-local corrections. J. Phys.: Condens. Matter, 15:2535, 2003. 22. J.P.Hague. Superconducting states of the quasi-2d Holstein model: Effects of vertex and non-local corrections. J. Phys.: Condens. Matter, 17:5663, 2005. 23. L.S.Macarie and N.d’Ambrumenil. J. Phys.: Condens. Matter, 7:3237, 1995.

0704.0366

Generalized Nariai Solutions for Yang-type Monopoles

Generalized Nariai Solutions for Yang-type Monopoles Pablo Diaz∗, Antonio Segui† Departamento de Fisica Teorica, Universidad de Zaragoza, 50009-Zaragoza, Spain. October 28, 2018 Abstract A detailed study of the geometries that emerge by a gravitating generalized Yang monopole in even dimensions is carried out. In particular, those which present black hole and cosmological horizons. This two-horizon system is ther- mally unstable. The process of thermalization will drive both horizons to coalesce. This limit is what is profusely studied in this paper. It is shown that eventhough coordinate distance shrinks to zero, physical distance does not. So, there is some remaining space which geometry has been computed and identified as a generalized Nariai solution. The thermal properties of this new spacetime are then calculated. Topics, as the elliptical relation between radii of spheres in the geometry or a dis- cussion about whether a mass-type term should be present in the line element or not, are also included. Keywords: Yang monopole, Nariai geometry, Horizon, Black hole. ∗[email protected] †[email protected] http://arxiv.org/abs/0704.0366v2 Contents 1 Introduction 2 2 The gravitational coupling. Some geometrical features 3 3 The horizon coalecence geometry 4 3.1 Case m = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Case m 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4 Conclusions 11 A Proof of the finite nonzero physical distance 12 B Horizon coalescence as a flow on the line 13 1 Introduction Monopoles have been subject of deep study and controversy all over the last century. This is so because, although no experimental evidence of their existence has been found, many theoretical issues make them almost unavoidable. They already appeared as solutions of Maxwell equations as long as the null B-divergence condition was relaxed, that is, ∇·B 6= 0. It was Dirac [1] in the early thirties who first proposed the theoretical possibility of creating an experiment to actually produce a “fake” monopole, in a way that its fakeness, say, the Dirac string, was undetectable. As a consequence, the product of the electric and the magnetic charges was quantized. Many years later, in 1959, the quantization requirement was confirmed by the celebrated Aharonov-Bohm experiment [2]. Since 1954, owing to the papers by Yang and Mills [3] and by Utiyama [4], gauge theories of a group of symmetry larger than U(1), in particular non abelian symmetry groups SU(2) and SU(3) (which eventually would conform the Standard Model of par- ticle physics) where gradually developped. In 1969, Lubkin [5] realized that monopoles can be classified by the homotopy group of the gauge symmetry group of the theory, so that the magnetic charge is replaced by the topological charge of the field configura- tion. In the case of the Dirac monopole, the homotopy group π1 of U(1) is exactly Z. However, it was not till 1975 that Yang [6] generalized the abelian monopole to the case of an SU(2)-invariant gauge theory in six dimensions, see also [7]. Modern approaches use the formalism of fiber bundles for a suitable description of monopoles. It generalizes the traditional classification in terms of the homotopic group of the gauge theory. In this way, magnetic monopoles are identified with the different instanton configurations which come up basically as non trivial maps of the gauge group, usually SU(N), onto Sd, where d is the spatial dimension. That is, magnetic monopoles are all those non trivial principal bundles with group structure SU(N) that can be realized on the hypersurface Sd. The classification coincides, as said before, with the different classes of homotopy groups. The genalization of Yang monopoles to an arbitrary even dimension was carried out in [8]. Using slightly different methods similar analysis have recently been done [9]. The reader can find good reviews on the subject in [10], [11] and the references therein. As every existing object in nature, monopoles couple to gravity via their energy- momentum tensor. The resulting geometry is obtained by solving the Yang-Mills- Einstein equations, which get greatly simplified by imposing spherical symmetry (as expected from a magnetic monopole field configuration). This geometry is fully specified by choosing a point in the space of parameters {µ,m,Λ, k}, the meaning of which will be explained in detail later on. For a given range of parameters, it is easy to prove that the geometry presents both a cosmological and an event horizon. A full analogy with the Schwarzschild-de Sitter solution reveals that, in these cases, the geometry is dy- namically driven through the parameter space into a thermally stable point where both horizons coalesce [12], the final line element being the analogue of Nariai’s spacetime in four dimensions. This paper is organized as follows: the next section sets a general framework and fixes the notation used later. The main body of the article concerns the analysis of the coalescence solutions. This is achieved in two subsections corresponding to the massless and massive cases respectively. An explicit computation of the resulting geometry is carried out in each case. A final section includes some conclusions and comments. Two appendixes have been added to the article. They are topics which lie somehow out of the main line of the paper, either for being technical aspects of a computation (Appendix A) or for presenting a new idea the exposition of which would need a new section, as in Appendix B. the absence of B, in turn, would not have prevented the reader from a full understanding of the paper. 2 The gravitational coupling. Some geometrical fea- tures The gravitational effects of these monopoles have been recently studied [9]. It was done, as usual, by minimally coupling the Yang-Mills energy-momentum tensor to gravity. Variations of the Einstein-Hilbert action − det g (R− 2Λ)− 1 Tr|F |2 with respect to the metric tensor leads to Gmn = 8πGTmn − gmnΛ, (2) where Tmn = γ tr(F pm Fnp)− gmntr(FpqF is the energy momentum tensor of the YM strength field. The traces are taken in the colour index and γ is the YM coupling constant. Finding general solutions for (2) is a highly complicated problem. However, imposing spherical symmetry simplifies the task enormously. According to this, the ansatz will be a spatially spherically symmetric (2k + 2)-dimensional metric whose line element reads ds2 = −∆dt2 +∆−1dr2 + r2dΩ22k. (4) The last equation is consistent with (2) and (3) when [9] ∆(r) = 1− 2Gm r2k−1 , (5) where R = k(2k+1) is the de Sitter radius, µ2 is proportional to 1 and measures the magnetic charge of the monopole, m comes up as a constant of integration with dimensions of mass and G is the Newton constant in 2k + 2 spacetime dimension. At first sight, (4) with (5) look like a Schwarzschild-de Sitter geometry in 2k + 1 spatial dimensions with an extra term, the one involving µ, which seems to be independent of the dimension of spacetime. It seems reasonable to think of this term as a contribution of the magnetic monopole. This simple image, even if not exact1, is helpful and, unless we face the vanishing limits, it may be kept in mind in the following. The next step (and the next temptation) is to analyze how the causal structure of this spacetime depends on given values of the parameters. The main body of this work concerns a deep analysis of the solution in the case when parameters µ,Λ, m and k allow the existence of two horizons. Then, inspired by the Schwarzschild-de Sitter unstable solution, it is claimed that the system gets dynamically driven to a value of the parameters where both horizons coalesce. Eventhough coordinate distance shrinks to zero, physical distance does not. A generalized Nariai geometry “between” the horizons is then explicitly obtained. The Nariai line element [13] is a nonsingular solution of the Einstein’s vacuum equations with a positive cosmological constant, Rµν = Λgµν . It was first found by Kasner [14] and its electrically charged generalization dates of 1959 [15]. However, the important fact that it emerges as an extremal limit of Schwarzschild-de Sitter black holes was not noticed until 1983 [12]. Nariai spacetime in four dimensions is the direct product dS2 × S2, dS2 being no more than the hyperbolic version of S2 as we change t → iτ . In 2k + 2 dimensions, the solution gets generalized to dS2 × S2k. Again, it is the direct product of two constant curvature spaces and admits a 3 + k(2k+ 1) group of isometries SO(2, 1)× SO(2k+ 1). The space is homogeneous since the group acts transitively and is locally static, given that a global dS-type spacetime cannot be described by merely one static coordinate chart. In four dimensions, radii of curvature of the two product spaces are equal if the black hole is neutral, and different in the charged case. If the black hole is electrically charged, the respective radii a and b are different and related by the equation a−2 + b−2 = 2Λ (6) as shown in [15]. This relation will be generalized in the magnetic case, the object of our study. A short but instructive recent work on the four dimensional geometry can be found in [16]. 3 The horizon coalecence geometry Studying the horizons of a geometry like (4) is equivalent to searching the divergencies of grr for finite values of the coordinates. This leads us to analyze the zeroes of function ∆(r), where the horizons will be located. For a certain range of values of {µ,Λ, k,m} there will be two horizons. Finding this region in the parameter space will be the first 1The resulting geometry is, of course, not just the sum of terms of different geometries, but it casually coincides. Differences are bound to exist on the limit of vanishing of a given contribution. For instance, let us suppose that, given a set of parameters, say {m,µ,Λ, k }, we can switch off µ (by neglecting it with respect to the others). The resulting geometry is topologically different to the one obtained by not assuming any monopole at all at the beginning, that is, the limit does not coincide. However, in the cases studied here, this is no more than an enough-to-be-aware-of subtlety. task. After that, attention will be focused on the coalescence point of the horizons2. The analysis consists of two steps, first, the parameterization of the coordinate separation of the horizons (ǫ) and the calculation of the physical distance between them when coalescence takes place (ǫ → 0). Then, following the strategy in [12], the computation of the line element of the remaining geometry. This program is carried out on two cases: m = 0 and m 6= 0, which are treated in the next subsections, respectively. The massless case must be seen as a toy model of the massive one. This distinction is made not merely for simplicity but also because, as will be explained, the mass parameter comes out naturally for dynamical requirements. 3.1 Case m = 0 In the massless case, ∆(r) gets reduced to ∆(r) = 1− . (7) Solving ∆ = 0 is equivalent to finding the zeroes of a biquadratic equation as long as r = 0 is not considered. We perform the change z ≡ r2 and solve a second order ordinary equation. The horizons are found to be at 1− 4µ z++ = 1− 4µ . (9) R > 2µ guarantees the existence of two positive solutions and, therefore, four solutions for the quartic equation. Two of them, r+ = + z+ and r++ = + z++, correspond to the radial coordinate of the inner (black hole) and outer (cosmological) horizon respectively. If R = 2µ, both solutions coincide, which means that the horizons coalesce. As said before, this does not mean that the geometry vanishes as a naive observation (given a wrong choice of coordinates) would make one think. Physical distance between the horizons, on the contrary, remains finite at the limit. In order to prove this, let us compute it. For fixed time and angular coordinates, the physical distance is D(µ,R) = ∫ r++ [−r4 +R2r2 − µ2R2]1/2 ∫ z++ − µ2R2 z − R2 )2]1/2dz (10) The requirement R > 2µ implies R − µ2R2 > 0 so the above integral is exactly solved as an cos−1-type. The result is D(R) = πR. (11) Surprisingly, the physical distance does not depend on µ. It means that, given a cos- mological constant Λ, one could “switch on” the monopole and go on till the horizons coalesce but the distance would remain unchanged. However, because of quantization 2Coalescence as seen in Schwarzschild coordinates. requirements, monopole charge µ cannot be tuned, but needs to have, instead, a fixed value upto a sign. On the other hand, the cosmological constant, Λ, should be chosen when writing the lagrangian. It means that changing its value does not drive us from one model to another but implies an essential change in the theory [17]. Therefore, we are not free to adjust any parameter arbitrarily as done with the mass of the black hole in the Schwarzschild-de Sitter case. Then, eventhough physical reasons would lead the horizons to coalesce, the absence of any free parameter in our model makes it impossible. In the next section, m will come to our help as a free parameter for the model. Despite the last remark, one could wonder about the kind of geometry that re- mains when the horizons coalesce. This task, even if seems just a curious exercise now, will be useful for the next section. Applying a technique similar to the one Gingspar and Perry [12] used to study the geometry of Nariai’s solution, we proceed by, first, parametrizing the separation of horizons as R = 2µ(1 + ǫ2), (12) in a way that coalescence corresponds to taking ǫ = 0. Then, we define a “wise” change of coordinates χ = cos−1 (r2 − r20) τ = ǫ , (13) where A = 1− 4µ2 and r20 = , and the angular coordinates remain unchanged. The new coordinates (13) might seem randomly chosen at first sight. However, there are some reasons that justify such a functional dependence. For instance, χ is nothing but the physical distance between r+ and r. The timelike coordinate t is multiplied by i in order to work in the Euclidean region3 and by ǫ because ∆/ǫ2 is expected to have a finite limit when ǫ → 0. Now, we apply (12) and (13) and expand ∆(r(χ))dτ 2, ∆−1(r(χ))dχ2 and r2(χ) up to first order in ǫ. The line element (4) reads ds2 = µ2dχ2 + µ2 sin2(χ) 1 + ǫ 2 cos(χ) dτ 2 + + 2µ2 2 cos(χ)ǫ dΩ22k. (14) We take limit ǫ → 0 to obtain ds2 = µ2 dχ2 + sin2(χ)dτ 2 + 2µ2dΩ22k. (15) As seen in (15), the 2k-sphere decouples from the rest. The resulting geometry is S2×S2k for k ≥ 2. Notice the parallelism between this geometry and Nariai’s solution, which is S2 × S2. The “classical” relation between radii (6) gets also generalized to a−2 + b−2 = C0Λ, (16) where C0 = k(2k+1) . The geometry (15) can be viewed as a “degenerate” black hole, in which the two horizons have the same (maximum) size and are in thermal equilibrium. This could be interpreted by an observer as a bath of radiation coming from both horizons 3τ will be periodic at both horizons, although different in each case. Equality will hold at the coalescence point, when thermal stability is reached. at a precise temperature [19]. The temperature can be calculated by means of surface gravity κ, as computed in the new coordinates (13) k(2k + 1) . (17) The entropy can also be computed as a quarter of the sum of the two horizons [18], so k(2k + 1) , (18) where ω2k is the area of the 2k-dimension unit sphere. 3.2 Case m 6= 0 In the massive case we recover the full expression (5) for ∆. Since the singular point r = 0 is not to be considered, we better analyze the function r2k−1∆(r) ∆̃ ≡ r2k−1∆ = − r2k+1 + r2k−1 − µ2r2k−3 − 2Gm. (19) It is known that a polynomial equation with powers equal to or higher than five is not generally solvable in a symbolical way. This happens for k ≥ 2. So, the purpose of doing a study for the massive case analogous to that achieved in the first section is ruined. Nevertheless, some information can be extracted from (19). We should first remember the sign of the parameters: R2 > 0 (de Sitter), µ2 > 0 for k ≥ 2, and m will be free in principle. Derivating (19) and equating to zero leads to a biquadratic equation of the (2k + 1)r4 + (2k − 1)r2 − (2k − 3)µ2 = 0, (20) which, as long as Λµ2 ≤ k (2k − 1)2 2k − 3 , (21) has two positive (and two negative) roots, rmin and rmax ≡ rc. In terms of the cosmo- logical constant r2c ≡ k(2k − 1) 1− 4(2k − 3)Λµ k(2k − 1)2 , (22) rmin is obtained from (22) by swapping the sign of the square root. A quick look at (19) shows that the smallest root is a minimum and the largest is a maximum of function ∆̃. Now, let us plug rc into (19): 1. If m > 0, then (see fig.1) a) ∆̃(rc) ≥ 0 implies that there are two event horizons, the black hole and the cosmological horizon. The inequality gets saturated at the coalescence point. b) ∆̃(rc) < 0 means that no horizon is found. 2. If m < 0, then (see fig.2) a) ∆̃(rmin) < 0 together with ∆̃(rc) < 0 implies that there is just one Cauchy horizon. b) ∆̃(rmin) < 0 together with ∆̃(rc) > 0 assures the existence of a Cauchy horizon and both black hole and cosmological horizon. c) ∆̃(rmin) > 0 leaves us with the cosmological horizon only. The case we will study is ∆̃(rmin) < 0 and ∆̃(rc) > 0 which, independently of the sign of m, assures4 the existence of black hole and cosmological horizons. This corresponds to values of m within range (see fig.3) 1.a 1.b Figure 1: Case m > 0. The curve represents function ∆̃(r). Figure 1.a has two roots which correspond to the black hole (r+) and cosmological horizon (r++) respectively. Figure 1.b shows the absence of horizons. 2.a 2.b 2.c Figure 2: Case m < 0. This time ∆̃(r) permits the existence of one (Cauchy) horizon as in Figure 2.a, three horizons (Cauchy, black hole and cosmological) as in 2.b, or just the cosmological horizon as shown in 2.c. m− < m < m+, (23) where Gmc ≡ Gm+ = 1 + 2k r2k−3c (r c − 2µ2). (24) The value of Gm− is obtained by replacing rc → rmin. In terms of Λ and µ we get Gm± = (2Λ)−k+1/2 1 + 2k − k + 2k2 ± k2(1− 2k)2 − 4Λµ2(2k − 3)k ]k−3/2 − k + 2k2 − 4Λµ2 ± k2(1− 2k)2 − 4Λµ2(2k − 3)k . (25) 4The value of m can be negative. That is because m should not be thought of as an entity with physical meaning but as a geometrical parameter. Short calculation in (25) shows that m gets negative values for Λµ2 ≥ k (1 + 2k). Hr,m+L Hr,m-L Figure 3: This figure shows the range of “masses” which are consistent with the existence of both black hole and cosmological horizons. The curve ∆̃(r) “moves down” in the process of coalescence. The crucial point is that both horizons coalesce when rc is a root of (19) which happens at m = mc(k,Λµ 2,Λ). Two relations have been imposed so far: de∆(r;m) |rc= 0, that is, (20), which defines rc, and ∆(rc;mc) = 0 which leads to mc. In order for m to be real, the bound which must be impossed on Λµ2 coincides with (21) which, in turn, is nothing but the condition for the existence of two horizons. So, if a given a value for Λµ2 is low enough to produce two horizons, there always exists a real value of m which makes them coalesce. Again, as in the Schwarzschild-de Sitter example, the system is unstable and the equilibrium point is reached at m = mc. Unlike the massless case, plugging m gives us enough room for maneuvre to drive the system to equilibrium. At this point, we would like to remark that the procedure of horizon coalescence, as studied in detail below, may be seen as a flow in a line which undergoes a Pitchfork bifurcation at the coalescence point. Parameter m, moved by thermal instability, drives the system to the critical situation. For concreteness see Appendix B. Let us focus on the near coalescence point. This can be parameterized by r = rc + δr = rc(1 + ǫ cosχ) (26) m = mc − δm = mc(1 + bǫ2). Parameterization of r also involves a change of coordinates r → χ and should be taken as imposed at the moment although it will be justified later. The horizons will be symmetrically located at: r+ = rc(1−ǫ) and r++ = rc(1+ǫ) which correspond to χ+ = π and χ++ = 2π, respectively 5. The value of b as well as the absence of a linear term in ǫ of the parameterization of m may be explained as follows. Near the coalescence point one should Taylor expand ∆ around rc and have in mind that, for ≪ 1, ∆ is aproximately parabolic, so that second order expansion is enough. By definition ∆(r+) = ∆(r++) = 0 5For a small enough ǫ, it is expected that the parabolic approach holds and, then, both horizons are symmetrically located with respect to rc. and ∆ reaches a maximum at rc. So, 0 = ∆(r++) = ∆(rc, m) + ∆ ′(rc, m)(rcǫ) + ∆′′(rc, m)(rcǫ) r2k−1c ∆′′(rc;mc)r 2, (27) which means that ∆′′(rc;mc)r . (28) Calculating the physical distance near the coalescence point would, again, imply solving the integration D(ǫ) = ∫ r++ ∆1/2(r) , (29) where r++ = r+ + 2rcǫ. Although the exact result is not computed, an explicit proof of its finite nonzero value is given in Appendix A. The procedure of calculating the physical distance also brings us some light on which is the change of coordinates that should be made in order to understand the resulting geometry. It turns out to be χ = cos−1 (r − rc) t, (30) where k − 2k2 + 2Λr2c is a dimensionless factor. The coalescence of horizons takes place at ǫ = 0. In order to study the geometry at the limit we proceed by calculating −∆dt2, ∆−1dr2 and r2 in the new coordinates (30) and expand in ǫ around ǫ = 0. The new line element gets determined by taking the zero order of the expansion. The relations for r and m in (26) are in accordance with (30), where b takes the value of (28), by virtue of the parabolic approach. From (30), it is straightforward to see that r2 takes a constant value r2c . Surprisingly, as in the massless and Schwarzschild-de Sitter cases, the geometry splits in two disconnected parts which lead to a product manifold S2 × S2k. The line element reads ds2 = Br2c dχ2 + sin2(χ)dτ 2 + r2cdΩ 2k, (32) where χ ∈ [π, 2π] and τ is periodic6. As seen in (32), S2 has radius a2 = Br2c , and S2k has radius b2 = r2c . Now, the generalized Bertotti relation (6) is a−2 + b−2 = 2(1− k) = CΛ, (33) where C(k,Λµ2) is obtained by inserting (22) in (33). Note that C k,Λµ2 = k(2k + = C0, and then (33) turns into (16), that is, into the massless case. This is no 6τ is periodic on both horizon surfaces all over the process in order to avoid the conical singularity at the horizons. At the coalescence point, however, both periods equal. surprising since Λµ2 = k(2k + 1)/4 is the condition for coalescence in the massless case (equivalent to R = 2µ), and, at the same time, it makes mc = 0. So, the massive geometry is a consistent extension of the massless one. Now, fixing Λ does not determine uniquely the geometry. Another dimensionless variable Λµ2 is required. As in the last section, the geometry (32) can be viewed as a “degenerate” black hole, in which the two horizons have the same (maximum) size and are in thermal equilibrium. In the present case the temperature is given in terms of the surface gravity κ by . (34) In Planck units,the entropy associated with this solution may be calculated (given that it is not extreme7) by means of the total area of the horizons as c . (35) 4 Conclusions The spherically symmetric solution of gravity due to a magnetic monopole in arbitrary dimension has been studied, in particular, when the set of parameters {Λ, µ,m, k} allows the existence of two horizons. In these cases, thermal instabilities drive a process of horizon coalescence. Even though coordinate separation between the horizons shrinks to zero, it has been proven in both the massless and the massive case that the physical distance does not. The geometry of the remaining space between the horizons has been calculated in both cases. They turned out to be Nariai-type solutions, that is, the product of a 2-sphere and a 2k-sphere for a (2k + 2)-dimensional spacetime. In each solution, the radii of the spheres are not independent. They are related by an elliptical equation which should be understood as the generalization of the relation found by Bertotti. The unique generalized equation involving these radii for both the massless and the massive case has been given. After computing the line element in each case, the thermodynamical properties (Hawking temperature and entropy) due to the existence of horizons have been calculated. The Yang monopole corresponds to the six dimensional case, where k = 2. The geometry obtained after coalescence is S2 × S4 as can be explicitly read in (32). This case is especially interesting since it may be described in String Theory (a realization of the Yang monopole in Heterotic String Theory has recently been done [21] as well as another complementary picture in Type-IIA String Theory [20]). In the same context, it looks possible to find results (18) and (35) for the entropy by application of some attractor mechanism [22, 23]. We believe that this would be an interesting topic to be addressed in future research. 7A charge black hole is said to be extreme when it has the minimum mass. Then, as it cannot release any energy without losing charge, it is supposed not to emit, and its associated Hawking temperature is 0. The black hole we are dealing with in this paper is extreme in the sense of carrying the “maximum mass” allowed by the cosmological constant Λ. Obviously, the temperature will not be zero. A Proof of the finite nonzero physical distance Computing the physical distance is equivalent to performing the integration ∫ r++ ∆1/2(r) , (36) where, for small ǫ, r++ = r+ + 2rcǫ. Divergencies might appear at the points where ∆ → 0. The case we have been considering all along section (3.2) concerns the existence of two horizons which coalesce, that is, two single roots r+ and r++ of ∆ which join to form a double one. Function ∆̃ can always be expressed as ∆̃ = (r − r+)(r++ − r)g(r), where g(r) is a polynomial function of powers of degree 2k − 1 and no zeroes within the range [r+, r++] are to be found by construction. Explicitly, equation (36) is D(ǫ) = ∫ r++2rcǫ (r − r+)1/2(r+ + 2rcǫ− r)1/2 rk−1/2 g1/2(r) ︸ ︷︷ ︸ . (37) Now, h(r) is a continuous divergenceless strictly positive function in the compact [r+, r++], which means that it will reach a positive maximum and minimum for certain r′s. Let us call hmax and hmin the values of the function h in these points 8. Then ∫ r++2rcǫ (r − r+)1/2(r+ + 2rcǫ− r)1/2 ≤ D(ǫ) ≤ ≤ hmax ∫ r++2rcǫ (r − r+)1/2(r+ + 2rcǫ− r)1/2 . (38) The integration can be performed: ∫ r++2rcǫ (r − r+)1/2(r+ + 2rcǫ− r)1/2 = π. (39) D(ǫ → 0) = πh(rc), (40) where the value of rc is given in (22). Integrations of form (39) are solved exactly by a cos−1 type function, and a nonzero finite result is obtained. It is remarkable that the same can be said for any ∆ we would choose, as long as no more than two single roots were to join to form a double one. The key point is that (39), which could be problematic, is independent of ǫ and therefore the distance is finite in the limit, when ǫ → 0. So, eventhough (39) was neither exactly the physical distance in the massive case nor in Schwarzschild-de Sitter solution (however, it was in the massless case as we have already seen in the first section), it is closely related to it. This fact gives us a hint or, at least, justifies the change of coordinates we were performing once and again to study the geometry at the limit ǫ → 0. 8These, in principle, depend on ǫ but coincide when ǫ → 0: hmin = hmax ≡ h(rc). B Horizon coalescence as a flow on the line The main phenomenon that concerns this paper, as said before, can be described in terms of the dynamics of a vector field on the line. The coalescence point, in this picture, is no more than a supercritical Pitchfork bifurcation. Let us remember some general features of the dynamics of a one-dimensional flow. The equation of a general vector field on the line can be expressed as: ẋ = f(x, α) (41) where f is any real function with real support, the dot means differentiation with respect to t and α is a parameter of the model. Fixed points of (41) require ẋ = 0, which must be obtained by finding the roots of f , that is f(x∗, α) = 0. (42) Equation (42) is solved by an n-collection of fixed points x∗i for a given value of α. Let us suppose that f has three roots if α = α0. Fixed points come closer as α moves and get “condensed” in a “fat” fixed point (bifurcation point) at α = αc. A paradigmatic example of a Pitchfork bifurcation is shown by function f(x) = x(α− x2). (43) One question arises naturally now about the role the horizons play in this picture. Let us claim that horizons are fixed points and the role of α is played by m. We will justify this identification by constructing the vector flow. Constructing a flow in a manifold (in our case it will be a line) is equivalent to giving a family of curves r̄(t) which covers the manifold or part of it. Each of the curves gets specified by the initial condition, say, r̄(t = 0). Now, let us consider geodesic motions. Without loss of generality, the angular coordinates of our geometry will be frozen, θ and φ are constants, and only radial curves r(t) are to be regarded. Static coordinate system will serve us to describe the movement for any r ∈ (r+, r++). Let us invoke intuition at this point. If r(0) is near the cosmological or the black hole horizon it is clear that a test particle will move out of the region by approaching each horizon respectively. Then, there is a point r = rg where the test particle will not “feel” any force and, consequently, it will not move9. This is the first (unstable) fixed point. Let us move the origin by defining r′ = r− rg, after this, primes will be dropped out to simplify notation. The flow at each point will be determined by the physical velocity ṙ(r) (as measured by an observer placed at r = 0) that a test particle would adquire at r if it is dropped with ṙ = 0 at around r = 0 (as close as possible). It is not hard to see that the velocity of the test particle, as seen by the static geodesic observer, is bound to be zero at both horizons. So, horizons are fixed points. Now, our system can be treated as a vector flow ṙ(r) which covers the region between the horizons. The vector flow has three fixed points: {r+, r++, 0} where the first two are stable. As m runs towards mc, the system shrinks into a Pitchfork bifurcation. Near the bifurcation point the flow can be approximated by ṙ = βr(r − r+)(r++ − r), (44) 9rg in our geometry, plays the role the asymptotic infinity does in Schwarzschild solution,that is, the point where the time-like Killing vector should be normalized in order to define the horizon temperature. Note that rg ≡ rc at the coalescence point, that is, when ǫ = 0. where β is a positive constant which depends on µ, k and Λ. On the one hand, in the coordinate system {χ, τ}, and using (30), we have −→ ṙ = . (45) On the other hand, equation (44), expressed in the new coordinate system, reads ṙ = ǫ3βr3c cosχ sin 2 χ, and so = −iǫβr3cB cosχ sin2 χ. (46) As expected, in the new coordinate system, every point converts into a fixed point as horizons coalesce (ǫ → 0). Since the flux lines were identified with geodesics of test particles, this can be understood as the abscence of forces at the end of the process. Acknowledgment We thank P. K. Townsend and Adil Belhaj for helpful discussions and Jean Nuyts for critical reading of the manuscript. This work has been supported by MCYT ( Spain) under grant FPA 2003-02948. References [1] P. A. M. Dirac, Quantised singularities in the electromagnetic field, Proc. Roy. Soc. Lond. A 133, 60 (1931). [2] Y. Aharonov and D. Bohm, Significance of Electromagnetic Potentials in the Quan- tum Theory, Phys. Rev. 115, 485 (1959). [3] C. N. Yang and R. L. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev. 96, 191 (1954). [4] Ryoyu Utiyama, Invariant Theoretical Interpretation of Interaction, Phys. Rev. 101, 1597 (1956). [5] E. Lubkin, Geometric Definition of Gauge Invariance, Ann. Phys. 23, 233 (1963). [6] C. N. Yang, Generalization of Dirac’s monopole to SU2 gauge fields, J. Math. Phys. 19, 320 (1978). [7] P. Goddard, J. Nuyts and D.I. Olive, Gauge Theories And Magnetic Charge, Nucl. Phys. B 125, 1 (1977). [8] Zalan Horvath, Laszlo Palla, Spontaneous Compactification And ’Monopoles’ In Higher Dimensions., Nucl. Phys. B 142, 327 (1978). [9] G.W. Gibbons and P.K. Townsend, Self-graviting Yang monopoles in all dimensions, Class. Quantum Grav. 23, 4873 (2006). [10] S. Coleman, The magnetic monopole fifty years later, in The Unity of the Funda- mental Interactions, ed. A. Zichichi (Plenum, New York, 1983). [11] E. J. Weinberg and P. Yi, Magnetic Monopole Dynamics, supersymmetry, and Du- ality, Phys. Rept. 43, 65 (2007). hep-th/0609055. [12] Paul Ginsparg and Malcom J. Perry, Semiclassical perdurance of de Sitter space, Nuclear Physics B 222, 245 (1983). [13] H. Nariai, Sci. Rep. Tohoku Univ., Ser. 1 35, 62 (1951). [14] E. Kasner, Trans. Am. Math. Soc. 27, 101 (1925). [15] B. Bertotti, Uniform Magnetic Field in the Theory of General Relativity, Phys. Rev. 116, 1331 (1959). [16] Marcello Ortaggio, Impulsive waves in the Nariai Universe, Phys. Rev.D 65, 084046 (2002). [17] G. W. Gibbons and S. W. Hawking, Cosmological Event Horizons, Thermodynam- ics, And Particle Creation, Phys. Rev. D 15, 2738 (1977). [18] S. W. Hawking and Simon F. Ross, Duality between electric and magnetic black holes, Phys. Rev. D 52, 5865 (1995) [19] R. Bousso and S. W. Hawking, Pair creation of black holes during inflation, Phys. Rev. D 54, 6312 (1996). [20] A. Belhaj, P. Diaz, A. Segui, On the Superstring Realization of the Yang Monopole, (2007). hep-th/0703255. [21] E. A. Bergshoeff, G. W. Gibbons and P. K. Townsend, Open M5-branes, Phys. Rev. Lett. 97, 231601 (2006). hep-th/0607193. [22] S. Ferrara, R. Kallosh, A. Strominger, N=2 extremal black holes, Phys. Rev. D 52, 5412 (1995). [23] S. Ferrara, R. Kallosh, Supersymmetry and Attractors, Phys. Rev. D 54, 1514 (1996). http://arxiv.org/abs/hep-th/0609055 http://arxiv.org/abs/hep-th/0703255 http://arxiv.org/abs/hep-th/0607193 Introduction The gravitational coupling. Some geometrical features The horizon coalecence geometry Case m=0 Case m=0 Conclusions Proof of the finite nonzero physical distance Horizon coalescence as a flow on the line

0704.0367

Instanton representation of Plebanski gravity. Consistency of the initital value constraints under time evolution

arXiv:0704.0367v6 [gr-qc] 18 Mar 2011 Instanton representation of Plebanski gravity. Consistency of the initital value constraints under time evolution Eyo Eyo Ita III October 24, 2018 Department of Applied Mathematics and Theoretical Physics Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road Cambridge CB3 0WA, United Kingdom [email protected] Abstract The instanton representation of Plebanski gravity provides as equa- tions of motion a Hodge self-duality condition and a set of ‘general- ized’ Maxwell’s equations, subject to gravitational degrees of freedom encoded in the initial value constraints of general relativity. The main result of the present paper will be to prove that this constraint sur- face is preserved under time evolution. We carry this out not using the usual Dirac procedure, but rather the Lagrangian equations of mo- tion themsleves. Finally, we provide a comparison with the Ashtekar formulation to place these results into overall context. http://arxiv.org/abs/0704.0367v6 1 Introduction In [1] a new formulation of general relativity was presented, named the instanton representation of Plebanski gravity. The basic dynamical variables are an SO(3, C) gauge connection Aaµ and a matrix Ψae taking its values in two copies of SO(3, C).1 The consequences of the associated action IInst were determined via its equations of motion, which hinge crucially on weak equalities implied by the the initial value constraints. For these consequences to be self-consistent, the constraint surface must be preserved for all time by the evolution equations. The present paper will demonstrate that this is indeed the case. We will not use the usual Hamiltonian formulation for totally constrained systems [2], since we will not make use of any canonical structure implied by IInst. Rather, we will deduce the time evolution of the dynamical variables directly from the equations of motion of IInst. Sections 2 and 3 of this paper present the instanton representation ac- tion and derive the time evolution of the basic variables. Sections 4, 5 and 6 demonstrate that the nondynamical equations, referred to as the diffeomor- phism, Gauss’ law and Hamiltonian constraints, evolve into combinations of the same constraint set. The result is that the time derivatives of these con- straints are weakly equal to zero with no additional constraints generated on the system. While we do not use the usual Dirac method in this paper, the result is still that the instanton representation is in a sense Dirac consistent. We will make this inference clearer by comparison with the Ashtekar vari- ables in the discussion section. On a final note, the terms ‘diffeomorphism’ and ‘Gauss’ law’ constraints are used loosely in this paper, in that we have not specified what transformations of the basic variables these constraints generate. The use of these terms is mainly for notational purposes, due to their counterparts which appear in the Ashtekar variables. 2 Instanton representation of Plebanski gravity The starting action for the instanton representation of Plebanski gravity is given by [1] IInst = d3xΨaeB F a0i + ǫkjmB −iN(detB)1/2 Λ+ trΨ−1 , (1) 1Index labelling conventions for this paper are that symbols a, b, . . . from the begini- ing of the Latin alphabet denote internal SO(3, C) indices while those from the middle i, j, k, . . . denote spatial indices. Both of these sets of indices take takes 1, 2 and 3. The Greek symbols µ, ν, . . . refer to spacetime indices which take values 0, 1, 2, 3. where Nµ = (N,N i) are the lapse function and shift vector from metric general relativity, and Λ is the cosmological constant. The basic fields are Ψae and A i , and we action (1) is defined only on configurations restricted to (detB) 6= 0 and (detΨ) 6= 0.2 In the Dirac procedure one refers to Nµ as nondynamical fields, since their velocities do not appear in the action. While the velocity Ψ̇ae also does not appear, we will distinguish this field from Nµ since the action (1), unlike for the latter, is nonlinear in Ψae. The equation of motion for the shift vector N i, the analogue of the Hamilton equation for its conjugate momentum Π ~N , is given by δIInst = ǫmjkB eΨae = (detB)(B −1)diψd ∼ 0, (2) where ψd = ǫdaeΨae is the antisymmetric part of Ψae. This is equivalent to the diffeomorphism constraint Hi owing to the nondegeneracy of B a, and we will often use Hi and ψd interchangeably in this paper. The equation of motion for the lapse function N , the analogue of the Hamilton equation for its conjugte momentum ΠN , is given by δIInst = (detB)1/2 Λ+ trΨ−1 = 0. (3) Nondegeneracy of Ψae and the magnetic field B e implies that on-shell, the following relation must be satisfied Λ + trΨ−1 = 0, (4) which we will similarly take as synonymous with the Hamiltonian constraint. The equation of motion for Ψae is δIInst = BkeF 0k + ǫkjmB m + iN detΨ(Ψ−1Ψ−1)ea ∼ 0, (5) up to a term proportional to (4) which we have set weakly equal to zero. One could attempt to define a momentum conjugate to Ψae, for which (5) would be the associated Hamilton’s equation of motion. But since Ψae forms part of the canonical structure of (1), then our interpretation is that this is not technically correct.3 The equation of motion for the connection Aaµ is given by δIInst ∼ ǫµσνρDσ(ΨaeF eνρ)− 4ǫmjkN mBkeΨ[de] +N(B−1)dj Λ+ trΨ−1 , (6) 2The latter case limits the application of our results to spacetimes of Petroc Types I, D and O (See e.g. [3] and [4]. 3This is because (5) contains a velocity Ȧa within F a and will therefore be regarded as an evolution equation rather than a constraint. This is in stark contrast with (2) and (3), which are genuine constraint equations due to the absence of any velocities. where we have defined ea(x, y) ≡ δAai (x) Bje(y) = ǫ −δae∂k + fedaAdk δ(3)(x, y); D ea ≡ 0. (7) The terms in large round brackets in (6) vanish weakly, since they are pro- portional to the constraints (2) and (4) and their spatial derivatives. For the purposes of this paper we will regard (6) as synonymous with ǫµσνρDσ(ΨaeF νρ) ∼ 0. (8) In an abuse of notation, we will treat (5) and (8) as strong equalities in this paper. This will be justified once we have completed the demonstration that the constraint surface defined collectively by (2), (3) and the Gauss’ constraint from (8) is indeed preserved under time evolution. As a note prior to proceeding we will often make the identification N(detB)1/2 detΨ ≡ −g (9) as a shorthand notation, to avoid cluttering many of the derivations which follow in this paper. 2.1 Internal consistency of the equations of motion Prior to embarking upon the issue of consistency of time evolution of the initial value constraints, we will check for internal consistency of IInst, which entails probing of the physical content implied by (8) and (5). First, equation (8) can be decomposed into its spatial and temporal parts as Di(ΨbfB f ) = 0; D0(ΨbfB f ) = ǫ ijkDj(ΨbfF 0k). (10) The first equation of (10) is the Gauss’ law constraint of a SO(3) Yang– Mills theory, when one makes the identification of ΨbfB f ∼ Eib with the Yang–Mills electric field. The Maxwell equations for U(1) gauge theory with sources (ρ, ~J), in units where c = 1, are given by ~∇ · ~B = 0; Ḃ = −~∇× ~E = 0; ~∇ · ~E = ρ; ~̇E = − ~J + ~∇× ~B. (11) Equations (10) can be seen as a generalization of the first two equations of (11) to SO(3) nonabelian gauge theory in flat space when one: (i) identifies with the SO(3) generalization of the electric field ~E, and (ii) one chooses Ψae = kδae for some numerical constant k. When ρ = 0 and ~J = 0, then one has the vacuum theory and equations (11) are invariant under the transformation ( ~E, ~B) −→ (− ~B, ~E). (12) Then the second pair of equations of (11) become implied by the first pair. This is the condition that the Abelian curvature Fµν , where F0i = Ei and ǫijkFjk = Bi, is Hodge self-dual with respect to the metric of a conformally flat spacetime. But equations (10) for more general Ψae encode gravitational degrees of freedom, which as shown in [1] generalizes the concept of self- duality to more general spacetimes solving the Einstein equations. Let us first attempt to derive the analogue for (10) of the second pair of (11) in the vacuum case. Acting on the first equation of (10) with D0 yields D0Di(ΨbfB f ) = DiD0(ΨbfB f ) + [D0,Di](ΨbfB f ) = 0. (13) Substituting the second equation of (10) into the first term on the right hand side of (13) and using the definition of temporal curvature as the commutator of covariant derivatives on the second term we have ijkDj(ΨbfF 0k)) + fbcdF 0iΨdfB f = fbcd 0k +B Ψdf = 0 (14) where we have also used the spatial part of the commutator ǫijkDiDjva = fabcB b vc. Note that the term in brackets in (14) is symmetric in f and c, and also forms the symmetric part of the left hand side of (5) 0i + i −g(Ψ−1Ψ−1)fb + ǫijkBifB k = 0, (15) re-written here for completeness. To make progress from (14), we will sub- stitute (15) into (14). This causes the last term of (15) to drop out due to antisymmetry, which leaves us with −gfbcd Ψdf (Ψ −1Ψ−1)fc +Ψdf (Ψ −1Ψ−1)fc = −2i −gfbcdΨ−1dc . (16) The equations are consistent only if (16) vanishes, which is the requirement that Ψae = Ψea be symmetric. This of course is the requirement that the diffeomorphism constraint (2) be satisfied. So the analogue of the second pair of (11) in the vacuum case must be encoded in the requirement that Ψae = Ψea be symmetric. 3 The time evolution equations We must now verify that the initial value constraints are preserved under time evolution defined by the equations of motion (5) and (6). These equa- tions are respectively the Hodge duality condition 0k + i −g(Ψ−1Ψ−1)fb + ǫijkN iBjbB f = 0, (17) and one of the Bianchi identity-like equations ǫijkDj(ΨaeF ok) = D0(ΨaeB e). (18) Since the initial value constraints were used to obtain the second line of (17) from (1), then we must verify that these constraints are preserved under time evolution as a requirement of consistency. Using F b0i = Ȧ i −DiAb0 and defining −g(B−1)fi (Ψ −1Ψ−1)fb + ǫmnkN mBnb ≡ iHbk, (19) Then equation (17) can be written as a time evolution equation for the connection, which is not the same as a constraint equation as noted earlier F b0i = −iHbi −→ Ȧbi = DiAb0 − iHbi . (20) From equation (20) we can obtain the following equation governing time evolution equation for the magnetic field Ḃie = ǫ ijkDjȦ k = ǫ ijkDj 0 − iHek = febcB 0 − iǫijkDjHek = −δ~θB e − iǫijkDjHek, (21) which will be useful. On the first term on the right hand side of (21) we have used the definition of the curvature as the commutator of covariant derivatives. The notation δ~θ in (21) suggests that that B e transforms as a SO(3, C) vector under gauge transformations parametrized by θb ≡ Ab0.4 Since we have not specified anything about the canonical structure of IInst, then δ~θ as used in (21) and in (24) should at this stage simply be regarded as a definition useful for shorthand notation. We will now apply the Liebnitz rule in conjunction with the definition of the temporal covariant derivatives to (18) to determine the equation gov- erning the time evolution of Ψae. This is given by D0(ΨaeB e) = B eΨ̇ae +ΨaeḂ e + fabcA 0(ΨceB e) = ǫ ijkDj(ΨaeF 0k). (22) Substituting (21) and (20) into the left and right hand sides of (22), we have BieΨ̇ae +Ψae febcB 0 − iǫijkDjHek + fabcA 0(ΨceB e) = −iǫijkDj(ΨaeHek).(23) In what follows, it will be convenient to use the following transformation properties for Ψae as A i under SO(3, C) gauge transformations δ~θΨae = fabcΨce + febcΨac Ab0; δ~θA i = −DiAa0; δ~θB e = −febcBibAc0. (24) 4We will make the identification with SO(3, C) gauge transformations later in this paper when we bring in the relation of IInst with the Ashtekar variables. Then using (24), the time evolution equations for the phase space variables ΩInst can be written in the following compact form Ȧbi = −δ~θA i − iHbi ; Ψ̇ae = −δ~θΨae − iǫ ijk(B−1)ei (DjΨaf )H k . (25) We have found evolution equations for Ψae and A i from the covariant equa- tions of Aaµ and the Hodge-duality condition We have obtained these without using Poisson brackets, and by assuming that the Hamiltonian and diffeo- morphism constraints are satisfied. Therefore the first order of business is then to check for the preservation of the initial value constraints under the time evolution generated by (25). This means that we must check that the time evolution of the diffeomorphism, Gauss’ law and Hamiltonian con- straints are combinations of terms proportional to the same constraints and their spatial derivatives, and terms which vanish when the constraints hold.5 These constraints are given by we{Ψae} = 0; (detB)(B−1)diψd = 0; (detB)1/2 Λ+ trΨ−1 = 0(26) where (detB) 6= 0 and (detΨ) 6= 0. We will occasionally make the identifi- cation N(detB)1/2(detΨ)1/2 ≡ −g (27) for a shorthand notation. Additionally, the following definitions are provided for the vector fields appearing in the Gauss’ constraint we = B eDi; ve = B e∂i (28) where Di is the SO(3, C) covariant derivative with respect to the connection Aai . Equations (26) are the equations of motion for the auxilliary fields A N i and N . 4 Consistency of the diffeomorphism constraint un- der time evolution The diffeomorphism constraint is directly proportional to ψd = ǫdaeΨae, the antisymmetric part of Ψae. So to establish the consistency condition for this constraint, it suffices to show that the antisymmetric part of the second equation of (25) weakly vanishes. This is given by ǫdaeΨ̇ae = −δ~θ(ǫdaeΨae)− iǫdaeǫ ijk(B−1)ei (DjΨaf )H k , (29) 5This includes any nonlinear function of linear order or higher in the constraints, a situation which involves the diffeomorphism constraint. which splits into two terms. Using (24), one finds that the first term of (29) is given by −ǫdaeδ~θΨae = −ǫdae fabcΨce +Ψacfebc δebδdc − δecδbd Ψce + δdbδac − δdcδab Ψdb − δbdtrΨ + δdbtrΨ−Ψbd Ab0 = 2Ψ[bd]A 0 = −ǫdbhAb0ψh, (30) which is proportional to the diffeomorphism constraint. The second term of (30) has two contributions due to H k as defined in (19). The first contribu- tion reduces to −iǫdaeǫijk(B−1)ei (DjΨaf )(H(1)) = −iǫdaeǫijk(B−1)ei (DjΨaf ) −g(B−1)gk(Ψ −1Ψ−1)gf = iǫdae(detB) −1ǫegh(Ψ−1Ψ−1)gfB hDjΨaf = i(detB)−1(Ψ−1Ψ−1)gf a − δgaδhd vv{Ψaf} = i(detB)−1(Ψ−1Ψ−1)gf dva{Ψaf} − vd{Ψgf} = i(detB)−1 (Ψ−1Ψ−1)dfGf + vd{Λ+ trΨ−1} . (31) The first term on the final right hand side of (31) is the Gauss’ constraint and the second term is the derivative of a term direction proportional to the Hamiltonian constraint.6 The second contribution to the second term of (29) is given by ǫdaeǫ ijk(B−1)ei (DjΨaf )(H(2)) k = ǫdaeǫ ijk(B−1)ei (DjΨaf )ǫmnkN = ǫdac n − δinδjm (B−1)ei (DjΨaf )N = ǫdaeN i(B−1)eivf{Ψaf} −N jDj(ǫdaeΨae) = ǫdaeN i(B−1)efGa −N jDjψd.(32) The result is that the time evolution of the diffeomorphism constraint is directly proportional to ψ̇d = i(detB)−1(Ψ−1Ψ−1)da + ǫdaeN i(B−1)ei Ab0ǫbdh − δdhN jDj ψh + i(detB) vd{(−g)−1/2H}, (33) which is a linear combination of terms proportional to the constraints (26) and their spatial derivatives. The result is that the diffeomorphism con- straint Hi = 0 is consistent with respect to the Hamiltonian evolution gen- erated by the equations (25). So it remains to verify consistency of Gauss’ law and the Hamiltonian constraints Ga and H. 6We have added in a term Λ, which can be regarded as a constant of integration with respect to the spatial derivatives from vd. 5 Consistency of the Gauss’ constraint under time evolution Having verified the consistency of the diffeomorphism constraint under time evolution, we now move on to the Gauss’ constraint. Application of the Liebnitz rule to the first equation of (26) yields Ġa = Ḃ eDiΨae +B eDiΨ̇ae +B fabfΨfe + febgΨag Ȧai . (34) Upon substituion of (21) and (25) into (34), we have Ġa = −δ~θB e − iǫijkDjHek DiΨae +B −δ~θΨae − iǫ ijk(B−1)ei (DjΨaf )H fabfΨfe + febgΨag −δ~θA i − iHbi .(35) Using the Liebniz rule to combine the δ~θ terms of (35), we have Ġa = −δ~θGa − iǫ k)DiΨae +B e Dm((B −1)ei (DjΨaf )H fabfΨfe + febgΨag i . (36) The requirement of consistency is that we must show that the right hand side of (36) vanishes weakly. First, we will show that the third term on the right hand side of (36) vanishes up to terms of linear order and higher in the diffeomorphism constraint. This term, up to an insignificant numerical factor, has two contributions. The first contribution is fabfΨfe + febgΨag Bie(H(1)) fabfΨfe + febgΨag (Ψ−1Ψ−1)eb fabf (Ψ −1)fb + febg(Ψ −1Ψ−1)ebΨag ∼ δ(1)a (~ψ) ∼ 0, (37) which is directly proportional to a nonlinear function of first order in ψd which is proportional to the diffeomorphism constraint. The second contri- bution to the third term on the right hand side of (36) is fabfΨfe + febgΨag Bie(H(2)) fabfΨfe + febgΨag ǫkmnN kBme B fabfΨfe + febgΨag (detB)Nk(B−1)dkǫdeb = (detB)Nk(B−1)dk δfdδae − δfeδad Ψfe + 2δdgΨag = (detB)Nk(B−1)dk Ψda − δadtrΨ + 2Ψad ≡ δ(2)a ( ~N) (38) which does not vanish, and neither is it expressible as a constraint. For the Gauss’ law constraint to be consistent under time evolution, a necessary condition is that this δ a ( ~N) term must be exactly cancelled by another term arising from the variation. Let us expand the terms in square brackets in (36). This is given, using the Liebniz rule on the second term, by ǫijk(DjH k)(DiΨae) + ǫ ijkBme Dm((B −1)ei (DjΨae)H = ǫijk(DjH k)(DiΨae)− ǫijkBme (B−1)en(DmBng )(B−1) i (DjΨaf )H +ǫmjk(DmDjΨaf )H + ǫmjk(DjΨaf )(DmH ). (39) The first and last terms on the right hand side of (39) cancel, which can be seen by relabelling of indices. Upon application of the definition of curvature as the commutator of covariant derivatives to the third term, then (39) reduces to −ǫijk(DnBng )(B−1) i (DjΨaf )H fabcΨcf + ffbcΨac . (40) The first term of (40) vanishes on account of the Bianchi identity and the second term contains two contributions which we must evaluate. The first contribution is given by (H(2)) fabcΨcf + ffbcΨac = (detB)Nk(B−1)dkǫdbf fabcΨcf + ffbcΨac = (detB)Nk(B−1)dk δdaδfc − δdcδfa Ψcf − 2δdcΨac = (detB)Nk(B−1)dk δdatrΨ−Ψda − 2Ψad = −δ(2)a ( ~N ),(41) with δ a ( ~N) as given in (37). So putting the results of (39), (40) and (41) into (36), we have Ġa = −δ~θGa + δ ~N) + δ(1)a ( ~ψ) + δ(1)a ( ~ψ)− δ(2)a ( ~N) = −δ~θGa + 2δ (1)(~ψ).(42) The velocity of the Gauss’ law constraint is a linear combination of the Gauss’ constraint with terms of the diffeomorphism constraint of linear or- der and higher. Hence the time evolution of the Gauss’ law constraint is con- sistent in the sense that we have defined, since δ(1)(~ψ) vanishes for ψd = 0. 6 Consistency of the Hamiltonian constraint un- der time evolution The time derivative of the Hamiltonian constraint, the third equation of (26), is given by ((detB)1/2(detΨ)1/2 (Λ + trΨ−1) + (Λ + trΨ−1) (43) which has split up into two terms. The first term is directly proportional to the Hamiltonian constraint, therefore it is already consistent. We will nevertheless expand it using (21) and (25) (B−1)di Ḃ d + (Ψ −1)aeΨ̇ae (detB)1/2(detΨ)1/2(Λ + trΨ−1) (B−1)di −δ~θB d − iǫijkDjHdk +(Ψ−1)ae −δ~θΨae − iǫ ijk(B−1)ei (DjΨaf )H H. (44) We will be content to compute the δ~θ terms of (44). These are (B−1)di δ~θB d = (B −1)di fdbfB 0 = δdbfdbfA 0 = 0 (45) on account of antisymmetry of the structure constants, and (Ψ−1)eaδ~θΨae = (Ψ −1)ea fabfΨfe + febgΨag = 0, (46) also due to antisymmetry of the structure constants. We have shown that the first term on the right hand side of (43) is consistent with respect to time evolution. To verify consistency of the Hamiltonian constraint under time evolution, it remains to show that the second term is weakly equal to zero. It suffices to show this just for the second term, in brackets, of (43) (Λ + trΨ−1) = −(Ψ−1Ψ−1)feΨ̇ef = (Ψ−1Ψ−1)ef δ~θΨae − iǫ ijk(B−1)ei (DjΨaf )H , (47) where we have used (25). Equation (47) has split up into two terms, of which the first term is (Ψ−1Ψ−1)eaδ~θΨae = (Ψ −1Ψ−1)ea fAbfΨfe + febgΨag fabf (Ψ −1)fa + febg(Ψ −1)eg Ab0 = m( ~ψ) ∼ 0 (48) which vanishes weakly since it is a nonlinear function of at least linear order in ψd. The second term of (47) splits into two terms which we must evaluate. The first contribution is proportional to (Ψ−1Ψ−1)eaǫijk(B−1)ei (DjΨaf )(H(1)) −g(Ψ−1Ψ−1)eaǫijk(B−1)ei (DjΨaf )(B−1)dk(Ψ−1Ψ−1)df −g(Ψ−1Ψ−1)ea(Ψ−1Ψ−1)df (detB)−1ǫedgBjgDjΨaf −g(detB)−1ǫedg(Ψ−1Ψ−1)ea(Ψ−1Ψ−1)dfvg{Ψaf} ≡ v{~ψ} (49) for some vector field v. We have used the fact that the term in (49) quartic in Ψ−1 in antisymmetric in a and f due to the epsilon symbol. Hence Ψaf as acted upon by vg can only appear in an antisymmetric combination, and is therefore proportional to the diffeomorphism constraint ψd whose spatial derivatives weakly vanish. Hence (49) presents a consistent contribution to the time evolution of H, which leaves remaining the second contribution to the second term of (47). This term is proportional to (Ψ−1Ψ−1)eaǫijk(B−1)ei (DjΨaf )(H(2)) = (Ψ−1Ψ−1)eaǫijk(B−1)ei (DjΨaf )ǫmnkN n − δinδjm (B−1)eiB −1Ψ−1)ea(DjΨaf ) N i(B−1)eiB f − δefN (Ψ−1Ψ−1)ea(DjΨaf ) = (−g)−1/2N iHai vf{Ψaf} − (Ψ−1Ψ−1)fa(N jDjΨaf ) = (−1)−1/2N iHai Ga −N jDj(Λ + trΨ−1). (50) The first term on the final right hand side of (50) is proportional to the Gauss’ law constraint, and the second term is proportional to the derivative of the Hamiltonian constraint. To obtain this second term we have added in Λ as a constant of differentiation with respect to ∂j . Substituting (48), (49) and (50) into (47), then we have Ḣ =∼ Ô(~ψ) + (−g)−1/2N iHai Ga + T̂ ((−g)−1/2H), (51) where Ô and T̂ are operators consisting of spatial derivatives acting to the right and c numbers. The time derivative of the Hamiltonian constraint is a linear combination of the Gauss’ law and Hamiltonian constraints and its spatial derivatives, plus terms of linear order and higher in the diffeo- morphism constraint and its spatial derivatives. Hence the Hamiltonian constraint is consistent under time evolution. 7 Recapitulation The final equations governing the time evolution of the initial value con- straints are given weakly by ψ̇d = i(detB)−1(Ψ−1Ψ−1)da + ǫdaeN i(B−1)ei Ab0ǫbdh − δdhN jDj ψh + i(detB) vd{Λ + trΨ−1}; Ġa = −fabcAb0Gc + δ(1)a (~ψ); ǫijk(B−1)di (DjH k ) + ǫ ijk(B−1)ei (Ψ −1)ae(DjΨaf )H −N j∂j (Λ + trΨ−1) +(−g)−1/2N iHai Ga − −g(detB)−1ǫedg(Ψ−2Ψ−1)ea(Ψ−1Ψ−1)dfvg{ǫafhψh}+m(~ψ).(52) Equations (52) show that all constraints derivable from the the action (1) are preserved under time evolution, since their time derivatives yield linear combinations of the same set of constraints and their spatial derivatives. There are no additional constraints generated which implies that the action (1) is consistent in the Dirac sense. On the other hand, we have not defined the canonical structure of (52) or any Poisson brackets. Equations (52) can be written schematically in the following form ~̇H ∼ ~H + ~G+H; ~̇G ∼ ~G+Φ( ~H); Ḣ ∼ H + ~G+Φ( ~H), (53) where Φ is some nonlinear function of the diffeomorphism constraint ~H, which is of at least first order in ~H. In the Hamiltonian formulation of a theory, one identifies time derivatives of a variable f with via ḟ = {f,H} the Poisson brackets of the variable with the Hamiltonian H . So while we have not specified Poisson brackets, equation (53) implies the existence of Poisson brackets associated to some Hamiltonian HInst for the action (1), { ~H,HInst} ∼ ~H + ~G+H; {~G,HInst} ∼ ~G+Φ( ~H); {H,HInst} ∼ H +Φ( ~H) + ~G. (54) So the main result of this paper has been to demonstrate that the instanton representation of Plebanski gravity forms a consistent system, in the sense that the constraint surface is preserved under time evolution. As a direction of future research we will compute the algebra of constraints for (1) directly from its canonical structure. Nevertheless it will be useful for the present paper to think of equations (52) in the Dirac context, mainly for compari- son with other formulations of general relativity. This will bring us to the Ashtekar variables. 8 Discussion: Relation of the instanton represen- tation to the Ashtekar variables We will now provide the rationale for not following the Dirac procedure for constrained systems [2] with respect to (1), by comparison with the Ashekar formulation of GR. The action for the instanton representation (1) can be written in the following 3+1 decomposed form IInst = 0we{Ψae} − ǫijkN iBjaBkeΨae −iN(detB)1/2(detΨ)1/2 Λ+ trΨ−1 , (55) which regards Ψae and A i as phase space variables. But the phase space of (55) is noncanonical since its symplectic two form δθInst = δ d3xΨaeB d3xBieδΨae ∧ δAai + d3xΨaeǫ ijkDj(δA k) ∧ δAai , (56) is not closed owing to the presence of the second term on the right hand side. The initial stages of the Dirac procedure applied to (55) state that the momentum conjugate to Aai yields the primary constraint Πia = δIInst δȦai = ΨaeB e. (57) Then making the identification σ̃ia = Π a and upon substitution into (57) and into (55), one obtains the action IAsh = σ̃iaȦ 0Ga −N iHi − , (58) which is the action for the Ashtekar complex formalism of general relativity [5], [6], with σ̃ia being the densitized triad. This is a totally constrained sys- tem with (Aa0, N i, N), respectively the SO(3, C) rotation angle Aa0, the shift vector N i and the densitized lapse function N = N(detσ̃)−1/2 as auxilliary fields. The constraints in (58) smearing the auxilliary fields are the Gauss’ law, vector and Hamiltonian constraints Ga = Diσ̃ a; Hi = ǫijkσ̃ a ; H = ǫijkǫ abcσ̃iaσ̃ σ̃kc +B . (59) From (58) one reads off the symplectic two form ΩAsh given by ΩAsh = d3xδσ̃ia ∧ δAai = δ d3xσ̃iaδA = δθAsh, (60) which is the exact functional variation of the canonical one form θAsh. The actions (55) and (58) are transformable into each other only under the condition (detB) 6= 0 and (detΨ) 6= 0. In (58) it is clear that σ̃ia and Aai form a canonically conjugate pair, which suggests that (55) is a noncanonical version of (58). The constraints algebra for (59) is { ~H[ ~N ], ~H [ ~M ]} = Hk N i∂kMi −M i∂kNi { ~H[N ], Ga[θa]} = Ga[N i∂iθa]; {Ga[θa], Gb[λb]} = Ga fabcθ {H(N), ~H [ ~N ]} = H[N i∂iN {H(N ), Ga(θa)} = 0;[ H(N),H(M ) = Hi[ N∂jM −M∂jN H ij], (61) which is first class due to closure of the algebra, and is therefore consistent in the Dirac sense. Let us consider (61) for each constraint with the total Hamiltonian HAsh and compare with (54). This is given schematically by { ~H,HAsh} ∼ ~H + ~G+H; {~G,HAsh} ∼ ~G+ ~H; {H,HAsh} ∼ H + ~H. (62) Comparison of (62) with (54) shows an essentially similar structure for the top two lines involving ~H and ~G.7 But there is a marked dissimilarity with respect to the Hamiltonian constraint H. Note that there is a Gauss’ law constraint appearing in the right hand side of the last line of (54) whereas there is no such constraint on the corresponding right hand side of (62). This means that while the Hamiltonian constraint is gauge-invariant under SO(3, C) gauge-transformations as implied by (61) and (62), this is not the case in (54). This means that the action (1), which as shown in [1] describes general relativity for Petrov Types I, D and O, has a different role for the Gauss’ law and Hamiltonian constraints than the action (58), which also describes general relativity. Therefore IInst and IAsh at some level correspond to genuinely different descriptions of GR, a feature which would have been missed had we applied the step-by-step Dirac procedure. 9 Appendix: Commutation relations for IInst We will now infer the Poisson brackets for (55) by inference from the corre- sponding canonical Ashtekar Poisson brackets {Aai (x), σ̃ b (y)} = δ (3)(x,y) (63) along with the vanishing brackets {Aai (x), Abj(y)} = {σ̃ia(x), σ̃ (x)} = 0. (64) To find the analogue of (63) and (64) for (55), we will use the tranformation equation σ̃ia = ΨaeB e, (65) which corresponds to a noncanonical transformation. Substitution of (65) into (63) yields {Aai (x),Ψbf (y)Bif (y)} = δ (3)(x,y) {Aai (x),Ψbf (y)}B (y) + Ψbf (x){Aai (x), B (y)}. (66) The second term on the right hand side of (66) vanishes on account of the first relations of (64), and upon multiplying (66) by the inverse magnetic field (B−1)ei , assumed to be nondegenerate, we obtain {Aai (x),Ψbf (y)} = δab (B−1(y)) (3)(x,y). (67) 7The linearly versus nonlinearly of the diffeomorphism constraints on the right hand side is just a minor difference. This gives us the Poisson brackets {A,A} ∼ 0 and {A,Ψ} ∼ B−1, which leaves remaining the brackets {Ψ,Ψ}. To obtain these, we substitute (65) into the second equation of (64), yielding {σ̃ia(x), σ̃bj(y)} = {Ψae(x)Bie(x),Ψbf (y)B f (y)} = Ψae(x){Bie(x),Ψbf (y)}B (y) + {Ψae(x),Ψbf (y)}Bie(x)B +Ψbf (x)Ψae(x){Bie(x), B f (y)}+Ψbf (y){Ψae(x), B f (y)}B e(x) = 0. (68) Noting that the third term vanishes on account of the first equation of (64), equation (68) reduces to {Ψae(x),Ψbf (y)}Bie(x)B f (y) +Ψae(x){Bie(x),Ψbf (y)}B (y)−Ψbf (y){Bjf (y),Ψae(x)}B e(x) = 0. (69) The bottom two terms of (69) can be computed using (67) {Bie(x),Ψbf (y)} = ǫimnDxm{Aen(x),Ψbf (y)} = ǫimnDxm(δeb (B−1(y))fnδ(3)(x,y)).(70) Substituting (70) into (69) and cancelling a pair of magnetic fields, then we have that {Ψae(x),Ψbf (y)}Bie(x)B f (y) = ǫ Ψae(x)D m +Ψba(y)D δ(3)(x,y). (71) Left and right multiplying (71) by the inverse of the magnetic fields, we have {Ψae(x),Ψbf (y)} = ǫijm (B−1(y)) mΨab(x)(B −1(x))ei +(B−1(x))eiD mΨba(y)(B −1(y)) δ(3)(x,y). (72) One sees that the internal components of Ψae have nontrivial commutation relations with themselves. References [1] Eyo Ita ‘Instanton representation of Plebanski gravity. Gravitational instantons from the classical formulation.’ arXiv: gr-qc/0703057 [2] Paul Dirac ‘Lectures on quantum mechanics’ Yeshiva University Press, New York, 1964 [3] Hans Stephani, Dietrich Kramer, Maclcolm MacCallum, Cornelius Hoenselaers, and Eduard Herlt ‘Exact Solutions of Einstein’s Field Equations’ Cambridge University Press [4] R. Penrose and W. Rindler ‘Spinors and space-time’ Cambridge Mono- graphs in Mathematical Physics [5] Ahbay Ashtekar ‘New Hamiltonian formulation of general relativity’ Phys. Rev. D36(1987)1587 [6] Ahbay Ashtekar ‘New variables for classical and quantum gravity’ Phys. Rev. Lett. Volume 57, number 18 (1986)

0704.0368

Metal-insulator transition in the low-dimensional organic conductor (TMTSF)2FSO3 probed by infrared microspectroscopy

EPJ manuscript No. (will be inserted by the editor) Metal-insulator transition in the low-dimensional organic conductor (TMTSF)2FSO3 probed by infrared microspectroscopy A. Pashkin1a, K. Thirunavukkuarasu1, Y.-L. Mathis2, W. Kang3, and C. A. Kuntscher1b 1 Experimentalphysik II, Universität Augsburg, 86159 Augsburg, Germany 2 Institute for Synchrotron Radiation, Forschungszentrum Karlsruhe, P.O. Box 3640, 76021 Karlsruhe, Germany 3 Department of Physics, Ewha Womans University, Seoul 120-750, Korea Received: October 28, 2018 Abstract. We present measurements of the infrared response of the quasi-one-dimensional organic conduc- tor (TMTSF)2FSO3 along (E‖a) and perpendicular (E‖b ′) to the stacking axis as a function of temper- ature. Above the metal-insulator transition related to the anion ordering the optical conductivity spectra show a Drude-like response. Below the transition an energy gap of about 1500 cm−1 (185 meV) opens, leading to the corresponding charge transfer band in the optical conductivity spectra. The analysis of the infrared-active vibrations gives evidence for the long-range crystal structure modulation below the transi- tion temperature and for the short-range order fluctuations of the lattice modulation above the transition temperature. Also we report about a new infrared mode at around 710 cm−1 with a peculiar temperature behavior, which has so far not been observed in any other (TMTSF)2X salt showing a metal-insulator transition. A qualitative model based on the coupling between the TMTSF molecule vibration and the reorientation of electrical dipole moment of the FSO3 anion is proposed, in order to explain the anomalous behavior of the new mode. PACS. 71.30.+h Metal-insulator transitions and other electronic transitions – 74.70.Kn Organic super- conductors 1 Introduction The organic Bechgaard salts (TMTSF)2X consist of stacks of planar TMTSF (tetramethyltetraselenafulva- lene) molecules separated by anions (X = PF6, AsF6, ClO4, Br, etc.). The charge transport in these systems is restricted to the direction along the molecular stacks, making the Bechgaard salts prime examples of one- dimensional metals. However, on cooling down most of them undergo a metal-insulator transition which prevents the onset of a superconducting state [1]. In Bechgaard salts with noncentrosymmetric anions such as ReO4, BF4 or FSO3 the metal-insulator transition is related to the anion ordering [2]. It was furthermore demonstrated that in some cases the metal-insulator transition can be sup- pressed by the application of external pressure, leading to a superconducting ground state [3]. The case of the anions X=FSO3 in this class of ma- terials is particularly interesting, since these anions are noncentrosymmetric and in addition possess a permanent electrical dipole moment. The first study of the basic prop- erties of (TMTSF)2FSO3 has been reported by Wudl et al. in 1982 [4]. Further studies have shown that this com- a email: [email protected] b email: [email protected] pound has the highest superconducting transition temper- ature (2.5 K at 8.5 kbar) among the Bechgaard salts. It was proposed that this is due to the interaction of the conducting electrons with the FSO3 anion dipoles [5]. A recent detailed study [6] revealed a very rich pressure- temperature phase diagram of (TMTSF)2FSO3 with a va- riety of different phases, which have not been completely identified up to now. Furthermore, by magnetoresistance measurements a two-dimensional electronic behavior was found in (TMTSF)2FSO3 under a pressure of around 6.2 kbar [7]. The interaction of the FSO3 anions with each other via long-range Coulomb forces and with the centrosymmetric surrounding formed by the TMTSF cations tends to order the anions below a certain temperature. The first-order structural phase transition related to this anion ordering occurs at around TMI=89 K in (TMTSF)2FSO3 at ambi- ent pressure. The change of the crystal structure modifies the electronic band structure: The effective half-filled con- ducting band splits into one filled and one empty band separated by an energy gap, leading to a sharp metal- insulator transition [5]. The structural analysis suggested a modulation of the crystal structure with wavevector q = (1/2, 1/2, 1/2) below the phase transition, which implies an antiferroelectric state [8,2]. The ordering of the FSO3 anions modulates the lattice resulting in a new unit cell http://arxiv.org/abs/0704.0368v1 2 A. Pashkin et al.: Metal-insulator transition in (TMTSF)2FSO3 probed by infrared spectroscopy of size 2a × 2b × 2c. Thus, there are eight formula units of (TMTSF)2FSO3 per unit cell in the low temperature phase. Correspondingly, one can expect a splitting of each vibrational mode into up to eight components [9]. The ratio of the energy gap to the transition tem- perature in (TMTSF)2FSO3 is ∼ 12.5 [4], which is ap- preciably higher than the value 3.5 predicted by the mean-field theory for the Peierls transition. Therefore, the metal-insulator in the Bechgaard salts with non- centrosymmetric anions was attributed to a special type of Peierls instability which originates from the anion-electron coupling [10]. In this work we present the results of a temperature- dependent polarized infrared reflectivity study of (TMTSF)2FSO3 single crystals in the far- and mid- infrared frequency range, in order to characterize the change of electronic and vibrational properties during the metal-insulator transition at TMI=89 K. This is the first infrared spectroscopic investigation of the compound (TMTSF)2FSO3. Our results allow a direct determination of the charge gap in the insulating state. Furthermore, we determined and analyzed the behavior of the vibrational modes during the metal-insulator transition, which can clarify details of the dipolar ordering. 2 Experimental (TMTSF)2FSO3 single crystals were grown by standard electrochemical techniques from TMTSF molecules and tetrabutylammonium-FSO3. The studied samples have a needle-like shape, with a size of approximately 2 × 0.2 × 0.1 mm3. The samples were mounted on a cold-finger Cry- oVac Konti-Mikro cryostat. The actual measuring temper- ature was controlled by a sensor attached in direct vicin- ity of the sample. The measurements were performed at the infrared beamline of the synchrotron radiation source ANKA. The polarized infrared reflectivity was measured in the range 150 - 10000 cm−1 using a Bruker IRscope II microscope attached to a IFS66v/S spectrometer. The frequency resolution was 1 cm−1 for all measured spec- tra. Optically transparent TPX and KBr cryostat win- dows were used for the measurements in the far- and mid- infrared frequency range, respectively. 3 Results and discussion 3.1 Electronic properties The reflectivity spectra of (TMTSF)2FSO3 above and be- low the metal-insulator transition temperature of 89 K for both polarizationsE‖a and E‖b′ (along and perpendicular to the stacking axis, respectively) are shown in Fig. 1. The reflectivity data in the spectral region at around 450 cm−1 are affected by the absorption features of the far-infrared TPX cryostat window and are therefore not shown. At 290 K the reflectivity of the sample along the stacking axis E‖a demonstrates a typical Drude behavior E || a 290 K 45 K 8000250 E || b' Frequency (cm 290 K 45 K Energy (meV) E || a Fig. 1. Reflectivity spectra of (TMTSF)2FSO3 above and be- low the metal-insulator transition for E‖a and E‖b′. (growth up to 1 when frequency tends to zero). In con- trast, at 45 K , i.e., below TMI , the reflectivity is almost frequency independent below 1000 cm−1, which is typical for an insulating state. The interference fringes observed below 400 cm−1 in the spectra for both polarizations are due to the partial transparency of the sample in the insulating phase. Per- pendicular to the stacking axis (E‖b′), the optical reflec- tivity and conductivity is much lower than along the a axis. Nevertheless, the observed changes during the metal- insulator transition are similar to those of the E‖a direc- tion. These results demonstrate the opening of an energy gap at the Fermi level for both studied directions. The dramatic effect of the temperature decrease on the electronic properties of (TMTSF)2FSO3 are more directly seen in the optical conductivity spectra. The E‖a optical conductivity σ1(ω) of (TMTSF)2FSO3 in the insulating (at 45 K) and conducting phase (at 290 K) obtained by means of Kramers-Kronig analysis is shown in Fig. 2. The dominating feature of the spectrum at 45 K is a strong charge transfer band due to electronic transitions across the gap. The arrow shows the band gap (1500 cm−1) ob- tained from the published temperature-dependent dc re- sistivity measurements [4]. Obviously, the agreement of this value with the onset of the optical interband tran- A. Pashkin et al.: Metal-insulator transition in (TMTSF)2FSO3 probed by infrared spectroscopy 3 1000 10000 100 1000 2D = 1500 cm Frequency (cm 290 K 45 K E || a Energy (meV) Fig. 2. E‖a optical conductivity spectra of (TMTSF)2FSO3 above and below the metal-insulator transition at TMI= 89 K. Hatched area depicts the Drude model fit of the high temper- ature optical conductivity. sition is very good. On the other hand, the optical con- ductivity at room temperature is mostly dominated by the Drude response of the free carriers. The corresponding fit using the Drude model is shown as the hatched area in Fig. 2. Obviously, the Drude model provides a good description of the measured room-temperature spectrum excluding the electron-molecular vibration (emv) antires- onance modes. The plasma frequency ωp = 8660 cm and the scattering rate Γ ≃ 1450 cm−1 obtained from the fit agree well with the Drude model parameters reported for other TMTSF salts [11]. The obtained value of the dc conductivity, σdc ≃ 860 (Ωcm) −1, is in reasonable agree- ment with the dc and microwave conductivity values of 1600 and 300 (Ωcm)−1, respectively, reported by Wudl et al. [4]. 3.2 Vibrational modes The TMTSF molecule with the point group symmetry D2h has in total 72 local vibrational modes classified ac- cording to the following representations [12] ΓD2h = (12ag + 11b3g + 11b1u + 11b2u) +(6b1g + 7b2g + 7au + 7b3u), (1) where the vibrations in the first brackets are polarized in the molecular plane (perpendicular to the stacking a axis) and the vibrations in the second brackets are polarized out of the plane (along the stacking axis a). The symmetric (gerade) vibrations are Raman active and the asymmet- ric (ungerade) vibrations are infrared active excluding the au silent modes. Some of the totally symmetric ag Ra- man modes are expected to appear in the infrared spectra for E‖a due to efficient emv coupling in the modulated stacking structure [11,13]. Table 1. The eigenfrequencies and assignment of some vi- brational modes observed in (TMTSF)2FSO3 for E‖a at 45 K below TMI . All numbers are in cm 45 K calculated frequency1 assignment 580 571 ν3(a1) FSO3 728 702 ν51(b2u) 902, 911, 915, 916 ν8(ag) 917, 924, 932 1020, 1031, 1036 1060 ν7(ag) 1067, 1072 1060 ν7(ag) 1362, 14502 1469 ν4(ag) 1354, 1364, 1369 1369 ν6(ag) 1373, 1379, 1385 1550, 1584, 1606 1596 ν3(ag) 1847, 1854, 1863 1863 ν3(ag) + ν11(ag) The tetrahedral FSO3 anion has C3v point group sym- metry which gives in total nine vibrational modes ΓC3v = 3a1(z, x 2 + y2, z2) + 3e(x, y, x2 − y2, xy, yz, xz), where e species correspond to the doublets. Thus, in the infrared spectra one expects six modes, with the 3a1 and 3e modes being polarized along and perpendicular to the polar axis of the anion, respectively. In this section we want to concentrate on the changes in the infrared phonon spectra for both polarizations across the metal-insulator transition. For E‖a several ag vibrations of the TMTSF molecules become infrared ac- tive in the insulating phase. This is due to the effec- tive emv coupling of these vibrations to the on-chain charge transfer band in the structure modulated due to the anion ordering. The list of the new modes observed below the transition together with their tentative as- signment is given in Table 1. Most of them are emv coupled ag modes polarized in the molecular plane or their combination as a triplet at around 1850 cm−1. The ν4(ag) mode involving the central C=C bond stretching is known to have especially strong emv coupling and there- fore it appears as a strong antiresonance mode in the optical conductivity spectrum. It should be pointed out that the observed appearance of ag modes for E‖a in the ordered phase is typical only for (TMTSF)2X com- pounds with non-centrosymmetric anions. In comparison, (TMTTF)2X salts possess a stronger stack dimerization, resulting in the emv coupling of the ag modes already in the disordered phase, and therefore the anion ordering transition causes only a frequency shift and an intensity change of the emv coupled modes [15]. 4 A. Pashkin et al.: Metal-insulator transition in (TMTSF)2FSO3 probed by infrared spectroscopy 1264 1280 1296 560 570 580 590 600 1140 1150 1160 1358 1365 1372 Frequency (cm Frequency (cm Frequency (cm Frequency (cm Fig. 3. Reflectivity spectra (shifted for clarity) of some phonons which experience changes during the metal-insulator transition at 89 K: (a) vibration polarized along the a axis; (b)-(d) vibrations polarized along the b′ axis. The ν3(a1) vibrational mode of the FSO3 anion at 580 cm−1 is observed for the whole studied temperature range. However, the lineshape of this mode in the metallic phase above TMI is inverted with respect to the insulating phase [see Fig. 3(a)], since the background dielectric con- stant is negative as expected for highly conducting metals at low frequencies. Such a change is a clear evidence for the suppression of the Drude conductivity in the insulat- ing phase of (TMTSF)2FSO3. The mode at 728 cm−1 observed for temperatures below TMI is particularly interesting, since its intensity gradually increases on temperature decrease (see Fig. 4). A similar behavior is found for the polarization perpen- dicular to the stacks, E‖b′. Moreover, above the transi- tion temperature a strong asymmetric mode is seen at 710 cm−1. This mode shifts to lower frequencies and gets stronger with increasing temperature. This mode has not been observed in any other earlier study of the Bechgaard salts. Therefore, it would be natural to assign it to a vibra- tion of FSO3 anion. However, such an assignment would be in contradiction to the experimental observations, since: 680 700 720 740 760 E || b' E || a 200 K 200 K Frequency (cm Fig. 4. Reflectivity spectra (shifted for clarity) of the vibration at around 710 cm−1 at different temperatures for E||a and E||b′. (i) the ν5(e) and ν2(a1) vibrations of FSO3 located close to the observed mode have frequencies which are by more than 100 cm−1 higher or lower [14]; (ii) the intensity of the anion vibration should not vanish at the order-disorder transition point. Thus, one has to attribute the modes at around 710 and 728 cm−1 to vibrations of the TMTSF molecules. We suggest that both modes originate from the ν51(b2u) in-plane vibration of the TMTSF molecule. Ac- cording to the normal-coordinate analysis [12,16] its fre- quency for a free TMTSF0.5+ cation is 702 cm−1. The corresponding atomic movements involve stretch- ing of the Se-C side bond and rocking of the adjacent methyl group. For the b2u vibration the inversion symme- try of the molecule is not preserved, causing its infrared- activity for the polarization perpendicular to the stacks. However, it is known that in (TMTSF)2X salts the dipole moment corresponding to the ν51(b2u) vibration is very small, and therefore this mode can hardly be detected even for E‖b′ where it should have the strongest inten- sity [16]. Nevertheless, in (TMTSF)2FSO3 this mode is particularly strong even at room temperature. This find- ing can be explained by the electrical dipole of the FSO3 anion pointing towards the Se-F bond. Similar to other A. Pashkin et al.: Metal-insulator transition in (TMTSF)2FSO3 probed by infrared spectroscopy 5 Fig. 5. Schematic illustration of the ν51(b2u) vibration coupled to the reorientation of the FSO3 electrical dipole moment. The projection of the crystal structure on the b− c plane is shown. Only the Se (large open circles) and C (small filled circles) atoms of the TMTSF molecules are presented, together with the displacements of the Se atoms. The grey filled circles be- tween molecules denote the positions of the FSO3 anions, the bold arrows show the two possible orientations of the anion dipole moment (p1 and p2). Because of the symmetry prop- erties of the b2u vibration the reorientation of the electrical dipole moment leads to a change of polarization ∆p in the perpendicular direction for any orientation of the anion dipole moment. non-centrosymmetric anions (ReO4, ClO4 etc.) the FSO3 anion has two possible symmetrically equivalent orienta- tions for which the dipole moment points towards the Se atoms of the neighboring TMTSF molecules. This situa- tion is sketched in Fig. 5, where p1 and p2 are two possible orientations of the FSO3 electrical dipole moment. During the vibration the dipole moment of the anion follows the position of the Se atom. Due to the symmetry properties of the b2u vibration the nearest Se atoms on both sides of the anion move in the same direction. Thus, for both possible orientations of the dipole the b2u vibration results in a change of the average polarization along the direction of ∆p (Fig. 5). The described coupling mechanism between the b2u vibration and the dipole moment of the anion in (TMTSF)2FSO3 should lead to a strong enhancement of the infrared strength of the ν51(b2u) vibration for E‖b since ∆p has the largest projection along this direction. On the other hand, ∆p is perpendicular to the stacking axis and the b2u mode should not appear for E‖a. This is indeed observed in our experiment above the transition temperature. Below the transition the long-range order of the anion sublattice builds up. Then the anion dipole mo- ment orientation is determined by the modulation of the whole lattice and it is not dependent on the movement of neighboring TMTSF molecules, i.e., the ν51(b2u) vibration is decoupled from the FSO3 anions. Therefore, its inten- sity should drop abruptly below TMI , in agreement with our observations (see Fig. 4). Moreover, the observed de- crease of the intensity of the coupled b2u mode at around Table 2. The eigenfrequency, width (given in bracket), and assignment of some vibrational modes observed for E‖b′ at selected temperatures. All numbers are in cm−1. 95 K 80 K 45 K assignment 580 (1.3) 580 (0.9) 580 (0.8) ν3(a1) FSO3 710 (7.1) 728 728 (2.0) ν51(b2u) 1154 (5.0) 1150 (3.4) 1150 (3.3) ν48(b2u) 1157 (3.8) 1158 (2.5) 1280 (5.1) 1276 (4.2) 1276 (2.1) ν4(e) FSO3 1288 (16) 1286 (4.4) 1286 (1.7) 1363 (3.2) 1361 (2.8) 1361 (1.3) ν47(b2u) 1366 (4.3) 1367 (2.4) 1367 (1.8) 710 cm−1 at 95 K compared to higher temperatures can be explained by taking into account short-range order fluc- tuations above the transition, evidence for which is also given below. Indeed, in the large enough dynamical regions where the anions are ordered, the coupling is suppressed and therefore the strength of the ν51(b2u) should decrease. On cooling down below TMI a vibration appears again at somewhat higher frequency (728 cm−1) for E‖b′ and its strength gradually increases with decreasing temper- ature. We suggest that this is the same ν51(b2u) vibra- tion described above. Since it is decoupled from the anion sublattice, its frequency is expected to increase abruptly below the transition. The increase in strength for both polarizations should be obviously related to the tempera- ture dependence of the order parameter (i.e., the degree of lattice modulation). One of the possible mechanisms can be the emv coupling of the ν51(b2u) vibration to the charge transfer bands along a and b′ directions. However, the detailed picture of this emv coupling is not clear, since the symmetry of b2u mode does not allow such kind of coupling. One can speculate that the electric field of the FSO3 dipoles in the ordered phase distorts the TMTSF molecules making them non-centrosymmetric. Then the emv coupling may become allowed for the b2u(ν51) mode. Noticeable changes in the phonon mode spectra across the metal-insulator transition are observed for E‖b′. The list of the parameters of these modes at temperatures above and below TMI is given in Table 2. An obvious split- ting into two components is seen for the ν48(b2u) mode at 1154 cm−1 [see Figure 3(b)]. In addition, the damping of the split components directly below TMI is lower than the damping of the single component directly above the tran- sition (see Table 2). This difference is probably related to the precursor short-range order fluctuations above the transition, which can induce a small splitting already in the disordered phase. An evidence for such fluctuations was found in x-ray diffuse scattering experiments [8,2]. This effect is even more clearly seen in the splitting of two other modes: the doublet ν4(e) vibration of the FSO3 anion at 1280 cm−1 [Fig. 3(c)] and the ν47(b2u) mode at around 1365 cm−1 [Fig. 3(d)]. For each of these modes 6 A. Pashkin et al.: Metal-insulator transition in (TMTSF)2FSO3 probed by infrared spectroscopy above TMI one can resolve two weakly split components. However, below TMI the splitting abruptly increases and the damping decreases (see Table 2) indicating the onset of long-range order. Since the described effect is observed not only for the FSO3 anion vibration but also for two vi- brations of the TMTSF cation, we can conclude that the short-range order fluctuations involve the modulation of the whole (TMTSF)2FSO3 lattice and not only the anion sublattice. 4 Conclusion We have performed an infrared spectroscopic study of the metal-insulator transition in (TMTSF)2FSO3. The ob- tained optical conductivity spectra for E‖a show a Drude- like conductivity above the anion ordering temperature and a charge transfer band formed below the transition. The onset of this band is in agreement with the energy gap value of 1500 cm−1 obtained from transport measure- ments [4]. The analysis of the infrared-active vibrations leads to the following conclusions: (i) the crystal structure modula- tion below the metal-insulator transition leads to a strong emv coupling of several ag vibrations which therefore be- come infrared-active; (ii) short-range order fluctuations of the FSO3 anions and the corresponding lattice modulation exist above the transition temperature, as it is seen from the splitting of some infrared-active modes for E‖b′; (iii) a new infrared-active mode located at around 710 cm−1 with a peculiar temperature behavior is detected and as- signed to the coupling between the b2u TMTSF molecule vibration and the electrical dipole moment of the FSO3 anion. The latter feature has not been observed in any other (TMTSF)2X salt showing a metal-insulator transi- tion. This points out the important role of the electrical dipole moment of the anion on the structural and dynam- ical properties of the (TMTSF)2FSO3 salt. 5 Acknowledgements We acknowledge the ANKA Angströmquelle Karlsruhe for the provision of beamtime and thank M. Süpfle, D. Moss, and B. Gasharova for technical assistance at the ANKA IR beamline. The financial support of the DFG (Emmy Noether-program) is acknowledged. References 1. T. Ishiguro, K. Yamaji, G. Saito, Organic Superconductors (Springer, Berlin, 1998) 2. J.P. Pouget, S. Ravy, J. Phys. I France 6, 1501 (1996) 3. D. Jerome, Chem. Rev. 104, 5565 (2004) 4. F. Wudl, E. Aharon-Shalom, D. Nalewajek, J.V. Waszczak, J. W. M. Walsh, J. L. W. Rupp, P. Chaikin, R. Lacoe, M. Burns, T.O. Poehler et al., The Journal of Chemical Physics 76(11), 5497 (1982) 5. R.C. Lacoe, S.A. Wolf, P.M. Chaikin, F. Wudl, E. Aharon- Shalom, Phys. Rev. B 27(3), 1947 (1983) 6. Y.J. Jo, E.S. Choi, H. Kang, W. Kang, I.S. Seo, O.H. Chung, Phys. Rev. B 67, 014516 (2003) 7. W. Kang, O.H. Chung, Y.J. Jo, H. Kang, I.S. Seo, Phys. Rev. B 68, 073101 (2003) 8. R. Moret, J.P. Pouget, R. Comes, K. Bechgaard, J. Phys. Colloq. France 44, 957 (1983) 9. C.C. Homes, J.E. Eldridge, Phys. Rev. B 40, 6138 (1989) 10. C.S. Jacobsen, H.J. Pedersen, K. Mortensen, G. Rindorf, N. Thorup, J.B. Torrance, K. Bechgaard, J. Phys. C 15, 2651 (1982) 11. C.S. Jacobsen, D.B. Tanner, K. Bechgaard, Phys. Rev. B 28, 7019 (1983) 12. M. Meneghetti, R. Bozio, I. Zanon, C. Pelice, C. Ricotta, M. Zanetti, J. Chem. Phys. 80, 6210 (1984) 13. C.C. Homes, J.E. Eldridge, Phys. Rev. B 42, 9522 (1990) 14. K. Nakamoto, Infrared and Raman Spectra of Inorganic and Coordination Compounds (Wiley, New York, 1986) 15. C. Garrigou Lagrange, A. Graja, C. Coulon, P. Delhaes, J. Phys. C: Solid State Phys. 17, 5437 (1984) 16. J.E. Eldridge, C.C. Homes, Phys. Rev. B 43, 13971 (1991) Introduction Experimental Results and discussion Conclusion Acknowledgements

0704.0369

The effect of Topcolor Assisted Technicolor, and other models, on Neutrino Oscillation

arXiv:0704.0369v1 [hep-ph] 3 Apr 2007 August 9, 2021 18:21 WSPC - Proceedings Trim Size: 9in x 6in SCGT06-takeuchi OCHA-PP-270, YITP-07-09, VPI-IPNAS-07-02 The effect of Topcolor Assisted Technicolor, and other models, on Neutrino Oscillation Minako Honda1, Yee Kao2, Naotoshi Okamura3, Alexey Pronin2, and Tatsu Takeuchi2∗ 1Physics Department, Ochanomizu Women’s University, Tokyo 112-8610, Japan 2Physics Department, Virginia Tech, Blacksburg VA 24061, USA 3Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan New physics beyond the Standard Model can lead to extra matter effects on neutrino oscillation if the new interactions distinguish among the three flavors of neutrino. In Ref. 1, we argued that a long-baseline neutrino oscillation ex- periment in which the Fermilab-NUMI beam in its high-energy mode2 is aimed at the planned Hyper-Kamiokande detector3 would be capable of constraining the size of those extra matter effects, provided the vacuum value of sin2 2θ23 is not too close to one. In this talk, we discuss how such a constraint would translate into limits on the coupling constants and masses of new particles in models such as topcolor assisted technicolor.4 1. Introduction When considering matter effects on neutrino oscillation, it is customary to consider only the W -exchange interaction of the νe with the electrons in matter. However, if new interactions beyond the Standard Model (SM) that distinguish among the three generations of neutrinos exist, they can lead to extra matter effects via radiative corrections to the Zνν vertex which effectively violate neutral current universality, or via the direct exchange of new particles between the neutrinos and matter particles. For instance, topcolor assisted technicolor4 treats the third generation differently from the first two and the Z ′ in this class of models couples more strongly to the ντ than to the νe or νµ. In Extended Technicolor (ETC) Models, such as that of Appelquist, Piai, and Shrock,5 the neutral technimesons, which mix with the Z, couple to different generation fermions differently, distinguishing among νe, νµ, and ντ . The diagonal ETC gauge bosons also couple to the different generations differently, as well as the ∗Presenting Author http://arxiv.org/abs/0704.0369v1 August 9, 2021 18:21 WSPC - Proceedings Trim Size: 9in x 6in SCGT06-takeuchi large variety of leptoquark states in the model. Flavor distinguishing matter effects from diagonal ETC and leptoquarks are induced by ETC gauge boson mixing. The effective Hamiltonian that governs neutrino oscillations in the pres- ence of neutral-current lepton universality violation, or new physics that couples to the different generations differently, is given by1 H = Ũ λ1 0 0 0 λ2 0 0 0 λ3  Ũ † = U 0 0 0 0 δm221 0 0 0 δm231 U † + a 0 0 0 0 0 0 0 0 be 0 0 0 bµ 0 0 0 bτ where U is the MNS matrix, a = 2EVCC , VCC = 2GFNe = Ne , (2) is the usual matter effect due to W -exchange between νe and the electrons, and be, bµ, bτ are the extra matter effects which we assume to be non-equal. We define the parameter ξ as bτ − bµ = ξ . (3) Then, the effective Hamiltonian can be rewritten as H = Ũ λ1 0 0 0 λ2 0 0 0 λ3  Ũ † = U 0 0 0 0 δm221 0 0 0 δm231 U †+a 1 0 0 0 −ξ/2 0 0 0 +ξ/2  , (4) where we have absorbed the extra b-terms in the (1, 1) element into a. The extra ξ-dependent contribution in Eq. (4) can manifest itself when a > |δm231| (i.e. E & 10GeV for typical matter densities in the Earth) in the νµ and ν̄µ survival probabilities as P (νµ → νµ) ≈ 1− sin2 2θ23 − δm231 P (ν̄µ → ν̄µ) ≈ 1− sin2 2θ23 + δm231 , (5) where ∆ ≈ ∆31c213 −∆21c212 , ∆ij = δm2ij L , cij = cos θij , (6) and the CP violating phase δ has been set to zero. As is evident from these expressions, the small shift due to ξ will be invisible if the value of sin2 2θ23 is too close to one. However, if the value of sin 2 2θ23 is as low August 9, 2021 18:21 WSPC - Proceedings Trim Size: 9in x 6in SCGT06-takeuchi as sin2 2θ23 = 0.92 (the current 90% lower bound), and if ξ is as large as ξ = 0.025 (the central value of from CHARM/CHARM II6), then the shift in the survival probability at the first oscillation dip can be as large as ∼ 40%. If the Fermilab-NUMI beam in its high-energy mode2 were aimed at a declination angle of 46◦ toward the planned Hyper-Kamiokande detector3 in Kamioka, Japan (baseline 9120 km), such a shift would be visible after just one year of data taking, assuming a Mega-ton fiducial volume and 100% efficiency. The absence of any shift after 5 years of data taking would constrain ξ to1 |ξ| ≤ ξ0 ≡ 0.005 , (7) at the 99% confidence level. In the following, we look at how this potential limit on ξ would translate into constraints on the Z ′ in topcolor assisted technicolor, and various types of leptoquarks. A more comprehensive analysis will be presented in Ref. 7. 2. Topcolor Assisted Technicolor Though there are several different versions of topcolor assisted technicolor,4 we consider here the simplest in which the quarks and leptons transform under the gauge group SU(3)s × SU(3)w × U(1)s × U(1)w × SU(2)L (8) with coupling constants g3s, g3w, g1s, g1w, and g, respectively. It is assumed that g3s ≫ g3w and g1s ≫ g1w. SU(2)L is the usual weak-isospin gauge group of the SM. The first and second generation fermions are assumed to be charged only under SU(3)w×SU(2)L×U(1)w, while the third generation fermions are assumed to be charged only under SU(3)s × SU(2)L ×U(1)s. The U(1) charges for both cases are set equal to the SM hypercharge. At scale Λ ∼ 1 TeV, technicolor, which is included in the model to generate the W and Z masses, is assumed to become strong and generate a condensate (of something which is left unspecified) which breaks the two SU(3)’s and the two U(1)’s to their diagonal subgroups: SU(3)s × SU(3)w → SU(3)c , U(1)s × U(1)w → U(1)Y , (9) which we identify with the usual SM color and hypercharge groups. The massless unbroken U(1) gauge boson Bµ and the massive broken U(1) gauge boson Z ′µ are related to the original U(1)s × U(1)w gauge fields Ysµ and Ywµ by Z ′µ = Ysµ cos θ1 − Ywµ sin θ1 August 9, 2021 18:21 WSPC - Proceedings Trim Size: 9in x 6in SCGT06-takeuchi Bµ = Ysµ sin θ1 + Ywµ cos θ1 (10) where tan θ1 = . (11) The currents to which the Bµ and Z µ couple to are: 1sYsµ+g1wJ 1wYwµ = g ′ (cot θ1J 1s − tan θ1J 1s + J 1w)Bµ , where . (13) The current J 1s + J 1w is the SM hypercharge current, and g ′ is the SM hypercharge coupling constant. The exchange of the Z ′ leads to the current-current interaction (cot θ1J1s − tan θ1J1w) (cot θ1J1s − tan θ1J1w) , (14) the J1sJ1s part of which does not contribute to neutrino oscillations on the Earth, while the J1wJ1w part is suppressed relative to the J1wJ1s part by a factor of tan2 θ1 ≪ 1. Therefore, we only need to consider the J1sJ1w interaction which only affects the propagation of ντ . The effective potential felt by ντ due to this interaction is Vντ = , (15) and the effective ξ is ξTT = Vντ − Vνµ (g′/MZ′) (g/MW )2 tan2 θW sin2 θW The limit |ξTT | ≤ ξ0 = 0.005 then translates into: MZ′ ≥ MZ sin2 θW ≈ 440GeV . (17) Unfortunately, this potential limit from the measurement of ξ is weaker than what is already available from precision electroweak data,8 and from direct searches for pp̄ → Z ′ → τ+τ− at CDF.9,10 August 9, 2021 18:21 WSPC - Proceedings Trim Size: 9in x 6in SCGT06-takeuchi 3. Generation Non-diagonal Leptoquarks The interactions of leptoquarks with ordinary matter can be described in a model-independent fashion by an effective low-energy Lagrangian as dis- cussed in Ref. 11. Assuming the fermionic content of the SM, the most gen- eral dimensionless SU(3)C × SU(2)L ×U(1)Y invariant couplings of scalar and vector leptoquarks satisfying baryon and lepton number conservation are given by: L = LF=2 + LF=0 , (18) where LF=2 = ejL − dciLνjL) + g γµejL) + g γµejR) V +2µ γµνjL) + g γµejR) V −2µ γµejL)Ṽ 2µ + (u γµνjL)Ṽ ejL)S 3 − (uciLejL + dciLνjL)S νjL)S , (19) LF=0 = (uiRejL) + h (uiLejR) (uiRνjL)− hij2R(diLejR) (diRejL)S̃ 2 + (diRνjL)S̃ (uiLγ µνjL + diLγ µejL) + h (diRγ µejR) V 01µ 2(uiLγ µejL)V 3µ + (uiLγ µνjL − diLγµejL)V 03µ + 2(diLγ µνjL)V Here, the scalar and vector leptoquark fields are denoted by S and V , their subscripts indicating the dimension of their SU(2)L representation, and the superscripts indicating the sign of the weak-isospin of each component. We allow for generation non-diagonal couplings with the indices i and j indicat- ing the quark and lepton generation numbers, respectively. The subscript L or R on the coupling constants indicate the chirality of the lepton involved in the interaction. For simplicity, color indices have been suppressed. The leptoquarks S1, ~S3, V2, Ṽ2 carry fermion number F = 3B + L = −2, while the leptoquarks S2, S̃2, V1, ~V3 have F = 0. The interactions that affect neu- trino oscillation are those with (ij) = (12) or (13). August 9, 2021 18:21 WSPC - Proceedings Trim Size: 9in x 6in SCGT06-takeuchi Table 1. Constraints on the leptoquark couplings with all the leptoquark masses set to 300 GeV. To obtain the bounds for a different leptoquark mass MLQ, simply rescale these numbers with the factor (MLQ/300 GeV) LQ CLQ δλ upper bound from |ξ| ≤ ξ0 S1 +3 |g |2 − |g13 |2 0.01 ~S3 +9 |g |2 − |g13 |2 0.003 S2 −3 |h |2 − |h13 |2 0.01 S̃2 −3 |h̃ |2 − |h̃13 |2 0.01 V2 +6 |g |2 − |g13 |2 0.005 Ṽ2 +6 |g̃ |2 − |g̃13 |2 0.005 V1 −6 |h |2 − |h13 |2 0.005 ~V3 −18 |h |2 − |h13 |2 0.002 It is straightforward to calculate the effective potentials due to the ex- change of these leptoquarks, as well as the effective values of ξ.7 Assuming a common mass for leptoquarks in the same SU(2)L weak-isospin multiplet, the effective ξ due to the exchange of any particular type of leptoquark can be written in the form ξLQ = CLQ δλ2LQ/M g2/M2 δλ2LQ . (21) Here, CLQ is a constant prefactor, and δλ LQ represents δλ2LQ = |λ12LQ|2 − |λ13LQ|2 , (22) where λ is a generic coupling constant. The values of CLQ and δλ for the different types of leptoquark are listed in Table 1. The constraint |ξLQ| ≤ ξ0 translates into: MLQ ≥ |CLQ||δλ2LQ| 2GF ξ0 |CLQ||δλ2LQ| × (1700GeV) . (23) Alternatively, one can fix the leptoquark mass and obtain upper bounds on the leptoquark couplings: ∣δλ2LQ 2GF ξ0 M2LQ ≈ 300GeV . (24) The values when MLQ = 300GeV are listed in the rightmost column of Table 1. Thought it is often stated that generation non-diagonal couplings of leptoquarks are strongly constrained by the absence of flavor chang- ing neutral currents, it is only the products of the (ij) = (12) and (13) couplings with other couplings that are constrained.12 The limits on the August 9, 2021 18:21 WSPC - Proceedings Trim Size: 9in x 6in SCGT06-takeuchi individual couplings can be improved considerably. The current leptoquark mass bounds from direct searches at the Tevatron, LEP, and HERA are in the 200∼300 GeV range assuming generation diagonal couplings set equal 4πα. At the LHC, leptoquarks, if they exist, can be expected to be pair-produced copiously through gluon-gluon fusion. The expected sensi- tivity is up to about 1.5 TeV.13 Depending on the value assumed for δλ2LQ, the bound from Eq. (23) can be competitive. Acknowledgments We would like to thank Drs. Andrew Akeroyd, Mayumi Aoki, Masafumi Kurachi, Robert Shrock, and Hiroaki Sugiyama for helpful discussions. This research was supported in part by the U.S. Department of Energy, grant DE–FG05–92ER40709, Task A (Kao, Pronin, and Takeuchi). References 1. M. Honda, N. Okamura, and T. Takeuchi, arXiv:hep-ph/0603268. 2. NUMI Technical Design Handbook, available at http://www-numi.fnal.gov/numwork/tdh/tdh index.html 3. Y. Itow et al., arXiv:hep-ex/0106019; updated version available at http://neutrino.kek.jp/jhfnu/. 4. C. T. Hill, Phys. Lett. B 345, 483 (1995); G. Buchalla, G. Burdman, C. T. Hill, and D. Kominis, Phys. Rev. D 53, 5185 (1996). 5. T. Appelquist, M. Piai and R. Shrock, Phys. Rev. D 69, 015002 (2004). 6. J. Dorenbosch et al. [CHARM Collaboration], Phys. Lett. B 180, 303 (1986); P. Vilain et al. [CHARM-II Collaboration], Phys. Lett. B 320, 203 (1994). 7. M. Honda, Y. Kao, N. Okamura, A. Pronin, and T. Takeuchi, in preparation. 8. R. S. Chivukula and J. Terning, Phys. Lett. B 385, 209 (1996); W. Loinaz and T. Takeuchi, Phys. Rev. D 60, 015005 (1999). 9. D. Acosta et al. [CDF Collaboration], Phys. Rev. Lett. 95, 131801 (2005). 10. W. M. Yao et al. [Particle Data Group], J. Phys. G 33 (2006) 1. 11. W. Buchmüller, R. Rückl and D. Wyler, Phys. Lett. B 191, 442 (1987); M. Tanabashi, p.412 of Ref. 10. 12. S. Davidson, D. C. Bailey and B. A. Campbell, Z. Phys. C 61, 613 (1994); M. Leurer, Phys. Rev. D 49, 333 (1994); M. Leurer, Phys. Rev. D 50, 536 (1994); E. Gabrielli, Phys. Rev. D 62, 055009 (2000). 13. ATLAS detector and physics performance. Technical design report. Vol. 2, CERN-LHCC-99-15, ATLAS-TDR-15; CMS physics : Technical Design Re- port v.2 : Physics performance, CERN-LHCC-2006-021, CMS-TDR-008-2; V. A. Mitsou, N. C. Benekos, I. Panagoulias and T. D. Papadopoulou, Czech. J. Phys. 55, B659 (2005).

0704.0370

Shaped angular dependence of the spin transfer torque and microwave generation without magnetic field

The magnetization of a ferromagnetic body can be manipulated by transfer of spin angular momentum from a spin-polarized curren Shaped angular dependence of the spin transfer torque and microwave generation without magnetic field O. Boulle1, V. Cros1, J. Grollier1, L. G. Pereira1,*, C. Deranlot1, F. Petroff1, G. Faini2, J. Barnaś3, A. Fert1 1 Unité Mixte de Physique CNRS/Thales and Université Paris Sud XI, Route départementale 128, 91767 Palaiseau, France 2 Laboratoire de Photonique et de Nanostructures LPN-CNRS, Route de Nozay, 91460 Marcoussis, France 3 Department of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland Abstract: The generation of oscillations in the microwave frequency range is one of the most important applications expected from spintronics devices exploiting the spin transfer phenomenon. We report transport and microwave power measurements on specially designed nanopillars for which a non-standard angular dependence of the spin transfer torque (wavy variation) is predicted by theoretical models. We observe a new kind of current-induced dynamics that is characterized by large angle precessions in the absence of any applied field, as this is also predicted by simulation with such a wavy angular dependence of the torque. This type of non-standard nanopillars can represent an interesting way for the implementation of spin transfer oscillators since they are able to generate microwave oscillations without applied magnetic field. We also emphasize the theoretical implications of our results on the angular dependence of the torque. The magnetization of a ferromagnetic body can be manipulated by transfer of spin angular momentum from a spin-polarized current. This is the concept of spin transfer introduced by Slonczewski [1] and Berger [2] in 1996. In most experiments, a spin-polarized current is injected from a spin polarizer into a “free” magnetic element, for example in pillar-shaped magnetic trilayers [3-6]. The phenomenon of spin transfer has a great potential for applications. It can be used either to switch a magnetic configuration (the configuration of a magnetic memory for example) [3-5] or to generate magnetic precessions and voltage oscillations in the microwave frequency range[6-7]. In the most usual situations, such oscillations are observed in the presence of a magnetic field. From a fundamental point of view, spin transfer effects raise two different types of problems [8]. First the spin transfer torque acting on a magnetic element is related to the transverse spin polarisation of the current (transverse meaning perpendicular to the magnetization axis of the element) and can be derived from spin-dependent transport equations [8-17]. On the other hand, the description of the magnetic excitations generated by the spin transfer torque raises problems of non-linear dynamics [8,18-20]. For example, in the simple limit where the excitation is supposed to be a uniform precession of the magnetization (macrospin approximation), this precession can be determined by introducing the spin transfer torque into a Landau-Lifshitz- Gilbert (LLG) equation for the motion of the magnetic moment. However, the determination of the spin transfer torque and the description of the magnetization dynamics cannot be regarded as independent problems. In standard trilayered structures with in-plane magnetizations and with the usual angular dependence, a switching regime is found at zero and low magnetic field and the precession regime with generation of voltage oscillations is mainly observed above some threshold field [8]. We will show that a new behavior, characterized by large angle precessions in the absence of any magnetic field, can be obtained in specially designed structures presenting a non-standard dependence of the spin transfer torque as a function of the angle between the fixed magnetization of the polarizer and the magnetization of the free layer. This non-standard angular dependence of the torque, that we call “wavy”, is obtained by choosing materials with different spin diffusion lengths for the “fixed” and “free” magnetic layers, which changes the distribution of the spin currents and spin accumulations in the structure. The observation of spin transfer oscillations at zero field in structures with a “wavy” angular dependence of the torque can represent a new way to obtain spin transfer oscillators operating without any applied field, an other possible way being the use of exchange interactions or anisotropy to generate local effective fields or non-collinear equilibrium configurations [21]. In addition, the observation of a wavy angular dependence of the torque represents a valuable test of the theory and shows that realistic predictions of the spin transfer torque and its angular dependence in a given structure are now possible. As we will see, in the models we consider here [15-16], the torque is calculated from parameters which, for most of them, can be derived from former CPP-GMR experiments [22-23]. The usual behaviour observed in pillars with in-plane magnetizations along an anisotropy axis corresponds to the standard angular dependence of the inset of Fig.1a, in which the torque starts from zero at ϕ = 0 (P equilibrium state with parallel magnetizations of the fixed and free magnetic layers) and keeps the same sign till it comes back to zero at ϕ = π (AP antiparallel state). At zero field and starting from a P state for example (Fig.1b), a negative current (electrons going from the free to the fixed layer in our convention) will destabilize the P state and stabilize the AP state, i.e. can switch the system from P to AP. In the presence of a large enough applied field favouring the P configuration, the torque cannot stabilize the AP state and leaves the system in an intermediate precession state. This is what we call the standard behaviour with irreversible switching at low field and precession at high field, as illustrated by Fig.2a (remark: in some low field experiments however, the irreversible switching is preceded by precessions in a very narrow current range just below the switching current). The non-standard behaviour with precession at zero and/or low field presented in this article is related to the existence of a wavy angular dependence of the torque acting on the free magnetic layer. This oscillatory angular dependence, with an inversion of the torque between ϕ = 0 and ϕ = π, is shown in Fig. 1a. We present the results of calculations in the models of Fert et al [15] and Barnaś et al [16-17] for a Py(8)/Cu(10)/Co(8) pillar. With respect to standard structures like Co/Cu/Co or Py/Cu/Py, the difference we have introduced is a large asymmetry between the spin diffusion lengths (SDL) in the magnetic layers, with a long SDL in Co ≈ 38 nm (at room temperature) and a short SDL in Py ≈ 4 nm [22-23]. The smaller spin asymmetry of the resistivity in Co could also affect the angular dependence but we have checked by additional calculations that the wavy variation comes primarily from the shorter SDL in the Py free layer and not from the different spin asymmetry coefficients, as this has been mistakenly written in Ref.[24]. The solid curves in Fig.1a correspond to the calculation in the model of Barnaś et al [16]. A wavy angular dependence is also predicted by the model of Fert et al [15] which gives the terms of first order in ϕ and (π-ϕ) in the vicinity of the colinear P or AP states (the solid straight lines at the left and right edges of the graph in Fig.1). Due to the inversion at small values of ϕ, a negative current (Fig. 1c) now stabilizes not only the AP state but also the P one and should be ‘inactive’. This can be a solution, for example, to reduce the spin-transfer-induced noise that is detrimental to read heads. In contrast, an appropriate positive current can destabilize both the P and AP states, leading to a precessional solution the motion equation, even at zero field. To validate these predictions, we have performed transport and microwave power measurements at room temperature on Py(8)/Cu/Co(8) elliptical nanopillars of approximate dimensions 100x155 nm². Only the top Py layer (free layer) and the Cu spacer are etched through. The unetched Co layer (“fixed” layer) lies directly on the Ta/Cu bottom electrode. Very similar results have been obtained on Py(8)/Cu/Co(4)/IrMn nanopillars in which the extended Co layer is exchange biased by the IrMn one. We show in Fig. 2b the GMR signal of a Py(8)/Cu/Co(8) sample. Starting, for example, from large negative fields, the switching to an AP state at about 40 Oe is related to the magnetization reversal of the free layer (Py) to the positive direction, as this can be found from subsequent CIMS experiments in which the current-induced return to P is made harder by a larger positive field (consistently with a positive orientation of the Py magnetization in the switching to the AP state). From the GMR minor cycles of the Py layer (see Supplementary Information), we find that the coercive field of the Py layer is 90 Oe and the dipolar field acting on it is 43 Oe. The different behaviours observed for standard and wavy angular dependences are first illustrated in Fig.2a and 2c. In Fig. 2a, we show the standard variation of differential resistance (dV/dI) versus I measured on a Py(4 nm)/Cu(10 nm)/Py(15 nm) pillar: starting from a P state, a negative current induces an irreversible switching from P to AP at low field and a reversible variation with the characteristic peak of steady precessions at high field. In contrast, starting again from a P magnetic configuration with magnetizations in the positive field direction but now with a Py(8 nm)/Cu(10 nm)/Co(8 nm) pillar for which a wavy angular dependence is expected, we detect (Fig. 2c) reversible peaks of dV/dI for positive currents and at very small fields on both sides of Happ= 0. The peak current increases with increasing applied positive field as expected since the P state becomes more stable. We have also performed experiments with an AP initial state. We find that dV/dI first drops to the level of a P state at some positive current and then, at higher current, exhibits the same characteristic precession peak we observe in measurements with a P initial state (data not presented). In Fig. 3, we present microwave power spectra recorded with the same P initial state and for several values of the current. Fig.3a is for zero applied field (actually, Happ ≈ 2 Oe) and Fig.3b for zero effective in-plane field (after subtracting the dipolar field). Coloured dots in the insets indicate the values of the current on the corresponding dV/dI vs I curves. A peak in the microwave power spectrum turns out approximately in some current range above the maximum of dV/dI. The frequency f of the microwave peak increases with the current (blue shift), in contrast with the red shift generally observed in standard pillars with in-plane magnetization. Actually, with the standard angular dependence of the torque, the theoretical prediction is a succession of red and blue shift regimes at increasing current but, in experiments with in-plane applied fields, the crossover to a blue shift regime has been seldom observed [25]. In macrospin simulations, a blue shift in f is predicted for the regime of out-of-plane (OP) precessions and is also associated with a decrease of f with increasing in-plane field. As shown in Fig.3 c, we observe this decrease of f with Happ. In Fig.4a, we present the current-field diagram of the microwave power. Microwave signals are emitted only in the top left corner of the diagram, i.e. at low field and in a zone which is also a region of increased resistance (Fig.4b). No excitation is observed at higher field. We can therefore put forward two main results from our microwave power data: i) Pillars in which a wavy angular dependence of the spin transfer torque is expected, generate microwave oscillations, but, in contrast with the standard behavior, when there are excited by positive currents and at zero field; ii) These microwave oscillations present a blue shift of their frequency with current, a behaviour generally associated with out-of-plane precessions. We first want to exclude that the effects described in the preceding paragraphs could arise from other origins than the wavy angular dependence of the STT. Could they arise from excitations of the Co “fixed” layer? We can first argue that the same behaviour is also observed when the 4nm thick and extended Co layer is pinned by an IrMn layer and that an excitation of a thin Co layer in the presence of such a strong pinning is quite improbable. We can also point out that, for un-pinned continuous magnetic layers, the switching current densities obtained by Chen et al. [26] are about one order of magnitude larger than ours. In addition, whereas a reduction of the thickness of the Co layer to 4 nm for the same 8nm Py thickness should make the excitation of Co easier (smaller current), our experimental results are in the opposite direction. The sample of Fig.2-3 exhibits the relatively simple behaviour predicted for a wavy angular dependence of the torque in a macrospin picture, i.e. precessions at zero and low field in positive current. However, in a series of five similar samples (with or without pinning by IrMn), we have also observed additional features in transport measurements. For example, in some samples and with an initial P state, we see not only peaks in dV/dI in positive current at zero or low field but also partial or total switchings in negative current. These excitations can be ascribed to a non- uniform distribution of the magnetization [27]. For a part of the sample, the angle ϕ between the magnetizations of the two layers is above ϕc, the angle of torque inversion, and can be excited by a negative current. However, we emphasize that these additional excitations observed in transport measurements are never associated with peaks in the emitted power in the Gigahertz range. All the samples share the same main features with microwave emission only at low field in positive current. We now present the theoretical implications of our experimental results and first comment briefly on the origin of the wavy angular dependence of the spin transfer torque in our samples. The physics governing this angular dependence can be discussed simply by considering that, in all the models [8,13-17] based on interfacial absorption of the transverse spin component and boundary conditions of the mixing conductance type (the language can be different in different formalisms), the spin transfer torque is proportional to the transverse component of the spin accumulation in the spacer layer. The key point is that the spin accumulation in a nonmagnetic conductor is directly related to the gradient of the spin current along the current axis z, dzjdm m /)(−∝ [28]. In configurations close to the P state of a standard pillar, with a thick fixed layer and a thin free layer made of the same material, the spin polarization of the current in the spacer decreases from the fixed layer to the free layer. This corresponds to a given sign of the spin accumulation. But an opposite sign is expected if, in the same configuration, the spin polarization of the current increases from the fixed layer to the free layer. This is what occurs for our Py(8nm)/Cu(10nm)/Co(8nm) pillars in an angular range close to the P configuration, as this can be seen from the spin accumulation calculated in the Section Methods. As shown in Fig 1a, calculations of STT based on two different models reflect this inversion of the spin accumulation by an inversion of the torque on the left part of the figure with respect to the standards case. However, as shown the figure, the inversion is a little less pronounced (less steep slope) in the model of Ref.[15] which goes beyond the simple mixing conductance approximation of Ref.[16]. For a further understanding, we have performed additional macrospin simulations of the current-induced precessions by solving a Landau-Lifschitz Gilbert equation including a spin transfer term using parameters compatible with the actual structure of the measured samples (see Methods, the simulations have been performed by two of the co-authors, O.B. and J.G., independently of those published in Ref.[24]). The simulated current-field diagram at T = 0 K is presented on Fig 4.d with a colour scale corresponding to the change of resistance. At high field (Happ larger than the anisotropy field) and in the current range we have considered, the only excitations are in-plane (IP) precessions occurring above a threshold current Ic1 and associated with a small change of resistance (which also corresponds to a small microwave power). At low field, the IP precessions above Ic1 (black and blue trajectories in Fig. 4c) are followed by out-of- plane (OP) precessions (orange and red trajectories) above a second current threshold Ic2. There is a general good agreement between the main features of the experimental and calculated phase diagrams. In particular, the zone of OP precessions in the top left corner of the diagram of Fig. 4d turns out to be also the zone where we measure the larger DC resistance increase (Fig. 4b) and also detect microwave excitations (Fig.4a). Quantitatively, if one compares the colours in Fig.4 b and c, one can see that the distribution of the resistance change in the diagram is well reproduced and that the experimental ΔR in the OP zone is only somewhat smaller than the calculated one (by about 20% in average). The simulations also give a distribution of microwave power (not shown) concentrated in OP top-left zone as in the experimental plot of Fig.4a but with a power which is about 80 times larger than the experimental one. This could be due to several reasons. First, there are certainly technical factors, like a large impedance mismatch in the detection circuit. Second, for the OP excitations, the limits of a macrospin approach for a quantitative prediction [6,30], have been put forward by several publications. Finally, for the IP precessions we could not detect in the microwave spectra, it can be pointed out that a very small variation of GMR is expected for angles between P and an angle similar to our ϕc in structures with our type of torque angular dependence [29]. This has also led us probably to overestimate the resistance change and the microwave power, since our calculation is based on a standard angular dependence of the GMR as sin2(ϕ/2). A confirmation that the zone of maximum resistance and microwave excitations in the top- left corner (positive currents and low fields) of the diagrams in Fig.4a-b can be identified with the zone of Out-of-Plane precessions in the calculated diagram (Fig. 4c) comes from the current and field dependence of the frequency. As shown in the inset of Fig. 4c, the simulations predict that a decrease of the frequency at increasing current for IP precessions is followed by an increase at the transition to OP precessions. This is in agreement with the frequency blue shift of the microwave excitations detected in the same zone of the phase diagram. The simulations also predict correctly the red shift for the variation with the field. Our simulations therefore support the picture of a non-standard behavior induced by a wavy angular dependence of the STT torque and characterized by out-of-plane precessions excited by positive current at zero and low field. During the submission process, we learned that oscillations of vortex structures in thick Py layers excited by STT have been observed at relatively low field [31,32]. However this leads to oscillations at relatively low frequency, below 1 GHz for layers in our aspect ratio [33], and the oscillations above 3 GHz we observe cannot be explained by this mechanism. Leaning on recent theoretical models of spin transfer torque, our experimental results should help designing more efficient spin transfer oscillators operating in a very small or even without an applied magnetic field. This is a necessary step (among others) on the implementation of these new spintronics-based oscillators in a microwave receiver system for telecommunication applications. Methods : The multilayers are grown by sputtering onto oxidized Si substrates. Two types of stacks were deposited : structure 1 = Au(20 nm)/Cu(5 nm)/Py(8 nm)/Cu(10 nm)/Co(8 nm)/Ta(10 nm)/Cu(80 nm)/Ta(10 nm) and structure 2 = Au(25 nm)/Py(8 nm)/ Cu(8 nm)/Co(4 nm)/IrMn(15 nm)/Ru(15 nm)/Cu(35 nm). Py stands for Permalloy. The results we present, are on a nanopillar with structure 1, but very similar results are observed with structure 2 when the fixed Co layer is pinned with an IrMn layer. This indicates that, even without an IrMn pinning layer, the magnetization of the extended Co layer is similarly fixed. For the nanofabrication process, we first defined (by e-beam lithography, evaporation deposition and lift-off) a Ti(15 nm)/Au (55 nm) elliptical mask on the magnetic multilayer. Then, the magnetic pillar is etched by ion milling with a real-time monitoring by mass spectroscopy down to the Cu/Co interface. The bottom electrode is defined by optical lithography and ion milling. The next step is a planarization of the pillar with a Su-8 resist layer that is also used to electrically isolate the bottom and the top electrode. The Su-8 layer on the top of the pillar is removed by reactive ion etching. Finally, the top Ti/Au electrode is defined by optical lithography, evaporation deposition and lift-off. We measured both the dc resistance and the differential resistance dV/dI using an additional 20µA ac current modulated at 5kHz. For the frequency-domain measurements, we applied a dc current on the sample through a bias-T. The high frequency voltage signal is then amplified (68 dB) and analysed on a commercial spectrum analyzer. The power spectra we show are extracted from the spectrum analyser (we do not correct them from a calibration done for quantities like the frequency-dependent amplifier gain, the attenuation in the transmission lines, and the impedance mismatches). They are only obtained by subtracting a reference spectra measured at Idc = 0 in the same magnetic field conditions. Note that the measured emitted power is therefore only a fraction of the actual emitted power from the pillars. Both transport and frequency measurements have been performed at room temperature and with in-plane magnetic field. The torques of Fig.1a have been calculated by introducing in the models of Refs.[16] and [17] parameters mostly derived from CPP-GMR experimental data [22-23]. For respectively Au, Py, Cu, Co and Ta, these parameters are: bulk resistivity ρ (μΩ.cm) = 2, 15, 2.9, 24, 170; bulk spin asymmetry coefficient β = 0, 0.76, 0, 0.46, 0; spin diffusion length lsf (nm) = 35, 4, 350, 38, 10. For the interface parameters, respectively Au/Cu, Cu/Py, Cu/Co, Co/Ta, Ta/Cu, the parameters are : interfacial resistance rb (fΩ.m²) = 0.17, 0.5, 0.51, 0.5, 0.5; interfacial spin asymmetry coefficient γ : 0, 0.7, 0.77, 0.7, 0; interfacial spin memory loss coefficient δ : 0.13, 0.25, 0.25, 0.25, 0.1. Note that the values of the Co and Ta resistivity have been measured on thin film we had grown in the same conditions. The unknown value of lsf in Ta has been estimated by fitting the calculated and experimental variations of resistance ΔR. We have also used the same parameters in routine programmes [34] developed for the CPP-GMR to calculate the spin accumulation in the spacer layer for our structure and in a standard structure Py (15 nm)/Cu (10 nm)/Py(2 nm), respectively – 2.2 and + 2.0 in arbitrary units and check the change of sign at the origin of the wavy angular dependence. For the simulations of the magnetization dynamics, we have solved a Landau Lifschitz Gilbert equation including a spin transfer term of the form M1x(M1xM2) with the angular dependence shown in Fig.1a. The calculations are performed at zero temperature. The saturation magnetization µ0Ms = 0.87 T has been derived from ferromagnetic resonance experiments performed on a Cu(6nm)/Py(7nm)/Cu(6nm) layer at room temperature. The other parameters are the anisotropy field Han= 0.009 T, the gyromagnetic factor γ0 = 2.21 105(s.A/m)-1, α = 0.011. The area of the pillars is about 1.38 104 nm², as derived by scanning electron microscopy. References: [1] Slonczewski, J.C. Current-driven excitation of magnetic multilayers. J. Magn. Magn. Mater. 159, L1-L7 (1996). [2] Berger, L. Emission of spin waves by a magnetic multilayer traversed by a current. Phys. Rev. B 54, 9353-9358 (1996). [3] Katine, J.A., Albert, F.J., Buhrman, R.A., Myers, E.B., Ralph, D.C., Current-driven magnetization reversal and spin-wave excitations in Co/Cu/Co pillars. Phys. Rev. Lett. 84, 3149- 3152, (2000). [4] Grollier, J., Cros, V., Hamzić, A., George, J.M., Jaffres, H., Fert, A., Faini, G., Ben Youssef, J., Le Gall, H., Magnetization reversal in Co/Cu/Co pillars by spin injection. Appl. Phys. Lett. 78, 3663-3665 (2001). [5] Urazhdin, S., Birge, N.O., Pratt, W.P., Bass, J. Switching current versus magnetoresistance in magnetic multilayer nanopillars. Appl. Phys. Lett. 84, 1516-1518 (2004). [6] Kiselev, S.I. et al., Microwave oscillations of a nanomagnet driven by a spin-polarized current. , Nature 425, 380-383 (2003). [7] Rippard, W.H., Pufall, M.R., Kaka, S., Russek, S.E., Silva, T.J., Direct-current induced dynamics in Co90Fe10/Ni80Fe20. Phys. Rev. Lett. 92, 027201 (2004). [8] Stiles, M.D. and Miltat, J. in Spin Dynamics in Confined Magnetic Structures, III, edited by B. Hillebrands and A. Thiaville (Springer, Berlin, 2006). [9] Waintal, X., Myers, E.B., Brouwer, P.W., Ralph, D.C., Role of spin-dependent interface scattering in generating current-induced torques in magnetic multilayers. Phys. Rev. B 62, 12317- 12327 (2000). [10] Slonczewski, J., Currents and torques in metallic magnetic multilayers. J. Magn. Magn. Mater. 247, 324-338 (2002). [11] Zhang, S., Levy, P.M., Fert, A., Mechanisms of spin-polarized current-driven magnetization switching. Phys. Rev. Lett. 88, 236601 (2002). [12] Stiles, M.D. and Zangwill, A., Anatomy of spin-transfer torque. Phys. Rev. B 66, 014407 (2002). [13] Brataas, A., Bauer, G. E. W., Kelly, P. J., Non-collinear magnetoelectronics, Physics Reports 427, 157-256 (2006) [14] Kovalev, A., Brataas, A., Bauer, G.E.W. Spin transfer in diffusive ferromagnet–normal metal systems with spin-flip scattering. Phys. Rev. B 66, 224424 (2002). [15] Fert, A., Cros, V., George, J.M., Grollier, J., Jaffres., H., Hamzić, A., Vaurès, A., Faini, G., Ben Youssef, J., Le Gall, H., Magnetization reversal by injection and transfer of spin. J. Magn. Magn. Mater. 272, 1706-1711 (2004). [16] Barnaś, J., Fert, A., Gmitra, M., Weymann, I., Dugaev, V.K., From giant magnetoresistance to current-induced switching by spin transfer. Phys. Rev. B 72, 024426 (2005) [17] Barnaś, J., Fert, A., Gmitra, M., Weymann, I., Dugaev, V.K., Macroscopic description of spin transfer torque. Mat. Sc. Engin. B 126, 271-274 (2006). [18] Rezende, S. M., de Aguiar, F. M., Azevedo, A., Magnon excitation by spin-polarized direct currents in magnetic nanostructures. Phys. Rev. B 73, 094402 (2006). [19] Bertotti, G. et al., Magnetization switching and microwave oscillations in nanomagnets driven by spin-polarized currents. Phys. Rev. Lett. 94, 127206 (2005). [20] Slavin, A.N., Kabos, P., Approximate theory of microwave generation in a current-driven magnetic nanocontact magnetized in an arbitrary direction. IEEE Trans. on Magnetics 41, 1264- 1273 (2005). [21] Rippard, W. H., Pufall, M. R., Silva, T. J. Quantitative studies of spin-momentum-transfer- induced excitations in Co/Cu multilayer films using point-contact spectroscopy Appl. Phys. Lett. 82, 1260-1262 (2003). [22] Bass, J., Pratt, W.P., Current-perpendicular (CPP) magnetoresistance in magnetic multilayers. J. Magn. Magn. Mat. 200, 274-289 (1999). [23] Bass, J., Pratt, W.P., Spin-Diffusion Lengths in Metals and Alloys, and Spin-Flipping at Metal/Metal Interfaces: an Experimentalist's Critical Review. cond-mat/0610085 (2006). [24] Gmitra, M. and Barnaś, J., Current-driven destabilization of both collinear configurations in asymmetric spin valves. Phys. Rev. Lett. 96, 207205 (2006). [25] Kiselev, S.I. et al., Spin-transfer excitations of permalloy nanopillars for large applied currents. Phys. Rev. B 72, 064430 (2005). [26] Chen, T. Y., Ji, Y., Chien, C. L., Switching by point-contact spin injection in a continuous film, Appl. Phys. Lett. 84, 380-382 (2004) [27] Acremann, Y. et al. Time-resolved imaging of spin transfer switching : beyond the macrospin concept. Phys. Rev. Lett. 96, 217202 (2006). [28] Valet, T., Fert, A., Theory of the perpendicular magnetoresistance in magnetic multilayers. Phys. Rev. B 48, 7099-7113 (1993). [29] Urazhdin S., Loloee R., Pratt Jr. W. P., Noncollinear spin transport in magnetic multilayers, Phys. Rev. B 71, 100401 (2005). [30] Berkov, D. V., Gorn, N. L., Magnetization precession due to a spin polarized current, Phys. Rev. B 72, 024455 (2005) [31] Pribiag, V. S., Krivorotov, I. N., Fuchs, G. D., Braganca, P. M., Ozatay, O., Sankey J. C., Ralph, D. C., Buhrman R. A., Magnetic vortex oscillator by dc spin-polarized current, cond- mat/0702253 (2007). [32] Pufall, M. R., Rippard, W. H ., Schneider, Russek M., S. E., Low Field, Current-Hysteretic Oscillations in Spin Transfer Nanocontacts, , cond-mat/0702416 (2007). [33] Novosad, V., Fradin, F. Y., Roy, P. E., Buchanan, K. S., Guslienko, K. Yu., Bader, S. D. Magnetic vortex resonance in patterned ferromagnetic dots Phys. Rev. B 72, 024455 (2005) [34] H. Jaffrès, http://www.trt.thalesgroup.com/ump-cnrs-thales Correspondence and requests for materials should be adressed to V. C. The authors declare they have no competing financial interests Acknowledgements The authors thank M. Gmitra for the calculations of Fig 1b based on the model of Ref[16]. We would like also to acknowledge H. Hurdequint for FMR measurements, L. Vila for assistance in fabrication, O. Copie and B. Marcilhac for assistance in transport and frequency measurements and M.R. Pufall for discussions. This work was partly supported by the french National Agency of Research ANR through the PNANO program (MAGICO PNANO-05-044-02) and the EU through the Marie Curie Training network SPINSWITCH (MRTN-CT-2006-035327). J. B. acknowledges support by funds from the Polish Ministry of Science and Higher Education as a research project (2006-2009). *Present address : Instituto de Física, UFRGS, 91501-970 Porto Alegre, RS, Brazil Figure captions Figure 1 Angular dependence of the spin transfer torque for a standard and a ‘wavy’ angular dependence. a, Variation of the spin transfer torque on the free Py layer of a Au(infinite)/Cu(5 nm)/Py(8 nm)/Cu(10 nm)/Co(8 nm)/Ta(10 nm)/Cu(infinite) multilayer as a function of the angle ϕ between the magnetizations of the free Py and fixed Co layers for positive and negative currents. The solid curves are calculated in the model of Barnaś et al [17], the solid straight lines represent the slopes of the torque variation as the angle tends to 0 and π and have been derived from the small angle expression of Fert et al [16]. The parameters used in the calculations and mainly derived from CPP-GMR data are listed in the Section Methods. Inset : typical variation of the spin transfer torque as a function of the angle between the magnetizations of the free and fixed layers for a standard trilayer structure (case of Co/Cu/Co from Ref.[10]). b- c, Sketches showing schematically the direction (blue arrow) of the spin transfer torque on the free layer for configurations close to the P and AP configurations of the free layer (m) and fixed layer (M) magnetizations for a standard (b) and a wavy (c) angular dependence of the torque. Figure 2 Transport measurements on nanopillars with standard or “wavy” angular dependence of the spin transfer torque. a, Differential resistance vs current measured for a nanopillar with a standard structure Py(15 nm)/Cu(10 nm)/Py(4 nm) at “low field” (H = 6 Oe) and “high field” (H = 133 Oe). In the latter case (precession), the applied field is larger than the coercive field equal to H = 133 Oe. Curves are offset for clarity. b-c : Transport data for a Co(8 nm)/Cu(10 nm)/Py(8 nm). nanopillar. b, Resistance vs field at low current (I = 200 µA). c, Differential resistance vs current for different applied fields around zero. These fields correspond to the coloured symbols in b. Figure 3. Microwave power spectra for the Co(8 nm)/Cu(10 nm)/Py(8 nm) nanopillar of Fig.2b-c. a, Microwave power spectra for an applied field close to zero (Happl = 2 Oe) at different currents corresponding to the coloured symbols in the inset. Inset: dV/dI vs I for Happl = 2 Oe. b, Microwave spectra for different applied currents corresponding to the symbols in inset for an effective (applied + dipolar) field of about zero (Happ = 43 Oe). Inset in b : dV/dI vs I for Happ = 43 Oe. c, Microwave spectra for I = 9 mA at different positive applied fields. Spectra are offset for clarity. Figure 4 Experimental and simulated spin-transfer-induced high frequency dynamics for a Co(8nm)/Cu10nm)/Py(8nm) nanopillar. a, Experimental integrated power between 0.1 to 8 GHz in colour scale as a function of field and current. b, Normalized experimental resistance in colour scale as a function of field and current (a reference curve has been subtracted to the experimental R vs I curves to remove the changes in resistance due to Joule heating). c-d : Simulated dynamics of the magnetization in a macrospin approach c, Results of macrospin numerical calculations of LLG equation as a function of current and field at T = 0K. The black line indicates the onset of current-induced precession. Inset in c : Variation of the calculated frequency as a function the current for Happ = 0 Oe. d Magnetization trajectories for Happ=0 (black arrow in c) at several increasing applied currents. π/2 π Standard angular dependence of the spin transfer torque I > 0 P stable AP unstable (H = 0) Wavy angular dependence of the spin transfer torque I < 0 P unstable AP stable (H = 0) I < 0 P stable AP stable (H = 0) I > 0 P unstable AP unstable (H = 0) Fig. 1 Boulle et al. -10 -5 0 5 10 Current (mA) - 19 Oe - 3 Oe 9 Oe 22 Oe -100 -50 0 50 100 12.95 12.97 12.99 Magnetic field (Oe) -6 -4 -2 0 2 4 6 Current (mA) Fig. 2 Boulle et al. 9,5 mA 8,5 mA 7,5 mA 6,5 mA 4 6 8 10 12.25 12.39 I (mA) I (mA) 1 2 3 4 36 Oe 24 Oe - 4 Oe Frequency (GHz) -19 Oe Happ ≈ 0 (2 Oe) a Heff ≈ 0 (Happ=43 Oe) I = 9 mA 11 mA 10 mA Fig. 3 Boulle et al. 9,5 mA 8,5 mA 7,5 mA 6,5 mA 4 6 8 10 12.25 12.39 I (mA) I (mA) 1 2 3 4 36 Oe 24 Oe - 4 Oe Frequency (GHz) -19 Oe Happ ≈ 0 (2 Oe) a Heff ≈ 0 (Happ=43 Oe) I = 9 mA 11 mA 10 mA Fig. 3 Boulle et al. 4 6 8 10 12.25 12.39 I (mA) I (mA) 1 2 3 4 36 Oe 24 Oe - 4 Oe Frequency (GHz) -19 Oe Happ ≈ 0 (2 Oe) a Heff ≈ 0 (Happ=43 Oe) I = 9 mA 11 mA 10 mA Fig. 3 Boulle et al. 0 50 100 150 Magnetic field (Oe) Power (pW)0 2.74 0 50 100 150 Magnetic field (Oe) 9.4 mA 6.9 mA 5.7 mA 3.7 mA 0 50 100 150 4 6 8 Current (mA) =0 Oe Magnetic field (Oe) plane In plane Parallel state ΔRdc(mΩ) ΔRdc(mΩ) Fig 4. Boulle et al. Article File #1 page 2 page 3 page 4 page 5 page 6 page 7 page 8 page 9 page 10 page 11 page 12 page 13 Figure 1 Figure 2 Figure 3 Figure 4

0704.0371

Dark energy interacting with neutrinos and dark matter: a phenomenological theory

Dark energy interacting with neutrinos and dark matter: a phenomenological theory G. M. Kremer∗ Departamento de F́ısica, Universidade Federal do Paraná Caixa Postal 19044, 81531-990 Curitiba, Brazil October 29, 2018 Abstract A model for a flat homogeneous and isotropic Universe composed of dark energy, dark matter, neutrinos, radiation and baryons is analyzed. The fields of dark matter and neutrinos are supposed to interact with the dark energy. The dark energy is considered to obey either the van der Waals or the Chaplygin equations of state. The ratio between the pressure and the energy density of the neutrinos varies with the red-shift simulating massive and non-relativistic neutrinos at small red-shifts and non-massive relativistic neutrinos at high red-shifts. The model can reproduce the expected red-shift behaviors of the deceleration parameter and of the density parameters of each constituent. The recent astronomical measurements of type-IA supernovae [1, 2, 3, 4] and the analysis of the power spectrum of the CMBR [5, 6, 7, 8, 9] provided strong evidence for a present accelerated expansion of the Universe [3, 10, 11, 12, 13, 14]; the nature of the responsible entity, called dark energy, still remains unknown. Furthermore, the measurements of the rotation curves of spiral galaxies [15] as well as other astronomical experiments suggest that the luminous matter represents only a small amount of the massive particles of the Universe, and that the more significant amount is related to dark matter. That offered a new setting for cosmological models with dark energy and dark matter and in these contexts many interesting phenomenological models appear in the literature analyzing the interaction of neutrinos [16, 17, 18] and dark matter [19, 20, 21, 22, 23, 24] with dark energy. With respect to dark energy some exotic equations of state were proposed in the literature and among others we quote the van der Waals [25, 26, 27, 28, 29] and the Chaplygin [30, 31, 32, 33] equations of state. In the present work a very simple cosmological model – for a homogeneous, isotropic and flat Universe composed by dark matter, dark energy, baryons, radiation and neutrinos – is investigated where the dark energy is modeled either by the van der Waals or the Chaplygin equations of state and interact with neutrinos and dark matter. Units have been chosen so that 8πG/3 = c = 1, whereas the metric tensor has signature (+,−,−,−). Let a homogeneous, isotropic and spatially flat Universe be characterized by the Robertson Walker metric ds2 = dt2 − a(t)2δijdxidxj , where a(t) denotes the cosmic scale factor. The sources of the gravitational field are related to a mixture of five constituents described by the fields of dark energy, dark matter, baryons, neutrinos and radiation. The components of the energy-momentum tensor of the sources is written as (T µν) = diag(ρ,−p,−p,−p), (1) where ρ and p denote the total energy density and pressure of the sources, respectively. In terms of the energy densities and pressures of the constituents it follows ρ = ρb + ρdm + ρr + ρν + ρde, p = pb + pdm + pr + pν + pde. (2) ∗[email protected] http://arxiv.org/abs/0704.0371v1 Above the indexes (b, dm, r, ν, de) refer to the baryons, dark matter, radiation, neutrinos and dark energy, respectively. The conservation law of the energy-momentum tensor T µν ;ν = 0 leads to the evolution equation for the total energy density of the sources, namely ρ̇+ 3 (ρ+ p) = 0, (3) where the dot refers to a differentiation with respect to time. The baryons and radiation are considered as non-interacting fields so that the evolution equa- tions for their energy densities read ρ̇b + 3 ρb = 0, ρ̇r + 4 ρr = 0, (4) once the baryons represent a pressureless fluid, i.e., pb = 0, and the radiation pressure is given in terms of its energy density by pr = ρr/3. According to a model proposed by Wetterich [19] the evolution equation for the energy density of a pressureless (pdm = 0) dark matter field which interacts with a scalar field φ is given by ρ̇dm + 3 ρdm = βρdmφ̇. (5) Here the scalar field plays the role of the dark energy and β is a constant which couples the fields of dark matter and dark energy. For interacting neutrinos with dark energy it is supposed that the evolution equation of the energy density is given by (see [17, 18]) ρ̇ν + 3 (ρν + pν) = α(ρν − 3pν)φ̇. (6) The coefficient α is connected with the mass of the neutrinos and for more details one is referred to [17, 18] and to the references therein. Here α will be consider a phenomenological coefficient that couples the dark energy field with the neutrinos. Note that if pν = ρν/3, there is no coupling between the fields of dark energy and neutrinos. Moreover, it is also important to note that the neutrinos in the past must behave as massless particles where the relationship between the pressure and the energy density is pν = ρν/3. Due to the coupling of the neutrinos with the scalar field they become massive and non-relativistic. For these reasons a barotropic equation of state for the neutrinos is proposed where the ratio between the pressure and the energy density wν = pν/ρν , given in terms of the red-shift z, reads K3(1/z) K2(1/z) K3(1/z) K2(1/z) . (7) Above K2(1/z) and K3(1/z) are modified Bessel functions of second kind. For small values of z, wν tends to the non-relativistic limit equal to 2/3, whereas for large values of z, wν tends to the relativistic limit equal to 1/3. It is noteworthy that for red-shifts z ≈ 10 this ratio reaches the value wν ≈ 1/3 and the coupling between the neutrinos and the dark energy is negligible. The expression given in (7) is motivated by the equation of the specific heat of a relativistic gas (see e.g. [34]). The evolution equation for the energy density of the dark energy field is obtained from equations (2) through (6), yielding ρ̇de + 3 (ρde + pde) = −αφ̇(ρν − 3pν)− βρdmφ̇. (8) The energy density and pressure of the dark energy are connected with the scalar field by φ̇ =√ ρde + pde. Since the purpose of this work is to develop a phenomenological theory, it is assumed 0 2 4 6 8 10 vw Ωb Figure 1: Density parameters as functions of red-shift: van der Waals fluid (solid lines) and Chap- lygin fluid (dashed lines). that the dark energy field behaves either as a van der Waals or a Chaplygin fluid with an equation of state given by [28, 29, 30, 31, 32, 33] pvw = 8wvwρvw 3− ρvw − 3ρ2vw, pch = − , (9) where wvw and A are positive free parameters in the van der Waals and Chaplygin equations of state, respectively. For the determination of the time evolution of the energy densities one has to close the system of differential equations by introducing the Friedmann equation = ρ. (10) From now on the red-shift will be used as a variable instead of time thanks to the following relationships ρ(1 + z) . (11) Equations (4) can be easily integrated leading to the well-known dependence of the energy densities of the baryons and radiation with the red-shift ρr(z) = ρr(0)(1 + z) 4, ρb(z) = ρb(0)(1 + z) 3, (12) whereas equations (5), (6) and (8) become a system of coupled differential equations for the energy densities ρdm, ρν and ρde, namely, (1 + z)ρ′dm − 3ρdm (ρde + pde)/ρ = −βρdm, (13) (1 + z)ρ′ν − 3(ρν + pν) (ρde + pde)/ρ = −α(ρν − 3pν), (14) (1 + z)ρ′de − 3(ρde + pde) (ρde + pde)/ρ = βρdm + α(ρν − 3pν). (15) In the above equations the prime refers to a differentiation with respect to the red-shift. In order to solve the coupled system of differential equations (13) – (15) one has to specify initial values for the energy densities at z = 0. The following initial values for the density parameters Ωi(z) = ρi(z)/ρ(z) taken from the literature (see [35] for a review) were chosen: Ωde(0) = 0.72, 0 500 1000 1500 2000 2500 3000 Figure 2: Density parameters as functions of red-shift for a van der Waals fluid as dark matter. Ωdm(0) = 0.229916, Ωb(0) = 5 × 10−2, Ωr(0) = 5 × 10−5, Ων(0) = 3.4 × 10−5. Moreover, one has to specify values for the coupling parameters α and β and for the parameters wvw and A which appear in the van der Waals and Chaplygin equations of state (9). One way to fix the two last parameters is through the use of the value of the deceleration parameter q = 1/2+ 3p/2ρ at z = 0. Indeed, by considering q(0) = −0.55 it follows wvw = 0.33851 and A = 0.50403. For the coupling parameters two sets of values were chosen, namely, (a) α = 5 × 10−5 and β = −5 × 10−5 for the van der Waals equation of state and (b) α = 10−1 and β = −10−2 for the Chaplygin equation of state. Its is also important to note that by increasing the value of the coupling parameter α (and/ or β) the transfer of energy between the dark energy and neutrinos (and/or dark matter) becomes more efficient. In Fig. 1 the density parameters are plotted as functions of the red-shift for values in the range 0 ≤ z ≤ 10. The straight lines refer to the case where the van der Waals equation of state is used to describe the dark energy field whereas the dashed lines correspond to the Chaplygin equation of state. The two density parameters that represent the dark energy field are denoted by Ωvw and Ωch. One can infer from this figure that the dark energy density parameter tends to zero for high red-shifts when the van der Waals equation of state is used, whereas it tends to a constant value for the Chaplygin equation of state. While for high red-shifts the van der Waals equation of state simulates a cosmological constant with pvw = −ρvw, the pressure of the Chaplygin fluid vanishes indicating that it becomes another component of the dark matter field (see also the behavior of the pressures indicated in Fig. 4). It is also important to note that the density parameters of the baryons and of the dark matter increase more with the red-shift for the van der Waals equation of state, since there is an accentuated decrease in the density parameter of the dark energy for this case. Note that the density parameters of the radiation and neutrinos are very small in this range of the red-shift and are not represented in this figure. The behavior of the density parameters for the cases of the van der Waals and Chaplygin equations of state are shown in Figs. 2 and 3, respectively, for red-shifts in the range 0 ≤ z ≤ 3000. One can conclude from these figures, as expected, that the density parameters of the neutrinos and radiation increase with the red-shift whereas those of the baryons and dark matter decrease. Furthermore, the equality between the “matter” and “radiation” fields occurs when z ≈ 3000 for the case where the dark matter field is modeled as a van der Waals fluid and z ≈ 4200 for the case of a Chaplygin fluid. This can be easily understood, since in the latter case the dark energy becomes dark matter for high red-shifts contributing for the density parameter of the “matter” field. In Fig. 4 are plotted the deceleration parameter and the ratio between the pressure and the energy density for both cases, the large frame corresponding to the van de Waals fluid whereas the small frame to the Chaplygin fluid. For both cases the deceleration parameter at z = 0 is equal to q(0) = −0.55, since this value was fixed in order to find the parameters wvw and A in the equations of state (9). The transition from the decelerated to the accelerated phase of the Universe occurs at zT = 0.73 and zT = 0.53 for the van der Waals and Chaplygin equations of state, respectively. It 0 500 1000 1500 2000 2500 3000 + Ων Figure 3: Density parameters as functions of red-shift for a Chaplygin fluid as dark matter. -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.751.5 2 -0.5 0 0.5 1 1.51.5 2 Figure 4: Deceleration parameter and ratio between the pressure and the energy density as functions of red-shift: large frame (van der Waals), small frame (Chaplygin). 0 2 4 6 8 10 with interactions without interactions Figure 5: Density parameters as functions of red-shift for a Chaplygin fluid with and without interactions. is interesting to note that while the Chaplygin equation of state simulates a cosmological constant with pch = −ρch for negative red-shifts which implies an accelerated phase of the Universe in the future, the van der Waals equation of state leads to a positive pressure and brings the Universe to another decelerated phase in the future. It is noteworthy to call attention that for positive values of the red-shift, the solution of the coupled differential equations (13) through (15) predicts that the van der Waals fluid behaves close to a cosmological constant with pvw ≈ −ρvw. This behavior does not lead to a new transition from a decelerated to an accelerated phase in the very early Universe, since the energy density of the radiation field increases so that the radiation pressure becomes larger than that of the van der Waals fluid. For high red-shifts the Universe first becomes dominated by the baryon and dark matter fields and for higher red-shifts by the radiation field. This model does not attempt to model the inflationary period, where the inflaton field dominates a short rapid evolution of the Universe. As final remarks we call attention to the fact that one expects that the coupling between dark energy, dark matter and neutrinos should be weak so that the parameters α and β are restricted to small values. The difference between the parameters adopted for the van der Waals and Chaplygin equations of state is due to stability conditions of the non-linear coupled system of differential equations (13) – (15), the van der Waals equation of state being more unstable for large values of these parameters than the Chaplygin equation of state. In Fig. 5 we have plotted the density parameters as functions of the red-shift for the case where a Chaplygin equation of state is used as dark energy. One can infer from this figure that the decay of the dark energy density parameter and the increase of the dark matter density parameter with the red-shift are more pronounced when there exists a coupling between the fields. The density parameter of the baryons remains unchanged since the baryons are uncoupled. As final comment it is important to note that even without couplings between the fields of dark energy, dark matter and neutrinos, this phenomenological model – with the equations of state of van der Waals and Chaplyging as dark energy – can describe satisfactorily the evolution of a Universe whose constituents are dark energy, dark matter, baryons, neutrinos and radiation. References [1] S. Perlmutter et al. Astrophys. J. 517, 565 (1999). [2] A. G. Riess et al. Astrophys. J. 560, 49 (2001). [3] M. S. Turner and A. G. Riess, Astrophys. J. 569, 18 (2002). [4] J. Tonry et al., Astrophys. J. 594, 1 (2003). [5] C. L. Bennett et al. Astrophys. J. Suppl. 148, 1 (2003). [6] H. V. Peiris et al., Astrophys. J. Suppl. 148, 213 (2003). [7] C. Netterfield et al. Astrophys. J. 571, 604 (2002). [8] N. Halverson et al. Astrophys. J. 568, 38 (2002). [9] D. N. Spergel et al. Astrophys. J. Suppl. 148, 175 (2003). [10] S. M. Carroll, astro-ph/0310342. [11] B. Schmidt et al., Astrophys. J. 507, 46 (1998). [12] G. Efstathiou, S. L. Bridle, A. N. Lasenby, M. P. Hobson and R. S. Ellis, astro-ph/9812226. [13] A. G. Riess et al., Astrophys. J. 516, 1009 (1998). [14] D. Huterer and M. S. Turner, Phys. Rev. D 60, 081301 (1999). [15] M. Persic, P. Salucci and F. Stel Mon. Not. Roy. Astron. Soc. 281, 27 (1996). [16] X.-J. Bi, B. Feng, H. Li and X. Zhang, Phys. Rev. D 72 123523 (2005). [17] A. W. Brookfield, C. van de Bruck, D. F. Mota and D. Tocchini-Valentini, Phys. Rev. Lett. 96, 061301 (2006). [18] A. W. Brookfield, C. van de Bruck, D. F. Mota and D. Tocchini-Valentini, Phys. Rev. D 73 083515 (2006). [19] C. Wetterich, Nucl. Phys. B 302, 645 (1988). [20] C. Wetterich, Astron. Astrophys. 301, 321 (1995). [21] L. Amendola, Phys. Rev. D 62, 043511 (2000). [22] J. B. Binder and G. M. Kremer, Braz. J. Phys. 35, 1038 (2005). [23] G. W. Anderson and S. M. Carroll, astro-ph/9711288 (1997) [24] J. B. Binder and G. M. Kremer, Gen. Relativ. Gravit. 38, 857 (2006). [25] S. Capozziello, S. De Martino and M. Falanga, Phys. Lett. A 299, 494 (2002). [26] S. Capozziello, V. F. Cardone, S. Carloni, S. De Martino, M. Falanga, A. Troisi and M. Bruni, J. Cosmol. Astropart. Phys. 04 (2005) 005. [27] V. F. Cardone, C. Tortora, A. Troisi and S. Capozziello, Phys. Rev. D 73 043508 (2006). [28] G. M. Kremer, Phys. Rev. D 68, 123507 (2003). [29] G. M. Kremer, Gen. Relativ. Gravit. 36, 1423 (2004). [30] A. Yu. Kamenshchik, U. Moschella and V. Pasquier, Phys. Lett. B 511, 265 (2001). [31] J. C. Fabris, S. V. B. Gonçalves and P. E. de Souza, Gen. Relativ. Gravit. 34, 53 (2002). [32] M. C. Bento, O. Bertolami and A. A. Sen A. A. Phys. Rev. D66, 043507 (2002). [33] G. M. Kremer, Gen. Relativ. Gravit. 35, 1459 (2003). [34] C. Cercignani and G. M. Kremer, The Relativistic Boltzmann Equation: Theory and Applica- tions (Birkhäuser, Basel, 2002). [35] M. Fukugita and P. J. E. Peebles, Astrophys. J. 616, 643 (2004). http://arxiv.org/abs/astro-ph/0310342 http://arxiv.org/abs/astro-ph/9812226 http://arxiv.org/abs/astro-ph/9711288

0704.0372

Levy-Lieb constrained-search formulation as a minimization of the correlation functional

Levy-Lieb constrained-search formulation as a minimization of the correlation functional. Luigi Delle Site∗ Max-Planck-Institute for Polymer Research Ackermannweg 10, D 55021 Mainz Germany. Abstract The constrained-search formulation of Levy and Lieb, which formally defines the exact Hohenberg-Kohn functional for any N -representable electron density, is here shown to be equivalent to the minimization of the correlation functional with respect to the N − 1 conditional probability density, where N is number of electrons of the system. The consequences and implications of such a result are here analyzed and discussed via a practical example. PACS numbers: 03.65. w, 71.10. w, 71.15.Mb ∗Electronic address: [email protected] http://arxiv.org/abs/0704.0372v1 mailto:[email protected] I. INTRODUCTION The Hohenberg-Kohn (HK) theorem [1] has opened new perspectives to the calculations of electronic-based properties of condensed matter [2], and, an aspect often disregarded, has given profound new insights into the general understanding of quantum mechanics. In fact the 3N -dimensional Schrödinger problem for the ground state of an electronic system: HNψ(r1, ...r2) = E0ψ(r1, ...r2);HN = i=1,N ∇2i ) + i=1,N v(ri) + where v(ri) is the external potential, is the electron-electron Coulomb term, E0 is the energy of the ground state and ψ(r1, ...rN) is the 3N -dimensional antisymmet- ric ground state wavefunction [3], is transformed into a ”manageable” variational prob- lem in three dimensions where the central role is played by the electron density: ρ(r) = ψ∗(r, r2, .....rN)ψ(r, r2, .....rN)dr2....drN , where ΩN−1 is the N − 1 spatial domain. In explicit terms the variational problems is written as: E0 =MinρE[ρ] (2) where Ω ρ(r)dr = N (Ω being the spatial domain of definition) and E[ρ] = T [ρ] + Vee[ρ] + Vext[ρ] is the energy functional composed respectively by the kinetic, electron-electron po- tential and the external potential functional. However in its original formulation the HK theorem and the related variational problem have got a restricted field of applicability; it is valid only if the electron density ρ(r) is v-representable, that is if ρ(r) is the density cor- responding to an antisymmetric wavefunction of the ground-state of an Hamiltonian of the form of Eq.1. It follows that the correct formulation of the variational problem becomes: E0 =MinρEv[ρ] (3) where v refers to the v-representability of ρ(r). As discussed in Ref.[2], there are no general conditions for a density to be v-representable and this makes the use of the HK theorem and its associated variational principle not practical. A generalization of the HK theorem which does not require ρ(r) to be v-representable was found, in parallel, by M.Levy [4] and E.Lieb [5] and it is usually known as the Levy constrained-search formulation or Levy-Lieb constrained-search formulation [6]; in this paper we adopt the latter terminology. We also notice that recently P.Ayers [7] has further clarified this concept and developed an axiomatic treatment of the Hohenberg-Kohn functional. In the following we briefly describe the crucial aspects of the abovementioned approach which are relevant for the current work. The starting point of the theory is the distinction between the ground state wavefunction, ψ, and a wavefunction ψλ that also integrates to the ground state electron density ρ(r). Since ψ is the ground state wavefunction, we have: 〈ψλ |HN |ψλ〉 ≥ 〈ψ |HN |ψ〉 = E0. (4) Taking into account that Vext[ρ] is a functional of ρ only, Eq.4 can be written as: 〈ψλ |T + Vee|ψλ〉 ≥ 〈ψ |T + Vee|ψ〉 (5) where T and Vee are respectively the kinetic and Coulomb electron-electron operator as defined in Eq.1. The meaning of Eq.5 is that ψ is the wavefunction that minimize the kinetic plus the electron-electron repulsion energy and integrates to ρ. It follows that the initial variational problem of Eq.2 can be transformed in a double hierarchical minimization procedure which formally allows for searching among all the ρ’s which are N -representable, i.e. it can be obtained from some antisymmetric wavefunction; this is a condition which is much weaker and more controllable than the v-representability. In explicit terms such a formulation is written as: E0 =Minρ Minψλ→ρ 〈ψλ |T + Vee|ψλ〉+ v(r)ρ(r)dr . (6) The inner minimization is restricted to all wave functions ψλ leading to ρ(r), while the outer minimization searches over all the ρ’s which integrate to N . The original HK formulation can then be seen as a part of this new one once its universal functional, F [ρ] = 〈ψ |T + Vee|ψ〉 is written as: F [ρ] =Minψ→ρ 〈ψ |T + Vee|ψ〉 . (7) The purpose of this work is to show that F [ρ] can be determined solely by a minimiza- tion with respect to the N − 1 conditional probability density of the electron correlation functional. This latter will be shown to be composed by the non local Fisher information functional [8] and the electron-electron two-particle Coulomb term. The advantage of this representation is manifold; it further clarifies the connection of electronic properties to the Fisher theory and shows that the knowledge of such a functional is the crucial ingredi- ent in density functional based approaches; it also identifies the Weizsacker kinetic term, ∫ |∇ρ(r)|2 dr, as necessary component of the universal functional F [ρ] and, in practical terms, offers an objective criterion of evaluation of ”approximate” exchange and correlation func- tional, i.e. among two functionals, the physically better founded is the ”smaller” one. In order to show the practical aspects of our idea we illustrate a potential application. II. THE NEW REPRESENTATION Before writing the functional in the conditional probability density formalism, we need to define such a quantity. Let us consider a generic fermionic wavefunction ψ(r1, ....rN), for simplicity we consider a real wavefunction, but the extension to a complex one can be also done [9]; we do not consider the spin dependence explicitly, however this will not influence the main conclusions. Then the N -particle probability density is [10, 11]: Nψ∗(r1, ....rN )ψ(r1, ....rN) = Θ 2(r1, ...., rN) (8) and this can formally decomposed as [10, 11]: Θ2(r1, ...., rN) = ρ(r1)f(r2, ......., rN/r1) (9) where ρ(r1) is the one particle probability density (normalized to N) and f(r2, ......., rN/r1) is the N − 1 electron conditional (w.r.t. r1) probability density, i.e. the probability density of finding an N −1 electron configuration, C(r2, ......., rN), for a given fixed value of r1. The function f satisfies the following properties: f(r2, ........, rN/r1)dr2.......drN = 1∀r1 (ii) f(r1, ..ri...rj−1, rj+1..., rN/rj) = 0; for i = j; ∀i, j = 1, N (iii) f(r1, .....ri.., rj...rk−1, rk+1.., rN/rk) = 0; for i = j; ∀i, j 6= k (10) The property (iii) of Eq.10 assures us that f reflects the fermionic character of an electronic wavefunction. In fact it says that if any two particles are in the same ’state’ ”r” the probability of that specific global configuration is zero. In principle, together with condition (ii), this is a way to mimic the antisymmetric character of the fermionic wavefunction since for fermions |ψ(r1, ...ri, ...rj, ...rN)| 2 = 0; for i = j, ∀i, j.It must be noticed that condition (iii) is complementary to (ii). With this formalism the term 〈ψ |T + Vee|ψ〉 can be written as (see Refs.[9, 10, 12]): 〈ψ |T + Vee|ψ〉 = |∇ρ(r)|2 , ...., rN/r)| , ....., rN/r) ....drN (N − 1) , ....., rN/r) |r− r ....drN dr(11) where we have identified r1 with r and made use of the property of electron indistinguisha- bility, thus r could be identified with any of the ri (and the same for r identified here with r2) without changing the results; a further consequence is that the Coulomb expression (last term on the r.h.s.) is written as the sum of N − 1 identical terms for the generic r and r particles. Using Eq.11 the Levy-Lieb constrained-search formulation can then be written as: E0 =Minρ Minf (Γ[f, ρ]) + |∇ρ(r)|2 v(r)ρ(r)dr where Γ[f, ρ] = , ...., rN/r)| , ....., rN/r) ....drN (N − 1) , ....., rN/r) |r− r ....drN dr. (13) In this way we have transferred the problem from from ψ to f which means that the focus is now on Γ[f, ρ], i.e., as discussed in Ref.[12], the correlation functional. III. A PRACTICAL EXAMPLE: THE PARAMETRIC EXPONENTIAL FORM In our previous work [12], we have proposed an approximation for f based on a two- particle factorization: f = ΠNi=2hi(EH(r, ri)) = Π (N−1)E(r)e−EH (r,ri) (14) where e−E(r) = e−EH (r,ri)dri. (15) here EH(r, ri) = ρ(r)ρ(ri) |r−ri| , N is the number of particle, and ω the volume corresponding to one particle. Such an approximation, due to its simplicity, allows us to write an analytic expression of the Fisher functional which can be used in a straightforward way in numerical calculations. However it does not satisfy the condition (iii) of Eq.10, and, for this reason, in order to use it into the Levy-Lieb constrained-search scheme it must be extended. The expression we propose here is the following: f(r2, ...rN/r) = Πn=2,Ne E(r)−γEH (r,rn) × Πi>j 6=1e −βEH(ri,rj) (16) with: e−E(r) = Πn=2,NΠi>j 6=1e −γEH (r,rn)−βEH (ri,rj)dr2.....drN (17) Here γ and β are two free parameters. As it can be easily verified this expression of f satisfies all the requirements of Eq.10. The meaning of f as expressed in Eq.16 is that the probability of finding a certain configuration for the N − 1 particles, having fixed particle r1 = r, depends not only on the fixed particle and its interaction with the N − 1 other particles as before, but also on the mutual arrangements of the N − 1 particles (it has also to be kept in mind that using the particle indistinguishability the formalism can be applied to any ri as a fixed particle). The parameters γ and β express how important the N − 1 mutual interactions are with respect to the interactions with r. Being now f a biparametric function, one can use the Levy-Lieb constrained search in our formulation and find the optimal values for γ and β. This practical example shows two different aspects of our formulation; basically we have shown that indeed it is possible to build a function f and actually it can be chosen in a way that its optimal expression can be determined via the constrained-search formulation. It must be noticed that this form of f is still rather simple since the spins are not explicitly considered when constructing the function and thus one cannot distinguish between the exchange and the correlation part of the electron-electron interaction as it is done in standard Density Functional Theory; as a consequence one should expect only an overall average description of these two terms which are here incorporated into the global correlation. However the construction of a more complete expression of f , which takes care of the effects of the spins, is the subject of current investigation. This emphasizes once more the merit of the general procedure shown here, that is different expressions of f , with different degrees of complexity, can be proposed and their relative validity checked by the constrained-search procedure. IV. DISCUSSION AND CONCLUSIONS As anticipated in the introduction, the consequences of Eqs.12,13 are rather interesting. The Levy-Lieb variational principle can be reformulated as: The universal functional F [ρ] is the one with the minimum correlation functional with respect to the electron conditional probability density. This new interpretation of the HK universal functional tells us that only an accurate description of the correlation effects, considering the Weizsacker term as a necessary term, leads to an accurate description of the whole energy functional; such a criterion is necessary and sufficient. It is obvious that it is necessary; without knowing Γ[f, ρ], F [ρ] cannot be known; it is sufficient because once Γ[f, ρ] or better f(r2, ...rN/r1) is (in principle) known than the whole energy functional is known explicitely. Clearly, the ”true” f(r2, ...rN/r1) is very difficult if not impossible to obtain [13], however it can be sufficiently well described on the basis of mathematical requirements and physical intuition as done for example in Ref.[12] and as shown in the previous section. From this point of view, Eqs.12,13, can be seen as an objective criterion to design, on the basis of physical intuition and fundamental mathematical requirements, valid energy functionals. In fact, as done in Ref.[12] and in the previous section, one can construct well-founded expressions for f keeping in mind the physical meaning of the electron correlation effects and the necessary related mathematical prescriptions of Eq.10. Next one can make use of Eqs.12,13 and choose among different functional forms of f , the one giving the ”smaller” Γ. It must be noticed that in this work we do not claim that finding a functional form of f is easier or more rigorous than to find an exchange-correlation functional in standard Density Functional Theory; it represents an alternative or complementary approach to the latter. However, the approach based on f allows one to express in a more direct way, via the choice of different forms of f , the physical principles related to the electron correlation effects and to have an explicit form of the correlation term for the kinetic functional which is of great advantage for kinetic functional based methods (see e.g. Refs.[14, 15]).An important aspect linked to the statement above is that the term: 1 |∇rf(r ,....,rN/r)| ,.....,rN/r) ....drNdr, is the well known non local Fisher information functional about which a vast literature is available (see e.g. [10, 16, 17] and references therein); this term is very often linked to the electron correlation functional and electronic properties(see Refs.[18, 19]), our work further clarifies this connection, suggesting that the results known from the analysis of the Fisher functional could be employed in this context. In conclusion we have shown an alternative view of the Levy-Lieb constrained search approach and provided an example which clarifies the practical advantage of our idea; in this sense the present work it is not merely a marginal new formal contribution to a rather well-known method, but gives a new powerful insight into the field of applicability for realistic systems. Acknowledgments I would like to thank Luca Ghiringhelli for a critical reading the manuscript. [1] P.Hohenberg and W.Kohn, Phys.Rev. 136, B864 (1964). [2] W.Yang and R.G.Parr, Density Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989. [3] We use atomic units where h̄, e and m are equal to one. [4] M.Levy, Proc.Natl.Acad.Sci.U.S.A. 76, 6062 (1979); see also: M.Levy, Phys.Rev.A 26, 1200 (1982). [5] E.Lieb, Int. Jour. Quant. Chem. 24, 243-277 (1983). An expanded version appears in Density Functional Methods in Physics, R. Dreizler and J. da Providencia eds., Plenum Nato ASI Series 123, 31-80 (1985). [6] M.H.Cohen and A.Wasserman, Phys.Rev.A 71, 032515 (2005) [7] P.W.Ayers, Phys.Rev.A 73, 012513 (2006); see also: P.W.Ayers, S.Golden and M.Levy, J.Chem.Phys. 124, 054101 (2006). [8] R.A.Fisher, Proc.Cambridge Philos.Soc. 22, 700 (1925). [9] L.Delle Site,J.Phys.A 38, 7893 (2005). [10] S.B.Sears, R.G.Parr and U.Dinur, Isr.J.Chem. 19, 165 (1980). [11] P.W.Ayers, J.Math.Phys. 46, 062107 (2005). [12] L.Delle Site,J.Phys.A 39, 3047 (2006). [13] M.Kohout, Int.J.Quant.Chem. 87, 12 (2002). [14] Y.A.Wang and E.A.Carter Orbital-free kinetic-energy density functional theory in Theoretical Methods in Condensed Phase Chemistry ed S D Schwartz (Dordrecht: Kluwer) chapter 5, pp 117 (2000). [15] N.Choly and E.Kaxiras, Solid State Comm. 121, 281 (2002). [16] A.Nagy, J.Chem.Phys. 119, 9401 (2003). [17] E.Romera and J.S.Dehesa, J.Chem.Phys. 120, 8906 (2004). [18] R. F. Nalewajski, Advances in Quantum Chemistry 43, 119, 2003. [19] R. F. Nalewajski, Chem. Phys. Lett. 386, 265 (2004). Introduction The new representation A practical Example: The parametric exponential form of f Discussion and Conclusions References

0704.0373

Reality of linear and angular momentum expectation values in bound states

Reality of linear and angular momentum expectation values in bound states Utpal Roy,∗ Suranjana Ghosh,† and T. Shreecharan‡ Physical Research Laboratory, Ahmedabad 380009, India Kaushik Bhattacharya§ Insituto de Ciencias Nucleares, Universidad Nacional Autónoma de Mexico, Circuito Exterior, C.U., A. Postal 70-543, C. Postal 04510, Mexico DF, Mexico ABSTRACT In quantummechanics textbooks the momentum operator is defined in the Cartesian coordinates and rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently one always generalizes the Cartesian prescription to other coordinates and falls in a trap. In this work we introduce the difficulties one faces when the question of the momentum operator in spherical polar coordinate comes. We have tried to point out most of the elementary quantum mechanical results, related to the momentum operator, which has coordinate dependence. We explicitly calculate the momentum expectation values in various bound states and show that the expectation value really turns out to be zero, a consequence of the fact that the momentum expectation value is real. We comment briefly on the status of the angular variables in quantum mechanics and the problems related in interpreting them as dynamical variables. At the end, we calculate the Heisenberg’s equation of motion for the radial component of the momentum for the Hydrogen atom. I. INTRODUCTION Quantum mechanics is a treasure house of peculiar and interesting things. Elementary textbooks of quantum mechanics [1, 2, 3] generally start with the postulates which are required to define the nature of the dynamical variables in the theory and their commutation relations. The choice of the dynamical variables is not that clear, as the coordinates in Cartesian system are all elevated to the status of operators where as time remains a parameter. More over in spherical polar coordinates only the radial component can be represented as an operator while the angles still remain as a problem. The difficulty of giving different status to the spatial coordinates and time is bypassed in quantum field theories where all the coordinates and time become parameters of the theory. But the problem with angles still remain a puzzle which requires to be understood in future. When we start to learn quantum mechanics, most of the time we begin with elementary calculations relating to the particle in a one dimensional infinite well, particle in a finite potential well, linear harmonic oscillator and so on. The main aim of these calculations is to solve the Schrödinger equation in the specific cases and find out the bound state energies and the energy eigenfunctions in coordinate space representation. While solving these problems we overlook the subtleties of other quantum mechanical objects as the definition of the momentum operator in various coordinates, the reality of its expectation value, etc.. In the last one or two decades there has been a number of studies regarding the self-adjointness of various operators [4]. The aim of these studies has been to analyze the self-adjointness of various operators like momentum, Hamiltonian etc. and find out whether these operators are really self-adjoint in some interval of space where the theory is defined, if not then can there be any mathematical method by which we can make these operators to be self-adjoint in the specified intervals ? In the present work we deal with a much elementary concept in quantum mechanics related to the reality of the expectation values of the momentum operator, be it linear or angular. We do not analyze the self-adjointness of the operators which requires different mathematical techniques. To test the self-adjointness of an operator we have to see whether the operator is symmetric in a specific spatial interval and the functional domain of the operator and its adjoint are the same. In the article we always keep in touch with the recent findings from modern research about self-adjoint extensions but loosely we assume that the operators which we are dealing with are Hermitian. If something is in contrary we point it out in the main text. A considerable portion of our article deals with the analysis ∗Electronic address: [email protected] †Electronic address: [email protected] ‡Electronic address: [email protected] §Electronic address: [email protected] http://arxiv.org/abs/0704.0373v1 mailto:[email protected] mailto:[email protected] mailto:[email protected] mailto:[email protected] of the fact that the expectation value of the momentum operator in various bound states are zero, a result which most of the textbooks only quote but never show. In the simpler cases the result can be shown by one or two lines of calculation, but in nontrivial potentials as the Morse potential, the Coulomb potential the result is established by using various properties of the special functions such as the associated Legendre and the associated Laguerre. The presentation of various materials in our article is done in the following way. Next section deals with the definition of the momentum operator and its properties. Section III deals with the intricacies of the definition of the momentum operator in spherical polar coordinates and the problems we face when we try to mechanically implement the quantization condition, which is invariably always written in Cartesian coordinates in most of the textbooks on quantum mechanics. In section IV we explicitly calculate the momentum expectation values in various potentials and show that in bound states we always get the expectation value of the linear momentum to be zero. Section V gives a brief discussion on the Ehrenfest theorem when we are using it to find out the time derivative of the expectation value of the radial component of momentum in the case of the Hydrogen atom. We end with the concluding section which summarizes the findings in our article. Before going into the main discussion we would like to mention about the convention. We have deliberately put a hat over various symbols to show that they are operators in quantum mechanics. Some times this convention becomes tricky when we are dealing with angular variables as there the status of these variables is in question. The other symbols have their conventional meaning. As we are always using the coordinate representation sometimes we may omit the hat over the position operator as in this representation the position operator and its eigenvalues can be trivially interchanged. II. DEFINITION OF THE MOMENTUM OPERATOR AND THE REALITY OF ITS EXPECTATION VALUE From the Poisson bracket formalism of classical mechanics we can infer: [x̂i, p̂j] = ih̄ δi j , (2.1) where δi j = 1 when i = j and zero for all other cases, and i, j = 1, 2, 3. In the above equation x̂i is the position operator and p̂j is the linear momentum operator in Cartesian coordinates. From the above equation we can also find the form of the momentum operator in position representation, which is: p̂i = −ih̄ . (2.2) It is interesting to note that the above expression of the momentum operator also gives us the form of the generator of translations. This is because of the property: [p̂x, F (x̂)] = −ih̄ dF (x̂) , (2.3) where F (x̂) is an arbitrary well defined function of x̂. The above equation ensures that the momentum operator generates translations along the x direction. Particularly in one-dimension the expression of the momentum operator becomes p̂x = −ih̄ ∂∂x . We know that the expectation value of the momentum operator must be real. If we focus on one-dimensional systems to start with, where the system is specified by the wave-function ψ(x, t), the expectation value of any operator Ô is defined by: 〈Ô〉 ≡ ψ∗(x, t) Ô ψ(x, t) dx , (2.4) where ψ∗ signifies complex conjugation of ψ and the extent of the system is taken as −∞ < x <∞. From the above equation we can write, 〈Ô〉∗ = ψ(x, t) Ô∗ ψ(x, t)∗ dx . (2.5) If 〈Ô〉 = 〈Ô〉∗ then the condition of the reality of the expectation value becomes: ψ∗(x, t) Ô ψ(x, t) dx = ψ(x, t) Ô∗ ψ(x, t)∗ dx . (2.6) For a three-dimensional system the above condition becomes, ψ∗(x, t) Ô ψ(x, t) d3x = ψ(x, t) Ô∗ ψ(x, t)∗ d3x . (2.7) Now if we take the specific case of the momentum operator in one-dimension we can explicitly show that its expectation value is real if the extent of the system is infinite and the wave-function vanishes at infinity. The proof is as follows. The expectation value of the momentum operator is: ψ∗(x, t) p̂x ψ(x, t) dx = −ih̄ ψ∗(x, t) ∂ψ(x, t) = −ih̄ ψ∗(x, t)ψ(x, t)|∞−∞ − ψ(x, t) ∂ψ(x, t)∗ , (2.8) If the wave functions vanish at infinity then the first term on the second line on the right-hand side of the above equation drops and we have, ψ∗(x, t) p̂x ψ(x, t) dx = −ih̄ ψ∗(x, t) ∂ψ(x, t) = ih̄ ψ(x, t) ∂ψ(x, t)∗ ψ(x, t) p̂∗x ψ(x, t) ∗ dx . (2.9) A similar proof holds for the three-dimensional case where it is assumed that the wave-function vanishes at the boundary surface at infinity. III. THE EXPECTATION VALUE OF THE MOMENTUM OPERATOR IN CARTESIAN AND SPHERICAL POLAR COORDINATES In non-relativistic version of quantum mechanics we know that if we have a particle of mass m which is present in a time-independent potential we can separate the Schrödinger equation: ∂ψ(x, t) ∇2 + V (x) ψ(x, t) , (3.1) into two equations, one is the time-dependent one which gives the trivial solution e− h̄ where E is the total energy of the particle, and the other equation is the time-independent Schrödinger equation: ∇2u(x) + 2m (E − V (x))u(x) = 0 , (3.2) where u(x) is the solution of the time-independent Schrödinger equation and the complete solution of the Eq. (3.1) ψ(x, t) = u(x)e− h̄ . (3.3) In the case of the free-particle, where V (x) = 0, we have u(x, t) = eik·x where E = k and k = |k|. The free- particle solution is an eigenfunction of the momentum operator with eigen value h̄k. Although if we try to find out the expectation value of the momentum operator as is done in the last section we will be in trouble as these wave-functions do not vanish at infinity, a typical property of free-particle solutions. But this problem is not related to the Hermiticity property of the momentum operator, it is related with the de-localized nature of the free-particle solution. In physics many times we require to solve a problem using curvilinear coordinate systems. The choice of our coordinate system depends upon the specific symmetry which we have at hand. Suppose we are working in spherical polar coordinates and the solution of Eq. (3.2) can be separated into well behaved functions of r, θ and φ as, u(x) = u(r, θ, φ) = R(r)Θ(θ)Φ(φ) . (3.4) If we try to follow the proof of the Hermiticity of the linear momenta components, as done in the last section, in spherical polar coordinates, then we should write: 〈p̂〉 = −ih̄ u∗(r, θ, φ)∇u(r, θ, φ) dτ , = −ih̄ R∗(r)Θ∗(θ)Φ∗(φ) r sin θ R(r)Θ(θ)Φ(φ)r2drdΩ , (3.5) where in the above equation er, eθ, eφ respectively are the unit vectors along r, θ and φ, and dΩ = sin θ dθ dφ. τ is the volume over which we integrate the expression in the above equation. From the last equation we can write: 〈p̂r〉 = −ih̄ |Θ(θ)|2|Φ(φ)|2dΩ r2R∗(r) dR(r) dr , (3.6) As Θ(θ) and Φ(φ) are normalized, the integration: |Θ(θ)|2|Φ(φ)|2dΩ = 1 and we can proceed as in Eq. (2.9) as: 〈p̂r〉 = −ih̄ r2R∗(r) dR(r) = −ih̄ r2R∗(r)R(r) 2rR∗(r) + r2 dR∗(r) R(r) dr . (3.7) If R(r) vanishes at infinity then the above equation reduces to, 〈p̂r〉 = r2R(r) dR∗(r) + 2ih̄ r|R(r)|2 dr , = 〈p̂r〉∗ + 2ih̄ r|R(r)|2 dr . (3.8) The above equation implies that 〈p̂r〉 is not real in spherical polar coordinates. The solution of the above problem lies in redefining p̂r as is evident from Eq. (3.8), and it was given by Dirac [5, 6]. The redefined linear momentum operator along r can be: p̂r ≡ −ih̄ = −ih̄ r . (3.9) This definition of the p̂r is suitable because in this form it satisfies the commutation relation as given in Eq. (2.1) where now the operator conjugate to r̂ is p̂r. The form of p̂r in Eq. (3.9) shows that for any arbitrary function of r as F (r) we must still have Eq. (2.3) satisfied. This implies that the modified form of p̂r is still a generator of translations along the r direction. Up to this point we were following what was said by Dirac regarding the status of the radial momentum operator. Still everything is not that smooth with the redefined operator as we can see that it turns out to be singular around r = 0, more over, although the radial momentum acts like a translation generator along r but near r = 0 it cannot generate a translation towards the left as the interval ends there. In this regard we can state that the issue of the reality of the radial component of the momentum in spherical polar coordinates is a topic of modern research in theoretical physics [7, 8]. It has been shown that the operator −ih̄ ∂ not Hermitian and more over it can be shown [4] that such an operator cannot be self-adjoint in the interval [0,∞]. In some recent work [8] the author claims that there can be an unitary operator which connects −ih̄ ∂ to −ih̄1 and as the former operator does not have a self-adjoint extension in the semi-infinite interval so the latter is also not self-adjoint in the same interval. If we further try to find out whether 〈p̂θ〉 and 〈p̂φ〉 are real, then we will face difficulties. Working out naively if we claim that p̂φ = rsinθ as suggested by the φ component of Eq. (3.5) we will notice that φ p̂φ does not have the dimension of action. This means p̂φ or p̂θ is not conjugate to φ or θ. This is a direct representation of the special coordinate dependence of the quantization condition. Only in Cartesian coordinates the variables conjugate to x, y and z are px, py and pz. Taking the clue from classical mechanics we know the proper dynamical variables conjugate to φ̂ and θ̂ are the angular momentum operators, namely L̂θ and L̂φ. In general L̂φ is given by: L̂φ = −ih̄ , (3.10) which can be shown to posses real expectation values by following a similar proof as is done in Eq. (2.8) and Eq. (2.9), if we assume Φ(0) = Φ(2π). In this form it is tempting to say that we can have a relation of the form, [φ̂, L̂φ] = ih̄ , (3.11) which looks algebraically correct. But the difficulty in writing such an equation is in the interpretation of φ̂ which has been elevated from an angular variable to a dynamical operator. In spherical polar coordinates both θ and φ are compact variables and consequently have their own subtleties. Much work is being done in trying to understand the status of angular variables and phases [9, 10], in this work we only present one example showing the difficulty of accepting φ̂ as an operator. From the solution of the time-independent Schrödinger equation for an isotropic potential we will always have: Φ(φ) = eiMφ , (3.12) where M = 0,±1,±2, ·, ·. Now if φ̂ is an operator we can find its expectation value, and it turns out to be: 〈φ̂〉 = 1 φeiMφe−iMφ dφ , = π , (3.13) and the expectation value of φ̂2 is: 〈φ̂2〉 = 1 φ2eiMφe−iMφ dφ , π2 . (3.14) Consequently ∆φ = 〈φ̂2〉 − 〈φ̂〉2 = π√ . Similarly calculating L̂φ we get: 〈L̂φ〉 = e−iMφeiMφ dφ , = Mh̄ , (3.15) as expected, and 〈L̂2φ〉 = M2h̄ . This implies ∆Lφ = 〈L̂2φ〉 − 〈L̂φ〉2 = 0. So we can immediately see that the Heisenberg uncertainty relation between φ̂ and L̂φ, ∆φ∆Lφ ≥ h̄/2 breaks down. This fact makes life difficult and we have no means to eradicate this problem. Taking the clue from the φ part we can propose that L̂θ is also of the form −ih̄ ∂∂θ . With this definition of L̂θ let us try to prove its Hermitian nature as done in Eq. (3.7). Taking R(r) and Φ(φ) in Eq. (3.4) separately normalized, we can write: 〈L̂θ〉 = −ih̄ Θ∗(θ) dΘ(θ) sin θdθ = −ih̄ sin θΘ∗(θ)Θ(θ)|π0 − cos θΘ∗(θ) + sin θ dΘ∗(θ) Θ(θ) dθ sin θΘ(θ) dΘ∗(θ) + ih̄ cos θ |Θ(θ)|2 dθ , = 〈L̂θ〉∗ + ih̄ cos θ |Θ(θ)|2 dθ . (3.16) The above equation shows that 〈L̂θ〉 is not real. The rest is similar to the analysis following Eq. (3.8) where now we have to redefine the angular momentum operator conjugate to θ as [11]: L̂θ ≡ −ih̄ cot θ . (3.17) Unlike the φ case, Θ(θ) are not eigenfunctions of L̂θ. But the difficulties of establishing θ as an operator still persists and in general θ is not taken to be a dynamical operator in quantum mechanics. It is known that both θ and φ are compact variables, i.e. they have a finite extent. But there is a difference between them. In spherical polar coordinates the range of φ and θ are not the same, 0 ≤ φ < 2π and 0 ≤ θ ≤ π. This difference can have physical effects. As φ runs over the whole angular range so the wave-function corresponding to it Φ(φ) is periodic in nature while due to the range of θ, Θ(θ) need not be periodic. Consequently there can be a net angular momentum along the φ direction while there cannot be any net angular momentum along θ direction. And this can be easily shown to be true. As the time-independent Schrödinger equation for an isotropic potential yields Φ(φ) as given in Eq. (3.12) similarly it is known that in such a potential the form of Θ(θ) is given by: Θ(θ) = Nθ P M (cos θ) , (3.18) where Nθ is a normalization constant depending on L, M and P M (cos θ) is the associated Legendre function, which is real. In the above equation L and M are integers where L = 0, 1, 2, 3, ·, · and M = 0,±1,±2,±3, ·, ·. The quantum number M appearing in Eq. (3.12) and in Eq. (3.18) are the same. This becomes evident when we solve the time-independent Schrödinger equation in spherical polar coordinates by the method of separation of variables. A requirement of the solution is −L ≤ M ≤ L. Now we can calculate the expectation value of L̂θ using the above wave-function and it is: 〈L̂θ〉 = −ih̄N2θ PLM (cos θ) dPLM (cos θ) cot θPLM (cos θ) sin θdθ = −ih̄N2θ PLM (cos θ) dPLM (cos θ) sin θdθ + PLM (cos θ)P M (cos θ) cos θ dθ (3.19) To evaluate the integrals on the right hand side of the above equation we can take x = cos θ and then the expectation value becomes: 〈L̂θ〉 = −ih̄N2θ PLM (x) dPML (x) (1− x2) 12 dx PLM(x)P 1− x2 . (3.20) The second term in the right hand side of the above equation vanishes as the integrand is an odd function in the integration range. For the first integral we use the following recurrence relation [12]: (x2 − 1) dPLM (x) =MxPLM (x) − (L+M)PLM−1(x) , (3.21) the last integral can be written as, 〈L̂θ〉 = ih̄N2θ x(1− x2)− 2PLM (x)P M (x) dx − (L+M) (1 − x2)− 2PLM (x)P M−1(x) dx . (3.22) PLM (x) = (−1)L+MPLM (−x) , (3.23) we can see immediately that both the integrands in the right hand side of the above equation is odd and consequently 〈L̂θ〉 = 0 as expected. A similar analysis gives 〈L̂φ〉 = Mh̄. It must be noted that the form of L̂θ still permits it to be the generator of rotations along the θ direction. As the motion along φ is closed so there can be a net flow of angular momentum along that direction but because the motion along θ is not so, a net momentum along θ direction will not conserve probability and consequently for probability conservation we must have expectation value of angular momentum along such a direction to be zero. In elementary quantum mechanics text books it is often loosely written that the solution of the time-independent Schrödinger equation is real when we are solving it for a real potential. But this statement is not correct. The reality of the solution also depends upon the coordinate system used. Specially for compact periodic coordinates we can always have complex functions as solutions without breaking any laws of physics. Before leaving the discussion on angular variables in spherical polar coordinates we want to point out one simple thing which is interesting. In Cartesian coordinates when we deal with angular momentum we know that: [L̂i, L̂j] = iǫijk L̂k , (3.24) where L̂i stands for L̂x, L̂y or L̂z. For this reason there cannot be any state which can be labelled by the quantum numbers of any two of the above angular momenta. But from the expressions of L̂φ and L̂θ we see that, [L̂φ, L̂θ] = 0 , (3.25) and consequently in spherical polar coordinates we can have wave-function solutions of the Schrödinger equation which are simultaneous eigenfunctions of both L̂φ and L̂θ as P M (θ). For real V (x), we expect the solution of the time-independent Schrödinger equation u(x) to be real, when we are solving the problem in Cartesian coordinates. In all these cases the expectation value of the linear momentum operators must vanish. The reason is simple and can be understood in one-dimensional cases where with real u(x) we directly see that the integral ∂u(x) dx is real and so ∗(x) p̂x u(x) dx becomes imaginary as p̂x contains i, as is evident from the first line in Eq. (2.9). So if the expectation value of the momentum operator has to be real then the only outcome can be that for all those cases where we have a time-independent solution in a bounded region of space, with a real potential and working in Cartesian coordinates, the expectation value of the momentum operator must vanish. The above statement is true in curvilinear coordinates also, but in those cases the definition of the momentum operators have to be modified. This fact becomes clear when we write the relationship between the probability flux and the expectation value of the momentum operator. The probability flux for a particle of mass m j(x, t) = − [ψ∗(x, t)∇ψ(x, t) − (∇ψ∗(x, t))ψ(x, t)] , Im (ψ∗(x, t)∇ψ(x, t)) , (3.26) where ‘Im’ implies the imaginary part of some quantity. Most of the elementary quantum mechanics books then proceeds to show that: d3x j(x, t) = , (3.27) which is obtained from Eq. (3.26) by integrating both sides of it over the whole volume. From Eq. (3.26) we immediately see that if the solution of the time-independent Schrödinger equation is real we will have j(x, t) = 0 and consequently from Eq. (3.27), 〈p̂〉 = 0. But this statement is also coordinate dependent, which is rarely said in elementary textbooks of quantum mechanics. Eq. (3.26) evidently does not hold in spherical polar coordinates. If we take Eq. (3.4) as the solution in a general isotropic central potential and use the general form of ∇ in spherical polar coordinates then it can be seen that jr(r, θ, φ, t) = 0 for a real potential. But then Eq. (3.27) does not hold as here p̂r is simply the radial component of ∇ and not as given in Eq. (3.9), and we know 〈 d 〉 is not zero. The reason why Eq. (3.26) is not suitable in spherical polar coordinates is related to the fact that in deriving Eq. (3.26) one assumes that the probability density of finding the quantum state within position x and x + dx at time t is |ψ(x, t)|2. But this statement is only true in Cartesian coordinates, in spherical polar coordinates the probability density of the system to be within a region r and r + dr, θ and θ + dθ, φ and φ+ dφ is not |ψ(r, θ, φ)|2 but |ψ(r, θ, φ)|2r2 sin θ and consequently the steps which follow leading to Eq. (3.26) in Cartesian coordinates are not valid in spherical polar coordinates. In general Eq. (3.26) will not be valid in any curvilinear coordinate system. The next section contains the actual calculations of the expectation values of the momentum operator in various cases where we have bound state solutions. In all the relevant cases discussed in this article it is seen that although 〈p̂x〉 = 0 but 〈p̂2x〉 is not zero as it is related to the Hamiltonian operator. In all the cases we must have, 〈(p̂x)s〉 = 0 , s = odd integer . (3.28) The above equation can be guessed from the reality of the expectation value of the momentum operator. IV. MOMENTUM EXPECTATION VALUES IN VARIOUS BOUND STATES In this section we will calculate the momentum expectation values in various bound states with stiff or slowly varying potentials. A. Particle in one-dimensional stiff potential wells 1. Infinite square well potential In this case we consider a particle to be confined in region −L along the x-axis where the potential is specified V (x̂) = ∞ , |x| ≥ = 0 , |x| < . (4.1) In this case the solution of the time-independent Schrödinger equation, Eq. (3.2), satisfies the boundary condition, = 0 , (4.2) and as the potential has parity symmetry about x = 0 we have two sets of solutions, the odd solutions: u(o)n (x) = , (4.3) and the even solutions: u(e)n (x) = (2n− 1)πx . (4.4) In the above equations n is a positive integer. Both of these functions, u n (x) for the odd case and u n (x) for the even case, are real and are not momentum eigenstates. But the momentum expectation values can be found out from the above solutions. For the odd solutions we have: 〈p̂x〉 = −ih̄ u(o)n (x) n (x) = −4inπh̄ = 0 , (4.5) as expected. Similarly for the even solutions it is also easy to show that the expectation value of the momentum operator vanishes. 2. Finite square well potential In this case, V (x̂) = 0 , |x| ≥ a , = −V0 , |x| < a , (V0 > 0) . (4.6) If we are not interested in the normalization constant of the bound state solution then the solution of the time- independent Schrödinger equation in this case is: u(x) ∼ e−κ|x| , |x| > a , ∼ cos(kx) , |x| < a , (even parity) ∼ sin(kx) , |x| < a , (odd parity) , (4.7) where, 2m(−|E|+ V0) , (4.8) 2m|E| . (4.9) In this case the expectation value of the momentum operator is: 〈p̂x〉 ∼ −ih̄ du(x) ∼ −ih̄ e2κxdx− e−2κxdx sin(kx) cos(kx) dx = 0 , (4.10) where the first two lines of the above equation holds up to a constant arising from the normalization of the wave- function. In deriving the last equation we have taken the odd parity solution, but the result remains unaffected if we take the even parity solution as well. 3. Dirac-delta potential In this case the potential is: V (x̂) = −V0 δ(x̂) , (V0 > 0) . (4.11) In this case there can be one bound state solution which is obtained after solving the Eq. (3.2). Demanding that the solution u(x) satisfies the boundary conditions: u(x = −ǫ) = u(x = +ǫ) , (4.12) = −2mV0 u(x = 0) , (4.13) where ǫ is an infinitesimal quantity tending to zero, we get the form of the solution which is: u(x) = κ eκx , x ≤ 0 , (4.14) κ eκx , x ≥ 0 , (4.15) where κ = mV0 and the energy of the bound state is E = −mV The expectation value of the momentum operator in this case is: 〈p̂x〉 = −ih̄ du(x) = −ih̄κ e2κx dx − e−2κx dx = 0 . (4.16) In this case, also from Hermiticity of the momentum operator we see that Eq. (3.28) holds true. B. Particle in one-dimensional slowly varying potentials 1. Linear harmonic oscillator potential In the case of the linear harmonic oscillator we have: V (x̂) = mω2x̂2 , (4.17) where ω is the angular frequency of the oscillator. The solution of Eq. (3.2) in this case, using the series solution method, yields: un(q) = Nn e 2 Hn(q) , (4.18) where n = 0, 1, 2, ·, · and q = αx where α = mω . Hn(q) are Hermite polynomials of order n and Nn is the normalization constant given by, π n! 2n . (4.19) The momentum expectation value in this case turns out to be, 〈p̂x〉 = −ih̄ un(q) dun(q) = −ih̄ Hn(q) dHn(q) q e−q H2n(q) dq = 0 . (4.20) The first integral on the right side of the second line of the last equation vanishes because, dHn(q) = 2nHn−1(q) and consequently the integral transforms into the orthogonality condition of the Hermite polynomials. The second integral on the second line of the right side of the above equation vanishes because the integrand is an odd function of q. The linear harmonic oscillator (LHO) has some very interesting properties. To unravel them we have to digress a bit from the wave-mechanics approach which we have been following and follow the Dirac notation of bra and kets. The Hamiltonian of the LHO in one-dimension is: mω2x̂2 , (4.21) which can also be written as: Ĥ = h̄ω â†â+ , (4.22) where â and ↠are the annihilation and the creation operators given by: , ↠≡ x̂− ip̂x . (4.23) It can be seen clearly from the above definitions that â is not an Hermitian operator. More over from the definition of the operators we see that, [â, â†] = 1 . (4.24) Conventionally the number operator is defined as: N̂ ≡ â†â , (4.25) and its eigen-basis are the number states |n〉 such that, N̂ |n〉 = n|n〉 . (4.26) The Hamiltonian of the LHO can be written in terms of the number operator and consequently the number states are energy eigenstates. In this basis the action of the annihilation and creation operators are as: â|n〉 = n |n− 1〉 , (4.27) â†|n〉 = n+ 1 |n+ 1〉 . (4.28) From the definitions of the annihilation and creation operators we can write the momentum operator as: p̂x = −i (â− â†) . (4.29) From Eq. (4.27), Eq. (4.28) and the above equation we can write the matrix elements of the momentum operator as: 〈n′|p̂x|n〉 = i n δn′, n−1 + n+ 1 δn′, n+1 . (4.30) The above equation shows that the momentum operator can connect two different energy eigenstates. In the case of LHO, except the number operator states, we can have another state which is an eigenstate of the annihilation operator â. This state is conventionally called the coherent state and it is given as: |α〉 = e− |n〉 , (4.31) where α is an arbitrary complex number. Now from Eq. (4.29) we can find the momentum expectation value of the coherent state and it is, 〈p̂x〉α ≡ 〈α|p̂x|α〉 = Im(α) . (4.32) From the above equation we can see that although the expectation value of the momentum operator is zero in the energy eigen-basis but it is not so when we compute the momentum expectation value in the coherent state basis, which is essentially a superposition of energy eigenstates. It must be noted that the momentum expectation value is non zero only when the parameter α has an imaginary part. 2. Pöschl-Teller potential Among the potentials belonging to the hypergeometric class the Pöschl-Teller potentials have been the most ex- tensively studied and used. This class of potentials consist of trigonometric as well as the hyperbolic type. The trigonometric versions have found applications in molecular and solid state physics and the hyperbolic variants have been used in various studies related to black hole perturbations. In the present work we use the trigonometric, symmetric Pöschl-Teller potential given by: V (x̂) = V0 tan 2(ax̂) , (4.33) where V0 can be parameterized as: λ(λ− 1) , (4.34) with for some positive number λ > 1 and a is some scaling factor. The energy eigenvalues of the bound state solutions En = − h̄2a2 (n2 + 2nλ+ λ) , (4.35) and the solution of the time-independent Schrödinger equation is, un(x) = Nn cos(ax)P 1/2−λ n+λ−1/2 (sin(ax)) , (4.36) where, a(n+ λ)Γ(n + 2λ) Γ(n+ 1) , (4.37) is the normalization constant and Pµν (x) is the associated Legendre function. At this point it is fair to point out that Pµν (x) is not the Legendre polynomial P M (x) appearing in Eq. (3.18), as µ and ν need not be integers as L and M . Pµν (x) is not a polynomial but the function appearing in the right hand side of Eq. (4.36) is a polynomial. Now as claimed in the text let us show that the momentum expectation value is indeed zero. Before we proceed let us simplify the notation a bit by calling µ = 1/2− λ and ν = n+ λ − 1/2. Substituting z = ax we can write the momentum expectation value as: 〈p̂x〉 = −ih̄N2n ∫ π/2 cos(z)Pµν (sin(z)) cos(z)Pµν (sin(z)) . (4.38) Note the limits of the integration range from π/2 to −π/2 since at this value the potential becomes infinity hence we need not consider the integration range to be the whole real line. For the sake of convenience let us make a change of variable; letting y = sin(z) the above integral becomes: 〈p̂x〉 = −ih̄N2n dy (1− y2)1/4Pµν (y) (1 − y2)1/4Pµν (y) . (4.39) Taking the derivative inside the integral we get: 〈p̂x〉 = −ih̄N2n (1− y2)1/2Pµν (y) dPµν (y) − y(1− y 2)−1/2 Pµν (y)P ν (y) . (4.40) It is known that for associated Legendre functions [13], Pµν (−x) = cos[(µ+ ν)π]Pµν (x)− sin[(µ+ ν)π]Qµν (x) , (4.41) where Qµν (x) is the other linearly independent solution of the associated Legendre differential equation. As in our case µ+ ν = n so Pµν (x) will have definite parity. As P ν (x) has definite parity so the contribution of the second term in the above integral vanishes since the total integrand is an odd function. The first integral is similar to the one in Eq. (3.20) and, due to the typical parity property of Pµν (x) as shown in Eq. (4.41), it also vanishes. Consequently we have 〈p̂x〉 = 0 as expected. 3. Morse potential Diatomic molecule is an exactly solvable system, if one neglects the molecular rotation. The most convenient model to describe the system, is the Morse potential [14]: V (x̂) = D(e−2βx̂ − 2e−βx̂) , (4.42) where x = r/r0 − 1, which is the distance from the equilibrium position scaled by the equilibrium value of the inter-nuclear distance r0. D is the depth of the potential, called dissociation energy of the molecule and β being a parameter which controls the width of the potential. In terms of the above scaled variable x, the time-independent Schrödinger equation becomes: d2u(x) +D(e−2βx − 2e−βx)u(x) = Eu(x) . (4.43) Here µ is the reduced mass of the molecule and the corresponding bound state eigen function comes out to be: uλn(ξ) = Ne −ξ/2ξs/2Lsn(ξ) , (4.44) where the variables are described as, ξ = 2λe−y; y = βx; 0 < ξ <∞ , (4.45) n = 0, 1, ..., [λ− 1/2] , (4.46) which is nothing but the quantum number of the vibrational bound states. Here [ρ] denotes the largest integer smaller than ρ, thus total number of bound states is [λ− 1/2] + 1. The parameters, 2µDr20 and s = − 8µr E , (4.47) satisfy the constraint condition s+2n = 2λ− 1. We note that the parameter λ is potential dependent and s is related to energy E. In Eq. (4.44), Lsn(y) is the associated Laguerre polynomial and N is the normalization constant [15]: β(2λ− 2n− 1)Γ(n+ 1) Γ(2λ− n)r0 . (4.48) We are looking for the expectation value of linear momentum for a vibrating diatomic molecule, and its expression 〈p̂x〉 = −ih̄ u∗n(ξ) un(ξ)dx . (4.49) In terms of the changed variable ξ = 2λe−βx the integration limit changes to ∞ to 0 and the expectation value becomes: 〈p̂x〉 = −ih̄ u∗n(ξ) un(ξ)dξ = ih̄N2 e−ξξs(Lsn(ξ)) 2dξ + e−ξξs−1(Lsn(ξ)) e−ξξsLsn(ξ) Lsn(ξ)dξ = ih̄N2 I2 + I3 . (4.50) Integral I1 is the orthogonality relation of the associated Laguerre polynomials, which is: e−ξξsLsn(ξ)L m(ξ)dξ = Γ(s+ n+ 1) Γ(n+ 1) δm,n . (4.51) To evaluate the second integral one uses the normalization integral of Morse eigenstates. The normalization relation u∗(ξ)u(ξ)dr = |N |2r0 e−ξξs−1(Lsn(ξ)) 2dξ = 1 . (4.52) The above integral involving ξ, is explicitly I2. N , being the normalization constant as given in Eq. 4.48. Thus it is very straight forward to evaluate I2 from the above relation as, Γ(n+ s+ 1) s Γ(n+ 1) . (4.53) The last integrand I3 includes a differentiation which can be written as [16]: Lsn(ξ) = −Ls+1n−1(ξ) . (4.54) Writing the right hand side of the above equation as a summation [17]: Ls+1n = Lsm , (4.55) and substituting the derivative term in integral I3 we obtain: I3 = − e−ξξsLsn(ξ)L m(ξ)dξ . (4.56) In the above integral m 6= n because m can go only upto (n− 1). Thus the integral vanishes. Now let us see what is the expectation value of momentum observable, after evaluating the three integrals above. Substituting the non-zero values I1 and I2 in Eq. 4.50, it is clear that the expectation value of momentum is zero as has been expected. C. Position expectation values for various potentials After a thorough discussion about the momentum expectation values for various solvable one-dimensional potentials, it is worth spending some time discussing about the average position of the particle inside the bound states. Among all the above examples, in each case we had V (x) = V (−x) except the Morse potential as Morse potential is not an example of a symmetric potential: V (x) 6= V (−x). In deriving the expectation values of momentum for above symmetric cases, we often considered that the integrals of odd functions over the symmetric limits vanishes. This result does not hold true for the asymmetric Morse potential. Already we have shown that the momentum expectation value: < p >= 0 for all the above potentials. When it comes to the expectation values of position, one can easily see that < x >= 0 for symmetric potentials whose centers are at the origin. On the other hand if this is not the case, suppose the infinite square well is defined in the range 0 ≤ x ≤ L also then the expectation value of position does not vanish. It becomes L/2. Thus, more accurately the average position of the particle is dependent on the symmetry of the potential where as the average momentum is solely guided by the reality of it’s eigenvalues and consequently it is zero always. Below we will briefly discuss how the asymmetry of the potential affects the expectation value of x in the case of the Morse potential. The expectation value of the position operator is: 〈x̂〉 = uλ∗n (ξ)xu n(ξ)dx. (4.57) The eigen function and the variables are respectively substituted from Eq. (4.44) and Eq. (4.45). We obtain 〈x̂〉 = N ln(2λ) e−ξξs−1(Lsn(ξ)) 2dξ + e−ξξs−1(Lsn(ξ)) 2 ln(ξ)dξ . (4.58) The first integral is already been obtained in Eq. (4.53). This result is independent of the quantum number n. The second integral (say I) is not that straight forward, because it contains associated Laguerre polynomial, logarithm, exponential and monomial functions. Here at best we can evaluate the integral atleast for some specific n as, n = 0 or n = 1, when the Laguerre polynomial is respectively replaced by 1 and (−ξ + s + 1). For the ground state wave function (n = 0), I would be In=0 = e−ξξs−1 ln(ξ)dξ, (4.59) which can be written in terms of Ψ(s) and Γ function [18]: In=0 = Γ(s)Ψ(s), (4.60) where, Ψ(s) is the logarithmic factorial function, defined as d(ln(s)! = Ψ(s). For n = 0, first integral reduces to Γ(s) from Eq. (4.53). Above two evaluations gives the ground state expectation value: 〈x̂〉n=0 = [ln(s+ 1)−Ψ(s)] . (4.61) For n = 1, one can proceed in the same way 〈x̂〉n=1 = e−ξξs+1 ln(ξ)dξ + (s+ 1)2 e−ξξs−1 ln(ξ)dξ2(s+ 1) e−ξξs ln(ξ)dξ (4.62) Γ(s+ 2)Ψ(s+ 2) + (s+ 1)2Γ(s)Ψ(s)− 2(s+ 1)Γ(s+ 1)Ψ(s+ 1) which simplifies to give the expectation value corresponding to the second eigen state: 〈x̂〉n=1 = ln(s+ 3)−Ψ(s+ 2) + 3 (s+ 2) . (4.63) Other expectation values for n > 1 can also be obtained in a similar fashion. The important point which is to be noted here is, though the average momentum vanishes, the average position is non-zero for Morse potential and remain so, irrespective of the choice of coordinate origin. This result is also true for all eigen states of the same Hamiltonian. D. Momentum expectation value for a three-dimensional slowly varying spherically symmetric potential In three dimensions, for a spherically symmetric potential the solution of the Schrödinger equation is given in Eq. (3.4). Here we have assumed that the variables can be separated. The expectation values of L̂θ and L̂φ have been evaluated in section III. In this section we take the case of the Hydrogen atom and calculate the expectation value of the radial component of the linear momentum. 1. The Hydrogen atom In this case, V (r̂) = −e . (4.64) where e is the electronic charge and r = x2 + y2 + z2. Now we have to write Eq. (3.2) in spherical polar coordinates and the solution of the time-independent Schrödinger equation is: unLM (r, θ, φ) = Nr RnL(r)YLM (θ, φ) , = Nr e −r/na0 L2L+1n−L−1 YLM (θ, φ) , (4.65) where a0 = is the Bohr radius and m is the reduced mass of the system comprising of the proton and the electron. n is the principal quantum number which is a positive integer, L2L+1n−L−1(x) are the associated Laguerre polynomials, YLM (θ, φ) are the spherical-harmonics, and Nr is the normalization arising from the radial part of the eigenfunction. The values which L and M can take is discussed in section III. The radial normalization constant is given by: (n− L− 1)! (n+ L)!2n . (4.66) The spherical-harmonics are given by, YLM (θ, φ) = 2L+ 1 (L −M)! (L +M)! PLM (cos θ)e iMφ , (4.67) where PML (cos θ) are the associated Legendre functions. It is noted that although the Coulomb potential is a real potential but the solution in spherical polar coordinates is not real, eiMφ, is complex. The spherical-harmonics are ortho-normalized according to the relation, dθ dφ sin θ YLM (θ, φ)YL̃ M̃ (θ, φ) = δLL̃ δMM̃ . (4.68) Let us write the eigenfunctions in terms of dimensionless quantity: ρ = 2r/na0 ≡ αr. Also we define k ≡ (2L + 1) and nr ≡ (n − L − 1) for the sake of convenience. With this amount of notational machinery the eigenfunctions can be written as: unLM (r, θ, φ) = Nr RnL(ρ)YLM (θ, φ) . (4.69) The radial momentum expectation value in this case is not given by −ih̄〈 ∂ 〉, its form is (already discussed in section III): 〈p̂ρ〉 = −ih̄Ñ2 dρ ρ2R∗nL(ρ) RnL(ρ) dΩ [YLM (θ, φ)] 2 . (4.70) Where Ñ2 = N2r /α 2. The integral for the spherical harmonics yields identity. The radial expectation value then becomes, 〈p̂ρ〉 = −ih̄Ñ2 e−ρρk+1[Lknr (ρ)] 2 + (L+ 1) e−ρρk[Lknr (ρ)] 2 + e−ρρk+1Lknr (ρ) [Lknr (ρ)] . (4.71) Using the recurrence relation [16]: Lknr (ρ) = ρ −1 [nr Lknr (ρ)− (nr + k)L nr−1(ρ) , (4.72) the expectation value integral acquires the form: 〈p̂ρ〉 = −ih̄Ñ2 e−ρρk+1[Lknr (ρ)] 2 + (nr + L+ 1) e −ρρk[Lknr(ρ)] 2 + e−ρρkLknr (ρ)L nr−1(ρ) . (4.73) The third contribution of the becomes zero from the orthogonality property of the associated Laguerre polynomials as given in Eq. (4.51). The contribution from the second term can also be found similarly. To find the share of the first term we make use of [19]: dρ e−ρ ρk+1 [Lknr (ρ)] (nr + k)! (2nr + k + 1). (4.74) Collecting all the contributions we get the radial expectation value to be zero as expected. V. A DISCUSSION ON HEISENBERG’S EQUATION OF MOTION AND EHRENFEST THEOREM The time evolution of any operator Ô in the Heisenberg picture is given by: [Ô, Ĥ ] , (5.1) where Ĥ is the Hamiltonian of the system. The Hamiltonian of a quantum system comprising of a particle of mass m is given by: + V (x̂) . (5.2) From the above two equations we can write the time evolution of the momentum operator in one dimension, in Cartesian coordinates as: [p̂x, Ĥ] = − V (x̂) , (5.3) which is the operator version of Newton’s second law in a time independent potential. Now if we take the expectation values of both sides of Eq. (5.3) in any basis we get: d〈p̂x〉 V (x̂) , (5.4) and historically the above equation is called the Ehrenfest theorem, which was deduced in a different way by P. Ehrenfest. Using the Ehrenfest theorem we can deduce that the rate of change of the expectation value of the momentum operator is zero in the case of the linear harmonic oscillator. In the case of the linear harmonic oscillator we have: V (x̂) = mω2x̂ , (5.5) and it can be trivially shown that 〈x̂〉 = 0. This directly implies that, d〈p̂x〉 = 0 , (5.6) for the linear harmonic oscillator. The above equation shows that the expectation value of the momentum along x direction is constant, and this constant is zero is known from other sources. Next we focus on the Hydrogen atom. The Hamiltonian of the Hydrogen atom is: Ĥ = − h̄ 2 − e , (5.7) where, 2 = −h̄2 sin θ sin θ sin2 θ , (5.8) whose eigenvalues are of the form h̄ L(L+ 1) in the basis YLM (θ, φ). In the expression of the Hamiltonian m is the reduced mass of the system comprising of the proton and electron. Next we try to apply Heisenberg’s equation to the radial momentum operator. Noting that the first term of the Hamiltonian is nothing but p̂2r the Heisenberg equation = − L̂ . (5.9) The above equation is the operator form of Newton’s second law in spherical polar coordinates. Next we evaluate the expectation value of both the sides of the above equation using the wave-functions given in Eq. (4.65). We know, n3 a20(L + , (5.10) a30 n 3L(L+ 1 )(L + 1) . (5.11) Using the above expectation values in Eq. (5.9) and noting that 〈L̂2〉 = h̄2L(L+1) we see that the time derivative of the expectation value of the radial momentum operator of the Hydrogen atom vanishes. The above analysis shows that the form of the Ehrenfest theorem as given in Eq. (5.4) is only valid in Cartesian coordinates. In the case of the Hydrogen atom if we used Eq. (5.4) we should have never got the correct result. VI. CONCLUSION In the present work we have emphasized on the reality of the momentum expectation value and using the reality of the expectation value as a bench mark we did find out the form of the momentum operator in spherical polar coordinate system. We found that most of the concepts which define the momentum operator in Cartesian coordinates do not hold good in spherical polar coordinates and in general in any other coordinate system. The reason being that whenever we do an integration in curvilinear coordinates the Jacobian of the coordinate transformation matrix comes inside the picture and the Cartesian results start to falter if we do not change the rules appropriately. The forms of the momentum along the radial direction and the form of the angular momentum operators are derived in section III. The status of the angular variables was briefly discussed in the same section. We explicitly calculated the expectation values of the momentum operator in various important cases and showed that the expectation value of the momentum operator do really come out to be zero as expected. Although the expectation value of the momentum operator vanishes in most of the bound states, with a real potential, the expectation value of the position is not required to vanish. The expectation value of the position operator is directly related with the parity property of the potential which was briefly discussed in subsection IVC. At the end we calculated the Heisenberg equation of motion for the radial momentum operator for the Hydrogen atom and showed its formal semblance with Newton’s second law. It was also shown that if we properly write the Heisenberg equation of motion in spherical polar coordinates then Ehrenfest’s theorem follows naturally. In short we conclude by saying: 1. the forms of the various momentum operators, in most of the coordinate systems, in quantum mechanics can be obtained by imposing the condition of the reality of their eigenvalues. The form of the probability conservation equation and Ehrenfest theorem must be modified in curvilinear coordinates to yield meaningful results. 2. There are obvious problems in elevating the status of angular variables to dynamical variables in quantum mechanics. 3. For compact variables, if the variable is periodic the expectation value of the angular momentum conjugate to it is non-zero. If the compact variable is not periodic then the angular momentum conjugate to it must vanish. 4. The momentum expectation values in cases of bound state motions vanish, whereas the position expectation values in those cases depends on the symmetry of the potential. Acknowledgements The authors thank Professors D. P. Dewangan, S. Rindani, J. Banerji, P. Panigrahi and Ms. Suratna Das for stimulating discussions and constant encouragements. [1] J. J. Sakurai, “Modern quantum mechanics”, International student edition, Addison-Wesley, 1999. [2] L I. Schiff, “Quantum mechanics”, McGraw-Hill International Editions, third edition. [3] R. Shankar, “Principles of quantum mechanics”, Plenum Press, New York 1994, second edition. [4] G. Bonneau, J. Faraut, G. Valent, Am. J. Phys. 69 322, (2001). [5] P. A. M Dirac. “The principles of quantum mechanics”, Fourth edition, Oxford university press, 1958. [6] S. Flügge. “Practical quantum mechanics I”, Springer-Verlag Berlin Heidelberg 1971. [7] G. Paz, Euro. J. Phys. 22 337, (2001). [8] G. Paz, J. Phys.A: Math. Gen. 35 3727, (2002). [9] P. Carruthers, M. M. Nieto, Rev.Mod. Phys. 40 411, (1968). [10] D. T. Pegg, S. M. Barnett, Phys.Rev.A 39 1665, (1989). [11] H. Essén, Am. J. Phys. 46 983, (1978). [12] I. S. Gradshteyn, I. M. Ryzhik. “Table of integrals, series, and products”. Academic Press, Harcourt India, sixth edition, page 955, 8.733 1 [13] I. S. Gradshteyn, I. M. Ryzhik. “Table of integrals, series, and products”. Academic Press, Harcourt India, sixth edition, page 956, 8.737 2. [14] P. M. Morse, Phys.Rev. 34 57, (1929). [15] S. Ghosh, A. Chiruvelli, J. Banerji and P. K. Panigrahi, Phys. Rev. A 73, 013411, (2006). [16] I. S. Gradshteyn, I. M. Ryzhik. “Table of integrals, series, and products”. Academic Press, Harcourt India, sixth edition, page 991, 8.971 2. [17] I. S. Gradshteyn, I. M. Ryzhik. “Table of integrals, series, and products”. Academic Press, Harcourt India, sixth edition, page 992, 8.974 3. [18] I. S. Gradshteyn, I. M. Ryzhik. “Table of integrals, series, and products”, sixth edition, (Academic Press, Harcourt India). [19] The specific integration result and other related expressions can be found in the web page: http://mathworld.wolfram.com/LaguerrePolynomial.html, Equation 24. http://mathworld.wolfram.com/LaguerrePolynomial.html Introduction Definition of the momentum operator and the reality of its expectation value The expectation value of the momentum operator in Cartesian and spherical polar coordinates Momentum expectation values in various bound states Particle in one-dimensional stiff potential wells Infinite square well potential Finite square well potential Dirac-delta potential Particle in one-dimensional slowly varying potentials Linear harmonic oscillator potential Pöschl-Teller potential Morse potential Position expectation values for various potentials Momentum expectation value for a three-dimensional slowly varying spherically symmetric potential The Hydrogen atom A discussion on Heisenberg's equation of motion and Ehrenfest theorem Conclusion References

0704.0376

Environmental noise reduction for holonomic quantum gates

Environmental noise reduction for holonomic quantum gates Daniele Parodi,1,2 Maura Sassetti,1,3 Paolo Solinas,4 and Nino Zangh̀ı1,2 1 Dipartimento di Fisica, Università di Genova, Genova, Italy 2 Istituto Nazionale di Fisica Nucleare (Sezione di Genova), Genova, Italy 3 INFM-CNR Lamia Via Dodecaneso 33, 16146 Genova, Italy 4 Laboratoire de Physique Théorique de la Matière Condensée, Université Pierre et Marie Curie, Place Jussieu, 75252 Paris Cedex 05, France (Dated: October 27, 2018) We study the performance of holonomic quantum gates, driven by lasers, under the effect of a dissipative environment modeled as a thermal bath of oscillators. We show how to enhance the performance of the gates by suitable choice of the loop in the manifold of the controllable parameters of the laser. For a simplified, albeit realistic model, we find the surprising result that for a long time evolution the performance of the gate (properly estimated in terms of average fidelity) increases. On the basis of this result, we compare holonomic gates with the so-called Stimulated Raman adiabatic passage (STIRAP) gates. PACS numbers: 03.67.Lx I. INTRODUCTION The major challenge for quantum computation is posed by the fact that generically quantum states are very del- icate objects quite difficult to control with the required accuracy—typically, by means of external driving fields, e.g., a laser. The interaction with the many degrees of freedom of the environment causes decoherence; more- over, errors in processing the information may lead to a wrong output state. Among the approaches aiming at overcoming these dif- ficulties are those for which the quantum gate depends very weakly on the details of the dynamics, in particu- lar, the holonomic quantum computation (HQC) [1] and the so-called Stimulated Raman adiabatic passage (STI- RAP) [2, 3, 4]. In the latter, the gate operator is obtained acting on the phase difference of the driving lasers dur- ing the evolution, while in the former the same goal is achieved by exploiting the non-commutative analogue of the Berry phase collected by a quantum state during a cyclic evolution. Concrete proposals have been put for- ward, for both Abelian [5, 6] and non-Abelian holonomies [7, 8, 9, 10, 11, 12]. The main advantage of the HQC is the robustness against noise deriving from a imperfect control of the driving fields [13, 14, 15, 16, 17, 18, 19, 20]. In a recent paper [21] we have shown that the dis- turbance of the environment on holonomic gates can be suppressed and the performance of the gate optimized for particular environments (purely superohmic thermal bath). In the present paper we consider a different sort of optimization, which is independent of the particular nature of the environment. By exploiting the full geometrical structure of HQC, we show how the performance of a holonomic gate can be enhanced by a suitable choice of the loop in the man- ifold of the parameters of the external driving field: by choosing the optimal loop which minimizes the “error” (properly estimated in terms of average fidelity loss). Our result is based on the observation that there are different loops in the parameter manifold producing the same gate and, since decoherence and dissipation crucially depend on the dynamics, it is possible to drive the system over trajectories which are less perturbed by the noise. For a simplified, albeit realistic model, we find the surprising result that the error decreases linearly as the gating time increases. Thus the disturbance of the environment can be drastically reduced. On the basis of this result, we compare holonomic gates with the STIRAP gates. In Sec. II the model is introduced and the explicit expression of the error is derived. In Sec. III we find the optimal loop, calculate the error, make a comparison with other approaches, and briefly sketch how to treat a different coupling with the environment. II. MODEL The physical model is given by three degenerate (or quasidegenerate) states, |+〉, |−〉, and |0〉, optically con- nected to another state |G〉. The system is driven by lasers with different frequencies and polarizations, acting selectively on the degenerate states. This model describes various quantum systems interacting with a laser radia- tion, ranging from semiconductor quantum dots, such as excitons [12] and spin-degenerate electron states [3], to trapped ions [8] or neutral atoms [7]. The (approximate) Hamiltonian modeling the effect of the laser on the system is (for simplicity, ~ = 1) [8, 12] H0(t) = j=+,−,0 ǫ|j〉〈j|+(e−iǫtΩj(t)|j〉〈G|+H.c) , (1) where Ωj(t) are the timedependent Rabi frequencies de- http://arxiv.org/abs/0704.0376v2 pending on controllable parameters, such as the phase and intensity of the lasers, and ǫ is the energy of the degenerate electron states. The Rabi frequencies are modulated within the adiabatic time tad, (which coin- cides with the gating time), to produce a loop in the pa- rameter space and thereby realize the periodic condition H0(tad) = H0(0). The Hamiltonian (1) has four time dependent eigen- states: two eigenstates |Ei(t)〉 , i = 1, 2, called bright states, and two eigenstates |Ei(t)〉, i = 3, 4, called dark states. The two dark states have degenerate eigen- value ǫ and the two bright states have timedependent energies λ±(t) = [ǫ ± ǫ2 + 4Ω2(t)]/2 with Ω2(t) = i=±,0 |Ωi(t)| 2 [22]. The evolution of the state is generated by Ut = Te dt′H0(t ′), (2) where T is the time-ordered operator. In the adiabatic approximation, the evolution of the state takes place in the degenerate subspace generated by |+〉, |−〉, and |0〉. This approximation allows to separate the dynamic con- tribution and the geometric contribution from the evolu- tion operator. Expanding Ut in the basis of instantaneous eigenstates of H0(t) (the bright and dark states), in the adiabatic approximation, we have Ut ∼= ′)dt′ |Ej(t)〉〈Ej(t)| Ut, (3) where Ut = Te dτV (τ), (4) here V is the operator with matrix elements Vij(t) = 〈Ei(t)|∂t|Ej(t)〉. The unitary operator Ut plays the role of timedependent holonomic operator and is the funda- mental ingredient for realizing complex geometric trans- formation whereas ′)dt′ |Ej(t)〉〈Ej(t)| is the dynamic contribution. Consider Ut for a closed loop, i.e., for t = tad, U = Utad . (5) If the initial state |ψ0〉 is a superposition of |+〉 and |−〉, then U|ψ0〉 is still a superposition of the same vectors (in general, with different coefficients)[12]. Thus the space spanned by |+〉 and |−〉 can be regarded as the “logi- cal space” on which the “logical operator” U acts as a “quantum gate” operator. Note that for t < tad, Ut|ψ0〉 has, in general, also a component along |0〉. However, as it is easy to show [22], at any instant t < tad, Ut|ψ0〉 can be expanded in the twodimensional space spanned by the dark states |E3(t)〉 and |E4(t)〉. It is important to observe that U depends only on global geometric fea- tures of the path in the parameter manifold and not on the details of the dynamical evolution [1, 12]. To construct a complete set of holonomic quantum gates, it is sufficient to restrict the Rabi frequencies Ωj(t) in such a way that the norm Ω of the vector ~Ω = [Ω0(t),Ω+(t),Ω−(t)] is time independent and the vector lies on a real three dimensional sphere [8, 12]. We parametrize the evolution on this sphere as Ω+(t) = sin θ(t) cos φ(t), Ω−(t) = sin θ(t) sin φ(t) and Ω0(t) = cos θ(t) with fixed initial (and final) point in θ(0) = 0, the north pole By straightforward calculation we obtain the analytical expression for V (t) in eq. (4), V (t) = iσy cos[θ(t)]φ̇(t), where σy is the usual Pauli matrix written in the basis of dark states. Thus, the oper- ator (4) becomes Ut = cos[a(t)] − iσy sin[a(t)], here a(t) = dτφ̇(τ) cos θ(τ). Accordingly, the logical op- erator U (5) is U = cos a− iσy sin a, (6) where a = a(tad) = ∫ tad dτφ̇(τ) cos θ(τ) (7) is the solid angle spanned on the sphere during the evolu- tion. Note that the are many paths on the sphere which generate the same logical operator U , and span the same solid angle a. In a previous work we have studied how interaction with the environment disturbs the logical operator U [21]. The goal of the present paper is to analyze whether and how such a disturbance can be minimized for a given U . To this end, we model the environment as a thermal bath of harmonic oscillators with linear coupling between system and environment [23]. The total Hamiltonian is H = H0(t) + α + cαxαA), (8) where A is the system interaction operator called, from now on, noise operator. We now consider the time evolution of the reduced density matrix of the system, determined by the Hamil- tonian (8). We rely on the standard methods of the “mas- ter equation approach,” with the environment treated in the Born approximation and assumed to be at each time in its own thermal equilibrium state at temperature T . This allows to include the effect of the environment in the correlation function (kB = 1) g(τ) = cos(ωτ) − i sin(ωτ) Here the spectral density is J(ω) = δ(ω − ωα), (10) at the low frequencies regimes, is proportional to ωs, with s ≥ 0, i.e., s = 1 describes a Ohmic environments, typ- ical of baths of conduction electrons, s = 3 describes a super-Ohmic environment, typical of baths of phonons [21, 24]. The asymptotic decay of the real part of g(τ) de- fines the characteristic memory time of the environment. Denoting with ρ̃(t) the time evolution of the reduced den- sity matrix of the system in the interaction picture, e.g., ρ̃(t) = U t ρUt, one has [24] ρ̃(tad) = ρ(0) + dτ{g(τ )[Ãà ρ̃(t− τ )− à ρ̃(t− τ )Ã] + g(−τ )[ρ̃(t− τ )à Ã− Ãρ̃(t− τ )à ]. (11) Here à and Ã′ stand for Ã(t) and Ã(t−τ), with the tilde denoting the time evolution in the interaction picture. In quantum information the quality of a gate is usually evaluated by the fidelity F , which measures the closeness between the unperturbed state and the final state, F = 〈ψ0(0)|U †ρ(tad)U|ψ0(0)〉, (12) where |ψ0(0)〉 is the initial state, and ρ(tad) = U ρ̃(tad)U is the reduced density matrix in the Schrödinger picture starting from the initial condition ρ(0) = |ψ0(0)〉〈ψ0(0)|. The average error is defined as the average fidelity loss, i.e., δ =< 1−F >= 1− < 〈ψ0(0)|ρ̃(tad)|ψ0(0)〉 >, (13) where < · · · > denotes averaging with respect to the uniform distribution over the initial state |ψ0(0)〉. The right-handside of Eq. (13) can be computed by the following steps: (1) solving Eq. (11) in strictly second order approxima- tion; this approximation corresponds to replace ρ̃(t − τ) with ρ(0); (2) using the adiabatic approximation U(t − τ, t) ≈ exp(iτH0(t)); (3) expanding the scalar product in Eq. (13) with respect to a complete orthonormal basis {|ϕn(t)〉}, n = 1, 2, 3, orthogonal to |ψ0(t)〉. In this way, one obtains ∫ tad dt G(t)|〈ψ0(t)|A|ϕn(t)〉| , (14) where G(t) = Re[g(τ)] cos(ω0nt) + Im[g(τ)] sin(ω0nt)) Here, ω0n = ω0−ωn are the energy differences associated to the transition ψ0 ↔ φn, with ω0 = ǫ, ω1 = λ+, ω2 = λ−, and ω3 = ǫ. The interaction between system and environment is ex- pressed by the noise operator A in Eq. (8). We shall now make the assumption that A = diag{0, 0, 0, 1} in the |G〉, |±〉, and |0〉 basis. In this case the transition between de- generate states are forbidden, however the noise breaks their degeneracy, shifting one of them. In spite of its sim- ple form, this A is nevertheless a realistic noise operator for physical semiconductor systems [4]. III. MINIMIZING THE ERROR The problem can be stated in the following way: given the noise operator A and the logical operator U , find a path on the parameter space (the surface of the sphere, described above) which minimizes the error δ. The total error δ, given by Eq. (14), can be decom- posed as δ = δtr + δpd, (16) where the transition error, δtr, is the contribution to the sum of the nondegenerate states (ω0n 6= 0) and the pure dephasing error δpd is the contribution of the degenerate states (ω0n = 0). Thus δpd = ∫ tad sin(ωt) sin2 2a(t) sin4 θ(t) (17) δtr = n=+,− 1 + [(λn − ǫ)/Ω]2 ∫ tad sin2 2θ(t)dt, where Γ0n = J(|ω0n|) |ω0n| − sgn(ω0n) correspond to the transition rates calculated by standard Fermi golden rules, supposing, as usual, G(t) ≈ G(∞) for g(τ) strongly peaked around τ = 0. In the following we define for simplicity n=+,− 1 + [(λn − ǫ)/Ω]2 Since we are interested at long time evolution, we start discussing the transition error which dominates in this regime [4, 25]. A. Transition rate As explained in Sec. II, the holonomic paths are closed curves on the surface of the sphere which start from the north pole. It turns out that the curve minimizing δtr can be found among the loops which are composed by a simple sequence of three paths (see the Appendix): evo- lution along a meridian (φ = const), evolution along a ����� ����� ��������� FIG. 1: The error δtr versus θM for two different a values: a = π/2 (dashed line) and a = π/4 (full line) correspond to NOT and Hadamard gate, respectively. parallel (θ = const) and a final evolution along a merid- ian to come back to the north pole. The error δtr in (18), depends on a given by Eq. (7), θM (the maximum angle spanned during the evolution along the meridian), ∆φ (the angle spanned along the parallel), and angular velocity v. We allow ∆φ ≥ 2π which corresponds to cover more than one loop along the parallel. The velocity along the parallel is v(t) = φ̇(t) sin θ and that along the meridian is v(t) = θ̇(t). In the following we assume that v is constant, and it cannot exceed the maximal value of vmax, fixed by adiabatic condition vmax ≪ Ω. The parameters a, θM , and ∆φ are connected by the relation a = ∆φ(1 − cos θM ). The error δtr is then δtr = δ tr + δ tr, (20) where δMtr = sin 4θM is the contribution along the meridian and δPtr = K sin θM sin 2 2θM 1− cos θM is the contribution along the parallel. In Fig. 1 δtr is plotted for a = π/2 and a = π/4 (corre- sponding to NOT and Hadamard gate, respectively) as a function of θM . One can see that δtr has a local minimum for θM = π/2 and a global minimum for θM = 0 where the error vanishes. This suggests that the best choice is to take θM as small as possible. It is interesting to consider the dependence of δtr also on the evolution time tad. For simplicity, we set the ve- locity v = vmax. In this case, changing θM (and then ∆φ) corresponds to a change in the evolution time. We obtain θM = arccos , (23) 5 10 15 20 25 30 vmaxtad FIG. 2: The error δtr versus vmaxtad for two different a val- ues: a = π/2 (dashed line) and a = π/4 (full line) correspond to NOT and Hadamard gate, respectively. The dotted-dashed line shows the value of the error at θ = π/2. The circles show the critical value of vmaxtad above which the best loop is the one with the minimal θM . where (vmaxtad) 2 + a2 . (24) Using these relations, δMtr and δ tr, given by (21) and (22) become functions of tad, vmax, and a. Note that m measures the space covered along the parallel, in fact ∆φ = 2πm. In Fig. 2 we see the behavior of δtr as a function of vmaxtad. The first minimum for both curves corre- sponds to θM = π/2, then the curves for long tad de- crease asymptotically to zero corresponding to the region in which θM → 0. In this regime we have δtr ∝ 1/tad which is drastically different from the results obtained with other methods where δtr ∝ tad, (see Refs [4, 25] and below Sec. III C). It should be observed that this surprising results is a merit of holonomic approach which allows to choose the loop in the parameter space, with- out changing the logical operation as long as it subtends the same solid angle. Observe that small θM and long tad mean large value of m, i.e., multiple loops around the north pole. Figure 2 shows that, for a given gate, there is a criti- cal value kc of vmaxtad which discriminate between the choice of θM (e.g., k = 6 for the Hadamard gate and k = 25 for the NOT gate). For vmaxtad < kc the best choice for the loop is θM = π/2; For vmaxtad > kc the best choice is the value of θM determined by eq. (23) and (24). Note that the region vmaxtad > kc is accessible with physical realistic parameters [12]. For example, if we choose the laser intensity Ω = 20 meV and vmax = Ω/50 (for which values the nonadiabatic transitions are forbid- den), the critical parameter corresponds to the critical time of 15 ps for the Hadamard gate and 42 ps for the NOT gate. B. Pure Dephasing Until now we have ignored the pure dephasing effect because we have assumed that it is negligible in com- parison with the transition error for long evolution time. Now, we check that the pure dephasing error contribu- tion can indeed be neglected. We can write the pure dephasing error using Eq. (17) and splitting to parallel and meridian part as δPpd = ∫ tad Q[a(t)] sin ωt sin4 θM (25) δMpd = Q[a(t)] sin ωt sin4(vmaxt) + sin vmaxt ,(26) where Q[a(t)] = 1 + 1/2 sin2[2a(t)]. To estimate δpd we assume that tad is longer with re- spect to the characteristic time of the bath. Remember- ing that J(ω) ∝ ωs, the pure dephasing error behavior along the parallel part at the temperature T is δPpd ∝ , T ≪ 1/tad , T ≫ 1/tad while the along meridian is δMpd ∝ . (28) Then, we can conclude that the pure dephasing can al- ways be neglected for long time evolution because it de- creases faster than the transition error. C. Comparison between HQC and STIRAP We make a comparison between holonomic quantum computation (HQC) and the STIRAP procedure which is an analogous approach to process quantum information. The STIRAP procedure ([2, 4]) is, in its basic points, very similar to the holonomic information manipulation. The level spectrum, the information encoding, the evolu- tion produced by adiabatic evolving laser are exactly the same. The fundamental difference is that in STIRAP the dynamical evolution is fixed (we must pass through a precise sequence of states) and then the correspond- ing loop in the parameter space is fixed. In particular, we go from the north pole to the south pole and back to the north pole along meridians. Since the loop, as in our model, is a sequence of meridian-parallel-meridian path, we can calculate the error and make a direct com- parison. In this case, the transition error results propor- tional to δtr ∝ tad and grows linearly in time while for HQC δtr ∝ 1/tad. Therefore, the HQC is fundamentally the favorite for long application times with respect to the STIRAP ones. Moreover, we can show that the freedom in the choice of the loop allows us to construct HQC which perform better than the best STIRAP gates. In Ref. [4] the mini- mum error (not depending on the evolution time) for STI- RAP was obtained reaching a compromise between the necessity to minimize the transition, pure dephasing error and the constraint of adiabatic evolution. With realistic physical parameters [21] (J(ω) = kω3e(−ω/ωc) , Ω = 10 meV, ǫ = 1eV, vmax = Ω/50, k = 10 −2(meV)−2, ωc = 0.5 meV and for low temperature), the total minimum error in Ref. [4] is δstirap = 10 −3. With the same parame- ters, we still have the possibility to increase the evolution time in order to reduce the environmental error. How- ever, for evolution time tad = 50 ps we obtain a total error δ = 1.5× 10−4 for the NOT gate and δ = 4× 10−5 for the Hadamard gate, respectively. As can be seen, the logical gate performance is greatly increased. D. More general noise Until now we have discussed the possibility to minimize the environmental error by choosing a particular loop in the parameter sphere but the structure of the error func- tional clearly depends on the system-environment inter- action. Then one might wonder if the same approach can be used for a different noise environment. For this reason, we now briefly analyze the case of noise matrix in the form A = diag{0, 1, 0,−1}. Again, for long evolution we can neglect the contribution of the pure de- phasing and focus on the transition error. In this case the interesting part of the error functional takes the form δtr = K[( sin 2θ cos 2θ)2 + (sin θ sin 2φ)2]. (29) Even if the analysis in this case is much more com- plicated, it can be seen that δtr has an absolute mini- mum for θM = 0. The long time behavior is the same (δtr ∝ 1/tad) such that the results are qualitatively anal- ogous to the above ones: for small θM loops (or long evolution at fixed velocity) the holonomic quantum gate presents a decreasing error. Then even in this case it is possible to minimize the environmental error. IV. CONCLUSIONS In summary, we have analyzed the performance of holonomic quantum gates in the presence of environmen- tal noise by focusing on the possibility to have small errors choosing different loops in the parameter mani- fold. Due to the geometric dependence, we can imple- ment the same logical gate with different loops. Since different loops correspond to different dynamical evolu- tions, we have used this freedom to construct an evolu- tion through “protected” or “weakly influenced” states leading to good holonomic quantum gates performances. This allows to select (once that the physical parameter are fixed) the best loop which minimizes the environ- mental effect. (Note that this optimization procedure is rather independent of the details of the simple model we have considered and arguably, it could be extended to more complicated systems without any substantial modification.) We have shown that for long time evo- lutions the noise decreases as 1/tad while in the other cases it increases linearly with adiabatic time. We also have shown that the same features can be found with different kinds of noise suggesting the possibility to find a way to minimize the environmental effect in the pres- ence of any noise. These results open a new possibility for implementation of holonomic quantum gates to build quantum computation because they seem robust against both control error and environmental noise. Acknowledgment The autors thank E. De Vito for useful discussions. One of the authors (P. S.) acknowledges support from INFN. Financial support by the italian MIUR via PRIN05 and INFN is acknowledged. APPENDIX A: MINIMIZING THEOREM Let us consider the family Cn composed of the closed curves generated by a sequence of n paths along a parallel (θ = const) alternated with paths along a meridian (φ = const). We call Cn a generic curve in this family. For example, the family C1 contains all the closed curves com- posed by the sequence of path meridian-parallel-meridian while the family C2 contains the curves meridian-parallel- meridian-parallel-meridian. We argue that the closed curve minimizing the error in Eq. (18) can be found in the C1 family. First, we show that any closed curve in C2 spanning a solid angle a on the sphere can be replaced by a closed curve in C1 spanning the same angle and producing a smaller error. In analogous way any closed curve in C3 can be replaced by a closed curve in C2 with smaller error and so on. By induction we obtain that any closed curve in Cn can be replaced by a curve in C1 spanning the same solid angle but producing smaller error. Since the curve belonging to Cn can approximate any closed curve on the sphere, the best curve can be found in C1. The crucial point is to show that any curve in C2 can be replaced by a curve in C1. Let us consider a generic curve C2 in C2 spanning a solid angle a: composed by a seg- ment of a meridian (with θ going from 0 to θ1), a parallel (spanning a ∆φ1 angle), meridian (with θ : θ1 → θ2), a parallel (spanning a ∆φ2 angle), and finally a segment to the north pole along a meridian. Let us consider two closed curves C11 and C 1 in C1 subtending the same solid angle a with, respectively, θ1 and θ2 as maximum angle spanned during the evolution along the meridian. First we analyze (20) along the meridian. Without losing gen- erality, we can take θ1 < θ2; it is clear from Eq. (21) that the value of δtr along the meridian for C 1 is smaller that for C21 : δ . We note from the Eq. (21), suitable extended to C2, that the two paths along the meridians depends only on θ2 and then produce the same error of C21 , < δMC2 = δMC2 . (A1) The difference between the contribution along the par- allel is δPC2−δ = ∆φ1 sin θ1 sin 2 2θ1− 1− cos θ1 1− cos θ2 sin θ2 sin 2 2θ2 δPC2−δ = ∆φ2 sin θ2 sin 2 2θ2− 1− cos θ2 1− cos θ1 sin θ1 sin 2 2θ1 Analysis of the positivity of the quantities given by Eqs. (A2) and (A3) shows that δPC2 cannot be at the same time smaller than δP and δP . In fact, there are two possibilities: If δPC2 > δ , from Eq. (A1) and (A3), δC2 = δ + δPC2 > δ + δPC1 = δC1 , (A4) and the best closed curve is C11 . If δ , from Eqs. (A1) and (A2), δC2 = δ + δPC2 > δ + δPC2 = δC2 , (A5) and the best closed curve is C21 . In the same way it can be shown that any closed curve in C3 can be replaced by a closed curve in C2 with smaller error. [1] P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999). [2] Z. Kis and F. Renzoni, Phys. Rev. A 65, 032318 (2002). [3] F. Troiani, E. Molinari, and U. Hohenester, Phys. Rev. Lett. 90, 206802 (2003). [4] K. Roszak, A. Grodecka, P. Machnikowski, and T. Kuhn, Phys. Rev. B 71, 195333 (2005). [5] J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, Na- ture (London) 403, 869 (2000). [6] G. Falci, R. Fazio, G. M. Palma, J. Siewert, and V. Ve- dral, Nature (London) 407, 355 (2000). [7] R. G. Unanyan, B.W. Shore, and K. Bergmann, Phys. Rev. A 59, 2910 (1999). [8] L.-M. Duan, J.I. Cirac, and P. Zoller, Science 292, 1695 (2001). [9] L. Faoro, J. Siewert, and R. Fazio, Phys. Rev. Lett. 90, 028301 (2003). [10] I. Fuentes-Guridi, J. Pachos, S. Bose, V. Vedral, and S. Choi, Phys. Rev. A 66, 022102 (2002). [11] A. Recati, T. Calarco, P. Zanardi, J. I. Cirac, and P. Zoller, Phys. Rev. A 66, 032309 (2002). [12] P. Solinas, P. Zanardi, N. Zangh̀ı, and F. Rossi, Phys. Rev. B 67, 121307(R) (2003). [13] A. Carollo, I. Fuentes-Guridi, M. F. Santos, and V. Ve- dral, Phys. Rev. Lett. 90, 160402 (2003). [14] G. De Chiara and G. M. Palma, Phys. Rev. Lett. 91, 090404 (2003). [15] A. Carollo, I. Fuentes-Guridi, M. F. Santos, and V. Ve- dral, Phys. Rev. Lett. 92, 020402 (2004). [16] V.I. Kuvshinov and A.V. Kuzmin, Phys. Lett. A, 316, 391 (2003). [17] P. Solinas, P. Zanardi, and N. Zangh̀ı, Phys. Rev. A 70, 042316 (2004). [18] S.-L. Zhu and P. Zanardi, Phys. Rev. A 72, 020301(R) (2005). [19] G. Florio, P. Facchi, R. Fazio, V. Giovannetti, and S. Pascazio, Phys. Rev. A 73, 022327 (2006). [20] I. Fuentes-Guridi, F. Girelli, and E. Livine, Phys. Rev. Lett. 94, 020503 (2005) [21] D. Parodi, M. Sassetti, P. Solinas, P. Zanardi, and N. Zangh̀ı, Phys. Rev. A 73, 052304 (2006). [22] The explicit expression for the bright states is |E1〉 = (Ω|e〉 + Ωi|i〉) and |E2〉 = (−Ω|e〉 + Ωi|i〉); for the dark states is |E3〉 = 1/(Ω |Ω+|2 + |Ω−|2)[Ω0(Ω+|+〉 + Ω−|−〉) − (Ω2 − |Ω0| 2)|0〉]) and |E4〉 = 1/ |Ω+|2 + |Ω−|2[Ω−|+〉− Ω+|−〉]. [23] A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46, 211 (1981). [24] U. Weiss, Quantum dissipative systems (World Scientific, Singapore, 1999). [25] R. Alicki, M. Horodecki, P. Horodecki, R. Horodecki, L. Jacak, and P. Machnikowski, Phys. Rev. A 70, 010501(R) (2004).

0704.0377

The lifetime of unstable particles in electromagnetic fields

arXiv:0704.0377v3 [hep-ph] 22 Dec 2008 The lifetime of unstable particles in electromagnetic fields Daniele Binosi1 and Vladimir Pascalutsa1, 2 1ECT* Trento, Villa Tambosi, Villazzano, I-38050 TN, Italy 2Institut für Kernphysik, Johannes Gutenberg Universität, Mainz D-55099, Germany (Dated: October 30, 2018) Abstract We show that the electromagnetic moments of unstable particles (resonances) have an absorptive contribution which quantifies the change of the particle’s lifetime in an external electromagnetic field. To give an example we compute here the imaginary part of the magnetic moment for the cases of the muon and the neutron at leading order in the electroweak coupling. We also consider an analogous effect for the strongly-decaying ∆(1232) resonance. The result for the muon is Imµ = eG2Fm 3/768π3, with e the charge and m the mass of the muon, GF the Fermi constant, which in an external magnetic field of B Tesla give rise to the relative change in the muon lifetime of 3 × 10−15 B. For neutron the effect is of a similar magnitude. We speculate on the observable implications of this effect. PACS numbers: 13.40.Em, 13.35.-r, 12.15.Lk, 23.40.-s http://arxiv.org/abs/0704.0377v3 I. INTRODUCTION The electromagnetic (e.m.) moments of a particle are among the few fundamental quan- tities which describe the particle properties and as such have thoroughly been studied. The most renowned examples are the magnetic moments of the electron and the muon which have been measured to unprecendented accuracy and yielded a number of physical insights, see[1] for recent reviews. What is far lesser known is that the e.m. moments of unstable particles are complex numbers in general [2, 3]. Their imaginary part reflects, of course, the unstable nature of the particle, however, the precise interpretation has been missing. In this paper we work out the relation, suggested first by Holstein [4], which should exist between the imaginary part of the magnetic moment and the effect of an external magnetic field on particle’s lifetime. The argument for such a relation is very simple. The (self-)energy of the particle with a lifetime τ has an absorptive part, which has an interpretation of the width Γ = 1/τ . The particle’s magnetic moment ~µ in the presence of magnetic field ~B induces the change in the energy: −~µ · ~B. The latter contribution can then also change the width, provided the magnetic moment has an absorptive part (Imµ 6= 0). The decay properties of unstable particles, such as muon or neutron are extremely well studied and are widely used for the precise determination of the Standard Model parameters[5, 6]. There are also a plethora of studies of how these particles behave in e.m. fields. A well-known example is the search for the neutron’s electric dipole moment[7]. In view of these studies it is compelling to investigate how the decay properties of unstable particles may be affected by e.m. fields. The lifetime of unstable quantum-mechanical systems is known to be affected by an e.m. field. Positronium provides a textbook example[8], where the effect arises due to the admixture of para- (S = 0) and ortho- (S = 1) positronium states with orbital momentum l = 0 by the magnetic field interacting with the magnetic moments of the constituents. As the result, already in the field of B = 0.2 Tesla, the lifetime of ortho-positronium decreases by almost a factor of 2. It is far from obvious how the same kind of an effect can arise for an elementary unstable particle, e.g., the muon. The above-mentioned relation between the imaginary part of the magnetic moment and the lifetime change may, therefore, provide us with both an interpre- µ νµ µ FIG. 1: The muon self-energy contributing to its decay width. tation for the imaginary part of the magnetic moment and the means to compute the effect of the lifetime change. In the following we examine in detail the case of the muon, compute the leading contri- bution to Imµ and the corresponding effect on the lifetime. Then we will briefly discuss the cases of the neutron and of the ∆-resonance. II. MUON DECAY (µ → e νeνµ) The leading contribution to the muon decay width arises at two-loop level, see Fig. 1. For our purposes, the W propagators in this graph can safely be assumed to be static — Fermi theory. We also neglect the mass of the electron in the loops, since it leads to an under- percent correction of O(me/m); here and in what follows, m is the muon mass. The graphs with other Standard Model fermions (e.g., quarks) in the loops need not to be considered here, because they cannot give any contribution to the muon width. Using dimensional regularization, we compute this graph in d = 4 − 2ǫ dimensions (in the limit ǫ → 0+),[14] Σ (p/) = (2π)d 2γµ(1− γ5) (p/− k/) γν (p− k)2 + iε Πµν(k). (1) where MW is the W -boson mass, g = |e|/ sin θW is the electroweak coupling related to the Fermi constant by GF/ 2 = g2/8M2W , e is the charge, θW is the Weinberg angle, and Πµν(k) = d (d− 2) (4π)d/2(d− 1) Γ (ǫ)Γ (1− ǫ)2 Γ (2− 2ǫ) × (−k2)−ǫ k2gµν − kµkν is the one-loop correction to the polarization tensor of the W boson. The decay width can then be found as Γ = −2 ImΣ (p/ = m). A brief calculation shows that the self-energy has the following form: Σ (p/) = v(s) p/ (1− γ5) , (3) with s = p2 and the scalar function v given by: v(s) = − G2F s 3(4π)4 − 2γE − 2 ln +O(ǫ) , (4) where γE = −Γ ′(1) is the Euler’s constant. The absorptive part of this function stems from the logarithm [ln(−s− iε) = ln s− iπ, for s > 0]: Im v(s) = − G2F s 384π3 . (5) Terefore, the width is Γ = −2m Im v(m2), and the muon lifetime: τ = 192π3/(G2Fm 5) ≃ 2.187× 10−6 sec, (6) This result is of course long-known due to the seminal work of Feynman and Gell-Mann on Fermi theory[9]. It is in a percent agreement with the experimental value[5]: τ (exp) = (2.19703± 0.00004) 10−6 sec, (7) The discrepancy is due the neglect of the electron mass and some radiative corrections, c.f.[10]. We now investigate the influence of the e.m. field on the leading contribution given by Eq. (6). Let us denote by Σ (x, y;Aµ) the self-energy in the presence of an external e.m. field Aµ. It is obtained by minimal substitution (∂µ → ∂µ − ieAµ) of the derivatives of all charged fields into the self-energy of Fig. 1. Expanding in the e.m. coupling, we obtain: Σ [x, y;Aµ] = Σ (i∂/ x) δ4(x− y) dz Λµ(x, y; z)Aµ(z) +O(e 2A2), (8) where Σ (i∂/) is the already computed self-energy in the vacuum, while Λ is the e.m. vertex correction of Fig. 2, with static W ’s. Denoting p (p′) the 4-momentum of the initial (final) muon and assuming the on-shell situation (p2 = p′ = p · p′ = m2), the vertex correction has in the momentum space the following general form: Λµ(p′, p) = e F γµ +G (p+ p′)µ + FA γ , (9) where F , G and FA are complex numbers. Note that eF/2m is the correction to the magnetic moment, and eF + eG is the correction to the electric charge. The Ward-Takahashi (WT) identity: (p′ − p) · Λ(p′, p) = e [Σ (p/)− Σ (p/′)] (10) µ µνµ FIG. 2: Electromagnetic correction to the muon decay. with the self-energy in Eq. (3) leads to the following conditions: F +G = −v(m2)− 2m2v′(m2), FA = v(m2) . (11) Therefore, the term FA is in fact necessary by the e.m. gauge invariance. The γ5 terms, in both self-energy and the vertex, are shown to vanish when summing over all the fermions in Standard Model[11]. However, this does not happen for the imaginary part because the heavier fermions do not contribute. The expression for the graph in Fig. 2 is (in Fermi theory) given by Λµ(p′, p) = − 64M4W (2π)d (2π)d γβ(1− γ5) 2γα(1− γ5) (p/′ − k/1) γµ (p/− k/1) γβ (k/1 − k/2) (k1 − k2)2 (p− k1)2 (p′ − k1)2 After a lengthy calculation we obtain the following result: ImF = 384π3 , ImG = , ImFA = − 384π3 , (13) hence satisfying the gauge-invariance conditions Eq. (11), for Im v given by Eq. (5). We would like to emphasize here that, of course, not only the magnetic moment, but also the charge operator receives an imaginary contribution, equal to e Im(F +G). However, through the WT identity, this contribution is completely fixed by the momentum dependence of the self-energy, and therefore is not independent. The same holds for FA. We thus discuss only the effect of the absorptive part of the magnetic moment, here given by Imµ = e ImF/2m = eG2Fm 3/768π3. The energy of the magnetic moment interacting with the magnetic field is equal to −µBz, with Bz being the projection of the field along the muon spin. Then the total energy, in the muon rest-frame, is given by: m − (i/2)Γ− µBz. We thus deduce that the absorptive part yields the following change in the muon width: ∆Γ = 2 ImµBz = 192π3 Bz , (14) while the change in the lifetime is ∆τ = −(∆Γ/Γ) τ , for ∆Γ/Γ ≪ 1. Given this result, we conclude that the positively-charged muons live shorter (longer) in a uniform magnetic field if their spin is aligned along (against) the field. For the relative change in the width we find: |eBz| <∼ 3× 10−15B T−1, (15) where B is the strength of the field in Tesla. Therefore, in moderate magnetic fields the change in the muon lifetime is tiny, well beyond the present experimental accuracy (which is at the ppm level). We will dwell on this more in the concluding part of the paper, but for now we turn to a more technical issue. It is interesting to observe that the result of Eq. (13), can simply be obtained by the minimal substitution into Eq. (3), rather than into the electron propagator in Eq. (1). To show this we go to coordinate space and hence write the self-energy as Σ (x, y) = Σ (i∂/ ) δ(x − y). The minimal substitution to the first order in e leads to the following vertex correction: Λ̃µ(x, y; z) = − δ/δAµ(z)Σ (i∂/ + eA/ ) δ(x− y) |A=0 . (16) Note that in general this is different from the vertex function in Eq. (8), since in the latter the minimal substitution is performed also in the internal lines. The general form of Eq. (9), of course, applies here as well, but now the scalar functions are completely specified by the self-energy: F̃ = −v(m2), G̃ = −2m2 v′(m2), F̃A = v(m2) . (17) Substituting the explicit form of Im v, we see that this method unambiguously leads to exactly the same result [Eq. (13)] as the full calculation. We emphasize though, that this method cannot always work (see, e.g., Ref.[12]), as will also be clear from the following examples. Nevertheless, it is worthwhile to investigate this method further, since knowing whether it is applicable a priori can enormously facilitate the calculations. III. NEUTRON DECAY AND THE ∆-RESONANCE We consider now the neutron β-decay. Assuming exact V − A interaction (gA = 1) and neglecting the electron mass (but not the proton mass, mp), the corresponding two-loop self-energy can still be written in the form of Eq. (3). We introduce δ = (s −m2p)/2s and treat it as a small parameter, since in the physical case (where s = m2n), δ ≃ 1.293 × 10−3. A simple calculation then yields: Im v(s) = −G F |Vud|2 s2 δ5, (18) where Vud is the relevant quark-mixing (CKM) matrix element. We note in passing that this result leads to the lifetime of τn ≈ 622 sec, to be compared with the experimental value of 886 sec. This 30% disagreement is largely due to the fact that in reality the axial coupling gA deviates from 1. However, for our order-of-magnitude estimate this discrepancy is unimportant. What is important is that the derivative of the self-energy is enhanced by one power of Im v′(s) = −(GF |Vud|) s δ4 . (19) and this opens the possibility for the enhancement of the effect in the lifetime. Namely, the relative change in the neutron width then goes as |∆Γn| ∼ µN |Bz| mn −mp <∼ 3× 10−14B T−1, (20) where µN ≃ 3.15× 10−14 MeV T−1 is the nuclear magneton. A more precise analysis of this effect for the neutron is beyond the scope of this paper. We focus instead on the example of the ∆-resonance, where such an enhancement will be shown to be even more dramatic, at least qualitatively. The ∆ resonance decays strongly into the pion and nucleon, ∆ → πN , and the cor- responding self-energy, to leading order in chiral effective-field theory, yields the following result for the absorptive part[3]: ImΣ∆(p/) = −23πλ 3C2 (α p/+mN ) , (21) where the isospin symmetry is assumed, e.g., mp = mn = mN . The constant C = hAm∆/8πfπ ≃ 1.5, where hA represents the πN∆ coupling and is fitted to the empira- cal width of the ∆, fπ ≃ 93 MeV is the pion-decay constant, and m∆ = 1232 MeV is the ∆ π(b)(a) FIG. 3: The leading chiral-loop correction to the magnetic moment of the ∆. mass. For simplicity we neglect the pion mass (i.e., take the chiral limit). Then, in Eq. (21), λ = (s−m2N)/2s, α = 1− λ. For s = m∆, λ ≈ (m∆ −mN)/mN ∼ 1/3 is a small parameter in the chiral effective-field theory with ∆’s (see Ref.[13] for a recent review), and will so be treated here too. The absorptive part of the magnetic dipole moment of the ∆ arises at this order from graphs in Fig. 3. These graphs, computed in Ref.[3], in the chiral limit yield the following result (upto λ4 terms): ImF (a) = 4πC2(λ− 3λ2 + 43 λ3) , ImG(a) = 4πC2(−λ+ 4λ2 − 71 λ3) , ImF (b) = 4πC2(λ2 + 1 λ3) , (22) ImG(b) = −32 πC2λ3 , where F and G correspond with the decomposition in Eq. (9), with the superscript referring to the corresponding graphs in Fig. 3; FA is absent in this case, of course. First of all, we observe that this result satisfies the WT conditions, Eq. (11), for each of the four charge states of the ∆, ∆++ : Im [F (a) +G(a) + F (b) +G(b)] = −2 ImΣ ′∆ , ∆+ : Im [1 (F (a) +G(a)) + 2 (F (b) +G(b))] = − ImΣ ′∆ , ∆0 : Im [−1 (F (a) +G(a)) + 1 (F (b) +G(b))] = 0 , (23) ∆− : − Im [F (a) +G(a)] = ImΣ ′∆ , where Σ ′∆ = ∂/∂p/Σ∆(p/)|p/=m∆, and hence ImΣ ′∆ = 4πC2(−λ2 + 73λ At the same time, the ‘naive’ minimal-substitution procedure [Eq. (16)], that happens to work for the muon, fails here miserably. It would predict that the magnetic moment contribution would go with the same power as the self-energy [Eq. (17)], which for the absorptive part means Imµ ∼ ImΣ (m∆) ∼ λ3. In reality it goes as λ. E.g., for the ∆+: Imµ∆+ = (e/2m∆) Im[ F (a) + 2 F (b)] π µNC 2 λ+O(λ2). (24) The fact that the self-energy goes as λ3, while Imµ as λ has as a consequence the enhance- ment of the lifetime change in the magnetic field by two powers of λ. Quantitatively such enhancements of the lifetime change over the lifetime by the phase- space volume do not make much difference in the above examples. However, it shows that it might be useful to look for manifestations of the lifetime change in the medium where the phase-space volume can be varied. IV. CONCLUSIONS AND OUTLOOK We have examined her a concept of the ‘absorptive magnetic moment’ — an intrinsic property of an unstable particle, together with the width or the lifetime. It manifests itself in the change of the particle’s lefetime in an external magnetic field, see Eq. (25) below. We have computed this quantity for the examples of muon, neutron and ∆-resonance to leading order in couplings. In all the three cosidered cases the effect on the lifetime is tiny for normal magnetic fields: in a uniform field of 1 Tesla the change in the lifetime is of order of 10−13 percent, at most. In the case of the muon we have computed this effect to the leading order in the elec- troweak coupling; the change in the lifetime is ∆τ = −2 ImµBz τ 2 = −96π3eBz/(G2Fm7) , (25) or, numerically, |∆τ | <∼ 6× 10−21 (B/T) sec. A direct measurement of this effect is therefore beyond the present experimental precision. Nevertheless, it is worthwhile to investigate the effect of the magnetic field on the differential decay rates, with the hope that some asymmetries could show a significantly bigger sensitivity. A notable feature of this effect is that the relative change of the lifetime is inversely proportional to the phase space. It goes as (mn −mp)−1 in the neutron case, and as (m∆ − −2 in the ∆-resonance case. (The difference in power is apparently because the neutron decays solely into fermions while the ∆ has a boson in the decay product.) One can expect that in the conditions where the phase-space is significantly reduced, e.g. for the neutron in nuclear medium, the effect of the lifetime change may become measurable. Especially interesting would be to evaluate the manifestations of this effect in neutron star formations. Not only the phase-space of the neutron decay is shrinking, the protons decay too, and all that occurs in magnetic fields as large as 1010 Tesla. Even larger fields can be achieved in atomic or nuclear systems. Finally, it is worthwhile to point out that in lattice QCD studies strong magnetic fields are standardly used to compute the electromagnetic properties of hadrons. Combined with the lattice techniques of extracting the width, the relation between the absorprive part and the lifetime change may allow to compute the former on the lattice for unstable hadrons. Acknowledgments We thank Barry Holstein and Marc Vanderhaeghen for a number of insightful discus- sions. The work of V.P. is partially supported by the European Community-Research In- frastructure Activity under the FP6 ”Structuring the European Research Area” programme (HadronPhysics, contract RII3-CT-2004-506078). [1] M. Passera, J. Phys. G 31, R75 (2005); J. P. Miller, E. de Rafael and B. L. Roberts, Rept. Prog. Phys. 70, 795 (2007). [2] L. V. Avdeev and M. Y. Kalmykov, Phys. Lett. B 436, 132 (1998). [3] V. Pascalutsa and M. Vanderhaeghen, Phys. Rev. Lett. 94, 102003 (2005); Phys. Rev. D 77, 014027 (2008). [4] B. R. Holstein, unpublished. [5] W. M. Yao et al. [Particle Data Group],“Review of particle physics,”J. Phys. G 33, 1 (2006). [6] D. Tomono [RIKEN RAL R77 Collaboration], AIP Conf. Proc. 842, 906 (2006); K. R. Lynch, AIP Conf. Proc. 870, 333 (2006); J. S. Nico, AIP Conf. Proc. 870, 132 (2006); A. P. Serebrov et al., arXiv:nucl-ex/0702009. [7] P. G. Harris et al., Phys. Rev. Lett. 82, 904 (1999); C. A. Baker et al., Phys. Rev. Lett. 97, 131801 (2006). [8] J. L. Basdevant and J. Dalibard, “Quantum Mechanics Solver,” (Springer, Berlin, 2005). [9] R. P. Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958). [10] T. van Ritbergen and R. G. Stuart, Phys. Rev. Lett. 82, 488 (1999). [11] A. Czarnecki and B. Krause, Nucl. Phys. Proc. Suppl. 51C, 148 (1996). [12] J. H. Koch, V. Pascalutsa and S. Scherer, Phys. Rev. C 65, 045202 (2002). [13] V. Pascalutsa, M. Vanderhaeghen and S. N. Yang, Phys. Rept. 437, 125 (2007). [14] Our conventions are: metric (+,−,−,−), ε0123 = +1, γ5 = iγ0γ1γ2γ3, γ’s stand for Dirac matrices and their totally-antisymmetric products: γµν = 1 [γµ, γν ], γµνα = 1 {γµν , γα}, γµναβ = 1 [γµνα, γβ].

0704.0378

An equilibrium problem for the limiting eigenvalue distribution of banded Toeplitz matrices

AN EQUILIBRIUM PROBLEM FOR THE LIMITING EIGENVALUE DISTRIBUTION OF BANDED TOEPLITZ MATRICES MAURICE DUITS AND ARNO B.J. KUIJLAARS Abstract. We study the limiting eigenvalue distribution of n×n banded Toeplitz matrices as n → ∞. From classical results of Schmidt-Spitzer and Hirschman it is known that the eigenvalues accumulate on a spe- cial curve in the complex plane and the normalized eigenvalue counting measure converges weakly to a measure on this curve as n → ∞. In this paper, we characterize the limiting measure in terms of an equilibrium problem. The limiting measure is one component of the unique vector of measures that minimes an energy functional defined on admissible vectors of measures. In addition, we show that each of the other com- ponents is the limiting measure of the normalized counting measure on certain generalized eigenvalues. 1. Introduction For an integrable function a : {z ∈ C | |z| = 1} → C defined on the unit circle in the complex plane, the n× n Toeplitz matrix Tn(a) with symbol a is defined by Tn(a) = aj−k, j, k = 1, . . . , n, (1.1) where ak is the kth Fourier coefficient of a, a(eiθ)e−ikθ dθ. (1.2) In this paper we study banded Toeplitz matrices for which the symbol has only a finite number of non-zero Fourier coefficients. We assume that there exist p, q ≥ 1 such that a(z) = k, ap 6= 0, a−q 6= 0. (1.3) Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium. ([email protected], [email protected]). The first author is a research assistant of the Fund for Scientific Research – Flanders. The authors were supported by the European Science Foundation Program MISGAM. The second author is supported by FWO-Flanders project G.0455.04, by K.U. Leuven research grant OT/04/21, by Belgian Interuniversity Attraction Pole NOSY P06/02, and by a grant from the Ministry of Education and Science of Spain, project code MTM2005- 08648-C02-01. http://arxiv.org/abs/0704.0378v1 2 MAURICE DUITS AND ARNO B.J. KUIJLAARS Thus Tn(a) has at most p + q + 1 non-zero diagonals. As in [1, p. 263], we also assume without loss of generality that g.c.d. {k ∈ Z | ak 6= 0} = 1. (1.4) We are interested in the limiting behavior of the spectrum of Tn(a) as n → ∞. We use spTn(a) to denote the spectrum of Tn(a): spTn(a) = {λ ∈ C | det(Tn(a)− λI) = 0} Spectral properties of banded Toeplitz matrices are the topic of the recent book [1] by Böttcher and Grudsky. We will refer to this book frequently, in particular to Chapter 11 where the limiting behavior of the spectrum is discussed. The limiting behavior of spTn(a) was characterized by Schmidt and Spitzer [10]. They considered the set lim inf spTn(a), (1.5) consisting of all λ ∈ C such that there exists a sequence {λn}n∈N, with λn ∈ spTn(a), converging to λ, and the set lim sup spTn(a), (1.6) consisting of all λ such that there exists a sequence {λn}n∈N, with λn ∈ spTn(a), that has a subsequence converging to λ. Schmidt and Spitzer showed that these two sets are equal and can be characterized in terms of the algebraic equation a(z) − λ = k − λ = 0. (1.7) For every λ ∈ C there are p+q solutions for (1.7), which we denote by zj(λ), for j = 1, . . . , p+ q. We order these solutions by absolute value, so that 0 < |z1(λ)| ≤ |z2(λ)| ≤ · · · ≤ |zp+q(λ)|. (1.8) When all inequalities in (1.8) are strict then the values zk(λ) are unambigu- ously defined. If equalities occur then we choose an arbitrary numbering so that (1.8) holds. The result by Schmidt and Spitzer [10], [1, Theorem 11.17], is that lim inf spTn(a) = lim sup spTn(a) = Γ0 (1.9) where Γ0 := {λ ∈ C | |zq(λ)| = |zq+1(λ)|}. (1.10) This result gives a description of the asymptotic location of the eigenvalues. The eigenvalues accumulate on the set Γ0, which is known to be a disjoint union of a finite number of (open) analytic arcs and a finite number of ex- ceptional points [1, Theorem 11.9]. It is also known that Γ0 is connected EIGENVALUES OF BANDED TOEPLITZ MATRICES 3 [13], [1, Theorem 11.19], and that C \ Γ0 need not be connected [1, Theo- rem 11.20], [2, Proposition 5.2]. See [1] for many beautiful illustrations of eigenvalues of banded Toeplitz matrices. The limiting eigenvalue distribution was determined by Hirschman [5], [1, Theorem 11.16]. He showed that there exists a Borel probability measure µ0 on Γ0 such that the normalized eigenvalue counting measure of Tn(a) converges weakly to µ0, as n → ∞. That is, λ∈spTn(a) δλ → µ0, (1.11) where in the sum each eigenvalue is counted according to its multiplicity. The measure µ0 is absolutely continuous with respect to the arclength mea- sure on Γ0 and has an analytic density on each open analytic arc in Γ0, which can be explicitly represented in terms of the solutions of the algebraic equation (1.7) as follows. Equip every open analytic arc in Γ0 with an orien- tation. The orientation induces ±-sides on each arc, where the +-side is on the left when traversing the arc according to its orientation, and the −-side is on the left. The limiting measure µ0 is then given by dµ0(λ) = zj+(λ) zj−(λ) dλ. (1.12) where dλ is the complex line element on Γ0 (taken according to the orien- tation), and where zj±(λ), λ ∈ Γ0, is the limiting value of zj(λ ′) as λ′ → λ from the ± side of the arc. These limiting values exist for every λ ∈ Γ0, with the possible exception of the finite number of exceptional points. Note that the right-hand side of (1.12) is a priori a complex measure and it is not immediately clear that it is in fact a probability measure. In the original paper [5] and in the book [1, Theorem 11.16], the authors give a different expression for the limiting density, from which it is clear that the measure is non-negative. We prefer to work with the complex expression (1.12), since it allows for a direct generalization which we will need in this paper. Note also that if we reverse the orientation on an arc in Γ0, then the ±- sides are reversed. Since the complex line element dλ changes sign as well, the expression (1.12) does not depend on the choice of orientation. The following is a very simple example, which however serves as a moti- vation for the results in the paper. Example 1.1. Consider the symbol a(z) = z+1/z. In this case we find that Γ0 = [−2, 2] and µ0 is absolutely continuous with respect to the Lebesgue measure and has density dµ0(λ) 4− λ2 , λ ∈ (−2, 2). (1.13) 4 MAURICE DUITS AND ARNO B.J. KUIJLAARS This measure is well-known in potential theory and is called the arcsine measure or the equilibrium measure of Γ0, see e.g. [9]. It has the property that it minimizes the energy functional I defined by I(µ) = |x− y| dµ(x) dµ(y), (1.14) among all Borel probability measures µ on [−2, 2]. The measure µ0 is also characterized by the equilibrium condition log |x− λ| dµ0(λ) = 0, x ∈ [−2, 2], (1.15) which is the Euler-Lagrange variational condition for the minimization prob- The fact that µ0 is the equilibrium measure of Γ0 is special for symbols a with p = q = 1. In that case one may think of the eigenvalues of Tn(a) as charged particles on Γ0, each eigenvalue having a total charge 1/n, that repel each other with logarithmic interaction. The particles seek to minimize the energy functional (1.14). As n → ∞, they distribute themselves according to µ0 and µ0 is the minimizer of (1.14) among all probability measures supported on Γ0. The aim of this paper is to characterize µ0 for general symbols a of the form (1.3) also in terms of an equilibrium problem from potential theory. The corresponding equilibrium problem is more complicated since it involves not only the measure µ0, but a sequence of p+ q − 1 measures µ−q+1, µ−q+2, . . . , µ−1, µ0, µ1, . . . , µp−2, µp−1 that jointly minimize an energy functional. 2. Statement of results 2.1. The energy functional. To state our results we need to introduce some notions from potential theory. Main references for potential theory in the complex plane are [8] and [9]. We will mainly work with finite positive measures on C, but we will also use ν1 − ν2 where ν1 and ν2 are positive measures. The measures need not have bounded support. If ν has unbounded support then we assume that log(1 + |x|) dν(x) < ∞. (2.1) In that case the logarithmic energy of ν is defined as I(ν) = |x− y| dν(x)dν(y) (2.2) and I(ν) ∈ (−∞,+∞]. EIGENVALUES OF BANDED TOEPLITZ MATRICES 5 Definition 2.1. We define Me as the collection of positive measures ν on C satisfying (2.1) and having finite energy, i.e., I(ν) < +∞. For c > 0 we define Me(c) = {ν ∈ Me | ν(C) = c}. (2.3) The mutual energy I(ν1, ν2) of two measures ν1 and ν2 is I(ν1, ν2) = |x− y| dν1(x)dν2(y). (2.4) It is well-defined and finite if ν1, ν2 ∈ Me and in that case we have I(ν1 − ν2) = I(ν1) + I(ν2)− 2I(ν1, ν2). (2.5) If ν1, ν2 ∈ Me(c) for some c > 0, then I(ν1 − ν2) ≥ 0, (2.6) with equality if and only if ν1 = ν2. This is a well-known result if ν1 and ν2 have compact support [9]. For measures in Me(c) with unbounded support, this is a recent result of Simeonov [11], who obtained this from a very elegant integral representation for I(ν1 − ν2). It is a consequence of (2.6) that I is strictly convex on Me(c), since ν1 + ν2 (I(ν1) + I(ν2))− I ν1 − ν2 (I(ν1) + I(ν2)) , for ν1, ν2 ∈ Me(c), with equality if and only if ν1 = ν2. Before we can state the equilibrium problem we also need to introduce the sets Γk := {λ ∈ C | |zq+k(λ)| = |zq+k+1(λ)|}, k = −q + 1, . . . , p− 1, (2.7) which for k = 0 reduces to the definition (1.10) of Γ0. We will show that each Γk is the disjoint union of a finite number of open analytic arcs and a finite number of exceptional points. All Γk are unbounded, except for Γ0 which is compact. The equilibrium problem will be defined for a vector of measures denoted by ~ν = (ν−q+1, . . . , νp−1). The component νk is a measure on Γk satisfying some additional properties that are given in the following definition. Definition 2.2. We call a vector of measures ~ν = (ν−q+1, . . . , νp−1) admis- sible if νk ∈ Me, νk is supported on Γk, and νk(Γk) = if k ≤ 0, if k ≥ 0, (2.8) for every k = −q + 1, . . . , p − 1. Now we are ready to state our first result. The proof is given in section 6 MAURICE DUITS AND ARNO B.J. KUIJLAARS Theorem 2.3. Let the symbol a satisfy (1.3) and (1.4), and let the curves Γk be defined as in (2.7). For each k ∈ {−q + 1, . . . , p − 1}, define the measure µk on Γk by dµk(λ) = zj+(λ) zj−(λ) dλ, (2.9) where dλ is the complex line element on each analytic arc of Γk according to a chosen orientation of Γk (cf. discussion after (1.12)). Then (a) ~µ = (µ−q+1, . . . , µp−1) is admissible. (b) There exist constants lk such that log |λ− x| dµk(x) = log |λ− x| dµk+1(x) + log |λ− x| dµk−1(x) + lk, (2.10) for k = −q + 1, . . . , p − 1, and λ ∈ Γk. Here we let µ−q and µp be the zero measures. (c) ~µ = (µ−q+1, . . . , µp−1) is the unique minimizer of the energy func- tional J defined by J(~ν) = k=−q+1 I(νk)− k=−q+1 I(νk, νk+1) (2.11) for admissible vectors of measures ~ν = (ν−q+1, . . . , νp−1). The relations (2.10) are the Euler-Lagrange variational conditions for the minimization problem for J among admissible vectors of measures. It may not be obvious that the energy functional (2.11) is bounded from below. This can be seen from the alternative representation J(~ν) = I(ν0) + k(k + 1) I ν−q+k − ν−q+k+1 k + 1 k(k + 1) I − νp−k−1 k + 1 . (2.12) We leave the calculation leading to this identity to the reader. Under the normalizations (2.8) it follows by (2.6) that each term in the two finite sums on the right-hand side of (2.12) is non-negative, so that J(~ν) ≥ I(ν0). Since ν0 is a Borel probability measure on Γ0 and Γ0 is compact, we indeed have that the energy functional is bounded from below on admissible vectors of measures ~ν. The alternative representation (2.12) will play a role in the proof of The- orem 2.3. EIGENVALUES OF BANDED TOEPLITZ MATRICES 7 Yet another representation for J is J(~ν) = j,k=−q+1 Ajk I(νj , νk) (2.13) where the interaction matrix A has entries Ajk = 1, if j = k, , if |j − k| = 1, 0, if |j − k| ≥ 2. (2.14) The energy functional in the form (2.13) and (2.14) also appears in the theory of simultaneous rational approximation, where it is the interaction matrix for a Nikishin system [7, Chapter 5]. It allows for the following physical interpretation: on each of the curves Γk one puts charged particles with total charge (q+k)/q or (p−k)/p, depending on whether k ≤ 0 or k ≥ 0. Particles that lie on the same curve repel each other. The particles on two consecutive curves interact in the sense that they attract each other but in a way that is half as strong as the repulsion on a single curve. Particles on different curves that are not consecutive do not interact with each other in a direct way. 2.2. The measures µk as limiting measures of generalized eigenval- ues. By (1.12) and Theorem 2.3 we know that the measure µ0 that appears in the minimizer of the energy functional J is the limiting measure for the eigenvalues of Tn(a). It is natural to ask about the other measures µk that appear in the minimizer. In our second result we show that the measures µk can be obtained as limiting counting measures for certain generalized eigenvalues. Let k ∈ {−q+1, . . . , p− 1}. We use Tn(z−k(a−λ) to denote the Toeplitz matrix with the symbol z 7→ z−k(a(z) − λ). For example, for k = 1, q = 1 and p = 2, we have −k(a−λ)) = a1 a0 − λ a−1 a2 a1 a0 − λ a−1 a2 a1 a0 − λ a−1 . . . . . . . . . . . . a2 a1 a0 − λ a−1 a2 a1 a0 − λ a2 a1 Definition 2.4. For k ∈ {−q + 1, . . . , p − 1} and n ≥ 1, we define the polynomial Pk,n by Pk,n(λ) = detTn(z −k(a− λ)) (2.15) and we define the kth generalized spectrum of Tn(a) by spk Tn(a) = {λ ∈ C | Pk,n(λ) = 0}. (2.16) 8 MAURICE DUITS AND ARNO B.J. KUIJLAARS Finally, we define µk,n as the normalized zero counting measure of spk Tn(a) µk,n = λ∈spk Tn(a) δλ (2.17) where in the sum each λ is counted according to its multiplicity as a zero of Pk,n. Note that λ ∈ spk Tn(a) is a generalized eigenvalue (in the usual sense) for the matrix pencil (Tn(z −ka), Tn(z −k)), that is, det(A − λB) = 0 with A = −ka) and B = Tn(z −k). If k = 0, then B = I and sp0 Tn(a) = spTn(a). If k 6= 0, then B is not invertible and the generalized eigenvalue problem is singular, causing that there are less than n generalized eigenvalues. In fact, since Tn(z −k(a−λ)) has exactly n−|k| entries a0−λ, we easily get that the degree of Pk,n is at most n− |k| and so there are at most n− |k| generalized eigenvalues. Due to the band structure of Tn(z −k(a−λ)) the actual number of generalized eigenvalues is substantially smaller. Proposition 2.5. Let k ∈ {−q+1, . . . , p−1}. Let Pk,n(λ) = γk,nλdk,n + · · · have degree dk,n and leading coefficient γk,n 6= 0. Then dk,n ≤ n, if k < 0, n, if k > 0. (2.18) Equality holds in (2.18) if either k > 0 and n is a multiple of p, or k < 0 and n is a multiple of q, and in those cases we have γk,n = (−1)(k+1)na|k|n/q−q , if k < 0 and n ≡ 0 mod q, (−1)(k+1)nakn/pp , if k > 0 and n ≡ 0 mod p. (2.19) We now come to our second main result. It is the analogue of the results of Schmidt-Spitzer and Hirschman for the generalized eigenvalues. Theorem 2.6. Let k ∈ {−q + 1, . . . , p− 1}. Then lim inf spk Tn(a) = lim sup spk Tn(a) = Γk, (2.20) φ(z) dµk,n(z) = φ(z) dµk(z) (2.21) holds for every bounded continuous function φ on C. The key element in the proof of Theorem 2.6 is a beautiful formula of Widom [14], see [1, Theorem 2.8], for the determinant of a banded Toeplitz matrix. In the present situation Widom’s formula yields the following. Let λ ∈ C be such that the solutions zj(λ) of the algebraic equation (1.7) are mutually distinct. Then Pk,n(λ) = detTn(z −k(a− λ)) = CM (λ) (wM (λ)) , (2.22) EIGENVALUES OF BANDED TOEPLITZ MATRICES 9 where the sum is over all subsets M ⊂ {1, 2, . . . , p+ q} of cardinality |M | = p− k and for each such M , we have wM (λ) := (−1)p−kap zj(λ), (2.23) and (with M := {1, 2, . . . , p+ q} \M), CM (λ) := zj(λ) (zj(λ)− zl(λ))−1. (2.24) The formula (2.22) shows that for large n, the main contribution comes from those M for which |wM (λ)| is the largest possible. For λ ∈ C \ Γk there is a unique such M , namely M = Mk := {q + k + 1, q + k + 2, . . . , p + q} (2.25) because of the ordering (1.8). 2.3. Overview of the rest of the paper. In section 3 we will state some preliminary results about analyticity properties of the solutions zj of the al- gebraic equation (1.7). These results will be needed in the proof of Theorem 2.3 which is given in section 4. In section 5 we will prove Proposition 2.5 and Theorem 2.6. Finally, we conclude the paper by giving some examples in section 6. 3. Preliminaries In this section we collect a number of properties of the curves Γk and the solutions z1(λ), . . . , zp+q(λ) of the algebraic equation (1.7). For convenience we define throughout the rest of the paper Γ−q = Γp = ∅, and µ−q = µp = 0. (the zero-measure). Occasionally we also use z0(λ) = 0, zp+q+1(λ) = +∞. 3.1. The structure of the curves Γk. We start with a definition, cf. [1, §11.2]. Definition 3.1. A point λ0 ∈ C is called a branch point if a(z) − λ0 = 0 has a multiple root. A point λ0 ∈ Γk is an exceptional point of Γk if λ0 is a branch point, or if there is no open neighborhood U of λ such that Γk ∩U is an analytic arc starting and terminating on ∂U . If λ0 is a branch point, then there is a z0 such that a(z0) = λ0 and a′(z0) = 0. Then we may assume that z0 = zq+k(λ0) = zq+k+1(λ0) for some k and λ0 ∈ Γk. For a symbol a of the form (1.3), the derivative a′ has exactly p+q zeros (counted with multiplicity), so that there are exactly p+q branch points counted with multiplicity. 10 MAURICE DUITS AND ARNO B.J. KUIJLAARS The solutions zk(λ) also have branching at infinity (unless p = 1 or q = 1). There are p solutions of (1.7) that tend to infinity as λ → ∞, and q solutions that tend to 0. Indeed, we have zk(λ) = −1/q(1 +O(λ−1/q)), for k = 1, . . . , q, 1/p(1 +O(λ−1/p)), for k = q + 1, . . . , p + q, (3.1) as λ → ∞. Here c1, . . . , cq are the q distinct solutions of cq = a−q (taken in some order depending on λ), and cq+1, . . . , cp+q are the p distinct solutions of cp = a−1p (again taken in some order depending on λ). The following proposition gives the structure of Γk at infinity. Proposition 3.2. Let k ∈ {−q+1, . . . , p−1}\{0}. Then there is an R > 0 such that Γk ∩ {λ ∈ C | |λ| > R} is a finite disjoint union of analytic arcs, each extending from |λ| = R to infinity. Proof. The proof is similar to the proof of [1, Proposition 11.8] where a similar structure theorem was proved for finite branch points. We omit the details. � It follows from Proposition 3.2 that the exceptional points for Γk are in a bounded set. Since the set of exceptional point is discrete we conclude that there are only finitely many exceptional points. Then we have the following result about the structure of Γk. Proposition 3.3. For every k ∈ {−q + 1, . . . , p − 1}, the set Γk is the disjoint union of a finite number of open analytic arcs and a finite number of exceptional points. The set Γk has no isolated points. Proof. This was proved for k = 0 in [10] and [1, Theorem 11.9]. For general k, there are only finitely many exceptional points and the proof follows in a similar way. � 3.2. The Riemann surface. From Proposition 3.3 it follows that the curves Γk can be taken as cuts for the p + q-sheeted Riemann surface of the alge- braic equation (1.7). We number the sheets from 1 to p+ q, where the kth sheet of the Riemann surface is Rk = {λ ∈ C | |zk−1(λ)| < |zk(λ)| < |zk+1(λ)|} = C \ (Γ−q+k−1 ∪ Γ−q+k). (3.2) Thus zk is well-defined and analytic on Rk. The easiest case to visualize is the case where consecutive cuts are disjoint, that is, Γ−q+k−1 ∩ Γ−q+k = ∅ for every k = 2, . . . , p+ q− 2. In that case we have that Rk is connected to Rk+1 via Γ−q+k in the usual crosswise manner, and zk+1 is the analytic continuation of zk across Γ−q+k. The general case is described in the following proposition. Proposition 3.4. Suppose A is an open analytic arc such that A ⊂ Γ−q+k, for k = k1, . . . , k2, and A ∩ (Γ−q+k1−1 ∪ Γ−q+k2+1) = ∅. Then for k = EIGENVALUES OF BANDED TOEPLITZ MATRICES 11 k1, . . . , k2+1, we have that the analytic continuation of zk across A is equal to zk1+k2−k+1. Thus across A, we have that Rk is connected to Rk1+k2−k+1. Proof. We have that |zk1(λ)| = |zk1+1(λ)| = · · · = |zk2(λ)| = |zk2+1(λ)| for λ ∈ A, with strict inequalities (<) for λ on either side of A. Choose an orientation for A. Then there is a permutation π of {k1, . . . , k2 + 1} such that zπ(k) is the analytic continuation of zk from the +-side of A to the −-side of A. Assume that there are k, k′ ∈ {k1, . . . , k2 + 1} such that k < k′ and π(k) < π(k′). Take a regular λ0 ∈ A and a small neighborhood U of λ0 such that A∩U = Γ−q+k ∩U = Γ−q+k′ ∩U and A∩U is an analytic arc starting and terminating on ∂U . Then we have a disjoint union U = U+∪U−∪(A∩U) where U+ (U−) is the part of U on the +-side (−-side) of A. The function φ defined by φ(λ) = zk(λ) zk′ (λ) , for λ ∈ U+, zπ(k)(λ) zπ(k′)(λ) , for λ ∈ U−, has an analytic continuation to U , and satisfies |φ(λ)| < 1 for λ ∈ U+ ∪ U− and |φ(λ)| = 1 for λ ∈ A ∩ U . This contradicts the maximum principle for analytic functions. Therefore π(k) > π(k′) for every k, k′ ∈ {k1, . . . , k2 + 1} with k < k′, and this implies that π(k) = k1 + k2 − k + 1 for every k = k1, . . . , k2 + 1, and the proposition follows. � 3.3. The functions wk(λ). A major role is played by the functions wk, which for k ∈ {−q + 1, . . . , p − 1}, are defined by wk(λ) = zj(λ), for λ ∈ C \ Γk. (3.3) Note that wk = (−1)p−ka−1p w{1,...,k} in the notation of (2.23). Proposition 3.5. The function wk is analytic in C \ Γk. Proof. Since zj is analytic on Rj = C \ (Γ−q+j−1 ∪ Γ−q+j), see (3.2), we obtain from its definition that wk is analytic in C \ j=1 Γ−q+j. Let A be an analytic arc in Γ−q+j \ Γk for some j < k + q. Choose an orientation on A. Since the arc is disjoint from Γk, we have that zj+(λ) = zπ(j)−(λ), for λ ∈ A and j = 1, . . . , q+k, where π is a permutation of {1, . . . , q+k}. Since wk is symmetric in the zj ’s for j = 1, . . . , q + k, it then follows that wk+(λ) = wk−(λ), for λ ∈ A, which shows the analyticity in C \Γk with the possible exception of isolated singularities at the exceptional points of Γ−q+1, Γ−q+2, . . . , Γk−1. However, each zj , and therefore also wk, is bounded near such an exceptional point, so that any isolated singularity is removable. � 12 MAURICE DUITS AND ARNO B.J. KUIJLAARS In the rest of the paper we make frequently use of the logarithmic de- rivative w′k/wk of wk. By the fact that wk does not vanish on C \ Γk and Proposition 3.5, it follows that w′k/wk is analytic in C \ Γk. By Proposition 3.4 it moreover has an analytic continuation across every open analytic arc A ⊂ Γk. Near the exceptional points that are no branch points w′k/wk re- mains bounded. At the branch points it can however have singularities of a certain order. Proposition 3.6. Let λ0 ∈ Γk be a branch point of Γk. Then there exists an m ∈ N such that w′k(λ) wk(λ) (λ− λ0)−m/(m+1) , (3.4) as λ → λ0 with λ ∈ C \ Γk. Proof. Let 1 ≤ j ≤ q+k. We investigate the behavior of zj(λ) when λ → λ0 such that λ remains in a connected component of C \ (Γj−1 ∪ Γj). Then zj(λ) → z0 for some z0 ∈ C with a(z0) = λ0. Let m0 + 1 be the multiplicity of z0 as a solution of a(z) = λ0. Then a(z) = λ0 + c0(z − z0)m0+1(1 +O(z − z0)), z → z0, (3.5) for some nonzero constant c0. Therefore, zj(λ) = z0 +O((λ− λ0)1/(m0+1)), (3.6) z′j(λ) = O((λ− λ0)−m0/(m0+1)), (3.7) for λ → λ0 such that λ remains in the same connected component of C \ (Γj−1 ∪ Γj). Let m be the maximum of all the multiplicities of the roots of a(z) = λ0. Then it follows from (3.6) and (3.7) that z′j(λ) zj(λ) = O((λ− λ0)−m/(m+1)) as λ → λ0 with λ ∈ C \ Γk. Then we obtain (3.4) in view of (3.3). � We end this section by giving the asymptotics of w′k/wk for λ → ∞. Proposition 3.7. As λ → ∞ with λ ∈ C \ Γk, we have w′k(λ) wk(λ) − q+k λ−1 +O λ−1−1/q , for k = −q + 1, . . . ,−1, −λ−1 +O(λ−2), for k = 0, λ−1 +O λ−1−1/p , for k = 1, . . . , p − 1. (3.8) Proof. This follows directly from (3.1) and (3.3). � EIGENVALUES OF BANDED TOEPLITZ MATRICES 13 4. Proof of Theorem 2.3 We use the function wk introduced in (3.3). We define µk by the formula (2.9) and we note that dµk(λ) = w′k+(λ) wk+(λ) w′k−(λ) wk−(λ) dλ. (4.1) Proposition 4.1. For each k = −q + 1, . . . , p − 1, we have that µk is a measure on Γk with total mass µk(Γk) = (q + k)/q if k ≥ 0, and µk(Γk) = (p− k)/p if k ≥ 0. Proof. We first show that µk is a measure, i.e., that it is non-negative on each analytic arc of Γk. Let A be an analytic arc in Γk consisting only of regular points. Let t 7→ λ(t) be a parametrization of A in the direction of the orientation of Γk. Then dµk(λ) = w′k+(λ(t)) wk+(λ(t)) w′k−(λ(t)) wk−(λ(t)) λ′(t)dt wk+(λ(t)) wk−(λ(t)) To conclude that µk is non-negative on A, it is thus enough to show that Re log wk+(λ) wk−(λ) = 0, for λ ∈ A, (4.2) Im log wk+(λ) wk−(λ) increases along A. (4.3) Since |wk+(λ)| = |wk−(λ)| for λ ∈ A, we have (4.2) so that it only remains to prove (4.3). There is a neighborhood U of A such that U \ Γk has two components, denoted U+ and U−, where U+ is on the +-side of Γk and U− on the −-side. It follows from Proposition 3.4 that wk has an analytic continuation from U− to U , which we denote by ŵk, and that |wk(λ)| < |ŵk(λ)| for λ ∈ U+, and equality |wk+(λ)| = |ŵk(λ)| holds for λ ∈ A. Thus it follows that Re log wk(λ) ŵk(λ) ≤ 0, for λ ∈ A, where ∂ denotes the normal derivative to A in the direction of U+. Then by the Cauchy-Riemann equations we have that Im log wk+(λ) ŵk+(λ) is increasing along A. Since ŵk+(λ) = wk−(λ) for λ ∈ A, we obtain (4.3). Thus µk is a measure. Next we show that µk is a finite measure, which means that we have to show that w′k+(λ) wk+(λ) w′k−(λ) wk−(λ) (4.4) 14 MAURICE DUITS AND ARNO B.J. KUIJLAARS Figure 1. Illustration for the proofs of Propositions 4.1 and 4.2. The solid line is a sketch of a possible contour Γk. The dashed line is the contour Γ̃k,R and the dotted line is the boundary of a disk of radius R around 0. is integrable near infinity on Γk and near every branch point on Γk. This follows from Propositions 3.7 and 3.6. Indeed, from Proposition 3.7 it follows w′k+(λ) wk+(λ) w′k−(λ) wk−(λ) λ−1−δ as λ → ∞, λ ∈ Γk. (4.5) where δ = 1/q if k < 0 and δ = 1/p if k > 0. Since δ > 0 we see that (4.4) is integrable near infinity. For a branch point λ0 of Γk, we have from Proposition 3.6 that there exist an m ≥ 1 such that w′k+(λ) wk+(λ) w′k−(λ) wk−(λ) (λ− λ0)−m/(m+1) as λ → λ0, λ ∈ Γk. (4.6) This shows that (4.4) is integrable near every branch point. Thus µk is a finite measure. Finally we compute the total mass of µk. LetD(0, R) = {z ∈ C | |z| < R}. Then for R large enough, so that D(0, R) contains all exceptional points of Γk and all connected components of C \ Γk (if any), µk(Γk ∩D(0, R)) = Γk∩D(0,R) w′k+(λ) wk+(λ) Γk∩D(0,R) w′k−(λ) wk−(λ) (4.7) where we have used the behavior (4.6) near the branch points in order to be able to split the integrals. Again using (4.6) we can then turn the two integrals into a contour integral over a contour Γ̃k,R as in Figure 1. The contour Γ̃k,R passes along the ±-sides of Γk ∩D(0, R) and if we choose the orientation that is also shown in Figure 1 (and which is independent of the EIGENVALUES OF BANDED TOEPLITZ MATRICES 15 choice of orientation for Γk), then µk(Γk ∩D(0, R)) = Γ̃k,R w′k(λ) wk(λ) dλ. (4.8) The parts of Γ̃k,R that belong to bounded components of C \Γk form closed contours along the boundary of each bounded component. By Cauchy’s theorem their contribution to the integral (4.8) vanishes. The parts of Γ̃k,R that belong to the unbounded components of C \Γk can be deformed to the circle ∂D(0, R) with the clockwise orientation. Thus if we use the positive orientation on ∂D(0, R) as in Figure 1, then we obtain from (4.8) µk(Γk ∩D(0, R)) = − ∂D(0,R) w′k(λ) wk(λ) Letting R → ∞ and using Proposition 3.7, we then find that µk is a measure on Γk with total mass µk(Γk) = (q + k)/q if k ≤ 0, and µk(Γk) = (p− k)/p if k ≥ 0. � The following proposition is the next step in showing that the measures µk from (2.9) satisfy the equations (2.10). Proposition 4.2. For k = −q + 1, . . . , p− 1, we have that dµk(x) x− λ = w′k(λ) wk(λ) , for λ ∈ C \ Γk, (4.9) log |λ− x| dµk(x) = − log |wk(λ)|+ αk, for λ ∈ C, (4.10) where αk is the constant log |a−q|+ kq log |a−q|, if k ≤ 0, log |a−q| − kp log |ap|, if k ≥ 0. (4.11) Proof. To prove (4.9), we follow the same arguments as in the calculation of µk(Γk) in the end of the proof of Proposition 4.1. Let λ ∈ C \ Γk, and choose R > 0 as in the proof of Proposition 4.1. We may assume R > |λ|. Then similar to (4.7) and (4.8) we can write Γk∩D(0,R) dµk(x) x− λ = Γ̃k,R w′k(x) wk(x)(x− λ) where Γ̃k,R has the same meaning as in the proof of Proposition 4.1, see also Figure 1. As in the proof of Proposition 4.1 we deform to an integral over ∂D(0, R), but now we have to take into account that the integrand has a pole at x = λ with residue w′k(λ)/wk(λ). Therefore, by Cauchy’s theorem Γk∩D(0,R) dµk(x) x− λ = w′k(λ) wk(λ) ∂D(0,R) w′k(x) wk(x)(x− λ) dx. (4.12) Letting R → ∞ and using Proposition 3.7 gives (4.9). 16 MAURICE DUITS AND ARNO B.J. KUIJLAARS Next we integrate (4.9) over a Jordan curve J in C \ Γk from λ1 to λ2. x− λ dµk(x) dλ = − ∫ ∫ λ2 x− λ dλ dµk(x) (log |λ1 − x| − log |λ2 − x|+ i∆J [arg(λ− x)]) dµk(x), (4.13) where ∆J [arg(λ − x)] denotes the change in argument of λ − x as when λ varies over J from λ1 to λ2. By (4.9) the integral (4.13) is equal to w′k(λ) wk(λ) dλ = log |wk(λ2)| − log |wk(λ1)|+ i∆J [argwk(λ)]. (4.14) Equating the real parts of (4.13) and (4.14) we get (log |λ1 − x| − log |λ2 − x|) dµk(x) = − log |wk(λ1)|+ log |wk(λ2)|. (4.15) Since λ1 and λ2 can be taken arbitrarily in a connected component of C\Γk, we find that there exists a constant αk ∈ R (which a priori could depend on the connected component) such that log |λ− x| dµk(x) = − log |wk(λ)|+ αk, (4.16) for all λ in a connected component of C \ Γk. By continuity the equation (4.16) extends to the closure of the connected component, which shows that the same constant αk is valid for all connected components. Thus (4.16) holds for all λ ∈ C. The exact value of αk can then be determined by expanding (4.16) for large λ. Suppose for example that k < 0. Then by (3.1) and (3.3) |wk(λ)| = |zj(λ)| = |a−q|(q+k)/q|λ|−(q+k)/q 1 +O(λ−1/q) as λ → ∞. Thus − log |wk(λ)| = q + k log |λ| − q + k log |a−q|+O(λ−1/q). (4.17) Since log |λ− x| dµk(x) = log |λ|µk(Γk) + o(1) = q + k log |λ|+ o(1), (4.18) as λ → ∞, the value (4.11) for αk follows from (4.16), (4.17), and (4.18). The argument for k > 0 is similar. This completes the proof of the proposition. To prove part (c) of Theorem 2.3 we also need the following lemma. EIGENVALUES OF BANDED TOEPLITZ MATRICES 17 Lemma 4.3. Let ~ν1 = (ν1,−q+1 . . . , ν1,p−1) and ~ν2 = (ν2,−q+1 . . . , ν2,p−1) be two admissible vectors of measures. Then J(~ν1 − ~ν2) is well defined and J(~ν1 − ~ν2) ≥ 0, (4.19) with equality if and only if ~ν1 = ~ν2. Proof. Since both ~ν1 and ~ν2 have finite energy, we find that J(~ν1 − ~ν2) is well defined. According to the alternative representation (2.12), we have J(~ν1 − ~ν2) = I(ν1,0 − ν2,0) k(k + 1)I ν1,−q+k − ν2,−q+k − ν1,−q+k+1 k + 1 ν2,−q+k+1 k + 1 k(k + 1)I ν1,p−k ν2,p−k ν1,p−k−1 k + 1 ν2,p−k−1 k + 1 (4.20) Using (2.6) and (2.8), we see that all terms in (4.20) are non-negative and therefore (4.19) holds. Suppose now that J(~ν1 − ~ν2) = 0. Then all terms in the right-hand side of (4.20) are zero, so that ν1,0 = ν2,0, (4.21) ν1,−q+k ν2,−q+k+1 k + 1 ν1,−q+k+1 k + 1 ν2,−q+k , for k = 1, . . . , q − 1, (4.22) ν1,p−k ν2,p−k−1 k + 1 ν1,p−k−1 k + 1 ν2,p−k , for k = 1, . . . , p− 1. (4.23) Using (4.21) in (4.22) with k = q − 1, we find ν1,−1 = ν2,−1. Proceeding inductively we then obtain from (4.22) that ν1,k = ν2,k for all k = −q + 1, . . . , 0. Similarly, from (4.21) and (4.23) it follows that ν1,k = ν2,k for k = 0, . . . , p− 1, so that ~ν1 = ~ν2 as claimed. � Now we are ready for the proof of Theorem 2.3. Proof of Theorem 2.3. (a) In view of Proposition 4.1 it only remains to show that µk ∈ Me for every k = −q + 1, . . . , p − 1. The decay estimate (4.5) implies that log(1 + |λ|) dµk(λ) < ∞. The fact that I(µk) < +∞ follows from (4.10). Indeed, I(µk) = − log |λ− x|dµk(x)dµk(λ) = (log |wk(λ)| − αk)dµk(λ) 18 MAURICE DUITS AND ARNO B.J. KUIJLAARS and this is finite since µk is a finite measure on Γk with a density that decays as in (4.5) and log |wk(λ)| is continuous on Γk and grows only as a constant times log |λ| as λ → ∞. Thus ~µ is admissible and part (a) is proved. (b) According to (4.10) we have log |λ− x| dµk(x)− log |λ− x| dµk+1(λ)− log |λ− x| dµk−1(λ) = −2 log |wk(λ)|+ 2αk + log |wk+1(λ)| − αk+1 + log |wk−1(λ)| − αk−1 = log wk+1(λ)wk−1(λ) wk(λ) + 2αk − αk+1 − αk−1 = log zq+k+1(λ) zq+k(λ) + 2αk − αk+1 − αk−1. (4.24) Since |zq+k(λ)| = |zq+k+1(λ)| for λ ∈ Γk, we see from (4.24) that (2.10) holds with constant lk = 2αk − αk−1 + αk+1. (4.25) Note that for k = −q + 1 and k = p − 1, we are using the convention that µ−q = µp = 0, and we also have put α−q = αp = 0. This proves part (b). (c) Let ~ν = (ν−q+1, . . . , νp−1) be any admissible vector of measures. From the representation (2.13) we get J(~ν) = J(~µ+ ~ν − ~µ) = J(~µ) + J(~ν − ~µ) + 2 j,k=−q+1 AjkI(µj, νk − µk). (4.26) Using (2.14), we find from (4.26) J(~ν) = J(~µ) + J(~ν − ~µ) + k=−q+1 I(2µk − µk−1 − µk+1, νk − µk) (4.27) For each k = −q + 1, . . . , p − 1, we have I(2µk − µk−1 − µk+1, νk − µk) log |λ− x| d(2µk − µk−1 − µk+1)(x) d(νk − µk)(λ) (4.28) By (2.10) the inner integral in the right-hand side of (4.28) is constant for λ ∈ Γk. Since νk and µk are finite measures on Γk with νk(Γk) = µk(Γk), we find from (4.28) that I(2µk − µk−1 − µk+1, νk − µk) = 0, for k = −q + 1, . . . , p− 1. Then (4.27) shows that J(~ν) = J(~µ)+J(~ν−~µ), which by Lemma 4.3 implies that J(~ν) ≥ J(~µ) and equality holds if and only if ~ν = ~µ. This completes the proof of Theorem 2.3. � EIGENVALUES OF BANDED TOEPLITZ MATRICES 19 5. Proofs of Proposition 2.5 and Theorem 2.6 5.1. Proof of Proposition 2.5. We will now prove Proposition 2.5, which follows by a combinatorial argument. Proof of Proposition 2.5. We prove (2.18) and (2.19) for k > 0. The case k < 0 is similar. Let us first expand the determinant in the definition of Pk,n(λ) = detTn(z −k(a− λ)) = (a− λ)j−π(j)+k. (5.1) Here Sn denotes the set of all permutation on {1, . . . , n}. By the band struc- ture of Tn(z −k(a − λ)) it follows that we only have non-zero contributions from permutations π that satisfy k − p ≤ π(j) − j ≤ q + k, for all j = 1, . . . , n. (5.2) Define for π ∈ Sn, Nπ = {j | π(j) = j + k}. (5.3) and denote the number of elements of Nπ by |Nπ|. For each π ∈ Sn we have j=1(a−λ)j−π(j)+k is a polynomial in λ of degree at most |Nπ|. So by (5.1) dk,n = degPk,n ≤ max |Nπ| (5.4) where we maximize over permutations π ∈ Sn satisfying (5.2). Let π ∈ Sn satisfying (5.2). We prove (2.18) by giving an upper bound for |Nπ|. Since j=1(π(j) − j) = 0 we obtain (π(j) − j)+ = (j − π(j))+, (5.5) where (·)+ is defined as (a)+ = max(0, a) for a ∈ R. Each j ∈ Nπ gives a contribution k to the left-hand side of (5.5). Therefore the left-hand side is at least k|Nπ|. By (5.2) we have that each term in the right hand side is at most p− k. Moreover, there are at most n− |Nπ| non-zero terms in this sum. Combining this with (5.5) leads to k|Nπ| ≤ (π(j) − j)+ = (j − π(j))+ ≤ (n− |Nπ|)(p − k). (5.6) Hence, if π is a permutation satisfying (5.2) |Nπ| ≤ n(p− k) . (5.7) Now (2.18) follows by combining (5.7) and (5.4). To prove (2.19), we assume that n ≡ 0 mod p. We claim that there exists a unique π such that equality holds in (5.7). Then equality holds in both 20 MAURICE DUITS AND ARNO B.J. KUIJLAARS inequalities of (5.6) and the above arguments show that this can only happen π(j) = j + k, or π(j) = j − p+ k, (5.8) for every j = 1, . . . , n. We claim that there exists a unique such permutation, namely π(j) = j + k, if j ≡ 1, . . . , (p − k) mod p, j − p+ k, if j ≡ (p− k + 1), . . . , p mod p. (5.9) To see this let π be a permutation satisfying (5.8). The numbers 1, . . . , p− k can not satisfy π(j) = j−p+k and thus satisfy π(j) = j+k. On the other hand, the numbers 1, . . . , k can not be the image of numbers j satisfying π(j) = j + k, and thus π(j) = j − p + k for j = p − k + 1, . . . , p. So (5.9) holds for j = 1, . . . , p. This means in particular that the restriction of π to {p + 1, . . . , n} is again a permutation, but now on {p + 1, . . . , n}. By the same arguments we then find that (5.9) holds for j = p + 1, . . . , 2p, and so on. The result is that (5.9) is indeed the only permutation that satisfies (5.8). Finally, a straightforward calculation shows that the coefficient of λ(p−k)n/p j=1(a − λ)j−π(j)+k with π as in (5.9) is nonzero and given by (2.19). This proves the proposition. � 5.2. Proof of Theorem 2.6. Before we start with the proof of Theorem 2.6 we first prove the following proposition concerning the asymptotics for Pk,n for n → ∞. Proposition 5.1. Let Mk = {q + k + 1, . . . , p+ q}. We have that Pk,n(λ) = (wMk(λ)) nCMk(λ) (1 +O(exp(−cKn)) , n → ∞, (5.10) uniformly on compact subsets K of C \ Γk. Here cK is a positive constant depending on K. Proof. First rewrite (2.22) as Pk,n(λ) = (wMk(λ)) nCMk(λ) (1 +Rk,n(λ)) . (5.11) with Rk,n defined by Rk,n(λ) = M 6=Mk (wM (λ)) nCM (λ) (wMk(λ)) nCMk(λ) . (5.12) Let K be a compact subset of C \ Γk. If K does not contain branch points then there exists A,B > 0 such that A < |CM (λ)| < B (5.13) for all λ ∈ K and M . Moreover, we have wM (λ) wMk(λ) zq+k(λ) zq+k+1(λ) ≤ sup zq+k(λ) zq+k+1(λ) < 1, (5.14) EIGENVALUES OF BANDED TOEPLITZ MATRICES 21 for all λ ∈ K and M 6= Mk. Therefore one readily verifies from (5.11) that there exist cK such that |Rk,n(λ)| ≤ exp(−cKn) for all λ ∈ K and n large enough. This proves the statement in case K does not contain branch points. Suppose that K does contain branch points. Without loss of generality we can assume that all branch points lie in the interior of K (otherwise we replace K by a bigger compact set). The boundary ∂K of K is a com- pact set with no branch points and therefore (5.10) holds for ∂K by the above arguments. Since wMk and CMk are analytic in K, we find by (5.11) that Rk,n is analytic in K. The maximum modulus principle for analytic functions states that supz∈K |Rk,n(z)| = supz∈∂K |Rk,n(z)| and thereby we obtain that (5.10) also holds for K with the same constant cK = c∂K . � We now state two particular consequences of (5.10). Corollary 5.2. Let k ∈ {−q + 1, . . . , p − 1}. For every compact set K ⊂ C \ Γk we have that µk,n(K) = 0 for n large enough. Proof. Let K be a compact subset of C \ Γk. By (5.10) it follows that Pk,n has no zeros in K for large n. Since nµk,n(K) equals the number of zeros of Pk,n in K the corollary follows. � Corollary 5.3. Let k ∈ {−q + 1, . . . , p− 1}. We have that dµk,n(x) x− λ = dµk(x) x− λ , (5.15) uniformly on compact subsets of C \ Γk. Proof. Let K be a compact subset of C \ Γk. Note that dµk,n(x) x− λ = λi∈spk Tn(a) λi − λ P ′k,n(λ) nPk,n(λ) , (5.16) for all λ ∈ K. With Mk and cK as in Proposition 5.1 we obtain from (5.10) P ′k,n(λ) nPk,n(λ) w′Mk(λ) wMk(λ) +O(1/n), n → ∞, (5.17) uniformly on K. Let us rewrite the right-hand side of (5.17). By expanding both sides of zq(a(z)− λ) = ap j=1(z − zj(λ)) and collecting the constant terms we obtain (−zj(λ)) = . (5.18) Since λ /∈ Γk, we can split this product in two parts, take the logarithmic derivative and use (3.3) and (2.23) to obtain z′j(λ) zj(λ) j=q+k+1 z′j(λ) zj(λ) w′k(λ) wk(λ) w′Mk(λ) wMk(λ) . (5.19) 22 MAURICE DUITS AND ARNO B.J. KUIJLAARS Combining (5.16), (5.17) and (5.19), we obtain dµk,n(x) x− λ = w′k(λ) wk(λ) (5.20) uniformly on K. Then (5.15) follows from (5.20) and (4.9). � Now we are ready for the proof of Theorem 2.6. Proof of Theorem 2.6. First we prove (2.21). By Proposition 2.5 and the fact that ~µ is admissible, we get (see (2.8)) µk,n(C) = degPk,n ≤ µk(C), (5.21) for every n ∈ N. Let C0(C) be the Banach space of continuous functions on C that vanish at infinity. The dual space C0(C) ∗ of C0(C) is the space of regular complex Borel measures on C. By (5.21) the sequence (µk,n)n∈N belongs to the ball in C0(C) ∗ centered at the origin with radius µk(C), which is weak ∗ compact by the Banach-Alaoglu theorem. Let µk,∞ be the limit of a weak ∗ convergent subsequence of (µk,n)n∈N. By weak∗ convergence and Corollary 5.2 we obtain that µk,∞ is supported on Γk. Combining this with (5.15) and the weak ∗ convergence leads to dµk(x) x− λ = dµk,∞(x) x− λ , (5.22) for every λ ∈ C \ Γk. The integrals in (5.22) are known in the literature as the Cauchy transforms of the measures µk and µk,∞. The Cauchy transform on Γk is an injective map that maps measures on Γk to functions that are analytic in C \ Γk (one can find explicit inversion formulae, see for example the arguments in [9, Theorem II.1.4] or the Stieltjes-Perron inversion formula in the special case Γk ⊂ R). Thus it follows from (5.22) that µk,∞ = µk. Therefore µk,n = µk (5.23) in the sense of weak∗ convergence in C0(C) ∗. Thus (2.21) holds if φ is a continuous function that vanishes at infinity. From (5.21) and (5.23) it also follows that µk,n(C) = µk(C), (5.24) Then the sequence (µk,n)n∈N is tight. That is, for every ε > 0 there exists a compact K such that µk,n(C \K) < ε for every n ∈ N. By a standard ap- proximation argument one can now show that (2.21) holds for every bounded continuous function φ on C. Having (2.21) and Proposition 5.1, we can prove (2.20) as in [1, Theo- rem 11.17]. Indeed, the sets lim infn→∞ spk Tn(a) and lim supn→∞ spk Tn(a) equal the support of µk, which is Γk. � EIGENVALUES OF BANDED TOEPLITZ MATRICES 23 –2 –1.5 –1 –0.5 0.5 1 1.5 lambda –2 –1.5 –1 –0.5 0.5 1 1.5 lambda Figure 2. Illustration for Example 1: The densities of the measures µ0 (left) and µ1 (right) for a = 4(z+1)3 6. Examples 6.1. Example 1. As a first example consider the symbol a defined by a(z) = 4(z + 1)3 . (6.1) In this case we have p = 2 and q = 1. So we obtain two contours Γ0 and Γ1 with two associated measures µ0 and µ1. This example appeared in [3], in which the authors gave explicit expressions for Γ0 and µ0. The following proposition also contains expressions for Γ1 and µ1. In what follows we take the principal branches for all fractional powers. Proposition 6.1. With a as in (6.1), we have that Γ0 = [0, 1] and dµ0(λ) = dλ. (6.2) Moreover, Γ1 = (−∞, 0] and dµ1(λ) = )1/3 − 1− λ− 1 (−λ)2/3 dλ. (6.3) Proof. A straightforward calculation shows that λ = 0 and λ = 1 are the branch points. Let λ ∈ Γ0 ∪ Γ1 and assume that λ is not a branch point. There exist y1, y2 ∈ C such that y1 6= y2, |y1| = |y2| and a(y1) = a(y2) = λ. Then it follows from (6.1) that |y1+1| = |y2+1|. Therefore y1 and y2 are intersection points of a circle centered at −1 and a circle centered at the origin. Since y1 6= y2, this means that y1 = y2 and therefore λ = a(y1) = a(y2) = a(y1) = λ, so that λ ∈ R. A further investigation shows that a(z)−λ has 3 different real zeros if λ > 1. If λ < 1 and λ 6= 0 then a(z) − λ has precisely 1 real zero and 2 conjugate complex zeros. Therefore, Γ0 ∪ Γ1 = (−∞, 1]. 24 MAURICE DUITS AND ARNO B.J. KUIJLAARS −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 k = 0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 k = 1 Figure 3. Illustration for Example 1: The spectrum spT50(a) (top) and the generalized spectrum sp1T50(a) (bot- tom), for the symbol a = 4(z+1)3 Now we will show that Γ0 = [0, 1] and Γ1 = (−∞, 0]. By Cardano’s formula the solutions of the algebraic equation a(z) = λ are given by zj(λ) = −1− 3λ1/3 1 + (1− λ)1/2 + ω−j 1− (1− λ)1/2 (6.4) for λ ∈ [0, 1] and zj(λ) = −1+ 3(−λ)1/3 1 + (1− λ)1/2 − ω−j−2 (1− λ)1/2 − 1 (6.5) for λ ∈ (−∞, 0]. Here ω = e2πi/3. One can check that |z1(λ)| = |z2(λ)| < |z3(λ)| for λ ∈ (0, 1] and |z1(λ)| < |z2(λ)| = |z3(λ)| for λ ∈ (−∞, 0). More- over, for λ = 0 we have z1(0) = z2(0) = z3(0) = −1. Therefore Γ0 = [0, 1] and Γ1 = (−∞, 0]. The density (6.2) was already given in [3] and (6.3) follows in a similar way. � In Figure 2 we plotted the densities of µ0 and µ1. Note that, due to the interaction between µ0 and µ1 in the energy functional, there is more mass of µ0 near 0 than near 1. We also see that the singularities of the densities for µ0 and µ1 are of order O(|λ|−2/3) for λ → 0, whereas the typical nature of a singularity in each of the measures is a square root singularity. The stronger singularity is due to the fact that a(z) − λ has a triple root for λ = 0. EIGENVALUES OF BANDED TOEPLITZ MATRICES 25 –4 –2 2 4 lambda –4 –2 2 4 lambda Figure 4. Illustration for Example 2: The densities of the measures µ0 (left) and µ1 = µ−1 (right) for a(z) = z 2 + z + z−1 + z−2. In Figure 3 we plotted the eigenvalues and generalized eigenvalues for n = 50. It is known that the eigenvalues are simple and positive [3, §2.3], which we also see in Figure 3. 6.2. Example 2. For the symbol a defined by a(z) = z2 + z + z−1 + z−2. (6.6) we have p = q = 2. From the symmetry a(1/z) = a(z) it follows that Γ−1 = Γ1 and µ−1 = µ1. The interesting feature of this example is that the contours Γ0 and Γ±1 overlap. To be precise, the interval (−9/4, 0) is contained in all three con- tours Γ−1,Γ0 and Γ1. This can be most easily seen by investigating the image of the unit circle under a. Consider a(eit) = 2 cos 2t+ 2cos t, for t ∈ [0, 2π). (6.7) A straightforward analysis shows that for every λ ∈ (−9/4, 0), the equation a(eit) = λ has four different solutions for t in [0, 2π). This means that the four solutions of the equation a(z) = λ are on the unit circle, and so in particular have the same absolute value. The equation a(z) − λ = 0 can be explicitly solved by introducing the variable y = z + 1/z. In exactly the same way as in the previous example one can obtain the limiting measures. We will not give the explicit formulas, but only plot the densities in Figure 4. The branch points are λ = −9/4, λ = 0 and λ = 4. The contours are given by Γ0 = [−9/4, 4], Γ−1 = Γ1 = (−∞, 0]. (6.8) The densities have singularities at the branch points in the interior of their supports. The singularities are only felt at one side of the branch points. Consider first µ0, whose density has a singularity at 0. However the limiting value when 0 is approached from the positive real axis is finite. The change in behavior of µ0 has to do with the fact that z1 is analytic on (0, 4) but not 26 MAURICE DUITS AND ARNO B.J. KUIJLAARS on (−9/4, 0). Therefore we find by (1.12) that dµ0(λ) = z1+(λ) z2+(λ) z1−(λ) z2−(λ) dλ (6.9) on (−9/4, 0), and dµ0(λ) = z2+(λ) z2−(λ) dλ (6.10) on (0, 4). For µ−1 = µ1 a similar phenomenon happens at λ = −9/4. This is a consequence of the fact that z1 has an analytic continuation into z2 when we cross (−∞,−9/4), but it has an analytic continuation into z4 when we cross (−9/4, 0). 6.3. Example 3. As a final example, consider the symbol a(z) = zp + z−q, (6.11) with p, q ≥ 1 and gcd(p, q) = 1. This example appeared in [10], where the authors mentioned that Γ0 is given by the star Γ0 = {rωj | j = 1, . . . , p+ q, 0 ≤ r ≤ R} (6.12) with ω = e2πi/(p+q) and R = (p + q)p−p/(p+q)q−q/(p+q). The other contours also have a star shape, namely Γk = {(−1)krωj | j = 1, . . . , p+ q, 0 ≤ r < ∞} (6.13) for k 6= 0. Note that the star Γk for k 6= 0 is unbounded. In Figure 5 we plotted the eigenvalues and the generalized eigenvalues for p = 2, q = 3 and n = 50. All the (generalized) eigenvalues appear to lie exactly on the contours. In the special case p = 1 it is known that the eigenvalues of Tn(a) lie indeed precisely on the star (6.12) and are all simple (possibly except for 0) [4, Theorem 3.2], see also [6] for a connection to Chebyshev-type quadrature. 6.4. Numerical stability. In Figure 3 and Figure 5 the eigenvalues and the generalized eigenvalues of T50(a) were computed numerically. To control the stability of the numerical computation of the eigenvalues one needs to analyze the pseudo-spectrum. For banded Toeplitz matrices the pseudo- spectrum is well understood [12, Th. 7.2]. To this date, a similar analysis of the pseudo-spectrum for the matrix pencil (Tn(z −ka), Tn(z −k)) has not been carried out. See [12, §X.45] for some remarks on the pseudo-spectrum for the generalized eigenvalue problem. EIGENVALUES OF BANDED TOEPLITZ MATRICES 27 −5 0 5 k = −2 −5 0 5 k = −1 −5 0 5 k = 0 −5 0 5 k = 1 Figure 5. Illustration for Example 3: The contours Γk and the eigenvalues and generalized eigenvalues for T50(a) for the symbol a = z2 + z−3. References 1. A. Böttcher and S. M. Grudsky, Spectral Properties of Banded Toeplitz Matrices, SIAM, Philadelphia, PA, 2005. 2. A. Böttcher and S. M. Grudsky, Can spectral values sets of Toeplitz band matrices jump?, Linear Algebra Appl., 351-352 (2002), pp. 99-116. 3. E. Coussement, J. Coussement and W. Van Assche, Asymptotic zero distribution for a class of multiple orthogonal polynomials, Trans. Amer. Math. Soc., (to appear) 4. M. Eiermann and R. Varga, Zeros and local extreme points of Faber polynomials associated with hypocycloidal domains, Electron. Trans. Numer. Anal., 1 (1993), pp. 49-71. 5. I. I. Hirschman, Jr., The spectra of certain Toeplitz matrices, Illinois J. Math., 11 (1967), pp. 145-159. 6. A. Kuijlaars, Chebyshev quadrature for measures with a strong singularity, J. Comput. Appl. Math., 65 (1995), pp. 207-214. 7. E. Nikishin and V. Sorokin, Rational Approximations and Orthogonality, Translations of Mathematical Monographs 92, American Mathematical Society, Providence, RI, (1991). 8. T. Ransford, Potential Theory in the Complex Plane, London Mathematical Society Student Texts 28, Cambridge University Press, Cambridge, 1995. 9. E.B. Saff and V. Totik, Logartihmic Potentials with External Fields, Grundlehren der Mathematischen Wissenschaften 316, Springer-Verlag, Berlin, 1997. 10. P. Schmidt and F. Spitzer, The Toeplitz matrices of an arbitrary Laurent polynomial, Math. Scand., 8 (1960), pp. 15-38. 28 MAURICE DUITS AND ARNO B.J. KUIJLAARS 11. P. Simeonov, A weigthed energy problem for a class of admissible weights, Houston J. Math., 31 (2005), pp. 1245-1260. 12. L.N. Trefethen and M. Embree, Spectra and Pseudospectra, Princeton University Press, Princeton, NJ, 2005. 13. J.L. Ullman, A problem of Schmidt and Spitzer, Bull. Amer. Math. Soc., 73 (1967), pp. 883-885. 14. H. Widom, On the eigenvalues of certain Hermitean operators, Trans. Amer. Math. Soc., 88 (1958), pp. 491-522. 1. Introduction 2. Statement of results 2.1. The energy functional 2.2. The measures k as limiting measures of generalized eigenvalues 2.3. Overview of the rest of the paper 3. Preliminaries 3.1. The structure of the curves k 3.2. The Riemann surface 3.3. The functions wk() 4. Proof of Theorem 2.3 5. Proofs of Proposition 2.5 and Theorem 2.6 5.1. Proof of Proposition 2.5 5.2. Proof of Theorem 2.6 6. Examples 6.1. Example 1 6.2. Example 2 6.3. Example 3 6.4. Numerical stability References

0704.0379

Capturing knots in polymers

FIG. 1. Knotted bead-spring polymer: Starting configuration with N=16384 beads; after 6 reduction steps (N=265); final configuration after 15 iterations (N=8) with the knotted (trefoil) region circled in red; and magnified. (enhanced online) Capturing knots in polymers Peter Virnau, Mehran Kardar Department of Physics, MIT, Cambridge, MA 02139-4307, USA Yacov Kantor School of Physics and Astronomy, Tel Aviv University, 69978 Tel Aviv, Israel (received, published) [DOI: 10.1063/1.2130690] Visualizing topological properties is a particularly challenging task. Although algorithms can usually determine if a loop contains a knot, finding its exact location is difficult (and not necessarily well-defined). Here, we apply a reduction method by Koniaris and Muthukumar , which was originally proposed to simplify polymers before calculating knot invariants. We start with one end and consider consecutive triangles formed by three adjacent monomers. If the triangle is not crossed by any of the remaining bonds, the particle in the middle is removed. Going back and forth between both ends we proceed until the configuration cannot be reduced any further (see Fig.1). Although the method is not perfect (sometimes entangled, but unknotted regions remain), it provides us with a valuable impression on the typical number of knots, their respective location and sizes This work was supported by the DFG grant Vi237/1. P. Virnau, Y. Kantor, and M. Kardar, J. Am. Chem. Soc., in press (2005). W. G. Taylor, Nature 406, 916 (2000). K. Koniaris and M. Muthukumar, J.Chem.Phys. 95, 2873 (1991). Pictures and movie were generated using the VMD visualization package; see W. Humphrey, A. Dalke, and K. Schulten,, J. Molec. Graphics 14, 33 (1996). Copyright (2005) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in the Gallery of Images in Chaos 15, 041103 (2005) and may be found at http://chaos.aip.org/chaos/gallery/toc_Dec05.jsp This version also contains a movie of the algorithm.

0704.0380

Exponential growth rates in a typed branching diffusion

Exponential growth rates in a typed branching diffusion The Annals of Applied Probability 2007, Vol. 17, No. 2, 609–653 DOI: 10.1214/105051606000000853 c© Institute of Mathematical Statistics, 2007 EXPONENTIAL GROWTH RATES IN A TYPED BRANCHING DIFFUSION By Y. Git, J. W. Harris1 and S. C. Harris Cambridge University, University of Bristol and University of Bath We study the high temperature phase of a family of typed branch- ing diffusions initially studied in [Astérisque 236 (1996) 133–154] and [Lecture Notes in Math. 1729 (2000) 239–256 Springer, Berlin]. The primary aim is to establish some almost-sure limit results for the long- term behavior of this particle system, namely the speed at which the population of particles colonizes both space and type dimensions, as well as the rate at which the population grows within this asymp- totic shape. Our approach will include identification of an explicit two-phase mechanism by which particles can build up in sufficient numbers with spatial positions near −γt and type positions near t at large times t. The proofs involve the application of a variety of martingale techniques—most importantly a “spine” construction involving a change of measure with an additive martingale. In ad- dition to the model’s intrinsic interest, the methodologies presented contain ideas that will adapt to other branching settings. We also briefly discuss applications to traveling wave solutions of an associ- ated reaction–diffusion equation. 1. Introduction. In this article we will consider a certain family of typed branching diffusions that have particles which move (independently of each other) in space according to a Brownian motion with variance controlled by the particle’s type process. The type of each particle evolves as an Ornstein– Uhlenbeck process and this type also controls the rate at which births occur. The particular form of this model permits many explicit calculations, but throughout we will strive to develop techniques that rely on general prin- ciples as much as possible, so they might readily adapt to other situations. This model was previously considered in [12, 13]; these papers form essential foundations for this work, although we will recall various results as necessary. Received December 2004; revised November 2006. 1Supported in part by an EPSRC studentship. AMS 2000 subject classification. 60J80. Key words and phrases. Spatial branching process, branching diffusion, multi-type branching process, additive martingales, spine decomposition. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Probability, 2007, Vol. 17, No. 2, 609–653. This reprint differs from the original in pagination and typographic detail. http://arxiv.org/abs/0704.0380v1 http://www.imstat.org/aap/ http://dx.doi.org/10.1214/105051606000000853 http://www.imstat.org http://www.imstat.org http://www.imstat.org/aap/ http://dx.doi.org/10.1214/105051606000000853 2 Y. GIT, J. W. HARRIS AND S. C. HARRIS We will make some significant applications of the spine theory for branch- ing processes. Inspired by the series of papers Lyons, Pemantle and Peres [19], Lyons [18] and Kurtz, Lyons, Pemantle and Peres [16], spine techniques have been instrumental in recent years in providing intuitive and elegant proofs of many important classical and new results in the theory of branch- ing processes. In this article we use the recent reformulation of the spine method presented in [8], which follows in similar spirit to the branching Brownian motion study of Kyprianou [17]. For a selection of other applica- tions of spine techniques, for example, see [1, 6, 7, 23] and references therein. 1.1. The branching model. We define Nt to be the set of particles alive at time t ≥ 0. For a particle u ∈ Nt, Xu(t) ∈ R is its spatial position, and Yu(t) ∈ R is the type of u. We will label offspring using the Ulam–Harris convention where, for example, if u=∅21 then particle u is the first child of the second child of the initial ancestor, and we will write v > u if particle v is a descendant of particle u. The configuration of the branching diffusion at time t is given by the point process Xt := {(Xu(t), Yu(t)) :u ∈Nt}. A particle’s type evolves as an Ornstein–Uhlenbeck process with an invari- ant measure given by the standard normal density φ(y) and an associated differential operator (generator) Qθ := − y ∂ where θ > 0 is considered to be the temperature of the system. The spatial motion of a particle of type y is a driftless Brownian motion on R with variance A(y) := ay2, where a≥ 0. A particle of type y particle is replaced by two offspring at a rate R(y) := ry2 + ρ, where r, ρ≥ 0. Each offspring inherits its parent’s current type and spatial position, and then moves off independently of all others. We use P x,y and Ex,y with x, y ∈ R to represent probability and expectation when the Markov process starts with a single particle at position (x, y). We will find the almost sure rate of exponential growth, D(γ,κ), of par- ticles which are found simultaneously with spatial positions near −γt and type positions near κ t at large times t. From this we can deduce the speed of extremal particles and hence the asymptotic shape of the particle system. The main effort is required in identifying D(γ,κ) as the almost sure limit of t−1 log 1{Xu(t)≤−γt;Yu(t)≥κ A TYPED BRANCHING DIFFUSION 3 In particular, the convergence properties of two different families of addi- tive martingales associated with the branching diffusion will lead directly to the spatial exponential growth rates and an upper bound on the space-type growth. For the remaining lower bound, we describe an explicit two-phase mechanism for amassing the required number of particles with prescribed space-type positions. The first phase involves building up an “excess” num- ber of particles, each covering a certain proportion of the required spatial distance. During their second phase, enough of these particles must succeed in making a difficult and rapid ascent into the required position. The latter phase is proved using an intuitive change of measure technique that induces a spine construction. The family of models we are considering is specific but nevertheless have some features of fundamental significance that motivate the choices for Qθ, R and A. If the spatial motion is ignored, we have investigated a binary branching Ornstein–Uhlenbeck process in a quadratic breeding potential. In contrast, Enderle and Hering [5] considered a branching Ornstein–Uhlenbeck with constant branching rate but random offspring distribution. A quadratic breeding potential is a critical rate for explosions in the population of parti- cles. In a branching Brownian motion on R with binary splitting occurring at rate xp at position x, the population will explode almost surely in finite time if p > 2, whereas for p = 2 the expected number of particles explodes while the total population remains finite for all time with probability 1 (see [15], Chapter 5.12). The Ornstein–Uhlenbeck process is not only a canoni- cal ergodic diffusion, but this type-motion has exactly the right drift to help counteract the quadratic breeding rate. For high temperatures, θ > 8r, there is a sufficiently strong mean-reversion in the type processes to ensure that the expected total population size does not blow up; but for temperatures θ ≤ 8r, the quadratic breeding overpowers the pull toward the origin, the expected population blows up in a finite time and particles behave very dif- ferently. Throughout this paper we consider only high temperatures θ > 8r, deferring the low and critical temperature regimes to future work. Given other choices, the quadratic spatial diffusion coefficient now becomes very natural, enabling us to find explicit families of (fundamental) additive mar- tingales since the linearized traveling-wave equation can be linked to the classical harmonic oscillator equations from physics. The binary branching mechanism was taken for simplicity; in principle our approach could extend to general offspring distributions, although new features would arise from possible extinctions and necessary offspring moment conditions. All these choices make the models rich in structure, possessing some very challenging features whilst remaining sufficiently tractable. 4 Y. GIT, J. W. HARRIS AND S. C. HARRIS 1.2. Application to reaction–diffusion equations. Following in the foot- steps of McKean [20], the solution of the reaction–diffusion equation +R(y)u(u− 1) + θ − y∂u with initial condition f(x, y)∈ [0,1] for all x, y ∈R, can be represented by u(t, x, y) =Ex,y f(Xu(t), Yu(t)) Of great importance for reaction–diffusion equations are traveling-wave so- lutions (e.g., see [21]). In the present context, a solution to equation (1) of the form u(t, x, y) := w(x− ct, y) is said to be a traveling-wave of speed c, where w(x, y) solves the traveling-wave equation +R(y)w(w − 1) + θ − y∂w = 0.(3) Fundamental to our study of the branching diffusion are two families of “ad- ditive” martingales, Z±λ (t) [defined at (6)], which are linked to the lineariza- tion of (1). When θ > 8r, Harris and Williams [13] determined when Z−λ is uniformly integrable (see Theorem 17) and then wλ(x, y) :=E x,y exp(−Z−λ (∞)) yields a traveling wave of speed c−λ . This gives the existence of traveling waves for all speeds c greater than some threshold c̃(θ) := inf c−λ . Furthermore, combining the McKean representation (2) with the almost- sure convergence result established in [12] (look ahead to Theorem 18) can give results on the attraction toward traveling waves from given initial data. For example, if − lnf(x, y)∼ eλxg(y) uniformly in y as x→∞ for some suit- able g ∈ L2(φ), the solution u(x, y) to (1) with initial conditions f satisfies u(t, x− c−λ t, y)→wλ(x+ x̂, y) as t→∞, where x̂ is some constant that can be determined from g. In future work we hope to develop the approach used for standard BBM and the FKPP equation in [11], and prove that traveling waves of a given speed c > c̃(θ) are unique (up to translation) and that no traveling waves exist for speeds c < c(θ). We anticipate that our new results on the growth rates of particles will aid in establishing some difficult estimates on the tail behavior of any traveling wave, and hence assist in proving the conjectured uniqueness. In addition, we expect our growth rate results will be essential in obtaining broader classes of initial conditions that are attracted toward traveling waves. In each of these problems, difficulties arise from the un- bounded type space where, for example, some control must be gained over the possible contributions to u∈Nt log f(Xu(t) − ct, Yu(t)) from particles that have large type positions in addition to large spatial positions. A TYPED BRANCHING DIFFUSION 5 2. Main results. In this section, we will present our main results that identify the growth rates found within the branching diffusion. We will give an overview for our proofs, identifying the key ideas and techniques used, as well as introducing some intuition for the dominant behavior of particles that underpins our approach. 2.1. Martingales. The principal tools used throughout this paper are two fundamental families of “additive” martingales, which were introduced in [13]. Before defining the martingales we give some key definitions. Let λmin :=− θ− 8r Let λ ∈R, with the following convention which we always use for λ: λmin <λ< 0. Also, define µλ := θ(θ− 8r− 4aλ2), ψ±λ := E±λ := ρ+ θψ λ , c λ :=−E λ /λ.(5) Will will occasionally write E±λ as E ±(λ) in order to emphasize that E±λ are really functions of λ; the ± superscripts will always distinguish these from expectation operators. Note that λmin is the point beyond which µλ is no longer a real number. The martingales are Z−λ and Z λ , defined for λ ∈ (λmin,0] as Z±λ (t) := v±λ (Yu(t))e λXu(t)−E±λ t,(6) where v±λ (y) := exp(ψ 2) are strictly-positive eigenfunctions of the opera- Qθ + 12λ 2A+R, with corresponding eigenvalues E−λ <E λ and A,R are the functions defined in Section 1.1. This operator is self-adjoint on L2(φ) with the inner product 〈·, ·〉φ where 〈f, g〉φ := fgφdy and φ is the standard normal density. Note that v−λ ∈ L2(φ), whereas v λ /∈L2(φ) so is not normalizable. The calculations of Section 3 make it easy to see these are martingales, and throughout the paper we will need a variety of martingale convergence results which are gathered together in Section 8. In particular, we will need to know precisely when Z−λ is uniformly integrable with a strictly positive limit, some further strong convergence results for other closely related sums over particles (also identifying which particles contribute nontrivially to their limits), and the rate of convergence to zero of the Z+λ martingales. 6 Y. GIT, J. W. HARRIS AND S. C. HARRIS 2.2. The asymptotic growth-rate of particles along spatial rays. As an essential initial step toward determining the growth rate of particles in the two-dimensional space-type domain, we first look at the growth rate of par- ticles in the spatial dimension only. For γ ≥ 0 and C ⊂R, define Nt(γ;C) := 1{Xu(t)≤−γt;Yu(t)∈C}.(7) The limit giving the expected rate of growth, t−1 logE(Nt(γ;R)) can be shown to exist and its value can be calculated to be ∆(γ) := inf λ∈(λmin,0) {E−λ + λγ} a−1(θ − 8r)(4γ2 + θa). An outline for this expectation calculation is given in Section 3. It is now tempting to guess that the asymptotic speed of the spatially left-most particle, c̃(θ), is given by c̃(θ) := sup{γ :∆(γ)> 0} r+ ρ+ 2(2r+ ρ)2 θ− 8r Recall that c̃(θ) = infλ∈(λmin,0) c λ is also the minimum threshold for traveling waves. In this particular situation, the guess that “expectation” and “almost sure” right-most particle speeds agree was first proved rigorously using a martingale change of measure technique in [13]. In this paper, we extend this connection and prove that the “expected” and “almost sure” rates of growth of particles with given speeds (Theorem 1) and given space-type locations (Theorem 3) agree. Theorem 1. Let γ ≥ 0 and y0 < y1. Under each P x,y law, the limit D(γ) := lim t−1 logNt(γ; [y0, y1]) exists almost surely and is given by D(γ) = ∆(γ), if 0≤ γ < c̃(θ), −∞, if γ ≥ c̃(θ). A TYPED BRANCHING DIFFUSION 7 Note that symmetry in the process means there is a corresponding re- sult for particles with spatial velocities greater than +γ (corresponding to positive λ values). We may occasionally make use of such process symme- tries without further comment. Then, since Nt(γ;R) is integer valued, the asymptotic speed of the right-most particle follows immediately: Corollary 2. Almost surely, t−1 sup{Xu(t) :u ∈Nt}= c̃(θ). This spatial growth rate result is proved in Section 10 using the martin- gale results from Section 8. In fact, it is very easy to obtain the upper bound by first dominating the indicator function with exponentials to reveal that Nt(γ;R) ≤ exp{(E−λ + λγ)t}Z λ (t), recalling that Z λ is a convergent mar- tingale, and then optimizing over the choice of λ. For the lower bound, we will use a strong convergence result obtained in [12], combined with the idea that each uniformly integrable martingale Z−λ essentially “counts” only the particles of corresponding velocity −γ. 2.3. The asymptotic shape and growth of the branching diffusion. The main result of this paper is the almost-sure rate of growth of particles which are in the vicinity of −γt in space and near κ t in type position at large times t. For γ, κ≥ 0, it can be shown that the limit t−1 logE(Nt(γ; [κ t,∞)))(10) exists and takes the value ∆(γ,κ) := inf λ∈(λmin,0) {E−λ + λγ − κ 2ψ+λ } (θ− κ2) θ(θ− 8r)(4aθγ2 + a2(θ+ κ2)2). An outline of this expectation calculation is given in Section 3. Once again, we will find that the “almost sure” rate of growth of particles agrees with this “expected” rate exactly where there is growth in particle numbers. Theorem 3. Let γ,κ ≥ 0 with ∆(γ,κ) 6= 0. Under each P x,y law, the limit D(γ,κ) := lim t−1 logNt(γ; [κ t,∞)) exists almost surely and is given by D(γ,κ) = ∆(γ,κ), if ∆(γ,κ)≥ 0, −∞, if ∆(γ,κ)< 0.(12) 8 Y. GIT, J. W. HARRIS AND S. C. HARRIS To prove the tricky lower bound of Theorem 3, which amounts to the major work of this paper, we will exhibit an explicit two-phase mechanism by which the branching diffusion can build up at least the required exponential number of particles near to −γt in space and κ t in type position by large times t. During the first phase, over a large time t the process builds up an ini- tial excess of approximately exp(∆(α)t) particles with spatial position at least −αt, as is already known from Theorem 1. In this “ergodic” phase, “typical” particles found near −αt in space will have drifted with a steady spatial speed of α whilst their type histories will have behaved roughly like OU processes with inward drift of µλy for a certain optimal choice λ(α) of parameter λ. For the second phase, we will show that the probability any individual particle has at least one descendant that makes a “rapid ascent” in both space and type dimensions from initial position (0,0) to final position near (−βt,κ t) is approximately exp(−Θ(β,κ)t), where Θ(β,κ) = θ(θ− 8r)(a2κ4 +4aθβ2) ,(13) and the time taken for this “rapid ascent” is an interval [0, τ ]. We show that this time τ can be chosen such that 2µλτ ∼ log t, and hence the additional time is asymptotically negligible in comparison with t. Intuitively, we will see that given an offspring that has successfully made such a difficult “rapid ascent,” it will most likely have roughly had its type process behaving like an OU process with an outward drift of µλy and the Brownian motion driving its spatial motion will have had a drift λ [corresponding to a real time spatial drift λA(y) that increases in strength as the type position y increases], for some optimal choice λ(β,κ) of parameter λ. The precise result required will be formulated rigorously as a large-deviation lower bound in Theorem 7 of Section 5, and is proved using a “spine” change of measure technique intimately related to the Z+λ martingales. Combining these two phases and using independence of the particles, we can see that the number of particles near (−αt,0) at time t that subse- quently proceed to have at least one descendant near (−(α + β)t, κ t) is approximately Poisson with mean exp({∆(α)−Θ(β,κ)}t). Optimizing for a fixed overall spatial speed γ, some calculus reveals that α+β=γ α,β>0 {∆(α)−Θ(β,κ)}=∆(ᾱ)−Θ(β̄, κ) = ∆(γ,κ),(14) A TYPED BRANCHING DIFFUSION 9 with optimal parameters ᾱ= γ θ+ κ2 and β̄ = γ θ+ κ2 .(15) Thus we will be able to demonstrate an explicit two-phase mechanism pro- ducing the required number of particles, with this outline argument later guiding our rigorous proof. In addition, it is interesting to note that the optimal choices for λ over each phase then also coincide at a single value λ̄= λ(ᾱ) = λ(β̄, κ). An informative large deviation heuristic for the rapid ascent can also be found in Section 4, with this section also containing some essential optimal path calculations. We actually prove the two-phase mechanism for the lower bound of Theorem 3 in Section 5, although we defer proving the large- deviation lower bound until Section 7 after presenting the necessary “spine” background in Section 6. We prove the upper bound for the space-type growth rate in Section 9, again making crucial use of martingale results from Section 8. Similarly to the spatial growth case, we can find an upper bound using the Z+λ mar- tingales, that is to say Nt(γ; [κ t,∞)) ≤ exp{(E+λ + λγ − κ2ψ λ )t}Z λ (t). However, as each Z+λ martingale converges to zero, we must show that its exponential decay rate is (E+λ −E λ ) before being able to optimize over the choice of λ to obtain the required upper bound. Given Theorem 3, and noting symmetries, it becomes straightforward to retrieve the following: Corollary 4. For any F ⊂R2, define Nt(F ) := 1{(Xu(t)/t,Yu(t)/ t)∈F}. If B ⊂R2 is any open set and C ⊂R2 is any closed set, then almost surely under any P x,y lim inf logNt(B)≥ sup (γ,κ)∈B D(γ,κ), lim sup logNt(C)≤ sup (γ,κ)∈C D(γ,κ), with the growth rate D(γ,κ) given at equation (12). We can also recover the almost sure asymptotic shape of the region occu- pied by the particles in the branching diffusion. 10 Y. GIT, J. W. HARRIS AND S. C. HARRIS Corollary 5. Let B ⊂R2 be any open set. Almost surely, under each P x,y law, Nt(B)→ 0, if S ∩B =∅, +∞, if S ∩B 6=∅, where S ⊂R2 is the set given by S := {(γ,κ) ∈R2|∆(γ,κ)> 0}. 3. Some expectation calculations. This section discusses how the ex- pected growth rates given in the previous section may be obtained. For this, we use the “many-to-one” lemma (see, e.g., [8]) and one-particle changes of measure. In the process we shall start to gain valuable intuition into how particles within the branching diffusion behave, as well as seeing hints as to which are the “correct” martingales to use to prove the almost-sure growth rate results. For simplicity, we assume throughout this section that the branching dif- fusion starts with one particle at the origin in both space and type at time zero, unless otherwise stated. We also introduce a family of single particle probability measures Pµ,λ with associated expectations Eµ,λ where, under Pµ,λ, η is an Ornstein–Uhlenbeck process with variance θ and drift µ, and ξt =B( 0 A(ηs)ds) where B is a Brownian motion with drift λ. Lemma 6 (Many-to-one). If f :R2 7→R is Borel measurable then f(Xu(t), Yu(t)) = Eθ/2,0 R(ηs)ds f(ξt, ηt) .(16) Using the many-to-one lemma, and changing measure to alter the drift of Brownian motion, we see that f(Xu(t), Yu(t)) = Eθ/2,0 R(ηs)ds f(ξt, ηt) = Eθ/2,0 e−λξt exp R(ηs) + A(ηs) × f(ξt, ηt) · eλξt−λ A(ηs)ds = Eθ/2,λ −λξt + R(ηs) + A(ηs) f(ξt, ηt) A TYPED BRANCHING DIFFUSION 11 To perform a further change of measure on the OU process to get rid of the time integrals in the exponential of the expectation, we recall that dPµλ,· dPθ/2,· µλ,θ/2 := exp ψ−λ η t −E−λ t+ R(ηs) + λ2A(ηs) and then f(Xu(t), Yu(t)) = Eθ/2,λ(exp(−λξt −ψ−λ η λ t)f(ξt, ηt) ·M µλ,θ/2 t )(17) = Eµλ,λ(exp(−λξt − ψ λ t)f(ξt, ηt)). Note that the many-to-one lemma, combined with the branching prop- erty, immediately suggests how to get “additive” martingales for the branch- ing diffusion from single particle martingales—for example, taking f(x, y) = exp{λx+ ψ−λ y2} in equation (17) quickly leads to the martingale Z We may now proceed to calculate the expected growth rates. However, for both clarity and brevity we will leave rigorous details to the interested reader, noting that the intuition we will gain from our rough calculations will later be invaluable in guiding our rigorous proof of the corresponding almost-sure growth rates. 3.1. The expected rate of growth along spatial rays. We first give the outline of some calculations to find the rate of growth in the expected number of particles near −γt in space at time t. Using the formula from (17), for λ ∈ (λmin,0) and any ε > 0 we have 1{t−1Xu(t)+γ∈(−ε,ε)} = Eµλ,λ(e −λξt−ψ−λ η 1{t−1ξt+γ∈(−ε,ε)}) ≤ e(E +λγ−λε)tEµλ,λ + γ ∈ (−ε, ε) ≥ e(E +λγ+λε)tEµλ,λ η2t ; + γ ∈ (−ε, ε) where, with some abuse of notation that we shall continue to use throughout this section, we will abbreviate this to 1{Xu(t)∼−γt} = Eµλ,λ(e −λξt−ψ−λ η 1{ξt∼−γt}) ∼ e(E +λγ)t Eµλ,λ(e t ; ξt ∼−γt) 12 Y. GIT, J. W. HARRIS AND S. C. HARRIS with the understanding that any subsequent arguments to identify expo- nential growth rates can readily be made rigorous by using the appropriate upper and lower bounds, and so on. Now, considering E−(λ) :=E−λ as a function of λ, we have from (8) that ∆(γ) = infλ∈(λmin,0){E−(λ) + λγ}=E−(λγ) + λγγ, where λγ satisfies (λγ) =−γ, hence λγ =−γ (θ− 8r) θa2 +4aγ2 .(19) Of course, choosing this optimal λγ value in (18) means that we must have si- multaneously maximized the expectation Eµλ,λ(exp(−ψ t ); ξt ∼−γt), and to confirm that this value is not exponentially decaying in t is now relatively straightforward. Under Pµλ,λ, η is an Ornstein–Uhlenbeck process with an invariant measure given by the probability density, φλ, of the normal distri- bution N(0, θ/(2µλ)); and ξt =B( 0 A(ηs)ds), where B is a BM with drift λ. Note also that by differentiating 〈(Qθ + (1/2)λ2A+R−E−λ )v λ , v λ 〉φ = 0 with respect to λ, using self-adjointness, and observing that φλ ∝ (v−λ )2φ, we find that 〈λAv−λ , v 〈v−λ , v A(y)φλ(y)dy. Then almost surely under Pµλ,λ, 0 A(ηs)ds) 0 A(ηs)ds 0 A(ηs)ds Aφλ dy = ,(20) and so when we use the optimal λγ value we get exactly the desired drift, since ∂E (λγ) =−γ. Then Eµλγ ,λγ (e η2t ; ξt ∼−γt) → lim Eµλγ ,λγ(e η2t ) φλγ (y)dy. In this way, we can obtain the exact rate of exponential growth for the expectation, t−1 logE(Nt(γ;R)) = ∆(γ). The changes of measure used above are actually suggesting a great deal about the dominant particles that are found in the vicinity of a given ray in space. An alternative discussion of this expectation result, involving a dual approach via large deviation theory for occupation densities, can also be found in [13]. A TYPED BRANCHING DIFFUSION 13 3.2. The expected asymptotic shape. We give a rough outline of calcula- tions that will yield the correct exponential growth in the expected number of particles both near −γt in space and κ t in type at large times t. Using the formula from (17) and abusing notation throughout in the same way as Section 3.1, we find that 1{Xu(t)∼−γt;Yu(t)≥κ = Eµλ,λ(e −λξt−ψ−λ η 1{ξt∼−γt;ηt≥κ ∼ e(E +λγ−κ2ψ− Pµλ,λ(ξt ∼−γt;ηt ≥ κ Now, from standard bounds on the tail of the normal distribution, Pµλ,λ(ξt ∼−γt;ηt ≥ κ = Pµλ,λ(ηt ≥ κ t)Pµλ,λ(ξt ∼−γt|ηt ≥ κ t)(21) ∼ e−µλ/θκ2tPµλ,λ(ξt ∼−γt|ηt ≥ κ and, since ψ−λ + (µλ/θ) = ψ λ , this yields 1{Xu(t)∼−γt;Yu(t)≥κ ∼ e(E +λγ−κ2ψ+ Pµλ,λ(ξt ∼−γt|ηt ≥ κ Recalling that ∆(γ,κ) := infλ∈(λmin,0){E λ +λγ−κ2ψ λ }, simple calculus re- veals this infimum is attained at a λ value of λ̄(γ,κ) =−γ θ(θ− 8r) a2(κ2 + θ)2 + 4aγ2θ ∈ (λmin,0),(23) and using this optimal value in equation (22) will lead to the upper bound limsup t−1 logE 1{Xu(t)∼−γt;Yu(t)≥κ t} ≤∆(γ,κ). It is also clear from equation (22) that when minimizing E−λ + λγ − κ2ψ we simultaneously maximize the probability Pµλ,λ(ξt ∼ −γt |ηt ≥ κ t). In particular, to get a matching lower bound, we do not want this probability to have any exponential decay in time when we choose the optimal parameter for λ. In fact, at least up to the exponential decay rate in time, it can be shown using large-deviations arguments that Pµλ̄,λ̄ (ξt ∼−γt;ηt ≥ κ t)∼ exp 14 Y. GIT, J. W. HARRIS AND S. C. HARRIS Indeed, we immediately gain the required upper bound from (21). For the lower bound, consider the following heuristics where we break paths into two sections: normal ergodic behavior over large time period [0, t] followed by a rapid ascent out to type position κ t over a much shorter period [t, t+ τ ]. (i) Ergodic behavior. Over a large time t, the occupation density of η will most likely have settled close to the invariant measure. Hence for large t, almost surely under Pλ,µλ , η2s ds→ (ii) Rapid ascent. Over a large time τ , but where τ = o(t), the probability that η starts close to the origin and ends near to κ t, having followed close to the path y over the entire time period τ , is roughly given by {ẏ(s) + µλy(s)}2 ds under the Pλ,µλ law. See, for example [24], Chapter 6, or [4], Chapter 5.6. After some Euler–Lagrange optimization, the path y(s) = κ sinhµλs sinhµλτ gives 0 y(s) 2 ds≈ κ2t/(2µλ), with the probability of this path being roughly exp(−(µλ/θ)κ2t). Combining these two types of behavior, we can find paths with final positions ηt+τ ∼ κ t, ξt+τ ∼ λa ∫ t+τ η2s ds∼ λa and, moreover, when substituting the optimal λ value of λ̄(γ,κ) and sim- plifying, this actually gives ξt+τ ∼ −γt. Further, one of these paths occurs with a probability of roughly exp(−(µλ̄/θ)κ2t), and note that t+ τ ∼ t since τ = o(t). Thus we see that to exponential order, the probability Pµλ̄,λ̄ (ξt ∼ −γt;ηt ≥ κ t) must be at least exp(−(µλ̄/θ)κ2t), as required. This heuristic argument can be made rigorous to prove, as claimed, that t−1 logE 1{Xu(t)≤−γt;Yu(t)≥κ =∆(γ,κ). If we scale all spatial coordinates by t−1 and all type coordinates by t)−1 at time t, the expected asymptotic shape can be considered to be the region S := {(γ,κ) :∆(γ,κ) ≥ 0} where, on average, we have growth in the numbers of (scaled) particles. A TYPED BRANCHING DIFFUSION 15 4. Short climb large deviation heuristics. In this section, we give a heuris- tic calculation that suggests why the probability a single particle manages to have at least one descendant in the vicinity of (−βt,κ t ) near time τ is roughly exp(−Θ(β,κ)t) for very large t, where Θ(β,κ) is given at equa- tion (13). For these heuristics, we will think of τ as large and fixed, but of smaller order than t (later on, in our rigorous approach, we will choose τ proportional to log t). We emphasize that the heuristics in this section are neither meant to be precise nor made rigorous, yet they will provide invalu- able intuition, guidance and motivation for our rigorous approach later on. Of particular importance will be the optimization problem that the heuris- tics suggest. Indeed, many of the exact calculations in Sections 4.2 and 4.3 will be essential later in the paper. Suppose we start the branching diffusion with a single particle at (0,0). First, we wish to know the probability that there is at least one particle at time τ that has a spatial position near −βt having followed close to the path x(s) for 0≤ s≤ τ and a type position near κ t having closely followed the path y(s) for 0≤ s≤ τ for t arbitrarily large. We recall from large deviation theory of Ventcel–Freidlin (see [24], Chap- ter 6, or [4], Chapter 5.6) that the probability a single particle manages to follow closely both the type path y(s) and the spatial path x(s) for 0≤ s≤ τ is roughly given by ẏ(s) + ds− 1 ẋ(s)2 ay(s)2 when x(0) = 0, x(τ) = −βt, y(0) = 0, y(τ) = κ t and t is very large. This probability will typically be very small, but if such paths are followed by particles in the branching diffusion, we have to also take account of the large breeding rates that are found far from the type origin. If we let X(s) represent the numbers of particles in the branching diffusion that are alive at time s and have traveled “close” to the path (x(u), y(u)) for 0 ≤ u ≤ s, then we can get a rough idea of how X might behave by considering the following birth–death process. 4.1. A birth–death process. For given fixed paths x(·) and y(·), let M be a time-dependent birth–death process where at time s particles either give birth to single offspring with breeding rate λ(s) given by λ(s) = ρ+ ry(s)2, or particles die with death rate µ(s) given by µ(s) = ẏ(s) + ẋ(s)2 ay(s)2 16 Y. GIT, J. W. HARRIS AND S. C. HARRIS (Note that the probability the initial particle of this birth–death process sur- vives the entire time period [0, τ ] is consistent with the rough large deviation probability for the branching diffusion at equation (24).) An important quantity is the effective total death rate up to time t which is defined by ν(s) := 0 {µ(w)− λ(w)}dw, so here ν(s) = J(x, y, s) ẏ(w) + ẋ(w)2 ay(w)2 − ry(w)2 − ρ The distribution for total number of offspring surviving, M(τ), for the time-dependent birth–death process is well known, for example, see [14]. Then defining Wτ := e −ν(τ) µ(s) eν(s) ds Uτ := 1− e−ν(τ)W−1τ , Vτ := 1−W−1τ , we have P(M(τ) = 0) = Uτ , P(M(τ) = n) = (1−Uτ )(1− Vτ )V n−1τ , n= 1,2, . . . with EM(τ) = e−ν(τ) and E(M(τ)|M(τ)≥ 1) =Wτ . In our particular case, we have E(M(τ)) = exp(−J(x, y, τ)). Define the largest effective total death rate prior to time τ by L(x, y, τ) := sup s∈[0,τ ] J(x, y, s)≥ 0. If we are in a case when L(x, y, τ) is very large, suggesting a high chance of extinction, then P(M(τ)≥ 1) = 1 0 µ(s)e ν(s) ds ∼Kτ exp(−L(x, y, τ)),(25) where K−1τ := 0 µ(s) exp(−{L(x, y, τ) − J(x, y, s)})ds. If there is at least one particle alive, we would then expect to have E(M(τ)|M(τ)≥ 1)∼K−1τ exp(L(x, y, τ)− J(x, y, τ)). Thus, we might guess that the probability any particles in the branching diffusion manage to make the difficult, rapid ascent along path (x, y) to finish up near (−βt,κ t ) can, very roughly, be estimated by exp(−L(x, y, τ)). [To A TYPED BRANCHING DIFFUSION 17 help see this, try writing x(s) = tf(s) and y(s) = tg(s), thinking of f, g as fixed paths and recall that t is very large and τ = o(t), then the role of Kτ in (25) is insignificant next to exp(−L(x, y, τ)).] We might then further guess that the chance any particles manage to stay near position (−βt,κ t) during a very small interval of time close to τ should roughly look like − inf L(x, y, τ) where we permit all possible paths x and y satisfying x(0) = 0, x(τ) =−βt and y(0) = 0, y(τ) = κ t for the fixed time τ . (We will state and prove a precise lower bound that corresponds to this guess at Theorem 7.) 4.2. Finding the optimal path and probability. We proceed to calculate L(x, y, τ) over paths x and y satisfying x(0) = 0, x(τ) =−βt and y(0) = 0, y(τ) = κ for the fixed time τ . We first note that L(x, y, τ) = inf s∈[0,τ ] J(x, y, s)≥ inf J(x, y, τ)(26) and we now proceed to calculate infx,y J(x, y, τ). We can easily optimize over the choice of function x given y, finding that ẋ(s)∝ ay(s)2 ⇒ x(s) = λa y(u)2 du where λ is the constant of proportionality and must satisfy 0 y(s) ,(27) yielding ẋ(s)2 ay(s)2 0 y(s) This is exactly as anticipated since, when following the path y in type space, the spatial position of a particle is following a Brownian motion with total amount of variance over period τ given by a 0 y(s) 2 ds. Hence, the proba- bility that a particle following the path y in type space will also be found near to βt in space at time τ is roughly ( −β2t2 0 y(s) 18 Y. GIT, J. W. HARRIS AND S. C. HARRIS Introducing the notation I(y) := ẏ(s) + − ry(s)2 we are left to find I(y) + 0 y(s) = inf I(y)− 1 y(s)2 ds− λβt ≥ sup I(y)− 1 y(s)2 ds− λβt where the first equality is trivially true by maximizing the quadratic in λ, the introduction of which conveniently removes the awkward integral in the denominator. Some further Euler–Lagrange optimization now gives the optimal path as yλ(s) = κ sinhµλs sinhµλτ (0≤ s≤ τ),(29) where θ(θ− 8r− 4aλ2) and then I(y)− 1 y(s)2 ds− λβt = sup cothµλτ − λβt The optimal parameter choice λ̂ (which depends on τ as well as the model parameters) then satisfies = κ2t cothµ 2 sinh2 µ (s)2 ds.(30) Then we have shown that 0 yλ̂(s) ≥ inf I(y) + 0 y(s) = inf I(y)− 1 y(s)2 ds− λβt ≥ sup I(y)− 1 y(s)2 ds− λβt ≥ I(y (s)2 ds− λ̂βt, A TYPED BRANCHING DIFFUSION 19 and, in fact, we see that the left- and right-hand sides of the above are equal by recalling (30). It follows that the preceding supremum and infimum can be freely interchanged, actually preserving equality at the inequality (28). Then, with the optimal spatial path xλ(s) := λa yλ(u) 2 du=−βt sinh2µλs− 2µλs sinh2µλτ − 2µλτ ,(31) and defining x̂ := x , ŷ := y , we have J(x, y, τ) = J(x̂, ŷ, τ) = t sup cothµλτ cothµ − λ̂β − ρτ. Finally, it is easy to check that J(x̂, ŷ, τ) = L(x̂, ŷ, τ), whence J(x, y, τ)≥ inf L(x, y, τ), and, combining with equation (26), we have found that L(x, y, τ) = inf J(x, y, τ) = J(x̂, ŷ, τ). 4.3. An important note on the optimal paths. As τ →∞, we have cothµλτ ↑ sup {κ2ψ+λ − λβ}= κ − λ̄ β, where the optimizing parameters of the supremums also converge with λ̂→ λ̄=−β θ(θ− 8r) a2κ4 + 4aθβ2 κ2 + θ .(32) Note the agreement with previous optimal values at equations (23) and (15). Then letting Θ(β,κ) := sup {κ2ψ+λ − λβ} θ(θ− 8r)(a2κ4 +4aθβ2) and writing x̄ := xλ̄ and ȳ := yλ̄, we note that for all ε, δ > 0 there exist τ̃ , µ > 0 such that for all t > 0 and τ > τ̃ − inf J(x, y, τ) ≥ exp(−J(x̄, ȳ, τ)) = exp cothµλ̄τ − λ̄β ≥ exp(−t(Θ(β,κ) + ε)). 20 Y. GIT, J. W. HARRIS AND S. C. HARRIS Further (when κ > 0), for all s ∈ [τ − µ, τ ], ȳ(s)≥ (κ− δ) t, x̄(s)≤−(β − δ)t. In particular, the paths stay close to the required positions for some fixed length of time with corresponding probability at least as large as required. 5. Proof of Theorem 3. Lower bound. In this section we will state a pre- cise short climb probability result and show how to combine it with almost sure spatial (only) growth rates to prove the lower bound of the growth rate in Theorem 3. This will make rigorous the two-phase mechanism described in Section 2 and suggested by the expectation calculations in Section 3. The first phase requires knowledge of the almost-sure rates of growth of particles in the spatial dimension only. To this end, we will already make full use of Theorem 1 throughout this section, deferring its proof until Section 10. The second phase requires a lower bound for the probability that a single particle makes a rapid ascent in type-space over the time interval [0, τ ]. This is the lower bound found in the heuristics of Section 4, but we require some further notation before the precise result can be stated. Note, throughout this section, we will only be interested in the optimal parameter value λ= λ̄ as introduced in Section 4.3. We wish to fix the relationship between sufficiently large t and τ as θ/(2µλ̄) e µλ̄τ = κ t(34) and so define τ = τ(t) by τ(t) := (2µλ̄) −1 log(2µλ̄t/θ), for 2µλ̄t > θ, 0, otherwise. Recall the optimal paths (x̄, ȳ) over s ∈ [0, τ ], where ȳ(s) = κ sinhµλ̄s sinhµλ̄τ ,(36) x̄(s) = aλ̄ ȳ(w)2 dw=−βt sinh2µλ̄s− 2µλ̄s sinh2µλ̄τ − 2µλ̄τ ,(37) with fixed end points ȳ(τ) = κ t and x̄(τ) =−βt. For large times t and δ, ε > 0, let t (u) := s∈[0,τ(t)] |Yu(s)− ȳ(s)|< ε t; sup s∈[0,τ(t)] |Xu(s)− x̄(s)|< δt .(38) We will use the notation u∈Nτ(t) t (u)(39) A TYPED BRANCHING DIFFUSION 21 for the event that there exists a particle in the branching diffusion that makes the short climb. Finally, recalling Θ(β,κ) given at (33), we can now state the short climb theorem: Theorem 7. Fix any y1 > y0 > 0, x ∈ R, and let ε0 > 0. Then for any ε, δ > 0, there exists T > 0 such that for all y ∈ [y0, y1], t−1 logP x,y(Aε,δt )≥−(Θ(β,κ) + ε0) for all t > T . We will prove Theorem 7 using a spine change of measure. This requires us to introduce the notation for the spine set-up in detail before proceeding, so this and further technical issues are postponed to Sections 6 and 7. Remark 8. We note that Theorem 7 is actually a stronger result than needed to prove Theorem 3 because we identify the specific paths followed by particles that are near position (βt,κ t ) at time t+ τ , rather than just considering the particle’s positions close to time t+ τ . In combining the two phases, we will have a huge number of independent trials each with a small probability of success, intuitively giving rise to a Poisson approximation for a large number of successful particles. In fact, in our proof of the lower bound of Theorem 3 below, we will actually use the following result about the behavior of sequences of sums of independent Bernoulli random variables. Lemma 9. For each n, define the random variable Bn := u∈Fn 1En(u) where the events {En(u) :u ∈ Fn} are independent. Let pn(u) := P (En(u)) and Sn := u∈Fn pn(u) and suppose that, for some ν ∈ (1/2,1), (Sn)2ν−1 <∞.(40) Then the sequence of (possibly dependent) random variables {B1,B2, . . .} has |Bn − Sn|> (Sn)ν for only finitely many n, almost surely. In particular, for any ε > 0, there exists some (random) N ∈N such that, with probability one, > 1− ε for all n >N.(41) Proof. For ν ∈ (1/2,1), Chebyshev’s inequality yields P(|Bn − Sn|> (Sn)ν)≤ u∈Fn pn(u)(1− pn(u)) 2ν−1 , 22 Y. GIT, J. W. HARRIS AND S. C. HARRIS and hence the Borel–Cantelli lemmas, combined with hypothesis (40), imply |Bn − Sn|> (Sn)ν for only finitely many n, almost surely. Equation (41) now follows on division by Sn, and noticing the assumption (40) implies that limn→∞Sn =∞. � Proof of Theorem 3. Lower bound. Define f−1(t) := t− τ(t), not- ing that both f(t)/t→ 1 and f−1(t)/t→ 1 as t→∞. Also, for n ∈ N and µ > 0, define Tn := (n+ 1)µ. We want to estimate the number of particles that are near the large position (−(α + β)Tn, κ Tn) during time interval [Tn−1, Tn]. For this, we will consider particles that travel with a velocity −α over time period [0, f−1(Tn)] before commencing their rapid ascent of (relatively short) duration τ(Tn) to be in final position at time Tn. Then s∈[Tn−1,Tn] Ns((α+ β − δ)Tn; [(κ− δ) Tn,∞)) u∈NTn s∈[Tn−1,Tn] {Xu(s)≤−(α+β−δ)Tn ;Yu(s)≥(κ−δ) Tn}}(42) u∈Fαn 1{N̄β,κn (u)>0} where Fαn := {u ∈Nf−1(Tn) :Xu(f −1(Tn))≤−αTn, Yu(f−1(Tn)) ∈ [y0, y1]} and, for u ∈ Fαn , N̄β,κn (u) := v∈NTn s∈[Tn−1,Tn] {Xv(s)−Xv(f−1(Tn))≤−(β−δ)Tn ;Yv(s)≥(κ−δ) Tn}}. We will now show that the sum at (42) grows as fast as anticipated: Lemma 10. For any ε > 0, we may choose µ > 0 such that there exists a random N ∈N where u∈Fαn 1{N̄β,κn (u)>0} ≥∆(α)−Θ(β,κ)− ε for all n >N with probability one. Proof. We will be able to apply Lemma 9 given sufficient information about the growth of |Fαn | and decay of the probabilities pβ,κn (u) := P (N̄ n (u)> 0|Ff−1(Tn)), A TYPED BRANCHING DIFFUSION 23 where u ∈ Fαn ⊂Nf−1(Tn). It follows easily from Theorem 1, f−1(Tn)/Tn → 1 and the continuity of ∆(α) that log |Fαn | ≥∆(α)− ε for all sufficiently large n. The definition of N̄β,κn (u) and spatial translation invariance implies that, for each u ∈ Fαn , the rapid ascent probability pβ,κn (u) depends only on the initial type position Yu(f −1(Tn)). For δ,µ > 0, define t (u) := s∈[τ(t)−µ,τ(t)] {Xu(s)−Xu(0)<−(β − δ)t;Yu(s)≥ (κ− δ) u∈Nτ(t) t (u).(43) Recalling the comments of Section 4.3, there exist ε′, δ′ > 0 and we may choose µ> 0 sufficiently small, such that pβ,κn (u) = P 0,Yu(f −1(Tn))(B )≥ P 0,Yu(f−1(Tn))(Aε ) =: p̄n(u) for all u ∈ Fαn whenever n is sufficiently large. Together with Theorem 7 and since Yu(f −1(Tn)) ∈ [y0, y1] for u ∈ Fαn , this reveals log pβ,κn (u) ≥ log p̄n(u) ≥−Θ(β,κ)− ε for all for u ∈ Fαn and all sufficiently large n, almost surely. Then we may combine the observations above to obtain u∈Fαn pβ,κn (u)≥∆(α)−Θ(β,κ)− Taking this last line together the assertion of Lemma 9 at equation (41) gives the result. � It is now straightforward to combine Lemma 10 with the inequality at (42) to see that, given ε, δ > 0, there exists µ > 0 and a random time T such t−1logNt((α+ β − δ)t; [(κ− δ) t,∞))≥∆(α)−Θ(β,κ)− ε for all t > T , almost surely. Since ε and δ can be taken arbitrarily small, using the optimal ᾱ and β̄ according to equations (14)–(15), we find lim inf t−1 logNt(γ, [κ t,∞))≥∆(γ,κ) almost surely, 24 Y. GIT, J. W. HARRIS AND S. C. HARRIS as required. [It is also interesting to note that λ̄= λᾱ = λ̄(γ,κ) from equa- tions (19), (23) and (32), so the optimal parameters are in agreement with those of the expectation calculations in Section 3 and the path large devia- tions in Section 4.] � 6. The “spine” setup and results. In this section, we describe how to construct an enriched branching diffusion with an identified “spine” or “back- bone” particle and discuss how to perform some extremely useful changes of measure (closely related to the additive martingales) that will essentially “force” the spine perform the short climb, whilst giving birth at an accel- erated rate to offspring that behave as if under the original measure. These spine techniques are at the very heart of our proof of Theorem 7 in Sec- tion 5. Spine ideas were first seen for branching Brownian motion in [3] and developed for Galton–Watson processes in [16, 18, 19]. Kyprianou [17] and Englander and Kyprianou [6], developed the technique for some families of branching diffusions; and more recently the spine approach has been signif- icantly improved in [8]. This approach uses several different filtrations on an enlarged probability space carrying the branching diffusion, and permits some very useful techniques and results to be developed. For example, “addi- tive” (many-particle) martingales can be represented as suitable conditional expectations of “spine” (single-particle) martingales and consequently there are clear interpretations for any changes of measure and all measures in- volved in our “spine” setup are probability measures with intuitive construc- tions. Following Hardy and Harris [8], we will first outline the notation and then describe the changes of measure. The notation described in this section is generalized to allow each particle u to have 1 +Au offspring, where each Au is an independent copy of a random variable with values in {0,1,2, . . .}. The spine techniques developed in this paper could readily be generalized to such models. All probability measures are to be defined on the space T̃ of marked Galton–Watson trees with spines; before defining precisely what this space is we need to set up some other notation. We recall the set of Ulam–Harris labels, Ω, defined by Ω := {∅}∪ n∈N(N) n, where N := {1,2,3, . . .}. For two words u, v ∈Ω, uv denotes the concatenated word, where we take u∅=∅u= u. So Ω contains elements such as “∅412,” which represents “the individual being the 2nd child of the 1st child of the 4th child of the initial ancestor ∅.” For labels u, v ∈Ω the notation v < u means that v is an ancestor of u, and |u| denotes the length of u. We define a Galton–Watson tree to be a set τ ⊂Ω such that: (i) ∅ ∈ τ , so there is the unique initial ancestor; (ii) if u, v ∈Ω, then vu ∈ τ ⇒ v ∈ τ , so τ contains all of the ancestors of its nodes; A TYPED BRANCHING DIFFUSION 25 (iii) for all u ∈ τ , there exists Au ∈ {0,1,2, . . .} such that for j ∈N, uj ∈ τ if and only if 1≤ j ≤ 1 +Au. The set of all such trees is T, and we will use the symbol τ for a particular tree. As our work concerns branching diffusions we shall often refer to the labels of τ as particles. Note that for the binary branching mechanism in this paper, P (Au = 1) ≡ 1; of course, here there is only one τ ∈ T—the binary tree. A Galton–Watson tree by itself only records the family structure of the individuals, so to each individual u ∈ τ we give a mark (Xu, Yu, σu) which contains the following information: • σu ∈ [0,∞) is the lifetime of particle u, which also determines the fission time of the particle as Su := v≤u σv . We may also refer to the Su as death times; • the function Xu(t) : [Su − σu, Su)→R describes the particle’s spatial mo- tion in R during its lifetime; • the function Yu(t) : [Su− σu, Su)→R describes the evolution of the parti- cle’s type in R during its lifetime. For clarity we must decide whether or not a particle is in existence at its death time: our convention will be that a particle dies “infinitesimally be- fore” its death time—this is why Xu and Yu are defined on [Su−σu, Su) and not [Su−σu, Su]—so that at time Su the particle u has disappeared and has been replaced by its two children. We denote a particular marked tree by (τ,X,Y,σ), or the abbreviation (τ,M), and the set of all marked Galton–Watson trees by T . For each (τ,X,Y,σ) ∈ T , the set of particles alive at time t is defined as Nt := {u ∈ τ :Su − σu ≤ t < Su}. For any given marked tree (τ,M) ∈ T we can distin- guish individual lines of descent from the initial ancestor: ∅, u1, u2, u3, . . . ∈ τ , where ui is a child of ui−1 for all i ∈ {2,3, . . .} and u1 is a child of the initial individual ∅. We call such a line of descent a spine and denote it by ξ. In a slight abuse of notation we refer to ξt as the unique node in ξ that is alive at time t, and also for the position of the particle that makes up the spine at time t; that is, ξt :=Xu(t), where u ∈ ξ∩Nt. However, although the interpretation of ξt should always be clear from the context, we introduce the following notation for use where some ambiguity may still arise: • nodet((τ,M, ξ)) := u if u ∈ ξ is the node in the spine alive at time t. It is natural to think of the spine as a single diffusing particle ξt, or, strictly speaking, the pair (ξt, ηt), where ηt is the type of the spine at time t. We define nt to be a counting function that tells us which generation of the spine is currently alive, or equivalently the number of fission times there have been on the spine: nt = |nodet(ξ)|. 26 Y. GIT, J. W. HARRIS AND S. C. HARRIS The collection of all marked trees with a distinguished spine is the space T̃ on which our probability measures will eventually be defined, but first we define four filtrations on this space that contain different levels of information about the branching diffusion. • Filtration (Ft)t≥0. We define a filtration of T̃ made up of the σ-algebras Ft := σ((u,Xu, Yu, σu) :Su ≤ t; (u,Xu(s), Yu(s) : s ∈ [Su − σu, t]) : t ∈ [Su − σu, Su)), which means that Ft is generated by the information concerning all par- ticles that have lived and died before time t, and also those that are still alive at time t. Each of these σ-algebras is a subset of the limit F∞ := σ( t≥0Ft). • Filtration (F̃t)t≥0. We define the filtration (F̃t)t≥0 by augmenting the filtration Ft with the knowledge of which node is the spine at time t; that is, (F̃t)t≥0 := σ(Ft,nodet(ξ)) and F̃∞ := σ( t≥0 F̃t), so that this filtration knows everything about the branching diffusion and everything about the spine. • Filtration (Gt)t≥0. (Gt)t≥0 is a filtration of T̃ defined by Gt := σ(ξs : 0 ≤ s≤ t), and G∞ := σ( t≥0 Gt). These σ-algebras are generated only by the spine’s motion and so do not contain the information about which nodes of the tree τ make up the spine. • Filtration (G̃t)t≥0. As we did in going from Ft to F̃t we create (G̃t)t≥0 from (Gt)t≥0 by including knowledge of which nodes make up the spine: (G̃t)t≥0 := σ(Gt,nodet(ξ)) and G̃∞ := σ( t≥0 G̃t). This means that G̃t also knows when the fission times on the spine occurred, whereas Gt does not. Now that we have defined the underlying space and filtrations, we can de- fine the probability measures of interest. We let the typed branching diffusion be as described in Section 1.1, with the probability measures {P x,y :x, y ∈R} on (T̃ ,F∞) representing the law of this typed branching diffusion when ini- tially started with a single particle at (x, y). We recall from [18] that, if f is an F̃t-measurable function, we can write fu1{ξt=u},(44) where fu is Ft-measurable. Now we can extend P x,y to a measure P̃ x,y on (T̃ , F̃∞) by choosing the particle that continues the spine uniformly each time there is a birth on the spine; more precisely, for any f ∈ mF̃t with representation like (44), we have: f dP̃ x,y(τ,M, ξ) := dP x,y(τ,M). A TYPED BRANCHING DIFFUSION 27 We construct the F̃t-measurable martingale ζ̃(t) as ζ̃(t) := v+λ (ηt)e {R(ηs)+1/2λ2A(ηs)}ds−E+λ t × 2nte− R(ηs)ds × eλξt−1/2λ A(ηs)ds(45) = v+λ (ηt)2 nteλξt−E Observe that this is a product of single-particle martingales, details of which can be found in [17] or [10]. One can think of these as h-transforms of the P̃ - law of the spine: the first makes η an outward-drifting Ornstein–Uhlenbeck process with drift parameter µλ; the second increases the breeding rate on the spine to 2R(·); and the third adds a spatial drift to ξ. Using the martingale ζ̃(t) we may define a measure Q̃ λ on (T̃ , F̃∞) by dP̃ x,y ζ̃(t) ζ̃(0) v+λ (y) v+λ (ηt)2 nteλξt−E t.(46) And since ζ̃(t) is a product of h-transforms, under Q̃ λ the process may be re-constructed path-wise according to the following description: • starting from spatial position x and type y the spine (ξt, ηt) diffuses spa- tially as a Brownian motion with infinitesimal variance A(ηt) and infinites- imal drift λA(ηt); • the type of the spine, ηt, begins at y and moves in type space as an outward-drifting Ornstein–Uhlenbeck process with generator + µλy • the spine branches at rate 2R(ηt), producing 2 particles; • one of these particles is selected uniformly at random; • the chosen offspring repeats stochastically the behavior of its parent; • the other offspring particle initiates a P ·,·-BBM from its birth position and type. The change of measure (46) projects onto the sub-algebra Ft as a condi- tional expectation: dP̃ x,y v+λ (y) P̃ x,y(v+λ (ηt)2 nteλξt−E t|Ft), and it is a short calculation using the methods of, for example, Hardy and Harris [10] to show that: 28 Y. GIT, J. W. HARRIS AND S. C. HARRIS Theorem 11. If we define Q λ := Q̃ λ |F∞ , then Q λ is a measure on F∞ that satisfies dP x,y = Ẑ+λ (t) := Z+λ (t) Z+λ (0) Moreover under Q λ , the path-wise construction of the branching diffusion is the same as under Q̃λ. Although the path-wise construction of the branching diffusion is the same under Q λ and Q̃ λ , only the measure Q̃ λ “knows” about the spine. It is clear, however, that we have Q̃ λ (A) =Q λ (A) for any A ∈ F∞. Under the measure Q̃ λ only the behavior of the spine is altered, and combining this observation with conditioning on the spine’s path and fission- times gives us a very useful representation for Z+λ (t) under Q̃ λ that we shall refer to as the spine decomposition: λ (t)|G̃∞) = v+λ (ηSu)e λξSu−E Su + v+λ (ηt)e λξt−E+λ t.(47) Throughout the rest of this article we will refer to the two pieces of this de- composition as the “sum term” and the “spine term.” This decomposition is discussed in detail for a wide variety of branching diffusions in [9], but to derive it we simply note that the contributions to Z+λ (t) from the sub- trees that branch off the spine have constant Q̃ λ -expectation because they behave as if under the original measure P , and we know that Z+λ (t) is a P - martingale. The spine decomposition reduces many calculations about the behavior of Z+λ (t) under Q̃ λ to one-particle calculations about the spine, and this observation is exploited in the spine proofs of Lp-bounds for some families of additive martingales in [9]. 7. Proof of Theorem 7. The short climb probability. With the spine foundations firmly established in Section 6, we may proceed with the proof of the short climb probability lower bound from Theorem 7. First, recall definitions (38) and (39), where A t is the event that there exists a particle that makes the short climb along optimal path (x̄, ȳ), and t (ξ) is the event that the spine makes the short climb. Note that ε controls the proximity to x̄ and δ the proximity to ȳ. Importantly, we will only be interested in taking λ= λ̄ throughout this section, although we will usually just write λ for notational simplicity. Also recall throughout that t and τ are related through (θ/(2µλ)) exp(2µλτ) = κ A TYPED BRANCHING DIFFUSION 29 Proof of Theorem 7. The key step in the proof of this is the following use of the spine change of measure: for any function g :R+ →R+ we have P x,y(A t ) =Q Ẑ+λ (τ) Ẑ+λ (τ) ≥ Q̃x,yλ Ẑ+λ (τ) ≥ Q̃x,yλ Ẑ+λ (τ) ; sup s∈[0,τ ] Ẑ+λ (s)≤ g(τ) ≥ g(τ)−1Q̃x,yλ t (ξ); sup s∈[0,τ ] Ẑ+λ (s)≤ g(τ) Essentially we just have to make the “correct” choice for both λ and g in expression (48), although there will still remain a number of technicalities to resolve. The first idea is to ensure the (originally rare) event A t actually occurs under the new measure Q̃ λ by making the spine follow close to the required path (x̄, ȳ); this is achieved by choosing the optimal value λ̄ for λ and choos- ing τ to be on the natural time scale it would take the spine to reach position t. In particular, this choice will mean that in the first line of the above set of inequalities there is no significant loss of mass when replacing the event t with A t (ξ). Next, we wish to choose the smallest possible g that will still leave some positive probability on the last line of the above argument. Hence, we wish to identify the rate of growth of the martingale Z+λ under λ , and this will essentially be governed by the contribution from the spine itself. With this is mind, and recalling the various properties of the optimal paths and parameters from Section 4, for ε0 > 0 we define gε0(τ) := exp ψ+λ + e2µλτ − (ψ+λ y 2 + λx) and recall from (35) that the scaling between t and τ is fixed throughout, where κ2t= (θ/(2µλ̄))e 2µλ̄τ for large t, hence t+ τ ∼ t. Note that since we are only considering the optimal value λ= λ̄, we have e2µλ̄τ = (κ2ψ+ − λ̄β)t=Θ(β,κ). Then from (48) we have P x,y(A t )≥ gε0(τ)−1Q̃ t (ξ); sup s∈[0,τ ] Ẑ+λ (s)≤ gε0(τ) .(49) Our strategy for the rest of this proof is to show that the Q̃ λ -probability in (49) is at least some ε′ > 0 for all sufficiently large t, uniformly for y ∈ [y0, y1], 30 Y. GIT, J. W. HARRIS AND S. C. HARRIS so that the decay rate part of (49) matches the desired rate in the statement of the theorem. Conditioning on the spine’s path and birth times, G̃∞, and then making use of some standard properties of conditional expectation we have t (ξ); sup s∈[0,τ ] Ẑ+λ (s)≤ gε0(τ) t (ξ); sup s∈[0,τ ] Ẑ+λ (s)≤ gε0(τ) s∈[0,τ ] Ẑ+λ (s)≤ gε0(τ) since A t (ξ) is G̃∞-measurable. We next observe that, conditional on G̃∞, we can write Ẑ+λ (t) as Ẑ+λ (t) = e y2+λx) λ (t− Su) + f(t) ,(50) where the Z λ are independent copies of Z λ started from a single particle at (ξSu , ηSu); and f(t) is the contribution to Z λ (t) from the spine, which, conditional on G̃∞, is a known function of t. Now if we could show, for 0< ε̃0 < ε0, s∈[0,τ ] f̂(s)≤ gε̃0(τ) and sup s∈[0,τ ] (Ẑ+λ (s)− f̂(s))≤ gε0(τ) where f̂(t) := e−(ψ y2+λx)f(t), we would have sups∈[0,τ ] Ẑ λ (s)≤ gε0(τ). Hence, defining Ẑ+λ (s) := Ẑ λ (s)− f̂(s), we have s∈[0,τ ] Ẑ+λ (s)≤ gε0(τ) ≥ Q̃x,yλ s∈[0,τ ] f̂(s)≤ gε̃0(τ) ; sup s∈[0,τ ] Ẑ+λ (s)≤ gε0(τ) 1{sups∈[0,τ ] f̂(s)≤gε̃0 (τ)/2} s∈[0,τ ] Ẑ+λ (s)≤ gε0(τ) since, conditional on G̃∞, the supremum of f̂ on [0, τ ] is known. We see from (50) that, conditional on G̃∞, Ẑ+λ (t) is a submartingale. This is because the Q̃ λ -conditional expectation of each of the Z λ in the sum y2+λx) λ (t− Su)(51) A TYPED BRANCHING DIFFUSION 31 is constant, so the expectation of the sum cannot decrease, and in fact this expectation increases every time there is a birth on the spine. Then by Doob’s submartingale inequality we have s∈[0,τ ] Ẑ+λ (s)≤ gε0(τ) = 1− Q̃x,yλ s∈[0,τ ] Ẑ+λ (s)≥ gε0(τ) ≥ 1− 2 gε0(τ) λ (Ẑ λ (τ)|G̃∞). We must note here that the expectation on the above line is not a priori finite. However, the expectation of each term in the sum (51) is bounded by sups∈[0,τ ] f̂(s), which we have control over via an indicator function and so we do not have to worry about this expectation blowing up. So we need to show that for all sufficiently large τ and all y ∈ [y0, y1], (ξ)∩{sups∈[0,τ ] f̂(s)≤gε̃0 (τ)/2} gε0(τ) λ (Ẑ λ (τ)|G̃∞) > ε′, and hence also t (ξ); sup s∈[0,τ ] Ẑ+λ (s)≤ gε0(τ) as required. This will follow by combining both parts of the following result. Lemma 12. Fix y1 > y0 > 0 and ε0 > ε̃0 > 0. (i) For all sufficiently small ε, δ > 0, there exists some ε′ > 0 and T̃ > 0 such that for all y ∈ [y0, y1] and all t > T̃ , t (ξ); sup s∈[0,τ ] f̂(s)≤ gε̃0(τ) > ε′. (ii) As τ →∞, gε0(τ) λ (Ẑ λ (τ)|G̃∞); sup s∈[0,τ ] f̂(s)≤ gε̃0(τ) uniformly over y ∈ [y0, y1]. Then we have shown that, for any ε0 > 0, y1 > y0 > 0, and sufficiently small ε, δ > 0, there exists a T > 0 such that, for all y ∈ [y0, y1] and all t > T , t−1 logP x,y(Aε,δt )≥−(Θ(β,κ) + ε0). 32 Y. GIT, J. W. HARRIS AND S. C. HARRIS Finally, we observe that the probability P x,y(A t ) is trivially monotone increasing in both ε and δ, and so it follows that if the result is true for all sufficiently small ε and δ, it is in fact true for all ε, δ > 0. This completes the proof of Theorem 7. � Proof of Lemma 12(i). We will prove Lemma 12(i) in a sequence of other lemmas, using a convenient coupling for the spine’s type process. First recall that, under Q̃ λ , ηs solves the SDE dηs = θ dBs + µληs ds, where Bs is a Q̃λ-Brownian motion. Noting that d(e −µλsηs) = e θ dBs, we can construct e−µλsηs as a time-change of a Brownian motion with e−µλsηs − η0 = e−µλw dBw = B̃(1− e−2µλs), where B̃ is also a Q̃ λ -Brownian motion started at the origin. In this way, we will construct ηy under P from a Brownian motion By started at y 2µλ/θ where, for s ∈ [0,∞), ηy(s) = eµλsBy(1− e−2µλs). To construct simultaneously all type processes ηy under the same measure P, we first construct the process By0 as an independent Brownian motion started at y0 2µλ/θ. Second, we construct the process B y1 by running an independent Brownian motion started at y1 2µλ/θ until it first hits the path of By0 , at which point we couple the two processes together. Next, for any other y ∈ (y0, y1), we run an independent Brownian motion By until it first meets with either the process By0 below or By1 above, at which point we couple it to the process it first hits. Finally, we construct all the corresponding spatial processes ξy under P from a single Brownian motion W by defining ξy(s) =W ηy(w)2 dw ηy(w)2 dw,(52) where W is started at x and is independent of the By processes. Constructed in this way, for each y ∈ [y0, y1], the P-law of (ξy, ηy) is the same as the Q̃ λ -law of (ξ, η). Fixing µ ∈ (0,1) and K >max{y1,1}, we define the events and stopping times Ayε := By(s) ∈ ,∀s ∈ (1− µ,1] T0 := inf{t :By0(t) = 0}, TK := inf{t :By1(t) =K}, Ãε,K :=A ε ∩Ay1ε ∩ {T0 > 1} ∩ {TK > 1}. A TYPED BRANCHING DIFFUSION 33 Then, clearly P(Ãε,K)> 0 and, on the event Ãε,K , the coupling gives 0< ηy0(s)≤ ηy(s)≤ ηy1(s)≤K eµλs, for all s≥ 0 and y ∈ [y0, y1]. Note that our construction also ensures that if event Ay0ε ∩Ay1ε occurs then so must Ayε for any y ∈ [y0, y1], hence Ayε ⊃ Ãε,K . Lemma 13. Let ε > 0. On event Ãε,K , there exists a deterministic time s0 = s0(ε)> 0 such that for all τ > s0, s∈[0,τ ] |ηy(s)− ȳ(s)| ≤ ε for all y ∈ [y0, y1]. Proof. Set s1 =− 12µλ logµ and then, on event Ãε,K , for all τ ≥ s > s1 we have ηy(s)− eµλs, for all y ∈ [y0, y1]. Writing ȳ(s) = 1− e−2µλs 1− e−2µλτ eµλs, we see that there exists s2 = s2(ε)> 0 such that, for τ ≥ s > s2, ȳ(s)− eµλs. Taking s3(ε) = max{s1, s2(ε)} now yields |ηy(s)− ȳ(s)|< ε eµλs ≤ ε t(53) for all τ ≥ s > s3 and all y ∈ [y0, y1]. Now consider s ∈ [0, s3]. On Ãε,K we have |ηy(s)− ȳ(s)| ≤ eµλs3(1 +K), and hence for some s4(ε)> 0 we have |ηy(s)− ȳ(s)| ≤ ε t for all τ > s4, all s ∈ [0, s3], and all y ∈ [y0, y1]. Taking s0(ε) =max{s3, s4} yields the result. 34 Y. GIT, J. W. HARRIS AND S. C. HARRIS Lemma 14. Let δ > 0. Then for all sufficiently small ε, there exists a deterministic τ0 = τ0(ε, δ)> 0 such that, on Ãε,K , we have s∈[0,τ ] ηy(w)2 dw− ȳ(w)2 dw < δt(54) for all τ > τ0 and all y ∈ [y0, y1]. Proof. Given any δ > 0, we first fix an ε > 0 sufficiently small such that ε(2+ ε ; this yields a corresponding s3 = s3(ε), which is chosen as at equation (53). Given this s3, we find τ1 = τ1(ε, δ)> 0 such that, for all τ > τ1, (K2 + 1) e2µλw dw < We now set τ0 = τ0(ε, δ) = max{s3, τ1}. With this choice of ε and τ0, we proceed to show that the inequality (54) is satisfied. Note that τ0 is deter- ministic and independent of y. From equation (53) we see that, on Ãε,K and for s > s3, ηy(w)2 dw ≥ ηy(w)2 dw+ ȳ(w)− ε ȳ(w)2 dw− ȳ(w)2 dw− 2 e2µλw dw ȳ(w)2 dw− e2µλw dw− (2ε) κt ȳ(w)2 dw− δ for all τ > τ0 and all y ∈ [y0, y1]. Similarly ηy(w)2 dw ≤ ηy(w)2 dw+ ȳ(w) + ȳ(w)2 dw+ ȳ(w)2 dw+K2 e2µλw dw+ ε ȳ(w)2 dw+ A TYPED BRANCHING DIFFUSION 35 for all τ > τ0 and all y ∈ [y0, y1]. Finally, for s ∈ [0, s3], on Ãε,K we have ηy(w)2 dw− ȳ(w)2 dw ηy(w)2 dw+ ȳ(w)2 dw ≤ (K2 + 1) e2µλw dw < δt for all τ > τ0 and all y ∈ [y0, y1]. � Lemma 15. Let δ > 0. Then for all sufficiently small ε > 0, there exists P-almost everywhere on Ãε,K a random time S0 = S0(δ, ε)<∞ such that s∈[0,τ ] ξy(s)− λa ηy(w)2 dw < δt, for all y ∈ [y0, y1] and all τ > S0. Proof. Given δ > 0, choose any δ′, δ′′ > 0 such that δ′(|β/λ|+ δ′′)< δ. Recalling the construction of ξy at (52), we see from standard properties of Brownian motion that there almost surely exists some S1 = S1(δ ′)<∞ such s∈[0,t] |W (s)|< δ′ for all t > S1. s∈[0,τ ] ηy(w)2 dw ηy(w)2 dw for all τ such that a y(w)2 dw > S1, and by the coupling construction, on Ãε,K this is true for all y ∈ [y0, y1] if a y0(w)2 dw > S1. Then there exists (P-almost everywhere on Ãε,K) a random time S2 = S2(δ ′)<∞, which depends on By0 and S1, such that a y(w)2 dw > S1 for all y ∈ [y0, y1] when τ > S2. Now by Lemma 14, given δ′′ and a sufficiently small ε, there exists a deterministic τ0 = τ0(ε, δ ′′)> 0 such that, on Ãε,K , ηy(w)2 dw ≤ a ȳ(s)2 ds+ δ′′t= + δ′′ t(56) for all τ > τ0 and all y ∈ [y0, y1]. Combining the inequalities at (55) and (56), we now see that, for τ > S0 = S0(ε, δ ′, δ′′) = max{S2, τ0}, s∈[0,τ ] ξy(s)− λa ηy(w)2 dw = sup s∈[0,τ ] ηy(w)2 dw for all y ∈ [y0, y1]. � 36 Y. GIT, J. W. HARRIS AND S. C. HARRIS On combining Lemmas 14 and 15 and recalling the definition of optimal path x̄ at (37), we obtain the following: Lemma 16. Let δ > 0. Then for all sufficiently small ε > 0, there exists P-almost everywhere on Ãε,K a random time S̃0 = S̃0(δ, ε)<∞ such that s∈[0,τ ] |ξy(s)− x̄(s)|< δt, for all y ∈ [y0, y1], and all τ > S̃0. We may now draw everything together to finish the proof of Lemma 12(i). First we observe that since λ < 0, on event A t (ξ), s∈[0,τ ] y2+λx) exp(ψ+λ η s + λξs −E+λ s) ≤ e−(ψ y2+λx) exp(ψ+λ (κ+ ε) 2t+ λ(−β − δ)t), and so, given ε̃0, we can choose first δ and then ε sufficiently small so that t (ξ)⊂ s∈[0,τ ] f̂(s)≤ gε̃0(τ) and, from Lemmas 13 and 16, there exists a random time T̃ = T̃ (δ, ε) <∞ such that on Ãε,K we have s∈[0,τ ] |ηy(s)− ȳ(s)|< ε s∈[0,τ ] |ξy(s)− x̄(s)|< δt for all τ > T̃ and all y ∈ [y0, y1]. That is, Ãε,K ∩ {T̃ < τ} ⊂Aε,δt (ξy) for each y ∈ [y0, y1], with the slight abuse of notation that s∈[0,τ(t)] |ηy(s)− ȳ(s)|< ε t; sup s∈[0,τ(t)] |ξy(s)− x̄(s)|< δt Note also that P(Ãε,K)> ε ′ for some ε′ > 0. Combining the above, for any y ∈ [y0, y1] we have t (ξ); sup s∈[0,τ ] f̂(s)≤ gε̃0(τ) t (ξ)) = P(A ≥ P(Ãε,K; T̃ < τ)→ P(Ãε,K) as τ →∞, as required. � A TYPED BRANCHING DIFFUSION 37 Proof of Lemma 12(ii). Consider the expectation of the “sum term.” We have λ (Ẑ λ (τ)|G̃∞) = e y2+λx) λ (t− Su) = e−(ψ y2+λx) λ (t− Su)|G̃∞) ≤ e−(ψ y2+λx)nτ max{eψ η(Su) 2+λξ(Su)−E+λ Su :u < ξτ} ≤ nτ sup s∈[0,τ ] f̂(s). Hence gε0(τ) λ (Ẑ λ (τ)|G̃∞); sup s∈[0,τ ] f̂(s)≤ gε̃0(τ) ≤ Q̃x,yλ gε̃0(τ) gε0(τ) ; sup s∈[0,τ ] f̂(s)≤ gε̃0(τ) ≤ e−(ε0−ε̃0)tQ̃x,yλ (nτ ), and we can now calculate Q̃ λ (nτ ) = Q̃ λ (Q̃ λ (nτ |G∞)), where G∞ the σ-algebra generated by the path of the spine (not including the birth times). Conditional on G∞, nτ is a Poisson random variable with mean given by 0 2(rη s + ρ)ds, and using Fubini’s theorem we have 2(rη2s + ρ)ds s)ds+2ρτ e2µλτ − − rθτ + 2ρτ 2y2κµλ t+ o(τ). So the Q̃ λ -expectation of nτ only grows linearly in t. Then since ε0− ε̃0 > 0, the expression at (57) tends to 0 as t→∞. Moreover, the expectation at (57) is bounded by the Q̃ λ -expectation, and hence the convergence is uniform over y ∈ [y0, y1], as claimed. � 8. Martingale results. In this section we recall some existing and prove some new martingale results that are intermediate steps in the proofs of 38 Y. GIT, J. W. HARRIS AND S. C. HARRIS Theorem 1 and the upper bound of Theorem 3. We recall from [13] that E−λ [also written E−(λ)] and ∆(γ) are Legendre conjugates with ∆(γ) = inf {E−(λ) + λγ}, E−(λ) = sup {∆(γ)− γλ}.(58) If, for λmin <λ< 0, we write γλ for the γ value which achieves the supremum on the right-hand side of equation (58), then the functions λ 7→ γλ from (−λmin,0) to (0,∞), and γ 7→ λγ from (0,∞) to (−λmin,0) are inverses of each other and, of course, λγ is the λ value which achieves the infimum on the left-hand side of equation (58). In addition, we note that γλ =− E−(λ) = θa2λ2 θ− 8r− 4aλ2 ,(59) that E−(λ) and ∆(γ) are convex functions, and that c̃(θ) = sup{γ :∆(γ)> 0}= inf{−E−(λ)/λ :λmin <λ< 0} = inf{c−λ :λmin < λ< 0}= c λ̃(θ) where c−λ :=−E λ /λ and λ̃(θ) :=− 2(θ− 8r)(θρ+ 2ρ2 + rθ) a(θ+4ρ)2 ∈ (λmin,0). A formula for c̃(θ) is given in equation (9). The following fundamental con- vergence result for the Z−λ martingale was first partly proved in [13], but also see [9] for a more complete proof using “spine” techniques. Theorem 17. Suppose λ ∈ (λmin,0]. (i) If λ ∈ (λ̃(θ),0], the martingale Z−λ is uniformly integrable and has an almost sure strictly positive limit. (ii) If λ≤ λ̃(θ), then Z−λ (∞) = 0 almost surely. The following convergence result was proved in [12] using martingales based on Hermite polynomials. Theorem 18. Let λ ∈ (λ̃(θ),0] and α< 1/4. For each P x,y starting law and every continuous bounded function f :R 7→R, we have f(Yu(t))e αYu(t) 2+λ(Xu(t)+c λ (∞), A TYPED BRANCHING DIFFUSION 39 where f0 := )1/4 ∫ f(y)eαy y2φ(y)dy(62) and φ(y) is the standard normal density. In this paper, we require a corollary to this theorem which specifies more precisely which particles contribute to the final limit. Corollary 19. Let λ ∈ (λ̃(θ),0] and α < 1/4. For each P x,y starting law and every continuous bounded function f :R 7→ R, we have for every ε > 0 f(Yu(t)) e αYu(t) 2+λXu(t)−E−λ t 1{|Xu(t)/t+γλ|<ε}−→a.s. f0Z λ (∞)(63) where γλ =− ∂∂λE −(λ) and f0 is given at equation (62). This last result will enable us to show in Section 10 that the almost sure growth rate is at least as large as the expected growth rate, D(γ)≥∆(γ). It is easy to see from Corollary 19 that when Z−λ (∞)> 0, there must exist at least one particle near to −γλt in space. Further, because of the decay rate of each term in the sum over particles at equation (63), it will be relatively straightforward to improve this to get the required exponential numbers of particles, exp(∆(γ)t), near −γλt for large times [as long as Z−λ (∞)> 0]. The following result concerns the rate at which the martingales Z+λ and Z−λ converge to zero. Theorem 20. Let λ ∈ (λmin,0). For every starting law, P x,y, logZ±λ (t) → λ(c±λ − c λ) a.s. where c±λ is given at (5), and c∗λ := c̃(θ), if λmin <λ≤ λ̃(θ), c−λ , if λ̃(θ)≤ λ < 0. Corollary 21. If λ ∈ (λmin,0), then Z+λ (t)→ 0 P x,y-almost surely. The rate of convergence of the Z+λ martingale in part (i) of Theorem 20 is crucial in Section 9 to obtain the upper bound on the almost sure growth rate, D(γ,κ) ≤ ∆(γ,κ). We also comment that if Corollary 19 were true for all α < ψ+λ , then we could have gained this upper bound at that point. 40 Y. GIT, J. W. HARRIS AND S. C. HARRIS Although Corollary 19 is only proven for α < 1/4 (where we can utilize suitable Hermite expansions), we conjecture that it holds for all α <ψ+λ . Proof of Corollary 19. Let ε > 0 be small, µ := λ−ε, λ,µ ∈ (λ̃(θ),0), f be a positive, continuous bounded function, α< 1/4 and note that γµ > γλ. Then we have f(Yu(t))e αYu(t) 2+λXu(t)−E−λ t 1{Xu(t)<−γµt} ≤ e(E µ −E−λ −εγµ)t f(Yu(t))e αYu(t) 2+µXu(t)−E−µ t 1{Xu(t)<−γµt} f(Yu(t))e αYu(t) 2+µXu(t)−E−µ t −E−µ +(λ−µ)γµ)t. Recall that E−(λ) is convex with ∂ E−(λ)≥ 0 and ∂ E−(λ) = γλ, so, from the Taylor expansion, E−λ −E µ + (µ− λ) E−(λ) (µ− λ)2 E−(λ) + o((µ− λ)2). Then taking ε > 0 small enough so that E−λ −E−µ +(λ−µ)γµ > 0, and using Theorem 18, we find that for any δ > 0 limsup f(Yu(t))e αYu(t) 2+λXu(t)−E−λ t 1{Xu(t)<−(γλ+δ)t} = 0. Similarly, we can show limsup f(Yu(t))e αYu(t) 2+λXu(t)−E−λ t 1{Xu(t)>−(γλ−δ)t} = 0, and hence the only contribution to the limit comes from the particles near −γλt in space. Combining this with Theorem 18 we have λ (∞) = limt→∞ f(Yu(t))e αYu(t) 2+λXu(t)−E−λ t = lim f(Yu(t))e αYu(t) 2+λXu(t)−E−λ t 1{|Xu(t)/t+γλ |<δ}. Proof of Theorem 20. We use a useful technique brought to our attention in [22]. Let p ∈ (0,1) so that, by Jensen’s inequality, Z±λ (t)p is a supermartingale; then for u, v > 0 we have (u+ v)p ≤ up + vp, A TYPED BRANCHING DIFFUSION 41 and hence Z±λ (t) Yu(t) 2+λ(Xu(t)+c Yu(t) 2+pλ(Xu(t)+c For any ε > 0, Doob’s supermartingale inequality says s≤w≤s+t Z±λ (w) p > εp λ (s) ≤ ε−p Yu(s) 2+pλ(Xu(s)+c and then s≤w≤s+t eδwZ±λ (w)> ε s≤w≤s+t Z±λ (w) p > e−pδ(s+t)εp ≤ ε−pepδt Yu(s) 2+pλ(Xu(s)+c p(λ(c± )+δ)s Now, if we can choose p ∈ (0,1) such that λ(c±λ − c pλ) + δ < 0 and pψ ψ+pλ, we must have e δuZ±λ (u)→ 0 almost surely by using a familiar Borel– Cantelli argument. [The condition pψ±λ < ψ pλ guarantees that the expec- tation in the last line above tends to a finite limiting value, hence stays bounded over all times s, as can be checked by using formula (17), for ex- ample.] For all 0 ≤ p < 1 we find pψ±λ < ψ pλ. Considering the graph of c quickly see that, for λ ∈ [λ̃(θ),0), taking p as close to 1 as we like gives the best rate. For λ ∈ [λmin, λ̃(θ)) we can choose p so that pλ= λ̃(θ), which gives the best rate. Recall from Theorem 17 that Z−λ (∞)> 0 when λ ∈ (λ̃(θ),0). Then, so far, we have proved the following: Lemma 22. For every starting law, P x,y, and for all ε > 0, if λ ∈ (λmin,0) e−εt e−λ(c )tZ±λ (t)→ 0 a.s. 42 Y. GIT, J. W. HARRIS AND S. C. HARRIS where c∗λ := c̃(θ), if λmin <λ≤ λ̃(θ), c−λ , if λ̃(θ)≤ λ < 0. It is clear that this gives the required upper bound of lim sup logZ±λ (t) ≤ λ(c±λ − c Now, for any ε > 0, if λ ∈ (λmin, λ̃(θ)] then Yu(t) 2+λ(Xu(t)+c̃(θ)t) ≥ eλ(Lt+c̃(θ)t)+εt →∞ a.s. since we know that Lt := inf{Xu(t) : u ∈ N(t)} satisfies Lt/t→ −c̃(θ) a.s. Otherwise, with λ∈ (λ̃(θ),0), Yu(t) 2+λ(Xu(t)+c t) ≥ eεtZ−λ (t)→∞ a.s. since here Z−λ (∞)> 0 a.s. Thus, in all cases, lim inf logZ±λ (t) ≥ λ(c±λ − c which completes the proof of Theorem 20. � 9. Proof of Theorem 3. Upper bound. The idea for the upper bound proof is to overestimate indicator function by exponentials, and then re- arrange the expressions to form martingale terms. Simply observe that for λ ∈ (λmin,0), Nt(γ, [κ t,∞)) = 1{Xu(t)≤−γt;Yu(t)≥κ 1{Xu(t)≤−γt;Yu(t)2≥κ2t}e (Yu(t) 2−κ2t)+λ(Xu(t)+γt) ≤ e(E −κ2ψ+ +λγ)t Yu(t) 2+λXu(t)−E+λ t ≤ e−λ(c )tZ+λ (t)e +λγ−κ2ψ+ where E±λ =−λc Recall from equations (11) and (32) that E−λ + λγ − κ2ψ λ has a minimal value of ∆(γ,κ) achieved when λ= λ̄(γ,κ). Since c̃(θ) is the minimal value of cλ, Theorem 20 implies that lim sup t−1 logZ+λ (t)≤ λ(c λ − c λ )(65) A TYPED BRANCHING DIFFUSION 43 almost surely for all λ ∈ (λmin,0). In cases where ∆(γ,κ)< 0, we can use the optimal value for λ, Theorem 20 and trivially note that Nt(γ, [κ t,∞)) is integer valued to deduce that 1{Yu(t)≥κ t;Xu(t)≤−γt} = 0 eventually, almost surely. Hence, D(γ,κ) =−∞ almost surely if ∆(γ,κ)< 0. Otherwise we have ∆(γ,κ)≥ 0, which in fact guarantees that γ ∈ (0, c̃(θ)] and hence λ̄(γ,κ) ∈ [λ̃(θ),0). Then since lim sup t−1 logNt(γ, [κ t,∞)) ≤ lim sup t−1 log(e−λ(c )tZ+λ (t)) + (E λ + λγ − κ 2ψ+λ ) we can again make use of Theorem 20 and the minimizing λ value, λ̄(γ,κ), to get the bound limsup t−1 logNt(γ, [κ t,∞))≤∆(γ,κ) almost surely, as desired. Notice that, when ∆(γ,κ) = 0, the right-hand side of the inequality at (64) will tend to infinity (see Corollary 19). Then, on the boundary, we have only shown that lim sup t−1 logNt(γ, [κ t,∞))≤ 0. 10. Proof of Theorem 1. The spatial growth rate. We first bound the spatial growth rate above. Suppose that C ⊂ R is Borel-measurable with y2φ(y)dy > 0. Let λ ∈ (λmin,0), then 1{Xu(t)≤−γt;Yu(t)∈C} ≤ 1{Yu(t)∈C}e λ(Xu(t)+γt) = e(E +λγ)t 1{Yu(t)∈C} e λXu(t)−E−λ t ≤ e(E +λγ)tZ−λ (t). Recalling equations (8) and (19), we therefore have 1{Xu(t)≤−γt;Yu(t)∈C} ≤ e ∆(γ)tZ−λγ (t). Now if γ ≥ c̃(θ), corresponding to λγ ∈ (λmin, λ̃(θ)] and having ∆(γ)≤ 0, we know from Theorem 17 that Z−λγ (∞) = 0 almost surely. Then, γ > c̃(θ) ⇒ 1{Xu(t)≤−γt;Yu(t)∈C} = 0 eventually, a.s. 44 Y. GIT, J. W. HARRIS AND S. C. HARRIS Otherwise, if γ ∈ (0, c̃(θ)), corresponding to λγ ∈ (λ̃(θ),0) and having ∆(γ)> 0, Theorem 17 tells us that Z−λγ (∞)> 0 almost surely, hence lim sup t−1 log 1{Xu(t)≤−γt;Yu(t)∈C} ≤∆(γ). Now we bound the growth rate from below. Let ε > 0 be small, λ̃(θ) < λ < 0, and µ = λ − ε. We recall now that E−λ is convex so ≥ 0 and γµ > γλ. Then eλXu(t)−E 1{−(γλ+ε)t≤Xu(t)≤−(γλ−ε)t;Yu(t)∈C} eλ(−(γλ+ε)t)−E 1{−(γλ+ε)t≤Xu(t)≤−(γλ−ε)t;Yu(t)∈C} = e(−λγλ−E −λε)t ∑ 1{−(γλ+ε)t≤Xu(t)≤−(γλ−ε)t;Yu(t)∈C} ≤ e(−λγλ−E −λε)t ∑ 1{Xu(t)≤−(γλ−ε)t;Yu(t)∈C}. t−1 log 1{Yu(t)∈C}e λXu(t)−E−λ t 1{|Xu(t)/t+γλ |<ε} ≤−λγλ−E−λ − λε+ t −1 log 1{Xu(t)≤−(γλ−ε)t;Yu(t)∈C}. Letting t→∞, using Corollary 19 and remembering that for λ̃(θ)< λ≤ 0 we have Z−λ (∞)> 0 a.s., we find 0≤−λγλ −E−λ − λε+ lim inft→∞ t −1 log 1{Xu(t)≤−(γλ−ε)t;Yu(t)∈C} and as ε > 0 can be arbitrarily small we have lim inf t−1 log 1{Xu(t)≤−γλt;Yu(t)∈C} ≥E λ + λγλ. Equivalently, lim inf t−1 log 1{Xu(t)≤−γt;Yu(t)∈C} ≥E + λγγ =∆(γ) and hence the lim sup and lim inf agree as required. We note that these proofs will easily adapt to cover a multi-type branching Brownian motion where the types evolve as a finite state Markov chain, such as found in [2], where it will also be possible to prove the analogous A TYPED BRANCHING DIFFUSION 45 convergence theorem required when we have a finite type space by adapting the proof of Theorem 18 found in [12]. In the standard branching Brownian motion case things are even simpler to adapt (where, of course, there is no need for any convergence result akin to Theorem 18). All the information necessary is contained in the martingales u∈Nt exp(λXu(t)− (λ 2/2 + r)t) studied by Neveu [22] and, as first came to our attention during discussions with J. Warren, the martingale with parameter λ can only be capable of “counting” particles near γλt in space at large times t, so when this martingale is uniformly integrable particles must perpetually be found with the corresponding speed. Of course, in this case more precise results, in the spirit of Watanabe [25], also exist. Acknowledgments. We would like to thank two anonymous referees for providing extremely helpful and thorough reviews of earlier incarnations of this manuscript. Their numerous invaluable comments led to a much improved presentation of this work. REFERENCES [1] Athreya, K. B. (2000). Change of measures for Markov chains and the L logL theorem for branching processes. Bernoulli 6 323–338. MR1748724 [2] Champneys, A., Harris, S. C., Toland, J., Warren, J. andWilliams, D. (1995). Algebra, analysis and probability for a coupled system of reaction-diffusion equa- tions. Philos. Trans. Roy. Soc. London 350 69–112. MR1325205 [3] Chauvin, B. and Rouault, A. (1988). KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees. Probab. Theory Related Fields 80 299–314. MR0968823 [4] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York. MR1619036 [5] Enderle, K. and Hering, H. (1982). Ratio limit theorems for branching Orstein– Uhlenbeck processes. Stochastic Process. Appl. 13 75–85. MR0662806 [6] Engländer, J. and Kyprianou, A. E. (2004). Local extinction versus local exponential growth for spatial branching processes. Ann. Probab. 32 78–99. MR2040776 [7] Geiger, J. (1999). Elementary new proofs of classical limit theorems for Galton– Watson processes. J. Appl. Probab. 36 301–309. MR1724856 [8] Hardy, R. and Harris, S. C. (2006). A new formulation of the spine approach to branching diffusions. Available at http://arxiv.org/abs/math.PR/0611054. [9] Hardy, R. and Harris, S. C. (2006). Spine proofs for Lp-convergence of branching diffusion martingales. Available at http://arxiv.org/abs/math.PR/0611056. [10] Hardy, R. and Harris, S. C. (2006). A spine proof of a large-deviations principle for branching Brownian motion. Stochastic Process. Appl. 116 1992–2013. [11] Harris, S. C. (1999). Travelling-waves for the FKPP equation via probabilistic ar- guments. Proc. Roy. Soc. Edinburgh Sect. A 129 503–517. MR1693633 [12] Harris, S. C. (2000). Convergence of a “Gibbs–Boltzmann” random measure for a typed branching diffusion. Séminaire de Probabilités XXXIV. Lecture Notes in Math. 1729 239–256. Springer, Berlin. MR1768067 http://www.ams.org/mathscinet-getitem?mr=1748724 http://www.ams.org/mathscinet-getitem?mr=1325205 http://www.ams.org/mathscinet-getitem?mr=0968823 http://www.ams.org/mathscinet-getitem?mr=1619036 http://www.ams.org/mathscinet-getitem?mr=0662806 http://www.ams.org/mathscinet-getitem?mr=2040776 http://www.ams.org/mathscinet-getitem?mr=1724856 http://arxiv.org/abs/math.PR/0611054 http://arxiv.org/abs/math.PR/0611056 http://www.ams.org/mathscinet-getitem?mr=1693633 http://www.ams.org/mathscinet-getitem?mr=1768067 46 Y. GIT, J. W. HARRIS AND S. C. HARRIS [13] Harris, S. C. and Williams, D. (1996). Large deviations and martingales for a typed branching diffusion. I. Astérisque 236 133–154. MR1417979 [14] Harris, T. E. (2002). The Theory of Branching Processes. Dover, Mineola, NY. MR1991122 [15] Itô, K. and McKean, H. P. (1965). Diffusion Processes and Their Sample Paths. Academic Press, New York. MR0199891 [16] Kurtz, T., Lyons, R., Pemantle, R. and Peres, Y. (1997). A conceptual proof of the Kesten–Stigum theorem for multi-type branching processes. In Classi- cal and Modern Branching Processes (Minneapolis, MN, 1994 ) (K. B. Athreya and P. Jagers, eds.). IMA Vol. Math. Appl. 84 181–185. Springer, New York. MR1601737 [17] Kyprianou, A. E. (2004). Travelling wave solutions to the K–P–P equation: Alter- natives to Simon Harris’ probabilistic analysis. Ann. Inst. H. Poincaré Probab. Statist. 40 53–72. MR2037473 [18] Lyons, R. (1997). A simple path to Biggins’ martingale convergence for branching random walk. In Classical and Modern Branching Processes (Minneapolis, MN, 1994 ) (K. B. Athreya and P. Jagers, eds.). IMA Vol. Math. Appl. 84 217–221. Springer, New York. MR1601749 [19] Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of L logL cri- teria for mean behavior of branching processes. Ann. Probab. 23 1125–1138. MR1349164 [20] McKean, H. P. (1975). Application of Brownian motion to the equation of Kolmogorov-Petrovskĭı–Piskunov. Comm. Pure Appl. Math. 28 323–331. MR0400428 [21] Murray, J. D. (2003). Mathematical Biology. II, 3rd ed. Springer, New York. MR1952568 [22] Neveu, J. (1988). Multiplicative martingales for spatial branching processes. In Sem- inar on Stochastic Processes 1987 (E. Çinlar, K. L. Chung and R. K. Getoor, eds.) 223–242. Birkhäuser, Boston. MR1046418 [23] Olofsson, P. (1998). The x logx condition for general branching processes. J. Appl. Probab. 35 537–544. MR1659492 [24] Varadhan, S. R. S. (1984). Large Deviations and Applications. SIAM, Philadelphia. MR0758258 [25] Watanbe, S. (1967). Limit theorem for a class of branching processes. In Markov Processes and Potential Theory (J. Chover, ed.) 205–232. Wiley, New York. MR0237007 Y. Git Statistical Laboratory Cambridge University 22 Mill Street Cambridge CB1 2HP E-mail: [email protected] J. W. Harris Department of Mathematics University of Bristol University Walk Bristol BS8 1TW E-mail: [email protected] S. C. Harris Department of Mathematical Sciences University of Bath Bath BA2 7AY E-mail: [email protected] URL: http://people.bath.ac.uk/massch/ http://www.ams.org/mathscinet-getitem?mr=1417979 http://www.ams.org/mathscinet-getitem?mr=1991122 http://www.ams.org/mathscinet-getitem?mr=0199891 http://www.ams.org/mathscinet-getitem?mr=1601737 http://www.ams.org/mathscinet-getitem?mr=2037473 http://www.ams.org/mathscinet-getitem?mr=1601749 http://www.ams.org/mathscinet-getitem?mr=1349164 http://www.ams.org/mathscinet-getitem?mr=0400428 http://www.ams.org/mathscinet-getitem?mr=1952568 http://www.ams.org/mathscinet-getitem?mr=1046418 http://www.ams.org/mathscinet-getitem?mr=1659492 http://www.ams.org/mathscinet-getitem?mr=0758258 http://www.ams.org/mathscinet-getitem?mr=0237007 mailto:[email protected] mailto:[email protected] mailto:[email protected] http://people.bath.ac.uk/massch/ Introduction The branching model Application to reaction–diffusion equations Main results Martingales The asymptotic growth-rate of particles along spatial rays The asymptotic shape and growth of the branching diffusion Some expectation calculations The expected rate of growth along spatial rays The expected asymptotic shape Short climb large deviation heuristics A birth–death process Finding the optimal path and probability An important note on the optimal paths Proof of Theorem 3. Lower bound The ``spine'' setup and results Proof of Theorem 7. The short climb probability Martingale results Proof of Theorem 3. Upper bound Proof of Theorem 1. The spatial growth rate Acknowledgments References Author's addresses

0704.0381

The collision velocity of the bullet cluster in conventional and modified dynamics

Mon. Not. R. Astron. Soc. 000, 1–8 (2007) Printed 27 October 2018 (MN LATEX style file v2.2) The collision velocity of the bullet cluster in conventional and modified dynamics G. W. Angus1⋆, S. S. McGaugh2† 1SUPA, School of Physics and Astronomy, University of St. Andrews, Scotland KY16 9SS 2Department of Astronomy, University of Maryland, College Park, MD 20742-242 USA Accepted ... Received ... ; in original form ... ABSTRACT We consider the orbit of the bullet cluster 1E 0657-56 in both CDM and MOND using accurate mass models appropriate to each case in order to ascertain the maximum plausible collision velocity. Impact velocities consistent with the shock velocity (∼ 4700 km s ) occur naturally in MOND. CDM can generate collision velocities of at most ∼ 3800 km s−1, and is only consistent with the data provided that the shock velocity has been substantially enhanced by hydrodynamical effects. Key words: gravitation - dark matter - galaxies: clusters: individual (1E 0657-56) 1 INTRODUCTION Many lines of observational evidence now oblige us to be- lieve that the universe is filled with a novel, invisible form of mass that dominates gravitationally over normal baryonic matter. In addition, a dark energy component which exerts negative pressure to accelerate the expansion of the universe is also necessary (Chernin et al. 2007). Though this ΛCDM paradigm is well established, we still have only ideas about what these dark components might be, and no laboratory detections thereof. One possible alternative to ΛCDM is the Modified New- tonian Dynamics (MOND; Milgrom 1983a,b,c). This hy- pothesis has been more successful than seems to be widely appreciated (McGaugh & de Blok 1998; Sanders & Mc- Gaugh 2002), and has received a theoretical boost from the introduction of generally covariant formulations (Bekenstein 2004; Sanders 2005; Zlosnik, Ferreira, & Starkman 2006,7). The dark matter and alternative gravity paradigms are rad- ically different, so every observation that might distinguish between them is valuable. ΛCDM is known to work well on large scales (e.g., Spergel et al. 2006) while MOND is known to work well in in- dividual galaxies (Sanders & McGaugh 2002). This success, incorporating the tight correlation between dark and lumi- nous mass in the DM framework (McGaugh 2005; Famaey et al. 2007b) extends over five decades in mass (Fig.1) rang- ing from tiny dwarfs (e.g., Milgrom & Sanders 2007) through spirals of low surface brightness (de Blok & McGaugh 1998), our own Milky Way (Famaey & Binney 2005) and other high ⋆ email: [email protected] † email: [email protected] surface brightness (Sanders 1996; Sanders & Noordermeer 2007) to massive ellipticals (Milgrom & Sanders 2003). The recent observations of tidal dwarf galaxies by Bournaud et al. (2007) provides a severe challenge to CDM but is natu- rally explained in MOND with zero free parameters (Mil- grom 2007; Gentile et al. 2007). Having said that, MOND persistently fails to completely explain the mass discrepancy in rich clusters of galaxies. Consequently, clusters require substantial amounts of non-luminous matter in MOND. That rich clusters contain more mass than meets the eye in MOND goes back to Milgrom’s original papers (Mil- grom 1983c). At the time, the discrepancy was very much larger than it is today, as it was not then widely appreci- ated how much baryonic mass resides in the intra-cluster medium. Further work on the X-ray gas (e.g., Sanders 1994, 1999) and with velocity dispersions (McGaugh & de Blok 1998) showed that MOND was at least within a factor of a few, but close inspection revealed a persistent discrepancy of a factor of two or three in mass (e.g., Gerbal et al. 1992; The & White 1998; Pointecouteau & Silk 2005, Buote & Canizares 1994). Weak gravitational lensing (Angus et al. 2007a; Takahashi & Chiba 2007; Famaey, Angus et al. 2007) provides a similar result. To make matters worse, the distribution of the unseen mass does not trace that of either the galaxies or the X- ray gas (Aguirre et al. 2001; Sanders 2003; Angus et al. 2007b; Sanders 2007). In Fig. 1 we plot the baryonic mass of many spiral galaxies and clusters against their circular velocity together with the predictions of MOND and CDM. MOND is missing mass at the cluster scale. CDM suffers an analogous missing baryon problem on the scale of individual galaxies. The colliding bullet cluster 1E-0657-56 (Clowe et al. c© 2007 RAS http://arxiv.org/abs/0704.0381v2 2 G. W. Angus and S. S. McGaugh 2004,2006, Bradac et al. 2006, Markevitch et al. 2004, Markevitch & Vikhlinin 2007) illustrates in a spectacular way the residual mass discrepancy in MOND. While cer- tainly problematic for MOND as a theory, it does not con- stitute a falsification thereof. Indeed, given that the need for extra mass in clusters was already well established, it would have been surprising had this effect not also manifested it- self in the bullet cluster. The new information the bullet cluster provides is that the additional mass must be in some collision-less form. It is a logical fallacy to conclude that because extra mass is required by MOND in clusters, that dark matter is required throughout the entire universe. While undeni- ably problematic, the residual mass discrepancy in MOND is limited to groups and rich clusters of galaxies: these are the only systems in which it systematically fails to remedy the dynamical mass discrepancy (see discussion in Sanders 2003). Could we be absolutely certain that we had accounted for all the baryons in clusters, then MOND would indeed be falsified. But CDM suffers an analogous missing baryon problem in galaxies (Fig.1) in addition to the usual dynam- ical mass discrepancy, yet this is not widely perceived to be problematic. In either case we are obliged to invoke the ex- istence of some dark mass which is presumably baryonic (or perhaps neutrinos) in the case of MOND. In neither case is there any danger of violating big bang nucleosynthesis constraints. The integrated baryonic mass density of rich clusters is much less than that of all baryons; having the required mass of baryons in clusters would be the proverbial drop in the bucket with regards to the global missing baryon problem. A pressing question is the apparently high relative ve- locity between the two clusters that comprise the bullet cluster 1E 0657-56 (Clowe et al. 2006, Bradac et al. 2006, Markevitch et al. 2004, Markevitch & Vikhlinin 2007). The relative velocity derived from the gas shockwave is vrel = 4740+710 −550 kms −1 (Clowe et al. 2006). Taken at face value, this is very high, and seems difficult to reconcile with ΛCDM (Hyashi & White 2006). The problem is sufficiently large that it has been used to argue for an additional long range force in the dark sector (Farrar & Rosen 2007). Here we ex- amine the possibility of such a large velocity in both CDM and MOND. One critical point that has only very recently been ad- dressed is how the shock velocity relates to the collision ve- locity of the clusters. Naively, one might expect the dissipa- tional collision of the gas clouds to slow things down so that the shock speed would provide a lower limit on the colli- sion speed. Recent hydrodynamical simulations (Springel & Farrar 2007; Milosavljevic et al. 2007) suggest the opposite. A combination of effects in the two hydrodynamical sim- ulations show that the shock velocity may be higher than the impact velocity. The results of the two independent hy- drodynamical simulations do not seem to be in perfect con- cordance, and the precise result seems to be rather model dependent. Nevertheless, it seems that the actual relative velocity lies somewhere in the range 3500-4500 kms−1. The difficulties posed by a high collision velocity for CDM have been discussed previously by Hayashi & White (2006) and Farrar & Rosen (2007). Whereas Springel & Far- rar (2007) and Milosavljevic et al. (2007) consider the com- plex hydrodynamic response of the two gas clouds during Figure 1. Shows baryonic mass against circular velocity. Rotat- ing galaxies (blue circles) are from McGaugh (2005) and clusters (green triangles) are from Sanders (2003) using the measured tem- perature to estimate the circular velocity assuming isothermality. The solid orange line is the CDM M-V relation (Steinmetz & Navarro 1999) assuming Mb = fbMvir with fb = 0.17 (Spergel et al. 2006) and the dashed red line is the MOND prediction. The spirals lie directly on the MOND prediction, but the clusters are generally 2-3 times in mass below it. The CDM expectation is nicely consistent with clusters, but implies many dark baryons in spirals in addition to the non-baryonic dark matter. the ongoing collision, here we investigate the ability of two clusters like those comprising the bullet cluster to accelerate to such a high relative velocity in the case of both CDM and MOND prior to the merger. We compute a simple free fall model for the two clusters in an expanding universe with realistic mass models, and ask whether the observed colli- sion velocity can be generated within the time available. We take care to match the mass models to the specific observed properties of the system appropriate to each flavor of grav- ity in order to realistically evaluate the orbit of the clusters prior to their collision. 2 MODELING THE FREEFALL We wish to address a simple question. Given the observed masses of the two clusters, is it possible to account for the measured relative velocity from their gravitational freefall? The expansion of the universe mitigates against large veloc- ities, since the clusters must decouple from the Hubble flow before falling together. Presumably it takes some time to form such massive objects, though this is expected to occur earlier in MOND than in ΛCDM (Sanders 1998, 2001; Mc- Gaugh 1999, 2004; Nusser 2002; Stachniewicz & Kutschera 2002; Knebe & Gibson 2004; Dodelson & Liguori 2006). The clusters are observed at z=0.3, giving at most 9Gyrs for them to accelerate towards each other. This imposes an up- per limit to the velocity that can be generated gravitation- ally. Without doing the calculation, it is not obvious whether the larger masses of the clusters in CDM or the stronger long range force in MOND will induce larger relative velocities. Since we know the state of the system directly prior to collision, it makes sense to begin our simulations from the final state and work backwards in time towards when the relative velocity was zero. This point, where the clus- c© 2007 RAS, MNRAS 000, 1–8 The collision velocity of the bullet cluster in conventional and modified dynamics 3 ters have zero relative velocity, is when they turned around from the Hubble flow and began their long journey gravi- tating towards each other. Working backwards in time leads to potentially counter-intuitive discussions (such as Hubble contraction), which we try to limit. We must account for the Hubble expansion in a manner representing the universe before z=0.3. The detailed form of the expansion history of the universe a(t) is not known in the case of MOND, so we take the scale factor of ΛCDM in both models da(t) + ΩΛa . (1) Where we take Ho = 72 km s −1Mpc−1, Ωm=0.27 and ΩΛ = 0.73. The important aspect is the basic fact that the uni- verse is expanding and the mutual attraction of the clusters must overcome this before they can plunge together at high velocity. We implement the scale factor in the simulations through the equation of motion [a(t)v] = g. (2) Computing this numerically, from time step to time step we calculate the ratio of the scale factor in the previous time step to the current time step (i.e. ǫ = a(ti−1)/a(ti); we use negative time steps to move backwards in time from the presently known configuration, so ǫ > 1 (higher i means earlier universe). We then have v(ti) = v(ti−1)ǫ+ g∆t. The right hand side of Eq. 2 differs in MOND and CDM not only because the law of gravity is altered, but also be- cause the gravitating masses are higher in CDM. The initial conditions are the crux of the problem, with at least 4 unknowns. These include the masses of the two clusters, the relative velocity of the clusters, and the dis- tance of separation between the two when they had this relative velocity. The separation is the same in MOND and Newtonian gravity, but the Newtonian mass is higher. The relative velocity of the two clusters can be mea- sured because, in the last few 100Myrs, the less massive sub cluster has passed through the centre of the more mas- sive main cluster. The ram pressure has imposed a smooth bow shock (Markevitch et al. 2004, Markevitch & Vikhlinin 2007) on the gas of the sub cluster. Since the relative veloc- ity is the foundation of the problem we leave it free and try to estimate it by fixing other variables. In our simulation, we think it sensible to consider the separation of the two clusters (i.e. of the two centres of mass) when they had the calculated relative velocity to be when the leading edge of the sub cluster’s gas cloud began to pass through the dense region of gas belonging to the main cluster and separate from the dark matter. It appears that the centre of the sub cluster’s gas cloud (the location of the bullet) is preceeded by the bow shock by around 200kpc further in the direc- tion of travel. We take 200kpc to also be the radius within which the gas of cluster 1 was dense enough to imprint the bow shock. Indeed, the gas mass of the main cluster (sub cluster) is only measured out to 180kpc (100kpc) and could not be found further detailed in the literature. However, the uncertainty is large enough that we wished to clarify the impact of different initial separations by always using a range of initial separations of between 350-500kpc. This Figure 2. The total enclosed masses for the main cluster (black) and sub cluster (red) for CDM (solid), MOND with standard µ (dashed) and simple µ (dotted). separation is defined as when the two pre-collision clusters had the relative velocity of vrel related to the shock velocity 4740+710 −550 kms −1. Now of course, they are on the opposite sides on the sky after having passed through each other and the gas has been offset from the DM. 3 THE COLLISION IN CDM In the CDM framework, it is no problem to generate two clusters in an N-body simulation and calculate all gravi- tational accelerations exactly. However, in MOND we are dealing with non linear gravity and the tools for such pur- poses are only now being developed (Nipoti et al. 2007a). Additionally, since we begin our simulations with the over- lap of the two clusters, it is not guaranteed that the clusters preserve their shapes as they separate. Furthermore, it was not possible to simply include accretion history (Wechsler et al. 2002) in the N-body simulations or easily vary the truncation radius of the DM halo as was necessary. Therefore, a better method was to semi-analytically ac- count for these aspects in a simulation where gravity of one cluster acting on the other is just the mass enclosed by a sphere around the gravitating body’s centre of mass with radius equal to the separation of the two cluster’s centres of mass. For this procedure, the only two unknowns are the sep- aration, which is initially known and computed each time- step; and the mass enclosed. The enclosed mass depends on the density profile of the two clusters and was fitted by Clowe et al. (2006) using NFW profiles of the form (see Angus & Zhao 2007) ρ(r) = m200r r + r200c = ln(1+c)− c 1 + c where c is the concentration. The enclosed mass goes as c© 2007 RAS, MNRAS 000, 1–8 4 G. W. Angus and S. S. McGaugh Figure 3. The total enclosed masses for the main cluster as a function of time, where 9.3Gyr marks the collision of the two clusters and 0Gyr represents the Big Bang. The three lines corre- spong to different mass assembly rates α=0.5 (black), 1.0 (blue) and 1.5 (red). The mass loss is halted at m200/50. m(r) = Am200 − 1 + For the main cluster they give m200 = 1.5 × 1015M⊙, r200 = 2100kpc and concentration c=1.94. For the sub clus- ter, m200 = 1.5 × 1014M⊙, r200 = 1000kpc and c=7.12. We augment the DM with a baryon fraction of 17% (Spergel et al. 2006) which was part of the total mass during freefall. We ran dynamical time steps (negative) such that ∆t = 10 d[pc]Myr (5) Where d is the separation of the two centres of mass and ∆t has a maximum value of 1Myr. Since initially d∼ 400kpc, the starting time steps are ∼0.06Myr. The simulations were run until 9Gyr had elapsed. The mass distributions as functions of radius for the two clusters in CDM and the ones used in the MOND sim- ulations are shown in Fig.2. A subtle point about the total masses of the two clusters is that we do not expect the mass to remain constant as we go back in time. Presumably they grew from a seed of negligible mass at high redshift (see discussion in Cameron & Driver 2007). This tends to im- pede their freefall, reducing the maximum collision velocity to ∼ 2900 kms−1 by the estimate of Farrar & Rosen (2007). To include this, without the impedance, we use the proce- dure of Wechsler et al. (2002) who used the relation M(z) = M(z = 0.3)e −α(z−0.3) where α obviously encodes the speed of the accretion or assembly of the halo. Typical values used in their work are 0.5< α <2.0. In Fig.3 we plot the mass enclosed within r200 for the two clusters as functions of reshift for α=0.5,1.0 and 1.5. Note we always keep a floor value of cluster mass of m200/50 so the halo is never completely disassembled. Another important point is whether mass integrated out beyond r200 should be included, since the actual virial ra- dius depends on both cosmology and redshift (Bullock et al. 2001). Indeed, the internal gravity of the main cluster has not reached ao by r200 = 2100kpc meaning the MOND dy- namical mass has not yet saturated. However, recall that it takes ∼ (r200−d)/vrel = (2100kpc−400kpc)/3400 kms−1 ∼ 500Myr for the clusters to separate enough for this extra matter to even begin to manifest itself. The cluster is also losing mass (backwards in time) due to accretion coupled with the fact there must be overdensities on the opposite side of the universe countering the influence of these overdensi- ties. However, using Eq.4 it is straight-forward to include all the enclosed mass out to any radius because the parameters r200 and m200 do not explicitly force the enclosed mass to truncate at r200, they simply define the shape of the profile. 4 THE COLLISION IN MOND In MOND, the basic modification of purely Newtonian dy- namics is µ(x)g = gN, (7) where gN is the Newtonian acceleration computed in the usual way from the baryonic mass distribution, g is the ac- tual acceleration (including the effective amplification due to MOND conventionally ascribed to DM), ao is the char- acteristic acceleration at which the modification becomes effective (∼ 10−10 m s−2), x = g/ao, and µ(x) is an inter- polation function smoothly connecting the Newtonian and MOND regimes. In the limit of large accelerations, g ≫ ao, µ → 1 and the Newtonian limit is obtained: everything be- haves normally. The MOND limit occurs only for exceed- ingly low accelerations, with µ → x for g ≪ ao. We im- plement two possible versions of the interpolation function: the ‘standard’ function traditionally used in fitting rotation curves: 1 + x2 (e.g., Sanders & McGaugh 2002), and the ‘simple’ function found by Famaey & Binney (2005) to provide a good fit the terminal velocity curve of the Galaxy: 1 + x . (9) A well known problem with implementing the MOND force law in numerical computations is that the original for- mulation (Eq 7) does not conserve momentum (Felten 1984; Bekenstein 2007). This was corrected with the introduction of a Lagrangian formulation of MOND (Bekenstein & Mil- grom 1984; Milgrom 1986) which has the modified Poisson equation ∇ · [µ(|∇Φ|/ao)∇Φ] = 4πGρ. (10) This formulation has been shown to obey the necessary conservation laws (Bekenstein & Milgrom 1984; Bekenstein 2007). With some rearrangement, it leads to µ(x)g = gN +∇× h, (11) c© 2007 RAS, MNRAS 000, 1–8 The collision velocity of the bullet cluster in conventional and modified dynamics 5 which we recognize as Eq 7 with the addition of a curl field. Unfortunately, implementing a numerical formulation of the modified Poisson equation is not a simple one-line change to typical N-body codes: this fails to obey the conservation laws. Instead, one needs an entirely different numerical ap- proach than is commonly employed. Progress has been made along these lines (e.g., Brada & Milgrom 1995, 1999; Ciotti, Londrillo, & Nipoti 2006; Nipoti, Londrillo & Ciotti 2007ab; Tiret & Combes 2007; see also Nusser 2002; Knebe & Gib- son 2004), but we do not seek here a full N-body treatment of complex systems. Rather, we wish to develop and apply a simple tool (Angus & McGaugh, in preparation) that can provide some physical insight into basic problems. For the specific case of the large collision velocity of the bullet clus- ter, it suffices to treat the curl field as a small correction to the center of mass motion (Milgrom 1986). The external field effect (see Milgrom 1983a, Bekenstein 2007) is crudely approximated as a constant of appropriate magnitude (Mc- Gaugh 2004). We checked the effect of varying the external field, which is modest. It is not possible to do better without complete knowledge of the mass distribution in the environ- ment of the clusters. When modeling the bullet cluster in MOND, Angus et al. (2007a) fitted the convergence map of Clowe et al. (2006) using spherical potential models for the four mass compo- nents. Their best fit gives masses for all four components in MOND and standard gravity. Unfortunately, the map is only sensitive out to 250kpc from the respective centres which ne- glects an over large portion of the dynamical mass. So, in or- der to remain consistent with the CDM simulations, we take the NFW profile and calculate what the MOND dynamical masses for the two commonly used interpolating functions (Eq.8 & 9) are, as shown in Fig.2. The Newtonian mass for the main cluster is twice that of the MOND dynamical mass with the standard µ and three times when the simple µ is used. The mutual gravity imposed upon the sub cluster by the main cluster is |gsub + gex| gsub = gn,sub = −GMmain(d)/d2 (12) and we simply swap the subscripts around to find the mutual gravity of the sub cluster upon the main cluster. Following on from above, d is the distance between the two centres of mass and Mmain(d) is the enclosed mass within a radius d from the centre of mass of the main cluster. The gex is the external field limiting the MOND correction which comes from large scale structure and is always assumed orthogonal to the direction of gsub, making the argument of the µ func- tion more easily expressed as (g2sub+g . The direction and amplitude of gex is unknown at all times. The MON- Dian additional acceleration becomes minor when the accel- eration drops below gex. We use gex = ao/30 (Aguirre et al. 2001; McGaugh 2004) which is roughly the external field imposed on the Milky Way by M31 and vice versa (Famaey et al. 2007a, Wu et al. 2007). 5 N-BODY COLLISION Our first attempt at simulating the collision in Newtonian gravity was using a standard N-body tree code. The benefit it gives is that in principal, we can more accurately compute the mutual gravity at the beginning of the simulation when the two clusters overlap. However, this is fraught with diffi- culties and inconsistencies. The first being that tidal effects undoubtedly stretch the two clusters and 2-body interac- tions may eject particles from the two halos. Therefore, it makes better sense to begin such a simulation from high redshift where the clusters are greatly separated and tidal effects are negligible and let them freefall in the expanding universe and when they collide, the tidal effects will be well accounted for. Of course, the problem is that it is not triv- ial to then sample collision velocities because the separation and time at which the two clusters began their freefall is not simply related. Moreover, the truncation of the two halos and different mass models are not easily varied. Neverthe- less, we did attempt a CDM N-body model with truncation at r200 for both halos. We found a similar result to that from the semi-analytical models of 3800 kms−1. 6 RESULTS The ability of the two clusters that comprise the bullet to bring each other to a halt at a finite time in the past is sen- sitive to both the flavour of gravity at work and the true relative velocity. For velocities larger than the maximum, the relative velocity never reaches zero and increase sharply at early times (large z). The two clusters do not gravitate strongly enough to generate such high velocities and would have to have had a huge relative velocity towards each other in the early universe in order to overcome the Hubble expan- sion and fall together with such a high relative velocity at z=0.3. Fig.4 shows how the relative velocity of the two clus- ters varies with time for a large sample of initial (meaning collisional) relative velocities for a CDM and MOND sample simulation. A difference of just 100 km s−1 can have a signif- icant impact on the time required to generate such a large velocity and by the same token, the longer the two clusters free-fall, the larger a velocity they can generate. Sadly, there is only a finite time (∼ 9Gyr)) since the Big Bang for this to happen. In Table 1 we’ve put the key results of the simulations so as to give the reader a feel for what the maximum rela- tive velocity that can be achieved is. Each velocity is that achieved with an initial separation of 425kpc, where taking 350 or 500kpc induces an increase or decrease of 100 kms−1 which we take as the minimum error. The most extreme CDM model is to have no truncation of the DM halos, ex- tending them out to r1. This absurd extreme allows a maxi- mum relative velocity of 4500 kms−1. Then, if we still allow the halos to extend to r1, but account for some assembly of the halos with α = 1 then the relative velocity reduces to 4200 kms−1. More realistically, if we truncate the halos at r200 and try four different halo assembly rates such that α =0.0, 0.5, 1.0 & 1.5 we get respective maximum relative velocities of 4000, 3900, 3800 and 3800 km s−1. These numbers repre- sent the plausible maximum relative velocities in the CDM framework. For the MOND case we ran simulations with both the simple (Eq.9) and standard (Eq.9) µ functions. The stan- dard function leads to higher dynamical masses from the c© 2007 RAS, MNRAS 000, 1–8 6 G. W. Angus and S. S. McGaugh Figure 4. Shows the relative velocity of the two clusters plotted against time (a) CDM and (b) MOND. Time=0Myr is the current (z=0.3) relative velocity of the two clusters with larger times corresponding to higher redshifts. Black lines correspond to relative velocities that are achievable, whereas red lines are not. In (a) we use the simulation (CDM2c) for which α=1.0, d=425kpc and we truncate the halos at r200. The relative velocities used are vrel=3500-4200 km s −1 in intervals of 100 km s−1. In (b) we use the simulation (MONDst2) which uses the standard µ function and α=0.5, d=425kpc. The relative velocities used are vrel=4100-4800 km s −1 in intervals of 100 km s−1. The 4 dashed lines are the predicted relative velocities according to the mean and 1σ error of the original relative velocity from Markevitch & Vikhlinin (2007) in blue, the simulations of Milosavljevic (2007) in green and Springel & Farrar (2007) in turquoise. The high observed collision velocity is more readily obtained in MOND than CDM. Model Max Vrel [ km s −1] Truncation Radius α Gravity CDM1a 4500 r1 0.0 Newtonian CDM1b 4200 r1 1.0 Newtonian CDM2a 4000 r200 0.0 Newtonian CDM2b 3900 r200 0.5 Newtonian CDM2c 3800 r200 1.0 Newtonian CDM2d 3800 r200 1.5 Newtonian MONDst1 4800 r200 0.0 MOND-standard µ MONDst2 4500 r200 0.5 MOND-standard µ MONDsi1 4600 r200 0.0 MOND-simple µ MONDsi2 4500 r200 0.5 MOND-simple µ Table 1. Shows the parameters used in the different models and gives the maximum attainable relative velocity for each. NFW profile, but lower 2-body gravity. The standard (sim- ple) function with no accretion and with α = 0.5 gener- ate 4800 (4600) and 4600 (4500) kms−1 respectively and for comparison, the maximum CDM velocity with those reduced masses is just 2700 (2300) kms−1. This is a clear demonstra- tion of the expectation in MOND for larger peculiar veloc- ities. We use the lower assembly parameter α = 0.5 be- cause structure is expected to form more swiftly in MOND (Sanders 1998, 2001). An important factor is that of the fitted NFW density profile to the convergence map, in which matter is extrapo- lated to 2100kpc and 1000kpc for the main and sub cluster respectively. Presumably the significance of the detection of this mass is negligible and the NFW fit has been made as- suming if we know the details in the central 250kpc, then we know the density out to r200. The mass sheet degeneracy is broken by constraining the mass at the edges of the fit based on the slope of the profile in the inner regions - but if the mass profile is wrong then it could lead to the completely wrong measurement for the value of the mass sheet (Clowe, de Lucia and King 2004). All of this means that the density profiles of the two clusters could be moderately different in reality. However, the actual shape of any profile is less important to the rela- tive velocity than simply the normalisation of the total mass. To this end we have simulated the collision with 10% more and 10% less mass for both clusters (with assembly param- eter α = 1). The effect is to increase (10% more mass) or decrease (10% less mass) the relative velocity by 200 kms−1 from 4800 kms−1 for model MONDst1. Another concern is that the clusters are unlikely to be spherically symmetric (Buote & Canizares 1996) and are presumably elongated in the direction of motion. Again this could lead to an incorrect density profile, whereas ellipticity itself would have little effect on our results. c© 2007 RAS, MNRAS 000, 1–8 The collision velocity of the bullet cluster in conventional and modified dynamics 7 7 SUMMARY We have constructed specific mass models for the bullet cluster in both CDM and MOND. We integrate backwards from the observed conditions to check whether the large (∼ 4700 km s−1) apparent transverse velocity can be at- tained in either context. We find that it is difficult to achieve vrel > 4500 kms −1 under any conditions. Never- theless, within the range of the uncertainties, the appropri- ate velocity occurs fairly naturally in MOND. In contrast, ΛCDM models can at most attain ∼ 3800 kms−1 and are more comfortable with considerably smaller velocities. Taken at face value, a collision velocity of 4700 km s−1 constitutes a direct contradiction to ΛCDM. Ironically, this cluster, widely advertised as a fatal observation to MOND because of the residual mass discrepancy it shows, seems to pose a comparably serious problem for ΛCDM. It has often been the case that observations which are claimed to falsify MOND turn out to make no more sense in terms of dark matter. Two critical outstanding issues remain to be clarified. The first is the exact density profiles and virial masses of the two clusters and the second is how the observed shock ve- locity relates to the actual collision velocity of the two grav- itating masses. The recent simulations of Springel & Farrar (2007) and Milosavljevic et al. (2007) seem to suggest that, contrary to naive expectations, hydrodynamic effects reduce the relative velocity of the mass with respect to the shock. A combination of effects is responsible, being just barely sufficient to reconcile the data with ΛCDM. Hydrodynam- ical simulations are notoriously difficult, and indeed these two recent ones do not agree in detail. It would be excellent to see a fully self-consistent simulation including both hy- drodynamical effects and a proper mass model and orbital computation like that presented here. There are a number of puzzling aspects to the hydrody- namical simulations. First of all, Springel & Farrar (2007) use Hernquist profiles for the DM distribution in the clus- ters and not NFW halos. Furthermore, they find that the morphology of the bullet is reproduced only for a remark- ably dead head-on collision. If the impact parameter is even 12kpc — a target smaller than the diameter of the Milky Way — quite noticeable morphological differences ensue. This can be avoided if the separation of mass centres hap- pens to be along our line of sight — quite a coincidence in a system already remarkable for having the vector of its collision velocity almost entirely in the plane of the sky. Fur- thermore, the mass models require significant tweaking from that infered from the convergence map and are unable to re- produce the currently observed, post merger positions of the gas and DM. It appears to us that only the first rather than the last chapter has been written on this subject. Getting this right is of the utmost importance, as the validity of both paradigms rests on the edge of a knife, separated by just a few hundred km s−1. More generally, the frequency of bullet-like clusters may provide an additional test. The probability of high collision velocities drops with dramatic rapidity in ΛCDM (Hyashi & White 2006). In contrast, somewhat higher velocities seem natural to MOND. Naively it would seem that high impact velocity systems like the bullet would be part and parcel of what might be expected of a MOND universe. With this in mind, it is quite intriguing that many bullet cluster like systems have been detected (although none quite as unique). The dark ring around Cl0024+17 tentatively observed by Jee et al. (2007; see also Famaey et al. 2007c), the dark core created by the “train wreck” in Abell 520 by Mahdavi et al. (2007), Cl0152+1357 (Jee et al. 2005a), MS1054+0321 (Jee et al. 2005b) and the line of sight merger with > 3000 kms−1 relative velocity observed by Dupke et al. (2007) for Abell 576 may all provide examples and potential tests. ACKNOWLEDGEMENTS We acknowledge discussions with Benoit Famaey, Tom Zlos- nik, Douglas Clowe, HongSheng Zhao, Greg Bothun, Moti Milgrom, Bob Sanders and Ewan Cameron. GWA thanks Steve Vine for his N-body tree code. GWA is supported by a PPARC scholarship. The work of SSM is supported in part by NSF grant AST0505956. REFERENCES Aguirre A., Schaye J., Quataert E., 2001, ApJ, 561, 550 Angus G.W., Famaey B., Zhao H.S., 2006, MNRAS, 371, Angus G.W., Zhao H.S., 2007, MNRAS, 375, 1146 Angus G.W., Shan H.Y., Zhao H.S., Famaey B., 2007a, ApJ, 654, L13 Angus G.W., Famaey B., Buote D.L., 2007b, submitted Angus G.W., McGaugh S.S., 2007, submitted Bekenstein J.D., 2004, PhRvD, 70, 083509 Bekenstein J.D., 2006, Contemporary Physics, 47, 387 Bekenstein J., Milgrom M., 1984, ApJ, 286, 7 Bournaud F., et al., 2007, Science, 311, 1166 Brada R., Milgrom M., 1995, MNRAS, 276, 453 Brada R., Milgrom M., 1999, ApJ, 519, 590 Bradac M., et al., 2006, ApJ, 652, 937 Bullock J.S., Kolatt T.S., Sigad Y., Somerville R.S., Kravtsov A.V., Klypin A.A., Primack J.R., Dekel A., 2001, MNRAS, 321, 559 Buote D.A., Canizares C.R., 1994, ApJ, 427, 86 Buote D.A., Canizares C.R., 1996, ApJ, 457, 565 Cameron E., Driver S.P., 2007, submitted Chernin A.D., et al. 2007, (arXiv:0706.4171) Ciotti L., Londrillo P., Nipoti C., 2006, ApJ, 640, 741 Clowe D., de Lucia G., King L., 2004, MNRAS, 350, 1038 Clowe D., et al., 2004, ApJ, 604, 596 Clowe D., et al., 2006, ApJ, 648, L109 de Blok, W.J.G., McGaugh, S.S., 1998, ApJ, 508, 132 Dodelson S., Liguori M., 2006, Phys. Rev. Lett, 97, 231301 Dupke R.A.,Mirabal N., Bregman J.N., Evrard A.E., 2007, (arXiv:0706.1073) Famaey B., Binney J., 2005, MNRAS, 363, 603 Famaey B., Bruneton J.P., Zhao H.S., 2007a, MNRAS, 377, Famaey B., Gentile G., Bruneton J-P., Zhao, H.S., 2007b, PhRvD, 75, 063002 Famaey B., Angus G.W., Gentile G., Zhao, H.S., 2007c, (arXiv:0706.1279) Farrar G.R., Rosen R.A., 2007, Phys. Rev. Lett, 98, 171302 Felten J.E., 1984, ApJ, 286, 3 c© 2007 RAS, MNRAS 000, 1–8 http://arxiv.org/abs/0706.4171 http://arxiv.org/abs/0706.1073 http://arxiv.org/abs/0706.1279 8 G. W. Angus and S. S. McGaugh Gerbal, D., Durret, F., Lachieze-Rey, M., Lima-Neto, G., 1992, A&A, 262, 395 Gentile G., Famaey B., Combes F., Kroupa P., Zhao, H.S., Tiret O., 2007, (arXiv:0706.1976) Hayashi E., White S., 2006, MNRAS, 370, L38 Jee M.J. et al., 2005a, ApJ, 618, 46 Jee M.J. et al., 2005b, ApJ, 634, 813 Jee M.J. et al., 2007, ApJ, 661, 728 Knebe A., Gibson B.K. 2004, MNRAS, 347, 1055 Mahdavi A., Hoeskstra H., Babul A., Balam D., Capak P., 2007, (arXiv:0706.3048) Markevitch M., et al., 2004, ApJ, 606,819 Markevitch M., Vikhlinin A., 2007, PhR, 443, 1 Milgrom M., 1983a, ApJ, 270, 365 Milgrom M., 1983b, ApJ, 270, 371 Milgrom M., 1983c, ApJ, 270, 384 Milgrom M., 1986, ApJ, 302, 617 Milgrom M., 1994, Ann. Phys. 229, 384 Milgrom M., 1995, ApJ, 455, 439 Milgrom M., 2007, (arXiv:0706.0875) Milgrom M., Sanders, R.H., 2003, ApJ, 599, L25 Milgrom M., Sanders, R.H., 2007, ApJ, 658, L17 Milosavljevic M., Koda J., Nagai D., Nakar E., Shapiro P.R., 2007, ApJ, 661, L131 McGaugh S.S., 1999, AIPC, 470, 72 McGaugh S.S., 2004, ApJ, 611, 26 McGaugh S.S., 2005, ApJ, 632, 859 McGaugh S.S., de Blok W.J.G., 1998, ApJ, 499 Nipoti C., Londrillo P., Ciotti L., 2007a, ApJ, 660, 256 Nipoti C., Londrillo P., Ciotti L., 2007b, MNRAS in press (arXiv:0705.4633) Nusser A., 2002, MNRAS, 331, 909 Pointecouteau E., Silk J., 2005, MNRAS, 364, 654 Sanders R.H., 1994, A&A, 284, L31 Sanders R.H., 1996, ApJ, 473, 117 Sanders R.H., 1998, MNRAS, 296, 1009 Sanders R.H., 1999, ApJ, 512, L23 Sanders R.H., 2001, ApJ, 560, 1 Sanders R.H., 2003, MNRAS, 342, 901 Sanders R.H., 2005, MNRAS, 363, 459 Sanders R.H., 2007, MNRAS in press, (astro-ph/0703590) Sanders R.H., McGaugh S.S., 2002, ARAA, 40, 263 Sanders R.H., Noordermeer E., 2007, MNRAS, 379, 702 Spergel D.N. et al., 2006, ApJS, 170, 377 Springel V., Farrar G.R., 2007, ApJ in press (astro-ph/0703232) Stachniewicz, S., Kutschera M., 2002, Acta physica Polonica B32, 362 Steinmetz M., Navarro J.F., 1999, ApJ, 513, 555 Takahashi R., Chiba T., 2007, (astro-ph/0701365) The, L.H., White, S.D.M., 1988, AJ, 95, 1642 Tiret O., Combes F., 2007, A&A, 464, 517 Wechsler R.H., Bullock J.S., Primack J.R., Kravtsov A.V., Dekel A., 2002, ApJ, 568, 52 Wu X., Zhao H.S., Famaey B. et al., 2007, 665, L101 Zlosnik T.G., Ferreira P.G., Starkman G.D., 2006, PhRvD, 74, 044037 Zlosnik T.G., Ferreira P.G., Starkman G.D., 2007, PhRvD, 75, 044017 c© 2007 RAS, MNRAS 000, 1–8 http://arxiv.org/abs/0706.1976 http://arxiv.org/abs/0706.3048 http://arxiv.org/abs/0706.0875 http://arxiv.org/abs/0705.4633 http://arxiv.org/abs/astro-ph/0703590 http://arxiv.org/abs/astro-ph/0703232 http://arxiv.org/abs/astro-ph/0701365 Introduction Modeling the freefall The collision in CDM The collision in MOND N-body collision Results Summary

0704.0382

On Some Subgroup Chains Related to Kneser's Theorem

On some subgroup chains related to Kneser’s theorem Yahya O. Hamidoune∗ Oriol Serra† Gilles Zémor‡ March 28, 2007 Abstract A recent result of Balandraud shows that for every subset S of an abelian group G there exists a non trivial subgroup H such that |TS| ≤ |T |+ |S| − 2 holds only if H ⊂ Stab(TS). Notice that Kneser’s Theorem only gives {0} 6= Stab(TS). This strong form of Kneser’s theorem follows from some nice properties of a certain poset investigated by Balandraud. We consider an analogous poset for nonabelian groups and, by using classical tools from Additive Number Theory, extend some of the above results. In particular we obtain short proofs of Balandraud’s results in the abelian case. 1 Introduction In order to avoid switching from multiplicative to additive notation, all groups will be written multiplicatively. Kneser’s addition theorem states that if S, T are finite subsets of an abelian group G then |ST | ≤ |S| + |T | − 2 holds only if ST is periodic (i.e, there is a non trivial subgroup H such that HST = ST .) Kneser’s Theorem is a fundamental tool in Additive number Theory. Proofs of this result may be found in [4, 5, 6, 7, 9]. In all previously known proofs of Kneser’s Theorem, the subgroup H depends crucially on both sets S and T . With the goal of breaking this double depen- dence in S and T , Balandraud investigated in recent work [1, 2] the properties of a combinatorial poset that we now present. Let S be a finite subset containing 1 of a group G. Following Balandraud, let us define a cell of S as a finite subset X such that, for all z /∈ X, it holds that zS 6⊂ XS. This notion is defined in [1, 2] and it is equivalent to the notion of Université Pierre et Marie Curie, Paris 6, Combinatoire et Optimisation - case 189, 4 place Jussieu, 75252 Paris Cedex 05. [email protected] Universitat Politècnica de Catalunya, Matemàtica Aplicada IV, Campus Nord - Edif. C3, C. Jordi Girona, 1-3, 08034 Barcelona, Spain. [email protected] Université de Bordeaux 1, Institut de Mathématiques de Bordeaux, 351 cours de la Libération, 33405 Talence. [email protected] http://arxiv.org/abs/0704.0382v1 nonextendible subset used in [3]. Throughout the paper, by a cell we always mean a cell of S. A cell X is called a u-cell if |XS| − |X| = u. A u-cell with minimal cardinality is called a u-kernel (of S). Balandraud showed that, for a finite set S in an abelian group G, in the poset of j–cells containing the unity ordered by inclusion with 1 ≤ j ≤ |S| − 2, the set of kernels form a chain of subgroups. Moreover, if there exists a u–cell, then there is a unique u–kernel containing the unit element which is contained in all u–cells containing the unit element. One of the consequences of this work is a new proof and the following strength- ening of Kneser’s Theorem: Theorem 1 (Balandraud) For any non-empty finite subset S of an abelian group G, there exists a finite subgroup H of G such that for any finite subset T of G one of the following conditions hold : • |TS| ≥ |T |+ |S| − 1 • HTS = TS and |TS| ≤ |HS|+ |HT | − |H| As far as the authors are aware this is a surprising and strong formulation that was not observed before and does not follow straightforwardly from the classical forms of Kneser’s Theorem. The purpose of the present note is to give a short proof for the nonabelian case that, in the poset of j–cells that are subgroups ordered by inclusion with 0 ≤ j ≤ |S| − 1, the set of kernels form a chain of subgroups. Moreover, each u-kernel of this poset is unique and contained in all u–cells of this poset. From this statement Kneser’s theorem allows one to deduce Balandraud’s re- sults for the abelian case, and in particular Theorem 1. Kneser’s Theorem has several equivalent forms. We use the following one; see e.g [4, 7]: Theorem 2 (Kneser [5]) Let G be an abelian group and X,Y ⊂ G be finite subsets such that |XY | ≤ |X| + |Y | − 2. Then |XY | = |HX|+ |HY | − |H|, where H = stab(XY ) = {x : xXY = XY }. Our main tool is the following Theorem of Olson[8, Theorem 2]. We give an equivalent formulation here where we use left–cosets instead of right–cosets. Theorem 3 (Olson [8]) Let X,Y be finite subsets of a group G, and let H and K be subgroups such that HX = X, KY = Y and KX 6= X, HY 6= Y . |X \ Y |+ |Y \X| ≥ |H|+ |K| − 2|H ∩K|. In particular either |X \ Y | ≥ |H| − |H ∩K| or |Y \X| ≥ |K| − |H ∩K|. We shall use the following lemma. Lemma 4 ([1, 2]) Let G be a group and 1 ∈ S ⊂ G be a finite subset. Then the intersection of two cells M1,M2 of S is a cell of S. Proof. Let x /∈ M1 ∩M2. There is i with x /∈ Mi. Then xS 6⊂ MiS. Hence xS 6⊂ (M1 ∩M2)S. We can now state our main result, namely Theorem 5 below. 2 An application of Olson’s Theorem Balandraud [1, 2] proved that, in the abelian case, the set of kernels containing the unit element and ordered by inclusion is a chain of subgroups. In the non abelian case we can prove only that the set of kernels that are subgroups forms a chain. The abelian case can then be easily recovered, since Kneser’s Theorem implies (as we shall see below) that a kernel containing the unit element is a subgroup. Theorem 5 Let S be a finite subset containing 1 of a group G. Let M be a u–kernel of S which is a subgroup. Let N be a subgroup which is a v–cell and suppose u, v ≤ |S| − 1. (i) If either N is a v–kernel or u = v then M ⊂ N or N ⊂ M . (ii) If N is a v–kernel and v ≤ u then M ⊂ N . Proof. Suppose that M 6⊂ N and N 6⊂ M . Note that, since M is a cell, if NMS = MS then NM = M , thus N ⊂ M against our assumption. Hence we may assume NMS 6= MS and similarly MNS 6= NS. By Theorem 3 we have one of the two following cases. Case 1: |MS| − |(MS) ∩ (NS)| = |(MS) \ (NS)| ≥ |M | − |M ∩N |. It follows that |(M ∩N)S| − |M ∩N | ≤ |(MS)∩ (NS)| − |M ∩N | ≤ |MS| − |M |. On the other hand we have u = |MS| − |M | < |S| ≤ |(M ∩N)S|. Since |MS| − |M | is a multiple of |M ∩N | we have u = |MS| − |M | = |(M ∩N)S| − |M ∩N |. By Lemma 4, M ∩N is a cell. Since M is a u–kernel, we have M ∩N = M, a contradiction. Case 2: |NS| − |(NS) ∩ (MS)| = |(NS) \ (MS)| ≥ |N | − |N ∩M |. It follows that |(N ∩ M)S| − |N ∩ M | ≤ |(NS) ∩ (MS)| − |N ∩ M | ≤ |NS| − |N |. On the other hand we have |NS| − |N | < |S| ≤ |(N ∩M)S|. Since |NS| − |N | is a multiple of |N ∩M | we have |NS| − |N | = |(N ∩M)S| − |N ∩M |. (1) Assume first u = v. Then u = |MS|−|M | = |NS|−|N | = |(N∩M)S|−|N∩M |. Since M is a u–kernel, we have M ∩N = M, a contradiction. Assume that N is a v–kernel. Then (1) implies N ∩M = N, a contradiction. This proves (i). Assume now that v ≤ u. Suppose M 6⊂ N . By (i) we have N ⊂ M , which implies in particular that |MS| − |M | is a multiple of N . Therefore, from u = |MS| − |M | < |S| ≤ |NS| we have u = |MS| − |M | ≤ |NS| − |N | = v which gives u = v. But then M 6⊂ N and N ⊂ M imply |N | < |M |, and since N is now a u–cell, this contradicts M being a u–kernel. We can now deduce Balandraud’s description for kernels and cells : Corollary 6 (Balandraud [1, 2]) Let G be an abelian group and S ⊂ G be a finite subset with 1 ∈ S. Let M be a u–kernel of S containing 1 with 1 ≤ u ≤ |S| − 2. Then, (i) M is a subgroup. (ii) Each u-cell is M–periodic. (iii) Each v–kernel with u < v ≤ |S| − 2 is a proper subgroup of M . Proof. Let X be a u-cell with u ≤ |S| − 2. By Kneser’s Theorem, the inequality |XS| − |X| = u ≤ |S| − 2 implies u = |XS| − |HX| = |HS| − |H|, (2) where H is the stabilizer of XS. Since X is a cell and HXS = XS, we have X = HX. Note that, since G is abelian, ({y} ∪ H)S = HS implies y ∈ Stab(HS) ⊂ Stab(XS), so that y ∈ H. This observation and (2) imply that H is an u–cell. In particular, by taking X = M , the period K of MS is a u–cell. Since KMS = MS and M is a u–cell, we have K ⊂ KM ⊂ M . Since M is a u–kernel we have M = K. This proves (i). Now let H be the stabilizer of XS, where X is a u–cell. As shown in the preceding paragraph H is also a u–cell. By Theorem 5 we have M ⊂ H and thus MH = H. Since X is a cell and HXS = XS, we have X = HX = MHX. Hence X ⊂ MX ⊂ MHX = X implies X = MX. This proves (ii). Finally, by (i), a v–kernel N is a subgroup. By Theorem 5 we have N ⊂ M . From Corollary 6, one can deduce Theorem 1. Proof of Theorem 1: We may assume without loss of generality that 1 ∈ S. Case 1: There is no m–cell for any 1 ≤ m ≤ |S| − 2. • either we have |TS| ≥ |S|+ |T | − 1 for any non-empty finite T , in which case the theorem clearly holds with H = {1}. • or there exists some non-empty finite T such that |TS| ≤ |S| + |T | − 2. Without loss of generality, we may also suppose 1 ∈ T . Now T must be contained in an m–cell with m ≤ |S| − 2, but since no such cell exists for 1 ≤ m, we have that T itself must be a cell (a 0-cell) i.e. |TS| = |T |. We therefore have HT = TH = T = TS = HTS where H is the (necessarily finite) subgroup generated by S. We have just proved that the theorem holds in this case with H = 〈S〉. Case 2: There exists an m–cell with 1 ≤ m ≤ |S| − 2. We may therefore consider the largest integer u ≤ |S| − 2 for which S admits a u–cell. Let H be the u–kernel containing 1. Note that u ≤ |S|−2 implies that H is different from {1}. Now let T be any finite non-empty subset such that |TS| − |T | ≤ |S| − 2. We shall prove that HTS = TS. By adding elements to T as long as necessary, we can find a cell X that contains T and such that XS = TS. Note that we then have |XS| − |X| ≤ |TS| − |T | ≤ |S| − 2, so that X is a v–cell for some v ≤ u. By Corollary 6 (ii) we have TS = XS = MXS = MTS where M is the v-kernel containing 1. By part (i) of Corollary 6, H is a subgroup of M so that TS = XS = HTS as well. Finally, |ST | ≤ |HS|+ |HT | − |H| follows from |ST | being a multiple of |H|. References [1] E. Balandraud, Une variante de la méthode isopérimetrique de Hamidoune, appliquée au theoreme de Kneser, Preprint, december 2005. [2] E. Balandraud, Quelques résultats combinatoires en Théorie Additive des Nombres, Thèse de Doctorat de l’Université de Bordeaux I, May 2006. [3] D. Grynkiewicz, A step beyond Kempermann structure Theorem, Preprint May 2006. [4] J. H. B. Kemperman, On small sumsets in Abelian groups, Acta Math. 103 (1960), 66–88. [5] M. Kneser, Summenmengen in lokalkompakten abelesche Gruppen, Math. Zeit. 66 (1956), 88–110. [6] H.B. Mann, Addition Theorems, R.E. Krieger, New York, 1976. [7] M. B. Nathanson, Additive Number Theory. Inverse problems and the ge- ometry of sumsets, Grad. Texts in Math. 165, Springer, 1996. [8] J.E. Olson, On the symmetric difference of two sets in a group. European J. Combin. 7 (1986), no. 1, 43–54. [9] T. Tao and V.H. Vu, Additive Combinatorics, Cambridge Studies in Ad- vanced Mathematics 105 (2006), Cambridge University Press. Introduction An application of Olson's Theorem

0704.0383

The Exact Boundary Condition to Solve the Schrodinger Equation of Many Electron System

Microsoft Word - arxiv_prasadtext.doc ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 1 of 21) The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System Rajendra Prasad Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] In an attempt to bypass the sign problem in quantum Monte Carlo simulation of electronic systems within the framework of fixed node approach, we derive the exclusion principle “Two electrons can’t be at the same external isopotential surface simultaneously” using the first postulate of quantum mechanics. We propose the exact Coulomb-Exchange nodal surface i.e. the exact boundary condition to solve the non- relativistic Schrödinger equation for the non-degenerate ground state of atoms and molecules. This boundary condition was applied to compute the ground state energies of N, Ne, Li2, Be2, B2, C2, N2, O2, F2, and H2O systems using diffusion Monte Carlo method. The ground state energies thus obtained agree well with the exact estimate of non-relativistic energies. INTRODUCTION An ideal target of a quantum chemist/physicist is to solve the non-relativistic Schrödinger equation exactly as it describes much of the world of chemistry. If we can solve this equation at a realistic cost, we can make very precise predictions. At present, only the full-CI method is available for obtaining the exact wave function within a given basis set, but this method is too demanding computationally and therefore not affordable even for a small system. In recent years increasing attention has been drawn to the random walk approach called diffusion Monte Carlo (DMC) method1 2 3 4 for solving Schrödinger equation. The ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 2 of 21) attractiveness of DMC method lies in that it can treat many body problems exactly. The DMC method is a projection method based on the combination of the imaginary time Schrödinger equation, generalized stochastic diffusion process, and Monte Carlo integration. The solution, it yields has only statistical error, which can be properly estimated and in principle, made as small as desired. Since in the DMC method the wave function has to be a population density, therefore, the DMC method can only describe the constant sign solution of the Schrödinger equation. This poses a serious problem if one is interested in the solution of a many electron system where the wave function is known to be antisymmetric (i.e. both positive and negative) with respect to interchange of two electrons. This situation is known as fermion sign problem in the quantum Monte Carlo literature1-4. The solution of this problem is one of the most outstanding in all of the computational physics/chemistry. This problem is often (mis)understood as a technical detail that defeating the numerical simulators. To the best of our knowledge no methodology is available to handle this problem in a systematic and controlled fashion. However, we think that it is essentially a problem of exact boundary, which is not known for many electron systems for obtaining well-behaved solutions of non-relativistic Schrödinger equation. It is our understanding that the boundary must be derived from the link between the formal mathematics and the physics of the real world. In this article, we will derive the boundary condition for atomic and molecular systems to obtain well-behaved solutions (i.e. bound state solution is single valued, continuous, quadratically integrable, and differentiable) of non-relativistic electronic Schrödinger equation. To start with, we are dealing with situations in which the ground state is non-degenerate only. ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 3 of 21) THE EXACT BOUNDARY CONDITION We have the time independent Schrödinger equation: ˆ Ψ=Ψ EH …………..(1) where Ĥ is the time independent non-relativistic electronic Hamiltonian operator in the Born-Oppenheimer approximation, E0 is the eigenvalue of the full many-electron ground state 0Ψ . The Ĥ is defined in atomic units as follows: +−∇−= electrons electrons electrons 2 1)( 1ˆ r ………..(2) where the external potential, ∑= Nuclei rV )( , ………….(3) 2∇ is Laplacian, ZI denotes nuclear charge, and rIi and rij symbolize the electron-nucleus and electron-electron distance, respectively. Following Hohenberg-Kohn theorem I, a proof only of existence5, the electron density )(0 r ρ in the ground state 0Ψ is a functional of )(rV , i.e. )(0 r ρ = )]([0 rV ρ . ………………………. (4) Further, the full many electron ground state 0Ψ is unique functional of )(0 r ρ , i.e. 0Ψ = )]([ 00 r ρΨ . ……………………(5) Evidently we can say that 0Ψ is a functional of )(rV i.e. 0Ψ = )]([0 rV Ψ . …………………. (6) We have a choice to express the exact density: ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 4 of 21) )]([0 rV ρ = ∑ 2 )]([ φ ………………..(7) Where N denotes number of electrons. The functionals, { }NirVi .....1)],([ = φ are exact ortho-normal one electron functions of the function )(rV , which give exact )]([0 rV (Caution to reader!! At the moment, here is nothing to do with so-called s, p, d, f, ..etc. type orbitals . The functionals { }NirVi .....1)],([ = φ are entirely different from those orbitals obtained from Kohn-Sham6 or similar formalisms.) Now we can write the exact N electron ground state wave function as a functional of N exact one electron functionals { }NirVi .....1)],([ = 0Ψ = )]]([)],......,([)],([[ 10 NN rVrVrV φφφΨ ……(8) or 0Ψ = )]]([)],......,([)],([[ 22110 NN rVrVrV φφφΨ ……..(9) Since each one electron functional in { }NirVi .....1)],([ = φ is a function of external potential )(rV , we can also write the exact N electron ground state wave function in functional form as follows: 0Ψ = )](),......,(),([ 210 NrVrVrV Ψ …………………...(10) Thus the exact N electron non-degenerate ground state wave function is a unique functional of external potential experienced by each electron i.e. functional of )(),......,(),( 21 NrVrVrV So far, it is not clear: • Whether the wave function is symmetric or antisymmetric with respect to interchange of any two electrons. • What are the analytical forms of { }NirVi .....1)],([ = • What is the analytical form of the exact wave function? ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 5 of 21) • How to get the exact wave function from the exact density. However, we get some idea about the topology of a well behaved ground state non- degenerate wave function and distribution functions in a given external potential )(rV In particular: “The probability of n-electrons (where n = 2..N) being found simultaneously on the isopotential surface of an external potential )(rV is same irrespective of positions of the electrons on the surface.” Now we proceed to decide the nature (symmetric or antisymmetric with respect to interchange of any two electrons) of a well behaved many electron wave function. Defining the local energy, EL: ∑∑ ∑ ∑ electrons electrons electrons Nuclei 11 10 ………….(11) The terms IiI rZ and ijr1 in the equation (11) will blow up if 0→Iir and 0→ijr unless so-called cusp conditions are obeyed by 0Ψ . The 0Ψ is exact and obeys electron nucleus (e-N) and electron-electron (e-e) cusp conditions. The wave function for a system of N identical particles must be symmetric or antisymmetric with regard to interchange of any two of the identical particles, i and j. Since the N particles are all identical, we could not have the wave function symmetric with regard to some interchanges and antisymmetric with regard to other interchanges. Thus the wave function of N identical particles must be either symmetric or antisymmetric with regard to every possible interchange of any two particles. ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 6 of 21) Let us assume 0Ψ is symmetric with regard to interchange of electrons i and j. There is a cusp in 0Ψ at rij = 0. This implies that 0Ψ is not differentiable at rij = 0. Therefore, 0Ψ , symmetric with respect to interchange of any two electrons is not a well- behaved solution. To make 0Ψ a well-behaved wave function, 0Ψ must be zero when rij = 0 and also it must change sign with respect to the interchange of two electrons, i.e. if ji rr = then 0Ψ = 0. This condition is universal and independent of kind of external potential. However, we are interested in a well-behaved solution of a bound state in a given external potential )(rV . From the previous argument, we know that the simultaneous probability of finding two electrons is same everywhere at the isopotential surface. Therefore, if )( irV - )( jrV = 0 then 0Ψ = 0. Extending to N electron system, we have If ( ) 0)()( =−= Π rVrVf then 0Ψ = 0. We can also express f as Vandermonde determinant: )()(....)()( )()(....)()( )()(....)()( 11....11 rVrVrVrV rVrVrVrV rVrVrVrV rVrVf =−= Π …(12) Consequently we have exclusion principle in the following form: “Two electrons can’t be at the same external isopotential surface simultaneously.” We see that if we are interested in a well behaved solution of the time independent Schrödinger equation, the boundary condition (12) (i.e. antisymmetric wave function) is obtained naturally due to singularity in the e-e interaction potential, which respects Pauli’s exclusion principle. If electrons i and j are of opposite spin then we say ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 7 of 21) that )( irV - )( jrV = 0 represents Coulomb (nodal) surface. If electrons i and j are of same spin then )( irV - )( jrV = 0 represents Coulomb-Exchange nodal surface. All together, the ( ) 0)()( =−= Π rVrVf represents the Coulomb-Exchange nodal surface of N electron system. Hereafter, we will call f as ExchangeCoulombf − nodal surface. However, the solution obtained for the Hamiltonian (2) within the boundary ExchangeCoulombf − =0 does not tell us about the spin multiplicity of the N electron system. Further, we can rewrite the functional f in terms of Hermite polynomials, )]([ rVH k )]([)]([....)]([)]([ )]([)]([....)]([)]([ )]([)]([....)]([)]([ )]([)]([....)]([)]([ 1112111 2122212 1112111 0102010 NNNNNN rVHrVHrVHrVH rVHrVHrVHrVH rVHrVHrVHrVH rVHrVHrVHrVH −−−−− = …..(13) In particular, if we multiply an optimizable one electron functional )]([ rVψ to the equation (13) and we obtain an N electron wave function: )]([)]([)]......([)]([)]([ 21 rVfrVrVrVNormrV N rrrrr ψψψ=Ψ ……………………………(14) The one-electron density functional corresponds to the wave function (14): ′′=′ ∑ 1111 )]([)]([)]([)]([)]();([ kkk rVHrVHArVrVrVrV rrrrrr ψψρ ……………….(15) where Ak is normalization constant of )]([)]([ rVHrV k The two-electron density functional in terms of one-electron density functional: )]();([)]();([)]();([)]();([ )](),();(),([ 21122211 rVrVrVrVrVrVrVrV rVrVrVrV ′′−′′= rrrrrrrr ……(16) ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 8 of 21) Here it appears that we can obtain exact ground state energy by optimizing only a one- electron functional )]([ rVψ in the equation (14). A very interesting and new physics obtained from the equation (13) is that each row in the determinant represents different level (k) of Kamalpur breathing (anharmonic quantum breathing) of electron cloud in a given external potential )(rV and each level, k is occupied by one electron (the elementary particle). BYPASSING THE SIGN PROBLEM We can bypass the fermion sign problem in the electronic structure diffusion Monte Carlo (DMC) method using fixed node approach. Here one assumes a prior knowledge of the nodal surface i.e. 0(R) = 0. Due to tiling property 7 of the exact ground state wave function, the Schrödinger equation is solved in the volume embraced by the nodal surface, where the wave function has a constant sign and in this way the fermion sign problem is bypassed. The exact knowledge of Coulomb Exchange nodal surface allows us for an exact stochastic solution of the Schrödinger equation. The restriction in the random walk RR ′→ during the electronic structure diffusion Monte Carlo simulation is as follows: reject accept RfRf ExchangeCoulombExchangeCoulomb )()( ………….(17) We have applied the boundary condition (17) for the ground state electronic structure diffusion Monte Carlo simulation of N, Ne, Li2, Be2, B2, C2, N2, O2, F2, and H2O systems. Monte Carlo calculations can be carried out using sets of random points picked from any arbitrary probability distribution. The choice of distribution obviously makes a ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 9 of 21) difference to the efficiency of the method. If the Monte Carlo calculations are carried out using uniform probability distributions, very poor estimates of high-dimensional integrals are obtained, which is not a useful method of approximation. These problems are handled by introducing the importance sampling approach8 9. In this approach the sampling points are chosen from a trial distribution, which concentrates on points where the trial function, ΦT(R) is large. In the present DMC calculations, we have chosen the trial function, ΦT in the form: ΦT = Φ.F , ….(18) where Φ denotes the Hartree Fock (HF) or multi configuration self consistent field (MCSCF) wave function and F is a correlation function that depends on inter-particle distances. The HF and MCSCF wave functions were obtained using the GAMESS package10 and employed Dunning’s cc-VTZ atomic basis set 11. In order to satisfy the electron nucleus (e-N) cusp condition, all Gaussian type s basis functions were replaced with eight Slater-type s basis functions. The exponents of Slater-type s functions were taken from Koga et al. 12 and satisfy the e-N cusp condition. We have chosen the Schmidt, Moskowitz, Boys, and Handy (SMBH) correlation function FSMBH 13. For the SMBH correlation function, Eqn. (19), we have included terms up to 2nd order, where order, s is defined as s = l + m + n. ( )  +−= ∑∑ ∑ µµµµµ atoms electrons iAASMBH rrrrrcF exp ………….(19) where ………….(20) ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 10 of 21) and r denotes inter-particle distance. Six non-redundant parameters out of the total ten were optimized keeping b = 1.0 as follows: 1) First we obtained optimal parameters by minimizing the energy and variance at the variational Monte Carlo (VMC) level. 2) Using this VMC optimal trial function, the trial function fixed node DMC calculation was carried out and walkers were collected after each 2000 steps. Further, correlation parameters were reoptimized to minimize the variance with ~100,000 walkers. Here reference energy was set to the trial function fixed node DMC energy. These optimized trial functions were used for importance sampling in the DMC simulation and a random walk RR ′→ was accepted if 0)()( >′−− RfRf ExchangeCoulombExchangeCoulomb . The DMC calculations were performed using the open source quantum Monte Carlo program, ZORI14. Around 10,000 walkers were used for the systems studied. The Umrigar et al.15 algorithm was used for DMC walks and Caffarel Assaraf et al.16 algorithm for population control. We have allowed only one electron walk at a time. The DMC calculations were done at several time steps. We report only those energies extrapolated to zero time step. We present the ground state DMC energies of N, Ne, Li2, Be2, B2, C2, N2, O2, F2, and H2O systems in Table I. The DMC energies obtained using our newly derived boundary ExchangeCoulombf − = 0 are far better than the trial function fixed node DMC energies17 and compare well with the experimental counterpart. However, present simulations were noisy and unpleasant compared to conventional trial function fixed node DMC simulations. It is worth noting that we have obtained DMC energy even lower than the exact value at smaller time steps for the atoms of relatively larger atomic ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 11 of 21) radius perhaps due to failure of the distributions to reach the steady state or equilibrium distributions in a finite number of steps. This problem can be handled in the Green’s function quantum Monte Carlo (GFQMC) method as it takes the advantage of the properties of Green’s functions in eliminating time-step entirely in treating the steady state equation. The GFQMC is well suited if boundaries are exactly known18. If the trial function boundary and the ExchangeCoulombf − = 0 does not coincide and also non-zero values of trial function are very much different from the exact solution, which could lead to large statistical fluctuations from poor sampling and possibly to an effective non-ergodic diffusion process due to the finite projection time in practical calculations. Therefore, we are looking for an alternative well behaved trial function whose boundary coincides with those of ExchangeCoulombf − . CONCLUSION This article presents a progress of the author's research in order to get exact solution of non-relativistic Schrödinger equation of many electron systems. A conclusion of this on going research is that we have derived the exclusion principle “Two electrons can’t be at the same external isopotential surface simultaneously” using the first postulate of quantum mechanics. We propose the exact Coulomb-Exchange nodal surface i.e. the exact boundary to solve the non-relativistic Schrödinger equation for non-degenerate ground state of atoms and molecules. Using this newly derived boundary condition, one can bypass the fermion sign problem in the electronic structure Quantum Monte Carlo simulation and hence the exact ground state energy as well as the exact electron density. ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 12 of 21) REFERENCES 1. Anderson, J. B. A random-walk simulation of the Schrödinger equation: +3H . J. Chem. Phys. 63, 1499-1503 (1975) 2. Ceperley, D. M. & Mitas, L. in New Methods in Quantum Mechanics, I. Prigogine, S. A. Rice, Eds.,(John Wiley and Sons, New York, 1996), Vol. 93. 3. Hammond, B. L., Lester, Jr, W. A. & Reynolds, P. J. Monte Carlo Methods in Ab Initio Quantum Chemistry; World Scientific: Singapore, 1994. 4. Ceperley, D. & Alder, B. Quantum Monte Carlo. Science 231, 555-560 (1986) 5. Hohenberg, P. & Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. 136, B864- B871 (1964) 6. Kohn, W. & Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev. 140, A1133-A1138 (1965) 7. Cepereley, D. M. Fermion nodes. J. Stat. Phys. 63, 1237-1267 (1991) 8. Metropolis, N. A., Rosenbluth, W., Rosenbluth, M. N., Teller, A. H. & Teller, E. Equation of State Calculations by Fast Computing Machines. J. Chem. Phys. 21, 1087- 1092 (1953) 9. Grimm, R. C. & Storer, R. G. Monte-Carlo solution of Schrödinger's equation. J. Comput. Phys. 7, 134-156 (1971) 10. Schmidt, M. W. et al. General atomic and molecular electronic structure system. J Comput Chem 14, 1347-1363 (1993) 11. Dunning, Jr., T. H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 90, 1007-1023 (1989) ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 13 of 21) 12. Koga, T., Kanayama, K., Watanabe, S. & Thakkar, A. J. Analytical Hartree-Fock wave functions subject to cusp and asymptotic constraints: He to Xe, Li+ to Cs+, H- to I-. Int. J. Quantum Chem. 71, 491-497 (1999) 13. Schmidt, K. E. & Moskowitz, J. W. Correlated Monte Carlo wave functions for the atoms He through Ne. J. Chem. Phys. 93, 4172-4178 (1990) 14. Aspuru-Guzik , A. et al. Zori 1.0: A parallel quantum Monte Carlo electronic structure package. J. Comp. Chem. 26, 856-862 (2005) 15. Umrigar, C. J., Nightingale, M.P. & Runge, K.J. A diffusion Monte Carlo algorithm with very small time-step errors. J. Chem. Phys. 99,2865-2890 (1993) 16. Assaraf, R., Caffarel, M. & Khelif, A. Diffusion Monte Carlo methods with a fixed number of walkers. Phys. Rev. E. 61, 4566-4575 (2000) 17. Filippi, C. & Umrigar, C. J. Multiconfiguration wave functions for quantum Monte Carlo calculations of first-row diatomic molecules. J. Chem. Phys. 105, 213-226 (1996) 18. Kalos, M. H., Monte Carlo Calculations of the Ground State of Three- and Four- Body Nuclei. Phys. Rev. 128, 1791-1795 (1962) ACKNOWLEDGEMENTS The QMC calculations were carried out at the Lawrence Berkeley National Laboratory, Berkeley. The author gratefully acknowledges Professor W. A. Lester for his support during the stay at Berkeley. The author is indebted to Professor P. Chandra of Banaras Hindu University, Varanasi for his interest and helpful discussion during the preparation of the manuscript. Professor S. K. Sengupta of Banaras Hindu University, Varanasi is acknowledged for careful reading of the manuscript. ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 14 of 21) TABLE I. The total ground state energies obtained from fixed-node DMC calculation. Atom / Molecule Bond length CSF,D ETFN-DMC (Ref. 17) ECEN-DMC (Extrapolated to τ=0) E0 -, 111 -54.5753(3) -54.5841(5) -54.5902(11) -54.5892 Ne 1,1 -128.9216(15) -128.938(1) -128.9375 Li2 5.051 -14.9911(1) -14.9938(1) -14.9955(5) -14.9954 Be2 4.63 5,16 -29.3176(4) -29.3301(2) -29.3378(15) -29.33854(5) B2 3.005 6,11 -49.3778(8) -49.3979(6) -49.41655(45) -49.415(2) C2 2.3481 4,16 77, 314 -75.8613(8) -75.8901(7) -75.9035(9) -75.9229(19) -75.923(5) 2.068 4,17 -, 552 -109.487(1) -109.505(1) -109.520(3) -109.5424(15) -109.5423 O2 2.282 -150.268(1) -150.277(1) -150.3274(15) -150.3268 F2 2.68 -199.478(2) -199.487(1) -199.5289(25) -199.5299 -199.52891(4) H2O -, 300 -76.4175(4) -76.429(1) -76.4376(11) -76.438(3) -76.4376 Bond lengths and energies are in atomic units. In the third column, we list the number of configuration state functions (CSFs) and number of determinants (D) in the trial function (ΦT). ETFN-DMC denotes the DMC energy with ΦT =0. ECEN-DMC denotes the DMC energy with ExchangeCoulombf − = 0. E0 denotes the exact, non-relativistic, infinite nuclear mass energy. The numbers shown in bracket are error bar. ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 15 of 21) Supplementary note for reviewers: 1. The proposed theory is to deal only real interacting many electron systems. To start with only non-degenerate ground state of many electron systems are considered. Author is neither interested nor intended to deal any kind of non- interacting systems such as free fermion, free electron gases, or free particles because author think that none of the real system belong to either of these classes. Author has chosen to construct the boundary condition from the link between the formal mathematics and the physics of many electron systems. 2. Difference between spatial nodes and Coulomb-Exchange nodal surface: I hope that people can distinguish spatial nodes and Coulomb Exchange nodes and the physics behind the different kind of nodes. Whatever I have discussed in this paper is only about Coulomb-Exchange nodal surfaces. There is no analogy with a particle in a box node and Coulomb-Exchange nodal surfaces. For example: The function f(r1,r2)=(r1-1)(r2-1)(r1-r2)exp(-2 r1-2 r2) is antisymmetric with respect to interchange of two electrons. However, the node (r1-1)(r2-1) is symmetric with respect to interchange of two electrons and fixed and this node can be compared with nodes of particle in a box. The Coulomb-Exchange node (r1-r2) is antisymmetric with respect to interchange of two electrons and responsible for removal of singularity in e-e interaction potential. The Coulomb- Exchange nodal surfaces only occur in a system of more than one electron systems. Author understands that the Coulomb-Exchange nodal surfaces are directly responsible for the existence of real many electron systems. 3. A consequence of proposed solution of the sign problem is that the ground state of Helium atom in the non-relativistic limit has a nodal surface. However, it is understood that He ground state wave function is symmetric and has no such node. ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 16 of 21) A consequence of proposed solution for the sign problem is that the ground state of the Helium atom in the non-relativistic limit has Coulomb-Exchange nodal surface, r1-r2=0. An understanding that He atom ground state wave function is symmetric and has no such node is an illusion only. This illusion arises due to a practice that the QMC people using phi(1)*phi(2)*Jastrow trial function, where phi(r) is 1s orbital. The trial function is symmetric with respect to exchange of two electrons. The trial function also satisfies electron nucleus cusp condition. We also expect that the final solution will satisfy e-e cusp condition. Since the trial function is symmetric, people got accurate energy and assumed that the final solution is also symmetric and it does not have any node also wave function is non-zero at the point of coincidence of two electrons. Where is Coulomb hole? However, it can be proven that a symmetric solution is not acceptable. Proofs are as follows: “Proof for the existence of Coulomb-Exchange node in He ground state exact wave function” A.) Let assume ),( 21 rrsym Ψ is an exact symmetric wave function. i.e. ),( 21 rrsym Ψ = ),( 12 rrsym Since ),( 21 rrsym Ψ is exact, it must satisfy the cusp condition at 21 rr = . Clearly there is a cusp at 21 rr Since there is a cusp at 21 rr = in ),( 21 rrsym Ψ , the second derivative 2 ),( xrrsym ∂Ψ∂ is not defined at 21 rr Therefore ),( 21 rrsym Ψ is not a well behaved solution and hence it is not an acceptable wave function. Only option left is antisymmetric solution. B.) Another proof: ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 17 of 21) ),( 21 rr Ψ = ∑∑ νµµν φφ )()( 21 rrc ………...(S-1) Where { })(rrµφ forms a real infinite one-electron basis. Since ),( 21 rr Ψ is expanded over infinite basis set and hence it is exact. This implies that ),( 21 rr Ψ satisfies the cusp condition at 21 rr ),( 21 Ψ∇ =∑∑ νµµν φφ )()( 21 1 rrc . ………...(S-2) The second derivative ),( 21 Ψ∇ is continuous at 21 rr = because each term in the expansion is continuous (the rules of continuity for algebraic combinations). This implies that there is no cusp in ),( 21 rr Ψ at 21 rr BUT ),( 21 rr Ψ has to satisfy the cusp condition at 21 rr This is only possible if ),( 21 rr Ψ changes the sign at 21 rr And hence the exact ),( 21 rr Ψ has exchange node irrespective of its spin multiplicity. C.) More illustrative example: Hamiltonian for He atom: H +−−∇−∇−= ………...(S-3) and )2,1()2,1( Ψ=Ψ EH ………...(S-4) Let expand νµµν φφ )2()1()2,1( c ………...(S-5) Where { })(rµφ is complete set of eigen functions of the Hamiltonian 1ˆ 2 −∇−= with eigenvalue equation )()(ˆ rrh µµµ φεφ = . Rewriting the He Hamiltonian: ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 18 of 21) H +−−∇−∇−= = hh ++ ………...(S-6) )2,1( )2()1()ˆˆ( )2,1( )2,1() )2,1( )2,1(ˆ EL +Ψ νµµν φφ ………...(S-7) Since )(rµφ is an eigen function of ĥ . We can write )2,1( )2()1()( EL +Ψ νµνµµν φφεε ………...(S-8) µννµ εε dE ++= µννµ εε dE −=+ ………...(S-9) )2,1( )2()1()( EL +Ψ νµµνµν φφ ………...(S-10) )2,1( )2()1( )2,1( )2()1( EL +Ψ ∞ ∞∞ ∞ νµµνµν νµµν φφφφ ………...(S-11) )2,1( )2()1( )2,1( )2,1( EL +Ψ νµµνµν φφ ………...(S-12) )2,1( )2()1( EEL +Ψ νµµνµν φφ ………...(S-13) If )2,1(Ψ is exact then EEL = ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 19 of 21) and )2,1( )2()1( νµµνµν φφ ………...(S-14) Now let assume that the exact solution is symmetric with respect to interchange of two electrons. )2,1(Ψ and )(rµφ are well behaved and differentiable. From the rules of continuity for algebraic combinations, the term in equation (S-14), )2,1( )2()1( νµµνµν φφdc is continuous and finite and it should not diverge when 012 →r . Therefore, symmetric solution is not acceptable. However, if )2,1(Ψ = 0 at r12=0 then )2,1( )2()1( νµµνµν φφdc will also diverge and can compensate the divergence in 1/r12 term. Thus the only acceptable solution is antisymmetric (with respect to interchange of two electrons) solution. D.) Another example: Almost all QMC people believe (their believe is based on some assumptions and approximations) that He atom ground state wave function is symmetric. This is an illusion. This can be understood as follows: Let us take trial functions of two electron system: ( )221 1 xxb xxg 21 211 ),( −−= ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 20 of 21) ( ) eeexx Uxxxxg 21 22212 221),( ( ) eeexx Uxxxxg 21 22213 21),( − −−= If someone claims that He ground state is symmetric, what kind of exact symmetric wave functions they are getting finally? The functions g1and g2 are symmetric with respect to interchange of two electrons. The functions like g1, g2, and g3 can satisfy cusp condition. The functions g1 and g2 are not differentiable at x1=x2 and therefore these are not acceptable. The antisymmetric function g3 are differentiable at x1=x2. However, people have got very accurate ground state energy for He atom using HF*Jastrow trial function and they concluded that He atom has no node without examining the simultaneous probability of finding two electrons at exactly same place. I think, they got good results due to inherent beauty of DMC technique. The functions g2 is symmetric and g3 is antisymmetric with respect to interchange of two electrons. However, g2*g2 and g3*g3 give the exactly same probability distribution i.e. same physics. Functions g2 and g3 vanish when x1=x2. Further, the VMC calculation for g2 and g3 will give exactly same energy. Can anyone predict that the VMC energies obtained from g2 and g3 represent singlet or triplet state? I am sure it is not possible. An antisymmetric wave function can satisfy cusp condition as well as it’s derivative will be continuous simultaneously at the point of coincidence. Here symmetric and antisymmetric wave functions serve same distribution. Why I should not prefer antisymmetric wave function for which a boundary condition can be imposed easily? 4. Further, if I assume the argument “He ground state wave function is symmetric and has no such node” is correct. The end will be a nonsense, which is as follows: ____________________________________________________________________________________________________________ The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: [email protected] 4/3/2007 (Page 21 of 21) It is very much common practice in QMC calculation to take Hartree-Fock trial function as a product of alpha-beta determinants. For example N atom: PSIT(1,2,3,4,5,6,7) = Det(1,2,3,4,5,6,7). PSIT(1,2,3,4,5,6,7) = Detα (1,2,3,4,5)*Detβ(6,7). PSIT(1,2,3,4,5,6,7) = Detα(1,2,3,4,5)*Detβ(6,7). Detα(1,2,3,4,5)*Detβ(6,7) ≠ Detα(1,2,3,4,7)*Detβ(6,5) The trial function PSIT is clearly neither symmetric nor antisymmetric with respect to interchange of alpha-beta electrons. It is not clear to me what kind of final solution (i.e. symmetric or antisymmetric) we will obtain with this trial function fixed node DMC? The fact that we can not write PSIexact = Phiα*Phiβ. 5. It is natural to ask what is node of 3S He atom and why people are getting very accurate energy with exchange node r1-r2=0? At the moment, I can only say that this is due to artifact of importance sampling because people have used HF*Jastrow trial function. The correlation energy for He(3S) atom is around 2mH and the overlap of HF trial function with exact wave function can be anticipated to be more than 99%. Perhaps due to technical reasons final DMC solution may have converged to He(3S). I have seen some recent papers on the node of He(3S) system. It is widely claimed that the node r1-r2=0 belongs to He(3S) system and it is exact. I differ with their argument and I proved that the exchange node r1-r2=0 belongs to He ground state. I do not know if anyone have performed DMC calculation with a trial function like psit(r1,r2)=(r1- r2)*exp(-2*r1)*exp(-2*r2) and reported the energy for He( 3S). Anyway, at present I am interested only in the non-degenerate ground state of atoms and molecules. Author welcomes further comments, questions, and suggestions if any. [email protected]

0704.0384

Clustering features of $^9$Be, $^{14}$N, $^7$Be, and $^8$B nuclei in relativistic fragmentation

Clustering features of 9Be, 14N, 7Be, and 8B nuclei in relativistic fragmentation D. A. Artemenkov,∗ T. V. Shchedrina, R. Stanoeva, and P. I. Zarubin† Joint Insitute for Nuclear Research, Dubna, Russia (Dated: November 10, 2021) Abstract Recent studies of clustering in light nuclei with an initial energy above 1 A GeV in nuclear treack emulsion are overviewed. The results of investigations of the relativistic 9Be nuclei fragmentation in emulsion, which entails the production of He fragments, are presented. It is shown that most precise angular measurements provided by this technique play a crucial role in the restoration of the excitation spectrum of the α particle sysytem. In peripheral interactions 9Be nuclei are dissociated practically totally through the 0+ and 2+ states of the 8Be nucleus. The results of investigations of the dissociation of a 14N nucleus of momentum 2.86 A GeV/c in emulsion are presented as example of more complicated system. The momentum and correlation characteristics of α particles for the 14N→3α +X channel in the laboratory system and the rest systems of 3α particles were considered in detail. Topology of charged fragments produced in peripheral relativistic dissociation of radioactive 8B, 7Be nuclei in emulsion is studied. PACS numbers: 21.45.+v, 23.60+e, 25.10.+s ∗Electronic address: [email protected] †Electronic address: [email protected]; URL: http://becquerel.lhe.jinr.ru http://arxiv.org/abs/0704.0384v1 mailto:[email protected] mailto:[email protected] http://becquerel.lhe.jinr.ru I. INTRODUCTION The peripheral fragmentation of light relativistic nuclei can serve as a source of infor- mation about their exitations above particle decay thresholds including many-body final states. The interactions of this type are provoked either in electromagnetic and diffraction processes, or in nucleon collisions at small overlapping of the colliding nucleus densities. A fragmenting nucleus gains an excitation spectrum near the cluster dissociation thresholds. In the kinetic region of fragmentation of a relativistic nucleus there are produced nuclear fragment systems the total charge of it is close to the parent-nucleus charge. A relative intensity of formation of fragments of various configurations makes it possible to estimate the importance of different cluster modes. The opening angle of the relativistic fragmentation cone is determined by the Fermi- momenta of the nucleon clusters in a nucleus. Being normalized to the mass numbers they are concentrated with a few percent dispersion near the normalized momentum of the primary nucleus. When selecting events with dissociation of a projectile into a narrow fragmentation cone we see that target-nucleus non-relativistic fragments either are absent (“white”stars in Ref.[1]), or their number is insignificant. The target fragments are easily separated from the fragments of a relativistic projectile since their fraction in the angular relativistic fragmentation cone is small and they possess non-relativistic momentum values. In the peripheral fragmentation of a relativistic nucleus with charge Z the ionization induced by the fragments can decrease down to a factor Z, while the ionization per one track – down to Z2. Therefore experiment should provide an adequate detection range. In order to reconstruct an event, a complete kinematic information about the particles in the relativistic fragmentation cone is needed which, e.g., allows one to calculate the invariant mass of the system. The accuracy of its estimation decisively depends on the exactness of the track angular resolution. To ensure the best angular resolution, it is necessary that the detection of relativistic fragments should be performed with the best spacial resolution. The nuclear emulsion technique, which underlies the BECQUEREL project at the JINR Nuclotron [2], well satisfies the above-mentioned requirements. It is aimed at a systematic search for peripheral fragmentation modes with statistical provision at a level of dozens events, their classification and angular metrology. Emulsions provide the best spacial res- olution (about 0.5 µm) which allows one to separate the charged particle tracks in the FIG. 1: An event of the type of “white”star from the fragmentation of a relativistic 9Be nucleus into two He fragments in emulsion. The photograph was obtained on the PAVIKOM(FIAN) complex. three-dimensional image of an event within one-layer thickness (600 µm) and ensure a high accuracy of angle measurements. The tracks of relativistic H and He nuclei are separated by sight. As a rule, in the peripheral fragmentation of a light nucleus its charge can be determined by the sum of the charges of relativistic fragments. Multiple-particle scattering measurements on the light fragment tracks enable one to separate the H and He isotopes. The analysis of the products of the relativistic fragmentation of neutron-deficient isotopes has some additional advantages owing to a larger fraction of observable nucleons and min- imal Coulomb distortions. Irradiation details and a special analysis of interactions in the BR-2 emulsion are presented in Ref. [3, 4]. In what follows, we give the first results of the study of the 9Be,8B, 7Be 14N nuclei fragmentation with a few A GeV energy which are obtained with the use of a part of the material analyzed. II. FRAGMENTATION OF 9BE NUCLEI The 9Be nucleus is a loosely bound n+α+α system. The energy threshold of the 9Be→n+α+α dissociation channel is 1.57 MeV. The study of the 9Be fragmentation at relativistic energies gives the possibility of observing the reaction fragments, which are the decay products of unbound 8Be and 5He nuclei. The method of nuclear emulsions used in the present paper allows one to observe the charged component of the relativistic 9Be→2He+n fragmentation channel. Owing to a good angular resolution of this method it is possible to separate the 9Be fragmentation events, which accompanied by the production of an unstable 8Be nucleus with its subsequent breakup to two a particles. In this case, the absence of a combinatorial background (of three and more α particles) for 9Be, which is typical for heavier Nα nuclei 12C and 16O makes it possible to observe distinctly this picture. Nuclear emulsions were exposed to relativistic 9Be nuclei at the JINR Nuclotron. A beam of relativistic 9Be nuclei was obtained in the 10B→9Be fragmentation reaction using a polyethylene target. The 9Be nuclei constituted about 80% of the beam, the remaining 20% fell on Li and He nuclei.[5] Events were sought by microscope scanning over the emulsion plates. In total 362 events of the 9Be fragmentation involving the two He fragment production in the forward fragmen- tation cone within a polar angle of 6◦(0.1 rad) were found. The requirement of conservation of the fragment charge in the fragmentation cone was fulfilled for the detected events. In event selection 5 - 7 tracks of various types were allowed in a wide (larger than 6◦) cone to increase statistics. An example of the 9Be→2He fragmentation event in emulsion is given in Fig. 1 [2]. This event belongs to the class of “white”stars as far as it contains neither target nucleus fragments, nor produced mesons. This event sample includes 144 “white ”stars. The angles of the tracks in emulsion for the detected events were obtained using a fine measuring microscope. Angular measurements for the 362 events were carried out with an accuracy not worse then 4.5×10−3 rad. In analyzing the data both He fragments observed in the 9Be→2He+n channel were supposed to be a particles. This assumption is motivated by the fact that at small angles the 9Be→ 24He+n fragmentation channel with an energy threshold of 1.57 MeV must dominate the 9Be→3He+4He+n channel whose energy threshold is 22.15 MeV. The 3He fraction will not exceed a few percent in this energy range [6] and all the He fragments in the detected events may be thought of as α particles. In Fig. 2a the PT transverse momentum distribution of α particles in the laboratory frame of reference is calculated without the account of particle energy losses in emulsion by the equation PT = p0 · A · sinθ (1) where p0, A and θ are the momentum per nucleon, the fragment mass and the polar emission angle, respectively. The outer contour corresponds to all events. The inner histogram is obtained for events accompanied by protons recoil of emulsion target (dashed area). The mean value of the transverse momentum for the total event sample in the laboratory system is equal to < PT >≈103 MeV/c with FWHM σ ≈72 MeV/c. This may be an indication of the fact that the experimental data are not of the same kind which can be pronounced when going over to the c.m.s. of two α particles. The P∗T transverse momentum distribution of α particles in the c.m.s. of two α particles , MeV/cTP 0 50 100 150 200 250 300 350 , MeV/c 0 50 100 150 200 250 300 350 FIG. 2: The PT transverse momentum distribution of α particles in the laboratory system (a), and the P∗T momentum distribution in the c.m.s. of an α particle pair (b). The outer contour corresponds to all events. The inner histogram is obtained for events, which accompanied by protons recoil of emulsion target (dashed area). described by the equation ∼= PT i − where PT i is the transverse momentum of an i-th α particle in the laboratory system nα=2 is given in Fig. 2b. There is observed a grouping of events around two peaks with the values < P ∗T i >≈24 MeV/c and < P T i >≈101 MeV/c. In Ref [7] the appropriate mean values of the α fragment transverse momenta are < P ∗T i >≈121 MeV/c for 16O→4α,< P ∗T i >≈141 MeV/c [8] for 12C→3α and < P ∗T i >≈200 MeV/ for 22Ne→5α (processing of the available data). There by we clearly see a tendency toward an increase of the mean α particle momentum with increasing their multiplicity. This implies a growth of the total Coulomb interaction of alpha clusters arising in nuclei. In the opening angle Θ distribution (Fig. 3) one can also see two peaks with mean values 4.6×10−3rad. and 26.8×10−3rad. The ratio of the numbers of the events in the peaks is close to unity. The Θ distribution entails the invariant energy Q2α distribution, which is calculated as a difference between the effective invariant mass M2α of an α fragment pair and the doubled α particle mass by the equations M22α = −( Q2α = M2α − 2 ·mα (3) rad.-310×, Θ 0 10 20 30 40 50 60 70 80 FIG. 3: The opening Θ angle distribution of α particles in the 9Be→2α fragmentation reaction at 1.2 A GeV energy. The outer contour corresponds to all events. The inner histogram is obtained for events, which accompanied by protons recoil of emulsion target (dashed area). where Pj is the α particle 4-momentum. In the invariant energy Q2α distribution (Fig. 4) there are two peaks in the ranges 0 to 1 MeV and 2 to 4 MeV. The shape of the distribution does not contradict the suggestion about the 9Be fragmentation involving the production of an unstable 8Be nucleus which decays in the 0+ and 2+ states. The values of the peaks of the invariant energy Q2α and the transverse momenta P∗T in the c.m.s. relate to each other. To the Q2α range from 0 to 1 MeV with a peak at 100 keV there corresponds a peak P∗T with < P T i >≈24 MeV/c , and to the Q2α range from 2 to 4 MeV there corresponds a peak with < P T i >≈101 MeV/c. III. FRAGMENTATION OF 14N NUCLEI A stack of layers of BR-2 emulsion was exposed to a beam of 14N nuclei accelerated [9] to a momentum of 2.86 A GeV/c at the Nuclotron of the Laboratory of High Energy Physics (JINR). Already been found amoung 950 inelastic events in which the total fragment charge was equal to the Z0=7 fragment charge and there were no produced particles. Events were sought by viewing over the track length which provided the accumulation of statistics , MeVα2Q 0 2 4 6 8 10 12 , keV 0 100 200 300 400 500 600 700 800 900 10000 FIG. 4: The invariant energy Q2α distribution of α particle pairs in the 9Be→2α fragmentation reaction at 1.2 A GeV energy. On the intersection: the Q2α range from 0 to 1 MeV. Arrows mark the 8Be nucleus levels (0+ and 2+). The outer contour corresponds to all events. The inner histogram is obtained for events, which accompanied by protons recoil of emulsion target (dashed area). without selection. The selected events are divided in two classes. The events of the type of “white”star and the interactions involving the production of one or a few target-nucleus fragments belong to the first class. Table I shows the charge multi-fragmentation topology which was studied for the events satisfying the above-mentioned conditions. The upper line is the Z>2 fragment charge, the second line is the number of single-charged fragments , the third one the number of two- charged fragments, and the fourth and fifth lines are the number of the detected events with a given topology for “white”stars and events with target-nucleus excitation for each channel, respectively. The two last lines present the total number of interactions calculated in absolute values and in percent. The analysis of the data of Table I shows that the number of channels involving Z>3 fragments for the “white”stars is larger by about a factor of 1.5 than that for the events accompanied by a target breakup. On the contrary, for the 2+2+2+1 charge configuration TABLE I: The charge topology distribution of the “white”stars and the interactions involving the target-nucleus fragment production in the 14N dissociation at 2.86 A GeV/c momentum. Zfr 6 5 5 4 3 3 – – NZ=1 1 – 2 1 4 2 3 1 NZ=2 – 1 – 1 – 1 2 3 NW.S. 13 4 3 1 1 1 6 17 Nt.f. 15 1 3 3 – 2 5 32 NP 28 5 6 4 1 3 11 49 NP,% 26 5 5 4 1 3 10 46 channel this number is smaller by about a factor of 1.5. Thus, in the events with target breakup, the projectile fragments more strongly than in the “white”stars. The data of Table I points to the predominance of the channel with the 2+2+2+1 charge configuration (49 events) which has been studied in more detail. The obtained results show that the 14N nucleus constitutes a very effective source for the production of 3α system. In order to estimate the energy scale of production of 3α particle systems in the 14N→3α+X channel, we present the invariant excitation energy Q3α distribution with re- spect to the 12C ground state: M23α = −( Q3α = M 3α −M( 12C) (4) where M(12C) is the mass of the ground state corresponding to the charge and the weight of the system being analyzed, M∗3α the invariant mass of the system of fragments. Statistics was increased to 132 events 14N→3α+X including 50 “white ”stars by scanning over the emulsion plates. The main part of the events is concentrated in the Q3α area from 10 to 14 MeV, covering the known 12C levels (Fig. 5). Softening of the conditions of the 3He + H selection, for which the target fragment production is allowed, does not result in a shift of the 3α excitation peak. This fact suggests the universality of the 3α state population mechanism. To estimate the fraction of the events involving the production of an intermediate 8Be nucleus in the reactions 14N→8Be+X→3α+X we present the invariant excitation energy Q3α, MeV 0 10 20 30 40 50 FIG. 5: The invariant excitation energy Q3α distribution of three α particles with respect to the 12C ground state for the process 14N→3α+X. The following notation is used: 1) all the events of the given dissociation, 2) “white”stars. distribution for an α particle pair with respect to the 8Be ground state (Fig. 6). The first distribution peak relates to the value to be expected for the decay products of an unstable 8Be nucleus in the ground state 0+. The distribution centre is seen to coincide well with the decay energy of the 8Be ground state. The fraction of the α particles originating from the 8Be decay is 25-30%. IV. FRAGMENTATION OF 7BE, AND 8B NUCLEI The results of investigations dealing with the charge topology of the fragments produced in peripheral dissociation of relativistic 8B, 7Be nuclei in emulsion are presented in Ref [2, 10, 11, 12]. Table II presents the numbers of the events detected in various channels of the 7Be , MeVα2Q 0 2 4 6 8 10 12 14 16 18 20 , keVα2Q 0 50 100 150 200 250 300 350 400 450 500 FIG. 6: The invariant excitation energy Q2α distribution of α particle pairs for the process 14N→3α+X. In the inset: a fraction of the distribution at 0-500 keV. fragmentation. Of them, the 3He+4He channel noticeably dominates, the channels 4He+d+p and 6Li+p constitute 10% each. Two events involving no emission of neutrons in the three- body channels 3He+t+p and 3He+d+d were registered. The reaction of charge-exchange of 7Be nuclei to 7Li nuclei was not detected among the events not accompanied by other secondary charged particles.The events involving no target fragments (nb=0) are separated from the events involving one or a few fragments (nb >0). For the first time, nuclear emulsions were exposed to a beam of relativistic 8B nuclei. We have obtained data on the probabilities of the 8B fragmentation channels in peripheral interactions. 55 events of the peripheral 8B dissociation which do not involve the produc- tion of the target-nucleus fragments and mesons (“white” stars ) were selected. A leading contribution of the 8B→7Be+p mode having the lowest energy threshold was revealed on the basis of these events. Information about a relative probability of dissociation modes with larger multiplicity have been obtained. Among the found events there are 320 stars in which the total charge of the relativistic fragments in a 8◦ fragmentation cone ΣZfr satisfies the condition ΣZfr >3. These stars were attributed to the number of peripheral dissocia- tion events Npf . The Npf relativistic fragment distribution of over charges NZ is given in TABLE II: 7Be fragmentation channel (number of events) Channel 2He 2He He+2H He+2H 4H 4H Li+H Li+H Sum nb =0 nb >0 nb =0 nb >0 nb =0 nb >0 nb =0 nb >0 3He+4He 30 11 41 3He+3He 11 7 18 4He+2p 13 9 22 4He+d+p 10 5 15 3He+2p 9 9 18 3He+d+p 8 10 18 3He+2d 1 1 3He+t+p 1 1 3p+d 2 2 2p+2d 1 1 6Li+p 9 3 12 Sum 41 18 42 33 2 1 9 3 149 Table III. There are given the data for 256 events containing the target-nucleus fragments - Ntf , as well as for 64 events which contain no target-nucleus fragments (“white” stars )– Npf . The role of the channels with multiple relativistic fragments NZ >2 is revealed to be dominant for the N“white” stars. Of peripheral events, the “white” stars Nws (Table III) are of very particular interest. They are not accompanied by the target-nucleus fragment tracks and makes it possible to clarify the role of different cluster degrees of freedom at a minimal excitation of the nuclear structure. Table IV gives the relativistic fragment charge distribution in the “white” stars for 7Be and 8B nuclei. The 8B events are presented without one single-charged relativistic fragment, that is a supposed proton halo. The identical fraction of the two main 2He and He+2H dissociation channels is observed for 7Be and 8B nuclei which points out that the 8Be core excitation is independent of the presence of an additional loosely bound proton in the 8B nucleus. TABLE III: The charge topology distribution of the number of interactions of the peripheral Npf type (Npf=Ntf+Nws), which were detected in an emulsion exposed to a second 8B nucleus beam. Here Zfr is the total charge of relativistic fragments in a 8 ◦ angular cone in an event, NZ the number of fragments with charge Z in an event, Nws the number of “white”stars, Ntf the number of events involving the target fragments, Nws the number of “white” stars. Zfr N5 N4 N3 N2 N1 Ntf Nws 7 - - - 1 5 1 - 6 - - - 2 2 8 2 6 - - - 1 4 6 4 6 - - - - 6 1 - 5 - - - 1 3 61 14 5 - - - 2 1 44 12 5 - - 1 - 2 8 - 5 - - 1 1 - 1 - 5 - 1 - - 1 17 24 5 1 - - - - 17 1 5 - - - - 5 21 4 4 - - - - 4 5 1 4 - - - 2 - 24 1 4 - - - 1 2 42 - TABLE IV: The charged dissociation mode distribution of the “white” stars produced by the 7Be and 8B nuclei. To make the comparison more convenient, for the 8B nucleus one H nucleus is eliminated from the charged mode and the channel fractions are indicated. ΣZfr=4 7Be % 8B (+H) % 2He 41 43 12 40 He+2H 42 45 14 47 4H 2 2 4 13 V. CONCLUSIONS The degree of the dissociation of the relativistic nuclei in peripheral interactions can reach a total destruction into nucleons and singly and doubly charged fragments. The emulsion technique allows one to observe these systems to the smallest details and gives the possibility of studying them experimentally. New experimental observations are reported from the emulsion exposures to 14N, 9Be, 8B, 7Be nuclei with energy above 1 A GeV. The main features of 9Be→2He relativistic frag- mentation are presented. For the particular case of the relativistic 9Be nucleus dissociation it is shown that precise angular measurements play a crucial role in the restoration of the excitation spectrum of the alpha particle fragments. This nucleus is dissociated practically totally through the 0+ and 2+ states of the 8Be nucleus. The data obtained from 9Be angular measurements can be employed for the estimation of the role of 8Be in more complicated Nα systems. The results of the study of the dissociation of 14N nuclei of a primary momentum of 2.86 A GeV/c in their interactions with the emulsion nuclei are also presented. The present investigation indicates the leading role of the 2+2+2+1 charge configuration channel. The energy scale of the 3α system production has been estimated. According to the avail- able statistics 80% of interactions are concentrated at 10-14 MeV. The fraction of the 14N→8Be+X→3α+X channel involving the production of an intermediate 8Be nucleus is about 25%. Advantages of emulsion technique are exploited most completely in the study of peripheral fragmentation of light stable and neutron deficient nuclei. The results of investigations dealing with the charge topology of the fragments produced in peripheral dissociation of relativistic 7Be, 8B nuclei in emulsion are presented. Information on the relative probability of dissociation modes with a larger multiplicity was obtained. The dissociation of a 7Be core in 8B indicates an analogy with that of the free 7Be nucleus. [1] N. P. Andreeva, et al., Phys. At. Nucl. 68, 455–465 (2005). [2] Web site of the BECQUEREL Project: http://becquerel.jinr.ru (2006). [3] M. I. Adamovich, et al., Phys. At. Nucl. 62, 1378–1387 (1999). [4] M. I. Adamovich, et al., Phys. At. Nucl. 62, 514–517” (2004). [5] D. A. Artemenkov, arXiv:nucl-ex/0605018 (2006). [6] V. V. Belaga, et al., Phys. At. Nucl. 59, 869–877 (1996). [7] F. A. Avetyan, et al., Phys. At. Nucl. 59, 110–116 (1996). [8] V. V. Belaga, et al., Phys. At. Nucl. 58, 2014–2020 (1995). [9] T. V. Shchedrina, et al., arXiv:nucl-ex/0605022 (2006). [10] R. Stanoeva, et al., arXiv:nucl-ex/0605013 (2006). [11] N. G.Peresadko, et al., arXiv:nucl-ex/0605014 (2006). [12] N. P. Andreeva, et al., arXiv:nucl-ex/0604003 (2006). http://becquerel.jinr.ru Introduction Fragmentation of 9Be nuclei Fragmentation of 14N nuclei Fragmentation of 7Be, and 8B nuclei Conclusions References

0704.0385

Super-shell structures and pairing in ultracold trapped Fermi gases

Super-shell stru tures and pairing in ultra old trapped Fermi gases Magnus Ögren and Henning Heiselberg Mathemati al Physi s, Lund Institute of Te hnology, P.O. Box 118, SE-22100 Lund, Sweden University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark (Dated: April 3, 2007) We al ulate level densities and pairing gaps for an ultra old dilute gas of fermioni atoms in harmoni traps under the in�uen e of mean �eld and anharmoni quarti trap potentials. Super- shell stru tures, whi h were found in Hartree-Fo k al ulations, are al ulated analyti ally within periodi orbit theory as well as from WKB al ulations. For attra tive intera tions, the underlying level densities are ru ial for pairing and super-shell stru tures in gaps are predi ted. PACS numbers: 03.75.Ss, 05.30.Fk Ultra old atomi gases have re ently been used to re- ate novel quantum many-body systems su h as strongly intera ting high temperature super�uids of fermions, Bose-Einstein ondensates, Mott insulators in opti al lat- ti es, et . These lab phenomena have a strong over- lap with ondensed matter [1℄, nu lear [2℄ and neutron star physi s [3℄. Finite fermion systems su h as atoms in traps, nu lei, helium and metal lusters, semi ondu tor quantum dots, super ondu ting grains, et ., have addi- tional interesting quantum stru tures su h as level spe - tra, densities and pairing. These will be observable as temperatures are further lowered in atomi trap experi- ments. The high degree of ontrol over physi al param- eters, in luding intera tion strength and density, makes the atomi traps marvelous model systems for general quantum phenomena. The purpose here is to al ulate the level spe tra, densities and pairing for zero-temperature Fermi gases in harmoni os illator (HO) traps with anharmoni and mean �eld perturbations, and to show that novel super- shell stru tures appear in both level densities and pairing. In al ulating level spe tra by analyti al periodi orbit theory and WKB as well as numeri al Hartree-Fo k, we also relate these di�erent theoreti al approa hes to one another. We treat a gas ofN fermioni atoms of massm in a HO potential at zero temperature, intera ting via a two-body intera tion with s-wave s attering length a. We shall mainly dis uss a spheri ally symmetri trap and a dilute gas (i.e. where the density ρ obeys the ondition ρ|a|3 ≪ 1) of parti les with two spin states of equal population. The Hamiltonian is then given by mω2r2i + U(ri) , (1) We will onsider both external anharmoni potentials of the form U = εr4 and parti le intera tions: U(ri) = (2π~2a/m) j 6=i δ 3(ri−rj). When intera tions are weak, the latter an be approximated by the mean �eld poten- U(r) = 2π~2a ρ(r) . (2) For a large number of parti les and U = 0 the Fermi energy is EF = ñF~ω where nF = ñF − 3/2 ≃ (3N)1/3 is the HO quantum number at the Fermi surfa e. The HO shells are highly degenerate with states having an- gular momenta l = nF , nF −2, ...,mod(nF , 2), due to the U(3) symmetry of the 3D spheri ally symmetri HO po- tential. However, intera tions split this degenera y. In the Thomas-Fermi (TF) approximation (see, e.g., [4℄) the Fermi energy is 2k2F (r) mω2r2 + U(r) . (3) The density ρ(r) = k3F (r)/3π ρ(r) = ρ0 1− r2/R2TF , (4) inside the loud r ≤ RTF = aosc 2ñF , where ρ0 = (2ñF ) 3/2/3π2a3osc is the entral density [5℄. For onve- nien e we set the os illator length aosc = ~/mω = 1 in the following. Taylor expanding the density and thereby also the mean �eld of Eq. (2) around the enter gives ρ(r) ≃ ρ0 r2/R2TF + r4/R4TF + ... , (5) the �rst term will simply in orporate a onstant shift in energies whereas the term quadrati in radius renor- malizes the HO frequen y as ωeff = ω 1− 6πaρ0/R2TF . The third term is quarti in radius and is therefore also of the same form as the external potential U(r) ≃ εr4 , (6) with ε = (3π~2a/4m)ρ0/R TF . Both the pure quarti potential and the mean �eld potential of Eq. (2) are an- harmoni and hange the level density by splitting the l degenera y of the HO shell nF at the Fermi surfa e. We will now al ulate analyti ally the level spe tra from perturbative periodi orbit theory for the quarti potential and subsequently within semi lassi al WKB wavefun tions for both the quarti and the mean �eld potential of Eq. (2). We will start with repulsive inter- a tions where pairing is not present. http://arxiv.org/abs/0704.0385v1 In periodi orbit theory [6℄, the level density an be written (to leading order in ~ ) in terms of a perturba- tive HO tra e formula [7, 8℄ g(E) = 1 +Re (−1)k M ei2πkE/~ω . (7) For the unperturbed HO (U = 0) the modulation fa tor is M = 1. For a quarti perturbed potential, as in Eq. (6), the modulation fa tor was al ulated in [8℄ e−i2kσ/~−iπ/2 + e−i3kσ/~+iπ/2 , (8) with σ = επE2/~2ω3, being a small lassi al a tion. The two terms arise from the hange in a tions for the ir le and diameter orbits respe tively due to the quarti po- tential [8℄. The resulting level density an be written in the fa torised form [9℄ g(E) = (~ω)3 (−1)k . (9) Here, the �rst term is the average level density, the osine fa tor gives the rapid HO shell os illations (mod- i�ed by the perturbation) whi h, however, are slowly modulated by the sine fa tor resulting in a beating pat- tern. Moreover, the non-perturbed HO limit, equivalent to M = 1 in Eq. (7), is re overed in the limit of |ε| → 0, where the U (3) symmetry is restored. The k = 1 term in Eq. (9) gives the major os illations in the level density and is shown in Fig. 1 (a). The beating pattern or super- shells is learly observed. The shell os illations vanish when the argument of the sine in Eq. (9) is an integer S = 1, 2, 3, ... times π, i.e. |ε|E2/2(~ω)3 = S. This gives the supernode ondition nF = E/~ω = 2S~ω/|ε| . (10) We now turn to an alternative al ulation of the level density with WKB. The splitting of the HO shells degen- erate levels l = nF , nF −2, ...,mod (nF , 2) in the shell nF by the mean-�eld potential an be al ulated perturba- tively in the dilute limit. An ex ellent approximation for the radial HO wave fun tion with angular momentum l and (nF − l)/2 radial nodes in the HO shell when nF ≫ 1 is the WKB one [10, 11℄: RnF l(r) ≃ sin(kl(r)r + θ) l (r)r , (11) between turning points r2± = ñF ± ñ2F − l(l + 1). Here, ñF = nF + 3/2 and the WKB wave number kl(r) is k2l (r) = 2ñF − r2 − l(l+ 1)/r2 . (12) When nF ≫ 1 the wave fun tion has many nodes 1 ≪ l ≪ nF and the os illations in R2nl(r) an be averaged 〈sin2(kl(r)r)〉 = 1/2 [10℄. The phase θ is then unim- portant. The single-parti le energies for the anharmoni potential of Eq. (6) are simply EnF ,l − ñF~ω = U(r)|RnF l(r)|2r2dr (13) πkl(r) 3ñ2F − l(l + 1) .(14) It is spe ial for the quarti perturbation that the level energies are linear in l(l+1). The resulting level spa ing in reases as (2l+1) just as the level degenera y for SO(3) symmetry. Therefore the level density is onstant within the bandwidth D ≡ |EnF ,l≡0 − EnF ,l=nF | = εn2F /2 (15) on energy s ales larger than 2D/nF but smaller than D. The level density vanishes between the bandwidths of two neighbouring n shells and therefore it generally has a strong os illatory behavior as shown in Fig. 1 (a). Its amplitude is largest when D ∼ ~ω/2. However, when D ≃ ~ω the level density is onstant and the os illa- tory behavior vanishes. This phenomenon repeats when D = S~ω sin e the level spe tra then overlap S times. With the bandwidth of Eq. (15) under this ondition, we obtain exa tly the same supernode ondition as for pe- riodi orbit theory, Eq. (10). We on lude that Craig's perturbative periodi orbit theory [7℄ is in exa t agree- ment with perturbative WKB for a quarti ally perturbed spheri al symmetri HO in three dimensions. We now turn to the slightly more ompli ated mean �eld potential of Eq. (2). Its level spe trum an also be al ulated from the WKB wave fun tions of Eq. (11). Inserting them in Eq. (13), we obtain EnF ,l − ñF~ω = 2/ F ~ω I . (16) Here, the integral I is 1− l(l+ 1)/ñ2F 1− x2 where x = (r2−ñF )/ ñ2F − l(l + 1). This integral is I = π for l ≃ nF and I = 8 2/3 for l = 0. The bandwidth is therefore D = 2/ 2/3− π . (18) Inserting this bandwidth in the supernode ondition D = S~ω gives 2/3− π F = S . (19) For example in the ase 2πa = 1 the supernodes S = 1, 2, 3, .. should o ur when nF ∼ 28, 44, 58, et . The Hartree-Fo k (HF) al ulations of the os illating part of the total energy, whi h is proportional to the level den- sity at the Fermi level [6℄, result in slightly higher su- pernodes, as in Fig 1 (b). The di�eren es arise be ause the WKB al ulations are perturbative in the intera tion strength, whereas in the HF al ulation the MF poten- tial U in ludes a large s attering length whi h, e.g., leads to orre tions for the e�e tive os illator frequen y. Also for the purely quarti term the perturbative approa h underestimates the exa t supernodes (see Fig. 3 of [8℄). For weaker intera tions 2πa = 0.1, the �rst supernode S = 1 should o ur at nF = 130 a ording to the ondi- tion of Eq. (19), in loser agreement with the HF result of Fig. 1 ( ). For omparison, the Taylor expansion of the mean �eld potential leads to the supernode ondition of Eq. (10) with ε = (3π~2a/4m)ρ0/R TF . It di�ers from Eq. (19) by the prefa tor, whi h is ∼ 34% smaller. It is a better approximation to expand e.g. around r = RTF /2 nF /2, where the orresponding prefa tor is only ∼ 8% smaller, su h that the supernode in Fig 1 ( ) is predi ted to nF = 137. Now expanding I of Eq. (17) for small l ≪ nF , one �nds I = (8/3) 2− l2/ 2n2F , (20) resulting in the level spe trum [10℄ EnF ,l − ñ~ω = − l(l+ 1) . (21) This level density is onstant at low l as for the potential in Eq. (14). However, near l ∼ nF the density of lev- els is slightly smaller as an be seen from the bandwidth orresponding to Eq. (21), whi h is ∼ 12% larger, for a given nF , than the bandwidth of Eq. (18). That the level density is not ompletely onstant within the bandwidth has the e�e t that a small periodi ity remains even at the super-shell ondition D = S~ω. Therefore the shell os illations do not disappear ompletely at the supern- odes, as an be seen in Fig. 1 (b, ), whereas for the purely quarti ase (a) the os illations disappear ompletely at the supernodes. Most atomi traps are not spheri al but igar shaped (prolate) with ωz <∼ω⊥. The unperturbed HO energies E = nz~ωz + n⊥~ω⊥ will generally lead to a onstant level density for energy s ales larger than ~ωz but smaller than ~ω⊥. When the os illator frequen y ratio ω⊥/ωz is a rational number, level degenera ies and larger os illations will o ur on the s ale ~ωz. Intera tions will, however, smear this level density. In any ase, super-shell stru ture is not expe ted as in the spheri al symmetri ase. In very oblate traps ωz ≫ ω⊥ the mean �eld potential is e�e tively two-dimensional and quadrati , i.e. it does not split the HO shells [10, 13℄. Thus we may expe t strong os illations in the level density on the s ale ~ω⊥, but again no super-shell stru ture. 0 20 40 60 80 100 −1000 1000 (a) 0 20 40 60 80 100 120 140 200 (c) 0 20 40 60 80 100 40 (b) Figure 1: ( olor online) The upper �gure (a) shows the leading term (k = 1) of the os illating part of the perturbative level density of Eq. (9) as a fun tion of nF = E/~ω, for the ase of an external potential V = VHO+εr with ε = 2/402 ≈ 0.0013. The middle and lower �gures (b, ) show the os illating part of the total energy a ording to a numeri al HF al ulation [12℄, with intera tion strength 2πa = 1 and 2πa = 0.1, as a fun tion of the HO shell number (~ = ω = 1). This illustrates qualitatively that a supernode, e.g. at nF = 40, an be due to intera tion (b) and/or an additional quarti term to the HO potential (a). Attra tive intera tions lead to pairing by an amount that is exponentially sensitive to the underlying level den- sity near the Fermi surfa e [2, 10, 11, 14℄. The level den- sity is the same for repulsive and attra tive intera tions ex ept that the levels are reversed when the sign of ε (Eqs. (9)) and (13)) or a is hanged (Eq. (16)). Therefore we an use the level densities and bandwidths al ulated above for pairing al ulations. Pairing in �nite systems is des ribed by the Bogoliubov-de Gennes (BdG) equa- tions [15℄ and take pla e between time-reversed states. As shown in [14℄ these states an be approximated by HO wave fun tions in dilute HO traps as long as the gap does not ex eed the os illator energy, ∆<∼~ω. Solving BdG for su h �nite systems is numeri ally ompli ated and we shall therefore apply further simplifying approximations, namely that the pairing gap ∆nl and the wavefun tion overlap matrix elements vary slowly with level l in a shell n. Both approximations are fair for the trapped atoms as argued in [11℄ and deviations an be understood. As result we arrive at a mu h simpli�ed gap equation ∫ ∼2nF g(E) dE (E − µ)2 +∆(µ)2 . (22) Here, the supergap G = 32 2nF |a|~ω/15π2 was al- ulated in [10℄ as the pairing gap when all states in a shell an pair; this is the ase for a region of in- tera tion strengths and parti le number where the gap is large as ompared to the level splitting, yet small 0 10 20 30 40 50 60 =(3N)1/3 38 40 42 44 Figure 2: ( olor online) Multi-shell pairing gaps for a HO trap with an additional quarti term in the potential with ε = 2/402, i.e. for the level density of Fig. 1 (a) with supernodes at nF ≃ 40, 40 2 ≈ 57, etc. The intera tion strength is a = −0.05 (top red urve), a = −0.03 (middle blue urve, with the inset �gure around the �rst supernode) and a = −0.01 (lower green urve). In the inset plot it is learly seen that the lo al minima for l ∼ nF and l ∼ 0 before the supernode turns into lo al maxima after the supernode, as a onsequen e of overlapping shells. The dashed (red) line is the multi-shell gap ∆ = G/(1−2G ln(nF )/~ω) for a = −0.05 and the upper/lower thin solid line (bla k) are the single mid-/end-shell pairing for a = −0.01 (see text). ompared to the shell splitting ~ω. ∆(µ) = ∆nl is the gap at the Fermi surfa e. g(E) = n2F /D is the level density within ea h bandgap D around ev- ery shell n = 0, 1, ...,∼2nF but vanishes between the bandgaps. The gap equation thus redu es to 1 = (G/D) ∑∼2nF (E + n~ω − µ)2 +∆2. The hemi al potential µ an be determined from the level spe trum; as we gradually �ll parti les into the shell nF at the Fermi surfa e, µ in reases from nF~ω to nF~ω+D. The ut-o� n<∼2nF in the sum of the gap equation models as a �rst approximation the more rigorous regularization pro edure des ribed in Ref. [16℄ that is required for a delta-fun tion pseudo-potential. By solving this simpli�ed gap equation of Eq. (22), we �nd that it still ontains and displays the essential interplay between the variation in level density and pair- ing. To illustrate the super-shell stru ture in pairing, we take the strongly anharmoni trap potential used for the level spe tra in Fig. 1 (a), and al ulate the pairing arising from a weak attra tive s attering length a < 0. For su� iently weak intera tions su h that pairing only takes pla e in the shell at the Fermi surfa e, we obtain the expe ted result from the gap equation: ∆ = G when D ≪ ∆, whereas for D ≫ ∆ we get ∆ = D exp(−D/2G) midshell (µ = nF~ω + D/2) and ∆ = 2D exp(−D/G) endshell (µ = nF~ω or µ = nF~ω +D). Pairing is thus stronger at midshell than at endshell, where there are fewer states to pair [11℄, and strong shell os illations fol- low as shown in Fig. 2. For stronger intera tions, pairing also takes pla e between states in shells around the Fermi shell and Eq. (22) gives: ∆ = G/ (1− 2G ln (nF ) /~ω) for small bandwidth [14℄. In Fig. 2 this urve is ompared with the �nite bandwidth result, whi h has strong os il- lations ex ept at the supernodes where the level density is ontinuous. At a supernode D = ~ω and the gap equa- tion (22) leads to a gap ∆ = 2nF~ω exp(−~ω/2G) [11℄. In summary, level densities, shell-os illations and super-shell stru tures in anharmoni traps al ulated from numeri al Hartree-Fo k and analyti al periodi or- bit theory as well as WKB were found to mat h to lead- ing order. Analogous super-shell stru tures were found in pairing from an approximated BdG al ulation. The mean �eld in atomi nu lei also have a large anharmoni potential and the HO shells start to overlap (the �rst supernode) already for heavy nu lei with nF ∼ 5 − 6. The interplay of level spe tra and multishell pairing is, however, di� ult to disentangle in nu lear pairing due to strong spin-orbit e�e t and small parti le number. Ul- tra old atomi traps, however, provide ideal systems for observing the ri h quantum stru tures su h as level den- sities and pairing. Dis ussions with Matthias Bra k on periodi orbit the- ory, Ben Mottelson on (nu lear) shell theory and pairing, and proof reading by Joel Corney, are gratefully a knowl- edged. [1℄ J. Bardeen, L. N. Cooper, J. R. S hrie�er, Phys. Rev. 108, 1175 (1957). [2℄ A. Bohr and B. R. Mottelson, Nu lear Stru ture Vols. I+II, Benjamin, New York 1969. [3℄ A. Bohr, B. R. Mottelson, D. Pines, Phys. Rev. 110, 936 (1958). [4℄ C. J. Pethi k and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge Univ. Press 2002. [5℄ For a �nite number of parti les the fa tor ñ = nF + 3/2 in ludes a orre tion to nF , whi h has been he ked nu- meri ally to improve the TF approximation and slightly hange the predi tion of supernodes. [6℄ M. Bra k and R. K. Bhaduri, Semi lassi al Physi s, re- vised edn (Boulder, CO: Westview) (2003). [7℄ S. C. Creagh, Ann. Phys., NY 248 60 (1996). [8℄ M. Bra k et al., J. Phys. A 38, 9941 (2005). [9℄ M. Ögren, unpublished (2006): www.magnus.ogren.se/notes/pot/derivationofgpert.pdf [10℄ H. Heiselberg and B. R. Mottelson, Phys. Rev. Lett. 88, 190401 (2002). [11℄ H. Heiselberg, Phys. Rev. A 68, 053616 (2003). Note that the square root of kl was missing in Eq. (6) of this Ref. as ompared to Eq. (11). [12℄ Y. Yu et al., Phys. Rev A 72, 051602(R) (2005). [13℄ B. P. van Zyl et al., Phys. Rev. A 67, 023609 (2003). [14℄ G. M. Bruun and H. Heiselberg, Phys. Rev. A 65, 053407 (2002). [15℄ P. G. de Gennes, Super ondu tivity of Metals and Alloys (Addison-Wesley, New York, 1989). [16℄ G. M. Bruun et al., Eur. Phys. J. D9, 433 (1999).

0704.0386

Quantum non-local effects with Bose-Einstein condensates

Quantum non-local effects with Bose-Einstein condensates F. Laloë a and W. J. Mullin b Laboratoire Kastler Brossel, ENS, UPMC, CNRS; 24 rue Lhomond, 75005 Paris, France Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003 USA We study theoretically the properties of two Bose-Einstein condensates in different spin states, represented by a double Fock state. Individual measurements of the spins of the particles are per- formed in transverse directions, giving access to the relative phase of the condensates. Initially, this phase is completely undefined, and the first measurements provide random results. But a fixed value of this phase rapidly emerges under the effect of the successive quantum measurements, giving rise to a quasi-classical situation where all spins have parallel transverse orientations. If the number of measurements reaches its maximum (the number of particles), quantum effects show up again, giving rise to violations of Bell type inequalities. The violation of BCHSH inequalities with an arbitrarily large number of spins may be comparable (or even equal) to that obtained with two spins. PACS numbers: 03.65.Ta,03.65.Ud,03.75.Gg,03.75.Mn The notion of non-locality in quantum mechanics (QM) takes its roots in a chain of two theorems, the EPR (Einstein Podolsky Rosen) theorem [1] and its log- ical continuation, the Bell theorem. The EPR theorem starts from three assumptions (Einstein realism, locality, the predictions of quantum mechanics concerning some perfect correlations are correct) and proves that QM is incomplete: additional quantities, traditionally named λ, are necessary to complete the description of physical re- ality. The Bell theorem [2, 3] then proves that, if λ exists, the predictions of QM concerning other imperfect corre- lations cannot always be correct. The ensemble of the three assumptions: Einstein realism, locality, all predic- tions of QM are correct, is therefore self-contradictory; if Einstein realism is valid, QM is non-local. Bohr [4] rejected Einstein realism because, in his view, the no- tion of physical reality could not correctly be applied to microscopic quantum systems, defined independently of the measurement apparatuses. Indeed, since EPR con- sider a system of two microscopic particles, which can be “seen” only with the help of measurement apparatuses, the notion of their independent physical reality is open to discussion. Nevertheless, it has been pointed out recently [5, 6] that the EPR theorem also applies to macroscopic sys- tems, namely Bose-Einstein (BE) condensates in two dif- ferent internal states. The λ introduced by EPR then cor- responds to the relative phase of the condensates, i.e. to macroscopic transverse spin orientations, physical quan- tities at a human scale; it then seems more difficult to deny the existence of their reality, even in the absence of measurement devices. This gives even more strength to the EPR argument and weakens Bohr’s refutation. It is then natural to ask whether the Bell theorem can be transposed to this stronger case. The purpose of this article is to show that it can. We consider an ensemble of N+ particles in a state defined by an orbital state u and a spin state +, and N− particles in the same state with spin orientation −. The whole system is described quantum mechanically by a double Fock state, that is, a “double BE condensate”: | Φ > = (au,+) ]N+ [ (au,−) | vac. > (1) where au,+ and au,− are the destruction operators asso- ciated with the two populated single-particle states and |vac. > is the vacuum state. We introduce a sequence of transverse spin measurements that leads to quantum predictions violating the so called BCHSH [7, 8] Bell in- equality. This is reminiscent of the work of Mermin [9], who finds exponential violations of local realist inequal- ities with N -particle spin states that are maximally en- tangled. By contrast, here we consider the simplest way in which many bosons can be put in two different in- ternal levels, with a N -particle state containing only the minimal possible correlations, those due to statistics. We find violations of inequalities that are the same order of magnitude as with the usual singlet spin state and may actually saturate the Cirel’son bound [10]. Double Fock states are experimentally more accessi- ble and much less sensitive to dissipation and decoher- ence than maximally entangled states [11]. Considering a system in a double Fock state, we assume that a se- ries of rapid spin measurements can be performed and described by the usual QM postulate of measurement, without worrying about decoherence between the mea- surements, thermal effects, etc. The operators associated with the local density of par- ticles and spins can be expressed as functions of the two fields operators Ψ±(r) associated with the two in- ternal states ± as: n(r) = Ψ†+(r)Ψ+(r) + Ψ −(r)Ψ−(r), σz(r) = Ψ +(r)Ψ+(r)−Ψ −(r)Ψ−(r), while the spin com- ponent in the direction of plane xOy making an angle ϕ with Ox is: σϕ(r) = e +(r)Ψ−(r)+ e −(r)Ψ+(r). Consider now a measurement of this component per- formed at point r and providing result η = ±1. The http://arxiv.org/abs/0704.0386v4 corresponding projector is: Pη=±1(r, ϕ) = [n(r) + η σϕ(r)] (2) and, because the measurements are supposed to be per- formed at different points (ensuring that these projectors all commute) the probability P(η1, η2, ...ηN ) for a series of results ηi± 1 for spin measurements at points ri along directions ϕi can be written as: < Φ | Pη1(r1, ϕ1)× Pη2(r2, ϕ2)× ....PηN (rN , ϕN ) | Φ > We now substitute the expression of σϕ(r) into (2) and (3), exactly as in the calculation of ref. [5], but with one difference: here we do not assume that the number of measurements is much smaller than N±, but equal to its maximum value N = N+ + N−. In the product of projectors appearing in (3), because all r’s are different, commutation allows us to push all the field operators to the right, all their conjugates to the left; one can then easily see that each Ψ±(r) acting on | Φ > can be re- placed by u(r) × au,± , and similarly for the Hermitian conjugates. With our initial state, a non-zero result can be obtained only if exactly N+ operators au,+ appear in the term considered, and N− operators au,−; a similar condition exists for the Hermitian conjugate operators. To express these conditions, we introduce two additional variables. As in [5], the first variable λ ensures an equal number of creation and destruction operators in the in- ternal states ± through the mathematical identity: einλ = δn,0 (4) The second variable Λ expresses in a similar way that the difference between the number of destruction operators in states + and − is exactly N+−N−, through the integral: e−inΛ ei(N+−N−)Λ = δn,N+−N− (5) The introduction of the corresponding exponentials into the product of projectors (2) in (3) provides the expres- sion (c.c. means complex conjugate): |u(rj)|2 eiΛ + e−iΛ + ηj ei(λ−ϕj+Λ) + c.c. where, after integration over λ and Λ, the only surviving terms are all associated with the same matrix element in state | Φ > (that of the product of N+ operators a†u,+ and N− operators a u,− followed by the same sequence of destruction operators, providing the constant result N+!N−!). We can thus write the probability as: P(η1, η2, ...ηN ) ∼ ei(N+−N−)Λ |u(rj)|2 eiΛ + e−iΛ + ηj ei(λ−ϕj+Λ) + c.c. or, by using Λ parity and changing one integration variable (λ′ = λ+ Λ), as: P(η1, η2, ...ηN ) = cos [(N+ −N−)Λ] {cos (Λ) + ηj cos (λ′ − ϕj)} (8) The normalization coefficient CN is readily obtained by writing that the sum of probabilities of all possible sequences of η’s is 1 (this step requires discussion; we come back to this point at the end of this article): cos [(N+ −N−)Λ] [cos (Λ)]N (9) Finally, we generalize (8) to any number of measurements M < N . A sequence of M measurements can always be completed by additional N −M measurements, leading to probability (8). We can therefore take the sum of (8) over all possible results of the additional N −M measurements to obtain the probability for any M as: P(η1, η2, ...ηM ) = cos [(N+ −N−)Λ] [cosΛ]N−M {cos (Λ) + ηj cos (λ′ − ϕj)} (10) The Λ integral can be replaced by twice the integral between ±π/2 (a change of Λ into π −Λ multiplies the function by (−1)N+−N−+N−M+M = 1). If M ≪ N , the large power of cosΛ in the first integral concentrates its contribution around Λ ≃ 0, so that a good approximation is Λ = 0. We then recover the results of refs [5, 6], with a single integral over λ defining the relative phase of the condensates (Anderson phase), initially completely undetermined, so that the first spin measurement provides a completely random result. But the phase rapidly emerges under the effect of a few measurements, and remains constant [12, 13, 14]; it takes a different value for each realization of the experiment, as if it was revealing the pre-existing value of a classical quantity. Moreover, when cosΛ is replaced by 1, each factor of the product over j remains positive (or zero), leading to a result similar to that of stochastic local realist theories; the Bell inequalities can then be obtained. However, when N − M is small or even vanishes, cosΛ can take values that are smaller than 1 and the factors may become negative, opening the possibility of violations. In a sense, the additional variable Λ controls the amount of quantum effects in the series of measurements. We now discuss when these standard QM predictions violate Bell inequalities. We need the value of the quantum average of the product of results, that is the sum of η1, η2, ...ηM × P(η1, η2, ...ηM ) over all possible values of the η’s, which according to (10) is given by: E(ϕ1, ϕ2, ..ϕM ) = cos [(N+ −N−)Λ] [cosΛ]N−M Consider a thought experiment where two condensates in different spin states (two eigenstates of the Oz spin component) overlap in two remote regions of space A and B , with two experimentalists Alice and Bob; they measure the spins of the particles in arbitrary transverse directions (perpendicular to Oz) at points of space where the orbital wave functions of the two condensates are equal. All measurements performed by Alice are made along a single direction ϕa, which plays here the usual role of the “setting” a, while all those performed by Bob are made along angle ϕb. We assume that Alice retains just the product A of all her measurements, while Bob retains only the product B of his; A and B are both ±1. We now assume two possible orientations ϕa and ϕ for Alice, two possible orientations ϕb and ϕ b for Bob. Within deterministic local realism, for each realization of the experiment, it is possible to define two numbers A, A′, both equal to ±1, associated with the two possible products of results η that Alice will observe, depending of her choice of orientation; the same is obviously true for Bob, introducing B and B′. Within stochastic local realism [8, 15], A and A′ are the difference of probabilities associated with Alice observing +1 or −1, i.e. numbers that have values between +1 and −1. In both cases, the following inequalities (BCHSH) are obeyed: − 2 ≤ AB +AB′ ± (A′B − A′B′) ≤ 2 (12) In standard quantum mechanics, of course, “unper- formed experiments have no results” [16], and several of the numbers appearing in (12) are undefined; only two of them can be defined after the experiment has been performed with a given choice of the orientations. Thus, while one can calculate from (11) the quantum average value < Q > of the sum of products of results appearing in (12), there is no special reason why < Q > should be limited between +2 and −2. Situations where the limit is exceeded are called “quantum non-local”. We have seen that the most interesting situations oc- cur when the cosines do not introduce their peaking effect around Λ = 0, i.e. when N+ = N− and M has its maxi- mum value N . Then, for a given N , the only remaining choice is how the number of measurements is shared be- tween Na measurements for Alice and Nb for Bob. Assume first that Na = 1 (Alice makes one measure- ment) and therefore Nb = N − 1 (Bob makes all the oth- ers). Since we assume that N+ = N− and M = N , the Λ integral in (11) disappears, and the λ integral contains only the product of cos (λ′ − ϕa) by the (N − 1)th power of cos (λ′ − ϕb), which is straightforward and provides cos (ϕa − ϕb) times the normalization integral CN . The quantum average associated with the product AB is thus merely equal to cos (ϕa − ϕb), exactly as the usual case of two spins in a singlet state. Then it is well-known that, when the angles form a “fan” [17] spaced by χ = π/4, a strong violation of (12) occurs, by a factor 2, sat- urating the Cirel’son bound [10]. A similar calculation can be performed when Alice makes 2 measurements and Bob N − 2, and shows that the quantum average is now equal to 1 1 + 1 + (1 − 1 ) cos 2 (ϕa − ϕb) no longer independent of N. If N = 4, the maximum of < Q > is 2.28 < 2 2, and rises to 2.41 as N → ∞. An expression for the generalization of the quantum av- erage to any number P and N − P of measurements by Alice and Bob, respectively, is (with χ = ϕa − ϕb): E(χ) = {P/2} P !(N − 2k)! k!(P − 2k)!(N − k)! sin2k χ cosP−2k χ where {P/2} is the integer part of P/2. The maximum of < Q > can then be found using a numerical Mathematica routine. Results are shown for several values of P in Fig. 1. The angles maximizing the quantum Bell quantity always occur in the fan shape, although the basic angle χ changes with P and N. All of the curves where P is held fixed have a finite < Q > limit with increasing N , and the optimum values of the angles approach constants. For the curve P = N/2, the limit is 2.32 when N → ∞, and the fan opening decreases as 1/ 10080604020 P FIG. 1: The maximum of the quantum average < Q > for Alice doing P experiments and Bob N − P , as a function of the total number of particles N . The usual Bell situation is obtained for N = 2, P = 1. Local realist theories predict an upper limit of 2; large violations of this limit are obtained, even with macroscopic systems (N → ∞). If P = 1, the violation saturates the Cirel’son limit for any N . We can also study cases where the number of measure- ments is M < N : if Bob makes all his measurements, but ignores one or two of them (independently of the or- der of the measurements), when he correlates his results with Alice, the BCHSH inequality is never violated. All measurements have to be taken into account to obtain violations. Furthermore, if the number of particles in the two condensates are not equal, no violation occurs either. Finally, it is possible to consider cases where we gener- alize the angles considered: experimenter Carole makes measurements at ϕc and ϕ c, and David at ϕd and ϕ We then find that a maximization of < Q > reduces to the cases already studied, where the new angles collapse to the previous angles ϕa, · · · , ϕ′b. For the sake of simplicity, we have not yet discussed some important issues that underlie our calculations. One is related to the so called “sample bias loophole” (or “detection/efficiency loophole”) and to the normalization condition (9), which assumes that one spin is detected at each point of measurement. A more detailed study (see second ref. [5]) should include the integration of each r in a small detection volume and the possibility that no particle is detected in it. This is a well-known dif- ficulty, which already appears in the usual two-photon experiments [8], where most photons are missed by the detectors. If this loophole still raises a real experimen- tal challenge, the difficulty can be resolved in theory by assuming the presence of additional spin-independent de- tectors [2, 8], which ensure the detection of one particle in each detector and create appropriate initial conditions (see for instance [18] for a description of an experiment with veto detectors). We postpone this discussion to an- other article [19]. A second issue deals with the definition of the local realist quantities A, B, etc. For two conden- sates, we have a slightly different situation than in the usual EPR situation: the local realist reasoning leads to the existence of a well-defined phase λ between the con- densates [5], not necessarily to deterministic properties of the individual particles. Fortunately, Bell inequalities can also be derived within stochastic local realist theories [3, 8] (see also for instance [9] or appendix I of [15]), and this difference is not a problem [19]. In conclusion, strong violations of local realism may occur for large quantum systems, even if the state is a simple double Fock state with equal populations; within present experimental techniques, this seems reachable with N ∼ 10 or 20. We have assumed that the mea- sured quantity is the product of many microscopic mea- surements, not their sum, which would be macroscopic; a product of results remains sensitive to the last measure- ment, even after a long sequence of others. Curiously, for very few measurements only the results are quantum, for many measurements they can be interpreted in terms of a classical phase, but become again strongly quantum when the maximum number of measurements is reached, a sort of revival of quantum-ness of the system. Laboratoire Kastler Brossel is “UMR 8552 du CNRS, de l’ENS, et de l’Université Pierre et Marie Curie”. [1] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935). [2] J.S. Bell, Physics 1, 195 (1964), reprinted in [3]. [3] J.S. Bell, “Speakable and unspeakable in quantum me- chanics”, Cambridge University Press (1987). [4] N. Bohr, Phys. Rev. 48, 696 (1935). [5] F. Laloë, Europ. Phys. J. D, 33, 87 (2005); see also cond-mat/0611043. [6] W.J. Mullin, R. Krotkov and F. Laloë, Phys. Rev. A74, 023610 (2006). [7] J.F. Clauser, M.A. Horne, A. Shimony and R.A. Holt, Phys. Rev. Lett. 23, 880 (1969). [8] J.F. Clauser and A. Shimony, Rep. on Progress in Phys. 41, 1883 (1978). [9] N.D. Mermin, Phys. Rev. Lett. 65, 1838 (1990). [10] B.S. Cirel’son, Letters in math. phys. 4, 93 (1980). [11] J.A. Dunningham, K. Burnett and S.M. Barnett, Phys. Rev. Lett. 89, 150401 (2002). [12] J. Javanainen and Sung Mi Yoo, Phys. Rev. Lett. 76, 161 (1996). [13] Y. Castin and J. Dalibard, Phys. Rev. A55, 4330 (1997). [14] I. Cirac, C. Gardiner, M. Naraschewski and P. Zoller, Phys. Rev. A54, R3714 (1996) and references in [6] [15] F. Laloë, Am. J. Phys. 69, 655 (2001). [16] A. Peres, Am. J. Phys. 46, 745 (1978). [17] The term “fan” refers to the angles arranged as ϕba = ϕa′b = ϕab′ and ϕa′b′ = 3χ where ϕab ≡ ϕa − ϕb. [18] J.S. Bell, Comments on at. and mol. phys. 9, 121 (1979); reprinted in [3]. [19] W.J. Mullin and F. Laloë, to be published http://arxiv.org/abs/cond-mat/0611043

0704.0387

Low mass visual binaries in the solar neighbourhood: The case of HD141272

Astron. Nachr. / AN Volume, No. Issue, 0 – 5 (Year of publication) / DOI DOI Low mass visual binaries in the solar neighbourhood: The case of HD141272⋆ T. Eisenbeiss1 ⋆⋆, A. Seifahrt1,2, M. Mugrauer1, T. O. B. Schmidt1, R. Neuhäuser1, and T. Roell1 1 Astrophysikalisches Institut und Universitäts-Sternwarte Jena, Schillergässchen 2-3, 07745 Jena, Germany 2 European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748, Garching, Germany Received 15.08.06, accepted 29.03.07 Published online . . . Key words binaries: visual – stars: late-type, low mass – astrometry We search for stellar and substellar companions of young nearby stars to investigate stellar multiplicity and formation of stellar and substellar companions. We detect common proper-motion companions of stars via multi-epoch imaging. Their companionship is finally confirmed with photometry and spectroscopy. Here we report the discovery of a new co-moving (13σ) stellar companion ∼ 17.8 arcsec (350AU in projected sepa- ration) north of the nearby star HD141272 (21 pc). With EMMI/NTT optical spectroscopy we determined the spectral type of the companion to be M3±0.5V. The derived spectral type as well as the near infrared photometry of the companion are both fully consistent with a 0.26+0.07−0.06M⊙ dwarf located at the distance of HD141272 (21 pc). Furthermore the photometry data rules out the pre-main sequence status, since the system is consistent with the ZAMS of the Pleiades. c© Year of publication WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim 1 Introduction HD141272 is a nearby G8 dwarf with a mass of 0.83+0.07 −0.03M⊙ (Nordström et al. 2004) located in the constellation Serpens Caput (αJ2000.0 = 15 h 48m 09.4s, δJ2000.0 = +01 ◦ 34′ 18′′). Its proper motion (µα cos δ = −176.19 ± 1.08mas/yr, µδ = −166.72 ± 1.13mas/yr) and parallax (π = 46.84± 1.05mas, i.e. 21 pc) are both well determined by the European astrometry satellite Hipparcos (Perryman et al. 1997). While Montes et al. (2001) list HD 141272 as a member of the Local asso- ciation with an age of ∼ 120Myr (Mart́ın et al. 2001), Fuhrmann (2004) suggested that this star belongs to the young Her-Lyr moving group, according to its UV- velocities. The age of some Her-Lyr members is esti- mated by Fuhrmann (2004) to approximately 100Myr (e.g. HR857, HD 82443, HD113449 and HR5829) which recently reached their main sequence position, while others seemed to be older than ∼ 200Myr (Fuhrmann 2004). Also Fuhrmann (2004) argued that HD141272, with an effective temperature of Teff = (5270±80)K, an absolute bolometric magnitudeMbol = (5.54±0.07)mag and metallicity of [Fe/H ] = (−0.08± 0.07) dex appears slightly too bright for its main se- quence position, indicating that it might be non single or young. ⋆ Based on observations obtained on La Silla in ESO programs 77.C-0572(A) and Calar Alto project number F06-3.5-016. ⋆⋆ E-mail: [email protected] On the other hand Gaidos, Henry & Henry (2000) measured a Fe corrected Li-equivalent width ofW6708 = 3.9 ± 1.9mÅ and a rotational velocity of v sin i ≈ 4.0 km/s, which might be too small for a 100Myr old star. Furthermore Chen et al. (2005) ob- served HD141272 using the infrared space telescope Spitzer and did not find any IR-excess at 24µm and 70µm indicating that HD141272 is not surrounded by an optically thick disk. Finally López-Santiago et al. (2006) revised the list of Her-Lyr members and candidates of Fuhrmann (2004) and classified HD141272 as an doubtful member, due to its lithium depletion. In our program we search for companions to Her- Lyr members and candidates and first results are pre- sented here. We found a co-moving companion of HD 141272 by a combination of archival first epoch im- ages and recent observations. We present our imaging, the astrometric data and reduction techniques in sec- tion 2 and 3, followed by a description of the spectro- scopic and photometric analysis of the new companion in section 4. The results are discussed in section 5. 2 Archival first epoch data Astrometry is an effective method to find companions of stars, by comparing two images taken with suffi- ciently long epoch difference. In order to find late-type stellar and substellar objects, we concentrate our search c© Year of publication WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim http://arxiv.org/abs/0704.0387v1 Astron. Nachr. / AN (Year of publication) 1 Fig. 1 POSS-I E image of HD141272 from 17 June 1950. The star is located at αJ2000.0 = 15 h 48m 09.4s, δJ2000.0 = +01 ◦ 34′ 18′′. A faint object is located in the north of HD141272, which is hardly recognizable due to the diffraction spikes of the primary star induced by saturation. With a pixel size of ∼ 10microns the pixel scale of the plate is ∼ 6.72 arcsec/pixel. on companions of young stars. Young objects are still in contraction and are brighter than older objects of the same mass hence, low mass objects are easier to detect. We found HD141272 in three epochs of the Super- COSMOS-Sky-Survey, namely a POSS-I (Palomar Ob- servatory Sky Survey) plate from 1950, as well as in UKST (United Kingdom Schmidt Telescope) infrared and red observations from 1981 and 1992. On all three plates we detected by eye inspection a faint object, located approximately 18 arcsec north of HD141272, which was not detected by the SuperCOSMOS ma- chine due to its small angular separation to the much brighter star and due to its overlap with the diffraction spike (Fig. 1). The diffraction spike of HD141272 intersects the northern object on all three plates hence, the detec- tion of this object would be inaccurate by means of most common detection techniques. Nevertheless, we obtained a position measurement of the companion can- didate on the POSS-I plate, using the Source Extractor package (Bertin & Arnouts 1996), included in the Star- link application GAIA (Gray et al. 2004). The source extractor uses thresholding and deblending of point- spread functions hence the method is more accurate than other detection techniques (e.g. Gaussian fitting) under the circumstances in Fig. 1. However, an system- atical error is possible, due to the perturbation of the primary’s spike. This error is larger in right accession than in declination and would affect the measurement of the position angle rather than the separation (see section 3, Fig. 4), due to the orientation of the system (Fig. 1 and 3). Due to its brightness HD141272 saturates the POSS- I plate. Furthermore the PSF (point spread function) is contaminated by the stray light of the companion candidate hence, position measurement via PSF cen- tering does not work sufficiently. We used the diffrac- tion spikes of the saturated primary to determine its position, since they are unaffected by the companion. We determined the intensity center of a spike taking ∼ 30 measurements for each spike using the data re- duction and analysis package ESO-MIDAS. The appli- cation of a linear regression gives the position of the star as intersection of the two spikes and leads to very small astrometric uncertainties (∆αH = 0.047 arcsec and ∆δH = 0.050 arcsec). In addition to the detection on the POSS-I plate HD 141272 and its companion-candidate are also de- tected in 2MASS images from observing epoch 2000. The 2MASS point source catalog (Cutri et al. 2003) lists the position of both objects with accurate astro- metric precision, see Tbl. 1. Equipped with these data we determined the proper motion of all stars in a 15 arcmin box around HD141272 which are detected at the POSS-I plate and listed in the 2MASS point source catalog (see Fig. 2). We de- rived the proper motion of all stars in the field by comparing the positions of all detected objects. The majority of sources only shows small proper motion following a normal distribution, since these stars are most probably at high distances. Using the Lilliefors test for normal distribution we derived the subsample of stars belonging to the background stars, since their proper motion follows a normal distribution (non mov- ing background stars). The standard deviation of the background stars gives the statistically derived proper motion error (σp.m., α = 8.8mas/yr, σp.m., δ = 6.8 mas/yr). Objects not belonging to the background stars are considered as companion candidates, if they are ly- ing within a 5-σ vicinity of HD141272 (ellipse in Fig. 2). Other objects are omitted, since these are either false detections or high-proper-motion stars moving in other directions. The proper motion of the nearby star HD141272 is clearly separated from the background stars. The companion candidate clearly shares the proper motion of HD 141272 and will be denoted HD141272B, here- after. Fig. 2 shows with high confidence (∼ 13σ) that HD141272A and B are co-moving over roughly 50 years. Due the above discussed astrometric uncertainties of HD 141272B this analysis gives a first indication of a new nearby young double star system www.an-journal.org c© Year of publication WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim 2 T. Eisenbeiss et al.: Low mass visual binaries in the solar neighbourhood: The case of HD141272 −300−250−200−150−100−50050 ∆ Ra/Year [mas/yr] standard error search radius HD141272 companion candidate non moving background stars Fig. 2 Proper motion plot of HD141272 (cross) and its companion candidate (circle) and non moving back- ground stars (upper left). X- and Y-axis show the change of the positions (in mas/yr). The plot is based on POSS-I Schmidt plate (17 June 1950) and 2MASS catalog data (29 April 2000). Error estimates are taken as 2-σ errors from the background stars. Data points lying outside the background stars and outside a 5-σ vicinity of HD141272 (large ellipse) are omitted, since these are either false detections or high proper motions stars moving in other directions. The statistical error of all data points is shown by the thick error cross in the lower left. The diagram shows the common proper motion of HD141272 and its new companion with a confidence of ∼ 13σ. Moreover, we used the non-moving background stars to estimate the positional error of the detections in the POSS-I plate. The mean of the distribution shows the systematic error of the POSS-I measurements (∆sys, α = −4.5mas/yr and ∆sys, δ = −4.9mas/yr as offset to (0, 0). The whole set of data points in Fig. 2 is shifted by that offset to correct for calibration errors between POSS-I and 2MASS data. The standard deviation shows the statistical measurement error (∆stat = σp.m.) hence, can be applied as standard detection error. The total detection error derived for the POSS-I plate is ∆α = 0.29 arcsec and ∆δ = 0.25 arcsec. The additional sys- tematic error for the companion candidate due to the diffraction spike of HD141272 is not included in this error analysis. Fig. 3 H-band image of HD 141272 and its compan- ion candidate taken with the near infrared camera Ω- Cass at the 3.5m telescope of the Calar Alto obser- vatory in Spain. The separation between HD141272 and its companion candidate is ∼ 17.8 arcsec at a position angle of ∼ 352.62◦ with a pixel scale of ∼ 0.2 arcsec/pixel. Note that HD141272 is slightly satu- rated. 3 Follow-up observations In order to get a third epoch on our astrometric re- sult and to detect or rule out further companions we observed HD141272 again in April 2006 (Fig. 3). We carried out H-band as well as narrow-band observations (1.644µm) with the near infrared camera Ω-Cass, in- stalled at the Cassegrain focus of the 3.5m telescope of the Calar Alto observatory in Spain. Ω-Cass is equipped with a 1024 × 1024 HgTeCd-detector with a pixel scale of ∼0.2 arcsec per pixel. We always used the short- est possible detector integration time (0.84 s) to limit strong saturation effects due to the bright star. For background subtraction we applied the standard jitter technique and chose 12 jitter-positions. On each jitter position 49 integrations (0.84 s) were co-added, yield- ing a total integration time in the H-band of 8.2min. All images were flatfielded with a skyflat image taken during twilight. The whole data reduction (background subtraction, flatfielding, and shift+add) was carried out with the ESO data-reduction package Eclipse (Devil- lard 2001). We calibrated our Ω-Cass image for relative astrom- etry, using the well known binary systems HIP 63322 and HIP 82817, which we observed during the same night and with the same instrumentation as our sci- ence image. Using the Hipparcos astrometry (Perry- c© Year of publication WILEY-VCH Verlag GmbH&Co.KGaA, Weinheim www.an-journal.org Astron. Nachr. / AN (Year of publication) 3 Table 1 Separation and position angle of the co-moving companion HD141272B relative to its primary HD141272A for all observing epochs. We also show the expected change of separation and position angle in case that the companion is a non-moving background source, derived with the well known proper and parallactic motion of the primary. epoch telescope/ pixel scale band sepobs. sepifback PAobs. PAifback [dd/mm/yyy] catalogs [arcsec] [arcsec] [arcsec] [◦] [◦] 17/07/1950 POSS-I 1.0 E (6442Å) 17.85±0.31 − 353.6±1.1 − 29/04/2000 2MASS 0.7 JHKS 17.83±0.150 26.92±0.33 352.42±0.48 14.61±0.75 20/04/2006 3.5m CA 0.2 H 17.851±0.041 28.12±0.31 352.62±0.18 16.48±0.68 man et al. 1997) and considering the maximal orbital motion of the calibration binaries we estimated the pixel scale (192 ± 0.43mas/pixel) and the orientation (−1.86±0.18 ◦) of the Ω-Cass images. This yields to the relative astrometric parameters of the system (Tbl. 1). For the detection of both objects we used the Gaussian centroiding technique, implemented in ESO-MIDAS. Further co-moving companions could be ruled out around HD141272 within an angular separation of ∼ 5 to 73 arcsec (1500AU of projected separation) with H- band magnitudes down to 18.3mag (S/N= 3). HD 141272A and B are separated by ∼ 17.8 arcsec (Fig. 3), hence the projected separation of the system is approximately 380AU and its orbital period can be estimated with Kepler’s third law to be roughly 7000 years (we use 0.83M⊙ for HD 141272A and 0.26M⊙ for B). During 56 years of epoch difference between the POSS-I and our H-band observation, this yields maximal orbital motion as large as ∼0.5 arcsec in sep- aration (edge-on orbit assumed) or ∼3◦ in position an- gle (face-on orbit assumed). Therefore, we derived the separation and the position angle of the companion for all three observing epochs which are summarized in Tab. 1. These results are also visualized in Fig. 4. Note that absolute calibrated astrometric data, derived for the POSS-I image as described in section 2, as well as catalog data from the 2MASS catalog is used in Fig. 4, while the third epoch data is based on relative astrom- etry, hence the uncertainties of that data point are sig- nificantly smaller. While the separation between HD141272A and B did not change during 56 years, we found a slight de- crease of its position angle. This effect is most likely due to the perturbation of the companions PSF by the diffraction spike of the primary (see section 2 and Fig. 1). Nevertheless Fig. 4 ensures the companionship of HD 141272B, since all data points are lying within the given error bars of the first epoch. 4 Photometry and Spectroscopy The infrared colors of both components of the new bi- nary system HD141272AB are listed in the 2MASS point source catalog, i.e. accurate J, H, and KS band 2.43 2.44 2.45 2.46 JD−2400000.5 2.43 2.44 2.45 2.46 JD−2400000.5 Fig. 4 Separation (sep) and position angle (PA) for HD 141272A and B from 1950 to 2006 (three data points). Upper lines show the changes of the proper- ties under the assumption HD141272B was a back- ground star (including parallactic motion of A) while the straight, opening lines give the range of the bi- nary movement, considering maximal orbital move- ment. While the separation stays approximately con- stant there is a change in the position angle, caused by the perturbation of the companions PSF due to the diffraction spikes of the primary. Table 2 2MASS Photometry of HD141272A and B Comp. J H KS [mag] [mag] [mag] A 5.991±0.021 5.610±0.027 5.501±0.018 B 9.298±0.020 8.725±0.055 8.456±0.023 photometry is available for the primary and its co- moving companion, which is summarized in Tab. 2. Ad- ditionally the I-band magnitude of both components (mI = 8.59 ± 0.02mag for A and mI = 10.572 ± 0.02mag for B) is measured in the second release of the DENIS database, while the accuracy for HD141272A is limited due to saturation effects, hence the given er- ror is probably underestimated. In order to obtain also unsaturated images of the primary we observed the binary system with Ω-Cass www.an-journal.org c© Year of publication WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim 4 T. Eisenbeiss et al.: Low mass visual binaries in the solar neighbourhood: The case of HD141272 in the FeII (1.644µm) narrow-band filter. Thereby, we used again the 12 point jitter pattern but co-added 15 integrations (4 s) per jitter position, yielding a total in- tegration time of 12min. The bright primary as well as its fainter co-moving companion are both well detected in this narrow-band image and their fluxes did not ex- ceed the linearity level of the Ω-Cass detector. Hence, we could use this image to derive the magnitude differ- ence between the primary star and its companion and obtained ∆HFeII = 3.166± 0.005mag, fully consistent with the magnitude difference derived from the 2MASS data in H-band (∆H = 3.115± 0.061mag.) Furthermore we acquired a low-resolution optical spectrum with EMMI at the NTT on La Silla to de- termine the spectral type of HD141272B and prove its common distance with HD141272A. The spectrum was taken in RILD and REMD mode covering a wavelength of 400-900nm with a resolution of R ≈ 3000 at 600nm. The data reduction followed the standard procedure for low-resolution optical spectra: After bias subtraction, flat fielding and wavelength calibration with a HeAr arc spectrum we corrected for the instrumental response and for telluric features using a spectrum of HR5501 taken at the same airmass as HD141272B. We determined the spectral type by comparing our spectrum with a standard sequence of M dwarfs in the same spectral range and with comparable spectral res- olution (Bochanski et al. 2006), see Fig. 5. The best fit resulted in a spectral type of M3.25 ± 0.25 which is consistent with a spectral type of M3.0± 0.5 deter- mined from the TiO5 spectral index of 0.49 following Cruz & Reid (2002). Adopting the latter spectral type as final we derived a spectrophotometric distance of 24.4±4.2 pc from the MJ relation given in Cruz & Reid (2002) and the J magnitude from 2MASS, assuming that the companion is on the Main sequence. The determined distance is in excellent agreement with the HIPPARCOS measured distance of 21.35±0.48 pc for HD 141272A, confirming their common distance. Hence, we call the companion HD141272B. 5 Conclusions With the astrometric data reduction and analysis tech- niques presented in this work, we could verify the com- mon proper motion of both components of the binary system HD141272AB during 56 years of epoch differ- ence between the first successful observation of this sys- tem on the POSS-I plates taken in July 1950 and our H-band imaging obtained with Ω-Cass in April 2006. Furthermore we obtained an optical spectrum of the companion and derived its spectral type to range be- tween M2.5V and M3.5V. The infrared apparent mag- nitudes of the co-moving companion are fully consis- tent with a M3 dwarf which is located at the distance 400 500 600 700 800 900 HD 141272 B Wavelength [nm] Fig. 5 Relative flux of the spectral sequence from M1 to M5 (Bochanski et al. 2006) in comparison to the EMMI spectrum of HD141272B, ranging from 400 to 900 nm. The resolutions are comparable (R ∼ 3000 for the EMMI spectrum and R ∼ 6000 for the standard spectra at 600 nm). HD 141272B shows good agree- ment with an M3 star. of HD141272A which finally confirms the companion- ship of this new binary system. The companion is an addition to the Catalog of Nearby Stars within 25 pc (Gliese & Jahreiß 1991). In order to get an estimation of the system age we compared the infrared photometry of HD 141272A and B with ≈ 1300 members of the Pleiades cluster which are listed in the WEBDA database (Mermilliod 1998). All objects are plotted in a J-K vs. MH color-magnitude diagram (Fig. 6). The colors of all objects are obtained from the 2MASS catalog and we derived the absolute H-Band magnitudes of all comparison stars using their 2MASS H-band photometry and a mean distance mod- ule of the Pleiades of 5.97mag (WEBDA database). The expected distance uncertainty of the cluster mem- bers which results in an uncertainty of their absolute H- band magnitudes was approximated with the angular diameter of the Pleiades cluster on the sky, assuming a similar extension of the cluster also in the radial direc- tion. The absolute H-band magnitudes of HD141272A and B are derived with 2MASS photometry and the Hipparcos parallax of the binary system. Compared to the Pleiades of the same J-K color HD141272A and B appear a little fainter, indicating that the system is already on the ZAMS, which is similar to the results of earlier works (Gaidos 1998; Wright et al. 2004). If we assume that both components of the binary system have already reached the ZAMS we can deter- mine the mass of the secondary using equation (11) from Kirkpatrick & McCarthy (1994) with the given c© Year of publication WILEY-VCH Verlag GmbH&Co.KGaA, Weinheim www.an-journal.org Astron. Nachr. / AN (Year of publication) 5 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 J−K [mag] 0.75 0.8 0.85 0.9 0.95 1 J−K [mag] HD 141272 A HD 141272 B Fig. 6 J-K vs. MH diagram for the Pleiades and HD141272A and B (rectangles symbolize the error boxes). The inserted plot shows HD141272B and the surrounding Pleiades stars drawn to a larger scale. The main sequence of the cluster can be seen although there are some outliers due to the mean distance module (5.97mag for Pleiades) applied. The mean error of the Pleiades is shown by the error cross in the lower left. HD 141272A and B appear a little fainter than Pleiades stars of the same J-K color. This indicates, that the system already reached the ZAMS. errors for the constants a and b and the range of the spectral type. We derived a mass of M∗ = 0.26 +0.07 −0.06M⊙. Future work should ascertain the age of the system and derive more properties of the M dwarf, which en- larges the list of nearby low mass stars bound in binary systems. Acknowledgements. We would like to thank the technical staff of the ESO NTT at La Silla as well as of the Centro Astronómico Hispano Alemán (CAHA) at Calar Alto for all their help and assistance in carrying out the observations. In addition we would like to thank John Bochanski, An- drew West, Suzanne Hawley and Kevin Covey for providing the electronic sequence of M-stars composite spectra. T.O.B. Schmidt acknowledges support from a Thur- ingian State Scholarship and from a Scholarship of the Evan- gelisches Studienwerk e.V. Villigst. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Process- ing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administra- tion and the National Science Foundation. We use imaging data from the SuperCOSMOS Sky Sur- vey, prepared and hosted by the Wide Field Astronomy Unit, Institute for Astronomy, University of Edinburgh, which is funded by the UK Particle Physics and Astron- omy Research Council. This research has made use of the VizieR catalogue access tool and the Simbad database, both operated at the Observatoire Strasbourg, as well as of the WEBDA database, operated at the Institute for Astronomy of the University of Vienna. The DENIS project has been partly funded by the SCI- ENCE and the HCM plans of the European Commission under grants CT920791 and CT940627. It is supported by INSU, MEN and CNRS in France, by the State of Baden- Württemberg in Germany, by DGICYT in Spain, by CNR in Italy, by FFwFBWF in Austria, by FAPESP in Brazil, by OTKA grants F-4239 and F-013990 in Hungary, and by the ESO C&EE grant A-04-046. Jean Claude Renault from IAP was the Project man- ager. Observations were carried out thanks to the contri- bution of numerous students and young scientists from all involved institutes, under the supervision of P. Fouqué, sur- vey astronomer resident in Chile. References Bertin, E., Arnouts, S.: 1996, A&AS 117, 393 Bochanski, J.J., West, A.A., Hawley, et al.: 2007, AJ 133, Chen, C.H., Patten, B.M., Werner, M.W., et al.: 2005, ApJ 634, 1372 Cruz, K.L., Reid, I.N.: 2002, AJ 123, 2828 Cutri, R.M. Skrutskie, M.F., van Dyk, S., et al.: 2003, 2MASS All Sky Catalog of point sources. (The IRSA 2MASS All-Sky Point Source Catalog, NASA/IPAC In- frared Science Archive. http://irsa.ipac.caltech.edu) Devillard, N.: 2001, in ASP Conf. Ser. 238: Astronomical Data Analysis Software and Systems X, ed. F. R. Harn- den, Jr., F. A. Primini, & H. E. Payne, 525–+ Fuhrmann, K.: 2004, AN 325,3 Gaidos, E.J.: 1998, PASP 110, 1259 Gaidos, E.J. Henry, G.W., Henry, S.M.: 2000, AJ 120, 1006 Gliese, W., Jahreisß, H.: 1991, Preliminary Version of the Third Catalogue of Nearby Stars, Tech. rep. Gray, N., Jenness, T. Allan, A., et al.: 2005, in Astronomical Society of the Pacific Conference Series, ed. P. Shopbell, M. Britton, & R. Ebert, 119–+ Kirkpatrick, J.D., Mc Carthy, Jr., D.W.: 1994, AJ 107,333 Lilliefors, H.W.: 1967, Journal of the American Statistical Association 62, 399 López-Santiago, J., Montes, D., Crespo-Chacón, I. et al.: 2006, ApJ 643, 1160 Mart́ın, E. L., Dahm, S., Pavlenko, Y.: 2001, ASP Conf. Ser. 245: Astrophysical Ages and Times Scales, 349 Mermilliod, J.-C.: ed. 1998, WEB Acces to the Open Clus- ter Database Montes, D., López-Santiago, J., Gálvez, M.C., et al.: 2001, MNRAS 328, 45 Nordström, B., Mayor, M., Andresen, J., et al.: 2004, A&A 418, 989 Perryman, M.A.C., Lindgren, L., Kovalevsky, J., et al.: 1997, A&A 323, L49 Wright, J.T., Marcy, G.W., Butler, R.P., Vogt, S.S.: 2004, ApJS 152, 261 www.an-journal.org c© Year of publication WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim http://irsa.ipac.caltech.edu Introduction Archival first epoch data Follow-up observations Photometry and Spectroscopy Conclusions

0704.0388

Sterile neutrinos at the CNGS

IFT-UAM/CSIC-07-16 Sterile neutrinos at the CNGS Andrea Donini, a Michele Maltoni, a Davide Meloni, b Pasquale Migliozzi, c Francesco Terranova d aInstituto F́ısica Teórica UAM/CSIC, Cantoblanco, E-28049 Madrid, Spain bI.N.F.N., Sezione di Roma I and Dip. Fisica, Univ. Roma “La Sapienza”, Pl. A. Moro 2, I-00185, Rome, Italy cI.N.F.N., Sezione di Napoli, I-80126, Naples, Italy dI.N.F.N., Laboratori Nazionali Frascati, Via E. Fermi 40, I-00044, Frascati, Italy PACS: 14.60.Pq, 14.60.Lm Abstract We study the potential of the CNGS beam in constraining the parameter space of a model with one sterile neutrino separated from three active ones by an O(eV2) mass- squared difference, ∆m2 . We perform our analysis using the OPERA detector as a reference (our analysis can be upgraded including a detailed simulation of the ICARUS detector). We point out that the channel with the largest potential to constrain the sterile neutrino parameter space at the CNGS beam is νµ → ντ . The reason for that is twofold: first, the active-sterile mixing angle that governs this oscillation is the less constrained by present experiments; second, this is the signal for which both OPERA and ICARUS have been designed, and thus benefits from an extremely low background. In our analysis we also took into account νµ → νe oscillations. We find that the CNGS potential to look for sterile neutrinos is limited with nominal intensity of the beam, but it is significantly enhanced with a factor 2 to 10 increase in the neutrino flux. Data from both channels allow us, in this case, to constrain further the four-neutrino model parameter space. Our results hold for any value of ∆m2 & 0.1 eV2, i.e. when oscillations driven by this mass-squared difference are averaged. We have also checked that the bound on θ13 that can be put at the CNGS is not affected by the possible existence of sterile neutrinos. http://arxiv.org/abs/0704.0388v2 1 Introduction The results of solar [1,2,3,4,5,6], atmospheric [7,8], reactor [9,10,11,12] and accelera- tor [13,14,15] neutrino experiments show that flavour mixing occurs not only in the hadronic sector, as it has been known for long, but in the leptonic sector as well. The full understanding of the leptonic mixing matrix constitutes, together with the dis- crimination of the Dirac/Majorana character of neutrinos and with the measurement of their absolute mass scale, the main goal of neutrino physics for the next decade. The experimental results point to two very distinct mass-squared differences, ∆m2 7.9 × 10−5 eV2 and |∆m2 | ≈ 2.4 × 10−3 eV2. On the other hand, only two out of the four parameters of the three-family leptonic mixing matrix UPMNS [16,17,18,19] are known: θ12 ≈ 34 ◦ and θ23 ≈ 43 ◦ [20]. The other two parameters, θ13 and δ, are still unknown: for the mixing angle θ13 direct searches at reactors [9,10,11] and three-family global analysis of the experimental data give the upper bound θ13 ≤ 11.5 ◦, whereas for the leptonic CP-violating phase δ we have no information whatsoever (see, however, Ref. [20]). The LSND data [21,22,23], on the other hand, would indicate a ν̄µ → ν̄e oscillation with a third neutrino mass-squared difference: ∆m2 ∼ 0.3 − 6 eV2, about two orders of magnitude larger than ∆m2 . Given the strong hierarchy among the solar, atmospheric and LSND mass-squared splittings, ∆m2 ≪ ∆m2 ≪ ∆m2 , it is not possible to explain all these data with just three massive neutrinos, as it has been shown with detailed calculations in Ref. [24]. A necessary condition to explain the whole ensemble of data in terms of neutrino oscillations is therefore the introduction of at least a fourth light neutrino state. This new light neutrino must be an electroweak singlet [18] in order to comply with the strong bounds on the Z0 invisible decay width [25,26]. For this reason, the LSND signal has often been considered as an evidence of the existence of a sterile neutrino. In recent years, global analyses of solar, atmospheric, short-baseline [27,28,29,30] exper- iments and LSND data have been performed to establish whether four-neutrino models can really reconcile the data and solve the puzzle [31,32,33,34,35,36,37,38]. The point is that providing a suitable mass-squared difference to each class of experiments is not enough: it is also necessary to show that the intrinsic structure of the neutrino mixing matrix is compatible with all the data. This turned out to be very hard to accomplish. In Ref. [39] it was shown that four-neutrino models were only marginally allowed, with best fit around ∆m2 ≃ 1 eV2 and sin2 2θLSND ≃ 10 −3. Generically speaking, the global analysis indicated that a single sterile neutrino state was not enough to reconcile LSND with the other experiments. For this reason, to improve the statistical compat- ibility between the LSND results and the rest of the oscillation data, models with two sterile neutrino states have been tested (see, for example, Ref. [40] and references therein). Although a slightly better global fit was achieved, a strong tension between the LSND data and the results from atmospheric and short-baseline experiments was still present. So far, the LSND signal has not been confirmed by any other experiment [41]. It is therefore possible that the LSND anomaly arises from some some yet unknown problem in the data set itself. To close the issue, the MiniBooNE collaboration [42] at FermiLab has recently performed a search for νµ → νe appearance with a baseline of 540 m and a mean neutrino energy of about 700 MeV. The primary purpose of this experiment was to test the evidence for ν̄µ → ν̄e oscillation observed at LSND with a very similar L/E range. No evidence of the expected signal has been found, hence ruling out once and for all the four-neutrino interpretation of the LSND anomaly. However, MiniBooNE data are themselves not conclusive: although no evidence for νµ → νe oscillation has been reported in the spectrum region compatible with LSND results, an unexplained excess has been observed for lower energy neutrinos. Furthermore, within a five-neutrino model this excess can be easily explained, and even reconciled with LSND and all the other appearance experiments [43]. On the other hand, a post-MiniBooNE global analysis including also disappearance data show that five-neutrino models suffer from the same problems as four-neutrino schemes, and in particular they are now only marginally allowed – a situation very similar to that of four-neutrino models before MiniBooNE data. Adding a third sterile neutrino 1 does not help [43], and in general global analyses seem to indicate that sterile neutrinos alone are not enough to reconcile all the data. Models with sterile neutrinos and exotic physics have been therefore proposed (see, for example, Ref. [46]). In summary, the present experimental situation is still confused. It is therefore worth- wile to understand if, aside of MiniBooNE, new neutrino experiments currently running or under construction can investigate the existence of sterile neutrinos separated from the active ones by O(eV2) mass-squared differences. In this paper we explore in detail the capability of the CNGS beam to perform this search. For definiteness we focus on the simplest case with only one extra sterile neutrino. Note that this model is perfectly viable once the LSND result is dropped, as it contains as a limiting case the usual three-neutrino scenario. Furthermore, it is easily generalizable by adding new sterile neutrino states, and it can be used as a basis for models with extra “sterile” states strongly decoupled from active neutrinos (such as in extra-dimensions models with a right-handed neutrino in the bulk [47]). The CNGS beam [48] has been built to test the (supposedly) dominant oscillation in at- mospheric neutrino data, νµ → ντ . In order to make possible τ production through CC interactions, the mean neutrino energy, 〈Eν〉 = 17 GeV, is much above the atmospheric oscillation peak for the CERN to Gran Sasso baseline, L = 732 Km. Two detectors are illuminated by the CNGS beam: OPERA (see Ref. [49] and refs. therein) will start data taking with the lead-emulsion target in 2007; ICARUS-T600 (see Ref. [50] and refs. therein) will start operating in 2008. Both detectors have been especially designed to look for τ ’s produced through νµ → ντ oscillation and to minimize the corresponding 1 A quite interesting scenario is, in our opinion, that in which three right-handed Majorana neutrinos are added to the three weakly-interacting ones. If the Majorana mass term M is O(eV), (3+3) light Majorana neutrinos are present at low-energy [44,45]. backgrounds. The expected number of τ events after signal selection in an experiment such as OPERA (after five years of data taking with nominal CNGS luminosity) is O(10) events with O(1) background event. At the CNGS distance and energy, neutrino oscillations mediated by an O(eV2) mass difference will appear as a constant term in the oscillation probability. In four-neutrino models, fluctuations induced by this term over the atmospheric νµ → ντ oscillation can be as large as 100% for specific points of the allowed parameter space. This is due to the fact that the leading angle for this oscillation is the less constrained one. The νµ → ντ channel, therefore, is extremely promising as a “sterile neutrino” smoking gun, as it has been commented elsewhere (see, for example, Refs. [51,52] and refs. therein). To test the model we will also make use of the νµ → νe channel. Notice that the background to this signal coming from τ → e decay is modified in four-neutrino models with respect to standard three-family oscillations. In fact, since νµ → ντ oscillations are depleted by active-sterile mixing with respect to standard ones, the τ → e background to νµ → νe oscillations gets depleted, too. A combined analysis of the two channels in four-neutrino models at the OPERA detector has been performed, taking into account properly all of the backgrounds. We stress, however, that the same analysis could be performed at ICARUS, as well. The previous considerations hold for any facility operating well beyond the kinematical threshold for τ production. In the specific case of the CNGS beam, the limited flux implies a modest improvement in the parameter space exclusion, see Sec. 6. An increase in the exposure of such facilities, however, would permit to improve the present bounds on the parameters of four-neutrino models and, in particular, to constrain the leading angle in νµ → ντ oscillations at the level of the other mixing parameters. The paper is organized as follows. In Sec. 2 we briefly review the main features of four-neutrino models and we introduce our parametrization of the mixing matrix. In Sec. 3 we compute the vacuum oscillation probabilities in the atmospheric regime and we review the present bounds on the active-sterile mixing angles. In Sec. 4 we recall the most relevant parameters of CNGS. In Sec. 5 we study theoretically the expectations of the νµ → ντ and νµ → νe channels at the CNGS. In Sec. 6 we present our results using these channels at the OPERA detector and the CNGS beam. Finally, in Sec. 7 we draw our conclusions. 2 Four neutrino mass schemes In four-neutrino models, one extra sterile state is added to the three weakly interacting ones. The relation between the flavor and the mass eigenstates is then described by a 4×4 unitary matrix U , which generalizes the usual 3×3 matrix UPMNS [16,17,18,19]. As stated in the introduction, in this work we only consider the case when the fourth mass eigenstate is separated by the other three by an O(eV2) mass-squared gap. There are six possible four-neutrino schemes, shown in Fig. 1, that can accommodate the results                   (3+1) (2+2) Fig. 1. The two classes of four–neutrino mass spectra, (3+1) and (2+2). from solar and atmospheric neutrino experiments and contain a third much larger ∆m2. They can be divided in two classes: (3+1) and (2+2). In the (3+1) schemes, there is a group of three close-by neutrino masses that is separated from the fourth one by the larger gap. In (2+2) schemes, there are two pairs of close masses separated by the large gap. While different schemes within the same class are presently indistinguishable, schemes belonging to different classes lead to very different phenomenological scenarios. A characteristic feature of (2+2) schemes is that the extra sterile state cannot be simultaneously decoupled from both solar and atmospheric oscillations. To understand why, let us define i∈ sol |Usi| 2 and cs = j ∈ atm |Usj| 2 (1) where the sums in i and j run over mass eigenstates involved in solar and atmospheric neutrino oscillations, respectively. Clearly, the quantities ηs and cs describe the fraction of sterile neutrino relevant for each class of experiment. Results from atmospheric and solar neutrino data imply that in both kind of experiments oscillation takes place mainly between active neutrinos. Specifically, from Fig. 46 of Ref. [20] we get ηs ≤ 0.30 and cs ≤ 0.36 at the 3σ level. However, in (2+2) schemes unitarity implies ηs + cs = 1, as can be easily understood by looking at Fig. 1. These models are therefore ruled out at a very high confidence level [53], and in the rest of this work we will not consider them anymore. On the other hand, (3+1) schemes are not affected by this problem. Although the experimental bounds on ηs and cs quoted above still hold, the condition ηs + cs = 1 no longer applies. For what concerns neutrino oscillations, (3+1) models are essentially unfalsifiable, since they reduce to the conventional three-neutrino scenario when the mixing between active and sterile states are small enough. The mixing matrix U can be conveniently parametrized in terms of six independent rotation angles θij and three (if neutrinos are Dirac fermions) or six (if neutrinos are Majorana fermions) phases δi. In oscillation experiments, only the so-called “Dirac phases” can be measured, the effect of the “Majorana phases” being suppressed by factors of mν/Eν . The Majorana or Dirac nature of neutrinos can thus be tested only in ∆L = 2 transitions such as neutrino-less double β-decay [54] or lepton number violating decays [25]. In the following analysis, with no loss in generality, we will restrict ourselves to the case of 4 Dirac-type neutrinos only. A generic rotation in a four-dimensional space can be obtained by performing six dif- ferent rotations along the Euler axes. Since the ordering of the rotation matrices Rij (where ij refers to the plane in which the rotation takes place) is arbitrary, plenty of different parametrizations of the mixing matrix U are allowed. The large parameter space (6 angles and 3 phases, to be compared with the standard three-family mixing case of 3 angles and 1 phase) is however reduced to a subspace whenever some of the mass differences become negligible. If the eigenstates i and j are degenerate, ro- tations in the ij-plane become unphysical and the corresponding mixing angle should drop from oscillation probabilities. If the matrix Rij is the rightmost one the angle θij automatically disappears, since the matrix commutes with the vacuum hamilto- nian. The parameter space gets therefore reduced to the physical angles and phases. If a different ordering of the rotation matrices is taken, no angle explicitly disappears from the oscillation formulas, but the physical parameter space is still smaller than the original one. In this case, a parameter redefinition is needed to reduce the parameter space to the observable sector. In Refs. [55,56] it was shown how the one-mass domi- nance (∆sol → 0 and ∆atm → 0, where ∆ = ∆m 2L/4E [57]) and two-mass dominance (∆sol → 0) approximations can be implemented in a transparent way (in the sense that only the physical parameters appear in oscillation probabilities) using a parametriza- tion in which rotations are performed in the planes corresponding to smallest mass difference first: USBL = R14(θ14) R24(θ24) R34(θ34) R23(θ23, δ3) R13(θ13, δ2) R12(θ12, δ1) . (2) This parametrization was shown to be particularly useful when maximizing oscilla- tions driven by a O(eV2) mass difference. The analytical expressions for the oscillation probabilities in the (3+1) model in the one-mass dominance approximation in this parametrization have been presented in Ref. [51]. In this paper, however, we are interested in a totally different regime: the “atmospheric regime”, with oscillations driven by the atmospheric mass difference, ∆m2 L/E ∼ π/2. We will then make use of the following parametrization, adopted in Ref. [43]: Uatm = R34(θ34) R24(θ24) R23(θ23, δ3) R14(θ14) R13(θ13, δ2) R12(θ12, δ1) . (3) It is convenient to put phases in R12 (so that it automatically drops in the two-mass dominance regime) and R13 (so that it reduces to the “standard” three-family Dirac phase when sterile neutrinos are decoupled). The third phase can be put anywhere; we will place it in R23. Note that in the one-mass dominance regime all the phases disappear from the oscillation probabilities. 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 Fig. 2. Allowed regions at 90%, 95%, 99% and 3σ CL in the (θ13, θ14) plane (left) and in the (θ24, θ34) plane (right) from the results of present atmospheric, reactor and LBL neutrino experiments. The undisplayed parameters θ23 and δ3 are marginalized. 3 Oscillation probabilities and allowed parameter space Let us first consider νe disappearance at L/E such that ∆sol can be safely neglected with respect to ∆atm and ∆sbl. We get for this probability in vacuum: Pee = 1− sin 2 2θ14 sin ∆sblL sin2 2θ13 sin ∆atmL , (4) where cij = cos θij and sij = sin θij . It is clear from Eq. (4) that reactor experiments such as Bugey and Chooz can put stringent bounds to θ13 and θ14, in this parametriza- tion. This is depicted in Fig. 2(left), where 90%, 95%, 99% and 3σ CL contours in the (θ13 − θ14)-plane are shown for ∆sol → 0 and ∆m = 2.4 × 10−3 eV2. The third mass difference, ∆m2 , is free to vary above 0.1 eV2. Notice that the νe disappearance probability does not depend on θ23, θ24 and θ34. It can be clearly seen that the three- family Chooz bound on θ13 is slightly modulated by θ14. Both angles, however, cannot be much larger than 10◦. We will therefore expand in these two parameters to deduce the other relevant oscillation probabilities. At the CNGS beam atmospheric oscillations are large, solar oscillations can be ne- glected and O(eV2) oscillations are extremely fast and can be averaged. It is useful to write down the oscillation probability (in vacuum) at typical atmospheric L/E, in the approximation ∆sol → 0, ∆sbl → ∞. In this regime: P (να → νβ) = δαβ − 4ℜ β3 (δαβ − U α3Uβ3 − U α4Uβ4) δαβUα4U β4 − |Uα4| 2|Uβ4| α4Uβ4 sin∆23L , where + stands for neutrinos and − for antineutrinos, respectively. Up to second order in θ13 and θ14 we get for the νµ disappearance oscillation probability: Pµµ = 1− 2c (1− c2 )− s2 − 2c3 s23(1− 2s )s13s14s24 cos(δ2 − δ3) ∆atmL A “negative” result in an atmospheric L/E νµ disappearance experiment (such as, for example, K2K), in which νµ oscillations can be very well fitted in terms of three-family oscillations, will put a stringent bound on the mixing angle θ24. The bound from such experiments on θ24 can be seen in Fig. 2(right), where 90%, 95%, 99% and 3σ CL contours in the (θ24 − θ34)-plane are shown for ∆sol → 0 and ∆m = 2.4× 10−3 eV2. The third mass difference, ∆m2 , is free to vary above 0.1 eV2. The mixing angles not shown have been fixed to: θ23 = 45 ◦; θ13 = θ14 = 0 (in this hypothesis, Pµµ does not depend on phases). Notice that the νµ disappearance probability does not depend on From the figure, we can see that θ24 cannot be much larger than 10 ◦, either. We will consider, therefore, the three mixing angles θ13, θ14 and θ24 being of the same order and expand in powers of the three. At second order in θ13, θ14 and θ24, we get: Pµµ = 1− 2s − 4s2 (1− 2s2 )− s2 ∆atmL . (7) Since both νe and νµ disappearance do not depend on θ34, we should ask which mea- surements give the upper bound to this angle that can be observed in Fig. 2(right). This is indeed a result of indirect searches for νµ → νs conversion in atmospheric exper- iments, using the different interaction with matter of active and sterile neutrinos. This can be understood from the (vacuum) νµ → νs oscillation probability at atmospheric L/E for which, at second order in θ13, θ14 and θ24, we get: Pµs = 2c sin2 2θ23(c + 2c34 sin 2θ23s34 s24(1− 2s ) cos δ3 + 2s23s13s14 cos δ2 ∆atmL ± c34 sin 2θ23s24s34 sin δ3 sin∆atmL . As it can be seen, the bound on θ34 arises from a measurement of spectral distortion (i.e., from the “atmospheric” term proportional to sin2∆atmL/2). On the other hand, bounds on θ13, θ14 and θ24 are mainly drawn by a flux normalization measurement. As a consequence, the bound on θ34 that we can draw by non-observation of νµ → νs oscillation in atmospheric experiments is less stringent than those we have shown before. For this reason, θ34 can be somewhat larger than θ13, θ14 and θ24, but still bounded to be below 40◦. In the following, we will expand in powers of the four mixing angles θ13, θ14, θ24 and θ34, that will be considered to be comparably small. Up to fourth-order in θ13, θ14, θ24 and θ34, the νµ → νe appearance probability in the atmospheric regime is: Pµe = 4 [1− s2 ] + s23s13s14s24 cos(δ2 − δ3) ∆atmL ± 2s23s13s14s24 sin(δ2 − δ3) sin∆atmL+ 2s Eventually, the νµ → ντ appearance probability up to fourth-order in θ13, θ14, θ24 and θ34 in the atmospheric regime is: Pµτ = 2s sin2 2θ23[c − 4 sin 2θ23s13s14[s23s34 cos δ2 + c23s24 cos(δ2 − δ3)] + 2 sin 2θ23s24s34c c34[c − 2c2 ] cos δ3 ∆atmL ∓ sin 2θ23s24s34c c34 sin δ3 sin∆atmL . As it was shown in Refs. [51,52], the νµ → ντ appearance channel is a good place to look for sterile neutrinos. This can be understood as follows: consider the νµ → ντ three-family oscillation probability in the atmospheric regime, up to fourth-order in P 3νµτ = Pµτ (θi4 = 0) ≃ c sin2 2θ23 sin ∆atmL . (11) The genuine active-sterile neutrino mixing effects are: ∆Pµτ ≡ Pµτ − P ) sin2 2θ23 + 2 sin 2θ23(1− 2s )s24s34 cos δ3 ∆atmL ∓ sin 2θ23s24s34 sin δ3 sin∆atmL+ . . . that is second-order in small angles θ13, θ14, θ24 and θ34. We would get a similar result for νµ disappearance, also. On the other hand, computing the corresponding quantity in the νµ → νe channel, we get: ∆Pµe ≡ Pµe − Pµe(θi4 = 0) = s23s13s14s24 cos(δ2 − δ3) sin ∆atmL ± 2s23s13s14s24 sin(δ2 − δ3) sin∆atmL+ . . . that is third-order in the same parameters. Notice, eventually, that all oscillation probabilities start with an energy-independent term and are, therefore, non-vanishing for L = 0, a result of our assumption that ∆sbl → ∞. 0 10 20 30 40 50 Eν(GeV) anti−νµ anti−νe Fig. 3. CNGS neutrino fluxes (in arbitrary units) as a function of the neutrino energy. Both muon and electron neutrino fluxes are illustrated. 4 The CNGS facility The CNGS is a conventional neutrino beam in which neutrinos are produced by the decay of secondary pions and kaons, obtained from collisions of 400 GeV protons from the CERN-SPS onto a graphite target. The resulting neutrinos are aimed to the under- ground Gran Sasso Laboratory (LNGS), located at 730 Km from CERN. This facility provided the first neutrinos in August 2006 [49]. Differently from other long baseline experiments, the neutrinos from CNGS can be exploited to search directly for νµ → ντ oscillations, since they have a mean energy well beyond the kinematic threshold for τ production. Moreover, the prompt ντ contamination (mainly fromDs decays) is negligi- ble. The expected νe contamination is also relatively small compared to the dominant νµ component, thus allowing for the search of sub-dominant νµ → νe oscillations through an excess of νe CC events. The energy spectra of the CNGS neutrino beam are shown (in arbitrary units) in Fig. 3 [58]. In the present paper we assume the nominal intensity for the CNGS, corresponding to 4.5× 1019 pot/year. OPERA has been designed to search for τ appearance through identification of the ντ CC interaction on an event-by-event basis. In particular, τ ’s are tagged identifying explicitly their decay kink through high resolution nuclear emulsions interleaved with lead sheets. For this detector, we can take advantage of the detailed studies of the νµ → ντ signal (see Ref. [59]) and of the νµ → νe signal (see Ref. [60]). The total non-oscillated CC event rates for a 1 Kton lead target with a neutrino flux νµ ν̄µ νe ν̄e 669.0 13.7 5.9 0.3 Table 1 Nominal performance of the CNGS reference beam [58]. The total non-oscillated CC event rates are calculated assuming 1019 pot and 1 Kton lead target mass. normalized to 1019 pot are shown in Tab. 1 and are evaluated according to dφνα(E) σνα(E) dE , (14) in which φνα is the flux of the neutrino flavour να and σνα the corresponding cross section on lead. 5 Appearance channels at the CNGS 5.1 νµ → ντ oscillations Since the CNGS experiments have been designed to search for νµ → ντ oscillation in the parameter region indicated by the atmospheric neutrino data, we can take full advantage of them in order to constrain (and, possibly, study) the four-family parameter space. The number of taus from νµ → ντ oscillations is given by the convolution of the νµ flux dφνµ(E)/dE with the ντ charged-current cross-section on lead, σ (E), weighted by the νµ → ντ oscillation probability, Pµτ (E), times the efficiency for the OPERA detector, εµτ : Nµτ = A dφνµ(E) Pµτ (E)σ (E) εµτ dE . (15) A is a normalization factor which takes into account the target mass and the nor- malization of the νµ flux in physical units. Specializing our analysis for the OPERA detector, we have considered an overall efficiency εµτ ∼ 13%, [59]. This efficiency takes into account that OPERA is able to exploit several decay modes of the final state τ , using both so-called short and long decays. The dominant sources of background for the νµ → ντ signal are charm decays and hadronic reinteractions. Both of them only depend on the total neutrino flux and not on the oscillation probabilities. The OPERA experiment at the CNGS beam has been designed precisely to measure this channel, and thus the corresponding backgrounds are extremely low. In Tab. 2 we report the expected number of τ events in the OPERA detector, according to Eq. (15), for different values of θ13, θ14, θ24 and θ34. Input points have been chosen according to the allowed regions in the parameter space shown in Sec. 3. The other (θ13; θ14; θ24; θ34) Nτ background (θ13; θ14; θ24; θ34) Nτ background (5◦; 5◦; 5◦; 20◦) 8.9 1.0 (10◦; 5◦; 5◦; 20◦) 8.5 1.0 (5◦; 5◦; 5◦; 30◦) 6.9 1.0 (10◦; 5◦; 5◦; 30◦) 6.5 1.0 (5◦; 5◦; 10◦; 20◦) 8.3 1.0 (10◦; 5◦; 10◦; 20◦) 7.9 1.0 (5◦; 5◦; 10◦; 30◦) 10.5 1.0 (10◦; 5◦; 10◦; 30◦) 10.3 1.0 3 families 15.1 1.0 3 families 14.4 1.0 Table 2 Event rates and expected background for the νµ → ντ channel in the OPERA detector, for different values of θ14, θ24 and θ34 in the (3+1) scheme. The other unknown angle, θ13 has been fixed to: θ13 = 5 ◦, 10◦. The CP-violating phases are: δ1 = δ2 = 0; δ3 = 90 ◦. As a reference, the expected value in the case of standard three-family oscillation (i.e., for θi4 = 0) is shown for maximal CP-violating phase δ. The rates are computed according to Eq. (15). parameters are: θ12 = 34 ◦; θ23 = 45 ◦; ∆m2 = 7.9× 10−5 eV2; ∆m2 = 2.4× 10−3 eV2 and ∆m2 = 1 eV2 (all mass differences are taken to be positive). Eventually, phases have been fixed to: δ1 = δ2 = 0; δ3 = 90 ◦. The expected background is also shown. Rates refer to a flux normalized to 4.5× 1019 pot/year (the nominal intensity of the CNGS), an active lead target mass of 1.8 Kton and 5 years of data taking. For comparison, we also report the expected number of events in the usual 3-family scenario. As it can be seen, in most part of the parameter space we expect a significant depletion of the signal with respect to standard three-neutrino oscillations. However, the differ- ence between (3+1) model νµ → ντ oscillations and standard ones is much bigger than the expected background. A good signal/noise separation can therefore be used to test the model. 5.2 νµ → νe oscillations The number of electrons from the νµ → νe oscillation is given by the convolution of the νµ flux dφνµ(E)/dE with the νe charged-current cross-section on lead, σ weighted by the νµ → νe oscillation probability, Pµe(E), times the efficiency for the OPERA detector, εµe(E) [60]: Nµe = A dφνµ(E) Pµe(E)σ (E) εµe(E) dE , (16) where A is defined as above. The overall signal efficiency εµe is the convolution of the kinematic efficiency εkinµe (that ranges from 60% to 80% for neutrino energies between 5 to 20 GeV) and several (nearly factorizable) contributions. Among them, the most relevant are trigger efficiencies, effects due to fiducial volume cuts, vertex and brick finding efficiencies and the electron identification capability. They result in a global constant factor εfactµe ∼ 48%. (θ13; θ14; θ24; θ34) Ne ν µ τ → e ν (5◦; 5◦; 5◦; 20◦) 3.5 19.4 5.3 2.8 0.9 (5◦; 5◦; 5◦; 30◦) 3.5 19.4 5.3 2.1 0.9 (5◦; 5◦; 10◦; 20◦) 2.4 19.4 5.3 2.3 0.9 (5◦; 5◦; 10◦; 30◦) 2.4 19.4 5.3 2.4 0.9 3 families 3.7 19.7 5.3 4.6 0.9 (10◦; 5◦; 5◦; 20◦) 10.6 19.4 5.3 2.7 0.9 (10◦; 5◦; 5◦; 30◦) 10.4 19.4 5.3 2.0 0.9 (10◦; 5◦; 10◦; 20◦) 8.8 19.4 5.3 2.2 0.9 (10◦; 5◦; 10◦; 30◦) 8.6 19.4 5.3 2.4 0.9 3 families 15.1 19.7 5.3 4.8 0.9 Table 3 Event rates and expected background for the νµ → νe channel in the OPERA detector, for different values of θ14, θ24 and θ34 in the (3+1) scheme. The other unknown angle, θ13, has been fixed to: θ13 = 5 ◦, 10◦. The CP-violating phases are: δ1 = δ2 = 0; δ3 = 90 ◦. As a reference, the expected value in the case of standard three-family oscillation(i.e., for θi4 = 0) is shown for maximal CP-violating phase δ. The rates are computed according to Eq. (16). Backgrounds have been computed following Ref. [60]. The dominant sources of background to the νµ → νe signal are, in order of importance: (1) νe beam contamination; (2) fake electrons due to π0 decays from νµ NC interactions; (3) electrons produced through τ decay, where the τ comes from νµ → ντ oscillations; (4) CC νµ events where the muon is lost and a track mimics an electron. Backgrounds (1), (2) and (4) depend very little on the oscillation parameters. On the other hand, the τ → e background depends strongly on the active-sterile mixing angles. As we have seen in Sec. 5.1, in the allowed region of the parameter space νµ → ντ oscillations are significantly depleted with respect to the standard three-neutrino ones. As a consequence, this background gets depleted, too. In Tab. 3 we report the expected number of electrons in the OPERA detector, according to Eq. (16), for different values of θ13, θ14, θ24 and θ34. Input points have been chosen according to the allowed regions in the parameter space shown in Sec. 3. The other parameters are: θ12 = 34 ◦; θ23 = 45 ◦; ∆m2 = 7.9× 10−5 eV2; ∆m2 = 2.4× 10−3 eV2 and ∆m2 = 1 eV2 (all mass differences are taken to be positive). Eventually, phases have been fixed to: δ1 = δ2 = 0; δ3 = 90 ◦. Backgrounds have been computed accordingly to Ref. [60]. Rates refer to a flux normalized to 4.5×1019 pot/year (the nominal intensity of the CNGS), an active lead target mass of 1.8 Kton and 5 years of data taking. For comparison, we also report the expected number of events in the usual 3-family scenario. As it can be seen from Tab. 3, the difference between the (3+1) model and the stan- dard three-neutrino oscillations are smaller in this channel than in the νµ → ντ one. Moreover, they linearly depends on θ13, as it is clear from Eq. (13). For θ13 = 5 ◦, this channel will be of no help to test the allowed parameter space of the (3+1) model. On the other hand, for θ13 saturating the Chooz-Bugey bound, both νµ → ντ and νµ → νe might cooperate. However, notice that backgrounds to this signal are much larger than the difference between (3+1) model and standard three-neutrino oscillations for any value of θ13. 6 Sensitivity to (3 + 1) sterile neutrinos at OPERA In this section we study the sensitivity to θ13 and to the active-sterile mixing angles θ14, θ24 and θ34 at the CNGS beam, using both the νµ → ντ and νµ → νe appearance channels at the OPERA detector. In the rest of this section, the known three-family subspace angles have been fixed to: θ12 = 34 ◦; θ23 = 45 ◦. The mass differences have been fixed to: ∆m2 = 7.9× 10−5 eV2 and ∆m2 = 2.4× 10−3 eV2. The CP-violating phases δ1 and δ2 have been kept fixed to δ1 = δ2 = 0. On the contrary, the CP-violating phase δ3 is fixed to two values: δ3 = 0 or 90 ◦. Notice that this phase is still present in the oscillation probabilities even when θ12 and θ13 vanish, see Eq. (10). At atmospheric L/E, oscillations driven by an O(eV2) mass difference are averaged. We have checked that our results apply for any value of ∆m2 ≥ 0.1 eV2. In Fig. 4 we show the sensitivity limit at 99% CL in the (θ13, θ14) plane (left) and in the (θ24, θ34) plane (right) from a null result of the OPERA experiment, assuming 1, 2, 3, 5 and 10 times the nominal intensity of 4.5 × 1019 pot/year. The coloured regions show the present bounds at 90% and 99% CL. We assume θ23 = 45 ◦ and δ3 = 0 ◦ (top) or δ3 = 90 ◦ (bottom). The sensitivity is defined as the region for which a (poissonian) 2 d.o.f.’s χ2 is compatible with a “null result” at the 99% CL. We refer to “null result” when θ13 and the three active-sterile mixing angles, θ14, θ24 and θ34 vanish simultaneously. Both νµ → ντ and νµ → νe oscillations have been considered, with the corresponding backgrounds treated properly as in Sec. 5. An overall systematic error of 10% has been taken into account. In the left panels of Fig. 4 we can see that OPERA can improve only a little the bound on θ13 after 5 years of data taking working at nominal CNGS beam intensity, both for δ3 = 0 (top panel) or δ3 = 90 ◦ (bottom panel). Increasing the nominal intensity, however, a significant improvement on the bound is achieved for any value of θ14. Notice that the limit on θ14 is almost unaffected by the OPERA data. This is because for the νµ → ντ and νµ → νe oscillation probabilities at atmospheric L/E, the θ14-dependence always arises at third-order in the small parameters θ13, θ14, θ24 and θ34 (see Eqs. (9) and (10) for the explicit expression in the adopted parametrization, Eq. (3)). On the contrary, the θ13-, θ24- and θ34-dependences in the same oscillation probabilities are quadratic in the small parameters. In case of vanishing active-sterile mixing angles, 0 2 4 6 8 10 12 14 = 45°, δ = 0° 0 2 4 6 8 10 12 14 = 45°, δ = 0° 0 2 4 6 8 10 12 14 = 45°, δ = 90° 0 2 4 6 8 10 12 14 = 45°, δ = 90° × 5× 10 Fig. 4. Sensitivity limit at 99% CL in the (θ13, θ14) plane (left) and in the (θ24, θ34) plane (right) from a null result of the OPERA experiment, assuming 1, 2, 3, 5 and 10 times the nominal intensity of 4.5 × 1019 pot/year. The coloured regions show the present bounds at 90% and 99% CL. We assume θ23 = 45 ◦ and δ3 = 0 ◦ (top) or δ3 = 90 ◦ (bottom). θi4 = 0, see Ref. [60]. In the right panels of Fig. 4 the sensitivity of OPERA to θ24 and θ34 is shown. First of all, notice that the sensitivity is strongly affected by the intensity of the beam. No improvement on the existing bounds on these two parameters is achieved after 5 years of data taking at nominal CNGS beam intensity, for any of the considered value of δ3. Already with a doubled flux intensity, some sensitivity to θ24, θ34 is achievable. The sensitivity enhancement strongly depends on the value of the CP-violating phase δ3, however. For δ3 = 0, OPERA can exclude a small part of the 99% CL allowed region, only. On the other hand, for δ3 = 90 ◦ twice the nominal CNGS flux suffices 0 2 4 6 8 10 12 14 = 45°, δ = 0° 0 2 4 6 8 10 12 14 = 45°, δ = 0° 0 2 4 6 8 10 12 14 = 45°, δ = 90° 0 2 4 6 8 10 12 14 = 45°, δ = 90° Fig. 5. Sensitivity limit at 99% CL in the (θ13, θ14) plane (left) and in the (θ24, θ34) plane (right) from the combined analysis of present data and a null result of the OPERA experiment, assuming 1, 2, 3, 5 and 10 times the nominal intensity of 4.5 × 1019 pot/year. The coloured regions show the present bounds at 90% and 99% CL. We assume θ23 = 45 ◦ and δ3 = 0 (top) or δ3 = 90 ◦ (bottom). to put a bound on θ34 ≤ 25 ◦ for θ24 ≥ 4 ◦ at 99% CL. For maximal CP-violating δ3, increasing further the CNGS flux can significantly constrain the (θ24, θ34) allowed parameter space. Notice, eventually, the strong correlation between θ24 and θ34 in the right panels of Fig. 4. This is an indication that the dominant channel that constrains these angles is νµ → ντ . As it can be seen in Eq. (10), the two angles always appear in combination, with an approximate exchange symmetry θ24 ↔ θ34. The allowed regions at 99% CL in the (θ13, θ14) plane (left) and in the (θ24, θ34) plane (right) from the combined analysis of present data and a null result of the OPERA experiment after 5 years of data taking (assuming 1, 2, 3, 5 and 10 times the nominal CNGS intensity of 4.5 × 1019 pot/year) are eventually shown in Fig. 5. The coloured regions refer to the present bounds at 90% and 99% CL, for θ23 = 45 ◦ and δ3 = 0 ◦ (top) or δ3 = 90 ◦ (bottom). As it can be seen, the sensitivity of OPERA strongly benefits from the complementary information on the neutrino parameters provided by other experiments. In this case, even with the nominal beam intensity the extension of the allowed regions is reduced by a moderate but non-negligible amount. 7 Conclusions The results of atmospheric, solar, accelerator and reactor neutrino experiments show that flavour mixing occurs not only in the quark sector, as it has been known for long, but also in the leptonic sector. Experimental data well fit into a three-family scenario. The existence of new “sterile” neutrino states with masses in the eV range is not excluded, however, provided that their couplings with active neutrinos are small enough. In this paper, we have tried to test the potential of the OPERA experiment at the CNGS beam to improve the present bounds on the parameters of the so-called four- neutrino models. The model, in which only one sterile neutrino is added to the three active ones responsible for solar and atmospheric oscillations, is the minimal extension of the standard three-family oscillation scenario. We have determined the presently allowed regions for all active-sterile mixing angles and studied the OPERA capability to constrain them further using both the νµ → νe and νµ → ντ channels. We have performed our analysis using the OPERA detector as a reference. It can be extended including a detailed simulation of the ICARUS detector at the CNGS beam. Our conclusions are the following: if the OPERA detector is exposed to the nominal CNGS beam intensity, a null result can improve a bit the present bound on θ13, but not those on the active-sterile mixing angles, θ14, θ24 and θ34. If the beam intensity is increased by a factor 2 or beyond, not only the sensitivity to θ13 increases accordingly, but a significant sensitivity to θ24 and θ34 is achievable. The (θ24, θ34) sensitivity strongly depends on the value of the CP-violating phase δ3, however, with stronger sensitivity for values of δ3 approaching π/2. Only a marginal improvement is achievable on the bound on θ14, that should be constrained by high-intensitiy νe disappearance experiments. Notice that our results hold for any value of ∆m2 ≥ 0.1 eV2, i.e. in the region of L/E for which oscillations driven by this mass difference are effectively averaged. Acknowledgements We acknowledge E. Fernández-Mart́ınez, P. Hernández, J. López-Pavón, M. Sorel and P. Strolin for useful discussions and comments. We thank T. Schwetz for pointing out to us an error in the first version of the paper and for useful comments on it. The work has been partially supported by the E.U. through the BENE-CARE networking activity MRTN-CT-2004-506395. A.D. received partial support from CiCYT through the project FPA2006-05423. M.M. received partial support from CiCYT through the project FPA2006-01105 and the MCYT through the Ramón y Cajal program. A.D. and M.M. acknowledge also financial support from the Comunidad Autónoma de Madrid through the project P-ESP-00346. D.M. would like to thank CERN, where part of this work has been accomplished. References [1] B. T. Cleveland et al., Astrophys. J. 496 (1998) 505. [2] J. N. Abdurashitov et al. [SAGE Collaboration], Phys. Rev. C 60 (1999) 055801 [arXiv:astro-ph/9907113]. [3] W. Hampel et al. [GALLEX Collaboration], Phys. Lett. B 447 (1999) 127. [4] S. Fukuda et al. [Super-Kamiokande Collaboration], Phys. Rev. Lett. 86 (2001) 5651 [arXiv:hep-ex/0103032]. [5] Q. R. Ahmad et al. [SNO Collaboration], Phys. Rev. Lett. 87 (2001) 071301 [arXiv:nucl-ex/0106015]. [6] S. N. Ahmed et al. [SNO Collaboration], Phys. Rev. Lett. 92 (2004) 181301 [arXiv:nucl-ex/0309004]. [7] Y. Fukuda et al. [Super-Kamiokande Collaboration], Phys. Rev. Lett. 81 (1998) 1562 [arXiv:hep-ex/9807003]. [8] M. Ambrosio et al. [MACRO Collaboration], Phys. Lett. B 517 (2001) 59 [arXiv:hep-ex/0106049]. [9] M. Apollonio et al. [CHOOZ Collaboration], Phys. Lett. B 466 (1999) 415 [arXiv:hep-ex/9907037]. [10] M. Apollonio et al. [CHOOZ Collaboration], Eur. Phys. J. C 27 (2003) 331 [arXiv:hep-ex/0301017]. [11] F. Boehm et al., Phys. Rev. D 64 (2001) 112001 [arXiv:hep-ex/0107009]. [12] K. Eguchi et al. [KamLAND Collaboration], Phys. Rev. Lett. 90 (2003) 021802 [arXiv:hep-ex/0212021]. http://arxiv.org/abs/astro-ph/9907113 http://arxiv.org/abs/hep-ex/0103032 http://arxiv.org/abs/nucl-ex/0106015 http://arxiv.org/abs/nucl-ex/0309004 http://arxiv.org/abs/hep-ex/9807003 http://arxiv.org/abs/hep-ex/0106049 http://arxiv.org/abs/hep-ex/9907037 http://arxiv.org/abs/hep-ex/0301017 http://arxiv.org/abs/hep-ex/0107009 http://arxiv.org/abs/hep-ex/0212021 [13] M. H. Ahn et al. [K2K Collaboration], Phys. Rev. Lett. 90 (2003) 041801 [arXiv:hep-ex/0212007]. [14] E. Aliu et al. [K2K Collaboration], Phys. Rev. Lett. 94 (2005) 081802 [arXiv:hep-ex/0411038]. [15] D. G. Michael et al. [MINOS Collaboration], Phys. Rev. Lett. 97 (2006) 191801 [arXiv:hep-ex/0607088]. [16] B. Pontecorvo, Sov. Phys. JETP 6 (1957) 429 [Zh. Eksp. Teor. Fiz. 33 (1957) 549]. [17] Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28 (1962) 870. [18] B. Pontecorvo, Sov. Phys. JETP 26 (1968) 984 [Zh. Eksp. Teor. Fiz. 53 (1967) 1717]. [19] V. N. Gribov and B. Pontecorvo, Phys. Lett. B 28 (1969) 493. [20] M. C. Gonzalez-Garcia and M. Maltoni, arXiv:0704.1800 [hep-ph]. [21] C. Athanassopoulos et al. [LSND Collaboration], Phys. Rev. C 54 (1996) 2685 [arXiv:nucl-ex/9605001]. [22] C. Athanassopoulos et al. [LSND Collaboration], Phys. Rev. Lett. 81 (1998) 1774 [arXiv:nucl-ex/9709006]. [23] A. Aguilar et al. [LSND Collaboration], Phys. Rev. D 64 (2001) 112007 [arXiv:hep-ex/0104049]. [24] G. L. Fogli, E. Lisi, A. Marrone and G. Scioscia, arXiv:hep-ph/9906450. [25] W. M. Yao et al. [Particle Data Group], J. Phys. G 33 (2006) 1. [26] LEP Collaborations (ALEPH, DELPHI, OPAL, L3) et al., Phys. Rept. 427 (2006) 257 [arXiv:hep-ex/0509008]. [27] J. Kleinfeller [KARMEN Collaboration], Nucl. Phys. Proc. Suppl. 87 (2000) 281. [28] F. Dydak et al., Phys. Lett. B 134 (1984) 281. [29] I. E. Stockdale et al., Z. Phys. C 27 (1985) 53. [30] Y. Declais et al., Nucl. Phys. B 434 (1995) 503. [31] W. Grimus and T. Schwetz, Eur. Phys. J. C 20 (2001) 1 [arXiv:hep-ph/0102252]. [32] S. M. Bilenky, C. Giunti and W. Grimus, Eur. Phys. J. C 1 (1998) 247 [arXiv:hep-ph/9607372]. [33] N. Okada and O. Yasuda, Int. J. Mod. Phys. A 12 (1997) 3669 [arXiv:hep-ph/9606411]. [34] V. D. Barger, S. Pakvasa, T. J. Weiler and K. Whisnant, Phys. Rev. D 58 (1998) 093016 [arXiv:hep-ph/9806328]. [35] S. M. Bilenky, C. Giunti, W. Grimus and T. Schwetz, Phys. Rev. D 60 (1999) 073007 [arXiv:hep-ph/9903454]. http://arxiv.org/abs/hep-ex/0212007 http://arxiv.org/abs/hep-ex/0411038 http://arxiv.org/abs/hep-ex/0607088 http://arxiv.org/abs/0704.1800 http://arxiv.org/abs/nucl-ex/9605001 http://arxiv.org/abs/nucl-ex/9709006 http://arxiv.org/abs/hep-ex/0104049 http://arxiv.org/abs/hep-ph/9906450 http://arxiv.org/abs/hep-ex/0509008 http://arxiv.org/abs/hep-ph/0102252 http://arxiv.org/abs/hep-ph/9607372 http://arxiv.org/abs/hep-ph/9606411 http://arxiv.org/abs/hep-ph/9806328 http://arxiv.org/abs/hep-ph/9903454 [36] O. L. G. Peres and A. Y. Smirnov, Nucl. Phys. B 599 (2001) 3 [arXiv:hep-ph/0011054]. [37] C. Giunti and M. Laveder, JHEP 0102 (2001) 001 [arXiv:hep-ph/0010009]. [38] M. Maltoni, T. Schwetz, M. A. Tortola and J. W. F. Valle, New J. Phys. 6 (2004) 122 [arXiv:hep-ph/0405172]. [39] M. Maltoni, T. Schwetz, M. A. Tortola and J. W. F. Valle, Nucl. Phys. B 643 (2002) 321 [arXiv:hep-ph/0207157]. [40] M. Sorel, J. M. Conrad and M. Shaevitz, Phys. Rev. D 70 (2004) 073004 [arXiv:hep-ph/0305255]. [41] E. D. Church, K. Eitel, G. B. Mills and M. Steidl, Phys. Rev. D 66 (2002) 013001 [arXiv:hep-ex/0203023]. [42] A. A. Aguilar-Arevalo et al. [The MiniBooNE Collaboration], Phys. Rev. Lett. 98 (2007) 231801 [arXiv:0704.1500 [hep-ex]]. [43] M. Maltoni and T. Schwetz, arXiv:0705.0107 [hep-ph], to appear in PRD. [44] A. de Gouvea, Phys. Rev. D 72 (2005) 033005 [arXiv:hep-ph/0501039]. [45] A. de Gouvea, J. Jenkins and N. Vasudevan, Phys. Rev. D 75 (2007) 013003 [arXiv:hep-ph/0608147]. [46] S. Palomares-Ruiz, S. Pascoli and T. Schwetz, JHEP 0509 (2005) 048 [arXiv:hep-ph/0505216]. [47] H. Pas, S. Pakvasa and T. J. Weiler, Phys. Rev. D 72 (2005) 095017 [arXiv:hep-ph/0504096]. [48] G. Giacomelli, arXiv:physics/0703247. [49] R. Acquafredda et al. [OPERA Collaboration], New J. Phys. 8 (2006) 303 [arXiv:hep-ex/0611023]. [50] S. Amerio et al. [ICARUS Collaboration], Nucl. Instrum. Meth. A 527 (2004) 329. [51] A. Donini and D. Meloni, Eur. Phys. J. C 22 (2001) 179 [arXiv:hep-ph/0105089]. [52] A. Donini, M. Lusignoli and D. Meloni, Nucl. Phys. B 624 (2002) 405 [arXiv:hep-ph/0107231]. [53] M. Maltoni, T. Schwetz, M. A. Tortola and J. W. F. Valle, Phys. Rev. D 67 (2003) 013011 [arXiv:hep-ph/0207227]. [54] S. M. Bilenky, S. Pascoli and S. T. Petcov, Phys. Rev. D 64 (2001) 113003 [arXiv:hep-ph/0104218]. [55] A. Donini, M. B. Gavela, P. Hernandez and S. Rigolin, Nucl. Phys. B 574 (2000) 23 [arXiv:hep-ph/9909254]. http://arxiv.org/abs/hep-ph/0011054 http://arxiv.org/abs/hep-ph/0010009 http://arxiv.org/abs/hep-ph/0405172 http://arxiv.org/abs/hep-ph/0207157 http://arxiv.org/abs/hep-ph/0305255 http://arxiv.org/abs/hep-ex/0203023 http://arxiv.org/abs/0704.1500 http://arxiv.org/abs/0705.0107 http://arxiv.org/abs/hep-ph/0501039 http://arxiv.org/abs/hep-ph/0608147 http://arxiv.org/abs/hep-ph/0505216 http://arxiv.org/abs/hep-ph/0504096 http://arxiv.org/abs/physics/0703247 http://arxiv.org/abs/hep-ex/0611023 http://arxiv.org/abs/hep-ph/0105089 http://arxiv.org/abs/hep-ph/0107231 http://arxiv.org/abs/hep-ph/0207227 http://arxiv.org/abs/hep-ph/0104218 http://arxiv.org/abs/hep-ph/9909254 [56] A. Donini, M. B. Gavela, P. Hernandez and S. Rigolin, Nucl. Instrum. Meth. A 451 (2000) 58 [arXiv:hep-ph/9910516]. [57] A. De Rujula, M. Lusignoli, L. Maiani, S. T. Petcov and R. Petronzio, Nucl. Phys. B 168 (1980) 54. [58] A.E. Ball et al., “CNGS: Update on secondary beam layout”, SL-Note-2000-063 EA. [59] P. Migliozzi, Nucl. Phys. Proc. Suppl. 155 (2006) 23. [60] M. Komatsu, P. Migliozzi and F. Terranova, J. Phys. G 29 (2003) 443 [arXiv:hep-ph/0210043]. http://arxiv.org/abs/hep-ph/9910516 http://arxiv.org/abs/hep-ph/0210043 Introduction Four neutrino mass schemes Oscillation probabilities and allowed parameter space The CNGS facility Appearance channels at the CNGS oscillations e oscillations Sensitivity to (3+1) sterile neutrinos at OPERA Conclusions References

0704.0389

Evolution of the Carter constant for inspirals into a black hole: effect of the black hole quadrupole

For reference, the following erratum corrects the published version of the paper. These errors have been fixed in this arxiv-version (the article starting on page 2 has the corrected expressions). Erratum: Evolution of the Carter constant for inspirals into a black hole: Effect of the black hole quadrupole [Phys. Rev. D 75, 124007 (2007)] Éanna É. Flanagan, Tanja Hinderer In Eqs. (3.16), (3.17), (3.18), (3.24), (3.25) and (3.26) of this paper, the variable r should be replaced everywhere by the variable r̃, and the variable θ should be replaced everywhere by the variable θ̃. The definitions of r̃ and θ̃ are given in Eq. (2.11). These replacements do not affect the any of the subsequent results in the paper. Also, the right hand side of Eq. (B3) is missing a term −4SLzr̃ and Eq. (2.24) is missing a factor of dϕ/dt̃ in front of Q. Some terms are missing in Eqs. (3.18), (3.26) and (3.30) - (3.33). The additional terms in Eqs. (3.18) and (3.26) 15r̃7 −75K2 + 2Kr̃(51r̃E + 50) + 8r̃2(r̃E + 1)(3r̃E + 5) 15p2r̃7 25p3(3p− 4r̃) + p2r̃2 11− 51e2 + 32pr̃3 1− e2 + 6r̃4 1− e2 respectively. These result in additional fractional corrections to Eq. (3.30) given by and the full expression replacing the O(Q) terms in Eq. (3.30) is then 〈K̇〉 = − (1− e2)3/2 cos(2ι) +O(S), O(S2)− terms. Equations (3.31), (3.32) and (3.33) contain typos in the O(S) and O(Q) terms, the corrected expressions are given below. We thank P. Komorowski for pointing this out. Equation (3.31) should be replaced by 〈ṗ〉 = −64 (1− e2)3/2 − S cos(ι) 96p3/2 1064 + 1516e2 + 475e4 149e2 469e2 227e4 cos(2ι) + e2 + [13− cos(2ι)] , (0.1) Equation (3.32) should be replaced by 〈ė〉 = −304 e(1− e2)3/2 121e2 Se(1− e2)3/2 cos(ι) 5p11/2 1172 + 932e2 + 1313e4 Q(1− e2)3/2 785e2 − 219e + 13e6 + 2195e2 + 251e4 + 218e6 cos(2ι) 2e(1− e2)3/2 2 + 3e2 + [13− cos(2ι)] , (0.2) and the corrected Eq. (3.33) is 〈ι̇〉 = S sin(ι)(1 − e 2)3/2 p11/2 1− e2 S2 sin(2ι) 240p6 8 + 3e2 8 + e2 Q cot(ι)(1 − e2)3/2 312 + 736e2 − 83e4 − 408 + 1268e2 + 599e4 cos(2ι) . (0.3) http://arxiv.org/abs/0704.0389v8 Evolution of the Carter constant for inspirals into a black hole: effect of the black hole quadrupole Éanna É. Flanagan1,2 and Tanja Hinderer1 Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853, USA Laboratory for Elementary Particle Physics, Cornell University, Ithaca, NY 14853, USA (Dated: November 4, 2018) We analyze the effect of gravitational radiation reaction on generic orbits around a body with an axisymmetric mass quadrupole moment Q to linear order in Q, to the leading post-Newtonian order, and to linear order in the mass ratio. This system admits three constants of the motion in absence of radiation reaction: energy, angular momentum along the symmetry axis, and a third constant analogous to the Carter constant. We compute instantaneous and time-averaged rates of change of these three constants. For a point particle orbiting a black hole, Ryan [15] has computed the leading order evolution of the orbit’s Carter constant, which is linear in the spin. Our result, when combined with an interaction quadratic in the spin (the coupling of the black hole’s spin to its own radiation reaction field), gives the next to leading order evolution. The effect of the quadrupole, like that of the linear spin term, is to circularize eccentric orbits and to drive the orbital plane towards antialignment with the symmetry axis. In addition we consider a system of two point masses where one body has a single mass multipole or current multipole of order l. To linear order in the mass ratio, to linear order in the multipole, and to the leading post-Newtonian order, we show that there does not exist an analog of the Carter constant for such a system (except for the cases of an l = 1 current moment and an l = 2 mass moment). Thus, the existence of the Carter constant in Kerr depends on interaction effects between the different multipoles. With mild additional assumptions, this result falsifies the conjecture that all vacuum, axisymmetric spacetimes posess a third constant of the motion for geodesic motion. PACS numbers: 04.25.Nx, 04.30.Db I. INTRODUCTION AND SUMMARY The inspiral of stellar mass compact objects with masses µ in the range µ ∼ 1 − 100M⊙ into massive black holes with masses M ∼ 105 − 107M⊙ is one of the most important sources for the future space-based gravitational wave detector LISA. Observing such events will provide a variety of information: (i) the masses and spins of black holes can be measured to high accuracy (∼ 10−4); which can constrain the black hole’s growth history [1]; (ii) the observations will give a precise test of general relativity in the strong field regime and unam- biguously identify whether the central object is a black hole [2]; and (iii) the measured event rate will give in- sight into the complex stellar dynamics in galactic nu- clei [1]. Analogous inspirals may also be interesting for the advanced stages of ground-based detectors: it has been estimated that advanced LIGO could detect up to ∼ 10 − 30 inspirals per year of stellar mass compact objects into intermediate mass black holes with masses M ∼ 102 − 104M⊙ in globular clusters [3]. Detect- ing these inspirals and extracting information from the datastream will require accurate models of the gravita- tional waveform as templates for matched filtering. For computing templates, we therefore need a detailed un- derstanding of the how radiation reaction influences the evolution of bound orbits around Kerr black holes [4–7]. There are three dimensionless parameters characteriz- ing inspirals of bodies into black holes: • the dimensionless spin parameter a = |S|/M2 of the black hole, where S is the spin. • the strength of the interaction potential ǫ2 = GM/rc2, i.e. the expansion parameter used in post- Newtonian (PN) theory. • the mass ratio µ/M . For LISA data analysis we will need waveforms that are accurate to all orders in a and ǫ2, and to leading order in µ/M . However, it is useful to have analytic results in the regimes a ≪ 1 and/or ǫ2 ≪ 1. Such approximate results can be useful as a check of numerical schemes that compute more accurate waveforms, for scoping out LISA’s data analysis requirements [1, 6], and for assessing the accuracy of the leading order in µ/M or adiabatic approximation [8–10]. There is substantial literature on such approximate analytic results, and in this paper we will extend some of these results to higher order. A long standing difficulty in computing the evolution of generic orbits has been the evolution of the orbit’s ”Carter constant”, a constant of motion which governs the orbital shape and inclination. A theoretical prescrip- tion now exists for computing Carter constant evolution to all orders in ǫ and a in the adiabatic limit µ ≪ M [9, 11–13], but it has not yet been implemented numer- ically. In this paper we focus on computing analytically the evolution of the Carter constant in the regime a≪ 1, ǫ ≪ 1, µ/M ≪ 1, extending earlier results by Ryan [14, 15]. We next review existing analytical work on the effects of multipole moments on inspiral waveforms. For non- spinning point masses, the phase of the l = 2 piece of the waveform is known to O(ǫ7) beyond leading order [16], while spin corrections are not known to such high order. To study the leading order effects of the central body’s multipole moments on the inspiral waveform, in the test mass limit µ ≪ M , one has to correct both the conservative and dissipative pieces of the forces on the bodies. For the conservative pieces, it suffices to use the Newtonian action for a binary with an additional multi- pole interaction potential. For the dissipative pieces, the multipole corrections to the fluxes at infinity of the con- served quantities can simply be added to the known PN point mass results. The lowest order spin-orbit coupling effects on the gravitational radiation were first derived by Kidder [17], then extended by Ryan [14, 15], Gergely [18], and Will [19]. Recently, the corrections of O(ǫ2) beyond the leading order to the spin-orbit effects on the fluxes were derived [20, 21]. Corrections to the waveform due to the quadrupole - mass monopole interaction were first considered by Poisson [22], who derived the effect on the time averaged energy flux for circular equatorial orbits. Gergely [23] extended this work to generic orbits and computed the radiative instantaneous and time averaged rates of change of energy E, magnitude of angular mo- mentum |L|, and the angle κ = cos−1(S ·L) between the spin S and orbital angular momentum L. Instead of the Carter constant, Gergely identified the angular average of the magnitude of the orbital angular momentum, L̄, as a constant of motion. The fact that to post-2-Newtonian (2PN) order there is no time averaged secular evolution of the spin allowed Gergely to obtain expressions for L̇ and κ̇ from the quadrupole formula for the evolution of the total angular momentum J = L+S. In a different paper, Gergely [18] showed that in addition to the quadrupole, self-interaction spin effects also contribute at 2PN order, which was seen previously in the black hole perturbation calculations of Shibata et al. [24]. Gergely calculated the effect of this interaction on the instantaneous and time-averaged fluxes of E and |L| but did not derive the evolution of the third constant of motion. In this paper, we will re-examine the effects of the quadrupole moment of the black hole and of the leading order spin self interaction. For a black hole, our analysis will thus contain all effects that are quadratic in spin to the leading order in ǫ2 and in µ/M . Our work will extend earlier work by • Considering generic orbits. • Using a natural generalization of the Carter-type constant that can be defined for two point particles when one of them has a quadrupole. This facilitates applying our analysis to Kerr inspirals. • Computing instantaneous as well as time-averaged fluxes for all three constants of motion: energy E, z-component of angular momentum Lz, and Carter-type constant K. For most purposes, only time-averaged fluxes are needed as only they are gauge invariant and physically relevant. However, there is one effect for which the time-averaged fluxes are insufficient, namely transient resonances that occur during an inspiral in Kerr in the vicin- ity of geodesics for which the radial and azimuthal frequencies are commensurate [10, 25]. The instan- taneous fluxes derived in this paper will be used in [10] for studying the effect of these resonances on the gravitational wave phasing. We will analyze the effect of gravitational radiation re- action on orbits around a body with an axisymmetric mass quadrupole moment Q to leading order in Q, to the leading post-Newtonian order, and to leading order in the mass ratio. With these approximations the adiabatic ap- proximation holds: gravitational radiation reaction takes place over a timescale much longer than the orbital pe- riod, so the orbit looks geodesic on short timescales. We follow Ryan’s method of computation [14]: First, we cal- culate the orbital motion in the absence of radiation re- action and the associated constants of motion. Next, we use the leading order radiation reaction accelerations that act on the particle (given by the Burke-Thorne formula [26] augmented by the relevant spin corrections [14]) to compute the evolution of the constants of motion. In the adiabatic limit, the time-averaged rates of change of the constants of motion can be used to infer the secular or- bital evolution. Our results show that a mass quadrupole has the same qualitative effect on the evolution as spin: it tends to circularize eccentric orbits and drive the orbital plane towards antialignment with the symmetry axis of the quadrupole. The relevance of our result to point particles inspi- ralling into black holes is as follows. The vacuum space- time geometry around any stationary body is completely characterized by the body’s mass multipole moments IL = Ia1,a2...al and current multipole moments SL = Sa1,a2...al [27]. These moments are defined as coefficients in a power series expansion of the metric in the body’s local asymptotic rest frame [28]. For nearly Newtonian sources, they are given by integrals over the source as IL ≡ Ia1,...al = ρx<a1 . . . xal>d 3x, (1.1) SL ≡ Sa1,...al = ρxpvqǫpq<a1xa2 . . . xal>d 3x.(1.2) Here ρ is the mass density and vq is the velocity, and ”< · · · >” means ”symmetrize and remove all traces”. For axisymmetric situations, the tensor multipole moments IL (SL) contain only a single independent component, conventionally denoted by Il (Sl) [27]. For a Kerr black hole of mass M and spin S, these moments are given by Il + iSl =M l+1(ia)l, (1.3) where a is the dimensionless spin parameter defined by a = |S|/M2. Note that Sl = 0 for even l and Il = 0 for odd l. Consider now inspirals into an axisymmetric body which has some arbitrary mass and current multipoles Il and Sl. Then we can consider effects that are linear in Il and Sl for each l, effects that are quadratic in the mul- tipoles proportional to IlIl′ , IlSl′ , SlSl′ , effects that are cubic, etc. For a general body, all these effects can be sep- arated using their scalings, but for a black hole, Il ∝ al for even l and Sl ∝ al for odd l [see Eq.(1.3)], so the ef- fects cannot be separated. For example, a physical effect that scales as O(a2) could be an effect that is quadratic in the spin or linear in the quadrupole; an analysis in Kerr cannot distinguish these two possibilities. For this reason, it is useful to analyze spacetimes that are more general than Kerr, characterized by arbitrary Il and Sl, as we do in this paper. For recent work on computing exact metrics characterized by sets of moments Il and Sl, see Refs. [29, 30] and references therein. The leading order effect of the black hole’s multipoles on the inspiral is the O(a) effect computed by Ryan [15]. This O(a) effect depends linearly on the spin S1 and is independent of the higher multipoles Sl and Il since these all scale as O(a2) or smaller. In this paper we compute the O(a2) effect on the inspiral, which includes the lead- ing order linear effect of the black hole’s quadrupole (lin- ear in I2 ≡ Q) and the leading order spin self-interaction (quadratic in S1). We next discuss how these O(a2) effects scale with the post-Newtonian expansion parameter ǫ. Consider first the conservative orbital dynamics. Here it is easy to see that fractional corrections that are linear in I2 scale as O(a2ǫ4), while those quadratic in S1 scale as O(a 2ǫ6). Thus, the two types of terms cleanly separate. We com- pute only the leading order, O(a2ǫ4), term. For the dissi- pative contributions to the orbital motion, however, the scalings are different. There are corrections to the radi- ation reaction acceleration whose fractional magnitudes are O(a2ǫ4) from both types of effects linear in I2 and quadratic in S1. The effects quadratic in S1 are due to the backscattering of the radiation off the piece of space- time curvature due to the black hole’s spin. This effect was first pointed out by Shibata et al. [24], who com- puted the time-averaged energy flux for circular orbits and small inclination angles based on a PN expansion of black hole perturbations. Later, Gergely [18] analyzed this effect on the instantaneous and time-averaged fluxes of energy and magnitude of orbital angular momentum within the PN framework. The organization of this paper is as follows. In Sec. II, we study the conservative orbital dynamics of two point particles when one particle is endowed with an ax- isymmetric quadrupole, in the weak field regime, and to leading order in the mass ratio. In Sec. III, we com- pute the radiation reaction accelerations and the instan- taneous and time-averaged fluxes. In order to have all the contributions at O(a2ǫ4) for a black hole, we include in our computations of radiation reaction acceleration the interaction that is quadratic in the spin S1. The ap- plication to black holes in Sec. IV briefly discusses the qualitative predictions of our results and also compares with previous results. The methods used in this paper can be applied only to the black hole spin (as analyzed by Ryan [14]) and the black hole quadrupole (as analyzed here). We show in Sec. V that for the higher order mass and current multipole moments taken individually, an analog of the Carter constant cannot be defined to the order of our approximations. We then show that under mild assump- tions, this non-existence result can be extended to exact spacetimes, thus falsifying the conjecture that all vac- uum axisymmetric spacetimes possess a third constant of geodesic motion. II. EFFECT OF AN AXISYMMETRIC MASS QUADRUPOLE ON THE CONSERVATIVE ORBITAL DYNAMICS Consider two point particles m1 and m2 interacting in Newtonian gravity, where m2 ≪ m1 and where the mass m1 has a quadrupole moment Qij which is axisymmetric: Qij = d3xρ(r) xixj − r2δij (2.1) ninj − . (2.2) For a Kerr black hole of mass M and dimensionless spin parameter a with spin axis along n, the quadrupole scalar is Q = −M3a2. The action describing this system, to leading order in m2/m1, is µv2 − µΦ(r) , (2.3) where v = ṙ is the velocity, the potential is Φ(r) = −M xixjQij , (2.4) µ is the reduced mass and M the total mass of the bi- nary, and we are using units with G = c = 1. We work to linear order in Q, to linear order in m2/m1, and to lead- ing order in M/r. In this regime, the action (2.3) also describes the conservative effect of the black hole’s mass quadrupole on bound test particles in Kerr, as discussed in the introduction. We shall assume that the quadrupole Qij is constant in time. In reality, the quadrupole will evolve due to torques that act to change the orientation of the central body. An estimate based on treating m1 as a rigid body in the Newtonian field ofm2 gives the scaling of the timescale for the quadrupole to evolve compared to the radiation reaction time as (see Appendix I for details) Tevol (2.5) Here, we have denoted the dimensionless spin and quadrupole of the body by S̄ and Q̄ respectively, and the last relation applies for a Kerr black hole. Since µ/M ≪ 1, the first factor in Eq. (2.5) will be large, and since 1/a ≥ 1 and for the relativistic regime M/r ∼ 1, the evolution time is long compared to the radiation re- action time. Therefore we can neglect the evolution of the quadrupole at leading order. This system admits three conserved quantities, the en- µv2 + µΦ(r), (2.6) the z-component of angular momentum Lz = ez · (µr× v), (2.7) and the Carter-type constant K = µ2(r× v)2 − 2Qµ (n · r)2 (n · v)2 − . (2.8) (See below for a derivation of this expression for K). A. Conservative orbital dynamics in a Boyer-Lindquist-like coordinate system We next specialize to units where M = 1. We also define the rescaled conserved quantities by Ẽ = E/µ, L̃z = Lz/µ, K̃ = K/µ 2, and drop the tildes. These spe- cializations and definitions have the effect of eliminating all factors of µ and M from the analysis. In spherical polar coordinates (r, θ, ϕ) the constants of motion E and Lz become (ṙ2 + r2θ̇2 + r2 sin2 θϕ̇2)− (1− 3 cos2 θ), (2.9) Lz = r 2 sin2 θϕ̇. (2.10) In these coordinates, the Hamilton-Jacobi equation is not separable, so a separation constant K cannot readily be derived. For this reason we switch to a different coordi- nate system (r̃, θ̃, ϕ) defined by r cos θ = r̃ cos θ̃ r sin θ = r̃ sin θ̃ . (2.11) We also define a new time variable t̃ by cos(2θ̃) dt̃. (2.12) The action (2.3) in terms of the new variables to linear order in Q is r̃2 sin2 θ̃ sin2 θ̃ . (2.13) However, a difficulty is that the action (2.13) does not give the same dynamics as the original action (2.3). The reason is that for solutions of the equations of motion for the action (2.3), the variation of the action vanishes for paths with fixed endpoints for which the time interval ∆t is fixed. Similarly, for solutions of the equations of motion for the action (2.13), the variation of the action vanishes for paths with fixed endpoints for which the time interval ∆t̃ is fixed. The two sets of varied paths are not the same, since ∆t 6= ∆t̃ in general. Therefore, solutions of the Euler-Lagrange equations for the action (2.3) do not correspond to solutions of the Euler-Lagrange equations for the action (2.13). However, in the special case of zero- energy motions, the extra terms in the variation of the action vanish. Thus, a way around this difficulty is to modify the original action to be µv2 − µΦ(r) + E . (2.14) This action has the same extrema as the action (2.3), and for motion with physical energy E, the energy com- puted with this action is zero. Transforming to the new variables yields, to linear order in Q: r̃2 sin2 θ̃ sin2 θ̃ + E − QE cos(2θ̃) . (2.15) The zero-energy motions for this action coincide with the zero energy motions for the action (2.14). We use this action (2.15) as the foundation for the remainder of our analysis in this section. The z-component of angular momentum in terms of the new variables (r̃, θ̃, ϕ, t̃) is Lz = r̃ 2 sin2 θ̃ sin2 θ̃ . (2.16) We now transform to the Hamiltonian: p2r̃ − − E − Q sin2 θ̃ +QE cos(2θ̃) (2.17) and solve the Hamiltonian Jacobi equation. Denoting the separation constant by K we obtain the following two equations for the r̃ and θ̃ motions: = 2E + , (2.18) = K − L sin2 θ̃ −QE cos(2θ̃). (2.19) Note that the equations of motion (2.18) and (2.19) have the same structure as the equations of motion for Kerr geodesic motion. Using Eqs. (2.18), (2.19) and (2.16) together with the inverse of the transformation (2.11) to linear order in Q, we obtain the expression for K in spherical polar coordinates: K = r4(θ̇2 + sin2 θϕ̇2) +Q(ṙ cos θ − rθ̇ sin θ)2 + Q (ṙ2 + r2 θ̇2 + r2 sin2 θϕ̇2)− 2Q cos2 θ. (2.20) This is equivalent to the formula (2.8) quoted earlier. B. Effects linear in spin on the conservative orbital dynamics To include the linear in spin effects, we repeat Ryan’s analysis [14, 15] (he only gives the final, time averaged fluxes; we will also give the instantaneous fluxes). We can simply add these linear in spin terms to our results because any terms of order O(SQ) will be higher than the order a2 to which we are working. The correction to the action (2.3) due to spin-orbit coupling is Sspin−orbit = −2µSn iǫijkxj ẋk . (2.21) We will restrict our analysis to the case when the unit vectors ni corresponding to the axisymmetric quadrupole Qij and to the spin Si coincide, as they do in Kerr. Including the spin-orbit term in the action (2.3) results in the following modified expressions for Lz and K: Lz = n · (µr× v)− [r2 − (n · r)2], (2.22) K = (r× v)2 − 4S n · (r× v)− 2Q (n · r)2 (n · v)2 − 1 . (2.23) In terms of the Boyer-Lindquist like coordinates, the con- served quantities with the linear in spin terms included Lz = r̃ 2 sin2 θ̃ sin2 θ̃ −Q sin4 θ̃ (2.24) K = r4(θ̇2 + sin2 θϕ̇2)− 4Sr sin2 θϕ̇ cos2 θ +Q(ṙ cos θ − rθ̇ sin θ)2 + QM (ṙ2 + r2θ̇2 + r2 sin2 θϕ̇2). (2.25) The equations of motion are = 2E+ − 4SLz , (2.26) = K − sin2 θ̃ −QE cos(2θ̃). (2.27) III. EFFECTS LINEAR IN QUADRUPOLE AND QUADRATIC IN SPIN ON THE EVOLUTION OF THE CONSTANTS OF MOTION A. Evaluation of the radiation reaction force The relative acceleration of the two bodies can be writ- ten as a = −∇Φ(r) + arr, (3.1) where arr is the radiation-reaction acceleration. Combin- ing this with Eqs. (2.6), (2.22) and (2.23) for E, Lz and K gives the following formulae for the time derivatives of the conserved quantities: Ė = v · arr, (3.2) L̇z = n · (r× arr), (3.3) K̇ = 2(r× v) · (r× arr)− n · (r× arr) +2Q(n · v) (n · arr)−Qv · arr. (3.4) The standard expression for the leading order radiation reaction acceleration acting on one of the bodies is [31]: ajrr = − jk xk + ǫjpqS pk xkxq + ǫjpqS pk xkvq ǫpq[jS xqvk. (3.5) Here the superscripts in parentheses indicate the number of time derivatives and square brackets on the indices denote antisymmetrization. The multipole moments Ijk(t) and Sjk(t) in Eq. (3.5) are the total multipole moments of the spacetime, i.e. approximately those of the black hole plus those due to the orbital motion. The expression (3.5) is formulated in asymptotically Cartesian mass centered (ACMC) co- ordinates of the system, which are displaced from the coordinates used in Sec. II by an amount [28] δr(t) = − µ r(t). (3.6) This displacement contributes to the radiation reaction acceleration in the following ways: 1. The black hole multipole moments Il and Sl, which are time-independent in the coordinates used in Sec. II, will be displaced by δr and thus will con- tribute to the (l + 1)th ACMC radiative multipole [28]. 2. The constants of motion are defined in terms of the black hole centered coordinates used in Sec. II, so the acceleration arr we need in Eqs. (3.2) – (3.4) is the relative acceleration. This requires calculat- ing the acceleration of both the black hole and the point mass in the ACMC coordinates using (3.5), and then subtracting to find arr = a rr − aMrr [14]. To leading order in µ, the only effect of the acceler- ation of the black hole is via a backreaction of the radiation field: the lth black hole moments couple to the (l+1)th radiative moments, thus producing an additional contribution to the acceleration. For our calculations at O(S1ǫ 3), O(I2ǫ 4), O(S21ǫ 4), we can make the following simplifications: • quadrupole corrections: The fractional corrections linear in I2 = Q that scale as O(a 2ǫ4) require only the effect of I2 on the conservative orbital dynamics as computed in Sec. IIA and the Burke-Thorne for- mula for the radiation reaction acceleration [given by the first term in Eq. (3.5)]. • spin-spin corrections: As discussed in the intro- duction, the fractional corrections quadratic in S1 to the conservative dynamics scale as O(a2ǫ6) and are subleading order effects which we neglect. At O(a2ǫ4), the only effect quadratic in S1 is the backscattering of the radiation off the spacetime curvature due to the spin. As discussed in item 1. above, the black hole’s current dipole Si = S1δi3 (taking the z-axis to be the symmetry axis) will contribute to the radiative current quadrupole an amount ij = − S1xiδj3. (3.7) The black hole’s current dipole Si will couple to the gravitomagnetic radiation field due to Sij as discussed in item 2. above, and contribute to the relative acceleration as [14]: aj spinrr = S1δi3S ij . (3.8) For our purposes of computing terms quadratic in the spin, we substitute S ij for Sij in Eq. (3.8). Evaluating these quadratic in spin terms requires only the Newtonian conservative dynamics, i.e. the results of Sec. II and Eqs. (3.2) – (3.4) with the quadrupole set to zero. • linear in spin corrections: Contributions to these effects are from Eq. (3.5) with the current quadrupole replaced by just the spin contribution (3.7), and from Eq. (3.8) evaluated using only the orbital current quadrupole. With these simplifications, we replace the expression (3.5) for the radiation reaction acceleration with ajrr = − jk xk + ǫjpqS (6) spin pk xkxq ǫjpqS (5) spin pk xkvq + ǫpq[jS (5) spin S1δi3 (5) orbit ij + S (5) spin . (3.9) To justify these approximations, consider the scaling of the contribution of black hole’s acceleration to the orbital dynamics. The mass and current multipoles of the black hole contribute terms to the Hamiltonian that scale with ∆H ∼ Slǫ2l+3 & Ilǫ2l+2. (3.10) Since the Newtonian energy scales as ǫ2, the fractional correction to the orbital dynamics scale as ∆H/E ∼ Slǫ2l+1 & Ilǫ2l. (3.11) To O(ǫ4), the only radiative multipole moments that con- tribute to the acceleration (3.5) are the mass quadrupole I2, the mass octupole I3, and the current quadrupole S2 (cf. [17]). Since we are focusing only on the leading or- der terms quadratic in spin (these can simply be added to the known 2PN point particle and 1.5PN linear in spin results), the only terms in Eq. (3.5) relevant for our pur- poses are those given in Eq. (3.9). The results from a computation of the fully relativistic metric perturbation for black hole inspirals [24] show that quadratic in spin corrections to the l = 2 piece compared to the flat space Burke-Thorne formula first appear at O(a2ǫ4), which is consistent with the above arguments. B. Instantaneous fluxes We evaluate the radiation reaction force as follows. The total mass and current quadrupole moment of the system are QTij = Qij + µxixj , (3.12) STij = S ij + xiǫjkmxkẋm, (3.13) where from Eq. (2.11) r̃ sin θ̃ cosϕ, r̃ sin θ̃ sinϕ, r̃ cos θ̃ . (3.14) Only the second term in Eq. (3.12) contributes to the time derivative of the quadrupole. We differentiate five times by using cos(2θ̃) , (3.15) to the order we are working as discussed above. Af- ter each differentiation, we eliminate any occurrences of dϕ/dt̃ using Eq. (2.24), and we eliminate any occurrences of the second order time derivatives d2r̃/dt̃2 and d2θ̃/dt̃2 in favor of first order time derivatives using (the time derivatives of) Eqs. (2.26) and (2.27). For computing the terms linear and quadratic in S1, we set the quadrupole Q to zero in all the formulae. We insert the resulting ex- pression into the formula (3.9) for the self-acceleration, and then into Eqs. (3.2) – (3.4). We eliminate (dr̃/dt̃)2, (dθ̃/dt̃)2, and (dϕ/dt̃) in favor of E, Lz, and K using Eqs. (2.24) – (2.27). In the final expressions for the in- stantaneous fluxes, we keep only terms that are of O(S), O(Q) and O(S2) and obtain the following results: 15r̃4 − 40K 272KE 196K2 + r̃2 − 3668 Kr̃ − 352KEr̃2 + 1024 Er̃3 + E2r̃4 −49K2 − 169KL2z + r̃ + 2r̃2 + 47KE + − 152 r̃3E − 16r̃4E2 −562K2 + Kr̃ − r̃2 + KEr̃2 − r̃3E − 160r̃4E2 cos(2θ̃) sin(2θ̃) 439K − 926 r̃ − 1528 θ ˙̃r −K2 + 22 Kr̃ − 28 r̃2 + KEr̃2 − 236 r̃3E − 32 r̃4E2 cos(2θ̃)− r̃3 sin(2θ̃) θ ˙̃r −49K2 + 6KL2z + 2r̃ 63K − 16 L2z − + r̃2 112KE − 48 − 1652 r̃3E − 224 r̃4E2 , (3.16) L̇z = 144LzE − 24KLz −50K2 + 240KL2z + Kr̃ − 7376 L2z r̃ + r̃2 + 56KEr̃2 − 1824 EL2z r̃ Er̃3 + E2r̃4 50K2 − 62 Kr̃ − 316 r̃2 − 56KEr̃2 − 624 Er̃3 − 128 E2r̃4 cos(2θ̃) −104K + 64r̃ + 64Er̃2 sin(2θ̃) ˙̃r 660Er̃2 + 753r̃− 360L2z − 435K + 1601r̃+ 1512r̃2E − 1185K cos 2θ̃ 174QLz sin(2θ̃) ˙̃r 2S2Lz Er̃2 + 16r̃ − 9K , (3.17) 20r̃ + 18r̃2E − 15K 280K2 − 14008 Kr̃ + r̃2 + Er̃3 − 2528 KEr̃2 + E2r̃4 −45K2 + r̃L2z(83 + 80r̃E)− 115KL2z + 14Kr̃(6 + 5r̃E) 15r̃7 cos(2θ̃) −2175K2 + 2975Kr̃+ 80r̃2 + 3012KEr̃2 − 112Er̃3 − 168E2r̃4 15r̃4 3075K − 20r̃ − 192Er̃2 sin(2θ̃) θ ˙̃r 7K − 2L2z −3K + 16 +K cos(2θ̃) 3K − 16 r̃ − 24 sin(2θ̃) −4K + θ ˙̃r. (3.18) C. Alternative set of constants of the motion A body in a generic bound orbit in Kerr traces an open ellipse precessing about the hole’s spin axis. For stable orbits the motion is confined to a toroidal region whose shape is determined by E, Lz, K. The motion can equivalently be characterized by the set of constants inclination angle ι, eccentricity e, and semi-latus rectum p defined by Hughes [32]. The constants ι, p and e are defined by cos ι = Lz/ K, and by r̃± = p/(1± e), where r̃± are the turning points of the radial motion, and r̃ is the Boyer-Lindquist radial coordinate. This param- eterization has a simple physical interpretation: in the Newtonian limit of large p, the orbit of the particle is an ellipse of eccentricity e and semilatus rectum p on a plane whose inclination angle to the hole’s equatorial plane is ι. In the relativistic regime p ∼M , this interpretation of the constants e, p, and ι is no longer valid because the orbit is not an ellipse and ι is not the angle at which the object crosses the equatorial plane (see Ryan [14] for a discussion). We adopt here analogous definitions of constants of motion ι, e and p, namely cos(ι) = Lz/ K, (3.19) = r̃±. (3.20) Here K is the conserved quantity (2.23) or (2.25), and r̃± are the turning points of the radial motion using the r̃ coordinate defined by Eq. (2.11), given by the vanishing of the right-hand side of Eq. (2.26). We now rewrite our results in terms of the new con- stants of the motion e, p and ι. We can use Eq. (2.26) together with the equations (3.19) and (3.20) to write E, Lz and K as functions of p, e and ι. To leading order in Q and S we obtain K = p 1− 2S cos ι 3 + e2 1 + e2 ) 2Q cos2 ι 3 + e2 , (3.21) E = − (1− e 2S cos ι 1− e2 1− e2 cos2 ι− 1 , (3.22) p cos ι 1− S cos ι (3 + e2)− 1 + e2 ) Q cos2 ι 3 + e2 . (3.23) As discussed in the introduction, the effects quadratic in S on the conservative dynamics scale as O(a2ǫ6) and thus are not included in this analysis to O(a2ǫ4). Inserting these relations into the expressions (3.16)– (3.18) gives, dropping terms of O(QS), O(Q2) and O(QS2): Ė = − 15p2r̃7 75p4 − 100p3r̃ + p2r̃2 11− 51e2 + 32pr̃3 1− e2 )− 6r̃4 1− e2 4S cos ι 15p7/2r̃9 735p6 − 2751p5r̃ + 10p4r̃2(365− 6e2)− 128pr̃5(1− e2)2 − 48r̃6(e2 − 1)3 64S cos ι 15p3/2r̃6 5p(−23 + 3e2)− 3r̃(−9 + e2 + 8e4) 15p4r̃9 4005p6 − 6499p5r̃ + 2p4r̃2 1577− 1977e2 − 24r̃6 1− e2 )3 − 32p3r̃3 8− 33e2 + 64pr̃5 1− 2e2 + e4 15p4r̃9 24p2r̃4 5− 27e2 + 22e4 − pr̃3 sin(2θ̃) 6585p2 − 4630pr̃+ 2292r̃2(1 − e2) θ ˙̃r 15p4r̃9 2p2 cos(2θ̃) 4215p4 − 7495p3r̃ + 4p2r̃2(1151− 951e2)− 1012pr̃3(1− e2) + 300r̃4(1− 2e2 + e4) 15p4r̃9 cos(2ι) 2535p6 − 3307p5r̃ + 12p4r̃2(37− 237e2)− 48r̃6(1− e2)3 + 800p3r̃3(1 + e2) + 128pr̃5(1− 2e2 + e4) 15p2r̃5 cos(2ι) 1 + 2e2 − 3e4 15p2r̃9 84r̃4(1− e2)2(1 + e2)2 + 345p4 − 905p3r̃ − 413pr̃3(1− e2) + 2p2r̃2(446− 201e2) 15p2r̃9 cos(2θ̃) 15p4 − 110p3r̃ + 4p2r̃2(47− 12e2)− 118pr̃3(1− e2) + 24r̃4(1− e2)2(1 + e2)2 15r̃9 cos(2ι) 45p2 − 80pr̃ + 36r̃2(1− e2) 15pr̃6 sin(2θ̃) ˙̃r 15p2 + 10pr̃ − 12r̃2(1− e2) , (3.24) L̇z = − 8 cos ι 15p2 − 20pr̃ + 9r̃2(1 − e2) 15p2r̃7 525p4 − 1751p3r̃ + 34p2r̃2(61− 6e2) + 12pr̃3(−69 + 29e2) + 6r̃4(17 + 2e2 − 19e4) 15p2r̃7 375p4 − 93p3r̃ + 468pr̃3(1− e2)− 10p2r̃2(58 + 21e2)− 48r̃4(1 − 2e2 + e4) cos(2θ̃) 15p2r̃7 450p4 − 922p3r̃ − 60pr̃3(3 + e2)− 9p2r̃2(−83 + 23e2) + 27r̃4(1 + 2e2 − 3e4) cos(2ι) 13p2 − 8pr̃ + 4r̃2(1− e2) sin(2θ̃) ˙̃r − Q cos ι 5p5/2r̃7 615p4 − 753p3r̃ + 15p2r̃2 19− 31e2 + 20pr̃3 1 + 3e2 + 9r̃4 1− 6e2 + 5e4 − Q cos ι 5p1/2r̃7 cos(2θ̃) 1185p2 − 1601pr̃+ 756r̃2(1− e2) − 2Q cos ι 5p5/2r̃7 2 cos(2ι) 45p4 − 18r̃4e2(1− e2)− 45p2r̃2(1 + e2) + 20pr̃3(1 + e2) − 435p3r̃3 sin(2θ̃) ˙̃θ ˙̃r 2 cos ι p1/2r̃7 9p2 − 16pr̃ + 36 r̃2(1− e2) , (3.25) 20pr̃ − 15p2 − 9r̃2(1− e2) 8S cos ι 15p3/2r̃7 525p4 − 1751p3r̃ + 2p2r̃2(1172− 57e2) + 12pr̃3(−99 + 19e2)− 24r̃4(−11 + 4e2 + 7e4) 5p2r̃7 −615p4 + 753p3r̃ + 30p2r̃2(17e2 − 9) + 72r̃4e2(1− e2)− 40pr̃3(1 + 3e2) 5p2r̃7 cos(2ι) −345p4 + 249p3r̃ − 160pr̃3(1 + e2) + 120p2r̃2(1 + 3e2) + 36r̃4(1 + 2e2 − 3e4) 15p2r̃7 2 cos(2θ̃) 2175p4 − 2975p3r̃ − 56pr̃3(1 − e2) + 2p2r̃2(713− 753e2) + 42r̃4(1 − 2e2 + e4) 15pr̃4 sin(2θ̃) 3075p2 − 20pr̃ + 96r̃2(1 − e2) −9p2 + 16pr̃ − 36 r̃2(1− e2) cos(2θ̃) + cos(2ι) 3p2 − 16 pr̃ + r̃2(1 − e2) sin(2θ̃) ˙̃r −2p2 + 7 pr̃ − 4 r̃2(1− e2) . (3.26) D. Time averaged fluxes In this section we will compute the infinite time- averages 〈Ė〉, 〈L̇z〉 and 〈K̇〉 of the fluxes. These averages are defined by 〈Ė〉 ≡ lim ∫ T/2 Ė(t)dt. (3.27) These time-averaged fluxes are sufficient to evolve or- bits in the adiabatic regime (except for the effect of res- onances) [12, 25]. In Appendix II, we present two dif- ferent ways of computing the time averages. The first approach is based on decoupling the r̃ and θ̃ motion us- ing the analog of the Mino time parameter for geodesic motion in Kerr [12]. The second approach uses the ex- plicit Newtonian parameterization of the orbital motion. Both averaging methods give the following results: 〈Ė〉 = −32 (1− e2)3/2 e4 − S cos(ι) cos(2ι) cos(2ι) ,(3.28) 〈L̇z〉 = − (1 − e2)3/2 cos ι e2 − S 2p3/2 cos ι + 7e2 + cos(2ι) 45 + 148e2 + cos(2ι) 1 + 3e2 + , (3.29) 〈K̇〉 = −64 (1 − e2)3/2 e2 − S 2p3/2 + 37e2 + cos(ι) cos(2ι) cos(2ι) . (3.30) Using Eqs. (3.21) and (3.23), we obtain from (3.28) – (3.30) the following time averaged rates of change of the orbital elements e, p, ι: 〈ṗ〉 = −64 (1− e2)3/2 − S cos(ι) 96p3/2 1064 + 1516e2 + 475e4 149e2 469e2 227e4 cos(2ι) + e2 + [13− cos(2ι)] , (3.31) 〈ė〉 = − e(1− e2)3/2 121e2 Se(1− e2)3/2 cos(ι) 5p11/2 1172 + 932e2 + 1313e4 Q(1− e2)3/2 785e2 219e4 + 13e6 + 2195e2 + 251e4 + 218e6 cos(2ι) S2e(1− e2)3/2 2 + 3e2 + [13− cos(2ι)] , (3.32) 〈ι̇〉 = S sin(ι)(1 − e 2)3/2 p11/2 1− e2 S2 sin(2ι) 240p6 8 + 3e2 8 + e2 Q cot(ι)(1 − e2)3/2 312 + 736e2 − 83e4 − 408 + 1268e2 + 599e4 cos(2ι) . (3.33) IV. APPLICATION TO BLACK HOLES A. Qualitative discussion of results The above results for the fluxes, Eqs. (3.31), (3.32) and (3.33) show that the correction terms at O(a2ǫ4) due to the quadrupole have the same type of effect on the evolution as the linear spin correction computed by Ryan: they tend to circularize eccentric orbits and change the angle ι such as to become antialigned with the symmetry axis of the quadrupole. The effects of the terms quadratic in spin are quali- tatively different. In the expression (3.28) for 〈Ė〉, the coefficient of cos(2ι) due to the spin self-interaction has the same sign as the quadrupole term, while the terms not involving ι have the opposite sign. The terms in- volving cos(2ι) in Eq. (3.30) for 〈K̇〉 of O(Q) and O(S2) terms have the same sign, while the terms not involving ι have the opposite sign. The fractional spin-spin cor- rection to 〈L̇z〉, Eq. (3.29), has no ι-dependence, and in expression (3.33) for 〈ι̇〉, the dependence on ι of the two effects O(Q) and O(S2) is different, too. This is not sur- prising as the O(Q) effects included here are corrections to the conservative orbital dynamics, while the effects of O(S2) that we included are due to radiation reaction. B. Comparison with previous results The terms linear in the spin in our results for the time averaged fluxes, Eqs. (3.28) – (3.33), agree with those computed by Ryan, Eqs. (14a) – (15c) of [15], and with those given in Eqs. (2.5) – (2.7) of Ref. [33], when we use the transformations to the variables used by Ryan given in Eqs. (2.3) – (2.4) in [33]. Equation (3.28) for the time averaged energy flux agrees with Eq. (3.10) of Gergely [23] and Eq. (4.15) of [18] when we use the following transformations: K = L̄2 Ā2 sin2 κ cos δ − (1− Ā2) cos2 κ = L̄2 E cos2 κ (1 + 2L̄2) sin2 κ cos δ , (4.1) cos ι = cosκ E cos2 κ (1 + 2L̄2) sin2 κ cos δ , (4.2) (δ + κ), (4.3) ξ0 = (ψ0 − ψi) + , (4.4) where Ā, L̄, κ, δ, ψ0 and ψi are the quantities used by Gergely. The first relation here is obtained from the turn- ing points of the radial motion as follows. We compute r̃± in terms of E and K and map these expressions back to r using Eqs. (2.11). The result can then be com- pared with the turning points in Gergely’s variables, Eq. (2.19) of [23], using the fact that E is the same in both cases. Instead of the evolution of the constants of motion K and Lz, Gergely computes the rates of change of the magnitude L of the orbital angular momentum and of the angle κ defined by cosκ = (L · S)/L. Using the trans- formations (4.1) – (4.4) and the definition of κ we verify that our Eq. (3.29) agrees with the 〈L̇z〉 computed using Gergely’s Eqs. (3.23) and (3.35) in [23] and Eq. (4.30) of [18]. In the limit of the circular equatorial orbits analyzed by Poisson [22], our Eq. (3.28) agrees with Poisson’s Eq. (22) when we use the transformations and specializations: , (4.5) ι = 0, (4.6) e2 = 0, (4.7) cosαA = 1, (4.8) where v and αA are the variables used by Poisson and the relation (4.5) is obtained by comparing the expressions for the constants of motion in the two sets of variables. The main improvement of our analysis over Gergely’s is that we express the results in terms of the Carter-type constant K, which facilitates comparing our results with other analyses of black hole inspirals. Our computations also include the spin curvature scattering effects for all three constants of motion; Gergely [18] only considers these effects for two of them: the energy and magnitude of angular momentum, not for the third conserved quan- tity. When we expand Eq. (3.28) for small inclination an- gles and specialize to circular orbits, then after converting p to the parameter v using Eq. (4.5), we obtain 〈Ė〉 = − 32 11Q− S = − 32 33− 527 . (4.9) This result agrees with the terms atO(a2v4) of Eq. (3.13) of Shibata et al. [24], whose calculations were based on the fully relativistic expressions. This agreement is a check that we have taken into account all the contribu- tions at O(a2ǫ4). The analysis in Ref. [24] could not dis- tinguish between effects due to the quadrupole and those due curvature scattering, but we can see from Eq. (4.9) that those two interactions have the opposite dependence on ι. Comparing (4.9) with Eq. (3.7) of [24] (which gives the fluxes into the different modes (l = 2,m, n), where m and n are the multiples of the ϕ and θ frequencies), we see that the terms in the (2,±2, 0) and the (2,±1,±1) modes are entirely due to the quadrupole, while the spin-spin in- teraction effects are fully contained in the (2,±1, 0) and (2, 0,±1) modes. V. NON-EXISTENCE OF A CARTER-TYPE CONSTANT FOR HIGHER MULTIPOLES In this section, we show that for a single axisymmetric multipole interaction, it is not possible to find an ana- log of the Carter constant (a conserved quantity which does not correspond to a symmetry of the Lagrangian), except for the cases of spin (treated by Ryan [15]) and mass quadrupole moment (treated in this paper). Our proof is valid only in the approximations in which we work – expanding to linear order in the mass ratio, to the leading post-Newtonian order, and to linear order in the multipole. However we will show below that with very mild additional smoothness assumptions, our non- existence result extends to exact geodesic motion in exact vacuum spacetimes. We start in Sec. VA by showing that there is no co- ordinate system in which the Hamilton-Jacobi equation is separable. Now separability of the Hamilton-Jacobi equation is a sufficient but not a necessary condition for the existence of a additional conserved quantity. Hence, this result does not yield information about the existence or non-existence of an additional constant. Nevertheless we find it to be a suggestive result. Our actual derivation of the non-existence is based on Poisson bracket compu- tations, and is given in Sec. VB. A. Separability analysis Consider a binary of two point masses m1 and m2, where the mass m1 is endowed with a single axisymmet- ric current multipole moment Sl or axisymmetric mass multipole moment Il. In this section, we show that the Hamilton-Jacobi equation for this motion, to linear order in the multipoles, to linear order in the mass ratio and to the leading post-Newtonian order, is separable only for the cases S1 and I2. We choose the symmetry axis to be the z-axis and write the action for a general multipole as ṙ2 + r2θ̇2 + r2 sin2 θϕ̇2 + f(r, θ) + g(r, θ)ϕ̇+ E] . (5.1) For mass moments, g(r, θ) = 0, while for current mo- ments f(r, θ) = 0. For an axisymmetric multipole of order l, the functions f and g will be of the form f(r, θ) = clIlPl(cos θ) , g(r, θ) = dlSl sin θ∂θPl(cos θ) (5.2) where Pl(cos θ) are the Legendre polynomials and cl and dl are constants. We will work to linear order in f and g. In Eq. (5.1), we have added the energy term needed when doing a change of time variables, cf. the discussion before Eq. (2.14) in section III. Since ϕ is a cyclic coordinate, pϕ = Lz is a constant of motion and the system has effectively only two degrees of freedom. Note that in the case of a current moment, there will be correction term in Lz: Lz = r 2 sin2 θϕ̇+ g(r, θ). (5.3) Next, we switch to a different coordinate system (r̃, θ̃, ϕ) defined by r = r̃ + α(r̃, θ̃, Lz), (5.4) θ = θ̃ + β(r̃, θ̃, Lz), (5.5) where the functions α and β are yet undetermined. We also define a new time variable t̃ by 1 + γ(r̃, θ̃, Lz) dt̃. (5.6) Since we work to linear order in f and g, we can work to linear order in α, β, and γ. We then compute the action in the new coordinates and drop the tildes. The Hamiltonian is given by p2r(1 + γ − 2α,r) + (1− 2α − 2β,θ + γ) (−α,θ − r2β,r)− E(1 + γ) 2r2 sin2 θ (1 + γ − 2α − 2β cot θ) (1− α + γ)− f − gLz r2 sin2 θ (5.7) and the corresponding Hamilton-Jacobi equation is Ĉ1 + + 2V̂ , (5.8) where we have denoted Ĉ1 = J(r, θ) [1 + γ − 2α,r] = 1 + γ − 2α,r + j, (5.9) Ĉ2 = J(r, θ) 1− 2α − 2β,θ + γ = 1− 2α − 2β,θ + γ + j, (5.10) Ĉ3 = J(r, θ) −α,θ − r2β,r = −α,θ − r2β,r, (5.11) V̂ = J(r, θ) 2r2 sin2 θ (1 + γ − 2α − 2β cot θ) + γ)− E(1 + γ) − f − gLz r2 sin2 θ 2r2 sin2 θ (1 + γ − 2α − 2β cot θ + j) −E(1 + γ + j)− 1 (1− α + γ + j) −f − gLz r2 sin2 θ . (5.12) The unperturbed problem is separable, so make the perturbed problem separable, we have multiplied the Hamilton-Jacobi equation by an arbitrary function J(r, θ), which can be expanded as J(r, θ) = 1 + j(r, θ), where j(r, θ) is a small perturbation. To find a solution of the form W =Wr(r)+Wθ(θ), we first specialize to the case where Ĉ3 = 0: − Ĉ3 = β,rr2 + α,θ = 0. (5.13) We differentiate Eq. (5.8) with respect to θ, using Eq. (5.8) to write (dWr/dr) 2 in terms of (dWθ/dθ) 2 and then differentiate the result with respect to r to obtain ∂θĈ2 ∂θĈ1 2V̂ ∂θĈ1 Ĉ1Ĉ2 . (5.14) Expanding Eq. (5.14) to linear order in the small quan- tities then yields the two conditions for the kinetic and the potential part of the Hamiltonian to be separable: 0 = ∂r∂θ 2α,r − − 2β,θ , (5.15) sin2 θ 2β,r cot 2 θ − 3β,rθ cot θ + β,r csc2 θ sin2 θ + α,rθ −∂r∂θ Pl(cos θ) + dlSlLz rl sin θ ∂θPl(cos θ) 2α,rθ − + 2Er2α,rθ , (5.16) where we have used Eq. (5.2) for f and g. Therefore, the following conditions must be satisfied: M4(θ)−N(r) = + β,θ − 2α,r, (5.17) M1(θ) = 2β cot 2 θ + β csc2 θ + β,θθ −3β,θ cot θ, (5.18) M2(θ) = r 2∂r(r 2β,r), (5.19) M3(θ) = 2rα,rθ − α,θ + ∂θPl(cos θ) −SlLz ∂θ(csc θ ∂θPl(cos θ)). (5.20) Here, the functions M and N are arbitrary integration constants. Solving the condition for the kinetic term to be sep- arable, Eq. (5.17), together with Eq. (5.13) gives the general solution that goes to zero at large r as cos(nθ + ν), (5.21) β = − A sin(nθ + ν), (5.22) where A and ν are arbitrary and n is an integer. These functions must satisfy the conditions (5.18) – (5.20) in order for the potential term to be separable as well. To see when this will be the case, we start by considering Eq. (5.20). Substituting the general ansatz α = a1(r)a2(θ) shows that a′2 = P l or a 2 = (cscθ P ′ depending on whether a mass or a current multipole is present. The function a1(r) is then determined from 0 = 2ra′1 − a1 + clIl/r (l−1) dlSlLz/r (5.23) Hence, [clIl/(2l)] r (1−l) [dlSlLz/(2l+ 1)] r (5.24) so that we obtain for mass moments Pl(cos θ) , β = P ′l (cos θ) (5.25) and for current moments dlSlLz 2l+ 1 csc θP ′l (cos θ) , (5.26) dlSlLz (2l+ 1)(l + 1) (csc θ P ′l (cos θ)) , (5.27) where we have used the condition (5.13) to solve for β. Substituting this in Eq. (5.19) determines that l = 2 for mass moments and l+1 = 2 for current moments. For an l = 2 mass moment, conditions (5.17) and (5.18) are satisfied as well, with n = 2 and ν = 0. For the case of an l = 1 current moment, the extra term inH is independent of θ anyway. But for any other multipole interaction, the Hamilton-Jacobi equation will not be separable. For example, for the current octupole Sijk, the last term in Eq. (5.7) is proportional to S3Lz(5 cos 2 θ − 1)/r5 and is therefore not separable. From Eq. (5.2) one can see that, for a general multipole, the functions f or g contain different powers of cos θ appearing with the same power of r since the Legendre polynomials can be expanded as [34]: Pl(cos θ) = (−1)n(2l− 2n)! 2ln!(l − n)!(l − 2n)! (cos θ)l−2n, (5.28) where N = l/2 for even l and N = (l + 1)/2 for odd l. It will not be possible to cancel all of these terms with (5.21) – (5.22) for l > 2. The case when Ĉ3 is non-vanishing will only be sepa- rable if all the coefficients are functions of r or of θ only, and if in addition, the potential also depends only on r or on θ. Achieving this for our problem will not be possible because the potential cannot be transformed to the form required for separability. B. Derivation of non-existence of additional constants of the motion In this subsection, we show using Poisson brackets that for a single axisymmetric multipole interaction, to linear order in the multipole and the mass ratio, a first integral analogous to the Carter constant does not exist, except for the cases of mass quadrupole and spin. Suppose that such a constant does exist. We write the Hamiltonian corresponding to the action (5.1) as H = H0 + δH and the Carter-type constant as K = K0 + δK(pr, pθ, Lz, r, θ), where 2r2 sin2 θ , (5.29) δH = − clIl Pl(cos θ)− dlSlLz rl+2 sin θ ∂θPl(cos θ),(5.30) K0 = p sin2 θ . (5.31) Computing the Poisson bracket gives, to linear order in the perturbations 0 = {H0, δK}+ {δH,K0} (5.32a) δK + {δH,K0}, (5.32b) where we have used that {H0,K0} = 0 and the fact that {H0, δK} = d(δK)/dt. Here, d/dt denotes the total time derivative along an orbit (r(t), θ(t), pr(t), pθ(t)) of H0 in phase space. The partial differential equation (5.32a) for δK thus reduces to a set of ordinary differential equa- tions that can be integrated along the individual orbits in phase space. The unperturbed motion for a bound orbit is in a plane, so we can switch from spherical to plane polar co- ordinates (r, ψ). In terms of these coordinates, we have H0 = p r/2+p ψ/2, K0 = p ψ, and cos θ = sin ι sin(ψ+ψ0), with cos ι = Lz/ K and the constant ψ0 denoting the angle between the direction of the periastron and the intersection between the orbital and equatorial plane. Then Eq. (5.32) becomes δK = η(t), (5.33) η(t) = − 2pψ dlSlLz sin ι rl+2(t) ∂ψPl(sin ι sin(ψ(t) + ψ0)) cos(ψ(t) + ψ0) 2pψ clIl rl+1(t) ∂ψPl(sin ι sin(ψ(t) + ψ0)). (5.34) For unbound orbits, one can always integrate Eq. (5.33) to determine δK. However, for bound periodic orbits there is a possible obstruction: the solution for the conserved quantity K0 + δK will be single valued if and only if the integral of the source over the closed orbit vanishes, ∮ Torb η(t)dt = 0. (5.35) Here, Torb is the orbital period. In other words, the par- tial differential equation (5.32) has a solution δK if and only if the condition (5.35) is satisfied. This is the same condition as obtained by the Poincare-Mel’nikov-Arnold method, a technique for showing the non-integrability and existence of chaos in certain classes of perturbed dy- namical systems [35]. Thus, it suffices to show that the condition (5.35) is violated for all multipoles other than the spin and mass quadrupole. To perform the integral in Eq. (5.35), we use the parameterization for the unperturbed motion, r = K/(1+ e cosψ) and dt/dψ = K3/2/(1+ e cosψ)2, so that the condition for the existence of a conserved quantity K0 + δK becomes clIl(1 + e cosψ) l−1∂ψPl(sin ι sin(ψ + ψ0))− dlSlLz K sin ι (1 + e cosψ)l∂ψ ∂ψPl(sin ι sin(ψ + ψ0)) cos(ψ + ψ0) (5.36) In terms of the variable χ = ψ + ψ0 − π/2, Eq. (5.36) can be written as dχclIl [1 + e(sinψ0 cosχ− cosψ0 sinχ)]l−1 Pl(sin ι cosχ) dlSlLz sin ι [1 + e(sinψ0 cosχ− cosψ0 sinχ)]l Pl(sin ι cosχ) . (5.37) Inserting the expansion (5.28) for Pl(cosχ), taking the derivatives, and using the binomial expansion for the first term in Eq. (5.37), we get 0 = clIl Alnjk e j(sin ι)l−2n(sinψ0) k(cosψ0) dχ (sinχ)j−k+1(cosχ)k+l−2n−1 dlSlLz Blnjk e j(sin ι)l−2n−1(sinψ0) k(cosψ0) dχ (sinχ)j−k+1(cosχ)k+l−2n−2. (5.38) The coefficients Alnkj and Blnkj are Alnkj = (−1)n+k+1(l − 1)!(2l − 2n)! 2ln!(l − 1− j)!k!(j − k)!(l − n)!(l − 2n− 1)! , Blnkj = (−1)n+kl!(2l− 2n)! 2ln!(l − j)!k!(j − k)!(l − n)!(l − 2n− 2)! . (5.39) The only non-vanishing contribution to the integrals in Eq. (5.38) will come from terms with even powers of both cosχ and sinχ. These can be evaluated as multiples of the beta function: 0 = clIl Clnjk e j(sin ι)l−2n(sinψ0) k(cosψ0) j−k δ(j−k+1),even δ(l+k−1),even dlSlLz Dlnjk e j(sin ι)l−2n−1(sinψ0) k(cosψ0) j−k δ(j−k+1),even δ(l+k),even. (5.40) Here, the coefficients are Clnjk = 2Γ( j + 1)Γ(k − n+ 1) Alnkj , Dlnjk = 2Γ( j + 1)Γ(k − n− 1 − n+ 3 Blnkj (5.41) Eq. (5.40) shows that for even l, terms with j =even (odd) and k =odd (even) give a non-vanishing contribu- tion for the case of a mass (current) multipole, and hence K0+δK is not a conserved quantity for the perturbed mo- tion. Note that terms with j =even and k =odd for even l occur only for l > 3, so for l = 2 the mass quadrupole term in Eq. (5.40) vanishes and therefore there exists an analog of the Carter constant, which is consistent with our results of Sec. II and our separability analysis. For odd l, terms with j =odd (even) and k =even (odd) are finite for Il (Sl). Note that for the case l = 1 of the spin, the derivatives with respect to χ in Eq. (5.37) evaluate to zero, so in this case there also exists a Carter-type con- stant. These results show that for a general multipole other than I2 and S1, there will not be a Carter-type constant for such a system. 1. Exact vacuum spacetimes Our result on the non-existence of a Carter-type con- stant can be extended, with mild smoothness assump- tions, to falsify the conjecture that all exact, axisymmet- ric vacuum spacetimes posess a third constant of the mo- tion for geodesic motion. Specifically, we fix a multipole order l, and we assume: • There exists a one parameter family (M, gab(λ)) of spacetimes, which is smooth in the parameter λ, such that λ = 0 is Schwarzschild, and each space- time gab(λ) is stationary and axisymmetric with commuting Killing fields ∂/∂t and ∂/∂φ, and such that all the mass and current multipole moments of the spacetime vanish except for the one of order l. On physical grounds, one expects a one parameter family of metrics with these properties to exist. • We denote by H(λ) the Hamiltonian on the tan- gent bundle overM for geodesic motion in the met- ric gab(λ). By hypothesis, there exists for each λ a conserved quantity M(λ) which is functionally independent of the conserved energy and angular momentum. Our second assumption is that M(λ) is differentiable in λ at λ = 0. One would expect this to be true on physical grounds. • We assume that the conserved quantity M(λ) is invariant under the symmetries of the system: L~ξM(λ) = L~ηM(λ) = 0, where ~ξ and ~η are the natural extensions to the 8 dimensional phase space of the Killing vectors ∂/∂t and ∂/∂φ. This is a very natural assumption. These assumptions, when combined with our result of the previous section, lead to a contradiction, showing that the conjecture is false under our assumptions. To prove this, we start by noting that M(0) is a con- served quantity for geodesic motion in Schwarzschild, so it must be possible to express it as some function f of the three independent conserved quantities: M(0) = f(E,Lz,K0). (5.42) Here E is the energy, Lz is the angular momentum, and K0 is the Carter constant. Differentiating the exact re- lation {H(λ),M(λ)} = 0 and evaluating at λ = 0 gives {H0,M1} = {E,H1}+ {Lz, H1}+ {K0, H1}, (5.43) where H0 = H(0), H1 = H ′(0), and M1 = M ′(0). As before, we can regard this is a partial differential equa- tion that determines M1, and a necessary condition for solutions to exist and be single valued is that the integral of the right hand side over any closed orbit must vanish: {E,H1}+ {Lz, H1}+ {K0, H1} (5.44) Now strictly speaking, there are no closed orbits in the eight dimensional phase space. However, the ar- gument of the previous section applies to orbits which are closed in the four dimensional space with coordinates (r, θ, pr, pθ), since by the third assumption above every- thing is independent of t and φ, and pt and pφ are con- served. Here (t, r, θ, φ) are Schwarzschild coordinates and (pt, pr, pθ, pφ) are the corresponding conjugate momenta. Next, we can pull the partial derivatives ∂f/∂E etc. outside of the integral. It is then easy to see that the first two terms vanish, since there do exist a conserved energy and a conserved z-component of angular momentum for the perturbed system. Thus, Eq. (5.44) reduces to {K0, H1} = 0. (5.45) Since M(0) is functionally independent of E and Lz, the prefactor ∂f/∂K0 must be nonzero, so we obtain {K0, H1} = 0. (5.46) The result (5.46) applies to fully relativistic orbits in Schwarzschild. We need to take the Newtonian limit of this result in order to use the result we derived in the previous section. However, the Newtonian limit is a lit- tle subtle since Newtonian orbits are closed and generic relativistic orbits are not closed. We now discuss how the limit is taken. The integral (5.46) is taken over any closed orbit in the four dimensional phase space (r, θ, pr, pθ) which cor- responds to a geodesic in Schwarzschild. Such orbits are non generic; they are the orbits for which the ratio be- tween the radial and angular frequencies ωr and ωθ is a rational number. We denote by qr and qθ the angle vari- ables corresponding to the r and θ motions [36]. These variables evolve with proper time τ according to qr = qr,0 + ωrτ, (5.47a) qθ = qθ,0 + ωθτ, (5.47b) where qr,0 and qθ,0 are the initial values. We denote the integrand in Eq. (5.46) by I(qr, qθ, a, ε, ι), where I is some function, and a, ε and ι are the parame- ters of the geodesic defined by Hughes [32] (functions of E, Lz and K0). The result (5.46) can be written as ∫ T/2 dτ I[qr(τ), qθ(τ), a, ε, ι] = 0, (5.48) where T = T (a, ε, ι) is the period of the r, θ motion. Since the variables qr and qθ are periodic with period 2π, we can express the function I as a Fourier series I(qr, qθ, a, ε, ι) = n,m=−∞ Inm(a, ε, ι)einqr+imqθ . (5.49) Now combining Eqs. (5.47), (5.48) and (5.49) gives n,m=−∞ Inm(a, ε, ι)einqr,0+imqθ,0 ×Si [(nωr +mωθ)T/2] , (5.50) where Si(x) = sin(x)/x. Since the initial conditions qr,0 and qθ,0 are arbitrary, it follows that Inm(a, ε, ι)Si [(nωr +mωθ)T/2] = 0 (5.51) for all n, m. Next, for closed orbits the ratio of the frequencies must be a rational number, so , (5.52) where p and q are integers with no factor in common. These integers depend on a, ε and ι. The period T is given by 2π/T = qωr = pωθ. The second factor in Eq. (5.51) now simplifies to (np+mq)π , (5.53) which vanishes if and only if n = n̄q, m = m̄p, n̄+ m̄ 6= 0, (5.54) for integers n̄, m̄. It follows that Inm(a, ε, ι) = 0 (5.55) for all n, m except for values of n, m which satisfy the condition (5.54) Consider now the Newtonian limit, which is the limit a → ∞ while keeping fixed ε and ι and the mass of the black hole. We denote by IN(qr , qθ, a, ε, ι) the Newtonian limit of the function I(qr , qθ, a, ε, ι). The integral (5.48) in the Newtonian limit is given by the above computation with p = q = 1, since ωr = ωθ in this limit. This gives dτIN = INn,−n(a, ε, ι) ein(qr,0−qθ,0), (5.56) where INnm are the Fourier components of IN. In the previous subsection, we showed that this function is non- zero, which implies that there exists a value k of n for which IN k,−k 6= 0. Now as a → ∞, we have ωr/ωθ → 1, and hence from Eq. (5.52) there exists a critical value ac of a such that the values of p and q exceed k for all closed orbits with a > ac. (We are keeping fixed the values of ε and ι). It follows from Eqs. (5.54) and (5.55) that Ik,−k(a, ε, ι) IN k,−k(a, ε, ι) = 0 (5.57) for all such values of a. However this contradicts the fact Ik,−k(a, ε, ι) IN k,−k(a, ε, ι) → 1 (5.58) as a→ ∞. This completes the proof. Hence, if the three assumptions listed at the start of this subsection are satisfied, then the conjecture that all vacuum, axisymmetric spacetimes possess a third con- stant of the motion is false. Finally, it is sometimes claimed in the classical dynam- ics literature that perturbation theory is not a sufficiently powerful tool to assess whether the integrability of a sys- tem is preserved under deformations. An example that is often quoted is the Toda lattice Hamiltonian [38, 39]. This system is integrable and admits a full set of con- stants of motion in involution. However, if one approx- imates the Hamiltonian by Taylor expanding the poten- tial about the origin to third order, one obtains a sys- tem which is not integrable. This would seem to indicate that perturbation theory can indicate a non-integrability, while the exact system is still integrable. In fact, the Toda lattice example does not invalidate the method of proof we use here. If we write the Toda lattice Hamiltonian as H(q,p), then the situation is that H(λq,p) is integrable for λ = 1, but it is not integrable for 0 < λ < 1. Expanding H(λq,p) to third order in λ gives a non-integrable Hamiltonian. Thus, the perturba- tive result is not in disagreement with the exact result for 0 < λ < 1, it only disagrees with the exact result for λ = 1. In other words, the example shows that pertur- bation theory can fail to yield the correct result for finite values of λ, but there is no indication that it fails in ar- bitrarily small neighborhoods of λ = 0. Our application is qualitatively different from the Toda lattice example since we have a one parameter family of Hamiltonians H(λ) which by assumption are integrable for all values of λ. VI. CONCLUSION We have examined the effect of an axisymmetric quadrupole moment Q of a central body on test parti- cle inspirals, to linear order in Q, to the leading post- Newtonian order, and to linear order in the mass ratio. Our analysis shows that a natural generalization of the Carter constant can be defined for the quadrupole inter- action. We have also analyzed the leading order spin self- interaction effect due to the scattering of the radiation off the spacetime curvature due to the spin. Combining the effects of the quadrupole and the leading order effects linear and quadratic in the spin, we have obtained ex- pressions for the instantaneous as well as time-averaged evolution of the constants of motion for generic orbits un- der gravitational radiation reaction, complete at O(a2ǫ4). We have also shown that for a single multipole interaction other than Q or spin, in our approximations, a Carter- type constant does not exist. With mild additional as- sumptions, this result can be extended to exact space- times and falsifies the conjecture that all axisymmetric vacuum spacetimes possess a third constant of motion for geodesic motion. VII. ACKNOWLEDGMENTS This research was partially supported by NSF grant PHY-0457200. We thank Jeandrew Brink for useful cor- respondence. Appendix A: Time variation of quadrupole: order of magnitude estimates In this appendix, we give an estimate of the timescale Tevol for the quadrupole to change. The analysis in the body of this paper is valid only when Tevol ≫ Trr, where Trr is the radiation reaction time, since we have neglected the time evolution of the quadrupole. We distinguish be- tween two cases: (i) when the central body is exactly non- spinning but has a quadrupole, and (ii) when the central body has finite spin in addition to the quadrupole. 1. Estimate of the scaling for the nonspinning case For the purpose of a crude estimate, the relevant in- teraction is the tidal interaction with energy QijEij ∼ − Q̄I cos2 θ, (A1) where Eij is the tidal field, θ is the angle between the symmetry axis and the normal to the orbital plane of m2, and we have written the quadrupole as Q ∼ Q̄I, where Q̄ is dimensionless and I is the moment of inertia. For small deviations from equilibrium, the relevant piece of the Lagrangian is schematically L ∼ Iψ̇2 + Q̄I m2 ψ2. (A2) We define the evolution timescale Tevol to be the time it takes for the angle to change by an amount of order unity, and since the amplitude of the oscillation scales roughly as ∼ m2/m1, the evolution time scales as T−2evol ∼ ω2orbit, (A3) where ω2orbit = M/r 3. Thus, the ratio of the evolution timescale compared to the radiation reaction timescale scales as Tevol/Trr ∼ . (A4) 2. Estimate of the scaling for the spinning case When the body is spinning the effect of the tidal cou- pling is to cause a precession. For the purpose of this estimate, we calculate the torque on m1 due to the com- panion’s Newtonian field. The torque N scales as Ni ∼ ǫimjQmkEjk. (A5) We assume that the precession is slow, i.e. ωprec ≪ S̄/m1 , (A6) where ωprec is the precession frequency and S̄ = S/m is the dimensionless spin. This gives the approximate scaling of the precession timescale as (cf. [37]) Tprec/Trr ∼ . (A7) and the evolution timescale is thus Tevol/Trr ∼ . (A8) Because of our assumption (A6) that the precession is slow, equation (A8) is valid only when ) S̄2 . (A9) When S̄ is sufficiently small that the condition (A9) is violated, the relevant timescale is instead given by Eq. (A3). 3. Application to Kerr inspirals For Kerr inspirals, S̄ ∼ a, Q̄ ∼ a2, µ/M ≪ 1 and r ∼M. (A10) Therefore, the condition (A9) is satisfied, and the pre- cession time is longer than the radiation reaction time Tprec/Trr ∼ . (A11) Note that for Kerr inspirals, since r ∼ M both formulas (A3) and (A7) give the same scaling. Moreover, for Kerr inspirals, the amplitude of the pre- cession will be small, of order the mass ratio µ/M . This is because of angular momentum conservation: in the rela- tivistic regime, the orbital angular momentum is a factor of µ/M smaller than the angular momentum of the black hole and can therefore not cause a large precession ampli- tude. Even if the orbital angular momentum at infinity is large, most of it will be radiated away as outgoing gravitational waves during the earlier phase of the inspi- ral. This factor of µ/M is taken into account when we consider the evolution timescale, which for Kerr inspirals reduces to Tevol/Trr ∼ . (A12) Since 1/a ≥ 1, M/r ∼ 1 and M/µ ≪ 1, the evolution time is long compared to the radiation reaction time and we can neglect the time variation of the quadrupole at leading order. Appendix B: Computation of time averaged fluxes 1. Averaging method that parallels fully relativistic averaging We start by noting that the differential equations (2.26) and (2.27) governing the r̃ and θ̃ motions decouple if we define a new time parameter t̂ by dt̂ = dt̃. (B1) This is the analog of the Mino time parameter for geodesic motion in Kerr [12]. The equations of motion (2.26)–(2.24) then become = V̂r̃(r̃), (B2) V̂r̃(r̃) = 2Er̃ 4 + 2r̃3 −Kr̃2 − 4SLzr̃ r̃ − 2L2z , (B3) = V̂θ̃(θ̃), (B4) V̂θ̃(θ̃) = K − sin2 θ̃ −QE cos 2θ̃, (B5) = V̂ϕr̃(r̃) + V̂ϕθ̃(θ̃), (B6) V̂ϕr̃(r̃) = , V̂ϕθ̃(θ̃) = sin2 θ̃ . (B7) The parameters t and t̂ are related by: = V̂tr̃(r̃) + V̂tθ̃(θ̃) (B8) V̂tr̃(r̃) = r̃ 2, V̂tθ̃(θ̃) = cos 2θ̃. (B9) It follows from Eqs. (B2) and (B4) that the functions r̃(t̂) and θ̃(t̂) are periodic; and we denote their periods by Λr̃ and Λθ̃. We define the fiducial motion associated with the constants of motion E, Lz and K to be the motion with the initial conditions r̃(0) = r̃min and θ̃(0) = θ̃min, where r̃min and θ̃min are given by the vanishing of the right-hand sides of Eqs. (B2) and (B4) respectively. The functions r̂(t̂) and θ̂(t̂) associated with this fiducial motion are given by ∫ r̂(t̂) r̃min V̂r̃(r̃) = t̂, (B10) ∫ θ̂(t̂) θ̃min V̂θ̃(θ̃) = t̂. (B11) From Eq. (B8) it follows that t(t̂) = t0 + V̂tr̃[r̃(t ′)] + V̂tθ̃[θ̃(t , (B12) where t0 = t(0). Next, we define the constant Γ to be the following average value: ∫ Λr̃ dt′V̂tr̃[r̂(t ′)] + dt′V̂tθ̃[θ̂(t ′)]. (B13) Then we can write t(t̂) as a sum of a linear term and terms that are periodic: t(t) = t0 + Γt̂+ δt(t̂), (B14) where δt(t̂) denotes the oscillatory terms in Eq. (B12). To average a function over the time parameter t̂, it is convenient to parameterize r̃ and θ̃ in terms of angular variables as follows. For the average over θ̃ we introduce the parameter χ by cos2 θ̂(t̂) = z− cos 2 χ, (B15) where z− = cos 2 θ̃− with z− being the smaller root of Eq. (B4): K + 3QE ± (K −QE)2 + 4QEL2z (B16) and where β = 2QE. Then from the definition (B11) of θ̂ together with Eq. (B4) and the requirement that χ increases monotonically with t̂ we obtain β (z+ − z− cos2 χ). (B17) Then we can write the average over t̂ of a function Fθ̃(t̂) which is periodic with period Λθ̃ in terms of χ as 〈Fθ̃〉t̂ = dt̂Fθ̃(t̂) Fθ̃[t̂(χ)] β (z+ − z− cos2 χ) , (B18) where Λθ̃ = β (z+ − z− cos2 χ) . (B19) Similarly, to average a function Fr̃(t̂) that is periodic with period Λr̃, we introduce a parameter ξ via 1 + e cos ξ , (B20) where the parameter ξ varies from 0 to 2π as r̃ goes through a complete cycle. Then, = P (ξ), (B21) P (ξ) ≡ V̂r̃[r̃(ξ)] pe | sin ξ | (1 + e cos ξ) (B22) The average over t̂ of Fr̃(t̂) can then be computed from 〈Fr̃〉t̂ = dξ Fr̃/P (ξ) dξ/P (ξ) . (B23) Now, a generic function Fr̃,θ̃[r̃(t̂), θ̃(t̂)] will be biperiodic in t̂: Fr̃,θ̃[r̃(t̂+Λr̃), θ̃(t̂+Λθ̃)] = Fr̃,θ̃[r̃(t̂), θ̃(t̂)]. Combin- ing the results (B18) and (B23) we can write its average as a double integral over χ and ξ as 〈Fr̃,θ̃〉t̂ = Λθ̃Λr̃ Fr̃,θ̃[r̃(ξ), θ̃(χ)] β (z+ − z− cos2 χ)P (ξ) (B24) To compute the time average of Ė, L̇z, and K̇, we need to convert the average of a function over t̂ calculated from (B24) to the average over t. As explained in detail in [9], in the adiabatic limit we can choose a time interval ∆t which is long compared to the orbital timescale but short compared to the radiation reaction time. From Eq. (B12) we have ∆t = Γt̂+ osc.terms. The oscillatory terms will be bounded and will therefore be negligible in the adiabatic limit, so we have to a good approximation 〈Ė〉t = 〈Ė V̂t〉t̂, (B25) where V̂t ≡ V̂tr̃ + V̂tθ̃, cf. Eq. (B8), and similarly for L̇z and K̇. The explicit results we obtain using this method are given in section III, Eqs. (3.28), (3.29), and (3.30). 2. Averaging method using the explicit parameterization of Newtonian orbits To perform the time-averaging using this method, we define a parameter ξ via 1 + e cos ξ , (B26) where the parameter ξ varies from 0 to 2π as r̃ goes through a complete cycle. Note that θ appears in Eqs. (3.16) – (3.18) only in terms that are linear in Q, so we can write θ in terms of ξ using the Newtonian relation x3 = r cos θ = r sin ι sin(ξ + ξ0). (B27) Here, ξ0 is the angle between the direction of the peri- helion and the intersection of the orbital and equatorial plane. Similarly, for the ṙθ̇ terms in Eqs. (3.17) and (3.26) we can use the Newtonian relations ṙ = e/ p sin ξ and ξ̇ = p/r2. From Eqs. (2.27) and (B20) it follows (1 + e cos ξ)2 −3 + e2 − 2e cos ξ + 2 cos2 ι(8 − e2 + 8e cos ξ + e2 cos 2ξ) , (B28) and from Eq. (2.12) (1 + e cos ξ) 2 sin2 ι sin2(ξ + ξ0)− 1 . (B29) Using these expressions, we compute the time-averaged fluxes from 〈Ė〉 = dξ Ė (dt/dt̃) (dt̃/dξ) dξ (dt/dt̃) (dt̃/dξ) (B30) and obtain: 〈Ė〉 = −32 (1− e2)3/2 e4 − S cos(ι) cos(2ι) cos(2ι) cos(2ξ0) sin cos(2ξ0) sin , (B31) 〈L̇z〉 = − (1− e2)3/2 cos ι e2 − S 2p3/2 cos ι + 7e2 + cos(2ι) −3− 45 45 + 148e2 + cos(2ι) 1 + 3e2 + e2 cos(2ξ0) sin , (B32) 〈K̇〉 = −64 (1 − e2)3/2 e2 − S 2p3/2 + 37e2 + cos(ι) cos(2ι) cos(2ι) e2 cos(2ξ0) sin . (B33) In the adiabatic limit, the terms involving cos(2ξ0) can be omitted because they average to zero. As explained by Ryan [15], the radiation reaction timescale for terms involving ξ0 is much longer than the precession timescale for most orbits, so the terms involving ξ0 will average away. This is consistent with our results for the adia- batic infinite time-averaged fluxes using the Mino time parameter. The Mino-time averaging method was based on the assumption that the fundamental frequencies are incommensurate and the motion fills up the whole torus, which is equivalent to averaging over ξ0. [1] L. Barack and C. Cutler, Phys. Rev. D 69, 082005 (2004) [2] K. Glampedakis and S. Babak, Class. Quantum Grav. 23, 4167 (2006) [3] D. A. Brown, et al. gr-qc/0612060 [4] E. Poisson, Living Rev. Relat. 7, 6 (2004), http://relativity.livingreviews.org/Articles/lrr-2004-6/index.html [5] Special Issue: Gravit. Rad. from Binary Black Holes: Ad- vances in the perturbative approach, Class. Quant. Grav. 22 (2005) [6] K. Glampedakis, Class. Quantum Grav. 22, S605 (2005) [7] S. Drasco, Class. Quantum Grav. 23, S769 (2006) [8] M. Favata and É. É. Flanagan Accuracy of adiabatic waveforms for eccentric orbits, (in preparation) [9] S. Drasco, É. É. Flanagan, and S. A. Hughes, Class. Quantum Grav. 22, S801 (2005) [10] É. É. Flanagan and T. Hinderer, Two timescale analysis of extreme mass ratio inspirals in Kerr. II. Numerical integration through resonances, (in preparation) [11] N. Sago, et al., Progr. Theor. Phys. 114, 509 (2005) [12] Y. Mino, Phys. Rev. D 67, 084027 (2003) [13] S. A. Hughes, S. Drasco, É. É. Flanagan, and J. Franklin, Phys. Rev. Lett. 94, 221101 (2005) [14] F. D. Ryan, Phys. Rev. D 52, R3159 (1995) [15] F. D. Ryan, Phys. Rev. D 53, 3064 (1996) [16] L. Blanchet, T. Damour, G. Farese, and B. Iyer, Phys. Rev. D 71, 124004 (2005) [17] L. Kidder, C. Will, and A. Wiseman, Phys. Rev. D 47, R4183 (1993) [18] L. A. Gergely, Phys. Rev. D 61, 024035 (1999) [19] C. Will, Phys. Rev. D 71, 084027 (2005) [20] G. Faye, L. Blanchet, and A. Buonanno, Phys. Rev. D 74, 104033 (2006) [21] L. Blanchet, A. Buonanno, and G. Faye, Phys. Rev. D 74, 104034 (2006) [22] E. Poisson, Phys. Rev. D 57, 5287 (1998) [23] L. A. Gergely and Z. Keresztes, Phys. Rev. D 67, 024020 (2003) [24] M. Shibata, M. Sasaki, H. Tagoshi, and T. Tanaka, Phys. Rev. D 51, 1646 (1995) [25] É. É. Flanagan and T. Hinderer, Two timescale analy- sis of extreme mass ratio inspirals in Kerr. I. General formalism, (in preparation) [26] C. Misner, K. Thorne, and J. Wheeler, Gravitation (W.H. Freeman and Co., San Francisco, 1973) [27] R. O. Hansen, J. Math. Phys. 15, 46 (1974) [28] K. S. Thorne, Rev. Mod. Phys 52, 300 (1980) [29] T. Bäckdahl, gr-qc/0612043 (2006); Class. Quantum Grav. 22, 3585 (2005) [30] C. Li and G. Lovelace, gr-qc/0702146 (2007) [31] L. Blanchet and T. Damour, Phys. Lett. 104A, 82 (1984) [32] S. A. Hughes, Phys. Rev. D 61, 084004 (2000) [33] K. Glampedakis, S. Hughes, and D. Kennefick, Phys. Rev. D 66, 064005 (2002) [34] G. Arfken, Mathematical Methods for Physicists (Aca- demic Press, CA, 1985) [35] V. K. Melnikov, Trans. Moscow Math. Soc. 12, 1-56 (1956) [36] W. Schmidt, Celestial mechanics in Kerr spacetime Class. Quantum Grav. 19 (2002) 2743-2764 [37] H. Goldstein, C. Poole, and J. Safko, Classical Mechanics (Addison-Wesley, 2002) [38] Ý. Birol and A. Hacinliyan, Phys. Rev. E 52, 4750 (1995) [39] H. Yoshida, Comm. Math. Phys. 116, 529 (1988) http://arxiv.org/abs/gr-qc/0612060 http://relativity.livingreviews.org/Articles/lrr-2004-6/index.html http://arxiv.org/abs/gr-qc/0612043 http://arxiv.org/abs/gr-qc/0702146

0704.0392

Simulation of Robustness against Lesions of Cortical Networks

Simulation of Robustness against Lesions of Cortical Networks Abbreviated title: Simulation of Robustness of Cortical Networks Marcus Kaiser1,2,3,a, Robert Martin2,4,a, Peter Andras1,2 and Malcolm P. Young2 1 School of Computer Science, University of Newcastle, Claremont Tower, Newcastle upon Tyne, NE1 7RU, UK 2 Henry Wellcome Building for Neuroecology, Institute of Neuroscience, University of Newcastle, Framlington Place, Newcastle upon Tyne, NE2 4HH, UK 3 Jacobs University Bremen, School of Engineering and Science, Campus Ring 6, 28759 Bremen, Germany 4 FR 2-1, NI, Informatik, Technische Universität Berlin, Franklinstr. 28/29, 10587 Berlin, Germany aAuthors contributed equally to this paper Correspondence: Marcus Kaiser; School of Computer Science, University of Newcastle, Claremont Tower, Newcastle upon Tyne, NE1 7RU, UK; E-mail: [email protected] Keywords: cat, macaque monkey, resilience, scale-free networks, small-world networks ABSTRACT Structure entails function and thus a structural description of the brain will help to understand its function and may provide insights into many properties of brain systems, from their robustness and recovery from damage, to their dynamics and even their evolution. Advances in the analysis of complex networks provide useful new approaches to understanding structural and functional properties of brain networks. Structural properties of networks recently described allow their characterization as small-world, random (exponential) and scale-free. They complement the set of other properties that have been explored in the context of brain connectivity, such as topology, hodology, clustering, and hierarchical organization. Here we apply new network analysis methods to cortical inter-areal connectivity networks for the cat and macaque brains. We compare these corticocortical fibre networks to benchmark rewired, small-world, scale-free and random networks, using two analysis strategies, in which we measure the effects of the removal of nodes and connections on the structural properties of the cortical networks. The brain networks’ structural decay is in most respects similar to that of scale-free networks. The results implicate highly connected hub-nodes and bottleneck connections as structural basis for some of the conditional robustness of brain systems. This informs the understanding of the development of brain networks’ connectivity. INTRODUCTION The brain can be remarkably robust to physical damage. Significant loss of neural tissue can be compensated for in a relatively short time by large-scale adaptation of remaining brain parts (e.g., Spear et al., 1988; Stromswold, 2000; Young, 2000). On the other hand, the removal of small amounts of tissue (e.g. in Broca’s area) can lead to a severe functional deficit. These findings provide a somewhat contradictory picture of the robustness of the brain and suggest a number of questions. Can we evaluate effective robustness given this variability in the effects of brain lesions? Are severity and nature of effects of localized damage predictable? We assess here how connectivity data of brain area connectivity can be brought to bear on these questions. The functionality of any system is grounded in its structural properties. For neurosciences, this has led to exploration of the structural properties of brain networks, such as topology, hodology, clustering, and hierarchical organization (e.g., Nicolelis et al., 1990; Felleman and van Essen, 1991; Young, 1992; Young et al., 1994; Hilgetag et al., 1996; Hilgetag et al., 2000; Sporns et al., 2000; Young, 2000; Petroni et al., 2001; Sporns et al., 2004; Kaiser and Hilgetag, 2006). Recent advances in the study of networks have extended those traditional structural descriptions (Strogatz, 2001), allowing to characterize networks as small-world (Watts and Strogatz, 1998), random and scale-free (e.g., Barabási and Albert, 1999; Albert et al., 2000). Small-world networks comprise well-connected local neighbourhoods with fewer long-range connections between neighbourhoods. The length of a path between two nodes, that is the number of connections that have to be crossed to go from one node to another, is comparable as low as for a randomly organized network. Scale-free networks are characterized by their specific distribution of connectivities or degrees—the number of connections that each node has. The degree distribution follows a power law. Whereas these networks can have highly-connected nodes or hubs also networks where nodes have maximally 20 connections have been described as scale-free based on the power-law degree distribution (Jeong et al., 2001). Small-world and scale-free properties are compatible, but not equivalent (see, e.g., Amaral et al., 2000). Scale-free networks have higher robustness than random ones against randomly located damage, whilst being sensitive to damage targeted at their most widely connected nodes (Barabási and Albert, 1999; Young, 2000). This is reminiscent of the properties of the brain described above. Previous studies have shown that functional networks of the human brain are scale-free (Eguiluz et al., 2005). However, at the level of resting state networks between cortical areas, it was argued that these networks are not scale-free (Achard et al., 2006). Here we analyze what pattern occurs at the level of structural connectivity. In order to establish, whether the brain has properties of scale-free networks, the integrity and robustness to damage of brain networks’ structure is compared to that of benchmark random and scale-free networks (an earlier version of this work had been presented as a conference abstract, see Martin et al., 2001). MATERIALS AND METHODS Brain structure connectivity data. We used macaque and cat cortical inter-areal connectivity data (Young, 1993; Young et al., 1994; Scannell et al., 1995) applying the CoCoMac database for the primate data (Stephan et al., 2001; Kötter, 2004). In both species, the data comprised connections among cortical regions of the neocortex. For the macaque brain, we considered 66 brain structures with 608 connections between them. In the case of the cat brain, we considered 56 structures and 814 connections. We excluded cross- hemispheric connections. The data was represented as the binary connectivity matrix of a graph. Nodes corresponded to the considered brain structures and edges to the reported connections between them. Note that due to the directed nature of brain connections, the connectivity matrix is not necessarily symmetric and the resulting graph has hence directed edges too. The edge density of the macaque brain graph, that is, the number of reported connections divided by the number of all possible connections, is 26.4% (Tab. 1). For the cat brain, the edge density is 14.2%. There are on average 9.2 connections for each structure in the macaque brain and 14.5 connections for the cat brain structures (see supplementary material for the connectivity matrices). [Table 1 near here] Benchmark networks for comparison. We constructed rewired, scale-free, small-world, and random networks to match the number of nodes and connections of the corresponding two brain networks (Tab. 1). Figure 1 shows examples of small random and scale-free networks to demonstrate differences in their topology. For random networks, the number of connections of a node is close to the average value over all nodes. For scale-free networks, however, nodes with a much higher number of connections can occur; see hub in Fig. 1b. [Figure 1 near here] Rewired networks. For rewired networks, each node has the same number of connections as in the original network, however, targets or sources of connections might have changed.Rewired networks were derived from the original cortical networks of the cat and macaque by exchanging connections so that the total number of connections of each node remained the same (the method for randomization is described in Milo et al., 2002). Whereas the degree distribution remains unchained, the cluster architecture is lost during rewiring. Thus, rewired networks allow looking at effects of the degree distribution alone. Scale-free networks. The algorithm to generate scale-free benchmark networks is based on Barabási and Albert (1999). However, in a modification of their approach we began with an initial graph of six and eight fully connected nodes respectively for the macaque and cat benchmark networks. This was necessary in order to ensure that the clustering coefficient (average percentage of connections between neighbours of a node; see definition below) of the initial graph matched the highest clustering coefficient found in the corresponding brain network. As proposed by Barabási and Albert (1999), further nodes were added one by one to the graph by preferential attachment. At the beginning of this process, the probability that a new node is connected to an existing node i is iP )( , where kj is the number of connections of the node j (Barabási and Albert, 1999). After establishing a connection to node i*, the probabilities are recalculated to reflect the nature of the scale-free networks: if i is connected to j, then it is more likely that i is connected to nodes which are already connected to j and it is less likely that i is connected to nodes which are not connected to j. The rescaling was undertaken according to connectednot areandif,)( connectedareandif),( iiiPk The probability for the connections in both directions is the same. We confirmed that this modified routine for generating scale-free networks was able to yield a power-law degree distribution (cf. supplementary material). Small-world networks. Small-world networks were generated by rewiring regular networks as described in the literature (Watts and Strogatz, 1998). The rewiring probability was adjusted so that the resulting networks had similar clustering coefficient than the respective cortical networks (Tab. Random networks. Whereas all benchmark networks are generated by a random process, we denote Erdös-Renyi random networks (Erdös and Rényi, 1960) as random networks in the remaining manuscript. Random networks were generated by establishing each potential connection between nodes with probability p. This probability was the desired connection density, that means, the connection density of the corresponding brain networks, 14.2% of the number of all possible connections for the macaque and 26.4% for the cat. The degree distribution in these random networks followed a binomial probability distribution. For large numbers of nodes this can be approximated by a Poisson distribution and hence the term ‘exponential degree distribution’ is also used (Bollobas, 1985). Graph similarity. To assess the discrepancy in connectivity between two networks, first their nodes are permuted according to their number of connections. Second, permutated cortical and benchmark matrices are compared by looking what ratio of directed edges in the adjacency matrix that occurred at the same position in both matrices and the total number of directed edges. This percentage is then the graph similarity S between graph A and B given the number of (directed) connections |E|: where  is element-by-element multiplication with an element in the resulting matrix non-zero if both elements are non-zero; Σ is the sum of all elements in the matrix and thus yields the number of directed edges existing in both matrices, as these are denoted by a value of one in the matrix. Note, that benchmark networks could be more similar than they appear for this measure as not all possible arrangements of nodes were tested. Testing all possibilities (1092 for the macaque and 1074 for the cat) would have been computationally unfeasible. Network characterisation. The clustering coefficient shows the fragmentation of the network. The coefficient is the ratio of the number of existing edges between neighbours of a node i and the number of possible edges between all these neighbours. We considered neighbouring nodes of node i to be all those nodes that have incoming or outgoing connections between them and node i. If a node i has ki neighbours, then the number of all possible in- and outgoing edges between the neighbouring nodes is ki * (ki – 1). The coefficient itself is a local property of each node and we define the average coefficient of all nodes to be the clustering coefficient of the graph. This is a measure of how well connected the nodes of the network are. Following Albert et al. (2000), we considered the average shortest path (ASP) or characteristic path length to characterize the network connectivity and integrity. The ASP between any two nodes in the network is the number of sequential connections that are necessary, on average, to link one node to another by the shortest possible route (Diestel, 1997). In case a network becomes disconnected in the process of removing edges/nodes and there is no path between two nodes, the pair of nodes is ignored. If no two connected points are left, the average shortest path is set to zero. We used Floyd’s algorithm to determine the matrix of the shortest paths between each pair of nodes (Cormen et al., 2001). Note that due to directed edges, the shortest path from node i to node j may not be the same as that from node j to node i. Target determination. In order to determine the importance of a node to the overall network structure, a simple metric has been used, namely the number of connections formed by this node. In experiments requiring the targeted removal of nodes from the networks, the most highly connected node was eliminated. To provide the corresponding metric for the targeted elimination of connections (edges) from the network, we chose edge betweenness (Girvan and Newman, 2002), that is, the number of shortest paths between all pairs of nodes that pass though the edge. Edges with high edge betweenness are chosen for targeted attack. Indeed, edge betweenness has been shown to highly correlate with structural network damage for cortical as well as other biological networks (Kaiser and Hilgetag, 2004). Analysis methods. We used the iterative random and targeted removal of nodes and connections to analyze the robustness of the networks against damage. Random removal means that we selected a node or connection and deleted it from the graph irrespective of the degree of the node. In the case of targeted removal, we selected the most important node or connection left in the network (see above). After each deletion, we calculated the ASP of the resulting graph. We continued the removal of nodes or connections until all nodes were removed from the network. To derive estimates of the variability in these connectivity measures, we considered 50 benchmark networks for each condition. In the cases of random removal, we repeated the analysis for the brain networks 50 times as well. RESULTS Degree distribution of cortical networks Fig. 2 shows the degree distributions of macaque and cat compared with a distribution of random networks. In comparison to random networks, the macaque cortical network has highly connected nodes but also more sparsely connected nodes, reminiscent of scale-free networks. This is also true for the cat network that shows a remarkable number of areas with few connections compared to random networks. Table 2 shows the five most-highly connected nodes for the cat and macaque networks. [Table 2 near here] The standard way of observing whether the cortical network resembles a scale-free network would be to search for a power-law in the degree distribution. However, this approach would be inappropriate for cortical networks for three reasons. First, the maximum number of connections of a node equals the number of regions in the network minus one, that means, 65 (macaque) or 55 (cat). Therefore, the degree distribution only consists of two scales. Second, where degree distributions with a low maximal degree had been studied before (Jeong et al., 2001), the number of nodes was considerably higher (>1,800). As less than 100 degrees form the degree distribution, results are unlikely to be robust. Third, there exists a sampling problem in that the amount of unknown or not included connections might change the shape of the degree distribution (Stumpf et al., 2005). Therefore, we will use indirect measures to determine whether cortical networks are similar to scale- free networks. [Figure 2 near here] Graph similarity Whereas the degree distribution is an abstraction of the underlying network, we looked at a direct comparison between the cortical and benchmark networks. Whereas a direct measure of network similarity was computationally unfeasible (see Methods), we compared the adjacency matrices after ordering nodes by their degrees (see methods). We then looked at the similarity of cortical networks with different benchmark networks (Fig. 3). For rewired cortical networks, the percentage of identical edges was 23% for rewired macaque and 38% for the rewired cat network. Interestingly, benchmark scale-free networks are as similar to the cortical networks as the rewired cortical networks. In contrast, the similarity of random and small-world networks is significantly lower. This can be attributed to the degree distribution of scale-free and cortical networks being comparable as the rewired network only has the degree distribution in common with the original cortical network. After these structural properties, we tested the effect of topological changes on general network properties. [Figure 3 near here] Sequential elimination of nodes We tested the influence of sequential node elimination on the network structure. Nodes were removed one by one from the network, either randomly or targeted. Plotting the ASP as a function of the fraction of deleted nodes illustrates the characteristic structural disintegration of each network type (See Fig. 4 for the example of targeted elimination of nodes from the Macaque benchmark networks. The complete set of curves for the different analysis types is available as supplementaryt material). [Figure 4 near here] Fig. 4A illustrates the effect of random and targeted removal of nodes from the Macaque brain network. Clearly, the specific decline in ASP is different for the two analysis strategies. Whilst the random removal causes only a slow rise in the ASP, targeted removal of highly connected nodes has a much stronger effect on the network structure of the brain network. After a steep rise in ASP the network fragments into smaller components. The remaining shortest paths, that is the paths between nodes within components, are smaller than in the original network. This process leads to a network with pairs of nodes that are connected to each other but not to other nodes of the network. In these cases, the shortest path decreases to a value of one. Finally, also nodes within pairs are removed leading to an ASP of zero. Fig. 4B–C contrast this specific curve to those observed when removing nodes from the different benchmark networks in a targeted fashion. Whilst the ASP in the random and small-world networks is hardly affected by the targeted elimination of a large proportion of nodes, in the scale-free, like in the brain networks, the effect of targeted node elimination manifests itself in a sharp rise in this measure. Moreover, both, the scale-free and the brain networks show a decline in the ASP around the fraction of deletions, and the characteristic behaviour of the brain network is within the 95% confidence interval encountered for the set of scale-free benchmark networks. This is not the case for the other benchmark networks considered (see Fig. 4). For the cat brain network (Fig. 5), the random and small-world networks show a different behaviour for targeted node removal than the original cortical network. Though the cat response to targeted node removal is largely within the 95% confidence interval for the scale-free benchmark networks, the peak ASP value and the fraction of deleted nodes where the peak occurs is lower for the cat cortical network. [Figure 5 near here] The decline in ASP at a later stage during the elimination process, as observed for the brain and scale-free networks may appear unusual and deserves some additional attention. It can have two reasons. First, it could be that the network gets fragmented into different disconnected components. Each of these is smaller, and likely to have a shorter ASP. Second, the overall decrease in network size with successive eliminations can lead to a decrease in shortest path. This is, however, likely to be a slow process, as it will usually be offset by an increase in ASP due to the targeted nature of the elimination. In order to quantitatively compare the different graphs, we consider two measures. The first is the maximal ASP measured during the removal of nodes; the second is the fraction of deleted nodes, for which the peak ASP occurs (Fig. 6). For the fraction of peak ASP, only the scale-free benchmark networks are close to the cortical fraction whereas all other benchmark networks show significantly higher fractions. This means that both in the cortical as well as the scale-free networks the removal of highly-connected nodes leads to a rapid increase of ASP so that the fraction of deleted nodes at which the maximum ASP occurs is earlier than for other networks. However, the peak value for scale-free networks is greater than that for cortical networks. [Figure 6 near here] Sequential elimination of connections We also tested the similarity of sequential connection elimination. Connections were eliminated one after another either randomly or targeted. Full details of the networks disintegration are shown in the supplementary material. Again we compare the maximal ASP measured during the removal of connections and the fraction of deleted connections, for which the peak ASP occurs (Fig. 7). [Figure 7 near here] Only the scale-free benchmark networks yield similar values for both the cat and macaque network whereas other networks yield similar values for just one of the cortical networks. DISCUSSION We have compared brain inter-area connectivity networks with different types of benchmark networks, including random, scale-free, and small-world networks, and found strong indications that the brain connectivity networks share some of their structural properties with scale-free networks. Besides a formal assessment of the network connectivity (degree distribution and graph similarity, see Figures 3 and 4), the analysis is based on a novel approach, which measures the effect of removal of components of the different networks on their structural integrity. In particular, we compared the effect that the removal of nodes and connections had on the ASP found in the brain connectivity networks and their benchmark counterparts. Note, however, that this analysis is based on cortical connectivity within one hemisphere. Connections between hemispheres and between the cortex and subcortical structures such as thalamic regions were not included. The reason for the lack of interhemispheric connections was that few tracing studies tested for and thus reported fibre tracts towards the contralateral hemisphere. Whereas information about thalamocortical connections would have been available, regions with available information about fibre tracts differed between the cat and macaque. To be consistent between species, the data was not included. For each species, an inclusion of these regions yielded similar results concerning the removal of nodes or edges (supplementary material). Simulated robustness and its relation to lesion studies How do our simulations relate to experimental lesion studies? Node elimination corresponds indirectly to inactivation or lesion of the corresponding brain areas, and from this perspective, we can interpret this analysis in terms of the brain’s robustness to regional damage. The elimination of connections corresponds indirectly to localized brain lesions that damage the white matter and interrupt communication between normally connected brain structures. The ASP yields a measure how well the brain is connected and how well different streams of information can be integrated. Analysing the spatial organisation of cortical networks shows that the brain is optimized towards a low ASP (Kaiser and Hilgetag, 2006). A recent clinical study of the EEG correlation network in Alzheimer patients suggests that increases in ASP lead to a reduced performance in memory tasks (Stam et al., 2007). In this study, the ASP of the EEG synchronization network has been higher in Alzheimer patients compared to the control group. Furthermore, there was a negative correlation between the patients’ ASP and their performance in a standard clinical memory test. Whereas the study was based on functional rather structural/anatomical networks, recent studies using diffusion tensor imaging have shown that changes in brain connectivity can be linked to diseases such as Schizophrenia and Alzheimer. All observations have been made equally during the analysis of the brain networks of cat and macaque, despite different edge densities in the two networks. It is therefore prudent to conclude that it may be extended to other mammalian brain networks. Hence, conditional robustness of brain function may be based to a large extent on two fairly simple structural properties of brain networks: firstly, the number of connections of individual nodes (Young, 2000), i.e., their scale-free nature, and secondly, the heavily connected local clusters with fewer important ‘bottlenecks’ between them (Kaiser and Hilgetag, 2004). Consequently, it appears feasible to determine the brain structures that are the most important to the maintenance of network function. Typically, brain networks should be able to function robustly in the face of damage to structures that have few connections and damage to connections that do not form part of many shortest connections between pairs of areas. On the other hand, functional effects should be dramatic when structures with very many connections (hubs) are damaged and when connections between structures with very different connectivity patterns (large edge betweenness, cf. Girvan and Newman, 2002) are damaged. Is the brain a scale-free network? One important feature of our approach is that the rigorous checking of a series of benchmark networks allows assessing the significance of any similarities to other network types found. In the study of a much simpler brain network, it has previously been established that the brain of C. elegans is small-world, but not scale-free (Amaral et al., 2000). However, we found that effects of damage on the modelled cat and macaque brain connectivity networks are largely similar to those observed in scale-free networks. Furthermore, the similarity of scale-free and original cortical networks, as measured by graph similarity, was higher than for other benchmark networks. This agrees with other findings: a scale-free network architecture has been found for functional brain networks in humans (Eguiluz et al., 2005). In addition, the human resting state network of 90 cortical and subcortical regions showed similar behaviour after the removal of nodes than our structural network (Achard et al., 2006). This could now be explained by the underlying structural connectivity. We note that this issue remains controversial. A study of the human resting state network between cortical areas (Achard et al., 2006), concluded that the resting state network is not a scale-free network as (a) it is more resilient towards targeted attack compared to a scale-free benchmark network, (b) the degree distribution is not a power-law, and (c) late developing areas such as the dorsolateral prefrontal cortex are among the hubs of the network. The structural network that we analyzed, however, differed from the resting state functional network. First, the resilience towards targeted attack was comparable with that of a scale-free network. Second, though the degree does not follow a power-law distribution this might be due to the small size of the network and incomplete sampling of connections between regions. A design for robustness or by-product of functional constraints? Is the brain optimized for high robustness or is robustness a by-product of other constraints? In our view, the emergence of highly-connected areas is more likely to be a side effect of brain evolution and development generating structures for efficient processing. For example, highly-connected areas (hubs) in the brain could play a functional role as integrators or spreaders of information (Sporns and Zwi, 2004). What could be developmental reasons for some regions having a higher connectivity than others? There are several potential developmental mechanisms for yielding brain networks with the highly- connected nodes. Work in brain evolution suggests that when new functional structures are formed by specialization of parts of phylogenetically older structures, the new structures largely inherit the connectivity pattern of the parent structure (e.g., Preuss, 2000). This means that the patterns are repeated and small modifications are added during the evolutionary steps that can arise by duplication of existing areas (Krubitzer and Kahn, 2003). Such inheritance of connectivity by copying of modules is proposed to lead to scale-free metabolic systems (Ravasz et al., 2002). A developmental mechanism for varying the edge degree of regions could be the width of the developmental time window for synaptogenesis at different regions (Kaiser and Hilgetag, 2007). In conclusion, we have introduced a quantitative method to characterize the robustness of brain networks and compare it to that of standard network types. We have shown that cortical networks are affected in ways similar to scale-free networks concerning the elimination of nodes or connections. However, a direct comparison of degree distributions has been impossible. Our analysis can be extended to employ more elimination strategies or use different properties to characterize the damaged networks. In the future, it would be interesting to compare the effect of experimental lesions with the simulated lesions of our approach. We therefore hope that this theoretical approach will prove useful in modelling robustness towards lesions. Acknowledgements Supported by the Wellcome Trust, EU Framework Five (R.M) as well as German National Merit Foundation and Fritz-ter-Meer-Foundation (M.K.). Abbreviations ASP, Average Shortest Path; ORI, Original (brain) network; RND, (Erdös-Renyi) Random network; SF, Scale-free network; SW, Small-world network; References Achard S, Salvador R, Whitcher B, Suckling J, Bullmore E (2006) A resilient, low-frequency, small- world human brain functional network with highly connected association cortical hubs. J Neurosci 26:63-72. Albert R, Jeong H, Barabási A-L (2000) Error and Attack Tolerance of Complex Networks. Nature 406:378-382. Amaral LAN, Scala A, Barthélémy M, Stanley HE (2000) Classes of small-world networks. Proc Natl Acad Sci 97:11149-11152. Barabási A-L, Albert R (1999) Emergence of Scaling in Random Networks. Science 286:509-512. Bollobas B (1985) Random Graphs. Cormen TH, Leiserson CE, Rivest RL, Stein C (2001) Introduction to Algorithms. Diestel R (1997) Graph Theory. New York: Springer. Eguiluz VM, Chialvo DR, Cecchi GA, Baliki M, Apkarian AV (2005) Scale-free brain functional networks. Phys Rev Lett 94:018102. Erdös P, Rényi A (1960) On the evolution of random graphs. Publ Math Inst Hung Acad Sci 5:17- Felleman DJ, van Essen DC (1991) Distributed hierarchical processing in the primate cerebral cortex. Cereb Cortex 1:1-47. Girvan M, Newman MEJ (2002) Community Structure in Social and Biological Networks. Proc Natl Acad Sci 99:7821-7826. Hilgetag CC, O'Neill MA, Young MP (1996) Indeterminancy of the visual cortex. Science 271:776- Hilgetag CC, Burns GAPC, O'Neill MA, Scannell JW, Young MP (2000) Anatomical Connectivity Defines the Organization of Clusters of Cortical Areas in the Macaque Monkey and the Cat. Phil Trans R Soc Lond B 355:91-110. Jeong H, Mason SP, Barabási A-L, Oltvai ZN (2001) Lethality and centrality in protein networks. Nature 411:41-42. Kaiser M, Hilgetag CC (2004) Edge vulnerability in neural and metabolic networks. Biol Cybern 90:311-317. Kaiser M, Hilgetag CC (2006) Nonoptimal Component Placement, but Short Processing Paths, due to Long-Distance Projections in Neural Systems. PLoS Computational Biology 2:e95. Kaiser M, Hilgetag CC (2007) Development of multi-cluster cortical networks by time windows for spatial growth. Neurocomputing:(in press). Kötter R (2004) Online Retrieval, Processing, and Visualization of Primate Connectivity Data from the CoCoMac Database. Neuroinformatics 2:127-144. Krubitzer L, Kahn DM (2003) Nature versus Nurture Revisited: An Old Idea with a New Twist. Prog Neurobiol 70:33-52. Martin R, Kaiser M, Andras P, Young MP (2001) Is the Brain a Scale-Free Network? In: Annual Conference of the Society for Neuroscience, p Paper 816.814. San Diego, US. Milo R, Shen-Orr S, Itzkovitz S, Kashtan N, Chklovskii D, Alon U (2002) Network Motifs: Simple Building Blocks of Complex Networks. Science 298:824-827. Nicolelis MAL, Yu CH, Baccalá LA (1990) Structural Characterization of the Neural Circuit responsible for Control of the Cardiovascular Functions in High Vertebrates. Comput Biol Med 20:379-400. Petroni F, Panzeri S, Hilgetag CC, Koetter R, Young MP (2001) Simultaneity of Responses in a Hierarchical Visual Network. Neuroreport 12:2753-2759. Preuss TM (2000) What's human about the human brain. In: The New Cognitive Neurosciences (Gazzaniga M, ed), pp 1219-1234. Cambridge, MA. Ravasz E, Somera AL, Mongru DA, Oltvai ZN, Barabási A-L (2002) Hierarchical Organization of Modularity in Metabolic Networks. Science 297:1551-1555. Scannell JW, Blakemore C, Young MP (1995) Analysis of Connectivity in the Cat Cerebral Cortex. J Neurosci 15:1463-1483. Spear PD, Tong L, McCall MA (1988) Functional influence of areas 17, 18 and 19 on lateral suprasylvian cortex in kittens and adult cats: implications for compensation following early visual cortex damage. Brain Res 447:79-91. Sporns O, Zwi JD (2004) The Small World of the Cerebral Cortex. Neuroinformatics 2:145-162. Sporns O, Tononi G, Edelman GM (2000) Theoretical Neuroanatomy: Relating Anatomical and Functional Connectivity in Graphs and Cortical Connection Matrices. Cereb Cortex 10:127- Sporns O, Chialvo DR, Kaiser M, Hilgetag CC (2004) Organization, development and function of complex brain networks. Trends Cogn Sci 8:418-425. Stam CJ, Jones BF, Nolte G, Breakspear M, Scheltens P (2007) Small-world networks and functional connectivity in Alzheimer's disease. Cereb Cortex 17:92-99. Stephan KE, Kamper L, Bozkurt A, Burns GA, Young MP, Kotter R (2001) Advanced database methodology for the Collation of Connectivity data on the Macaque brain (CoCoMac). Philos Trans R Soc Lond B Biol Sci 356:1159-1186. Strogatz SH (2001) Exploring complex networks. Nature 410:268-276. Stromswold K (2000) The cognitive neuroscience of language acquisition. In: The New Cognitive Neurosciences (Gazzaniga M, ed), pp 909-932. Cambridge, MA. Stumpf MPH, Wiuf C, May RM (2005) Subnets of Scale-Free Networks are Not Scale-Free: Sampling Properties of Networks. Proc Natl Acad Sci USA 102:4221-4224. Watts DJ, Strogatz SH (1998) Collective Dynamics of 'small-World' Networks. Nature 393:440-442. Young MP (1992) Objective Analysis of the Topological Organization of the Primate Cortical Visual System. Nature 358:152-155. Young MP (1993) The organization of neural systems in the primate cerebral cortex. Phil Trans R Soc 252:13-18. Young MP (2000) The architecture of visual cortex and inferential processes in vision. Spatial Vision 13:137-146. Young MP, Scannell JW, Burns GA, Blakemore C (1994) Analysis of connectivity: neural systems in the cerebral cortex. Rev Neurosci 5:227-250. Tables Table 1. Comparison of brain networks and benchmark networks. The table shows the average shortest path and the clustering coefficient statistics for the macaque and cat brain structure networks, and for the respective benchmark random, rewired, small-world, and scale-free networks. For the benchmark networks, the data shows the mean value and the standard deviation of 50 generated networks. Average shortest path Clustering coefficient Macaque 2.414 0.453 Random mean 2.093 ± 0.009 0.142 ± 0.004 Rewired mean 2.118 ± 0.010 0.239 ± 0.009 Small-world mean 2.439 ± 0.054 0.416 ± 0.022 Scale-free mean 2.078 ± 0.042 0.564 ± 0.042 Cat 1.961 0.542 Random mean 1.749 ± 0.002 0.265 ± 0.003 Rewired mean 1.803 ± 0.006 0.381 ± 0.006 Small-world mean 1.868 ± 0.017 0.461 ± 0.016 Scale-free mean 1.768 ± 0.014 0.535 ± 0.029 Table 2. Overview of the most highly-connected regions in the cat and macaque network. The table shows the total number of connections of the region (degree) as well as the number of incoming / afferent (in-degree) and outgoing / efferent (out-degree) connections. The maximal possible number of connections would have been 110 connections for the cat and 130 connections for the macaque. Rank Area Total Incoming Outgoing 1 AES 59 30 29 2 Ia 55 29 26 3 7 54 28 26 4 Ig 52 22 30 5 5al 49 30 19 Macaque Rank Area Total Incoming Outgoing 1 A7B 43 23 20 2 LIP 42 19 23 3 A46 42 23 19 4 FEF 38 19 19 5 TPT 37 18 19 Figures Figure 1. Examples of random and scale-free networks. Schematic view of network connectivity features. (A) Simple scale-free network having highly-connected nodes (hubs) here shown at the centre. (B) Simple random network; both networks have the same number of nodes and edges. 0 5 10 15 20 25 30 35 40 45 degree s Macaque Random 0 5 10 15 20 25 30 35 40 45 50 55 degree s Cat Random Figure 2. Direct comparison of degree distribution. (A) Histogram of the degree distribution of the macaque (gray) compared to the distribution of random networks (binomial distribution given the probability p=0.1417 that an edge occurs, black). (B) Histogram of the degree distribution of the cat (gray) compared to the distribution of random networks (binomial distribution given the probability p=0.2643 that an edge occurs, black). rewired scale-free random small-world Macaque Figure 3. Similarity of network connectivity. For each type of benchmark network, 1,000 networks were generated. As the cat network has a larger number of edges, the percentages of similar edges are also higher. The similarity with the cortical networks is as good for the scale-free networks as for the rewired cortical networks. In contrast, the similarity of random and small-world networks is significantly lower. Figure 4. Sequential node eliminations in Macaque cortical networks. The fraction of deleted nodes (zero for the intact network) is plotted against the average shortest path (ASP) after node removals. Nodes were removed in order of connectivity, starting with the most highly connected nodes (targeted elimination) or the node order was determined randomly (random elimination). (A) Cortical network during targeted (dashed) and random (solid line) elimination. In the subsequent plots B, C and D, the dashed line shows the average effect of targeted elimination and the thin dashed lines the 95% confidence interval for the generated networks. The solid line represents the average effect of random elimination. The dashed grey line represents targeted removal in the cortical network of A for comparison. (B) Small-world benchmark network. (C) Scale-free benchmark network. (D) Random benchmark network. (The complete set of figures for cat and macaque with node and edge elimination and the effect on ASP is available in the supplementary material). Figure 5. Sequential node eliminations in cat cortical networks. The fraction of deleted nodes (zero for the intact network) is plotted against the average shortest path (ASP) after node removals. Nodes were removed in order of connectivity, starting with the most highly connected nodes (targeted elimination) or the node order was determined randomly (random elimination). (A) Cortical network during targeted (dashed) and random (solid line) elimination. Lines in B-C have the same meaning as in Fig. 4. (B) Small-world benchmark network. (C) Scale-free benchmark network. (D) Random benchmark network. A cortical scale-free rewired random small-world Macaque cortical scale-free rewired random small-world Macaque Figure 6. Fraction and value of peak ASP for targeted node elimination. The average values and standard deviations are shown for the 50 generated benchmark networks. (A) Fraction of eliminated nodes, at which the largest ASP was attained. For the cat cortical network, only the fraction of peak ASP for the scale-free network is close to the cat network whereas the fractions of other benchmark networks are higher. The same is the case for the macaque cortical network. (B) Peak value of the ASP. It is higher for scale-free networks than for cortical networks, in contrast to more similar values for the other benchmark networks. A cortical scale-free rewired random small-world Macaque cortical scale-free rewired random small-world Macaque Figure 7. Fraction and value of peak ASP for targeted connection elimination. The average values and standard deviations are shown for the 50 generated benchmark networks. (A) Fraction of eliminated connections, at which the largest ASP was attained. For the cat network, scale-free and small-world fractions are similar to the cortical value whereas fractions of rewired and random networks are significantly higher. For the macaque network, however, all benchmark networks except for the small-world network show a similar fraction of peak ASP. (B) Peak values of the ASP. The peak value of the cat cortical network can be matched by the random and rewired networks, nearly by the scale-free but significantly not by the small-world network. For the macaque, all networks except for the scale-free network show significantly different values.

0704.0393

A thermodynamic model for the melting of supported metal nanoparticles

A thermodynamic model for the melting of supported metal nanoparticles S. C. Hendy∗ Industrial Research Ltd, Lower Hutt, New Zealand and MacDiarmid Institute for Advanced Materials and Nanotechnology, School of Chemical and Physical Sciences, Victoria University of Wellington, New Zealand (Dated: November 4, 2018) Abstract We construct a simple thermodynamic model to describe the melting of a supported metal nanoparticle with a spherically curved free surface both with and without surface melting. We use the model to investigate the results of recent molecular dynamics simulations, which suggest the melting temperature of a supported metal particle is the same as that of a free spherical particle with the same surface curvature. Our model shows that this is only the case when the contact angles of the supported solid and liquid particles are similar. This is also the case for the temperature at which surface melting begins. http://arxiv.org/abs/0704.0393v1 INTRODUCTION Despite decades of study, the melting of nanoparticles continues to generate interest [1, 2, 3, 4]. In general, the melting temperature of spherical nanoparticles has been found to decrease in proportion to the surface area to volume ratio of the particle [5], as the surface free energy of a molten droplet is less than that of the corresponding solid particle. Although free nanoparticle calorimetry has advanced considerably in recent years [6, 7], most experimental determinations of the melting points of nanoparticles are conducted with supported particles (gold [8], lead [9] and tin [10], for example). The melting of supported nanoparticles is also important in carbon nanotube growth and other catalytic processes [11, 12], and for the stability of devices assembled from nanoparticles [13, 14]. Thus it is of interest to study the effect of the substrate on the melting point of a supported nanoparticle. Recent molecular dynamics simulations [15] of supported iron nanoparticles with a strongly interacting substrate found that the melting point of the particles was reduced in inverse proportion to the equilibrium surface curvature that results as they relax to wet the substrate. This statement also holds in the free particle limit since the curvature of a free spherical particle of radius a is proportional to its surface to volume ratio, 3/a. Inter- estingly, the simulations in Ref. [15] found that the constant of proportionality between the decrease in melting point and the surface curvature did not depend on whether the particle was supported or free. In other words, the melting temperature of a supported particle that has a free surface with radius of curvature a, was found to be the same as that of a free spherical particle with the same surface curvature. The simulated nanoparticles in Ref. [15] also exhibited surface melting prior to complete melting. Surface melting is phenomena thought to occur both on certain bulk metal surfaces [16] and in certain metal nanoparticles [17, 18]. In this paper we use a simple thermodynamic model to investigate the role of the sub- strate in both melting and surface melting of metal nanoparticles. Our model suggests that the result in Ref. [15], that the relative decrease in melting point is proportional to the solid particle surface curvature, only holds when the contact angles of the supported solid and liquid particles with the substrate are similar. We also show that supported clusters will exhibit surface melting under certain circumstances, and that the surface melting tempera- ture in free and supported particles in clusters with same surface curvature is the same only when the contact angles of the supported solid and liquid phases coincide. GEOMETRY OF SUPPORTED PARTICLES We start by considering a solid nanoparticle, initially spherical with radius a, that is placed on a flat substrate. We neglect the effect of faceting, curvature dependent surface energies and internal strains due to epitaxial mismatch with the substrate. Furthermore, we will assume that the particle has relaxed to its equilibrium geometry i.e. that the nanopar- ticle has relaxed to ”wet” the substrate. Provided the nanoparticle is heated sufficiently slowly, the particle should relax to this geometry prior to melting. With these simplifying assumptions, the geometry of the relaxed particle can be approximated by a spherical cap, as shown in Fig. 1, with dimensions parameterised by either the cap height H , or radius of curvature of the free surface R, which minimizes the surface energy of the nanoparticle and substrate. The surface energy Γ of the system can be written as: Γ = 2πRHγs + πH(2R−H)(γsb − γb) + Γb (1) where γs is the surface energy density of the free particle surface, γb is the surface energy density of the substrate, γsb is the particle-substrate interfacial energy density and Γb is the total energy of the bare substrate. We will assume that the density of the particle ρs does not depend on the contact angle so that the volume of the supported particle remains the same as that of the free particle. Writing the volume of the particle as a function of H and R, it is then straightforward to show that Γ is minimized if H = − (∆γsb/γs)R where ∆γsb = γb − γs − γsb. We note that ∆γsb is often called the spreading parameter in the context of wetting phenomena: if ∆γsb > 0 then the particle will relax to fully wet the substrate. Here we are interested in the case where the particle does not fully wet the substrate (contact angles greater than zero) i.e. when ∆γsb < 0 and H/R = −∆γsb/γs > 0 at equilibrium. In fact this minimum value of Γ can be written as Γ∗ = 2πγs a2 + Γb, (2) where R∗s is the corresponding radius of curvature of the supported solid nanoparticle, given R∗s = (3 + ∆γsb/γs) . (3) Thus Γ∗ and R∗s are the equilibrium surface energy and radius of curvature of the particle respectively. Note that the contact angle of the particle can range from 0 to 180 degrees depending on the value of the spreading parameter ∆γsb. MELTING AND SURFACE MELTING In what follows we will assume that the density of the solid and liquid phases are identical i.e. ρs = ρl = ρ. We first consider the situation in which there is no surface melting. In this case, melting will occur at a temperature when the free energy of the solid particle wetting the substrate is equal to that of the corresponding liquid droplet wetting the substrate. If γl is the surface energy density of the free liquid droplet and R l is the corresponding equilibrium radius of curvature, then the difference in free energy between the solid and liquid is fs − fl + 3 where fs (fl) is the bulk free energy density of the solid (liquid). Now, using fl − fs = ρL (1− T/Tc), where L is the latent heat of melting and Tc is the bulk melting temperature, we find the melting temperature Tm of the supported particle is given by: Tm = Tc = T freem (R ρR∗sL Tc (5) Thus if R∗s = R l = R ∗ then we recover the result of Ref. [15], namely that Tm = T In other words, if the contact angles of the solid and liquid droplets are equal, the melting temperature of the supported particle is the same as that of a free particle with an identical surface curvature, a = R∗. However, if the curvature of the supported liquid particle is dif- ferent from that of the supported solid particle, it can be seen that the melting temperature will deviate from that found in Ref. [15]. Now we consider surface melting as illustrated in figure 1 which is thought to occur in many metals prior to melting [16]. We are interested in determining the onset of melting, when the solid particle is wet by a thin layer of melt (thickness δ) at the solid-vapor interface. We will assume that this melt forms a layer of uniform thickness with a geometry like that represented in figure 1 with δ = R− r = H − h. The total free energy of the surface melted particle is then a function of δ: F (δ) = Vs(δ)(fs − fl) + V fl +Γ(δ) where Vs(l) is the volume of the solid (liquid) and Γ is the thickness dependent surface energy. In particular Γ = π (2RHγl + r(2r − h)γsb + δ(2R− δ)γlb + 2rhγsl(δ)) where γsl(δ) = γsl+∆γsl exp (−δ/ξ) and ξ is a correlation length that describes the thickness dependence of the interfacial energy in thin metallic liquid films [16] (in Pb, for example, ξ has been measured to be ∼ 0.6 nm [19]). As the surface melting proceeds, the curvature of the particle will relax to minimize the free energy i.e R∗ = R∗(δ) where R∗ minimizes the free energy F for a given δ. In an isolated spherical nanoparticle of radius a, by minimizing the free energy F (δ) with respect to δ and setting δ = 0, one can show that surface melting begins at a temperature, Ts(a) given by T frees (a) = Tc (γs − γl) . (6) provided ∆γsl > 0 and a > ξ(γs − γl)/∆γsl (if a is less than this, full melting will precede surface melting i.e. T frees > T m [17], and equation (5) will hold). For surface melting to occur in a supported solid nanoparticle with equilibrium curvature Rs, a minimum in the free energy F (δ) must appear at δ = 0. It is straightforward to show that a minimum in F at δ = 0 occurs at the temperature Ts: Ts (Rs) = T s (Rs) + γs∆γlb − γl∆γsb = T frees (Rs) + cos θs − cos θl 1− cos θs where θs and θl are the contact angles for solid particle and liquid particle respectively (defined via Young’s relation γs(l) cos θs(l) = γb − γs(l)b). Once again, if the contact angles of the solid and liquid droplets are equal, then the temperature at which surface melting occurs is identical to that of a free particle with the same surface curvature, Rs i.e. Ts = T s (Rs). Further, if cos θs > cos θl, so that the substrate favors contact with the solid over that with the liquid, the corresponding Ts increases and vice versa. Complete melting will occur once the free energy of the surface melted particle, F (δ), equals that of the corresponding liquid droplet, Fl i.e. at the temperature Tm and liquid film thickness δm which satisfy F (δm) = Fl. It is not possible to obtain an analytic expression for δm or Tm, but numerical solutions to the resulting equations are shown in figure 2 as a function of Rs for Pb particles. The figures clearly show the strong dependence of the melting temperature on the liquid droplet contact angle: a difference of ∼ 10o in the molten particle contact angle can shift the melting point by ∼ 100 K. Note that the melting point of a free particle with radius Rs no longer coincides with that of a supported particle with radius of curvature Rs when cos θs = cos θl, as in general the radius of curvature of the critical surface melted droplet will not be that of the solid particle (although the curves lie close to each other). CONCLUSION We conclude that the melting temperature (and surface melting temperature, if the par- ticle exhibits surface melting) of supported nanoparticles depends on the radius of curvature (or the contact angle) of both the supported solid and liquid particles. In general, we do not expect these curvatures to be the same: on a non-ideal solid substrate for example, epitaxial effects may favor one phase over the other. It is likely that the ideal substrate used in Ref [15] resulted in very similar solid and liquid particle contact angles. We have shown that is only in this ”ideal” case that the melting temperature of free and supported particles with the same curvature is coincident, whether they exhibit surface melting or otherwise. Thus, results from free particle melting, where the curvature of the solid and liquid particles remain substantially the same, have only limited applicability to supported particle melting. ∗ Electronic address: [email protected] [1] H. Haberland, in Atomic Clusters and Nanoparticles: Les Houches Session LXXIII (Springer, Berlin, 2002). [2] H. Haberland, J. Donges, O. Kostko, M. Schmidt, and B. von Issendorff, Phys. Rev. Lett. 04, 035701 (2005). [3] G. A. Breaux, C. M. Neal, B. Cao and M. F. Jarrold, Phys. Rev. Lett. 94, 173401 (2005). [4] D. Schebarchov, and S. C. Hendy, Phys. Rev. Lett. 96, 256101 (2006). [5] Ph. Buffat and J-P. Borel, Phys. Rev. A 13, 2287 (1976). mailto:[email protected] [6] M. Schmidt, R. Kusche, T. Hippler, J. Donges, W. Kronmuller, B. von Issendorff, and H. Haberland, Phys. Rev. Lett. 86, 1191-1194 (2001). [7] G. A. Breaux, R. C. Benirschke, T. Sugai, B. S. Kinnear and M. F. Jarrold, Phys. Rev. Lett. 91, 215508 (2003). [8] T. Castro, R. Reifenberger, E. Choi and R. P. Andres, Phys. Rev. B 42, 8548 (1990). [9] T. Ben David Y. Lereah, G. Deutscher, R. Kofman, and P. Cheyssac, Phil. Mag. A 71, 1135-1143 (1995). [10] S. L. Lai J. Y. Guo, V. Petrova, G. Ramanath, and L. H. Allen, Phys. Rev. Lett. 77, 99 (1996). [11] M. H. Kuang, Z. L. Wang, X. D. Bai, J. D. Guo and E. G. Wang, Appl. Phys. Lett. 76, 1255 (2000). [12] F. Ding, A. Rosén and K. Bolton, Phys. Rev. B 70, 075416 (2004). [13] J. G. Partridge, S. A. Brown, A. D. F. Dunbar, M. Kaufmann, S. Scott, M. Schulze, R. Reichel, C. Seigert and R. Blaikie, Nanotechnology 15, 1382 (2004). [14] R. Reichel, J. G. Partridge, F. Natali, T. Matthewson, S. A. Brown, A. Lassesson, D. M. A. Mackensie, A. Ayesh, K. C. Tee, A. Awasthi and S. C. Hendy, Appl. Phys. Lett. 89, 213105 (2006). [15] F. Ding, A. Rosén, S. Curtarolo and K. Bolton, Appl. Phys. Lett. 88, 133110 (2006). [16] J. F. van der Veen, B. Pluis and A. W. Denier van der Gon, in Kinetics of Ordering and Growth at Surfaces 343-354, (Plenum Press, New York, 1990). [17] T. Bachels, H.-J. Güntherodt and R. Schäfer, Phys. Rev. Lett. 85, 1252 (2000). [18] U. Tartaglino, T. Zykova-Timan, F. Ercolessi and E. Tosatti, Phys. Rep. 411, 291321 (2005). [19] B. Pluis, T. N. Taylor, D. Frenkel and J. F. van der Veen, Phys. Rev. B 40, 1353 (1989). FIG. 1: The model for the geometry of a supported nanoparticle in equilibrium. We assume that the particle is a spherical cap of height H and radius of curvature R (left - the dashed lines simply illustrate the radius of curvature). At the onset of surface melting, we assume that the geometry is close to that of the solid particle in its equilibrium geometry and that that the solid particle (radius of curvature r and height h) is initially wet by a molten layer of uniform thickness δ = R−r = H−h (right). 2 4 6 8 10 12 14 γlb=0.05 γlb=0.15 γlb=0.10 Rs (nm) 2 4 6 8 10 12 14 cos θs=cos θl γlb=0.05 γlb=0.15 γlb=0.10 Rs (nm) 2 4 6 8 10 12 14 γlb=0.05 γlb=0.15 γlb=0.10 Rs (nm) 2 4 6 8 10 12 14 cosθs=cosθl γlb=0.05 γlb=0.15 γlb=0.10 FIG. 2: The melting temperature Tm and critical liquid film thickness δm for supported Pb clusters as a function of the radius of curvature Rs of the relaxed solid particle for γlb = 0.05, 0.10, 0.15 J m−2 and in the case where cos θs = cos θl (γlb ≃ 0.13 J m −2). Also shown is the melting temperature of a free particle with radius Rs. Other surface energies used are γsv = 0.61, γlv = 0.48, γsl = 0.05, γb = 0.25 and γsb = 0.1 J m −2 giving a contact angle of 75.8o for the solid supported cluster, and contact angles for the liquid droplets ranging from 78.0o to 65.4o respectively. Other parameters used were ξ = 0.63 nm, ρ = 10950 kg m−3, L = 22930 J kg−1 and Tc = 600.65 K [9]. Introduction Geometry of supported particles Melting and surface melting Conclusion References

0704.0394

Average optimality for risk-sensitive control with general state space

Average optimality for risk-sensitive control with general state space The Annals of Applied Probability 2007, Vol. 17, No. 2, 654–675 DOI: 10.1214/105051606000000790 c© Institute of Mathematical Statistics, 2007 AVERAGE OPTIMALITY FOR RISK-SENSITIVE CONTROL WITH GENERAL STATE SPACE1 By Anna Jaśkiewicz Wroc law University of Technology This paper deals with discrete-time Markov control processes on a general state space. A long-run risk-sensitive average cost crite- rion is used as a performance measure. The one-step cost function is nonnegative and possibly unbounded. Using the vanishing discount factor approach, the optimality inequality and an optimal stationary strategy for the decision maker are established. 1. Introduction and the model. This paper deals with discrete-time Markov control processes on a general state space. The one-step cost function is nonnegative and possibly unbounded. The decision maker is supposed to be risk-averse with a constant risk coefficient γ > 0. The risk-sensitive aver- age cost criterion is used as a performance measure. The aim of the work is to establish the optimality inequality for risk-sensitive dynamic programming and derive an optimal stationary policy. The result is proved under two different sets of compactness-continuity assumptions, namely, for Markov control processes with weakly continuous transition probabilities [Condition (W)], as well as transition probabilities that are continuous with respect to setwise convergence [Condition (S)]. A similar problem for risk-neutral stochastic control models has been examined in [27] using the vanishing dis- count factor approach. However, it is well known that, for risk-sensitive con- trol models, an analogous approximation of the average cost via a sequence of the corresponding discounted models does not work. Instead of this, fol- lowing [9, 15, 16], we introduce an auxiliary discounted minimax problem. A variational formula that expresses the mutual relationship between the relative entropy function and the logarithmic moment-generating function enables us to connect the discounted minimax model with the original one. Received March 2006; revised September 2006. 1Supported by MEiN Grant 1 P03A 01030. AMS 2000 subject classifications. Primary 60J05, 90C39; secondary 60A10. Key words and phrases. Risk-sensitive control, Borel state space, average cost optimal- ity inequality. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Probability, 2007, Vol. 17, No. 2, 654–675. This reprint differs from the original in pagination and typographic detail. http://arxiv.org/abs/0704.0394v1 http://www.imstat.org/aap/ http://dx.doi.org/10.1214/105051606000000790 http://www.imstat.org http://www.ams.org/msc/ http://www.imstat.org http://www.imstat.org/aap/ http://dx.doi.org/10.1214/105051606000000790 2 A. JAŚKIEWICZ Next, assuming that a certain family of functions is bounded [Condition (B)] and using Fatou’s lemma (for weakly or setwise convergent measures), we obtain the optimality inequality. The predecessor of our result is Theorem 4.1 in [16], where the optimality inequality for the risk-sensitive dynamic programming with a countable state space was established. Instead of boundedness assumption (B), Hernández- Hernández and Marcus [16] assume that there exists a stationary policy which induces a finite average cost that is equal some constant in each state. On the other hand, it is well known that an optimal risk-sensitive average cost may depend on the initial state (see Example 1). This behavior happens if the risk factor is too large. Instead of this restriction on the risk coefficient, we use Condition (B), which makes the process reach “good states” sufficiently fast. There is a rich literature in risk-sensitive control, going back at least to the seminal works of Howard and Matheson [18] and Jacobson [19], which covered the finite horizon case. The average cost criterion on the infinite horizon was studied in [5, 8, 14, 15, 16, 31] for a denumerable state space and in [10, 11, 20] for a general state space. It is also worth mentioning that risk-sensitive control finds natural applications in portfolio managment, where the objective is to maximize the growth rate of the expected utility of wealth; see [3, 4, 30] and the references cited therein. The paper is organized as follows. Below a Markov control model with the long-run average cost criterion as a performance measure is described, as well as some basic notation is set up. In Section 2 we introduce preliminaries and present the auxiliary discounted minimax problem, which is, in turn, solved in Section 3. The main result is established in Section 4. Section 5 contains a discussion of Condition (B), and in the Appendix a variational formula for the logarithmic moment-generating function is stated. A discrete-time Markov control process is specified by the following ob- jects: (i) The state space X is a standard Borel space (i.e., a nonempty Borel subset of some Polish space). (ii) A is a Borel action space. (iii) K is a nonempty Borel subset of X×A. We assume that, for each x ∈X , the nonempty x-section A(x) = {a ∈A : (x,a) ∈K} of K is compact and represents the set of actions available in state x. (iv) q is a regular conditional distribution from K to X. (v) The one-step cost function c is a Borel measurable mapping from K to [0,+∞]. RISK-SENSITIVE CONTROL 3 Then the history spaces are defined as H0 = X, Hk = (X ×A) k ×X and H∞ = (X ×A) ∞. As usual, a policy π = {πk, k = 0,1, . . .} ∈ Π is a sequence of transition probabilities from Hk to A such that πk(A(xk)|hk) = 1, where hk = (x0, a0, . . . , xk) ∈Hk. The class of stationary policies is identified with the class F of measurable functions f from X to A such that f(x) ∈A(x). It is well known that F is nonempty [6]. By the Ionescu–Tulcea theorem [24], for each policy π and each initial state x0 = x, a probability measure P and a stochastic process {(xk, ak)} are defined on H∞ in a canonical way, where xk and ak describe the state and the decision at stage k, respectively. By Eπx we denote the expectation operator with respect to the probability measure Pπx . Let γ > 0 be a given risk factor. For any initial state x ∈X and policy π ∈ Π, we define the following risk-sensitive average cost criterion: J(x,π) = lim sup logEπx exp c(xk, ak) Our aim is to minimize J(x,π) within the class of all policies and find a policy π∗, for which J∗(x) := inf J(x,π) = J(x,π∗). Throughout the paper the following assumption will be supposed to hold true even without explicit reference: ∃π̃ ∈ Π J(x, π̃) < +∞.(G) Remark 1. Throughout the remainder, we assume that the risk factor γ > 0 is arbitrary and fixed. Therefore, here and subsequently, we shall not indicate that some quantities depend on γ [e.g., we write J(x,π) instead of Jγ(x,π), dropping the index γ]. 2. Preliminaries. Let Pr(X) be the set of all probability measures on X. Fix ν ∈ Pr(X). The relative entropy function R(·‖ν) is a mapping from Pr(X) into R defined as follows: R(µ‖ν) := dµ, µ≪ ν, +∞, otherwise. It is well known that R(µ‖ν) is nonnegative for any µ ∈ Pr(X) and R(µ‖ν) = 0 if and only if µ = ν (consult Lemma 1.4.1 in [12]). We shall consider the following auxiliary minimax problem, associated with our original Markov control process. The set X is the state space, 4 A. JAŚKIEWICZ while A and Pr(X) are the action sets for the decision maker and op- ponent, respectively. The process then operates as follows. In a state xn, n = 0,1, . . . , the controller chooses an action an ∈ A(xn), while the oppo- nent selects µn(·)[xn, an] ∈ Pr(X). As a consequence, the controller pays γc(xn, an)−R(µn‖q(·|xn, an)) to his opponent, and the system moves to the next state according to the probability distribution µn(·)[xn, an]. We shall deal with the following classes of strategies. It will cause no confusion if we continue to use the same letters to denote strategies for the controller. Namely, π stands for a randomized control strategy (policy), whereas f denotes a stationary strategy. We write Π and F to denote the sets of corresponding strategies. For the opponent’s class of strategies, we confine to the stationary one, which is identified with the class P of stochastic kernels p on X given K. Let (Ω,F) be the measurable space consisting of the sample space Ω = (X × A)∞ and its product σ-algebra F . Then for an initial state x ∈ X, and strategies π and p, there exists a unique probability measure Pπpx and, again, a stochastic process {(xk, ak)} is defined (Ω,F) in a canonical way, where xk denotes the state at time k and ak is the action for the controller. With some abuse of notation, we let hk stand for the history of the process up to the kth state, that is, hk = (x0, a0, x1, . . . , ak−1, xk). The corresponding expectation operator is denoted by Eπpx . For fixed x ∈ X, π ∈ Π and p ∈ P , we define the following functional costs: Vβ(x,π, p) = βkEπpx [γc(xk, ak) −R(p(·|xk, ak)‖q(·|xk, ak))],(1) where β ∈ (0,1) is the discount factor, and j(x,π, p) = lim sup Eπpx [γc(xk, ak) −R(p(·|xk, ak)‖q(·|xk, ak))]. Note that, since the function R(·‖·) is lower semicontinuous on Pr(X) × Pr(X) and p and q are stochastic kernels [i.e., measurable functions of (x,a)], it follows that the mapping (x,a) 7→R(p(·|x,a)‖q(·|x,a)) is measurable (Lemma 1.4.3(f) in [12]). Observe that Vβ(x,π, p) and j(x,π, p) might be undetermined, because c can be unbounded. We thus restrict the set of admissible strategies for the opponent in the following way. RISK-SENSITIVE CONTROL 5 Definition 1. Given π = {πk} ∈ Π, we say that p ∈ P is a π-admissible strategy iff A(xk) R(p(·|xk, a)‖q(·|xk, a))πk(da|hk) < +∞,(2) and moreover, there exists a constant C ≥ 0, possibly depending on π and p, such that A(xk) [γc(xk, a) −R(p(·|xk, a)‖q(·|xk, a))]πk(da|hk) + C ≥ 0, for all histories of the process hk, k ≥ 0, induced by p and π. We denote this set by Q(π). [Note that this set is nonempty, since p = q ∈Q(π) for any π ∈ Π.] Let us introduce the following notation. For any π ∈ Π, p ∈ Q(π) and n≥ 1, define Jn(x,π) = logE x exp c(xk, ak) jn(x,π, p) = Eπpx [γc(xk, ak) −R(p(·|xk, ak)‖q(·|xk, ak))]. Now we are ready to present the result that was originally proved in [16] for Markov strategies. However, it still remains valid when arbitrary strategies for the decision maker are considered. Therefore, for the sake of clarity, we state the result with its proof. Proposition 1. Let x∈X and p ∈Q(π). Then: (a) supp∈Q(π) jn(x,π, p) ≤ Jn(x,π) for each n≥ 1, (b) lim supn→∞ supp∈Q(π) jn(x,π, p) ≤ γJ(x,π). Proof. (a) Let p ∈ Q(π) be any stochastic kernel. For n = 1, we con- clude j1(x,π, p) ≤ E x (γc(x,a0)) ≤ logE γc(x,a0) = J1(x,π), where the first inequality holds since the relative entropy is nonnegative, and the second one is due to Jensen’s inequality. Now assume that the hypothesis is true for some n≥ 1. Clearly, jn+1(x,π, p) = Eπpx [γc(xk, ak) −R(p(·|xk, ak)‖q(·|xk, ak))] = Eπpx [γc(xk, ak) −R(p(·|xk, ak)‖q(·|xk, ak))], n≥ 1. 6 A. JAŚKIEWICZ Denote by π(1) the “1-shifted” strategy, that is, (·|hk) = πk+1(·|x0, a0, hk), k ≥ 0. Then, we have jn+1(x,π, p) = Eπpx [γc(x,a0) + jn(x1, π (1), p) −R(p(·|x,a0)‖q(·|x,a0))] ≤ Eπpx (γc(x,a0)) + Eπpx (E x {[Jn(x1, π (1)) −R(p(·|x,a0)‖q(·|x,a0))]|a0}) = Eπx log e γc(x,a0) + Eπpx Jn(x1, π (1))p(dx1|x,a0) −R(p(·|x,a0)‖q(·|x,a0)) log eγc(x,a0)π0(da0|x) eJn(x1,π (1))q(dx1|x,a0)π0(da0|x) eγc(x,a0)+ γc(xk,ak)q(dx1|x,a0)π0(da0|x) ≤ log eγc(x,a0)+ γc(xk,ak)q(dx1|x,a0)π0(da0|x) = Jn+1(x,π). Clearly, the first inequality follows from the induction hypothesis. The third inequality is due to Jensen’s inequality, whilst the second one follows from Lemma A in the Appendix. Since p ∈Q(π) is arbitrary, we get the desired conclusion. Part (b) follows directly from part (a). � Remark 2. Note that in the proof of Proposition 1 we did not really have to use the fact that p ∈ Q(π). The only assumption which plays an essential role is condition (2). Namely, it guarantees that jn(x,π, p) is well defined for all n≥ 1, x ∈X and π ∈ Π. However, in Definition 1 we restrict the opponent’s class of strategies to the set Q(π) in order to be able to apply the Hardy–Littlewood theorem. In actual fact, later on it will be clear that the set Q(π), where π ∈ Π, is sufficiently large. Namely, the supremum of certain discounted functional costs over the set Q(π) will not change if we add new elements to Q(π); see the proofs of Lemmas 1 and 2. RISK-SENSITIVE CONTROL 7 Let π̃ be as in assumption (G) and let p ∈Q(π̃). Then from the Hardy– Littlewood theorem (Theorem H.2 in [13]), we get lim sup (1 − β)Vβ(x, π̃, p) ≤ lim sup jn(x, π̃, p) and from Proposition 1(b), lim sup p∈Q(π̃) jn(x, π̃, p) ≤ γJ(x, π̃). Combining these two inequalities, we conclude that lim sup (1− β)Vβ(x, π̃, p) ≤ γJ(x, π̃) for every p ∈Q(π̃). This in turn yields lim sup (1− β)Vβ(x) ≤ γJ(x, π̃),(4) where Vβ(x) is the upper value of functional cost (1), that is, Vβ(x) = inf p∈Q(π) Vβ(x,π, p). Consequently, inequality (4) and assumption (G) together lead to the fol- lowing: Vβ(x) < +∞(5) for each x ∈X and β ∈ (0,1). In addition, Vβ(x) ≥ 0. Now defining ρ := inf J(x,π), mβ := inf Vβ(x) and observing that lim sup (1 − β)mβ ≤ γρ,(6) one can deduce that there exists a sequence of discount factors {βn} con- verging to 1 for which (1− βn)mβn = l,(7) where l is a certain nonnegative constant. 8 A. JAŚKIEWICZ 3. A solution to the auxiliary discounted minimax problem. The main thrust of this section is to solve the auxiliary discounted minimax problem introduced in the previous section. In other words, we look for a discounted functional equation whose solution is the function Vβ . This is done by an ap- proximation of the above-mentioned minimax models by ones with bounded cost functions. These models in turn are solved by a fixed point argument in Proposition 1. Next, we show in Lemma 1 that the corresponding solutions equal the upper values of some discounted costs on the infinite horizon. Fi- nally, the limit passage in Lemma 2 gives the desired discounted functional equation with the function Vβ as a solution. We shall need the following two sets of compactness-semicontinuity as- sumptions, which will be used alternatively. Condition (S). (i) The set A(x) is compact. (ii) For each x ∈X and every Borel set D ⊂X, the function q(D|x, ·) is continuous on A(x). (iii) The cost function c(x, ·) is lower semicontinuous for each x ∈X. Condition (W). (i) The set A(x) is compact and the set-valued mapping x 7→ A(x) is upper semicontinuous, that is, {x ∈X : A(x) ∩ B 6= ∅} is closed for every closed set B in A. (ii) The transition law q is weakly continuous on K, that is, the function (x,a) 7→ u(y)q(dy|x,a), (x,a) ∈K, is continuous function for each bounded continuous function u. (iii) The cost function c is lower semicontinuous on K. By Lb(X) and Bb(X), we denote the set of all bounded lower semicontin- uous and bounded Borel measurable functions on X, respectively. Further, let N stand for the set of positive integers. Choose N ∈ N and define the truncated cost function cN (x,a) = min{N,c(x,a)}. The following result was proved under Condition (W) for bounded cost functions by a fixed point argument; see page 72 in [10]. However, a simple and obvious modification of the proof gives the conclusion under Condition (S) as well. RISK-SENSITIVE CONTROL 9 Proposition 2. Under (W) [(S)], for any discount factor β ∈ (0,1) and a number N ∈N, there exists a unique function wNβ ∈ Lb(X) [w β ∈Bb(X)] such that = min a∈A(x) N (x,a) q(dy|x,a) for each x ∈X, and 0 ≤ (1 − β)wNβ (x) ≤Nγ.(9) Moreover, there exists a stationary strategy f0 ∈ F (possibly depending on β and N) that attains the minimum in (8). Let β and N be fixed just in the next lemma. Lemma 1. Assume (W) or (S). Then, it holds wNβ (x) = inf p∈Q(π) Eπpx β k[γcN (xk, ak) −R(p(·|xk, ak)‖q(·|xk, ak))] for any initial state x ∈X. Proof. Note that (8) can be rewritten in the following equivalent form: wNβ (x) = min a∈A(x) γcN (x,a) + log q(dy|x,a) .(11) Applying Lemma A in the Appendix to (11), we get wNβ (x) = min a∈A(x) µ∈∆(x,a) γcN (x,a) −R(µ‖q(·|x,a)) + β wNβ (y)µ(dy) ∆(x,a) := {µ ∈ Pr(X) :R(µ‖q(·|x,a)) < +∞}, (x,a) ∈K. Moreover, the measure µ0(dy)[x,a] = q(dy|x,a) q(dy|x,a) achieves the supremum in (12). Put p0(dy|x,a) = µ0(dy)[x,a] for each (x,a) ∈K.(13) 10 A. JAŚKIEWICZ Note that p0 ∈Q(π) for any strategy π ∈ Π. This directly follows from the definition of R(p0(·|x,a)‖q(·|x,a)) and (9). Simple calculations give the up- per bound R(p0(·|x,a)‖q(·|x,a)) ≤ 2 1 − β for every (x,a) ∈K. Let p0 be defined as in (13). By (12), we then have wNβ (x) ≤ γc N (x,a) −R(p0(·|x,a)‖q(·|x,a)) + β wNβ (y)p 0(dy|x,a). By iteration of this inequality n times, it follows wNβ (x) ≤ βkEπp x [γc N (xk, ak) −R(p 0(·|xk, ak)‖q(·|xk, ak))] + βn+1Eπp β (xn+1), where π is any strategy for the controller. Now, letting n→∞ and making use of (9), we conclude wNβ (x) ≤ βkEπp x [γc N (xk, ak) −R(p 0(·|xk, ak)‖q(·|xk, ak))]. Since π is arbitrary, we get wNβ (x) ≤ inf βkEπp x [γc N (xk, ak) −R(p 0(·|xk, ak)‖q(·|xk, ak))] ≤ inf p∈Q(π) βkEπpx [γc N (xk, ak)(14) −R(p(·|xk, ak)‖q(·|xk, ak))]. Note that inequality (14) is valid because p0 ∈Q(π). On the other hand, by (12), we can write wNβ (x) ≥ γc N (x, f0(x)) −R(p(·|x, f0(x))‖q(·|x, f0(x))) wNβ (y)p(dy|x, f 0(x)), with f0 as in Proposition 2 and any p ∈Q(f0). Proceeding along the same line, we infer wNβ (x) ≥ x [γc N (xk, f 0(xk)) −R(p(·|xk, f 0(xk))‖q(·|xk, f 0(xk)))]. RISK-SENSITIVE CONTROL 11 Since p ∈Q(f0) is arbitrary, we easily deduce wNβ (x) ≥ sup p∈Q(f0) x [γc N (xk, f 0(xk)) −R(p(·|xk, f 0(xk))‖q(·|xk, f 0(xk)))] ≥ inf p∈Q(π) βkEπpx [γc N (xk, ak) −R(p(·|xk, ak)‖q(·|xk, ak))]. Finally, combining (14) with (15) completes the proof. � In the remainder of the paper, we shall use the following notation. Let L(X) denote the set of all lower semicontinuous functions on X, whereas B(X) stands for the set of all Borel measurable functions on X. Lemma 2. Let (W) [(S)] hold and β ∈ (0,1). Then, we have the follow- (a) The function wβ(x) := lim wNβ (x) is finite and nonnegative for each x ∈X. Moreover, wβ ∈L(X) [wβ ∈B(X)]. (b) The functional equation holds ewβ(x) = min a∈A(x) eγc(x,a) eβwβ(y)q(dy|x,a) for all x ∈X. Furthermore, there exists a Borel measurable selector fβ ∈ F of the minima in (16). (c) For any x ∈X, wβ(x) = Vβ(x). Proof. Let x ∈X and β ∈ (0,1) be fixed. From (10), it is easily seen that the sequence {wNβ (x)} is nondecreasing in N. Therefore, wβ(x) = limN→∞w β (x) exists and by (9), it is nonnegative. Clearly, under (S), wβ ∈B(X), whereas, under (W), wβ ∈L(X); see Proposition 10.1 in [26]. In order to prove that wβ(x) is finite for each x ∈X, observe first that, for any π ∈ Π, p ∈Q(π) and N ∈N, Vβ(x,π, p) = βkEπpx [γc(xk, ak) −R(p(·|xk, ak)‖q(·|xk, ak))] βkEπpx [γc N (xk, ak) −R(p(·|xk, ak)‖q(·|xk, ak))]. 12 A. JAŚKIEWICZ Moreover, from Lemma 1, we have Vβ(x) = inf p∈Q(π) Vβ(x,π, p) ≥ inf p∈Q(π) βkEπpx [γc N (xk, ak) −R(p(·|xk, ak)‖q(·|xk, ak))] = wNβ (x). Hence, letting N →∞, it follows Vβ(x) ≥ lim wNβ (x) = wβ(x).(17) By (5), Vβ(x) is finite for each x ∈X, so is wβ(x). This finishes the proof of part (a). In order to prove part (b), note that by (11) and part (a) the limit a∈A(x) γcN (x,a) + log q(dy|x,a) exists. Since the first and the second term in (18) are nondecreasing and (W) or (S) holds, then we may interchange the limit with the minimum (see Proposition 10.1 in [26]). Furthermore, making use of the Lebesgue monotone convergence theorem, we conclude (16). The existence of a Borel measurable selector fβ ∈ F follows from the compactness–semicontinuity assumptions and Proposition D.5 in [17]. We now turn to proving part (c). Again, taking a logarithm on both sides of (16), it follows wβ(x) = min a∈A(x) γc(x,a) + log eβwβ(y)q(dy|x,a) .(19) Applying Lemma A in the Appendix to (19), we easily obtain wβ(x) = min a∈A(x) µ∈∆(x,a) γc(x,a) −R(µ‖q(·|x,a)) + β wβ(y)µ(dy) ∆(x,a) = {µ ∈ Pr(X) :R(µ‖q(·|x,a)) < +∞}, (x,a) ∈K. Observe that by (20), for any p ∈Q(fβ), the following holds: wβ(x) ≥ γc(x, fβ(x)) −R(p(·|x, fβ(x))‖q(·|x, fβ(x))) wβ(y)p(dy|x, fβ(x)). RISK-SENSITIVE CONTROL 13 Iterating this inequality n times, we immediately obtain wβ(x) ≥ x [γc(xk, fβ(xk)) −R(p(·|xk, fβ(xk))‖q(·|xk, fβ(xk)))] + βn+1E x wβ(xn+1)(21) x [γc(xk, fβ(xk)) −R(p(·|xk, fβ(xk))‖q(·|xk, fβ(xk)))]. Now note that, by Definition 1, x [γc(xk, fβ(xk)) −R(p(·|xk, fβ(xk))‖q(·|xk, fβ(xk)))] ≥−C, for some C ≥ 0 and k ≥ 1. Thus, letting n→∞ in (21), it follows wβ(x) ≥ x [γc(xk, fβ(xk)) −R(p(·|xk, fβ(xk))‖q(·|xk, fβ(xk)))] = Vβ(x, fβ, p). Since p ∈Q(fβ) is arbitrary, we see that wβ(x) ≥ sup p∈Q(fβ) Vβ(x, fβ, p) ≥ Vβ(x).(22) Inequalities (17) and (22) combined conclude the proof of part (c). � 4. A solution to the risk-sensitive control problem. For any x ∈X and any discount factor β ∈ (0,1), define hβ(x) := Vβ(x) −mβ with mβ = infx∈X Vβ(x). Obviously, hβ is nonnegative. The following boundedness assumption is supposed to hold true. As men- tioned in the Introduction, we put off discussing it until Section 5: Condition (B). For any x ∈X , supβ∈(0,1) hβ(x) < +∞. Remark 3. A similar assumption and its equivalent variants were used to study the expected average cost criterion for Markov decision processes in the risk-neutral setting [17, 27, 28]. Roughly speaking, Hernández-Lerma and Lasserre [17], Schäl [27], and Sennott [28] assume that the family of the so-called normalized β-discounted cost functions is bounded. This assump- tion, however, simply holds for ergodic Markov decision processes. More 14 A. JAŚKIEWICZ precisely, if the n-step transition probabilities converge to the unique in- variant probability measure geometrically fast, and the cost functions are bounded (or more generally satisfy a certain growth hypothesis), then the aforementioned family of functions is pointwise relatively compact [21, 22]. It is worth pointing out that this requirement is crucial to obtain the opti- mality inequality in the risk-neutral case; see [27, 28]. In Section 5 we provide an example that illustrates that also in the risk-sensitive case Condition (B) cannot be weakened. We shall need the following two versions of Fatou’s lemma for converging measures. Lemma 3. Let {µn} be a sequence of probability measures converging to µ ∈ Pr(X) and let {hn} be a sequence of measurable nonnegative functions on X. Then, h(y)µ(dy) ≤ lim inf hn(y)µn(dy) in the following cases: (a) {µn} converges setwise to µ [i.e., f(y)dµn(y) → f(y)dµ(y)∀f ∈ Bb(X)], and h(x) = lim infn→∞ hn(x); (b) {µn} converges weakly to µ, and h(x) = inf{lim infn→∞ hn(xn) :xn → x}; moreover, h ∈ L(X). Proof. Part (a) is due to Royden [25], page 231, whereas part (b) was proved by Serfozo [29]. For the proof of lower semicontinuity of h, the reader is referred to Lemma 3.1 in [22]. � Now we are in a position to state the main result of the paper. This theo- rem concerns a study of the risk-sensitive average cost optimality inequality, which is sufficient to establish the existence of an optimal stationary policy. Theorem 1. Assume (B) and (W) [or (S)]. Then, for each risk factor γ > 0, there exist a constant l̂ and a nonnegative function h ∈ L(X) [h ∈ B(X)] and f̂ ∈ F such that h(x) + l̂ ≥ min a∈A(x) γc(x,a) + log eh(y)q(dy|x,a) = γc(x, f̂(x)) + log eh(y)q(dy|x, f̂(x)) RISK-SENSITIVE CONTROL 15 for all x ∈X. Moreover, = inf J(x,π) = J(x, f̂). In other words, l̂/γ is the optimal risk-sensitive average cost and f̂ is a risk-sensitive average cost optimal stationary policy. Remark 4. (a) There are two papers [16, 27] that can be treated as predecessors of our work. They both deal with the optimality inequality but within two different frameworks. The first work [16] establishes the optimal- ity equation for the risk-sensitive dynamic programming on a denumarable state space. In the other one, the result is obtained for Markov control pro- cesses on an uncountable state space for the risk factor γ = 0. From this point of view, our result is an extention of Theorem 4.1 in [16] to a general state space and Theorem 3.8 in [27] to the risk-sensitive case. Moreover, the common feature of the discussed results is that their proofs are based on the vanishing discount factor approach. Our proof also relies on this method, and similarly, as in [27] or [21, 22], makes use of the Fatou lemmas for setwise and weakly convergent measures. (b) Finally, it is also worth mentioning that there are papers studying the optimality equation in the risk-sensitive dynamic programming, which is of the following form: h(x) + l̂ = min a∈A(x) γc(x,a) + log eh(y)q(dy|x,a) .(24) The constant l̂ is (under suitable assumptions) an optimal cost with respect to the risk-sensitive average cost criterion. Let us mention and discuss a few representative papers that deal with equation (24). In [8, 15] Markov control models satisfying a simultaneous Doeblin condition, on a finite and countable state space, respectively, are considered. The cost functions are supposed to be bounded and the risk factor must be sufficiently small. Otherwise, as argued in [8], the optimality equation need not have a solution. In [10] Di Masi and Stettner extend the result to a general state space by retaining bounded cost functions and replacing a simultaneous Doeblin condition with a very strong assumption on transition probabilities. In [11], however, they replace this assumption by one imposed on the risk coeffi- cient. Finally, the class of Markov control models that requires neither any ergodicity conditions nor the smallness of the risk factor was pointed out by Jaśkiewicz in [20]. Fairly recently Borkar and Meyn [5] considered Markov decision processes with unbounded cost functions on a denumarable state space. Their result 16 A. JAŚKIEWICZ assumes the following: the state space is irreducible under all Markov poli- cies, the costs are norm-like, and there exists a policy that induces a finite average risk-sensitive cost. Moreover, their proof is based on a multiplicative ergodic theorem that was studied in more detail in [1]. Proof of Theorem 1. Let {βn} be a sequence of discount factors converging to 1 for which (7) holds. Defining l̂ := l = lim (1 − βn)mβn and applying (6), we note that ≤ inf J(x,π)(25) for any x ∈X. Assume for a while that inequality (23) is satisfied and there exists f̂ ∈ F as in the statement of Theorem 1. We prove that f̂ is an optimal policy. From (23), we have h(x) ≥ γc(x, f̂(x)) − l̂ + log eh(y)q(dy|x, f̂(x)). By iteration of this inequality n times, we obtain h(x) ≥ logEπx exp γc(xk, f̂(xk)) + h(xn+1) − (n+ 1)l̂. Since h is nonnegative, we infer + l̂≥ Jn+1(x, f̂) with Jn+1(x, f̂) defined in (3). Letting n→∞, it follows ≥ J(x, f̂), x ∈X.(26) Hence, (25) and (26) together imply = J(x, f̂) = inf J(x,Π) for each x ∈X. We next focus on showing inequality (23). Let n≥ 1 and put hn := hβn , fn := fβn. Note that (19) can be rewritten in the following form: (1 − βn)mβn + hn(x) = min a∈A(x) γc(x,a) + log eβnhn(y)q(dy|x,a) = γc(x, fn(x)) + log eβnhn(y)q(dy|x, fn(x)). RISK-SENSITIVE CONTROL 17 (i) Assume first (S) and define h(x) = lim inf hn(x). Taking the lim inf on both sides of (27), we get lim inf ((1 − βn)mβn + hn(x)) = l̂ + h(x) = lim inf a∈A(x) γc(x,a) + log eβnhn(y)q(dy|x,a) Making use of Lemma 3(a) and the measurable selection theorem (see Propo- sition D.5(a) in [17]), one can prove that there exists f̂ ∈ F such that (23) holds. (ii) Now assume (W). Fix x0 ∈X and choose any xn → x0, n→∞. Take a subsequence {nk} of positive integers such that lim inf hn(xn) = lim hnk(xnk). Then by (27), lim inf ((1− βn)mβn + hn(xn)) = l̂ + lim inf hn(xn) = l̂ + lim hnk(xnk) = lim a∈A(xnk ) γc(xnk , a) + log eβnkhnk (y)q(dy|xnk , a) = lim γc(xnk , fnk(xnk)) + log eβnkhnk (y)q(dy|xnk , fnk(xnk)) Note that G = {x0} ∪ {xn} is compact in X. From the upper semicontinu- ity of x 7→A(x), compactness of every A(z) and Berge’s theorem (see [2] or Theorem 7.4.2 in [23]), it follows that z∈GA(z) is compact in A. There- fore, {fnk(xnk)} has a subsequence converging to some a0 ∈A. By (W)(i), a0 ∈ A(x0), that is, (x0, a0) ∈ K. Without loss of generality, assume that fnk(xnk) → a0, k →∞. By the lower semicontinuity of the cost function c and (28), we have l̂ + lim inf hn(xn) ≥ γc(x0, a0) + lim eβnkhnk (y)q(dy|xnk , fnk(xnk)). This and Lemma 3(b) imply that l̂ + lim inf hn(xn) ≥ γc(x0, a0) + log eh̃(y)q(dy|x0, a0), where eh̃ is the generalized lim inf of the sequence eh̃k = ehnk . Clearly, h≤ h̃. By Lemma 3(b), h ∈L(X). Thus, l̂ + lim inf hn(xn) ≥ γc(x0, a0) + log eh(y)q(dy|x0, a0).(29) 18 A. JAŚKIEWICZ Since xn → x0 was chosen arbitrarily, we infer from (29) that l̂ + h(x0) ≥ γc(x0, a0) + log eh(y)q(dy|x0, a0). The last inequality shows that, for any x ∈X, there exists an ax ∈A(x) such l̂ + h(x) ≥ γc(x,ax) + log eh(y)q(dy|x,ax) ≥ min a∈A(x) γc(x,a) + eh(y)(y)q(dy|x,a) By our compactness–semicontinuity assumptions and Proposition D.5(b) in [17], there exists some f̂ ∈ F such that (23) holds. � 5. A discussion. This section is devoted to a discussion of Condition (B). We start with revisiting Example 3.1 in [8]. Example 1. Put X = {0,1}, A = {a}, c(x) := c(x,a) = x and the tran- sition matrix is as follows: ρ 1 − ρ where ρ ∈ (0,1). Recall that the following was proved. Let us consider three cases for the risk factor γ: (I) γ <− log(1− ρ), (II) γ = − log(1− ρ), (III) γ >− log(1− ρ). Then if (I) or (II) hold, the optimal risk-sensitive average cost equals 0 and is independent of the initial state. In case (III) we have J∗(0) = 0 and J∗(1) = 1 + log(1−ρ) > 0. In addition, it is interesting to observe that, for (II) and (III) cases, there does not exist a function h :X 7→ R such that optimality inequality (23) is satisfied. Indeed, to see this take x = 1 and consider (III). The optimality inequality is then as follows: γJ∗(1) + h(1) = γ + log(1 − ρ) + h(1) ≥ γ + log(eh(1)(1− ρ) + eh(0)ρ). Note that the right-hand side is strictly greater than γ + log(eh(1)(1 − ρ)), which equals to the left-hand side. Similar calculations for case (II) also lead to a contradiction. Hence, although an optimal cost is constant, the optimality inequality need not have a solution. Now we turn to checking Condition (B). Let Vβ be as in Lemma 2. Clearly, Vβ = w β for N ≥ 1 and Vβ(0) = 0. Then, by (8) under (I), we get Vβ(1) = γ + log[e βVβ(1)(1 − ρ) + ρ] < γ + log[eVβ(1)(1 − ρ) + ρ]. RISK-SENSITIVE CONTROL 19 Hence, Vβ(1) < log eγ(1− ρ) 1− eγ(1− ρ) ∀β ∈ (0,1), and consequently, supβ∈(0,1) hβ(x) < +∞. Now let the risk factor γ be as in (III). Then by (8), Vβ(1) > γ + log(1 − ρ) + βVβ(1), which in turn implies that Vβ(1) > γ + log(1− ρ) Thus, hβ(1) = Vβ(1) goes to the infinity when β ր 1. For case (II), we obtain Vβ(1) = − log(1− ρ) + log[e βVβ(1)(1− ρ) + ρ] = βVβ(1) + log 1 + e−βVβ(1) 1 − ρ If Vβ(1) ր +∞ when β ր 1, then the right-hand side of (31) also goes to the infinity. On the contrary, assume that supβ∈(0,1) Vβ(1) ≤C for some constant C > 0. Then, Vβ(1) ≥ log[1 + e−Cρ/(1 − ρ)] which leads to a contradiction when β ր 1. In consequence, in case (II) the family {hβ(1)} does not satisfy Condition (B) either. Therefore, the following conclusion can be drawn. Condition (B) is nec- essary to obtain a solution to the optimality inequality. For a verification of Condition (B), one can use Lemma 4 below. For a similar result in the risk-neutral, case we refer to [27, 28]. For some η ≥ 0, define the stopping time τ = τ(β) := inf{n≥ 0 :Vβ(xn) ≤mβ + η}. Lemma 4. For η ≥ 0, β ∈ (0,1) and x ∈X, hβ(x) ≤ η + inf logEπx exp γc(xk, ak) Proof. By Lemma 2(b), (c) and the fact that Vβ(y) ≥ 0, y ∈ X , we Vβ(x) = min a∈A(x) γc(x,a) + log eβVβ(y)q(dy|x,a) < γc(x,a) + log eVβ(y)q(dy|x,a) 20 A. JAŚKIEWICZ for each x ∈X. Subtracting mβ from both sides in (32), we obtain Vβ(x) −mβ < γc(x,a) + log e(Vβ(y)−mβ )q(dy|x,a). Iteration of this inequality up to the stopping time τ yields Vβ(x) −mβ < logE c(xk,ak)+η = η + logEπx exp c(xk, ak) Since π ∈ Π is an arbitrary policy, we easily get the conclusion. � Note that the fact Eπx exp γc(xk, ak) < +∞(33) has the following interpretation: before the process will reach “good states,” the incurred costs at “early stages” should not be too large. Indeed, let us define a set D as follows. We say that x ∈D iff Vβ(x) ≤mβ + η for a certain η ≥ 0. Clearly, D 6= ∅. Denote by τD the first return time of the process, governed by fβ, to set D. Certainly, if (33) holds with τ := τD, then Condition (B) is satisfied. In Example 1 we can take D = {0} and η = 0, since Vβ(0) ≤ 0 + 0. If γ is as in (I), then (33) holds: E1 exp τ0−1∑ γc(xk) enγ(1− ρ)n−1ρ = eγ(1− ρ) 1− eγ(1− ρ) In other cases (33) fails to hold and, in addition, the earlier calculations show that hβ(1) = +∞. Summing up, the presented example shows that, without Condition (B) imposed on the family of functions {hβ(x)}, β ∈ (0,1), a solution to the optimality inequality need not exist, and moreover, the optimal risk-sensitive average cost may depend on the initial state. In view of the above discussion, Condition (B) is designed to prevent the accrual of infinite expected costs. Namely, the costs incurred at transient states, that may be occupied only at “early stages,” have an important and definite influence on a long-run performance measure. Therefore, Condition (B) requires the model to be sort of communicating insofar as certain sets of “good states” to be reached sufficiently fast. Then, the optimal risk-sensitive average cost is constant and the optimality inequality takes place. In addition, it is worth mentioning that RISK-SENSITIVE CONTROL 21 the ergodicity itself of a Markov process/chain does not help so much as in the risk-neutral case. In other words, for an ergodic Markov chain, it may happen that the optimal risk-sensitive average cost depends on the initial state as in Example 1. Moreover, in this example one can even prove in a straightforward way that under case (I) [either under Condition (B) or for sufficiently small risk factors], the optimality equation (24) is satisfied. Therefore, it would be interesting to know whether Condition (B) (together with some compactness–continuity assumptions) is sufficient to obtain a solution to the optimality equation. There is a conjecture that, since in the risk-neutral case a counterpart of Condition (B) is not sufficient [7], neither is it in the risk-sensitive setting. But this question is beyond the scope of the paper and remains open. APPENDIX The lemma below establishes a variational formula for the logarithmic moment-generating function. The reader is referred to Theorem 4.5.1 and Proposition 1.4.2 in [12] for its proof. Lemma A. Let X be a Polish space, h a measurable function mapping on X into R, which is either bounded from below or bounded from above, and ν a probability measure on X . (a) Then, we have the variational formula ehdν = sup −R(µ‖ν) + where ∆ = {µ ∈ Pr(X ) :R(µ‖ν) < +∞}. (b) Let µ0 denote the probability measure on X , which is µ0 ≪ ν and satisfies (x) = eh(x)∫ eh dν Then, the supremum in the variational formula is attained uniquely at µ0. Acknowledgments. A part of this research was done while the author was a Humboldt research fellow and visiting the University of Ulm. The author gratefully acknowledges support from the Alexander von Humboldt Foundation. The second part of this paper was written at the Institute of Mathematics and Computer Science, Wroc law University of Technology. The author is greatly indebted to Professor Ulrich Rieder for drawing her attention to paper [16], suggesting the problem and for several helpful conversations. 22 A. JAŚKIEWICZ REFERENCES [1] Balaji, S. and Meyn, S. P. (2000). Multiplicative ergodicity and large deviations for an irreducible Markov chains. Stochastic Process. Appl. 90 123–144. MR1787128 [2] Berge, E. (1963). Topological Spaces. MacMillan, New York. [3] Bielecki, T., Hernández-Hernández, D. and Pliska, S. (1999). Risk-senisitive control of finite state Markov chains in discrete time, with applications to port- folio managment. Math. Methods Oper. Res. 50 167–188. MR1732397 [4] Bielecki, T. and Pliska, S. (1999). Risk-senisitive dynamic asset managment. Appl. Math. Optim. 39 337–360. MR1675114 [5] Borkar, V. S. and Meyn, S. P. (2002). Risk-sensitive optimal control for Markov decision processes with monotone cost. Math. Oper. Res. 27 192–209. MR1886226 [6] Brown, L. D. and Purves, R. (1973). Measurable selections of extrema. Ann. Statist. 1 902–912. MR0432846 [7] Cavazos-Cadena, R. (1991). A counterexample on the optimality equation in Markov decision chains with the average cost criterion. Systems Control Lett. 16 387–392. [8] Cavazos-Cadena, R. and Fernández-Gaucherand, E. (1999). Controlled Markov chains with risk-sensitive criteria: Average cost, optimal equations and optimal solutions. Math. Methods Oper. Res. 49 299–324. MR1687362 [9] Dai Pra, P., Meneghini, L. and Runggaldier, W. J. (1996). Some connections between stochastic control and dynamic games. Math. Control Signals Systems 9 303–326. MR1450355 [10] Di Masi, G. B. and Stettner, L. (2000). Risk-sensitive control of discrete-time Markov processes with infinite horizon. SIAM J. Control Optim. 38 61–78. MR1740607 [11] Di Masi, G. B. and Stettner, L. (2000). Infinite horizon risk sensitive control of discrete time Markov processes with small risk. Systems Control Lett. 40 15–20. MR1829070 [12] Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York. MR1431744 [13] Filar, J. and Vrieze, K. (1997). Competitive Markov Decision Processes. Springer, New York. MR1418636 [14] Fleming, W. H. and Hernández-Hernández, D. (1997). Risk-sensitive control of finite state machines on an infinite horizon. SIAM J. Control Optim. 35 1790– 1810. MR1466928 [15] Hernández-Hernández, D. and Marcus, S. I. (1996). Risk sensitive control of Markov processes in countable state space. Systems Control Lett. 29 147–155. [Corrigendum (1998) Systems Control Lett. 34 105–106.] MR1422212 [16] Hernández-Hernández, D. and Marcus, S. I. (1999). Existence of risk-sensitive optimal stationary policies for controlled Markov processes. Appl. Math. Optim. 40 273–285. MR1709324 [17] Hernández-Lerma, O. and Lasserre, J. B. (1993). Discrete-Time Markov Control Process: Basic Optimality Criteria. Springer, New York. MR1363487 [18] Howard, R. A. and Matheson, J. E. (1972). Risk-sensitive Markov decision pro- cesses. Management Sci. 18 356–369. MR0292497 [19] Jacobson, D. H. (1973). Optimal stochastic linear systems with exponential per- formance criteria and their relation to deterministic differential games. IEEE Trans. Automat. Control 18 124–131. MR0441523 http://www.ams.org/mathscinet-getitem?mr=1787128 http://www.ams.org/mathscinet-getitem?mr=1732397 http://www.ams.org/mathscinet-getitem?mr=1675114 http://www.ams.org/mathscinet-getitem?mr=1886226 http://www.ams.org/mathscinet-getitem?mr=0432846 http://www.ams.org/mathscinet-getitem?mr=1687362 http://www.ams.org/mathscinet-getitem?mr=1450355 http://www.ams.org/mathscinet-getitem?mr=1740607 http://www.ams.org/mathscinet-getitem?mr=1829070 http://www.ams.org/mathscinet-getitem?mr=1431744 http://www.ams.org/mathscinet-getitem?mr=1418636 http://www.ams.org/mathscinet-getitem?mr=1466928 http://www.ams.org/mathscinet-getitem?mr=1422212 http://www.ams.org/mathscinet-getitem?mr=1709324 http://www.ams.org/mathscinet-getitem?mr=1363487 http://www.ams.org/mathscinet-getitem?mr=0292497 http://www.ams.org/mathscinet-getitem?mr=0441523 RISK-SENSITIVE CONTROL 23 [20] Jaśkiewicz, A. (2006). A note on risk-sensitive control of invariant models. Technical Report, Wroc law University of Technology. [21] Jaśkiewicz, A. and Nowak, A. S. (2006). On the optimality equation for average cost Markov control processes with Feller transition probabilities. J. Math. Anal. Appl. 316 495–509. MR2206685 [22] Jaśkiewicz, A. and Nowak, A. S. (2006). Zero-sum ergodic stochastic games with Feller transition probabilities. SIAM J. Control Optim. 45 773–789. MR2247715 [23] Klein, E. and Thompson, A. C. (1984). Theory of Correspondences. Wiley, New York. MR0752692 [24] Neveu, J. (1965). Mathematical Foundations of the Calculus of Probability. Holden- Day, San Francisco, CA. MR0198505 [25] Royden, H. L. (1968). Real Analysis. MacMillan, New York. MR0151555 [26] Schäl, M. (1975). Conditions for optimality in dynamic programming and for the limit n-stage optimal policies to be optimal. Z. Wahrsch. Verw. Gebiete 32 179– 196. MR0378841 [27] Schäl, M. (1993). Average optimality in dynamic programming with general state space. Math. Oper. Res. 18 163–172. MR1250112 [28] Sennott, L. I. (1999). Stochastic Dynamic Programming and the Control of Queue- ing Systems. Wiley, New York. MR1645435 [29] Serfozo, R. (1982). Convergence of Lebesgue integrals with varying measures. Sankhyã Ser. A 44 380–402. MR0705462 [30] Stettner, L. (1999). Risk sensitive portfolio optimization. Math. Methods Oper. Res. 50 463–474. MR1731299 [31] Whittle, P. (1990). Risk-Sensitive Optimal Control. Wiley, Chichester. MR1093001 Institute of Mathematics and Computer Science Wroc law University of Technology Wybrzeże Wyspiańskiego 27 PL-50-370 Wroc law Poland E-mail: [email protected] http://www.ams.org/mathscinet-getitem?mr=2206685 http://www.ams.org/mathscinet-getitem?mr=2247715 http://www.ams.org/mathscinet-getitem?mr=0752692 http://www.ams.org/mathscinet-getitem?mr=0198505 http://www.ams.org/mathscinet-getitem?mr=0151555 http://www.ams.org/mathscinet-getitem?mr=0378841 http://www.ams.org/mathscinet-getitem?mr=1250112 http://www.ams.org/mathscinet-getitem?mr=1645435 http://www.ams.org/mathscinet-getitem?mr=0705462 http://www.ams.org/mathscinet-getitem?mr=1731299 http://www.ams.org/mathscinet-getitem?mr=1093001 mailto:[email protected] Introduction and the model Preliminaries A solution to the auxiliary discounted minimax problem A solution to the risk-sensitive control problem A discussion Appendix Acknowledgments References Author's addresses

0704.0395

A Study of $B_{d}^0 \to J/\Psi \eta^{(\prime)}$ Decays in the pQCD Approach

ZJOU-PHY-TH-07-02 NJNU-TH-07-11 A Study of B0d → J/Ψη(′) Decays in the pQCD Approach Xin Liua∗, Zhen-Jun Xiaob†, Hui-Sheng Wangc a. Department of Physics, Zhejiang Ocean University, Zhoushan, Zhejiang 316000, P.R. China b. Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, Jiangsu 210097, P.R. China and c. Department of Applied Mathematics and Physics, Anhui University of Technology and Science, Wuhu, Anhui 241000, P.R. China (Dated: November 4, 2018) Abstract Motivated by the very recent measurement of the branching ratio of B0d → J/ψη decay, we calculate the branching ratios of Bd 0 → J/ψη and Bd0 → J/Ψη′ decays in the perturbative QCD (pQCD) approach. The pQCD predictions for the branching ratios of considered decays are: BR(B0d → J/Ψη) = (1.96 +9.68 −0.65) × 10−6, which is consistent with the first experimental measurement within errors; while BR(B0d → J/Ψη′) = (1.09 +3.76 −0.25) × 10−6, very similar with B0d → J/Ψη decay and can be tested by the forthcoming LHC experiments. The measurements of these decay channels may help us to understand the QCD dynamics in the corresponding energy scale, especially the reliability of pQCD approach to these kinds of B meson decays. PACS numbers: 13.25.Hw, 12.38.Bx, 14.40.Nd ∗ [email protected] † [email protected] http://arxiv.org/abs/0704.0395v1 Very recently, the first observation of B0d → J/Ψη decay was reported by Belle Collab- oration [1], and the branching ratio measured is BR(B0d → J/Ψη) = (9.5± 1.7(stat)± 0.8(syst))× 10−6, (1) which is consistent with the currently available theoretical predictions [1, 2, 3]. Up to now, the theoretical calculations for the branching ratios of Bd → J/Ψη(′) decays were obtained by using the heavy quark factorization approximation in Ref. [2], or from the measured J/Ψπ0 and J/ΨK0 branching ratios[3, 4, 5] based on the assumption of the SU(3) flavor symmetry of strong interaction. In this paper, we will calculate the branching ratios of B0d → J/Ψη and B0d → J/Ψη(′) decays directly by employing the low energy effective Hamiltonian [6] and the perturbative QCD (pQCD) factorization approach [7, 8, 9]. The paper is organized as follows: we present the formalism used in the calculation of B0d → J/ψη(′) decays in Sec. I. In Sec. II, we show the numerical results and compare them with the measured values. A short summery and some conclusions are also included in this section. I. FORMALISM AND PERTURBATIVE CALCULATIONS The pQCD approach has been developed earlier from the QCD hard-scattering ap- proach [7], and has been used frequently to calculate various B meson decay channels [7, 8, 9, 10]. For two body charmless hadronic Bd,s → Mη(′) (here M stands for the pseudo-scalar or vector light mesons composed of the light quarks u, d, s) decays, the pQCD predictions generally agree well with the measured values [9, 10, 11]. In Refs. [12, 13], the authors calculated B → D∗sK,D s and Bs → D(∗)+D(∗)− decays and found that the pQCD approach works well for such decays. Here we try to apply the pQCD approach to calculate the B meson decays involving the heavier J/Ψ meson as one of the two final state mesons. A. Formulism In pQCD approach, the decay amplitude of B → J/ΨP (P = η, η(′) here) decay can bo written conceptually as the convolution, A(B →M1M2) ∼ d4k1d 4k3 Tr C(t)ΦB(k1)ΦJ/Ψ(k2)ΦP (k3)H(k1, k2, k3, t) , (2) where the term “Tr” denotes the trace over Dirac and color indices. C(t) is the Wilson coefficient which results from the radiative corrections at short distance. In the above convolution, C(t) includes the harder dynamics at larger scale thanMB scale and describes the evolution of local 4-Fermi operators from mW (the W boson mass) down to t ∼ Λ̄MB) scale, where Λ̄ ≡ MB −mb. The function H(k1, k2, k3, t) is the hard part and can be calculated perturbatively. The function ΦM is the wave function which describes hadronization of the quark and anti-quark to the mesonM . While the functionH depends on the process considered, the wave function ΦM is independent of the specific process. Using the wave functions determined from other well measured processes, one can make quantitative predictions here. Using the light-cone coordinates the B meson and the two final state meson momenta can be written as (1, 1, 0T ), P2 = (1, r2, 0T ), P3 = (0, 1− r2, 0T ), (3) respectively, where r = MJ/Ψ/MB, and the light meson masses m η have been ne- glected. The longitudinal polarization vector of the J/Ψ meson, ǫL, is given by ǫL = 2MJ/Ψ (1,−r2, 0T ). Putting the light (anti-) quark momenta in B, J/Ψ and η( ′) mesons as k1, k2, and k3, respectively, we can choose k1 = (x1P 1 , 0,k1T ), k2 = (x2P 2 , 0,k2T ), k3 = (0, x3P 3 ,k3T ). (4) Then, for B → J/Ψη decay for example, the integration over k−1 , k−2 , and k+3 in eq.(2) will lead to A(B → J/Ψη′) ∼ dx1dx2dx3b1db1b2db2b3db3 C(t)ΦB(x1, b1)ΦJ/Ψ(x2, b2)Φη(x3, b3)H(xi, bi, t)St(xi) e −S(t)] ,(5) where bi is the conjugate space coordinate of kiT , and t is the largest energy scale in functionH(xi, bi, t). The large logarithms ln(mW/t) are included in the Wilson coefficients C(t). The large double logarithms (ln2 xi) on the longitudinal direction are summed by the threshold resummation [14], and they lead to St(xi) which smears the end-point singularities on xi. The last term, e −S(t), is the Sudakov form factor which suppresses the soft dynamics effectively [15]. Thus it makes the perturbative calculation of the hard part H applicable at intermediate scale, i.e., MB scale. We will calculate analytically the function H(xi, bi, t) for the considered decays in the first order in αs expansion and give the convoluted amplitudes in next section. B. The B0d → J/Ψη( ′) Decays The low energy effective Hamiltonian for decay modes B0d → J/ψη( ′) can be written as Heff = [VcbV cd (C1(µ)O 1(µ) + C2(µ)O 2(µ))] , (6) with the four-fermion operators Oc1 = d̄αγ µ(1− γ5)cβ · c̄βγµ(1− γ5)bα , Oc2 = d̄αγµ(1− γ5)cα · c̄βγµ(1− γ5)bβ (7) where the Wilson coefficients Ci(µ) (i = 1, 2), we will use the leading order (LO) expres- sions, although the next-to-leading order (NLO) results already exist in the literature [6]. This is the consistent way to cancel the explicit µ dependence in the theoretical formulae. For the renormalization group evolution of the Wilson coefficients from higher scale to lower scale, we use the formulae as given in Ref.[16] directly. FIG. 1: Typical Feynman diagrams contributing to the Cabibbo- and color- suppressed B0d → J/Ψη( ′) decays. As for B meson wavefunction, we make use of the same parameterizations as used in the studies of different processes [16]. For vector J/ψ meson, in terms of the nota- tion in Ref. [17], we decompose the nonlocal matrix elements for the longitudinally and transversely polarized J/ψ mesons into ΦJ/Ψ(x) = mJ/ψǫ/ LΨ L(x) + ǫ/ LP/Ψ , (8) Here, ΨL denote for the twist-2 distribution amplitudes, and Ψt for the twist-3 distri- bution amplitudes. x represents the momentum fraction of the charm quark inside the charmonium. The J/ψ meson asymptotic distribution amplitudes read as [18] ΨL(x) = 9.58 x(1− x) x(1 − x) 1− 2.8x(1− x) Ψt(x) = 10.94 (1− 2x)2 x(1− x) 1− 2.8x(1− x) . (9) It is easy to see that both the twist-2 and twist-3 DAs vanish at the end points due to the factor [x(1− x)]0.7. From the effective Hamiltonian (6), the Feynman diagrams corresponding to the con- sidered decay are shown in Fig.1. With the meson wave functions and Sudakov factors, the hard amplitude is given as Feη = 8πCFm dx1dx3 b1db1b3db3 φB(x1, b1) (1− r2) (1 + x3(1− r2))φAη (x3, b3) + r0(1− 2x3) ·φPη (x3, b3) (1− 2x3) + r2(1 + 2x3) φTη (x3, b3) ·αs(t1e) he(x1, x3, b1, b3) exp[−Sab(t1e)] 1− (1− x1)r2φPη (x3, b3)− x1r2φAη (x3, b3) ·αs(t2e)he(x3, x1, b3, b1) exp[−Sab(t2e)] . (10) where r0 = m 0/mB; CF = 4/3 is a color factor. The function he, the scales t e and the Sudakov factors Sab are displayed in Appendix A. For the non-factorizable diagrams 1(c) and 1(d), all three meson wave functions are involved. The integration of b3 can be performed using δ function δ(b3 − b1), leaving only integration of b1 and b2. For the concerned operators, the corresponding decay amplitude Meη = dx1dx2 dx3 b1db1b2db2 φBs(x1, b1) 2rrcφ J/Ψ(x2, b2)φ η (x3, b2)− 4rr0rcφtJ/Ψ(x2, b2)φTη (x3, b2) 2 + x3(1− 2r2) φLJ/Ψ(x2, b2)φ η (x3, b2) x3r0 + (x2 − x3)r0r2 φLJ/Ψ(x2, b2)φ η (x3, b2) ·αs(tf )hf(x1, x2, x3, b1, b2) exp[−Scd(tf )]} . (11) where rc = mc/mB,mc is the mass for c quark. For the B0d → J/Ψη′ decay, the Feynman diagrams are obtained by replacing the η meson in Fig. 1 with the meson η′. The corresponding expressions of decay amplitudes will be similar with those as given in Eqs.(10-11), since the η and η′ are all light pseudoscalar mesons and have the similar wave functions. The expressions of B0d → J/Ψη′ decay can be obtained simply by the following replacements φAη −→ φAη′ , φPη −→ φPη′ , φTη −→ φTη′ , r0 −→ r′0. (12) For the η−η′ system, there exist two popular mixing basis: the octet-singlet basis and the quark-flavor basis [19, 20]. Here we use the quark-flavor basis [19] and define ηq = (uū+ dd̄)/ 2, ηs = ss̄. (13) The physical states η and η′ are related to ηq and ηs through a single mixing angle φ, = U(φ) cosφ − sinφ sin φ cos φ . (14) The three input parameters fq, fs and φ in the quark-flavor basis have been extracted from various related experiments [19, 20] fq = (1.07± 0.02)fπ, fs = (1.34± 0.06)fπ, φ = 39.3◦ ± 1.0◦, (15) where fπ = 130 MeV. In the numerical calculations, we will use these mixing parameters as inputs. It worth of mentioning that the effects of possible gluonic component of η′ meson will not considered here since it is small in size [10, 21, 22]. For B0d → J/Ψη decay, by combining the contributions from different diagrams, the total decay amplitude can be written as M(B0d → J/Ψη) = VcbV ∗cdF1(φ) FeηfJ/Ψ +MeηC2 where the relevant mixing parameter is F1(φ) = cos φ/ It should be mentioned that the Wilson coefficients Ci = Ci(t) in Eq. (16) should be calculated at the appropriate scale t using equations as given in the Appendices of Ref. [16]. Here the scale t in the Wilson coefficients should be taken as the same scale appeared in the expressions of decay amplitudes in Eqs. (10) and (11). This is the way in pQCD approach to eliminate the scale dependence. In order to estimate the effect of higher order corrections, however, we introduce a scale factor at = 1.0± 0.2 and vary the scale tmax as described in Appendix A. Similarly, the decay amplitudes for B0d → J/Ψη′ decay can be obtained easily from Eq.(16) by the following replacements of F1(φ) → F ′1(φ) = sinφ/ II. NUMERICAL RESULTS AND DISCUSSIONS In this section, we will calculate the branching ratios for those considered decay modes. The input parameters and the wave functions to be used are given in Appendix B. In numerical calculations, central values of input parameters will be used implicitly unless otherwise stated. With the complete decay amplitudes, we can obtain the decay width for the considered decays, Γ(B0d → J/ψη( ′)) = (1− r2) M(B0d → J/ψη( . (17) By employing the quark-flavor scheme of η−η′ system and using the mixing parameters as given in Eq. (15), one finds the branching ratios for the considered two decays with error bars as follows: Br( B0d → J/Ψη) = 1.96+0.71−0.50(ωb) +9.65 −0.39(at) +0.32 +0.13(a2) +0.14 −0.13(fJ/Ψ) × 10−6, (18) Br( B0d → J/Ψη′) = 1.09+0.32−0.24(ωb) +3.73 +0.01(at) +0.28 +0.01(a2) +0.08 −0.07(fJ/Ψ) × 10−6, (19) where the main errors are induced by the uncertainties of ωb = 0.40 ± 0.05 GeV, at = 1.0 ± 0.2, a2 = 0.115 ± 0.115 and fJ/Ψ = 0.405 ± 0.014 GeV , respectively. One can see that the pQCD predictions are sensitive to the variations of ωb and at. For B0d → J/Ψη decay, the central value of the pQCD prediction for Br(B0d → J/Ψη) is a factor of 4 smaller than the measured value as given in Eq. (1) [1]. But the pQCD prediction is in fact still consistent with Belle’s first measurement if we take the large theoretical and experimental errors into account. By varying the scale factor at in the range of at = [0.8, 1.0], for example, the central value of Br(B → J/Ψη) will change in the range of [0.2, 1.1]×10−5 accordingly. It is not difficult to understand such at dependence. Since the J/Ψ meson is much heavier than light mesons, and therefore moving not as fast as those light meson when B meson is decaying. So a small decrease of the scale ti will lead to a larger Wilson coefficients C1,2(t) and αs(ti), and consequently results in a larger decay rate. For B0d → J/Ψη′ decay, only experimental upper limit (at 90% C.L) is available now: BR(B0 → J/Ψη′) < 6.3 × 10−5 [4, 5]. The pQCD prediction for the branching ratio of B0d → J/Ψη′ decay is very similar in magnitude with that of B0d → J/Ψη, consistent with the upper limit and will be tested in the forthcoming LHC experiments. At the leading order, only the tree Feynman diagrams as shown in Fig. 1 contribute to B0d → J/Ψη(′) decays. There exists no CP violation in these decays within the standard model, since there is only one kind of Cabibbo-Kabayashi-Muskawa (CKM) phase involved in the corresponding decay amplitudes, as can be seen from eq. (16). In short, we calculated the branching ratios of B0d → J/Ψη and B0d → J/Ψη′ decays at the leading order by using the pQCD factorization approach. Besides the usual factoriz- able diagrams, the non-factorizable spectator diagrams are also calculated analytically in the pQCD approach. By keeping the transverse momentum kT , the end-point singularity disappears in our calculation. From our calculations and phenomenological analysis, we found the following results: • Using the quark-flavor scheme, the pQCD predictions for the branching ratios are Br(B0d → J/Ψη) = 1.96+9.68−0.65 × 10−6, (20) Br(B0d → J/Ψη′) = 1.09+3.76−0.25 × 10−6, (21) where the various errors as specified previously have been added in quadrature. • The major theoretical errors of the pQCD predictions are induced by the uncertain- ties of the hard energy scale ti’s and the parameters ωb. Acknowledgments X. Liu would like to acknowledge the financial support of The Scientific Research Start-up Fund of Zhejiang Ocean University under Grant No.21065010706. This work was partially supported by the National Natural Science Foundation of China under Grant No.10575052, and by the Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) under Grant No. 20050319008. APPENDIX A: RELATED FUNCTIONS We show here the function hi’s, coming from the Fourier transformations of the function H(0), he(x1, x3, b1, b3) = K0 x1x3(1− r2)mBb1 θ(b1 − b3)K0 x3(1− r2)mBb1 x3(1− r2)mBb3 + θ(b3 − b1)K0 x3(1− r2)mBb3 x3(1− r2)mBb1 St(x3), (A1) hf(x1, x2, x3, b1, b2) = θ(b2 − b1)I0(MB x1x3(1− r2)b1)K0(MB x1x3(1− r2)b2) + (b1 ↔ b2) K0(MBF(1)b2), for F (1) > 0 0 (MB | b2), for F 2(1) < 0 , (A2) where J0 is the Bessel function, K0 and I0 are the modified Bessel functions with K0(−ix) = −(π/2)Y0(x) + i(π/2)J0(x), and F(j)’s are defined by F 2(1) = (x1 − x2)x3(1− r2) + r2c , (A3) F 2(2) = (x1 − x2)x3(1− r2) + r2c . (A4) The threshold resummation form factor St(xi) is adopted from Ref. [17] St(x) = 21+2cΓ(3/2 + c)√ πΓ(1 + c) [x(1 − x)]c, (A5) where the parameter c = 0.3. This function is normalized to unity. The Sudakov factors used in the text are defined as Sab(t) = s x1mB/ 2, b1 x3mB/ 2, b3 (1− x3)mB/ 2, b3 ln(t/Λ) − ln(b1Λ) ln(t/Λ) − ln(b3Λ) , (A6) Scd(t) = s x1mB/ 2, b1 x2mB/ 2, b2 (1− x2)mB/ 2, b2 x3mB/ 2, b1 (1− x3)mB/ 2, b1 ln(t/Λ) − ln(b1Λ) ln(t/Λ) − ln(b2Λ) , (A7) where the function s(q, b) are defined in the Appendix A of Ref. [16]. The scale ti’s in the above equations are chosen as t1e = at ·max( x3(1− r2)MB, 1/b1, 1/b3), t2e = at ·max( x1(1− r2)MB, 1/b1, 1/b3), tf = at ·max( x1x3(1− r2)MB, (x1 − x2)x3(1− r2) + r2cMB, 1/b1, 1/b2), (A8) where at = 1.0±0.2 and r =MJ/Ψ/MB. The scale ti’s are chosen as the maximum energy scale appearing in each diagram to kill the large logarithmic radiative corrections. APPENDIX B: INPUT PARAMETERS AND WAVE FUNCTIONS The masses, decay constants, QCD scale and B0d meson lifetime are (f=4) = 250MeV, fπ = 130MeV, fJ/Ψ = 405MeV, 0 = 1.08GeV, MB0d = 5.28MeV, MJ/Ψ = 3.097GeV, MW = 80.41GeV, τB0 = 1.54× 10−12s. (B1) For the CKM matrix elements, here we adopt the Wolfenstein parametrization for the CKM matrix, and take λ = 0.2272, A = 0.818, ρ = 0.221 and η = 0.340 [4]. For the B meson wave function, we adopt the model φB(x, b) = NBx 2(1− x)2exp (ωbb) , (B2) where ωb is a free parameter and we take ωb = 0.40± 0.05 GeV in numerical calculations, and NB = 91.745 is the normalization factor for ωb = 0.40 for the B meson. The wave function for dd̄ components of η(′) meson is given by Φηdd̄(p, x, ζ) ≡ P/φAηdd̄(x) +m (x) + ζm 0 (v/n/− v · n)φTηdd̄(x) , (B3) where p and x are the momentum and the momentum fraction of ηdd̄ respectively, while φAηdd̄, φ and φTηdd̄ represent the axial vector, pseudoscalar and tensor components of the wave function respectively. We here assume that the wave function of ηdd̄ is same as the π wave function based on SU(3) flavor symmetry. The parameter ζ is either +1 or −1 depending on the assignment of the momentum fraction x. The explicit expression of chiral enhancement scale m 0 = m 0 is given by [21] [m2η cos 2 φ+m2η′ sin (m2η′ −m2η) cosφsinφ], (B4) and numerically m 0 = 1.07MeV for mη = 547.5 MeV, mη′ = 957.8 MeV, fq = 1.07fπ, fs = 1.34fπ and φ = 39.3 For the distribution amplitude φAηq , φ and φTηq , we utilize the results for π meson obtained from the light-cone sum rule [23] including twist-3 contributions: φAηq(x) = fqx(1 − x) 1 + a 5(1− 2x)2 − 1 21(1− 2x)4 − 14(1− 2x)2 + 1 , (B5) φPηq(x) = 30η3 − 3(1− 2x)2 − 1 −3η3ω3 − ρ2ηq − ρ2ηq(s)a 35(1− 2x)4 − 30(1− 2x)2 + 3 ,(B6) φTηq(x) = fq(1− 2x) + (5η3 − η3ω3 − ρ2ηq − ρ2ηqa 2 )(10x 2 − 10x+ 1) , (B7) with the updated Gegenbauer moments [24] 2 = 0.115, a 4 = −0.015, ρηq = 2mq/mqq, η3 = 0.015, ω3 = −3.0. (B8) [1] B. Aubert et al. (BaBar Collaboration), Phys. Rev. Lett. 98, 131803 (2007). [2] A. Deandrea et al., Phys. Lett. B 318, 549 (1993). [3] P.Z. Skands, J. High Energy Phys. 0101 (2001) 008. [4] W.-M. Yao et al. ( Particle Data Group), J. Phys. G 33, 1 (2006). [5] Heavy Flavor Averaging Group, E. Barberio et al., hep-ex/0603003; and online update at http://www.slac.stanford.edu/xorg/hfag. [6] G. Buchalla, A.J. Buras, and M.E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996). [7] G.P. Lepage and S.J. Brodsky, Phys. Rev. D 22, 2157 (1980). [8] C.-H. V. Chang and H.N. Li, Phys. Rev. D 55, 5577 (1997); T.-W. Yeh and H.N. Li, Phys. Rev. D 56, 1615 (1997). [9] H.N. Li, Prog.Part.& Nucl.Phys. 51, 85 (2003), and reference therein. H.N. Li and H.L. Yu, Phys. Rev. Lett. 74, 4388 (1995); Phys. Lett. B 353, 301 (1995); Phys. Rev. D 53, 2480 (1996). [10] X. Liu, H.S. Wang, Z.J. Xiao, L.B. Guo and C.D. Lü, Phys. Rev. D 73, 074002 (2006); H.S. Wang, X. Liu, Z.J. Xiao, L.B. Guo and C.D. Lü, Nucl. Phys. B 738. 243 (2006); Z.J. Xiao, X.F. Chen and D.Q. Guo, Eur.Phys.J. C 50 (2007) in press; Z.J. Xiao, D.Q Guo and X.F. Chen, Phys. Rev. D 75 , 014018 (2007); Z.J. Xiao, X. Liu and H.S. Wang, Phys. Rev. D 75034017 (2007); Z.J. Xiao, X.F. Chen and D.Q. Guo, hep-ph/0701146. [11] A. Ali, G. Kramer, Y. Li, C.D. Lü, Y.L. Shen, W. Wang, and Y.M. Wang, hep-ph/0703162. [12] Y. Li and C.D. Lü, J. Phys. G 29, 2115 (2003); High Energy & Nucl.Phys. 27, 1061 (2003). [13] Y. Li, C.D. Lü, and Z.J. Xiao, J. Phys. G 31, 273 (2005). [14] H.N. Li, Phys. Rev. D 66, 094010 (2002). [15] H.N. Li and B. Tseng, Phys. Rev. D 57, 443, (1998). [16] C.-D. Lü, K. Ukai and M.Z. Yang, Phys. Rev. D 63, 074009 (2001). [17] T. Kurimoto, H.N. Li, and A.I. Sanda, Phys. Rev. D 65, 014007 (2002); Phys. Rev. D 67, 054028 (2003). [18] A.E. Bondar and V.L. Chernyak, Phys. Lett. B 612, 215(2005). [19] Th. Feldmann, P. Kroll and B. Stech, Phys. Rev. D 58, 114006 (1998). [20] R. Escribano and J.M. Frere, J. High Energy Phys. 0506 (2005) 029; J. Schechter, A. Subbaraman and H. Weigel, Phys. Rev. D 48, 339 (1993) [21] Y.-Y. Charng, T. Kurimoto, H.N. Li, Phys. Rev. D 74, 074024 (2006). [22] R. Escribano, J. Nadal, hep-ph/0703187. [23] P. Ball, J. High Energy Phys. 9809, 005 (1998); P. Ball, J. High Energy Phys. 9901, 010 (1999). [24] P. Ball and R. Zwicky, Phys. Rev. D 71, 014015 (2005); P. Ball, V.M. Braun, and A. Lenz, J. High Energy Phys. 0605 (2006) 004. http://arxiv.org/abs/hep-ex/0603003 http://www.slac.stanford.edu/xorg/hfag http://arxiv.org/abs/hep-ph/0701146 http://arxiv.org/abs/hep-ph/0703162 http://arxiv.org/abs/hep-ph/0703187 Formalism and Perturbative Calculations Formulism The Bd0 J/(') Decays Numerical results and Discussions Acknowledgments Related Functions Input parameters and wave functions References

0704.0396

Finite-temperature phase transitions in a two-dimensional boson Hubbard model

Finite-temperature phase transitions in a two-dimensional boson Hubbard model Min-Chul Cha1 and Ji-Woo Lee2 Department of Applied Physics, Hanyang University, Ansan 426-791, Korea Department of Physics, Myongji University, Yongin 449-728, Korea We study finite-temperature phase transitions in a two-dimensional boson Hubbard model with zero-point quantum fluctuations via Monte Carlo simulations of quantum rotor model, and construct the corresponding phase diagram. Compressibility shows a thermally activated gapped behavior in the insulating regime. Finite-size scaling of the superfluid stiffness clearly shows the nature of the Kosterlitz-Thouless transition. The transition temperature, Tc, confirms a scaling relation Tc ∝ ρ with x = 1.0. Some evidences of anomalous quantum behavior at low temperatures are presented. PACS numbers: 73.43.Nq, 74.25.Dw, 05.30.Jp Recently quantum phase transitions[1, 2] have drawn a lot of attention in systems of interacting particles. Typ- ically strong interactions suppress the itineracy of par- ticles to induce a strongly correlated insulating phase, whereas with weak interactions a conducting phase is stable. The criticality of these zero-temperature phase transitions can be investigated at low, but finite, tem- peratures. How quantum fluctuations associated with a quantum critical point(QCP) have influence on phases at finite temperatures [3, 4, 5] is a theoretically interesting and an experimentally relevant question. At finite temperatures, it is expected that a quantum phase transition turns into a classical one with the same order parameter or disappears. Remnant quantum fluc- tuations near a QCP may bring anomalous properties[3], which can be captured by scaling relations, and lead to crossover behaviors as temperature rises. Some pos- sibilities such as reentrant behaviors due to the inter- play of quantum and thermal fluctuations have been proposed[6]. These issues can be clarified by direct investigations of a generic quantum mechanical model. So far most of the theoretical investigations heavily rely on the exact solu- tion of the quantum Ising model, available strictly in one dimension[4]. Interacting bosonic systems simulated via Monte Carlo methods, not suffering from negative sign problems, will be an ideal place to study these problems. In previous works, a quantum XY model, equivalent to hard-core bosons at half-filling, showed the Kosterlitz- Thouless(KT) transition[7] at finite temperature in two dimensions[8, 9]. In the model with nearest neighbor repulsion, destruction of the solid order as well as the superfluidity by thermal fluctuations was observed [10]. However, generic finite-temperature phase diagrams have not been constructed. In this work, we investigate the thermally driven phase transitions of a two-dimensional quantum rotor model, which is believed to share the same critical properties of a soft-core generic boson Hubbard model[11], via Monte Carlo simulations. The results are summarized in the phase diagram as shown in Fig. 1. Finite-size scaling properties of the superfluid stiffness confirm that the na- ture of the classical phase transition associated with the destruction of superfluidity is consistent with that of the KT transition, and clearly support the scenario of the universal jump at the critical point[12]. Finite tempera- ture, T , sets the size in the temporal direction, leading to a scaling behavior[4, 11] Tc ∝ ρ0x with x = 1.0, where Tc is the transition temperature and ρ0 is the superfluid stiff- ness at zero temperature. The compressibility diverges at the transition. In the insulating regime at low tempera- ture, thermally activated behavior of the compressibility with a finite energy gap is observed. Some anomalous de- pendence of energy and specific heat on T , possibly due to quantum fluctuations, are observed for T < 0.25U . The Hamiltonian of a boson Hubbard model reads nj(nj − 1)− µ nj − w ibj + b jbi),(1) where bj(b j) is the boson annihilation(creation) operator 0 0.01 0.02 0.03 0.04 0.05 µ=0.9 Normal fluid Mott Insulator Superfluid KT transition Gapped fluid FIG. 1: (Color online) Phase diagram on the space of hop- ping strength, t(= n0(n0 + 1)w), and temperature, T , in unit of U . The solid line denotes the classical phase transi- tions, which terminates at a QCP at T = 0. The dotted line represents crossover between gapped fluid and normal fluid. http://arxiv.org/abs/0704.0396v1 at the j-th site, and nj is the number operator. U and w stand for the strengths of the on-site repulsion and of the nearest neighbor hopping, respectively, and µ is the chemical potential. It is convenient to put µ/U + 1/2 = n0 + n̄ with an integer n0 and −1/2 < n̄ ≤ 1/2 so that n0 represents the background number of bosons per site and n̄ is a charge offset. When n̄ = 0, the density of bosons is fixed to a commensurate filling across the transition. For non- integer n̄, however, an integer filling in a Mott insulator shifts to a non-integer filling in a compressible fluid. We study the phase transition of the latter case in (2+1)- dimensional L×L×Lτ square lattices, where L denotes the size in a spatial dimension and Lτ in the temporal dimension. Since the phase transition of the model in Eq. (1) is characterized by the establishment of phase coher- ence, we may rewrite the Hamiltonian in terms of the phase angle θj of bosons by replacing bj(b j) =√ −iθj ( nj + 1e iθj ) with nj = . Under the as- sumption that the nature of the transition is governed only by the fluctuations of θj , not those of the hop- ping strength, we replace nj → n0 so that bj(b†j) =√ −iθj ( n0 + 1e iθj ). Then, the Hamiltonian is re- duced to a quantum rotor model nj(nj − 1)− µ nj − 2t cos(θi − θj),(2) where t = n0(n0 + 1)w. Here we take the number of bosons nj ≥ 0. Through a path integral mapping, we can construct the corresponding classical action[14] Jτr (J r − 1)− ǫµJτr − ln IJxr (2ǫt)− ln IJyr (2ǫt)(3) with the partition function ∇· ~J=0 { ~Jr} ~J], (4) where ǫ = β/Lτ is a lattice constant in the imaginary time axis for an inverse temperature β, ~Jr is an integer current at site r = (j, τ) with a spatial index j and a tem- poral index τ , which is conserved at each site as denoted by ∇ · ~J = 0, and Im(x) is the modified Bessel func- tion given by the relation eK cos θ = m=−∞ Im(K)e In this work, we investigate the properties of the model in Eq. 3 via Monte Carlo simulations using a recently proposed worm algorithm [13]. In order to reduce the systematic errors in discretizing the imaginary time axis, we need to take ǫ tU ≪ 1. We take Uǫ = 0.5 - 2 for t ≪ U and set the energy unit U = 1. The superfluid stiffness in a finite system is given by[14] ρL = β −1L2−d〈W 2x 〉, (5) 0.02 0.03 0.04 0.05 L=128 0 0.5 1 L exp(-b (t ) µ=0.9 =0.0409 b=1.85 0 10 20 30 40 50 60 0.005 0.015 L=128 0 0.1 0.2 0.3 0.4 L exp(-b (β ) µ=0.9 t=0.034 t=0.034 b=3.35 =28.8 FIG. 2: (Color online) Finite-size scaling behaviors of the su- perfluid stiffness as a function of (a) hopping strength and (b) temperature. For both cases, data collapsing onto a sin- gle curve works fine in terms of the scaling parameter L/ξ as shown in insets, consistent with the nature of the KT transi- tion and the universal jump at the critical point. where Wx = L r and 〈...〉 denotes the averages over the probabilites determined by the partition func- tion of Eq. (4), and d is the spatial dimensionality. Sim- ilarly the compressibility is κ = βL−d[〈N2〉 − 〈N〉2], (6) with N = L−1τ r . The energy expectation is given 〈H〉 = L−1τ 〈 〉 , (7) and the specific heat is CV = L −d(∂〈H〉/∂T ). We consider the case for µ = 0.9 so that n0 = 1 and n̄ = 0.4. Figure 2 shows the finite-size scaling behav- ior of the superfluid stiffness as a function of (a) t and (b) β. Finite-size scaling properties of the transition can be obtained by plotting the curves in terms of a scal- ing variable L/ξ, where ξ is the correlation length. Here we assume an essential singularity[15] ξ ∼ exp(bδ−1/2), where δ = t − tc (or β − βc) is a tuning parameter and b is a non-universal scaling factor. In terms of this scal- ing variable, we obtain high-quality data collapsing onto a single curve for different sizes, consistent with the na- ture of the KT transition. The scaling behavior also sup- ports the scenario of the universal jump of the superfluid stiffness[12], (π/2)βcρ∞ = 1, at the critical point in the thermodynamic limit. By extrapolating the single curves to the critical point, we find that (π/2)βcρ∞ ≈ (a)1.01 and (b)1.06. These numbers are, however, sensitive to fitting parameters b and tc(βc). Figure 3a shows the behavior of the compressibility. The finite-size scaling ansatz of the compressibility is written in the form κ = Lz−dX̃κ(L(t− tc)1/ν , β/Lz), (8) where X̃κ is a dimensionless scaling function and z is the dynamical critical exponent. For the generic superfluid- insulator transition(GSIT), z = 2 is expected[11]. The crossing behavior of the compressibility curves for differ- ent sizes at the critical point t0c = 0.023±0.001, therefore, represents the scaling properties near the QCP, where t0c is the critical hopping strength at zero temperature. For different values of µ, we have similar results with t0c just shifted. We find that the compressibility diverges at the transi- tion. In the superfluid side, κ ∼ 1/(t− t0c). This strongly supports that the longe-range density fluctuations drive the transition. In the insulating side, the compressibility has an activated form e−∆gap/T with a finite energy gap ∆gap. This dependence is shown in Fig. 3b for different t, from which we can calculate ∆gap as shown in the in- set. For small t, we need a large number of Monte Carlo steps to obtain equilibrium and have bigger error bars in determination of ∆gap. The gap vanishes around t = t as expected. Thus we have a so-called ’V-shaped’ phase diagram (Fig.1). In the insulating regime, the Mott insulator exists at T = 0, which turns into an activated gapped fluid with a finite energy gap at low temperature. It gradually disappears in a high-temperature normal fluid. This crossover line can be specified by the condition ∆gap/T ≈ 1. The phase coherence in a superfluid at T = 0 is destroyed by quantum fluctuations to form a QCP or by thermal fluctuations at T > 0 to define clas- sical phase transitions. The phase boundary in Fig. 1 is obtained by tuning t for given T (black circles) as well as by tuning β for a given t (red squares). Note that the phase boundary follows a scaling relation Tc ∝ |t− t0c |zν , which implies that β determines the correlation length in the temporal direction, where ν is the correlation length 0 0.01 0.02 0.03 0.04 0.05 L=12 Lτ=18 L=16 Lτ=32 L=20 Lτ=50 L=24 Lτ=72 L=28 Lτ=98 µ=0.9 0 50 100 150 200 1e-05 0.0001 0.001 0 0.01 0.02 µ=0.9 t=.005 t=.010 t=.015 t=.020 t=.021 t=.022 t=.023(b) FIG. 3: (Color online) (a) Compressibility of the boson Hub- bard model shows behavior of the GSIT with z = 2.0, diverg- ing at the transition. (b) In the insulating regime, we have thermally activated behaviors, κ ∼ e−∆gap/T , from which ∆gap can be evaluated. Inset: ∆gap as a function of t, van- ishing at the QCP. critical exponent. The boundary in Fig. 1 is consistent with the expectation zν = 1[11] for the GSIT. It is interesting to check the predicted scaling relation [4, 11] Tc ∝ ρ0x in this model. Figure 4 shows that the zero-temperature superfluid stiffness ρ0, denoted by dotted line, which obtained via extrapolation of values at T > 0, follows ρ0 ∝ |t− t0c |, implying that x = 1.0. It is consistent with the hyperscaling argument[11] suggesting x = z/(d+ z − 2). We expect that this quantum criticality disappears as temperature rises, which means quantum fluctuations possibly leave some tracks in bulk properties at low tem- peratures. Figure 5 shows the specific heat, CV , and the energy expectation values, 〈H〉, as a function of T for dif- ferent t. Sharp rises of CV in the conducting regime or round up-rises in the insulating regime are followed by indents, regions indicated by N, which apparently rep- resent anomalous behavior due to quantum fluctuations and disappear at high temperatures for T & 0.25. This feature strongly suggests a crossover in normal fluid from quantum mechanical to classical regime. Similarly the curves of 〈H〉 show bumps, indicated by H, only in the range where quantum critical fluctuations are expected to have effects. In summary, we have investigated the phase transi- tions at finite temperature in a two-dimensional quan- tum rotor model in which intrinsic zero-point fluctua- tions are present. Finite-size scaling of the superfluid stiffness shows an essential singularity of the KT phase transition and the universal jump at the critical point. The compressibility diverges at the transition. In the insulating regime, the compressibility shows a thermally activated behavior, κ ∼ e−∆gap/T , from which we can successfully evaluate the gap. This indicates that the in- sulating behavior at low temperature gradually crosses over to the behavior of normal fluid as temperature in- creases. The transition temperature Tc shows a scaling behavior Tc ∝ |t − t0c |, showing that finite T limits the length of quantum fluctuations in the temporal direction, and a hyperscaling relation Tc ∝ ρ0. The behavior of the specific heat and the energy suggests that, as tempera- ture rises, quantum critical regime near a QCP crosses over to classical regime. MCC would like to thank Gerardo Ortiz for helpful discussions and the hospitality of Department of Physics, Indiana University, where parts of this work were carried out. This work was supported by Korea Research Fund grant No. R05-2004-000-11004-0. 0.02 0.025 0.03 0.035 0.04 0.045 0.05 β=200 β=400 µ=0.9 FIG. 4: (Color online) Superfluid stiffness for different β. As β increases, the size dependence becomes smaller. This allows us to extrapolate the curves to obtain zero-temperature su- perfluid stiffness, ρ0, in the thermodynamic limit as denoted by dotted line. It shows that ρ0 ∝ |t− t c | with t c ≈ 0.22. [1] S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 1999). [2] S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Sahar, Rev. Mod. Phys. 69, 315 (1997). [3] P. Coleman and A. J. Schofield, Nature (London) 433, 226 (2005). [4] A. Kopp and S. Chakravarty, Nature Phys. 1, 53 (2005). [5] S. Sachdev, Phys. Rev. B 55, 142 (1997). [6] S. Kim and M. Y. Choi, Phys. Rev. B 41, 111 (1990). [7] J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973). [8] H.-Q. Ding and M. S. Makivić, Phys. Rev. B, 42, R6827 (1990). [9] K. Harada and N. Kawashima, J. Phys. Soc. Jpn. 67, 2768 (1998); A. W. Sandvik and C. J. Hamer, 60 6588 (1999). [10] G. Schmid, S. Todo, M. Troyer, and A. Dorneich, Phys. Rev. Lett., 88, 167208 (2002). [11] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys. Rev. B 40, 546 (1989). [12] D. R. Nelson and J. M. Kosterlitz, Phys. Rev. Lett., 39, 1201 (1977). [13] F. Alet and E. S. Sørensen, Phys. Rev. E 67, 015701(R) (2003); Phys. Rev. E 68, 026702 (2003). [14] M. Wallin, E. S. Sørensen, S. M. Girvin, and A. P. Young, Phys. Rev. B 49, 12115 (1994). [15] J. M. Kosterlitz, J. Phys. C 7, 1046 (1974). 0 0.1 0.2 0.3 0.4 0.5 t = 0.005 t = 0.020 t = 0.040 t = 0.050 t = 0.070 t = 0.100 -0.85 t = .005 t = .010 t = .020 t = .025 t = .030 t = .040 0 0.1 0.2 0.3 0.4 t = .040 t = .050 t = .060 t = .070 t = .080 t = .090 t = .010 FIG. 5: (Color online) Specific heat, CV , as a function of T for different t. Sharp rises in the conducting regime, signa- ture of the superfluid transition, or round up-rise of CV in the insulating regime are followed by indents which disappear in high temperature region, T & 0.25. Insets: The curves of the energy expectation values, 〈H〉, have bumps at low tempera- tures possibly due to the effects of quantum fluctuations.

0704.0397

Conditional generation of path-entangled optical NOON states

APS/123-QED Conditional generation of path-entangled optical NOON states Anne E. B. Nielsen and Klaus Mølmer Lundbeck Foundation Theoretical Center for Quantum System Research, Department of Physics and Astronomy, University of Aarhus, DK-8000 Århus C, Denmark (Dated: November 4, 2018) We propose a measurement protocol to generate path-entangled NOON states conditionally from two pulsed type II optical parametric oscillators. We calculate the fidelity of the produced states and the success probability of the protocol. The trigger detectors are assumed to have finite dead time, and for short pulse trigger fields they are modeled as on/off detectors with finite efficiency. Continuous-wave operation of the parametric oscillators is also considered. PACS numbers: 03.65.Wj, 03.67.-a, 42.50.Dv I. INTRODUCTION Nonclassical states of light have many applications, and a number of different protocols exist for the genera- tion of various classes of states. The two-mode maximally entangled N -photon states |NOON〉 = |N, 0〉+ eiφ|0, N〉 , (1) the so-called NOON states, are particularly interesting because a single-photon phase shift of χ induced in one of the two components changes the relative phase of the two terms by Nχ. This special property of NOON states may be utilized to enhance spatial resolution in (quantum) mi- croscopy and lithography [1], and in interferometry it has been shown that a certain measurement strategy, using NOON states, leads to a phase estimation error scaling as L−1/4N T if the phase to be estimated is known to lie within an interval from −π/L to π/L, where NT is the total number of photons used in the measurements [2]. This is better than the classical shot noise limit ofN and NOON states are thus useful to perform accurate measurements and may be a valuable field resource in sensors. NOON states are also a source of entanglement with applications in quantum information protocols and in fundamental studies such as tests of Bells inequality It is thus of great interest to be able to produce NOON states, and various NOON state generation schemes have been suggested theoretically [4, 5, 6, 7, 8, 9] and studied in experiments [7, 8, 10, 11]. The N = 1 and N = 2 NOON states may be generated by combining either a single photon and a vacuum state or two single-photon states on a 50 : 50 beam splitter, but this simple approach is not directly extendable to N > 2, and we shall thus mainly be concerned with generation of N = 3 NOON states in the present paper, even though the suggested protocol is, in principle, applicable for all N . Mitchell, Lundeen, and Steinberg have generated NOON states with N = 3 from a pair of down converted photons and a local oscillator photon using certain polarization transforming components and post-selection [10]. In this experiment, however, the successful generation of the NOON state is witnessed by a destructive detection of the state. In the present paper we propose and analyze in detail a nondestructive generation protocol, which con- ditions the successful generation of the N -photon NOON state on the registration of N photo detection events in other field modes, and which uses as resource only linear optics and the output from two optical parametric oscil- lators (OPOs). The protocol does not rely on efficient photo detection. The analysis is carried out in terms of Wigner function formalism, and effects of finite detector efficiency and finite detector dead time are considered. Conditional generation of nonclassical states occupying a single mode has been investigated both experimentally and theoretically [12, 13, 14, 15, 16, 17, 18, 19, 20]. With the correlated output from a single nondegenerate OPO it is, for instance, possible to generate n-photon Fock states of light in the signal beam conditioned on n photo detections in the idler (trigger) beam [16, 19, 20], and in principle the entanglement of a highly squeezed two- mode field from an OPO makes it possible to prepare any state in the signal beam that can be either measured as an eigenstate of a suitable observable of the idler beam or produced as the final state of a generalized measurement. The basic idea of the protocol proposed in the present paper is to mix the output from two OPOs and employ the entanglement to prepare a two-mode state in two of the output beams by detection of the desired output state in the remaining beams. In Sec. II we explain the NOON state generation pro- tocol in detail. In Sec. III we analyze the performance of the protocol quantitatively for pulsed OPO sources. We provide the fidelity of the generated states and the success probability. In Sec. IV we consider production of NOON states from continuous-wave OPO sources, and Sec. V concludes the paper. II. EXPERIMENTAL SETUP FOR NOON STATE GENERATION The experimental setup is illustrated in Fig. 1. Two pulses of two-mode squeezed states are generated by two identical OPOs via type II parametric down conversion. The field mode operators of the modes generated by the http://arxiv.org/abs/0704.0397v2 first OPO are denoted â+ and â−, respectively, while the field mode operators of the modes generated by the sec- ond OPO are denoted b̂+ and b̂−, respectively. For def- initeness, we assume that the plus modes are vertically polarized and that the minus modes are horizontally po- larized. The modes are separated spatially by the first two polarizing beam splitters, and the third polarizing beam splitter combines the â− and b̂+ modes, which are subsequently subjected to the NOON state measurement proposed in [21] and illustrated for N = 3 in Fig. 1. The idea behind this measurement is to apply the highly non- linear operator ÂN = â − − (b̂+eiθ)N to the state. The result is only nonzero if either the â− mode or the b̂+ mode contains at least N photons. On the other hand, if the squeezing is sufficiently small, it is unlikely to have more than a total of N photons in the two trigger modes, and by conditioning on the successful application of ÂN , we select the pulses of the system where N photon pairs are generated in one OPO and zero photon pairs in the other. It is equally probable that the photons originate from the first OPO or from the second OPO, and, as we shall see in detail below, the result is that a NOON state is generated conditionally in the output modes â+ and As stated in [21], ÂN can be rewritten as a simple product of single photon annihilation operators âN− − â− − ei2πn/N , (2) and it is thus possible to implement ÂN by means of beam splitters and photo detectors. We first consider odd values of N . Beam splitters are used to divide the input into N distinct spatial modes labeled by n = 1, . . . , N . The beam splitter reflectivities are chosen to obtain the same expectation value of the intensity in each of the modes. The vertically polarized modes are then phase shifted by the factor ei2πn/N+iπ relative to the horizon- tally polarized modes, i.e., b̂+ → −b̂+ei2πn/N , and finally polarizing beam splitters with principal planes oriented at 45◦ relative to the horizontal polarization transform â− and −b̂+ei2πn/N into (â−− b̂+ei2πn/N )/ 2 (the trans- mitted mode) and (â− + b̂+e i2πn/N )/ 2 (the reflected mode) [22]. The annihilation of a photon in each of the modes transmitted by the beam splitters witnesses the overall application of the operator ÂN . If one ob- serves both reflected and transmitted modes simultane- ously, one conditions on detection events in all the trans- mitted modes and no detection events in all the reflected modes. If detection events are instead observed in all the reflected modes and in none of the transmitted modes, an operator of the form (2) is also obtained, but θ is ef- fectively transformed into θ + π due to the phase shift at the polarizing beam splitter, and the value of φ of the generated NOON states is changed by Nπ (see below). The success probability is thus increased by a factor of two if both outcomes are accepted. FIG. 1: Experimental setup for NOON state generation. OPO, optical parametric oscillator; PBS, polarizing beam splitter; PS, phase shifter; and APD, avalanche photo diode. The part of the setup enclosed in the dashed box performs the NOON state measurement, and here it is shown for N = 3. Note that the polarizing beam splitters inside the box are ori- ented at 45◦. The numbers denote beam splitter reflectivities of 1/3 and 1/2, and the three phase shifters transform b̂+ into −b̂+e 2πin/3, where n = 1, 2, 3, respectively. See text for details. For even values of N a similar measurement scheme is applicable, but it is sufficient to divide the field into N/2 spatial modes initially, and in this case the NOON state generation is conditioned on detection events in both transmitted and reflected modes (see [21]). III. PERFORMANCE OF THE PROTOCOL After this presentation of the basic idea and the phys- ical setup we now consider the actual outcome of the detection process. For short pulse OPO output the dead time of the photo detectors may typically be longer than the pulse duration, and we shall thus assume that it is impossible to obtain more than a single detection event per detector per pulse, i.e., if the detector efficiency is unity, the detectors are only able to distinguish between vacuum and states different from the vacuum state. Such detectors are denoted on/off detectors, and they are dis- cussed in detail in Ref. [23]. The finite dead time of the detectors is not severe to the measurement procedure de- scribed in [21] because the on/off detector model and the conventional photo detector model, represented by the annihilation operator, lead to identical signal states if the total number of photons in the idler modes is guaranteed to be less than or equal to the number of conditioning detection events, i.e., N . We analyze the performance of the setup using Gaus- sian Wigner function formalism [15, 19, 20], which is applicable because the squeezed states generated by the OPOs and the vacuum states coupled into the system via the beam splitters are all Gaussian. In general, the Wigner function of an n-mode Gaussian state with zero mean field amplitude takes the form WV (x1, p1, . . . , xn, pn) = det(V ) TV −1y, (3) where y ≡ (x1, p1, . . . , xn, pn)T and V is the 2n × 2n covariance matrix. If ĉi denotes the field mode annihi- lation operator of mode i, the elements of V are given in terms of the real and imaginary parts of the expecta- tion values 〈ĉ†i ĉj〉 and 〈ĉiĉj〉. Note that for a multi-mode Gaussian state we are free to include only the modes of interest in (3) because the partial trace operation over un- observed modes is equivalent to integration over the cor- responding quadrature variables. A unit efficiency ‘on’ detection in mode i projects mode i on the subspace of Hilbert space that is orthogonal to the vacuum state, i.e., the Wigner function is multiplied by (1− 2πW0(xi, pi)), whereW0(x, p) = exp(−x2−p2)/π is the Wigner function of the vacuum state, the variables xi and pi are integrated out, and the state is renormalized. Since the Gaussian nature of a state is preserved under linear transforma- tions, and since a detector with single-photon efficiency η is equivalent to a beam splitter with transmission η fol- lowed by a unit efficiency detector [23], effects of non-unit detector efficiency are easily included in the covariance matrix. To calculate 〈ĉ†i ĉj〉 and 〈ĉiĉj〉 explicitly we note that the state generated by the OPOs is [24] |ψi〉 = (1− r2) rn+m|n, n,m,m〉, (4) where r is the squeezing parameter and the modes are listed in the order: â+, â−, b̂+, b̂−. We assume that N is odd and consider the transmitted trigger modes (which we number from 1 to N), the â+ mode (mode N + 1), and the b̂− mode (mode N + 2). By expressing the field operators of the trigger modes (those observed by the unit efficiency detectors) in terms of â−, b̂+, and field operators representing vacuum states we find 〈ĉ†j ĉk〉 = 〈ψi| − − e−2πij/N b̂ (â− − e2πik/N b̂+eiθ)|ψi〉 1 + e2πi(k−j)/N , (5) where j ∈ {1, 2, . . . , N}, k ∈ {1, 2, . . . , N}, λ ≡ ηr2/(1− r2), and we allow of a constant phase shift θ of b̂+ relative to â−. Furthermore 〈ĉ†N+1ĉN+1〉 = 〈ψi|â +â+|ψi〉 = r2/(1− r2), (6) 〈ĉ†N+2ĉN+2〉 = 〈ψi|b̂ −b̂−|ψi〉 = r2/(1− r2), (7) 〈ĉk ĉN+1〉 = 〈ψi| (â− − e2πik/N+iθ b̂+)â+|ψi〉 1− r2 , (8) 〈ĉk ĉN+2〉 = 〈ψi| (â− − e2πik/N+iθ b̂+)b̂−|ψi〉 1− r2 e2πik/N+iθ , (9) 〈ĉj ĉk〉 = 〈ĉN+1ĉN+1〉 = 〈ĉN+2ĉN+2〉 = 〈ĉN+1ĉN+2〉 = 〈ĉ†N+1ĉN+2〉 = 〈ĉ k ĉN+1〉 = 〈ĉ k ĉN+2〉 = 0. (10) For even values ofN the factors η/(2N) are replaced by η/N . Note that loss in the signal beam may be taken into account by performing the transformations â+ →√ ηsâ+ and b̂− → ηsb̂− in the above expressions, where 1− ηs is the loss. The NOON state fidelity FN of the signal state con- ditioned on N photo detection events in the transmitted trigger modes is WNOON(xN+1, pN+1, xN+2, pN+2) (1− 2πW0(xi, pi)) WV (x1, p1, . . . , xN+2, pN+2) dxidpi , (11) whereWNOON is the Wigner function of the NOON state (1), and (1− 2πW0(xi, pi)) WV (x1, p1, . . . , xN+2, pN+2) dxidpi , (12) is the success probability, i.e., the probability to obtain the conditioning detection events and produce the NOON state in a given pulse of the OPO system. We expand the product (1− 2πW0(xi, pi)) = (−2πW0(xi, pi))di , where the sum is over all d ≡ (d1, d2, . . . , dN ) with di ∈ {0, 1}, and define the diagonal matrix Jd = diag(d1, d1, d2, d2, . . . , dN , dN ) and the n×n identity ma- trix In. Furthermore, we divide the covariance matrix into four parts Vtt Vts V Tts Vss , (14) where Vtt is the 2N×2N covariance matrix of the trigger modes, Vss is the 4 × 4 covariance matrix of the signal modes, while Vts contains the correlations between the trigger and the signal modes, and we define the vector ys = (xN+1, pN+1, xN+2, pN+2) T and the matrix Ud = Vss − V Tts Jd(JdVttJd + I2N )−1JdVts. (15) This allows us to write Eqs. (11) and (12) in the following compact forms [25] det(I2N + JdVtt) WNOON(ys)WUd(ys)dys, (16) det(I2N + JdVtt) . (17) Since WNOON is a product of a polynomial and a Gaus- sian the integral in Eq. (16) may be evaluated analytically and for N = 3 and η = 1 we find (1− r2)2(2− r2)2(3− 2r2)(6 − 5r2) 18(4− 3r2) , (18) where the optimal value φ = Nθ + π + 2πn, n ∈ Z, is assumed. Expressions for PN are given in table I for N = 1, 2, 3, and 4, and FN and PN are plotted for N = 3 in Figs. 2 and 3, respectively. We observe that high prob- abilities are only found in the parameter regime, where the fidelity is low. If, for instance, we want a NOON state fidelity of at least 0.9, we choose r = 0.14, and if η = 0.25, P3 is of order 10 −8. With a repetition rate of order 106 s−1 (see [16]) one state is produced every second minute on average. The production rate is very dependent on detector efficiency, and if η is increased to unity, the rate is increased by approximately a factor of For odd values of N we may observe both reflected and transmitted trigger modes and condition on detec- tion events in all the transmitted trigger modes and no detection events in all the reflected trigger modes, or, vice versa. In this case we also include the reflected trigger modes in the covariance matrix, which we now denote by V +. By a similar analysis as above we obtain the success probability P+N = 2 2N (−2) det(I4N + J , (19) N PN P (λ+1)2 (λ+1)2 λ3(λ+4) (λ+2)2(λ+3)(λ+6) 2λ3(3λ+4) (λ+1)2(λ+2)2(2λ+3)(5λ+6) λ4(λ2+6λ+6) (λ+1)2(λ+2)2(λ2+8λ+8) TABLE I: Success probabilities calculated from Eqs. (17) and (19). λ ≡ ηr2/(1− r2). 0 0.2 0.4 0.6 0.8 1 FIG. 2: NOON state fidelity F3 (solid lines) and F 3 (dashed lines) as a function of squeezing parameter r for η = 1 (upper lines), η = 0.25 (middle lines), and η → 0 (lower lines). Note that in the latter case F3 = F where J+d ≡ diag(d1, d1, . . . , dN , dN , 1, 1, . . . , 1, 1), while the NOON state fidelity F+N is given by Eq. (11) with V replaced by the matrix V − (V +R ) T (V +RR + I2N ) −1V +R , (20) where V +RR is the covariance matrix of the reflected trig- ger modes and V +R consists of the correlations between the reflected trigger modes and the signal and transmit- ted trigger modes. Explicit results for P+N are given in table I for N = 1 and 3. F+3 and P 3 are compared to F3 and P3 in Figs. 2 and 3, and it is observed that F and P+3 are both larger than F3 and P3 if r is not large (and η > 0). For r → 1, P+3 → 0 because in this limit it is very unlikely to obtain no detection events in all the reflected or in all the transmitted trigger modes. In the limit of very small detector efficiency a simple expression for the NOON state fidelity for the case of N trigger detectors is easily derived without using Wigner function formalism. In general, if the state of interest is expressed in the photon number basis, the mathemat- ical operation corresponding to an ‘on’ detection is to multiply each term by 1− (1 − η)n, where n is the number of photons in the mode observed by the non- unit efficiency detector, trace out the detected mode, and renormalize. If nη ≪ 1 for all contributing terms, 1− (1− η)n ≈ √nη ∝ n, and the on/off detector 0 0.2 0.4 0.6 0.8 1 FIG. 3: Success probability P3 (solid lines) and P 3 (dashed lines) as a function of squeezing for η = 1 (upper lines) and η = 0.25 (lower lines). The dotted lines represent the approx- imate expression (23). model becomes equivalent to the photo detector model. In this case the density operator of the output state is obtained as ρ = M 〈p|〈q|(âN− − (b̂+eiθ)N )|ψi〉 〈ψi|((â†−)N − (b̂ −iθ)N )|q〉|p〉 (1− r2)N+2 2N !r2N (r2)n+m (n−N)! |n,m〉〈n,m| −e−iNθ n!(m+N)! (n−N)!m! |n,m〉〈n−N,m+N | (r2)n+m (m−N)! |n,m〉〈n,m| −eiNθ (n+N)!m! n!(m−N)! |n,m〉〈n+N,m−N | where M is a normalization constant and the traces are over the â− and b̂+ modes. This leads to the NOON state fidelity N = 〈NOON|ρ|NOON〉 = (1− r 2)N+2, (22) where again φ = Nθ + π + 2πn, n ∈ Z, is assumed. It is interesting to compare this result with the fidelity (1 − r2)N+1 obtained for production of N -photon states from a single two-mode squeezed state by conditioning on N detection events in the idler beam and using detec- tors with very small efficiency. If a single-photon state is produced by this method and transformed into an N = 1 NOON state as explained in the Introduction, the NOON state fidelity is F1,s = (1−r2)2, and the success probabil- ity is P1,s = λ/(λ + 1). Choosing squeezing parameters such that F1,s = F1, we find that P 1 = (4/3)P1,s in the high fidelity limit. It is thus possible to achieve a higher success probability using the scheme with two OPOs, but the price to pay is a more technically involved setup, and NOON states with two different values of φ are pro- duced. For N = 2 the present protocol and combination of two single-photon states on a 50 : 50 beam splitter, each produced conditionally from a single OPO, lead to identical fidelities and success probabilities. Finally we note that the photo detector model underestimates FN for η > 0 because 1 − (1 − η)n = η i=0 (1 − η)i < nη for n = 2, 3, . . . while 1− (1− η)n = nη for n = 0, 1, i.e., the ‘wrong’ terms containing more than N photons are given a too large weight. This is also what we observe in Fig. 2. In the limit of small r and for odd values of N the success probability is given approximately by the simple expression 〈ψi|((â†−)N − (b̂ −iθ)N ) (âN− − (b̂+eiθ)N )|ψi〉 = (2N)N λN (N odd). (23) Again η/(2N) must be replaced by η/N to obtain PN for even values of N . The approximation to P3 is shown in Fig. 3. IV. NOON STATES FROM CONTINUOUS-WAVE OPO SOURCES Our protocol is not limited to pulsed fields, and for completeness we now briefly consider NOON state gen- eration from continuously driven OPOs. We assume N = 3. For continuous-wave fields each of the three detected trigger beams and the two signal beams are described by time dependent field operators ĉi(t). The trigger detections take place in particular modes local- ized around the three detection times tc1, tc2, and tc3, and we want to determine the NOON state fidelity of an output state occupying one temporal mode in each signal beam. Following the general multimode formal- ism in Refs. [15, 20], the five relevant modes are spec- ified by the mode functions fi(t), and the correspond- ing five single mode operators are ĉi = fi(t)ĉi(t)dt. In general, we are free to choose the two output mode functions at will, and in the present case it is natural to choose the mode function which gives rise to the largest three-photon state fidelity when three-photon states are generated from a single type II continuous-wave OPO. Since we are mainly interested in the parameter region where the squeezing is small and the NOON state fidelity is large, we use the optimal three-photon state mode function derived for very low beam intensity in [20], i.e., f4(t) = f5(t) = k=1 ck γ/2 exp(−γ|t− tck|/2), where 0 0.05 0.1 0.15 FIG. 4: NOON state fidelity as a function of ǫ/γ for states generated from a pair of continuous-wave OPO sources when conditioning on three simultaneous trigger detection events tc1 = tc2 = tc3. 0 0.5 1 1.5 2 FIG. 5: Fidelity of NOON states from continuous-wave OPO sources as a function of separation between trigger detection events (tc3 − tc1)γ for N = 3, tc3 − tc2 = tc2 − tc1, and ǫ/γ = 0.01. the coefficients ck are functions of the intervals between the detection times and γ is the leakage rate of the OPO output mirror. We furthermore assume that the trigger mode functions are nonzero only in an infinitesimal time interval centered at the detection time, which is valid if the trigger detections take place on a time scale much shorter than γ−1. Since we consider a low intensity con- tinuous beam, and since we formally assume that the trigger detectors only register the light field in infinitesi- mal time intervals around the detection times, the anni- hilation operator detector model is perfectly valid in this case and detector dead time is insignificant. We may now proceed as above and eliminate all the irrelevant modes from the analysis by writing down the Gaussian Wigner function of the five interesting modes. The only difference is that this time 〈ĉ†i ĉj〉 and 〈ĉiĉj〉 are expressed in terms of the two time correlation functions of the OPO output. Also, the operators applied to the Wigner function to take conditioning into account are different because the annihilation detector model is used. The reader is referred to Refs. [15, 20] for details. The resulting fidelity is shown as a function of ǫ/γ in Fig. 4, where ǫ is the nonlinear gain in the OPO, and as a function of the temporal distance between the condition- ing detection events in Fig. 5. As in the pulsed case the fidelity decreases when the degree of squeezing increases. The fidelity also decreases when the temporal distance between the conditioning detection events increases from zero, but it is permissible to have a small time interval between the trigger detection events. We note that the curves represent a lower limit to the theoretically achiev- able fidelity since a better fidelity may be obtained for another choice of output state mode functions. V. CONCLUSION In conclusion we have analyzed a method to gener- ate path entangled NOON states from the output from two optical parametric oscillators. The method relies on the joint detection of photons in a number of trig- ger beams, and we presented a theoretical analysis of the role of detector efficiency and dead time for the fidelity of the states obtained and the success probability of the protocol. Our specific NOON state protocol applies for general photon numbers of the states, but in practice it is not realistic to go beyond the case of N = 3, studied here. This is due to the reduction of the fidelity due to unwanted contributions from higher number states, when the OPO output power gets too high, combined with the severe reduction of the probability to obtain the number of conditioning detection events needed when the OPO output power is too low. The N = 3 NOON states, which can be produced at 90% fidelity at the rate of one state produced every 10− 100 seconds, seem to be at the limit of realistic experiments of the proposed kind. Finally, we also determined the NOON state fidelity for continuous- wave fields, where the best NOON state occupies a pair of suitably selected temporal mode functions, and where we find high fidelities as long as the trigger events occur within a short time window compared to the lifetime of the OPO cavity field. We presented this analysis for the production of op- tical NOON states, but we note that recent theoreti- cal proposals and experiments with four wave mixing of matter waves [26], engineered quadratic interactions among trapped ions [27], and entanglement between field and atomic degrees of freedom [28, 29] bring promise for similar conditional generation of atomic and interspecies atom-field NOON states. This work was supported by the European Integrated project SCALA. [1] A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000). [2] L. Pezzé and A. Smerzi, quant-ph/0508158. [3] M. D’Angelo, A. Zavatta, V. Parigi, and M. Bellini, Phys. Rev. A 74, 052114. [4] P. Kok, H. Lee, and J. P. Dowling, Phys. Rev. A 65, 052104 (2002). [5] J. Fiurasek, Phys. Rev. A 65, 053818 (2002). [6] H. F. Hofmann, Phys. Rev. A 70, 023812 (2004). [7] P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gaspa- roni, and A. Zeilinger, Nature (London) 429, 158 (2004). [8] H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, Phys. Rev. Lett. 94, 090502 (2005). [9] N. M. VanMeter, P. Lougovski, D. B. Uskov, K. Kieling, J. Eisert, and J. P. Dowling, quant-ph/0612154. [10] M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, Nature (London) 429, 161 (2004). [11] A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, Phys. Rev. Lett. 98, 030502 (2007). [12] M. Dakna, T. Anhut, T. Opatrný, L. Knöll, and D.-G. Welsch, Phys. Rev. A 55, 3184 (1997). [13] A. B. URen, C. Silberhorn, J. L. Ball, K. Banaszek, and I. A. Walmsley, Phys. Rev. A 72, 021802(R) (2005). [14] A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, Science 312, 83 (2006). [15] K. Mølmer, Phys. Rev. A 73, 063804 (2006). [16] A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, Phys. Rev. Lett. 96, 213601 (2006). [17] J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, Phys. Rev. Lett. 97, 083604 (2006). [18] K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, Opt. Express 15, 3568 (2007). [19] A. E. B. Nielsen and K. Mølmer, Phys. Rev. A 75, 023806 (2007). [20] A. E. B. Nielsen and K. Mølmer, Phys. Rev. A 75, 043801 (2007). [21] F. W. Sun, Z. Y. Ou, and G. C. Guo, Phys. Rev. A 73, 023808 (2006). [22] P. Hariharan, N. Brown, and B. C. Sanders, J. Mod. Opt. 40, 1573 (1993). [23] P. P. Rohde and T. C. Ralph, J. Mod. Opt. 53, 1589 (2006). [24] A. Ekert and P. Knight, Am. J. Phys. 63, 415 (1995). [25] J. Eisert and M. B. Plenio, Int. J. Quantum Inf. 1, 479 (2003). [26] G. K. Campbell, J. Mun, M. Boyd, E. W. Streed, W. Ket- terle, and D. E. Pritchard, Phys. Rev. Lett. 96, 020406 (2006) [27] D. Porras and J. I. Cirac, Phys. Rev. Lett. 93, 263602 (2004). [28] B. B. Blinov, D. L. Moehring, L. M. Duan, and C. Mon- roe, Nature, 428, 153 (2004). [29] J. Volz, M. Weber, D. Schlenk, W. Rosenfeld, J. Vrana, K. Saucke, C. Kurtsiefer, and H. Weinfurter, Phys. Rev. Lett. 96, 030404 (2006). http://arxiv.org/abs/quant-ph/0508158 http://arxiv.org/abs/quant-ph/0612154

0704.0398

Renewals for exponentially increasing lifetimes, with an application to digital search trees

Renewals for exponentially increasing lifetimes, with an application to digital search trees The Annals of Applied Probability 2007, Vol. 17, No. 2, 676–687 DOI: 10.1214/105051606000000862 c© Institute of Mathematical Statistics, 2007 RENEWALS FOR EXPONENTIALLY INCREASING LIFETIMES, WITH AN APPLICATION TO DIGITAL SEARCH TREES By Florian Dennert and Rudolf Grübel Universität Hannover We show that the number of renewals up to time t exhibits distri- butional fluctuations as t→∞ if the underlying lifetimes increase at an exponential rate in a distributional sense. This provides a proba- bilistic explanation for the asymptotics of insertion depth in random trees generated by a bit-comparison strategy from uniform input; we also obtain a representation for the resulting family of limit laws along subsequences. Our approach can also be used to obtain rates of convergence. 1. Introduction. Let (Yk)k∈N be a sequence of independent, nonnegative random variables and let (Sn)n∈N0 , S0 := 0, Sn := Yk for all n ∈N, be the associated sequence of partial sums. Regarding the Yk’s as successive lifetimes and Sn as the time of the nth renewal, we interpret Nt := sup{n ∈N0 :Sn ≤ t} as the number of renewals up to and including time t; (Nt)t≥0 is the renewal process. Standard renewal theory assumes that the Yk’s all have the same distribution, in which case Nt, appropriately rescaled, is asymptotically nor- mal as t→∞. For this result, and for renewal theory in general, we refer the reader to Section XI in [3]. In this note we consider exponentially increasing lifetimes. We show that in such a case the distribution of Nt does not converge and that asymp- totic distributional fluctuations appear (Section 2). Such fluctuations occur frequently in the analysis of algorithms. The renewal theoretic framework Received January 2006. AMS 2000 subject classifications. Primary 60K05; secondary 68Q25. Key words and phrases. Asymptotic distributional behavior, limiting periodicities, re- newal processes. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Probability, 2007, Vol. 17, No. 2, 676–687. This reprint differs from the original in pagination and typographic detail. http://arxiv.org/abs/0704.0398v1 http://www.imstat.org/aap/ http://dx.doi.org/10.1214/105051606000000862 http://www.imstat.org http://www.ams.org/msc/ http://www.imstat.org http://www.imstat.org/aap/ http://dx.doi.org/10.1214/105051606000000862 2 F. DENNERT AND R. GRÜBEL provides a probabilistic view of this phenomenon in connection with digital search trees (Section 3). We also indicate how our approach can be used to obtain rates of convergence (Section 4). 2. Renewals for increasing lifetimes. We assume that the lifetimes in- crease exponentially with rate α, where α> 1 is fixed throughout the sequel, in the sense that α−kYk →distr Y∞ and α−kEYk →EY∞(1) for some random variable Y∞ and as k→∞. Here “→distr” denotes conver- gence in distribution, so that the first part of (1) means that Ef(α−kYk) =Ef(Y∞) for all bounded continuous functions f :R→ R. Below we will use the fact that in order to prove Xn →distr X it is sufficient to show that Ef(Xn)→ Ef(X) holds for all bounded and uniformly continuous functions. For details and a general treatment of convergence in distribution we refer the reader to [1]. Of course, only the distribution µ = L(Y∞) of Y∞ matters, so we will occasionally write α−kYk →distr µ instead. Finally, throughout this note a condition involving moments is meant to imply that these moments are finite. An important role will be played by S∞ := α−kY∞,k, where (Y∞,k)k∈N0 is a sequence of independent and identically distributed random variables with L(Y∞,0) = L(Y∞), Y∞ as in (1). From EY∞ <∞ we obtain ES∞ = α(α − 1)−1EY∞ <∞ and therefore P (S∞ <∞) = 1; more- over, we then also have that −kY∞,k converges almost surely and hence in distribution to S∞ as n→∞. We will also assume that L(Y∞) has no atoms, that is, P (Y∞ = y) = 0 for all y ∈R+.(2) Finally, it is an elementary analytic fact that, for a sequence (xn)n∈N of real numbers with limit x ∈R, α−kxn−k = The following lemma can be regarded as a random version of (3). Lemma 1. If (1) and (2) are satisfied, then α−nSn →distr S∞ as n→∞, and P (S∞ = y) = 0 for all y ∈R. RENEWALS FOR INCREASING LIFETIMES 3 Proof. Suppose that (Uk)k∈N is a sequence of independent random variables on some probability space (Ω,A, P ), all uniformly distributed on the unit interval. Let Fk be the distribution function of Yk, F the distribution function of Y∞. We use a variant of the quantile construction: Ỹk := F k (Uk), Ỹ∞,k := F −1(Uk) for all k ∈N. We then have L(Ỹ1, . . . , Ỹn) = L(Y1, . . . , Yn) for all n ∈N, which implies L(α−nSn) =L(α−nS̃n) with S̃n := Using α−nS̃n = k=0 α −k(α−(n−k)Ỹn−k) we obtain α−nS̃n − α−kỸ∞,n−k α−kE|α−(n−k)Ỹn−k − Ỹ∞,n−k|.(4) With Y ′k := F k (U1) and Y ∞ := F −1(U1) we have E|α−kỸk − Ỹ∞,k|=E|α−kY ′k − Y ′∞|.(5) From (1) it follows that α−kY ′k →distr Y ′∞ and Eα−kY ′k → EY ′∞. Because of Y ′k, Y ∞ ≥ 0 Theorem 5.4 in [1] applies and gives the L1-convergence of α−kY ′k to Y ∞, that is, E|α−kY ′k − Y ′∞| → 0 as k →∞. Using this together with (3), (4) and (5) we obtain α−nS̃n − α−kỸ∞,n−k = 0.(6) Now let f :R→R be bounded and uniformly continuous. We have |Ef(α−nSn)−Ef(S∞)|= Ef(α−nS̃n)−Ef α−kỸ∞,n−k α−kỸ∞,k α−kỸ∞,k f(α−nS̃n)− f α−kỸ∞,n−k α−kỸ∞,k α−kỸ∞,k For the first integral on the right-hand side we use (6), for the second an elementary estimate shows that the difference between the arguments of f converges to 0 in probability. In both cases we now use uniform continuity 4 F. DENNERT AND R. GRÜBEL when the arguments of f are close to each other and boundedness otherwise. This leads to Ef(α−nSn) =Ef(S∞), which gives the convergence in distribution. The statement about the atoms of S∞ follows from (2) and the fact that S∞ is equal in distribution to Y∞ + α −1S∞ with Y∞ and S∞ independent. � The above proof is based on classical weak convergence arguments. An alternative proof can be obtained via the Wasserstein distance dW (µ, ν) = inf{E|X − Y | :L(X) = µ,L(Y ) = ν}, its known relation to weak convergence and convergence of the first moments, and the same variant of the quantile construction, which in this context is known as the comonotone coupling. We write ⌊x⌋ for the greatest integer less than or equal to x and {x} for the fractional part of x ∈R. Theorem 2. Suppose that (1) and (2) are satisfied and let Qη := L(⌊− logαS∞ + η⌋), 0≤ η ≤ 1.(7) If (tn)n∈N is a sequence of real numbers with tn →∞ and such that {logα tn}→ η for some η ∈ [0,1], then Ntn − ⌊logα tn⌋→distr Qη as n→∞. Proof. We use the abbreviations kn := ⌊logα tn⌋ and ηn := {logα tn}. In particular, logα tn = kn + ηn. Further, let Z∞ := − logαS∞. By a standard renewal theoretic argument, P (Nt = j) = P (Sj ≤ t)−P (Sj+1 ≤ t) for all t≥ 0, j ∈N0, hence P (Ntn − kn = j) = P (Skn+j ≤ tn)− P (Skn+j+1 ≤ tn) = P (− logα(α−kn−jSkn+j) + ηn ≥ j) −P (− logα(α−kn−j−1Skn+j+1) + ηn ≥ j + 1) → P (⌊Z∞ + η⌋= j) as n→∞, where in the last step Lemma 1 and three general facts about convergence in distribution were used: First, the continuous mapping theorem, which implies that − logα(α−mSm)→distr − logαS∞ as m→∞; secondly, the in- terplay with convergence in probability, see Theorem 4.1 in [1], which yields RENEWALS FOR INCREASING LIFETIMES 5 − logα(α−nSn)+ ηn →distr − logαS∞+ η as n→∞; finally, that L(S∞) and therefore also L(− logαS∞+η) assign probability 0 to single points and that this implies P (− logα(α−nSn) + ηn ≥ z) = P (− logαS∞ + η ≥ z) for all z ∈R. A structural consequence of the representation (7) is the →distr-continuity of η 7→Qη on the open unit interval; at η = 0 this function is right continuous, at η = 1 it is left continuous. The extreme members are translates of each other in the sense that Q0({j}) =Q1({j + 1}) for all j ∈ Z. The total variation distance dTV of probability measures is defined by dTV(µ, ν) := sup |µ(B)− ν(B)|, for µ, ν concentrated on Z this can be written as dTV(µ, ν) = |µ({j})− ν({j})|.(8) For a sequence of probability measures that are concentrated on a fixed countable set Scheffé’s lemma implies that weak convergence is equivalent to convergence in total variation distance, hence (7) can be rewritten as dTV(L(Ntn − ⌊logα tn⌋),Q{log tn}) = 0. Because of the continuity of [0,1] ∋ η 7→Qη this in turn leads to a statement that avoids the use of subsequences, dTV(L(Nt − ⌊logα t⌋),Q{log t}) = 0.(9) In Section 4 we will investigate the rate of convergence in (9) in a particular case. 3. An application to digital search trees. The nodes of a (rooted, di- rected) binary tree can be represented by finite strings of 0’s and 1’s if we interpret 0 as a move to the left and 1 as a move to the right. The length of the string is the depth (or level) of the node it represents, the root node corre- sponds to the empty string and has level 0. The sequence (Tn)n∈N associated with a sequence (xn)n∈N of numbers from the unit interval by the DST (dig- ital search tree) algorithm is obtained as follows: For T1, we put x1 into the root node. If x1, . . . , xn have been stored into Tn then the position of xn+1 is determined by traveling through the tree with the direction given by the bi- nary expansion of xn+1 until an empty node has been found. This algorithm and its properties are discussed in the standard texts of the area, for exam- ple, [8, 10, 11]. As an example we consider the first ten numbers given in [8], 6 F. DENNERT AND R. GRÜBEL Fig. 1. Binary tree. Appendix A, ( 2, log 2, log 3, log 10). Let xi be the fractional part of the ith entry, 1≤ i≤ 10; the relevant first four bits of the respective binary expansions are given by (0110,1011,0011,0010, 0100,0111, 0011,1011,0001,0100). This leads to the binary tree given in Figure 1. Consider now the sequence (Tn)n∈N0 of random trees that the DST algo- rithm associates with a sequence (Un)n∈N of independent random variables, where we assume that the Un’s are uniformly distributed on the unit inter- val and that T0 is the empty tree. Let Xn(θ) be the depth of the first free node of Tn along the path determined by a sequence θ ∈ {0,1}N. Such a θ defines a family of nested intervals of length 2−k, k = 1,2,3, . . . , and it is easy to see that (Xn(θ))n∈N0 is a Markov chain with X0(θ) ≡ 0 and tran- sition probabilities pk,k+1 = 1− pk,k = 2−k for all k ∈ N0. Conditioning on the value of Un+1 we see that the distribution of Xn(θ) is the same as the distribution of Zn+1, the insertion depth of Un+1. This quantity is known as “unsuccessful search” in the literature on the analysis of algorithms. [Of course, this distributional equality does not hold for the joint distributions: n 7→Xn(θ) is increasing, n 7→ Zn+1 is not.] For example, the next number in Knuth’s list is x11 = 1/ log 2, the binary expansion of the fractional part {x11} begins with 011100 and hence x11 would be inserted at level 4 as the right child of x6. The Markov chain (Xn(θ))n∈N0 is of the simple birth type and can there- fore be described by its respective holding times Y1, Y2, Y3, . . . in the states k = 0,1,2, . . . . These are independent, and Yk has a geometric distribution with parameter pk−1,k, that is, for all k ∈N, P (Yk = j) = (1− 2−k+1)j−12−k+1 for all j ∈N. Here we interpret the case k = 1 as Y1, the holding time in 0, being constant and equal to 1. As a result of its simple stochastic dynamics, (Xn(θ))n∈N0 is equal to the renewal process N associated with the sequence (Yk)k∈N, observed at discrete times, that is, (Xn(θ))n∈N0 = (Nn)n∈N0 . It is easy to see that for this sequence (Yk)k∈N of lifetimes conditions (1) and (2) are satisfied and that L(Y∞) = Exp(2), with Exp(λ) the exponential distribution RENEWALS FOR INCREASING LIFETIMES 7 with parameter λ (and mean 1/λ). Hence Theorem 2 can be applied: If the sequence (n(m))m∈N ⊂ N is such that n(m) →∞ and {log2 n(m)} → η as m→∞, then Xn(m)(θ)− ⌊log2 n(m)⌋→distr Qη.(10) Here Qη , 0≤ η ≤ 1, is the distribution of ⌊− log2 S∞+η⌋, S∞ := k=0 2 −kY∞,k and Y∞,k, k ∈N0, are independent and identically distributed with L(Y∞,1) = Exp(2). Alternatively, we can write S∞ := k=1 Ỹk with Ỹk, k ∈N, again in- dependent and L(Ỹk) = Exp(2k) for all k ∈N. The explicit representation of the family of limit distributions on the basis of the convolution product of the distributions Exp(2k), k ∈N, can be used to obtain a series expansion for the distribution functions associated with Qη , 0 ≤ η ≤ 1. For this, we start with a partial fraction expansion: For all n ∈N and all z ∈C with |ℜ(z)|< 2, (1− 2−kz)−1 = an,k(1− 2−kz)−1,(11) where an,k := j=1(1− 2j)−1 j=1 (1− 2−j)−1. Reading (11) as an equality relating characteristic functions we obtain Exp(21) ⋆Exp(22) ⋆ · · · ⋆Exp(2n) = an,kExp(2 k).(12) Note, however, that the right-hand side in (12) is not the usual mixture of probability distributions as the coefficients alternate in sign. With ak := b (1− 2j)−1, b := (1− 2−j)−1, letting n→∞ in (12) leads to L(S∞) = k=1 akExp(2 k), so that Qη((−∞, x]) = P (⌊− log2(S∞) + η⌋ ≤ x) = P (S∞ > 2 η−1−x)(13) ak exp(−2k+η−1−x) for all x ∈ Z. This representation of the limiting distribution functions as an alternating series has already been obtained by Louchard [9] in the context of digital search trees and by Flajolet [4] in the context of approximate counting; see also Section 6.4 in [10] and Section 6.3 in [8] for related results. These authors use a completely different approach, more analytic in flavor and relying on combinatorial identities due to Euler. 8 F. DENNERT AND R. GRÜBEL Our main point here, however, is not a rederivation of (13) but the rep- resentation of the family {Qη : 0≤ η < 1} in terms of one particular random variable, which is first shifted by η and then discretized. This representation can, for example, be used to obtain information about the tail behavior of the limit distributions. Janson [7] notes that (13) by itself would only give an exponential rate of decrease for the tail probabilities, he then provides an analytic argument that improves this to a superexponential rate by show- ing that the associated Fourier transform is an entire function. Using the representation S∞ = k=1 2 −kZk with Zk independent and L(Zk) = Exp(1) together with the fact that Exp(1) has a density bounded by 1, we get P (S∞ ≤ 2−j)≤ P (Z1 ≤ 2−j+1)P (Z2 ≤ 2−j+2) · · ·P (Zj−1 ≤ 2−1) ≤ 2−j+12−j+2 · · ·2−1 = 2−j(j−1)/2 for all j ∈ N. Because of Qη([k,∞)) ≤ P (S∞ ≤ 2−k+1) for all k ∈ N, k ≥ 2, this leads to Qη([x,∞)) = o(exp(−ρx2)) as x→∞, for all ρ < (log 2)/2. The fact that a representation by discretization is possible in many situ- ations where fluctuations were first found by calculation seems to belong to the folklore of the subject, at least in simple instances such as the asymp- totic distributional behavior of the maximum of a sample from a geometric distribution. The geometric case together with some renewal theoretic tech- niques (for identically distributed lifetimes) was used in [5] to obtain results of the above type for von Neumann addition. In [2] a discretization represen- tation occurs on the level of stochastic processes, leading to a probabilistic approach to fluctuation phenomena in the context of multiplicities of the maximum in a random sample from a discrete distribution. In a recent pa- per, Janson [7] studies the effects of discretizing random variables and the resulting distributional fluctuations and gives a range of interesting exam- ples. Of course, the explanation for periodicities can be, and indeed often is, quite different and mechanisms other than discretization may be responsible; see, for example, [6] and the references given there. 4. Rates of convergence. The renewal theoretic approach can also be used to obtain rates of convergence. We sketch one of the possibilities, for a particular choice of distances, and give details for the DST situation from Section 3. Let, for t > 0, k(t) := ⌊logα t⌋ and η(t) := {logα t}. The Kolmogorov–Smirnov distance of two probability measures µ and ν on the real line is defined by dKS(µ, ν) := sup |µ((−∞, x])− ν((−∞, x])|. RENEWALS FOR INCREASING LIFETIMES 9 If X and Y are real random variables, then we abbreviate dKS(L(X),L(Y )) to dKS(X,Y ); if F and G are the associated distribution functions, then dKS(X,Y ) = ‖F − G‖∞, where the supremum norm for general bounded functions f :R → R is given by ‖f‖∞ := supx∈R |f(x)|. The Kolmogorov– Smirnov distance is obviously invariant under strictly monotone transfor- mations. For example, dKS(αX + β,αY + β) = dKS(X,Y ) for all α,β ∈R, α 6= 0, and for X,Y > 0, dKS(X,Y ) = dKS(logX, logY ). With the notation as in the proof of Theorem 2, |P (Nt − k(t) = j)−P (⌊− logα(S∞) + η(t)⌋= j)| ≤ |P (− logα(α−k(t)−jSk(t)+j) + η(t)≥ j)−P (− logα(S∞) + η(t)≥ j)| + |P (− logα(α−k(t)−j−1Sk(t)+j+1) + η(t)≥ j +1) − P (− logα(S∞) + η(t)≥ j + 1)|. With the auxiliary quantities Zt := ⌊− logα(S∞) + η(t)⌋, φ(m) := dKS(α−mSm, S∞) and the above properties of the Kolmogorov–Smirnov distance this leads to |P (Nt − k(t) = j)−P (Zt = j)| ≤ φ(k(t) + j) + φ(k(t) + j +1).(14) It is often possible to obtain an upper bound for negative j, say j ≤−k(t)/2, directly. In such cases the above elementary renewal theoretic argument leads to a bound for the ‖ · ‖∞-distance between the probability mass functions of Nt − k(t) and Zt, for example; note that the latter variable has distribution Qη(t) where Qη , 0≤ η ≤ 1, is the set of limit distributions along subsequences that appears in Theorem 2. The above argument covers the step from (α−mSm)m∈N to (Nt)t≥0. How- ever, in an application the starting point will usually be the convergence of the scaled lifetimes in (1), which means that we also need an analogue for Lemma 1 that gives rates of convergence. We carry this out in the specific context of digital search trees. The fol- lowing general bounds will turn out to be useful: If X has density fX and if P (|Y | ≤ c) = 1, then dKS(X,X + Y )≤ c‖fX‖∞.(15) Indeed: For all z ∈R, P (X ≤ z− c)≤ P (X+Y ≤ z)≤ P (X ≤ z+ c), so that |P (X + Y ≤ z)−P (X ≤ z)| ≤max{P (X ≤ z + c)−P (X ≤ z), P (X ≤ z)−P (X ≤ z − c)}, 10 F. DENNERT AND R. GRÜBEL and, of course, P (X ∈ (a, b]) ≤ (b − a)‖fX‖∞. This bound can easily be generalized to dKS(X,X + Y )≤ c‖fX‖∞ +P (|Y |> c) for all c > 0,(16) where we still assume that X has density fX , but Y may be arbitrary. Note that X and Y need not be independent in (15) and (16). If they are independent then it is easy to show that dKS(X,X + Y )≤ ‖fX‖∞E|Y |.(17) In (17) boundedness of Y is not needed but the bound obviously makes sense only if Y has finite first moment. Finally, in connection with density bounds the interplay with convolution is of interest: We have ‖f ⋆ g‖∞ ≤ ‖f‖∞ for all probability densities f, g. For example, if a sum of independent random variables contains a summand with distribution Exp(λ), then the density of the sum is bounded by λ. Lemma 3. With (Yk)k∈N and S∞ as in Section 3, dKS(2 −nSn, S∞) =O(n2 Proof. Let (Zk)k∈N be a sequence of independent random variables, all exponentially distributed with parameter 1. Then S∞ is equal in distribution k=1 2 −kZk. We recall that the kth lifetime Yk has a geometric distribu- tion with parameter 2−k+1. On the basis of (Zk)k∈N we define a sequence (Ỹk)k∈N by Ỹk := ⌊αkZk⌋+1 for all k ∈N, with α1 := 0, αk := (− log(1− 2−k+1))−1 for k > 1. It is easy to check that (Ỹk)k∈N =distr (Yk)k∈N, 2 2k−1Zk =distr 2−kZk. Hence, with φ(n) denoting the dKS-distance of 2 −nSn and S∞, φ(n)≤ φ1(n) + φ2(n) + φ3(n) for all n ∈N, with φ1, φ2, φ3 defined by φ1(n) := dKS Ỹk,2 φ2(n) := dKS αkZk,2 2k−1Zk φ3(n) := dKS 2−kZk, 2−kZk RENEWALS FOR INCREASING LIFETIMES 11 For the random variables in φ1 we have Vn ≤ 2−n Ỹk ≤ Vn + n2−n with Vn := 2−n αkZk. It is easy to show that the densities of Vn, n ∈N, can be uniformly bounded for all n by some finite constant C1, hence (15) implies that φ1(n)≤C1n2−n for all n ∈N. The elementary bounds − 1≤− log(1− x) for 0< x≤ together with α1 = 0 imply supk∈N |αk − 2k−1|= 1, hence we have αkZk − 2−n 2k−1Zk ≤ 2−n The familiar combination of Markov’s inequality and moment generating functions gives Zk ≥ (1 + κ)n =O(2−n) if κ is chosen large enough, so that we can use (16) with c = c(n) = (1 + κ)n2−n to obtain that φ2(n)≤C2n2−n for all n ∈N, for some finite constant For φ3 finally we use (17): For the densities of the finite sums we again have a finite uniform bound for all n, and k=n+1 2−kZk k=n+1 2−kEZk = 2 so that φ3(n) ≤ C32−n for all n ∈ N with some C3 <∞. Putting these to- gether we arrive at φ(n)≤Cn2−n for all n ∈N with some finite constant C. � In the application under consideration we obtain a rate of convergence result with respect to the total variation distance, which is stronger than a result for the supremum norm distance of the corresponding probability mass functions that we mentioned in connection with (14). Theorem 4. With (Xn(θ))n∈N and Qη as in Section 3, dTV(L(Xn(θ)− ⌊log2 n⌋),Q{log2 n}) = o(n −γ) for all γ < 1. 12 F. DENNERT AND R. GRÜBEL Proof. We use the abbreviations k(n) := ⌊log2 n⌋ and η(n) := {log2 n}. Let γ < 1 be given and choose ε > 0 such that ε < 1− γ. Lemma 3 together with (14) gives j≥−εk(n) |P (Nn − k(n) = j)−Qη(n)({j})| ≤C j≥(1−ε)k(n) for all n ∈ N with some finite constant C. Our choice of ε implies that the upper bound has the desired rate o(n−γ). For the remaining part of the infinite sum in (8) we replace the absolute difference of the probabilities by their sum, which means that it is now enough to show that P (Nn ≤ (1− ε)k(n)) = o(n−γ),(18) P (− log2(S∞)≤−εk(n) + 1) = o(n−γ).(19) Here we have used that Qη is the distribution of ⌊− log2(S∞)+ η⌋. It is easy to show that the moment generating function for S∞ exists in a neighbor- hood of 0, hence P (S∞ > x) = o(e −κx) for all x > 0(20) with some κ > 0. Straightforward manipulations show that (20) implies (19); indeed, the probability converges faster to 0 than any negative power of n. Using once again the relation between the number of renewals and the partial sums of the lifetimes we further obtain, with m(n, ε) := ⌊(1− ε)k(n)⌋, P (Nn ≤ (1− ε)k(n)) ≤ P (Sm(n,ε) ≥ n) = P (2−m(n,ε)Sm(n,ε) ≥ n2−m(n,ε)) ≤ dKS(2−m(n,ε)Sm(n,ε), S∞) +P (S∞ ≥ n2−m(n,ε)). For the Kolmogorov–Smirnov distance we use Lemma 3, for the tail of S∞ the desired rate follows with (20). This gives (18) and hence completes the proof. � REFERENCES [1] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. MR0233396 [2] Bruss, F. Th. and Grübel, R. (2003). On the multiplicity of the maximum in a discrete random sample. Ann. Appl. Probab. 13 1252–1263. MR2023876 [3] Feller, W. (1971). An Introduction to Probability Theory and Its Applications II, 2nd ed. Wiley, New York. MR0270403 [4] Flajolet, Ph. (1985). Approximate counting: A detailed analysis. BIT 25 113–134. MR0785808 [5] Grübel, R. and Reimers, A. (2001). On the number of iterations required by von Neumann addition. Theor. Inform. Appl. 35 187–206. MR1862462 http://www.ams.org/mathscinet-getitem?mr=0233396 http://www.ams.org/mathscinet-getitem?mr=2023876 http://www.ams.org/mathscinet-getitem?mr=0270403 http://www.ams.org/mathscinet-getitem?mr=0785808 http://www.ams.org/mathscinet-getitem?mr=1862462 RENEWALS FOR INCREASING LIFETIMES 13 [6] Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl. 110 177–245. MR2040966 [7] Janson, S. (2006). Rounding of continuous random variables and oscillatory asymp- totics. Ann. Probab. 34 1807–1826. [8] Knuth, D. E. (1973). The Art of Computer Programming 3. Sorting and Searching. Addison–Wesley, Reading, MA. MR0445948 [9] Louchard, G. (1987). Exact and asymptotic distributions in digital binary search trees. Theor. Inform. Appl. 21 479–496. MR0928772 [10] Mahmoud, H. M. (1992). Evolution of Random Search Trees. Wiley, New York. MR1140708 [11] Sedgewick, R. and Flajolet, Ph. (1996). An Introduction to the Analysis of Al- gorithms. Addison–Wesley, Reading, MA. Institut für Mathematische Stochastik Universität Hannover Postfach 60 09 D-30060 Hannover Germany E-mail: [email protected] [email protected] http://www.ams.org/mathscinet-getitem?mr=2040966 http://www.ams.org/mathscinet-getitem?mr=0445948 http://www.ams.org/mathscinet-getitem?mr=0928772 http://www.ams.org/mathscinet-getitem?mr=1140708 mailto:[email protected] mailto:[email protected] Introduction Renewals for increasing lifetimes An application to digital search trees Rates of convergence References Author's addresses

0704.0399

Hawking radiation of linear dilaton black holes

LAPTH-1178/07 Hawking radiation of linear dilaton black holes G. Clémenta∗, J.C. Fabrisb†and G.T. Marquesa,b‡ aLaboratoire de Physique Théorique LAPTH (CNRS), B.P.110, F-74941 Annecy-le-Vieux cedex, France b Departamento de F́ısica, Universidade Federal do Esṕırito Santo, Vitória, 29060-900, Esṕırito Santo, Brazil April 3, 2007 Abstract We compute exactly the semi-classical radiation spectrum for a class of non-asymptotically flat charged dilaton black holes, the so- called linear dilaton black holes. In the high frequency regime, the temperature for these black holes generically agrees with the surface gravity result. In the special case where the black hole is massless, we show that, although the surface gravity remains finite, there is no radiation, in agreement with the fact that massless objects cannot radiate. e-mail: [email protected] e-mail: [email protected] e-mail:[email protected] http://arxiv.org/abs/0704.0399v1 Quantum field theory in curved spacetime predicts new phenomena such as particle emission by a black hole [1]. This is due to the fact that the vac- uum for a quantum field near the horizon is different from the observer’s vacuum at spatial infinity. A distant observer thus receives from a black hole a steady flux of particles exhibiting, in the high frequency regime, a black body spectrum with a temperature proportional to the surface grav- ity [2]. Although Hawking’s original derivation of this black hole evaporation dealt with realistic collapsing black holes, Unruh [3] showed that the same results are obtained when the collapse is replaced by appropriate boundary conditions on the horizon of an eternal black hole. In the semi-classical ap- proximation, the black hole radiation spectrum may be evaluated by com- puting the Bogoliubov coefficients relating the two vacua. An equivalent procedure is to compute the reflection and absorption coefficients of a wave by the black hole. Usually, the wave equation cannot be solved exactly, and one must resort to match solutions in an overlap region between the near-horizon and asymptotic regions [4, 5]. In the special case of the (2+1)- dimensional BTZ black hole [6], an exact solution of the wave equation is available, which allows for an exact computation of the radiation spectrum, leading to the Hawking temperature [7, 8, 9]. In this Letter, we discuss another case of black holes also allowing for an exact semi-classical computation of their radiation spectrum, that of lin- ear dilaton black hole solutions to Einstein-Maxwell dilaton (EMD) theory in four dimensions. Linear dilaton black holes are a special case of the more general class of non-asymptotically flat black hole solutions to EMD [10, 11], which we first briefly present. We discuss the evaporation of these non-asymptotically flat black holes and show that they either collapse to a naked singularity in a finite time, or evaporate in an infinite time. We then specialize to linear dilaton black holes, and outline the analytical computa- tion of their radiation spectrum. For massive black holes, this computation leads, in the high frequency regime, to the same temperature which is ob- tained from the surface gravity. However in the case of massless extreme black holes, we find that, although the surface gravity remains finite, there is no radiation, in agreement with the fact that a massless object cannot radiate. EMD is defined by the following action R− 2∂µφ∂µφ− e−2αφFµνFµν , (1) where Fµν is the electromagnetic field, and φ is the dilatonic field, with cou- pling constant α. This theory admits static spherically symmetric solutions representing black holes. Among these black hole solutions there are asymp- totically flat ones [12, 13] as well as non-asymptotically flat configurations [10, 11]. In the present work, we are interested in the non-asymptotically flat black hole solutions ds2 = rγ(r − b) dt2 − rγ(r − b) dr2 + r(r − b)dΩ2 , (2) 1 + γ dr ∧ dt , e2αφ = ν2 . (3) 1− α2 1 + α2 . (4) The constants b and r0 are related to the mass and to the electric charge of the black hole through M = (1− γ)b/4 , Q = 1 + γ . (5) The solutions (2),(3) interpolate between the Schwarzschild solution for γ = −1 (α2 → ∞) and the Bertotti-Robinson solution for γ = +1 (α2 = 0). For b > 0 the horizon at r = b hides the singularity at r = 0, while in the extreme black hole case b = 0 the horizon coincides with the singularity. This is a curious case, with vanishing mass but a finite electric charge. For −1 < γ < 0 (α2 > 1) the central singularity is timelike and clearly naked [11]. On the other hand, for 0 ≤ γ < 1 (0 < α2 ≤ 1), the central singularity is null and marginally trapped [14], so that signals coming from the centre never reach external observers. Thus in this case, extreme black holes can be still considered as black holes indeed. The statistical Hawking temperature of the black holes (2), computed as usual by dividing the surface gravity by 2π is given by . (6) It is finite for all γ if b 6= 0. For b = 0 and −1 < γ < 0 (naked singularity). the temperature is infinite, while for b = 0 and 0 < γ < 1 (extreme black hole), the temperature vanishes. The case b = γ = 0 is intriguing. Although this an extreme black hole, the situation is different from that of asymptotically flat extreme black holes. The near-horizon Euclidean extreme Reissner-Nordström geometry is cylindrical, rather than conical, so that its statistical temperature is ar- bitrary, contrary to the zero value derived from surface gravity [15]. In the present case the two-dimensional Euclidean continuation of the metric (2) with γ = 0 clearly has a conical singularity at r = b for all values of b, including b = 0, leading for this particular extreme black hole to the finite temperature TH = 1/4πr0, in agreement with the value (6). However this result is questionable. A black hole with pointlike horizon and zero mass clearly cannot radiate, so one should rather expect its temperature to be zero. We will return to this question presently. As black holes (2) radiate, they loose mass according to Stefan’s law = −σAhT 4H , (7) where σ is Stefan’s constant, and Ah = 4πr 1−γ is the horizon area. Assuming that only electrically neutral quanta are radiated, (7) implies that the horizon area decreases according to (4π)3(1− γ) −3(1+γ) 1+3γ , (8) which is solved by b(t) = r0 t− t0 )−1/3γ (γ 6= 0) , b(t) = r0 exp t− t0 (γ = 0) , (9) where c = 3σ/16π3, and t0 is an integration constant. The outcome de- pends on the sign of γ. For γ < 0, the Hawking temperature increases with decreasing mass and the black hole collapses to a naked singularity (or evap- orates away altogether in the Schwarzschild case γ = −1) in a finite time according to b ∼ (t0 − t)1/3|γ|. On the other hand, for γ ≥ 0, the Hawking temperature decreases (or is constant for γ = 0) with decreasing mass, and the black hole evaporates in an infinite time, reaching the extreme black hole state b = 0 only asymptotically. We now proceed to a more precise evaluation of the temperature of non- asymptotically flat black holes from the study of wave scattering in these spacetimes. The wave equation ∇2φ = 0 (10) does not generically allow for an exact solution in the spacetimes (2). How- ever, it can be solved analytically [16] in the case of linear dilaton black holes with γ = 0 and b 6= 0, with the metric ds2 = r − b dt2 − r − b dr2 + r(r − b)dΩ2 , (11) Considering the harmonic eigenmodes φ(x) = ψ(r, t)Ylm(θ, ϕ) , ψ(r, t) = R(r)e −iωt , (12) we obtain the following radial equation: r(r − b)∂rR r − b − l(l + 1) R = 0 (13) (ω̄2 ≡ ω2r20). Putting , R = yiω̄f , (14) reduces (13) to the equation y(1−y)∂2yf+ 1+2iω̄−2(1+ iω̄)y ω̄2− iω̄− λ̄2−1/4 f = 0 , (15) λ̄2 = ω̄2 − (l + 1/2)2 . (16) This is a hypergeometric equation y(1− y)∂2yf + c− (a+ b+ 1)y ∂yf − abf = 0 , (17) + i(ω̄ + λ̄) , b = + i(ω̄ − λ̄) , c = 1 + 2iω̄ . (18) It follows that the general solution of equation (13) is R = C1 r − b + i(ω̄ + λ̄), + i(ω̄ − λ̄), 1 + 2iω̄; b− r r − b )−iω̄ − i(ω̄ + λ̄), 1 − i(ω̄ − λ̄), 1 − 2iω̄; b− r .(19) Putting r − b = ex/r0 , (20) the partial wave near the horizon (x→ −∞) is thus ψ ≃ C1eiω(x−t) +C2e−iω(x+t) . (21) To obtain the behavior of the partial wave near spatial infinity, we must expand the solutions of (15) in hypergeometric functions of argument 1/y. The relevant transformation is F (a, b, c; y) = Γ(c)Γ(b− a) Γ(b)Γ(c− a) (−y)−aF (a, a+ 1− c, a+ 1− b; 1/y) Γ(c)Γ(a − b) Γ(a)Γ(c − b) (−y)−bF (b, b+ 1− c, b+ 1− a; 1/y) . (22) This leads to the asymptotic behavior )−1/2( i(λx−ωt) +B2e −i(λx+ωt) (λ = λ̄/r0), where the amplitudes of the asymptotic outgoing and ingoing waves B1 and B2 are related to the amplitudes of the near-horizon outgoing and ingoing waves C1 and C2 by B1 = Γ(2iλ̄) Γ(1 + 2iω̄) Γ(1/2 + i(ω̄ + λ̄))2 Γ(1− 2iω̄) Γ(1/2 − i(ω̄ − λ̄))2 B2 = Γ(−2iλ̄) Γ(1 + 2iω̄) Γ(1/2 + i(ω̄ − λ̄))2 Γ(1− 2iω̄) Γ(1/2− i(ω̄ + λ̄))2 . (24) Hawking radiation can be considered as the inverse process of scattering by the black hole, with the asymptotic boundary condition B1 = 0 (the outgoing mode is absent). The coefficient for reflection by the black hole is then given by |C1|2 |C2|2 |Γ(1/2 + i(ω̄ + λ̄))2|2 |Γ(1/2 + i(ω̄ − λ̄))2|2 cosh2 π(ω̄ − λ̄) cosh2 π(ω̄ + λ̄) . (25) The resulting radiation spectrum is = (eω/TH − 1)−1 . (26) For high frequencies, λ̄ ≃ ω̄ = ω/r0, and we recover from (25) the Hawking temperature as computed from the surface gravity, . (27) The above computation fails in the linear dilaton vacuum case b = 0. The question of assigning a temperature to such massless black holes might be evacuated by arguing that they cannot be formed, either through cen- tral collapse of matter, or (as we have seen above) through evaporation of massive black holes. Nevertheless, as a matter of principle one should con- sider the possibility of primordial massless black holes. From the general temperature law (6) these should have a finite temperature. On the other hand, being massless they cannot radiate energy away, so their temperature should vanish. The question can be settled by solving the massless Klein-Gordon equa- tion in the metric (11) with b = 0, ds2 = dt2 − r0 dr2 − r0rdΩ2 . (28) This metric can be rewritten as ds2 = Σ2 dτ2 − dx2 − dΩ2 , (29) x = ln(r/r0) , τ = t/r0 , Σ = r0e x/2 , (30) showing that the linear dilaton vacuum metric is conformal to the product M2 × S2 of a two-dimensional Minkowski spacetime with the two-sphere. Performing also the redefinition φ = Σ−1ψ , (31) the Klein-Gordon equation (10) is reduced to ∇2φ = Σ−3 ∂ττ − ∂xx −∇2Ω + ψ = 0 , (32) where ∇2Ω is the Laplacian operator on the two-sphere. For a given spherical harmonic with orbital quantum number l, the re- duced Klein-Gordon equation is thus ∇22ψl + (l + 1/2)2ψl = 0 , (33) with ∇22 the Dalembertian operator on M2. Also, for a given spherical harmonic the four-dimensional Klein-Gordon norm reduces to theM2 norm: ‖φ‖2 = 1 |g|g0µφ∗ ∂µ φ = dxψ∗l ∂τ ψl . (34) Thus, the problem of wave propagation in the linear dilaton vacuum reduces to the propagation of eigenmodes of a free Klein-Gordon field in two dimen- sions, with effective mass µ = l+1/2. Clearly there is no reflection, so that the linear dilaton vacuum does not radiate and hence its Hawking temper- ature vanishes, contrary to the naive surface gravity value (6). A similar reasoning holds in 2+1 dimensions for the BTZ vacuum [6] (M = L = 0), which is conformal to M2 × S1. We have shown that a complete analytical computation of the radia- tion spectrum is possible for linear dilaton black hole solutions of EMD. For massive black holes, this leads in the high frequency regime to a Planckian distribution with a temperature independent of the black hole mass, in ac- cordance with the surface gravity value. On the other hand, we find that extreme, massless black holes do not radiate, thereby solving the paradox presented by apparently hot (if the surface gravity temperature is taken seriously) yet massless black holes. Acknowledgements: J.C.F. thanks the LAPTH for the warm hospitality during the elaboration of this work. He also thanks CNPq (Brazil) for partial support. J.C.F. and G.T.M. thank the French-Brazilian scientific cooperation CAPES/COFECUB for partial financial support. References [1] N.D. Birrell and P.C.W. Davies, Quantum fields in curved space, Cambridge University Press, Cambridge (1982). [2] S.W. Hawking, Commun. Math. Phys. 43 (1975) 199. [3] W.G. Unruh, Phys. Rev. D14 (1976) 870. [4] D. Page, Phys. Rev. D13 (1976) 198. [5] W.G. Unruh, Phys. Rev. D14 (1976) 3251. [6] M. Bañados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849. [7] K. Ghoroku and A.L. Larsen, Phys. Lett. B328 (1994) 28. [8] M. Natsuume, N. Sakai and M. Sato, Mod. Phys. Lett. A11 (1996) 1467. [9] D. Birmingham, I. Sachs and S. Sen, Phys. Lett. B413 (1997) 281. [10] K.C.K. Chan, J.H. Horne and R.B. Mann, Nucl. Phys. B447 (1995) [11] G. Clément and C. Leygnac, Phys. Rev. D70 (2004) 084018. [12] G.W. Gibbons and K. Maeda, Nucl. Phys. B298 (1988) 741. [13] D. Garfinkle, G.T. Horowitz and A. Strominger, Phys. Rev. D43 (1991) 3140. [14] S.A. Hayward, Class. Quantum Grav. 17 (2000) 4021. [15] S.W. Hawking, G.T. Horowitz and S.F. Ross, Phys. Rev. D51 (1995) 4302. [16] G. Clément, D. Gal’tsov and C. Leygnac, Phys. Rev. D67 (2003) 024012.

0704.0400

The S-Matrix of AdS/CFT and Yangian Symmetry

arXiv:0704.0400v4 [nlin.SI] 27 Mar 2008 arxiv:0704.0400 AEI-2007-019 The S-Matrix of AdS/CFT and Yangian Symmetry Niklas Beisert Max-Planck-Institut für Gravitationsphysik Albert-Einstein-Institut Am Mühlenberg 1, 14476 Potsdam, Germany [email protected] Abstract We review the algebraic construction of the S-matrix of AdS/CFT. We also present its symmetry algebra which turns out to be a Yangian of the centrally extended su(2|2) superalgebra. 1 Introduction and Overview Bethe’s ansatz [1] for solving a one-dimensional integrable model was and remains a powerful tool in contemporary theoretical physics: 75 years ago it solved one of the first models of quantum mechanics, the Heisenberg spin chain [2]; today it provides exact solutions for the spectra of certain gauge and string theories and thus helps us understand their duality [3] better. Since the discovery of integrable structures in planar N = 4 supersymmetric gauge theory [4] and in planar IIB string theory on AdS5×S5 [5] the tools for computing and comparing the spectra of both models have evolved rapidly. We now have complete asymptotic Bethe equations [6, 7] which interpolate smoothly between the perturbative regimes in gauge and string theory and which agree with all available data. In this note we will focus on the S-matrix [8] in the excitation picture above a ferro- magnetic ground state. We start by reviewing the algebraic construction of the S-matrix in Sec. 2. In Sec. 3 we subsequently show that this S-matrix has indeed a larger symmetry algebra: a Yangian. http://arxiv.org/abs/0704.0400v4 2 The Universal Enveloping Algebra U(su(2|2)⋉R2) In this section the results on the S-matrix of AdS/CFT shall be reviewed from an al- gebraic point of view. The applicable symmetry is a central extension h of the Lie superalgebra su(2|2) which we consider first. We continue by presenting the Hopf al- gebra structure of its universal enveloping algebra and its fundamental representation. Finally, we comment on the S-matrix and its dressing phase factor. Lie Superalgebra. The symmetry in the excitation picture for light cone string theory on AdS5×S5 and for single-trace local operators in N = 4 supersymmetric gauge theory is given by two copies of the Lie superalgebra [9, 10] h := su(2|2)⋉ R2 = psu(2|2)⋉ R3. (2.1) It is a central extension of the standard Lie superalgebras su(2|2) or psu(2|2), see [11]. It is generated by the su(2)× su(2) generators Rab, Lαβ, the supercharges Qαb, Saβ and the central charges C, P, K. The Lie brackets of the su(2) generators take the standard [Rab,R d] = δ d − δadRcb, [Lαβ ,Lγδ] = δ Lαδ − δαδ Lγβ, [Rab,Q d] = −δadQγb + 12δ d, [L d] = +δ d − 12δ [Rab,S δ] = +δ δ − 12δ δ, [L δ] = −δαδ Scβ + 12δ δ. (2.2) The Lie brackets of two supercharges yield {Qαb,Scδ} = δcbLαδ + δαδ Rcb + δcbδαδ C, {Qαb,Qγd} = εαγεbdP, {Saβ,Scδ} = εacεβδK. (2.3) The remaining Lie brackets vanish. Where appropriate, we shall use the collective symbol JA for the generators. The Lie brackets then take the standard form [JA, JB] = fABC J C . (2.4) For simplicity of notation, we shall pretend that all generators are bosonic; the general- isation to fermionic generators by insertion of suitable signs and graded commutators is straightforward. Hopf Algebra. Next we consider the universal enveloping algebra U(h) of h. The construction of the product is standard, and one identifies the Lie brackets (2.4) with graded commutators. For the coproduct one can introduce a non-trivial braiding [12,13] ∆JA = JA ⊗ 1 + U [A] ⊗ JA (2.5) ∆Rab = R b ⊗ 1 + 1⊗Rab, ∆Lαβ = L β ⊗ 1 + 1⊗ Lαβ, ∆Qαb = Q b ⊗ 1 + U+1 ⊗Qαb, ∆Saβ = S β ⊗ 1 + U−1 ⊗Saβ, ∆C = C⊗ 1 + 1⊗ C, ∆P = P⊗ 1 + U+2 ⊗P, ∆K = K⊗ 1 + U−2 ⊗ K, ∆U = U ⊗ U . Table 1: The coproduct of the braided universal enveloping algebra U(h). with some abelian1 generator U (a priori unrelated to the algebra) and the grading [R] = [L] = [C] = 0, [Q] = +1, [S] = −1, [P] = +2, [K] = −2. (2.6) The coproduct is spelt out in Tab. 1 for the individual generators. The above grading is derived from the Cartan charge of the sl(2) automorphism [11] of the algebra h and therefore the coproduct is compatible with the algebra relations. We should define the remaining structures of the Hopf algebra: the antipode S and the counit ε [12,13]. The antipode is an anti-homomorphism which acts on the generators S(1) = 1, S(U) = U−1, S(JA) = −U−[A]JA. (2.7) The counit acts non-trivially only on 1 and U ε(1) = ε(U) = 1, ε(JA) = 0. (2.8) Cocommutativity. This coproduct is in general not quasi-cocommutative as can eas- ily be seen by considering the central charges P, K in Tab. 1. To make it quasi-cocommu- tative we have to satisfy the constraints [12] 1− U+2 1− U+2 ⊗P, K⊗ 1− U−2 1− U−2 ⊗ K. (2.9) They are solved by identifying the central charges P, K with the braiding factor U as follows [13] P = gα 1− U+2 , K = gα−1 1− U−2 . (2.10) This leads to the following quadratic constraint PK− gα−1P− gαK = 0. (2.11) It was furthermore shown in [14] that the coproduct is quasi-triangular, at least at the level of central charges, see also [15]. 1Curiously, we can include the supersymmetric grading (−1)F in the generator U to manually impose the correct statistics. This is helpful for an implementation within a computer algebra system. In this case U would anticommute with fermionic generators. Fundamental Representation. The algebra h has a four-dimensional representation [10] which we will call fundamental. The corresponding multiplet has two bosonic states |φa〉 and two fermionic states |ψα〉. The action of the two sets of su(2) generators has to be canonical Rab|φc〉 = δcb |φa〉 − 12δ b |φc〉, Lαβ|ψγ〉 = δγβ |ψα〉 − 12δ β |ψγ〉. (2.12) The supersymmetry generators must also act in a manifestly su(2)×su(2) covariant way Qαa|φb〉 = a δba|ψα〉, Qαa|ψβ〉 = b εαβεab|φb〉, Saα|φb〉 = c εabεαβ|ψβ〉, Saα|ψβ〉 = d δβα|φa〉. (2.13) We can write the four parameters a, b, c, d using the parameters x±, γ and the constants g, α as g γ, b = , c = , d = . (2.14) The parameters x± (together with γ) label the representation and they must obey the constraint − x− − 1 . (2.15) The three central charges C,P,K and U are represented by the values C, P,K and U which read 1 + 1/x+x− 1− 1/x+x− , P = gα , K = , U = . (2.16) They furthermore obey the quadratic relation C2−PK = 1 . Note that the corresponding quadratic combination of central charges C2−PK is singled out by being invariant under the sl(2) external automorphism. Fundamental S-Matrix. In [10,14] an S-matrix acting on the tensor product of two fundamental representations was derived. It was constructed by imposing invariance under the algebra h [∆JA,S] = 0. (2.17) We will not reproduce the result here, it is given in [14]. Note that we have to fix the parameters ξ = U = x+/x− in order to make the action of the generators compatible with the coproduct (2.5).2 2This identification removes all braiding factors from the S-matrix in [14] which will thus satisfy the standard Yang-Baxter (matrix) equation, see also [10, 16, 17]. This S-matrix has several interesting properties. Firstly, it is not of difference form; it cannot be written as a function of the difference of some spectral parameters. Sec- ondly, the S-matrix could be determined uniquely up to one overall function merely by imposing a Lie-type symmetry (2.17) [10]. This unusual fact is related to an unusual feature of representation theory of the algebra h: The tensor product of two fundamental representations is irreducible in almost all cases [14]. Intriguingly this S-matrix is equivalent to Shastry’s R-matrix [18] of the one-dimen- sional Hubbard model [19]. Furthermore the Bethe equations [10] contain two copies of the Lieb-Wu equations for the Hubbard model [20]. These observations of [14] estab- lish a link between an important model of condensed matter physics and string theory (complementary to the one in [21]). Finally, let us note that one can derive (asymptotic) Bethe equations from the S- matrix and thus confirm the conjecture in [6]. So far this step has been performed in two different ways: by means of the nested coordinate [10] and the algebraic [17] Bethe ansatz. Phase Factor. The remaining overall phase factor of the S-matrix clearly cannot be determined by demanding invariance under h. The phase factor was computed to some approximation from gauge theory [22] and from string theory [23]. The problem of an algebraically undetermined phase factor is in fact generic. Usually one imposes a further crossing symmetry relation to obtain a constraint on it. Indeed the known string phase factor is consistent with crossing symmetry [24] as was shown in [25]. By substituting a suitable ansatz [26] for the phase factor into the crossing symmetry relation a conjecture for the all-orders phase factor at strong coupling was made in [27]. A corresponding all-orders expansion at weak coupling was presented in [7]. The latter conjecture was obtained by a sort of analytic continuation in the perturbative order of the series. Let us illustrate this principle by means of a very simple example: Consider the rational function f(x) = 1/(1−x). It has the following expansions at x = 0 and at x = ∞ n, f(x) −n (2.18) with an = 1 and bn = −1. When we consider an and bn as analytic functions of the index, we can make the observation (“reciprocity”) an = −b−n. (2.19) Of course there are various ways in which the two functions +1 and −1 could be related, but the choice (2.19) appears to work for a surprisingly large class of functions!3 It was proved in [30] that it does apply for the conjectured expansion of the phase factor. Very useful integral expressions for the phase have recently appeared in [31]. The analytic expression of the dressing phase can formally be obtained from the psu(2, 2|4) Bethe 3Among other physical examples, we have identified circular Maldacena-Wilson loops [28] and non- critical string theory [29] where this reciprocity can be applied. Furthermore, summation by the Euler- MacLaurin formula (also known as zeta-function regularisation) is consistent with it. I thank Curt Callan, Marcos Mariño and Tristan McLoughlin for discussions of this principle. equations [32] (see however appendix D in [33]) in analogy to the covariant approach of [34, 21, 35]. While this proposal may seem to be encouraging in general, it is at the same time strange from the Hopf algebra point of view to use an S-matrix which does not obey the crossing relation [32]. This calls for further investigations. Several tests of the phase have recently appeared, they are based on four-loop unitary scattering methods [36], numerical evaluation [37, 38], analytic methods [37, 30, 39] and on taking a certain highly non-trivial limit [40]. 3 The Yangian Y(su(2|2) ⋉ R2) In the section we investigate Yangian symmetry [41,42] for the above S-matrix. We will start with a very brief review of Yangian symmetry for generic S-matrices (see [43] for more extensive reviews), and then we apply the framework to the S-matrix discussed above. Yangians and S-Matrices. Typically the symmetries of rational S-matrices are of Yangian type. The Yangian Y(g) of a Lie algebra g is a deformation of the universal enveloping algebra of half the affine extension of g. More plainly, it is generated by the g-generators JA and the Yangian generators ĴA. Their commutators take the generic form [JA, JB] = fABC J [JA, ĴB] = fABC Ĵ C , (3.1) and they should obey the Jacobi and Serre relations J[A, [JB, JC]] J[A, [JB, ĴC]] Ĵ[A, [ĴB, JC]] 2fAGD f F fGHKJ {DJEJF}. (3.2) The symbol fABC = gADgBEf C represents the structure constants f C with two indices lowered by means of the inverse of the Cartan-Killing forms gAD and gBE . The brackets { } and [ ] at the level of indices imply total symmetrisation and anti-symmetrisation, respectively. Finally, ~ is a scale parameter whose value plays no physical role. The first two relations lead to a constraint on the structure constants fABC . The third relation a deformation of the Serre relation for affine extensions of Lie algebras. The Yangian is a Hopf algebra and the coproduct takes the standard form ∆JA = JA ⊗ 1 + 1⊗ JA, ∆ĴA = ĴA ⊗ 1 + 1⊗ ĴA + 1 ~fABCJ B ⊗ JC . (3.3) where fABC = gBDf C . The antipode S is defined by S(JA) = −JA, S(ĴA) = −ĴA + 1 ~fABCf D, (3.4) 4For g = su(2) it has to be replaced by a quartic relation. and the counit ε takes the standard form ε(1) = 1, ε(JA) = ε(ĴA) = 0. (3.5) For the study of integrable systems, the evaluation representations of the Yangian are of special interest. For these the action of the Yangian generators ĴA is proportional to the Lie generators ĴA|u〉 = ~uJA|u〉. (3.6) Here |u〉 is some state of the evaluation module with spectral parameter u. This Yangian representation is finite-dimensional if the g-representation is. One merely has to ensure that the Serre relation (3.2) is satisfied. This is indeed not the case for all representations of all Lie algebras. The power of the Yangian symmetry lies in the fact that tensor products of evaluation representations are typically irreducible (except for special values of their spectral parameters). This allows for simple proofs (e.g. for the Yang-Baxter relation) by representation theory arguments. Let us finally consider the connection to the S-matrix. The S-matrix is a permutation operator; it acts by interchanging two modules of the algebra S : V1 ⊗ V2 → V2 ⊗ V1. (3.7) In particular, for the tensor product of two evaluation modules one has S|u1, u2〉 ∼ |u2, u1〉. (3.8) Invariance of the S-matrix under the Yangian means [∆JA,S] = [∆ĴA,S] = 0 (3.9) for all generators JA, ĴA. The existence of such an S-matrix is equivalent to quasi- cocommutativity of Y(g). Note that only the difference of spectral parameters appears in the invariance condition: We can write the action of the coproduct of Yangian generators on the evaluation module |u1, u2〉 as ∆ĴA ≃ (u1 − u2)JA ⊗ 1 + u2∆JA + ~fABCJB ⊗ JC . (3.10) Here the first equation in (3.9) ensures that the term proportional to u2 drops out from the second equation. Therefore the S-matrix typically depends on the difference u1 − u2 of spectral parameters only. Yangians in AdS/CFT. Yangian symmetries for planar AdS/CFT have been inves- tigated in [44], both for classical string theory and for gauge theory at leading order, see also [45] Yangian symmetry also persists to higher perturbative orders in both mod- els [22, 46] and it is likely that it also exists at finite coupling. This Yangian can be understood as a symmetry of the Hamiltonian on an infinite world sheet or as an expan- sion of the full monodromy matrix. The Lie symmetry in this picture is psu(2, 2|4) and the Yangian would be Y(psu(2, 2|4)). Here we consider a different picture of well-separated excitations on a ferromagnetic ground state and of their scattering matrix. In this picture the Lie symmetry reduces to two copies of h and the corresponding Yangian would be Y(h). Our Yangian should arise as a subalgebra of the full Yangian Y(psu(2, 2|4)) when acting on asymptotic excitation states. Hopf Algebra. Let us now consider Y(h). We have already studied the universal enveloping algebra U(h). All we still need to do is to introduce one generator ĴA for each JA obeying the relations (3.1,3.2), and define its coproduct, antipode as well as counit. In (2.5) we have defined a braided coproduct for the universal enveloping algebra. For consistency with the Serre relations, we also have to apply an analogous braiding to the standard Yangian coproduct ∆ĴA = ĴA ⊗ 1 + U [A] ⊗ ĴA + ~fABCJBU [C] ⊗ JC . (3.11) Note that lowering an index requires to use the inverse Cartan-Killing form of the algebra. In the case of h the Cartan-Killing form is degenerate and we need to extend the algebra by the sl(2) outer automorphism, see [14]. Effectively, lowering an index leads to an interchange of the automorphism generators with the central charges. We refrain from spelling out the Cartan-Killing form or the modified structure constants. Instead we present the complete set of coproducts of Yangian generators in Tab. 2, where we also fix the value of ~. For the sake of completeness we state the antipode5 and the counit S(ĴA) = −U−[A]ĴA, ε(ĴA) = 0. (3.12) Cocommutativity. An important question is if this coproduct can be quasi-cocom- mutative.6 A first step is to consider the central generators Ĉ, P̂, K̂. For that purpose it is favourable to choose suitable combinations Ĉ′ = Ĉ+ 1 gα−1P− 1 P̂′ = P̂+ C P− 2gα K̂′ = K̂− C K− 2gα−1 , (3.13) for whom the coproduct almost trivialises ∆Ĉ′ = Ĉ′ ⊗ 1 + 1⊗ Ĉ′, ∆P̂′ = P̂′ ⊗ 1 + U+2 ⊗ P̂′, ∆K̂′ = K̂′ ⊗ 1 + U−2 ⊗ K̂′. (3.14) The combination Ĉ′ is already cocommutative, and in order to make the generators P̂′, K̂′ cocommutative we have to set as above in (2.9,2.10) P̂′ = iguPP, K̂ ′ = iguKK (3.15) with two universal constants uP and uK. With this choice, Ĉ, P̂, K̂ also become cocom- mutative because they differ from Ĉ′, P̂′, K̂′ only by central elements. 5Note that fA = 0 here, so there is no contribution from the Lie generators. 6The braiding factors in (3.11) turn out to be very important for the Yangian. It can easily be seen that without them the coproduct cannot be quasi-cocommutative. This is in contradistinction to the universal enveloping algebra where the braided as well as the unbraided coproduct are quasi- cocommutative. ∆R̂ab = R̂ b ⊗ 1 + 1⊗ R̂ab Rac ⊗Rcb − 12R b ⊗Rac SaγU+1 ⊗Qγb − 12Q bU−1 ⊗Saγ δab S γU+1 ⊗Qγd + 14δ dU−1 ⊗Sdγ , ∆L̂αβ = L̂ β ⊗ 1 + 1⊗ L̂αβ Lαγ ⊗ Lγβ + 12L β ⊗ Lαγ QαcU−1 ⊗Scβ + 12S βU+1 ⊗Qαc δαβ Q cU−1 ⊗Scδ − 14δ δU+1 ⊗Qδc, ∆Q̂αb = Q̂ b ⊗ 1 + U+1 ⊗ Q̂αb LαγU+1 ⊗Qγb + 12Q b ⊗ Lαγ RcbU+1 ⊗Qαc + 12Q c ⊗Rcb CU+1 ⊗Qαb + 12Q b ⊗ C εαγεbdPU−1 ⊗Sdγ − 12ε αγεbdS γU+2 ⊗P, ∆Ŝaβ = Ŝ β ⊗ 1 + U−1 ⊗ Ŝaβ RacU−1 ⊗Scβ − 12S β ⊗Rac LγβU−1 ⊗Saγ − 12S γ ⊗ Lγβ CU−1 ⊗Saβ − 12S β ⊗ C εacεβδKU+1 ⊗Qδc + 12ε acεβδQ cU−2 ⊗ K, ∆Ĉ = Ĉ⊗ 1 + 1⊗ Ĉ PU−2 ⊗ K− 1 KU+2 ⊗P, ∆P̂ = P̂⊗ 1 + U+2 ⊗ P̂ − CU+2 ⊗P+P⊗ C, ∆K̂ = K̂⊗ 1 + U−2 ⊗ K̂ + CU−2 ⊗ K− K⊗ C. Table 2: The coproduct of the Yangian generators in Y(h). Fundamental Evaluation Representation. For the fundamental evaluation repre- sentation we make the ansatz7 ĴA|X 〉 = ig(u+ u0)JA|X 〉. (3.16) By comparison with (3.13,3.15) we can infer that u has to be related to the parameters of the fundamental representation by u = x+ + = x− + (x+ + x−)(1 + 1/x+x−) . (3.17) Furthermore uP and uK in (3.15) have to both coincide with the universal constant u0 = uP = uK. As an aside we state the eigenvalue of the quadratic combination CĈ − 1 PK̂ − 1 KP̂ = 1 ig(u+ u0). (3.18) Fundamental S-Matrix. Using the coproducts in Tab. 2 we have confirmed that the S-matrix is also invariant under all of the Yangian generators [∆ĴA,S] = 0. (3.19) We have used a computer algebra system to evaluate the action of the Yangian gener- ators and the S-matrix.9 To show invariance requires heavy use of the identity (2.15). Superficially it is very surprising to find all these additional symmetries of the S-matrix. The deeper reason however should be that the coproduct is quasi-cocommutative. We have thus proved quasi-cocommutativity when acting on fundamental representations. It is interesting to see that the S-matrix is based on standard evaluation represen- tations of the Yangian. Nevertheless, it is not a function of the difference of spectral parameters. This unusual property traces back to the link between the spectral param- eter u and the h-representation parameters x± in (3.17). The latter is again related to the braiding in the coproduct (3.11). As our S-matrix is equivalent [14] to Shastry’s R-matrix, our Yangian is presumably an extension of the su(2)×su(2) Yangian symmetry of the Hubbard model found in [47]. 4 Conclusions and Outlook In this note we have reviewed the construction of the S-matrix with centrally extended su(2|2) symmetry that appears in the context of the planar AdS/CFT correspondence and the one-dimensional Hubbard model. We have furthermore shown that the S-matrix has an additional Yangian symmetry whose Hopf algebra structure we have presented. This Yangian is not quite a standard Yangian, but its coproduct needs to be braided in order to be quasi-cocommutative. This fact is intimately related to the existence of a 7We believe, but we have not verified that this is compatible with the Serre relations (3.2). 8It is conceivable that a further consistency requirement fixes the value of u0, presumably to zero. 9We have also confirmed the invariance of the singlet state found in [10]. triplet of central charges with non-trivial coproduct and leads to the wealth of unusual features of the S-matrix. In connection to the Yangian there are many points left to be clarified. Most im- portantly the representation theory needs to be understood. Which representations of h lift to evaluation representations of Y(h)? At what values of the spectral parameters do their tensor products become reducible? This information could be used to prove that the coproduct is quasi-cocommutative. Also the Yang-Baxter equation for the S-matrix should follow straightforwardly. It might also give some further understanding of bound states [48]. Then it would be highly desirable to construct a universal R-matrix for this Yangian and show that it is quasi-triangular. This would put large parts of the integrable structure for arbitrary representations of this algebra on solid ground much like for the case of generic simple Lie algebras. Some further interesting questions include: Is this Yangian the unique quasi-co- commutative Hopf algebra based on h? Does the double Yangian [42] exist and what is its structure? Can the sl(2) automorphism of the algebra be included at the Yangian level such that the coproduct is quasi-cocommutative? What would the representations be in this case? Acknowledgements. I am grateful to C. Callan, D. Erkal, A. Kleinschmidt, P. Ko- roteev, N. MacKay, M. Mariño, T. McLoughlin, J. Plefka, F. Spill and B. Zwiebel for interesting discussions. References [1] H. Bethe, Z. Phys. 71, 205 (1931). [2] W. Heisenberg, Z. Phys. 49, 619 (1928). [3] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998), hep-th/9711200. [4] J. A. Minahan and K. Zarembo, JHEP 0303, 013 (2003), hep-th/0212208. N. Beisert, C. Kristjansen and M. Staudacher, Nucl. Phys. B664, 131 (2003), hep-th/0303060. N. Beisert and M. Staudacher, Nucl. Phys. B670, 439 (2003), hep-th/0307042. [5] G. Mandal, N. V. Suryanarayana and S. R. Wadia, Phys. Lett. B543, 81 (2002), hep-th/0206103. I. Bena, J. Polchinski and R. Roiban, Phys. Rev. D69, 046002 (2004), hep-th/0305116. [6] N. Beisert and M. Staudacher, Nucl. Phys. B727, 1 (2005), hep-th/0504190. [7] N. Beisert, B. Eden and M. Staudacher, J. Stat. Mech. 07, P01021 (2007), hep-th/0610251. [8] M. Staudacher, JHEP 0505, 054 (2005), hep-th/0412188. [9] N. Beisert, Phys. Rept. 405, 1 (2004), hep-th/0407277. [10] N. Beisert, hep-th/0511082. [11] W. Nahm, Nucl. Phys. B135, 149 (1978). [12] C. Gómez and R. Hernández, JHEP 0611, 021 (2006), hep-th/0608029. [13] J. Plefka, F. Spill and A. Torrielli, Phys. Rev. D74, 066008 (2006), hep-th/0608038. [14] N. Beisert, J. Stat. Mech. 07, P01017 (2007), nlin.SI/0610017. [15] N. Beisert and P. Koroteev, arxiv:0802.0777. [16] G. Arutyunov, S. Frolov and M. Zamaklar, JHEP 0704, 002 (2007), hep-th/0612229. [17] M. J. Martins and C. S. Melo, Nucl. Phys. B785, 246 (2007), hep-th/0703086. [18] B. S. Shastry, Phys. Rev. Lett. 56, 2453 (1986). [19] J. Hubbard, Proc. R. Soc. London A276, 238 (1963). [20] E. H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445 (1968). [21] A. Rej, D. Serban and M. Staudacher, JHEP 0603, 018 (2006), hep-th/0512077. [22] D. Serban and M. Staudacher, JHEP 0406, 001 (2004), hep-th/0401057. [23] G. Arutyunov, S. Frolov and M. Staudacher, JHEP 0410, 016 (2004), hep-th/0406256. N. Beisert and A. A. Tseytlin, Phys. Lett. B629, 102 (2005), hep-th/0509084. R. Hernández and E. López, JHEP 0607, 004 (2006), hep-th/0603204. N. Gromov and P. Vieira, Nucl. Phys. B789, 175 (2008), hep-th/0703191. [24] R. A. Janik, Phys. Rev. D73, 086006 (2006), hep-th/0603038. [25] G. Arutyunov and S. Frolov, Phys. Lett. B639, 378 (2006), hep-th/0604043. [26] N. Beisert and T. Klose, J. Stat. Mech. 06, P07006 (2006), hep-th/0510124. [27] N. Beisert, R. Hernández and E. López, JHEP 0611, 070 (2006), hep-th/0609044. [28] J. K. Erickson, G. W. Semenoff and K. Zarembo, Nucl. Phys. B582, 155 (2000), hep-th/0003055. [29] D. J. Gross and N. Miljkovic, Phys. Lett. B238, 217 (1990). [30] A. V. Kotikov and L. N. Lipatov, Nucl. Phys. B769, 217 (2007), hep-th/0611204. [31] A. V. Belitsky, Phys. Lett. B650, 72 (2007), hep-th/0703058. N. Dorey, D. M. Hofman and J. Maldacena, Phys. Rev. D76, 025011 (2007), hep-th/0703104. [32] K. Sakai and Y. Satoh, Phys. Lett. B661, 216 (2008), hep-th/0703177. [33] A. Rej, M. Staudacher and S. Zieme, J. Stat. Mech. 0708, P08006 (2007), hep-th/0702151v2. [34] N. Mann and J. Polchinski, Phys. Rev. D72, 086002 (2005), hep-th/0508232. [35] N. Gromov, V. Kazakov, K. Sakai and P. Vieira, Nucl. Phys. B764, 15 (2007), hep-th/0603043. [36] Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower and V. A. Smirnov, Phys. Rev. D75, 085010 (2007), hep-th/0610248. [37] M. K. Benna, S. Benvenuti, I. R. Klebanov and A. Scardicchio, Phys. Rev. Lett. 98, 131603 (2007), hep-th/0611135. [38] M. Beccaria, G. F. De Angelis and V. Forini, JHEP 0704, 066 (2007), hep-th/0703131. [39] L. F. Alday, G. Arutyunov, M. K. Benna, B. Eden and I. R. Klebanov, JHEP 0704, 082 (2007), hep-th/0702028. I. Kostov, D. Serban and D. Volin, Nucl. Phys. B789, 413 (2008), hep-th/0703031. [40] J. Maldacena and I. Swanson, Phys. Rev. D76, 026002 (2007), hep-th/0612079. [41] V. G. Drinfel’d, Sov. Math. Dokl. 32, 254 (1985). [42] V. G. Drinfel’d, J. Math. Sci. 41, 898 (1988). [43] D. Bernard, Int. J. Mod. Phys. B7, 3517 (1993), hep-th/9211133. N. J. MacKay, Int. J. Mod. Phys. A20, 7189 (2005), hep-th/0409183. [44] L. Dolan, C. R. Nappi and E. Witten, JHEP 0310, 017 (2003), hep-th/0308089. [45] L. Dolan, C. R. Nappi and E. Witten, hep-th/0401243, in: “Quantum theory and symmetries”, ed.: P. C. Argyres et al., World Scientific (2004), Singapore. M. Hatsuda and K. Yoshida, Adv. Theor. Math. Phys. 9, 703 (2005), hep-th/0407044. L. Dolan and C. R. Nappi, Nucl. Phys. B717, 361 (2005), hep-th/0411020. [46] A. Agarwal and S. G. Rajeev, Int. J. Mod. Phys. A20, 5453 (2005), hep-th/0409180. N. Berkovits, JHEP 0503, 041 (2005), hep-th/0411170. B. I. Zwiebel, J. Phys. A40, 1141 (2007), hep-th/0610283. N. Beisert and D. Erkal, J. Stat. Mech. 0803, P03001 (2008), arxiv:0711.4813. [47] D. B. Uglov and V. E. Korepin, Phys. Lett. A190, 238 (1994), hep-th/9310158. [48] N. Dorey, J. Phys. A39, 13119 (2006), hep-th/0604175. H.-Y. Chen, N. Dorey and K. Okamura, JHEP 0611, 035 (2006), hep-th/0608047.

0704.0401

Modelling the Near-IR Spectra of Red Supergiant-dominated Populations

Stellar Populations as Building Blocks of Galaxies Proceedings IAU Symposium No. 241, 2007 A. Vazdekis et alr., eds. c© 2007 International Astronomical Union DOI: 00.0000/X000000000000000X Modelling the Near-IR Spectra of Red Supergiant-dominated Populations Ariane Lançon1, Jay S. Gallagher2, Richard de Grijs3, Peter Hauschildt4, Djazia Ladjal5, Mustapha Mouhcine6, Linda J. Smith7, Peter R. Wood8, Natascha Förster Schreiber9 1Observatoire de Strasbourg (UMR 7550), 11 rue de l’Université, 67000 Strasbourg, France email: [email protected] 2Dept. of Astronomy, 5534 Sterling, University of Wisconsin, Madison, WI 53706, USA 3Dept. of Physics & Astronomy, University of Sheffield, Hicks Building, Honusfield Rd., Sheffield S3 7RH, UK 4Hamburger Sternwarte, Gojenbergsweg 112, 21029 Hamburg, Germany 5Institute of Astronomy, Katholieke Universiteit, Celestijnenlaan 200B, 3001 Leuven, Belgium 6Astrophysics Research Institute, Liverpool John Moores University, Twelve Quays House, Egerton Wharf, Birkenhead, CH41 1LD, UK 7Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA 8RSAA, Mt Stromlo Observatory, Cotter Road, Weston Creek, ACT 2611, Australia 9MPI für Extraterrestrische Physik, Giessenbachstrasse, 85741 Garching, Germany Abstract. We report on recent progress in the modelling of the near-IR spectra of young stellar populations, i.e. populations in which red supergiants (RSGs) are dominant. First, we discuss the determination of fundamental parameters of RSGs from Phoenix model fits to their near-IR spectra; RSG-specific surface abundances are accounted for and effects of the microturbulence parameter are explored. New population synthesis predictions are then described and, as an example, it is shown that the spectra of young star clusters in M82 can be reproduced very well from 0.5 to 2.4µm. We warn of remaining uncertainties in cluster ages. Keywords. galaxies: stellar content, galaxies: starburst, galaxies: star clusters, galaxies: indi- vidual (M82), infrared: galaxies, infrared: stars, stars: supergiants 1. Introduction Red supergiant stars (RSGs) provide most of the near-IR light emitted by young stellar populations, such as those in starburst galaxies. As star forming environments tend to be dusty, rest-frame optical analyses are incomplete (highly obscured populations are missed) and it is crucial to improve our understanding of spectra at longer wavelengths. In the past, the near-IR analysis of young stellar populations has often been restricted to the determination of the average properties of the dominant stars, such as their spectral types or abundances. The subsequent interpretation of these results in terms of precise stellar population ages and star formation histories remains an enormous challenge, as it requires (i) a good understanding of the near-IR spectra of individual RSGs and (ii) adequate stellar evolution tracks. We have started a programme that aims at providing state of the art predictions for the emission of RSG-dominated populations and at characterizing remaining uncertainties. Currently, the project focuses on wavelengths between 0.81 and 2.4µm and spectral resolutions of order λ/δλ = 103. http://arxiv.org/abs/0704.0401v1 2 A. Lançon et al. 2. Empirical and synthetic spectra of red supergiants In principle, synthetic stellar spectra are more practical for the prediction of galaxy spectra than empirical ones, because theory allows us to sample parameter space without biases. Lançon et al. (2007) show that modern theoretical spectra can reproduce the near- IR (+optical) emission of giant stars well down to effective temperatures Teff ≃ 3400K, but that they are not yet satisfactory at lower temperatures and higher luminosities. They used new Phoenix models to compute spectra at the necessary resolution (0.1 Å before smoothing), with solar abundances and with the RSG-specific abundances obtained as the result of internal mixing along stellar evolution tracks; the models include some 109 individual molecular and atomic lines, assume spherical symmetry, and allow dust to form if conditions are fulfilled. Model limitations include the assumptions of local thermal equilibrium (LTE) and of hydrostatic equilibrium. A sample of 101 empirical spectra covering wavelengths between 0.51, 0.81 or 0.90µm and 2.4µm was used for comparison (Lançon & Wood 2000, Lançon et al. in preparation). The data were acquired with CASPIR on the 2.3m ANU Telescope at Siding Spring and with SpeX at IRTF, Hawaii. Below Teff ∼ 3400K, uncertain input line lists are a problem in the models (especially for molecular bands around 1µm). At high luminosity (luminosity class Ia and Iab), the main difficulty is to reproduce simultaneously extremely deep CN bands and the relative strengths of the CO bandheads around 1.7µm and at 2.3µm. RSG-specific abundances improve fits to the CN bands. Exploratory calculations show that values near 10 km/s for the “microturbulence” (a 1D-model parameter that hides poorly understood 3D physical phenomena) may be able to solve both problems (Fig. 1, top left). The calculation of a new grid has been launched to explore this further. In the mean time, the study shows that the population synthesis community still has to rely on empirical spectra for RSGs, and it warns that the lack of satisfactory stellar models implies large uncertainties on the derived fundamental parameters of the observed stars. 3. Population synthesis using averaged stellar spectra In order to compute spectra of synthetic populations, we have constructed three se- quences of average empirical spectra, corresponding to luminosity classes Ia, Iab and Ib/II. Each subset was sorted into bins according to the estimated Teff , the spectra were dereddened (an estimate of the reddening is provided by the model fits), and averages were computed. The sequences shown in Fig. 1 (top right) account for varying micro- turbulence in a preliminary way, based on the limited number of high microturbulence models available to us at the time of this writing. We chose to flag any star with an initial mass above 7M⊙ as a supergiant, which implies that the new spectra affect predictions up to the age of about 75Myr (Fig. 1, middle left). We note that predictions vary signif- icantly depending on the adopted evolutionary tracks; different authors predict different red supergiant lifetimes, and main sequence rotation affects both the surface abundances and the final red (and blue) supergiant numbers. 4. Star clusters in M82 The synthetic spectra of single stellar populations (SSPs) at solar metallicity are com- pared with those of young star clusters in starbursts, such as M82-L and M82-F (Smith & Gallagher 1999). The selected clusters are massive (well above 105M⊙), i.e. stochastic effects due to an underpopulated RSG-branch are avoided. A few have well determined optical ages (based on standard non-rotating evolutionary tracks). Figure 1 (middle right Modelling Red Supergiant Populations 3 3000 4000 5000 Age = 18 Myr Av = 1 0.5 1.0 1.5 2.0 2.5 Age=18 Av=3.7 Rv=2.4 chi2=1.58934 1.0 1.5 2.0 Wavelength (micron) 12 CO [FeII] 1.55 1.60 1.65 1.70 Wavelength (micron) bad pixels 2.1 2.2 2.3 Figure 1. Top left: Spectrum of an M0Ia RSG compared with models with vmicroturb=2km/s (top: 4200K, log(g)=-1, AV =4.4) and with vmicroturb=10 km/s (bottom: 4500K, log(g)=0, AV =4.7; note the improved CN at 1.1µm and CO around 1.6 and 2.3µm). Top right: Param- eters assigned to the new sequences of average spectra, superimposed on the solar metallicity tracks of Bressan et al. (1993). Middle left: Comparison of a new SSP spectrum (black) with the standard predictions of Pegase.2 (differences are largest between 10 and 20Myr). Middle right: Best near-IR fit to the spectrum of cluster M82-L. The extinction law of Cardelli et al. (1989) with RV =2.4 allows us to also reproduce the optical spectrum (from Smith & Gallagher 1999). The error spectrum and the χ2 weighting function are shown. Bottom: Zoom-ins of the H and K windows. 4 A. Lançon et al. and bottom) shows cluster L, the cluster observed with SpeX with the best signal-to- noise ratio: an excellent fit is obtained over the whole available range in wavelength. Such results make the new models valuable tools for purposes such as weak emission line measurements. The χ2-test restricted to near-IR wavelengths not affected by strong tel- luric absorption shows that age is formally determined to an accuracy of about ±10Myr. Because of strong reddening, the optical age of cluster L cannot be determined as well as that of cluster F: 50-70Myr (Gallagher & Smith 2001, McCrady et al. 2005, Bastian et al. 2007). For F, our current models provide a near-IR age range of 32 to 46Myr. This small but nevertheless significant disagreement calls for several comments. (i) Be- fore accounting for luminosity-dependent microturbulence, we found a near-IR age of 10 to 20Myr; we hope that our next generation of synthetic stellar spectra will significantly reduce uncertainties originating in uncertain fundamental parameters of stars. (ii) The spectrum used for optical age-dating and our near-IR spectrum have different slopes in the region of overlap. This suggests slightly different positions were observed: the ob- scuration across M82-F is not at all uniform. In addition, a younger cluster located at very small projected distance might contaminate the near-IR data. (iii) Modified stellar tracks (e.g. including rotation) might affect optical ages as well as near-IR ones. 5. Conclusions The spectra of young stellar populations at solar metallicity, observed at R∼ 103, can now be modelled well from the optical through the near-IR. Nevertheless, ages based on near-IR spectra remain severely affected by uncertainties. They are due mainly to system- atic errors, which further work needs to characterize and reduce. Errors are associated on one hand with the fundamental parameters of red supergiant stars (theoretical spectra, microturbulence, surface abundances of C, N and O, non-LTE, variability, winds, giant- supergiant transition), and on the other with evolutionary tracks (convection, opacities, rotation, binarity, effects of a dense environment). We expect rapid progress in stellar at- mosphere models to provide us with tools to test stellar tracks further. Complete optical and near-IR spectra of massive clusters such as those of M82 are useful test cases for the identification and correction of systematic errors, but even they are not trivial to exploit (due to inhomogeneous background populations and extinction, mass segregation, etc.). References Bastian, N., Konstantopoulos, I., Smith, L.J. & Gallagher, J.S. 2007, MNRAS in press Cardelli, J.A., Clayton, G.C. & Mathis, J.S. 1989 ApJ 345, 245 Gallagher, J.S. & Smith, L.J. 1999 MNRAS 304, 540 Lançon, A. & Wood, P.R. 2000, A&AS 146, 217 Lançon, A., Hauschildt, P., Ladjal, D. & Mouhcine, M. 2007, A&A in press McCrady, N., Graham, J.R. & Vacca, W.D. 2005 ApJ 621, 278 Smith, L. J. & Gallagher, J. S. 2001 MNRAS 326, 1027 Discussion Gustafsson: Do the models with high microturbulence include turbulent pressure in a consistent way? Lançon (after discussion with P.H. and H. Lamers): No. But the microturbu- lent velocities required to reproduce the spectra with 1D models are supersonic, which suggests that the actual process is not microturbulence... Therefore it is unclear how to relate this parameter of 1D models to pressure. Introduction Empirical and synthetic spectra of red supergiants Population synthesis using averaged stellar spectra Star clusters in M82 Conclusions

0704.0402

Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem

LOCATING THE PEAKS OF LEAST-ENERGY SOLUTIONS TO A QUASILINEAR ELLIPTIC NEUMANN PROBLEM YI LI AND CHUNSHAN ZHAO Abstract. In this paper we study the shape of least-energy solutions to the quasilinear problem εm∆mu−u m−1 + f (u) = 0 with homogeneous Neumann boundary condition. We use an intrinsic variation method to show that as ε → 0+, the global maximum point Pε of least-energy solutions goes to a point on the boundary ∂Ω at the rate of o(ε) and this point on the boundary approaches to a point where the mean curvature of ∂Ω achieves its maximum. We also give a complete proof of exponential decay of least-energy solutions. 1. Introduction and statement of results In this paper we study the shape of certain solutions to the following quasilinear elliptic Neumann problem: (1.1) εm∆mu− um−1 + f (u) = 0, u > 0 in Ω, = 0 on ∂Ω, where m (2 ≤ m < N) and 0 < ε ≤ 1 are constants and Ω ⊆ RN (N ≥ 3) is a smooth bounded domain. The operator ∆mu = div(|∇u|m−2 ∇u) is the m- Laplacian operator, and ν is the unit outer normal to ∂Ω. Problem (1.1) appears in the study of non-Newtonian fluids, chemotaxis and biological pattern formation. For example, in the study of non-Newtonian fluids, the quantity m is a characteristic of the medium: media with m > 2 are called dilatant fluids, and those with m < 2 are called pseudo-plastics. If m = 2, they are Newtonian fluids (see [3] and its bibliography for more backgrounds). For the case m = 2, (1.1) is also known as the stationary equation of the Keller– Segal system in chemotaxis [14] or the limiting stationary equation of the so-called Gierer–Meinhardt system in biological pattern formation (see [23]). First let us recollect some results related to our problem. In a series of remarkable papers, C.-S. Lin, W.-M. Ni and I. Takagi [14], Ni and Takagi [17], [18] studied the Neumann problem for certain elliptic equations, including (1.2) d∆u− u+ up = 0, u > 0 in Ω, = 0 on ∂Ω, where d > 0, p > 1 are constants, and p is subcritical, i.e., p < N+2 . First, Lin, Ni and Takagi [14] applied the mountain-pass lemma [1] to show the existence of Key words and phrases. Quasilinear Neumann problem, m-Laplacian operator, least-energy solution, exponential decay, mean curvature. http://arxiv.org/abs/0704.0402v1 2 YI LI AND CHUNSHAN ZHAO a least-energy solution ud to (1.2), by which is meant that ud has the least energy among all solutions to (1.2) with the energy functional Id (u) = |∇u|2 + 1 u2 − 1 defined on W 1,2 (Ω). Hereinafter u+ = max {u, 0} and u− = min {u, 0}. Then in [17], [18], Ni and Takagi investigated the shape of the least-energy solution ud as d becomes sufficiently small, and showed that ud has exactly one peak (i.e., local maximum of ud) at Pd ∈ ∂Ω. Moreover, as d tends to zero, Pd approaches a point where the mean curvature of ∂Ω achieves its maximum. See [15] for a review in this field. Also see [16] for the critical case p = N+2 , and [5], [6], [7], [8], [9] for existence and properties of multiple-peaks solutions to (1.2). From now on we make some hypotheses on f : R → R, as follows. (H2) f (t) ≡ 0 for t ≤ 0 and f ∈ C1 (R). (H3) f(t) = O (t p) as t→ ∞ with m− 1 < p < N (m− 1) +m N −m . (H4) Let F (t) = f (s) ds. Then there exists a constant θ ∈ that F (t) ≤ θtf (t) for t > 0. f (t) is strictly increasing for t > 0 and f (t) = O tm−1+δ as t→ 0+ with a constant δ > 0. (H6) Let g (u) = (m− 1)um−1 − uf ′ (u) um−1 − f (u) . Then g (u) is non-increasing on (uc,∞), where uc is the unique positive solution for f (t) = t Next we present some preliminary knowledge about least energy solutions of the following problem: (1.3) ∆mu− um−1 + f(u) = 0 in RN u > 0 in RN As before we define an “energy functional” I:W 1,m(RN ) −→ R associated with (1.3) by (1.4) I(ṽ) = (εm |∇ṽ|m + |ṽ|m)− F (ṽ+) Next let us give a remark on ground states to the problem 1.3. Here by a ground state we mean a non-negative nontrivial C1 distribution solution which tends to zero at ∞. For case m = 2, it is well known that the problem 1.3 has a unique ground state (up to translations) which is radially symmetric [4]. For case 2 < m < N uniqueness and radial symmetry of ground states are still open. But the Steiner symmetrization tells us the least-energy solutions must be radially symmetric (certainly least-energy solutions are ground states). Our assumptions guarantee that the uniqueness (up to translations) of radial ground states (see [20]), which implies the uniqueness of least-energy solutions of the problem (1.3). Exact exponential decay of radial ground states was given in [11], thus we have the following proposition about the unique radial least-energy solution to problem 1.3: Proposition 1.1. Under assumptions (H2)–(H6), there is a unique least energy solution w(x) for (1.3) satisfying: LOCATING THE PEAKS OF LEAST-ENERGY SOLUTIONS 3 ( i ) w is radial, i.e., w(x) = w(|x|) = w(r) and w ∈ C1(RN ) with w(0) = maxX∈RN w(x), w ′(0) = 0 and w′(r) < 0, ∀r > 0. (ii) limr−→∞ w(r)r m(m−1) e( m r = C0 > 0 for some constant C0 and limr−→∞ w′(r) Remark 1.1. A good example for f (t) which satisfies all hypotheses (H2)–(H6) is f (t) = tp for m− 1 < p < N (m− 1) +m Next we define an “energy functional” Jε : W 1,m (Ω) → R associated with (1.1) (1.5) Jε (v) = (εm |∇v|m + |v|m)− F (v+) with F (v+) = f (s) ds. Then the well-known mountain-pass lemma [1] implies (1.6) cε = inf t∈[0,1] Jε (h (t)) is a positive critical value of Jε, where Γ is the set of all continuous paths joining the origin and a fixed nonzero element e ∈ W 1,m (Ω) such that e ≥ 0 and Jε (e) ≤ 0. It turns out cε can also be characterized as follows: cε = inf Jε (u) u ∈ W 1,m (Ω) ; u ≥ 0, u 6≡ 0, (εm |∇u|m + um) dx = f (u)u dx (1.7) cε = inf M [u] | u ∈ W 1,m (Ω) , u 6≡ 0 and u ≥ 0 in Ω M [u] = sup Jε (tu) . Hence cε is the least positive critical value and a critical point uε of Jε with critical value cε is called a least-energy solution. Notice also that if we let c∗ = I(w) = (|∇w|m + wm) dx− F (w) dx, where w is the unique least energy solution of (1.3), then c∗ can also be characterized (1.8) c∗ = inf M∗ [v] | v ∈ W 1,m , v 6≡ 0 and v ≥ 0 in RN M∗ [v] = sup I (tv) . We refer to Lemma 2.1 of [13] for the above characterizations. Next we consider the following problem: 4 YI LI AND CHUNSHAN ZHAO v ∈W 1,m with RN+ = (x1, · · · , xN ) ∈ RN , xN ≥ 0 and satisfies (1.9) ∆mv − vm−1 + f (v) = 0, v > 0 in RN+ , = 0 on xN = 0. The solutions of (1.9) can be characterized as critical points of the functional defined over W 1,m as follows. (ṽ) = |∇ṽ|m + ṽm) dx− F (ṽ+) dx. Similarly as above the least positive critical value C∗ corresponding to least energy solutions of (1.9) can be characterized as (1.10) C∗ = inf ṽ∈W 1,m(RN+ ),ṽ≥0,ṽ 6≡0 (tṽ) and moreover (1.11) C∗ = due to the boundary condition in (1.9) and the fact that w is radial and hence = 0. We also refer to Lemma 2.1 of [13] for the above characterization of C∗. In Theorem 1.3 of [13], we proved the following theorem. Theorem 1.1. Under hypotheses (H2)–(H6), let uε be a least-energy solution of (1.1). Then all local maximum points(if more than one) of uε aggregate to a global maximum point Pε at a rate of o(ε) and dist(Pε, ∂Ω)/ε→ 0 as ε → 0+, where dist(·, ·) is the general distance function. Moreover, we have the following upper- bound estimate for cε as ε→ 0+: (1.12) cε ≤ εN c∗ − (N − 1) max H (P ) γε+ o (ε) where H (P ) denotes the mean curvature of ∂Ω at P , γ > 0 is a positive constant given by (1.13) γ = N + 1 |w′ (|z|)|m zN dz. Our goal in this paper is to locate the position on ∂Ω where the global maximum point Pε of uε in Ω approaches, provided ε is sufficiently small. For the case m = 2, Ni and Takagi [18] located the peak by linearizing the equation d∆u−u+f (u) = 0 around the ground state w. But this method fails for our problem with m 6= 2 due to the strong nonlinearity of the m-Laplacian operator ∆mu = div(|∇u|m−2 ∇u). So we have to use the intrinsic variational method created by Del Pino and Felmer in [2] to attack it. We also give a complete proof of the exponential decay of the least-energy solution uε. We remark that our proof is complete and does not require the non-degeneracy of the unique radial least energy solution w as stated in Proposition 1.1, and hence it is different from Ni’s and Takagi’s work [17]. Now our results can be stated as follows: Theorem 1.2. Under hypotheses (H2)–(H6), let uε be a least-energy solution of (1.1) and P̃ε ∈ ∂Ω with dist(Pε, P̃ε) = dist(Pε, ∂Ω). Then as ε→ 0+, after passing to a sequence P̃ε approaches P̄ ∈ ∂Ω with LOCATING THE PEAKS OF LEAST-ENERGY SOLUTIONS 5 (ii) H = max H (P ), where H (P ) denotes the mean curvature of ∂Ω at P as stated before, and moreover (iii) the associated critical value cε can be estimated as ε→ 0+ as follows : (1.14) cε = ε c∗ − (N − 1)H γε+ o (ε) where c∗, γ are as stated in Theorem 1.1. The organization of this paper is as follows: In Section 2, we will prove some lemmas which will be used in proving Theorem 1.2. The proof of Theorem 1.2 will be given in Section 3. 2. Some lemmas and exponential decay of uε First we prove the following lemma related to exponential decay of the least- energy solution uε. Lemma 2.1. Let ε be sufficiently small and that the least-energy solution uε achieves its global maximum at some point Pε. Then there exist two positive con- stants c3 and c4 independent of uε or ε such that (2.1) uε (x) ≤ c3 exp {−c4 |x− Pε| /ε} |∇uε(x)| ≤ c3ε−1 exp{−c4|x− Pε|/ε}. Before beginning to prove this lemma, we give a remark on it. Remark 2.1. For the case m = 2, under the assumption of non-degeneracy of the linearized operator ∆− 1 + f ′ (w), where w is the unique ground state of (1.3), Ni and Takagi [18] showed that uε (x) can be written as (2.2) uε (x) = w (x) + εφ1 (x) + o (ε) and φ1 (x) enjoys the exponential-decay property ([18]). Clearly we cannot derive exponential decay of uε (x) as stated in Lemma (2.1) from (2.2) even though both w (x) and εφ1 (x) have exponential decay property. Proof of Lemma 2.1. Since ∂Ω is a smooth compact submanifold of RN , it follows from the tubular neighborhood theorem [10] that there exists a constant ω (Ω) > 0 which depends only on Ω such that ΩI = x ∈ Ω, d (x, ∂Ω) < ω (Ω) is diffeomor- phic to the inner normal bundle I = {(x, y) : x ∈ ∂Ω, y ∈ (−ω (Ω) , 0] νx} , here νx is the unit outer normal of ∂Ω at x, and the diffeomorphism is defined as follows: ∀x ∈ ΩI , there exists an unique x̂ ∈ ∂Ω such that d (x, x̂) = d (x, ∂Ω) , then Φ∗ : x −→ (x̂,−d (x, x̂) νx̂) . Moreover this diffeomorphism satisfies Φ∗|∂Ω = Identity. Similarly, let ΩO = x ∈ RN \ Ω, d(x, ∂Ω) < ω (Ω) . Then ΩO is diffeo- morphic to the outer normal bundle O = {(x, y) : x ∈ ∂Ω, y ∈ [0, ω (Ω)) νx} , and the diffeomorphism is given as follows. ∀x ∈ ΩO, there exists an unique x̄ ∈ ∂Ω such that d(x, x̄) = d (x, ∂Ω) , and then Φ# : x −→ (x̄, d (x, x̄) νx̄) and Φ#|∂Ω = Identity. Note that (∂Ω)NI is clearly diffeomorphic to (∂Ω) O via the following reflection Φ∗ : (∂Ω) I −→ (∂Ω) O defined by Φ ∗ ((x, y)) = (x,−y) . 6 YI LI AND CHUNSHAN ZHAO Therefore, Φ = Φ−1∗ ◦ Φ∗−1 ◦ Φ# : ΩO −→ ΩI is the desired diffeomorphism and Φ|∂Ω = Identity. Moreover, if we let x = Φ(z) = (Φ1(z), · · · ,ΦN(z)) , z ∈ ΩO, and z = Ψ(x) = Φ−1(x) = (Ψ1(x), · · · ,ΨN(x)) , x ∈ ΩI , gij = gij = (Φ (z)) , we have gij |∂Ω = gij |∂Ω = δij with δij being the Kro- necker symbol. Denote G = and A = G− I with I being the N ×N identity matrix, g(x) = det (gij) and ûε(x) = uε (Φ (x)) for x ∈ ΩO. Then ûε(x) satisfies the following equations: εmLûε − gûm−1ε + gf (ûε) = 0, ûε > 0 in ΩO = 0, on ∂Ω, where Lûε = s,l=1 ∇ûεG (∇ûε)T g (∇ûε)G where Tr means taking the trace of a square matrix. For 0 < γ̃ ≤ ω(Ω), let ΩOγ̃ = x ∈ ΩO, d(x, ∂Ω) < γ̃ . We know ‖A‖C0 can be made arbitrarily small by making γ̃ sufficiently small. Next we define ūε = uε(x), x ∈ Ω ûε(x), x ∈ ΩO, g̃ij = δij , x ∈ Ω gij , x ∈ ΩO, g̃ij = δij , x ∈ Ω gij , x ∈ ΩO, and Ã(x, ξ) = Ã1(x, ξ), · · · , ÃN (x, ξ) for ξ = (ξ1, · · · , ξN ) with Ãi(x, ξ) = s,l=1 g̃slξsξl g̃ijξj and g̃ = det(g̃ij), B(x, u) = −um−1 + f(u) . Then ūε(x) satisfies (2.3) εm div Ã(x,∇ūε) +B(x, ūε) = 0 in Ω in the weak sense. LOCATING THE PEAKS OF LEAST-ENERGY SOLUTIONS 7 For any ball Br(x0) ⊂ Ω ΩO with radius r and center x0 ∈ Ω, let ρ = |x− x0|. Then for any smooth increasing function φ = φ(ρ) we have (∇φ)G(∇φ)T g(∇φ)G ∣∇φ(I +A)(∇φ)T det(I +A)−1∇φ(I +A) = |∇φ|m−2 ∇φ+ ∣∇φ(I + tA)(∇φ)T det(I + tA)−1∇φ(I + tA) = |∇φ|m−2 ∇φ ∣∇φ(I + tA)(∇φ)T (∇φ)A(∇φ)T det(I + tA)−1∇φ(I + tA)dt ∣∇φ(I + tA)(∇φ)T det(I + tA)−1 det(I + tA)−1 ∇φ(I + tA)dt ∣∇φ(I + tA)(∇φ)T det(I + tA)−1(∇φ)Adt. Therefore (2.4) (∇φ)G(∇φ)T g(∇φ)G 3(N − 1) φ′ +K by taking γ̃ sufficiently small, here K > 0 is a constant depending only on Ψ, hence only on Ω and φ′ = dφ(ρ) From now on γ̃ = γ̃(Ω) is fixed such that (i) 3 ≤ √g ≤ 5 , (ii) (2.4) holds for any smooth increasing radial function φ(ρ) and (iii) 3 |ξ|m ≤ Ã(x, ξ) · ξ ≤ 5 |ξ|m for any ξ = (ξ1, · · · , ξN ). Denote Ωγ̃ = Ω ∪ ΩOγ̃ . Let Ωε = (Ω− Pε) and uε(x) = uε(Pε + εx) for x ∈ Ωε. Then uε is a solution to the following problem: (2.5) ε − (uε)m−1 + f(uε) = 0, uε > 0 in Ωε = 0, on ∂Ωε, where n is the unit outer normal of ∂Ωε. Similarly, let Ω Ωγ̃ − Pε ūε(x) = ūε(Pε + εx) for x ∈ Ωγ̃ε . Since ūε converges to the unique radial least- energy solution w of (1.3) in C1loc(R N ) ∩W 1,m(RN ) as ε → 0+ (see the proof of Theorem 1.2 of [13]) and w satisfies: (i) w is radial, i.e.,w(x) = w(|x|) = w(r) > 0 (ii) lim w(r)r m(m−1) e( m−1 ) m r = C0 > 0 (see Theorem 1 of [11]) which yields w(r) ≤ κe−µr for a constant κ > 0 and . First we fix a constant η > 0 such that 1 tm−1 > f(t) for t ∈ (0, η]. From hypothesis (H5) it follows that such an η exists. Then there exist ε0 > 0 sufficiently small and R0 sufficiently large such that 4κ exp{−µR0} < η and 8 YI LI AND CHUNSHAN ZHAO ‖ūε − w‖C0(BR0(0)∩Ωε) ≤ κ exp{−µR0}, which yields uε|(∂BR0 (0))∩Ωε ≤ 2κ exp{−µR0}. Note that ε − 7 (uε)m−1 = 1 (uε)m−1 − f(uε) > 0 in Ωε \BR0(0), = 0 on ∂Ωε \BR0(0), uε ≤ 2κ exp{−µR0} on ∂BR0(0) ∩ Ωε. Then we have uε(x) ≤ 2κ exp{−µR0}, for x ∈ Ωε \BR0(0) due to the strong maximum principle ([22]). We get by scaling back that uε|Ω\BεR0 (0) ≤ 2κ exp{−µR0} (2.6) uε(x) ≤ w + κ exp{−µR0} ≤ κ exp{− }+ κ exp{−µR0} ≤ 2κ exp{−µ|x| for x ∈ Ω ∩BεR0(0). From definition of ūε we know ūε(x) ≤ 2κ exp{− µ (|x| − 2dist (Pε, ∂Ω)) } ≤ 4κ exp{−µ|x| } for x ∈ Ωγ̃∩BεR0(0) for ε ∈ (0, ε0] with ε0 sufficiently small due to the fact dist(Pε, ∂Ω) = o(ε) as ε→ 0+. Note that Ωγ̃\BεR0 (0) ūε ≤ 4κ exp{−µR0}. Choice of R0 and γ̃ tells us for any 0 < t ≤ 4κ exp{−µR0} B(x, t) = −tm−1 + f(t) tm−1. ∀x0 ∈ Ω \BεR0(0) and Br(x0) ⊂ Ωγ̃ \BεR0(0), define φ(x) = φ(ρ) = φ(|x− x0|) Ωγ̃\BεR0 (0) where λ∗ > 0 is a constant to be determined later. Simple calculations show that (i) φ′(ρ) > 0 and 3(N−1) φ′ +K (m− 1) (λ∗)m ))m−2 tanh(λ∗ρε ) ( λ∗ρε ) +ε (λ∗) ))m−1 LOCATING THE PEAKS OF LEAST-ENERGY SOLUTIONS 9 for any 0 < λ∗ ≤ λ̂, where λ̂ > 0 is a small constant depending only on m and Ω through K. We remark that we have used the fact maxr∈[0,∞) tanhr < ∞ for m ≥ 2. From now on we choose λ∗ = λ̂. Therefore we have εm div(Ã(x,∇ūε))− ūm−1ε ≥ 0 in Br(x0), εm div(Ã(x,∇φ))− m−1 ≤ 0 in Br(x0). Clearly φ|∂Br(x0) ≥ ūε|∂Br(x0). Then from the Comparison Theorem (Theorem 10.1 of [19]) it follows that φ(x) ≥ ūε(x) in Br(x0). In particular, φ(x0) ≥ ūε(x0). Thus we get uε(x0) ≤ Ωγ̃\BεR0(0) exp{−λ∗r Choosing r = d x0, ∂ Ωγ̃ \BεR0(0) we get uε(x0) ≤ 4κ exp{−µR0 − } ≤ 2κ exp{− λ̃(εR0 + r) with λ̃ = min{µ, λ∗}. Note that x0 belongs to one of the following two cases: (i) d x0, ∂ Ωγ̃ \BεR0(0) = d (x0, ∂BεR0(0)) , (ii) d x0, ∂ Ωγ̃ \BεR0(0) x0, ∂Ω For case (i) we have d(x0, Pε) ≤ εR0 + r and therefore (2.7) uε(x0) ≤ 4κ exp{− λ̃d(x0, Pε) For case (ii) we have r ≥ γ̃ and thus (2.8) uε(x0) ≤ 4κ exp{−λ̃ εR0 + r } ≤ 4κ exp{− λ̃γ̃ ≤ 4κ exp{−λ̃ γ̃ diam(Ω) · d(x0, Pε) Combining (2.6), (2.7) and (2.8) together and letting c̃3 = 4κ, c̃4 = min{µ, λ̃, λ̃γ̃diam(Ω)} yields (2.9) uε(x) ≤ c̃3 exp{− c̃4|x− Pε| Next we show the estimate for |∇uε| holds. First from (2.5) it follows that (2.10) ∆mu ε = (uε) m−1 − f(uε), uε > 0 in Ωε For x ∈ Ωε and dist(x, ∂Ωε) ≥ 1, consider (2.10) in the unit ball centered at x, i.e., B1(x). Then by an C 1,α estimate (see [21], for example) there exists two constants 10 YI LI AND CHUNSHAN ZHAO C > 0 and α∗ ∈ (0, 1) which are independent of ε such that (2.11) ‖uε‖C1,α∗ (B 1 (x)) ≤ C ‖uε‖L∞(B1(x)) + ‖ (u m−1 − f(uε)‖ L∞(B1(x)) ≤ c∗3 exp{−c∗4|x− Pε|}, where we have used (2.9) and the fact that uε(x) = uε(Pε + εx) for x ∈ Ωε. Especially we have (2.12) |∇uε(x)| ≤ c∗3 exp{−c∗4|x− Pε|}, for x ∈ Ωε and dist(x, ∂Ωε) ≥ 1. For x ∈ Ωε with dist(x, ∂Ωε) < 1. Let x0 ∈ ∂Ωε be a point such that dist(x, x0) = dist(x, ∂Ωε) and consider ū ε(x) = ūε(Pε + εx) in B2(x0), the ball of radius 2 centered at x0, then from (2.3) it follows that ū satisfies (2.13) div Ã(Pε + εx,∇ūε) +B(Pε + εx, ū ε) = 0 in B2(x0) in the weak sense. Then applying an C1,α estimate (see [21], for example) again yields as above that there exists two constants C > 0 and α∗ ∈ (0, 1) which are independent of ε such that ‖ûε‖C1,α∗(B1(x0)) ≤ C ‖ûε‖L∞(B2(x0)) + ‖B(Pε + εx, û L∞(B2(x0)) ≤ c∗3 exp{−c∗4|x− Pε|} by adjusting c∗3 and c 4 if it is necessary. Especially we have (2.14) |∇uε(x)| ≤ c∗3 exp{−c∗4|x− Pε|}, Thus combining (2.11) and (2.14) together and scaling back we have for x ∈ Ω |∇uε(x)| ≤ c∗3ε−1 exp{−c∗4 |x− Pε| Proof of Lemma 2.1 is completed by letting c3 = max{c̃3, c∗3} and c4 = min{c̃4, c∗4}. Remark 2.2. Our proof of the Lemma 2.1 with necessary minor modifications also works well for elliptic systems. Next we present a lemma related to extensions of uε. Lemma 2.2. There exists a C1-extension ũε of uε which has compact support in N and satisfies (ii) ‖ũε‖W 1,m(RN ) ≤ c5 ‖uε‖W 1,m(Ω) and ‖ũε‖C1(RN ) ≤ c5 ‖uε‖C1(Ω̄), (iii) ũε also has the exponential-decay property as stated in Lemma 2.1, i.e., there exists an absolute constant λ ≥ 1 such that (2.15) 0 ≤ ũε ≤ c3λ exp |x− Pε| |∇ũε(x)| ≤ c3λε−1 exp{− |x− Pε| LOCATING THE PEAKS OF LEAST-ENERGY SOLUTIONS 11 (iv) there exists a positive constant δ̃ = δ̃ (Ω) such that for any P ∈ ∂Ω, ũε|B (P )\Ω is the reflection of uε through ∂Ω. Proof. Let d̃ = d ∂Ω, ∂Ωγ̃ and 0 ≤ ̺(x) ≤ 1 be a smooth cut-off function such that ̺(x) ≡ 1 for x ∈ {x ∈ RN , d(x,Ω) ≤ d̃ } and ̺(x) ≡ 0 for x ∈ RN \ Then ũε = ̺ūε satisfies (ii), (iii) and (iv) automatically. The proof of this lemma is completed. � Similar to energy density introduced in [2], we define the energy density associ- ated with (1.1) as follows: E (w, y′) = (|∇w|m + wm)− F (w) (y′, 0) for y′ ∈ RN−1. Then we have the following lemma. Lemma 2.3. Let G be a C2 function in a neighborhood of the origin of RN−1. i,j=1 Gij (0) yiyjE (w, y ′) dy′ = 2∆G (0) γ, where γ is the constant defined in (1.13), and y′ = (y1, . . . , yN−1), and Gij (0) = ∂yi∂yj (0) . Proof. In Lemma 2.4 of [13], we showed that (2.16) γ = (|∇w|m) + wm − F (w) zN dz. Next we introduce the polar coordinates z1 = r sin θN−1 sin θN−2 · · · sin θ2 sin θ1, z2 = r sin θN−1 sin θN−2 · · · sin θ2 cos θ1, z3 = r sin θN−1 sin θN−2 · · · cos θ2, ... , zN = r cos θN−1, and notice that (r, θ1, . . . , θN−1) | r > 0, 0 ≤ θ1 < 2π, 0 ≤ θj < π for j = 2, . . . , N − 2, and 0 ≤ θN−1 < and that dz = rN−1 sin θ2 sin 2 θ3 · · · sinN−2 θN−1 dr dθ1 · · · dθN−1. After elementary computations one obtains (2.17) γ = |w′ (r)|m + wm (r) − F (w (r)) rN dr · ωN−2, where ωN−2 is the volume of the unit ball in R N−2. Here we used the fact that w is radially symmetric. 12 YI LI AND CHUNSHAN ZHAO Using the radial symmetry of w again, we obtain i,j=1 Gij (0) yiyjE (w, y ′) dy′(2.18) Gii (0) y iE (w, y ′) dy′ Gii (0) · N − 1 |y′|2E (w, y′) dy′ = ∆G (0) · E (w, r) rN dr · ωN−2, where E (w, r) = (1/m) |w′ (r)|m + wm (r) − F (w (r)) . Comparing (2.17) and (2.18) yields i,j=1 Gij (0) yiyjE (w, y ′) dy′ = 2∆G (0) γ. The proof of Lemma 2.3 is completed. � 3. Proof of Theorem 1.2 With the help of the lemmas in Section 2, now we can give the proof of Theorem Proof of Theorem 1.2. Since as ε→ 0+, Pε → ∂Ω at the rate of o(ε), it follows that d(Pε, P̃ε)/ε → 0, where P̃ε ∈ ∂Ω is the closest point on ∂Ω to Pε. then by passing to a sequence, P̃ε → P̄ ∈ ∂Ω. After an ε-dependent rotation and translation, we may assume that P̃ε is at the origin and Ω can be described in a fixed cubic neighborhood V of P̄ as the set { (x′, xN ) | xN > ψε (x′) } with x′ = (x1, . . . , xN−1) , where ψε is smooth, ψε (0) = 0, ∇ψε (0) = 0. Furthermore, we may assume that ψε converges locally in the C 2 sense to ψ, a corresponding parametrization at P̄ . Note that since P̃ε is the origin, so we have Pε/ε → 0 as ε → 0+. Thus we have ũε(x) = ũε(εx) = ũε x− Pε → w(x) in C1loc as ε → 0+. From the characterization of cε = Jε (uε) in Section 1, we have ε−NJε (uε) ≥ ε−NJε (tuε) = IΩε (tuε) for all t > 0. Hereinafter IΩ∗ (v) = (|∇v|m + |v|m) dx− F (v) dx. IΩε (tu ε) = IΩε (tũ ε) ≥ I (tũε) + I(Ωε∩Vε)\RN+ (tũε)− I(RN+∩Vε)\Ωε (tũ ε)(3.1) = I + II− III, LOCATING THE PEAKS OF LEAST-ENERGY SOLUTIONS 13 with Vε = V. Let us choose t = tε so that IRN (tũε) maximizes in t. Then from the definition of C∗ in (1.10), equality (1.11) and Lemma 2.2 it follows that I = I (tεũ ε) ≥ c∗ e−c6/ε for some constant c6 > 0 independent of ε. Next we give an estimate of tε. Lemma 3.1. There is a unique tε ∈ (0,∞) such that tmε (|∇ũε| + (ũε)m) dx − F (tεũ ε) dx = sup tm (|∇ũε|m + (ũε)m) dx− F (tũε) dx and moreover (3.2) tε = 1 + o (1) as ε→ 0+. Proof. Under assumption (H5), the existence and uniqueness of tε can be proved similarly to the proof of Lemma 2.1 of [13]. Here we only need show (3.2). Let (3.3) hε (t) = (|∇ũε|m + (ũε)m) dx− F (tũε) dx. (3.4) h′ε (t) = t (|∇ũε|m + (ũε)m) dx− ũεf (tũε) dx = tm−1 (|∇w|m + wm) dx− wf (tw) dx+ o(1), here we have used the exponential decay of ũε in Lemma 2.2, exponential decay of w and ũε → w in C1loc as ε → 0+. Moreover the term o (1) → 0 uniformly in t on each compact interval as ε → 0+. (3.3) tells us hε(1) = 12c∗ + o(1), which yields that tε is bounded and away from 0. Also from (3.4) it follows that (3.5) h′ε (t) = t wf (w) dx− w f (tw) dx+ o (1) = tm−1 f (w) − f (tw) dx+ o (1) . Therefore at t = tε we have (3.6) f (w) − f (tεw) (tεw) dx = o (1) . Since f(t)/tm−1 is strictly increasing (see (H5)) it follows from (3.6) that tε = 1 + o (1) . The proof of Lemma 3.1 is completed. � 14 YI LI AND CHUNSHAN ZHAO Proof of Theorem 1.2 continued. Using again the exponential decay of uε in Lemma 2.1 and the expansion of tε in Lemma 3.1, we obtain −II = − (RN−1×{0})∩Vε dy′(3.7) (ψε(εy′)) tmε (|∇ũε| + (ũε) )− F (tεũε) (y′, yN ) dyN = − (1 + o (1)) (RN−1×{0})∩(Ωε∩Vε) (ψε(εy′)) (|∇uε|m + (uε)m)− F (uε) (y′, yN) dyN . Similarly, (3.8) III = (1 + o (1)) Vε∩(RN−1×{0}) (ψε(εy (|∇ũε|m + (ũε)m)− F (ũε) (y′, yN) dyN . In above a+ = max{a, 0}, a− = min{a, 0}. Since ψε (0) = 0, ∇ψε (0) = 0 and ψε converges in the C 2 local sense to ψ, and ũε → w in the C1 local sense in RN with uniform exponential decay with respect to ε, it follows from the dominated convergence theorem that (−II + III) i,j=1 ψij (0) yiyj (|∇w|m + wm)− F (w) (y′, 0) dy′ = ∆ψ (0) γ = (N − 1)H γ (by Lemma 2.3). Thus we have cε ≥ εN c∗ − (N − 1)H γε+ o (ε) But (1.12) in Theorem 1.1 tells us cε ≤ εN c∗ − (N − 1) max H (P ) γε+ o (ε) Therefore we get (ii) H = max H (P ), which is (ii) of Theorem 1.2, (iii) cε = ε c∗ − (N − 1)H γε+ o (ε) as ε→ 0+, which is part (iii) of Theorem 1.2. The proof of Theorem 1.2 is completed. � Acknowledgement. The authors want to give their thanks to anonymous referee for some helpful comments. LOCATING THE PEAKS OF LEAST-ENERGY SOLUTIONS 15 References [1] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 51 #6412 [2] M. Del Pino and P.L. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J. 48 (1999), 883–898. MR 2001b:35027 [3] J.I. Dı́az, Nonlinear Partial Differential Equations and Free Boundaries, Vol. I: Elliptic Equations, Research Notes in Mathematics, vol. 106, Pitman Advanced Publishing Program, Boston, 1985. MR 88d:35058 [4] B. Gidas, W.M. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equa- tions in RN , Advances in Math, Supplementary Studies 7A (1981) 369-402. MR 84a:35083 [5] C. Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J. 84 (1996), 739–769. MR 1997i:35052 [6] C. Gui and N. Ghoussoub, Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent, Math. Z. 229 (1998), 443–474. MR 2000k:35097 [7] C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations 158 (1999), 1–27. MR 2000g:35035 [8] , On multiple mixed interior and boundary peak solutions for some singularly per- turbed Neumann problems, Canad. J. Math. 52 (2000), 522–538. MR 2001b:35023 [9] C. Gui, J. Wei, and M. Winter, Multiple boundary peak solutions for some singularly per- turbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), 47–82. MR 2001a:35018 [10] M.-M. Hirsch, Differential Topology, Graduate Texts in Mathematics, vol. 33, Springer- Verlag, New York, 1976. MR 56# 6669 [11] Y. Li and C. Zhao, A note on exponential decay properties of ground states for quasilinear elliptic equations, Proc. Amer. Math. Soc. 133 (2005), 2005–2012. MR 2006a:35091 [12] , On the structure of solutions to a class of quasilinear elliptic Neumann problems, J. Differential Equations 212 (2005), 208–233. MR 2006b:35107 [13] , On the shape of least-energy solutions for a class of quasilinear elliptic Neumann problems, IMA Journal of Applied Mathematics 2007; doi: 10.1093/imamat/hx1032. [14] C.-S. Lin, W.-M. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988), 1–27. MR 89e:35075 [15] W.-M. Ni,Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc. 45 (1998), no. 1, 9–18. MR 99a:35132 [16] W.-M. Ni, X.B. Pan, and I. Takagi, Singular behavior of least-energy solutions of a semilin- ear Neumann problem involving critical Sobolev exponents, Duke Math. J. 67 (1992), 1–20. MR 93j:35081 [17] W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), no. 7, 819–851. MR 92i:35052 [18] , Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), 247–281. MR 94h:35072 [19] P. Pucci and J. Serrin, The strong maximum principle revisted, J. Differential Equations 196 (2004), no. 1, 1-66. MR 2004k:35033 [20] J. Serrin and M.-X. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J. 49 (2000), no. 3, 897–923. MR 2002d:35072 [21] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differ- ential Equations 51 (1984), no. 1, 126–150. MR 85g:35047 [22] J. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), no. 3, 191–202. MR 86m:35018 [23] J. Wei, On the boundary spike layer solutions to a singularly perturbed Neumann problem, J. Differential Equations 134 (1997), 104–133. MR 98e:35076 Department of Mathematics, The University of Iowa, Iowa City, IA 52242 16 YI LI AND CHUNSHAN ZHAO Department of Mathematics, Hunan Normal University, Changsha, Hunan E-mail address: [email protected] Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460 E-mail address: [email protected] 1. Introduction and statement of results 2. Some lemmas and exponential decay of u0=x"0122 3. Proof of Theorem ?? References

0704.0403

Review: Semiconductor Quantum Light Sources

Microsoft Word - preprint.doc Preprint version of Nature Photonics 1, 215 (2007) Review: Semiconductor Quantum Light Sources Andrew J Shields Toshiba Research Europe Limited, 260 Cambridge Science Park, Cambridge CB4 0WE, UK Abstract Lasers and LEDs display a statistical distribution in the number of photons emitted in a given time interval. New applications exploiting the quantum properties of light require sources for which either individual photons, or pairs, are generated in a regulated stream. Here we review recent research on single-photon sources based on the emission of a single semiconductor quantum dot. In just a few years remarkable progress has been made in generating indistinguishable single-photons and entangled photon pairs using such structures. It suggests it may be possible to realise compact, robust, LED-like semiconductor devices for quantum light generation. Applications of Quantum Photonics Applying quantum light states to photonic applications allows functionalities that are not possible using ‘ordinary’ classical light. For example, carrying information with single-photons provides a means to test the secrecy of optical communications, which could soon be applied to the problem of sharing digital cryptographic keys.1 2 Although secure quantum key distribution systems based on weak laser pulses have already been realised for simple point-to-point links, true single-photon sources would improve their performance.3 Furthermore, quantum light sources are important for future quantum communication protocols such as quantum teleportation. 4 Here quantum networks sharing entanglement could be used to distribute keys over longer distance or through more complex topologies.5 A natural progression would be to use photons for quantum information processing, as well as communication. In this regard it is relatively straightforward to encode and manipulate quantum information on a photon. On the other hand, single-photons do not interact strongly with one-another, a prerequisite for a simple photon logic gate. In linear optics quantum computing67 (LOQC) this problem is solved using projective measurements to induce an effective interaction between the photons. Here triggered sources of single-photons and entangled pairs are required as both the qubit carriers, as well as auxiliary sources to test the successful operation of the gates. Although the component requirements for LOQC are challenging, they have recently been relaxed significantly by new theoretical schemes. 7 Quantum light states are also likely to become increasingly important for various types of precision optical measurement.8 For these applications we would ideally like light sources which generate pure single-photon states “on demand” in response to an external trigger signal. Key performance measures for such a source are the efficiency, defined as the fraction of photons collected into the experiment or application per trigger, and the second order correlation function at zero delay, see text box. The latter is essentially a measure of the two-photon rate compared to a classical source with random emission times of the same average intensity. In order to construct applications involving more than one photon, it is also important that photons emitted from the source (at different times), as well as those from different sources, are otherwise indistinguishable. In the absence of a convenient triggered single-photon source, most experiments in quantum optics rely on non-linear optical processes for generating quantum light states. Optically pumping a crystal with a χ(2) non-linearity has a finite probability of generating a pair of lower energy photons via parametric down conversion. This may be used to prepare photon pairs with time-bin entanglement,9 entangled polarisations,1011 or alternatively single-photon states ‘heralded’ by the second photon in the pair.12 A χ(3) non-linearity in a semiconductor has also been used to generate entangled pairs.13 As these non-linear processes occur randomly, there is always a finite probability of generating two pairs that increases with pump power. As double pairs degrade the fidelity of quantum optical gates, the pump laser power must be restricted to reduce the rate of double pairs to an acceptable level, which has a detrimental effect upon the efficiency of the source.14 This means that although down-conversion sources continue to be highly successful in demonstrating few photon quantum optical gates, scaling to large numbers may be problematic. Solutions have been proposed based on switching multiple sources,15 or storing photons in a switched fibre loop.16 Ideally we would like a quantum light source that generates exactly one single-photon, or entangled-pair, per excitation trigger pulse. This may be achieved using the emission of a single quantum system. After relaxation, a quantum system is by definition no longer excited and therefore unable to re-emit. Photon anti-bunching, the tendency of a quantum source to emit photons separated in time, was first demonstrated in the resonance fluorescence of a low density vapour of Na atoms,17 and subsequently for a single ion.18 Quantum dots are often referred to as “artificial atoms”, as their electron motion is quantised in all three spatial directions, resulting in a discrete energy level spectrum, like that of an atom. They provide a quantum system which can be grown within robust, monolithic semiconductor devices and can be engineered to have a wide range of desired properties. In the following we review recent progress towards the realisation of a semiconductor technology for quantum photonics. An excellent account of the early work can be found in Ref. 19. Space restrictions limit discussion of work on other quantised systems. For this we refer the reader to the comprehensive review in Ref 20. Optical Properties of Single Quantum Dots Nano-scale quantum dots with good optical properties can be fabricated using a natural growth mode of strained layer semiconductors.21 When InAs is deposited on GaAs it initially grows as a strained two-dimensional sheet, but beyond some critical thickness, tiny islands like those shown in Fig.1a form in order to minimize the surface strain. Overgrowth of the islands leads to the coherent incorporation of InxGa1-xAs dots into the crystal structure of the device, as can be seen in the cross-sectional image of Fig.1c. The most intensively studied are small InAs dots on GaAs emitting around 900-950nm at low temperatures, which can be conveniently measured with low noise Si single photon detectors. A less desirable feature of the self-organising technique is that the dots form at random positions on the growth surface. However, recently considerable progress has been made on controlling the dot position (Fig.1b) within the device structure by patterning nanometer sized pits on the growth surface.2223 As InGaAs has a lower energy bandgap than GaAs, the quantum dot forms a potential trap for electrons and holes. If sufficiently small, the dot contains just a few quantised levels in the conduction and valence bands, each of which holds two electrons or holes of opposite spin. Illumination by a picosecond laser pulse excites electrons and holes which rapidly relax to the lowest lying energy states either side of the bandgap. A quantum dot can thus capture two electrons and two holes to form the biexciton state, which decays by a radiative cascade, as shown schematically in Fig.2a. One of the trapped electrons recombines with one of the holes and generates a first photon (called the biexciton photon, X2). This leaves a single electron-hole pair in the dot (the exciton state), which subsequently also recombines to generate a second (exciton, X) photon. The biexciton and exciton photons have distinct energies, as can be seen in the low temperature photoluminescence spectrum of Fig.2a, due to the different Coulomb energies of their initial and final states. Often a number of other weaker lines can also be seen due to recombination of charged excitons which form intermittently when the dot captures an excess electron or hole.24 Larger quantum dots, with several confined electron and hole levels, have a richer optical signature due to the large number of exciton complexes that can be confined. High resolution spectroscopy reveals that the X2 and X transitions of a dot are in fact both doublets with linearly polarised components parallel to the [110] and [1-10] axes of the semiconductor crystal, labelled here H and V, respectively.2526 The origin of this polarisation is an asymmetry in the electron-hole exchange interaction of the dot which produces a splitting of the exciton spin states. The asymmetry derives from an elongation of the dot along one crystal axis and in-built strain in the crystal. It mixes the exciton eigenstates of a symmetric dot with total z-spin Jz = +1 and -1 into symmetric and anti-symmetric combinations, which couple to two H or two V polarised photons, respectively, as shown in Fig.2. The exciton state of the dot has a typical lifetime of ~1ns, which is due purely to radiative decay. As this is much longer than the duration of the exciting laser pulse, or the lifetime of the photo-excited carrier population in the surrounding semiconductor, only one X photon can be emitted per laser pulse. This can be proven, as first reported27 by Peter Michler, Atac Imamoglu and their colleagues in Santa Barbara, by measuring the second order correlation function, g(2)(τ) of the exciton photoluminescence,2829 see text box. In fact each of the exciton complexes of the dot generates at most one photon per excitation cycle, which allows single-photon emission from also the biexciton or charged exciton transitions.30 Cross-correlation measurements313233 between the X and X2 photons confirm the time correlation expected for the cascade in Fig.2a, ie the X photon follows the X2 one. Indeed the shape of the cross-correlation function for both CW and pulsed excitation can be accurately described with a simple rate equation model and the experimentally measured X and X2 decay rates. 34 Semiconductor Microcavities A major advantage of using self-assembled quantum dots for single-photon generation is that they can be easily incorporated into cavities using standard semiconductor growth and processing techniques. Cavity effects are useful for directing the emission from the dot into an experiment or application, as well as for modifying the photon emission dynamics. 3536 Purcell37 predicted enhanced spontaneous emission from a source in a cavity when its energy coincides with that of the cavity mode, due to the greater density of optical states to emit into. For an ideal cavity, in which the emitter is located at the maximum of the electric field with its dipole aligned with the local electric field, the enhancement in decay rate is given by Fp = (3/4π 2) (λ/n)3 Q/V, where Q is the quality factor, a measure of the time a photon is trapped in the cavity, and V is the effective mode volume. Thus high photon collection efficiency, and simultaneously fast radiative decay, requires small cavities with highly reflecting mirrors and a high degree of structural perfection. However, without controlling the location of the dot in the cavity, as discussed below, it may be difficult to achieve the full enhancement predicted by the Purcell formula. Figure 3 shows images of some of the single quantum dot cavity structures that have proven most successful. Pillar microcavities, formed by etching cylindrical pillars into semiconductor Bragg mirrors placed either side of the dot layer, have shown large Purcell enhancements and have a highly directional emission profile, thus making good single-photon sources.38394041 Purcell factors of around 6 have been measured directly,4041 through the rate of cavity-enhanced radiative decay compared to that of a dot without cavity, implying a coupling to the cavity mode of β=Fp/(1+Fp)>85%, if we assume the leaky modes are unaffected by the cavity. However, the experimentally determined photon collection efficiency, which is a more pertinent parameter for applications, is typically ~10%, due the fact that not all the cavity mode can be coupled into an experiment and scattering of the mode by the rough pillar edges. We can expect that the photon collection efficiency will increase with improvements to the processing technology or new designs of microcavity. Another means of forming a cavity is to etch a series of holes in a suspended slab of semiconductor, so as to form a lateral variation in the refractive index which creates a forbidden energy gap for photonic modes in which light cannot propagate.42 Photons can then be trapped in a central irregularity in this structure: usually an unetched portion of the slab. Such photonic bandgap defect cavities have been fabricated in Si with Q values approaching 106.4344 High quality active cavities have also been demonstrated in GaAs containing InAs quantum dots. 45464748 A radiative lifetime of 86 ps, corresponding to a Purcell factor of Fp~12, has been reported. 47 Very recently a lifetime of 60ps was measured for a cavity in the strong coupling regeme.48 If the Q-value is sufficiently large, the system enters the strong coupling regime where the excitation oscillates coherently between an exciton in the dot and a photon in the cavity. The spectral signature of strong coupling, an anti- crossing between the dot line and the cavity mode, has been observed for quantum dots in pillar microcavities,49 photonic bandgap defect cavities,50 microdisks51 and microspheres.52 It has been demonstrated for atom cavities that strong coupling allows the deterministic generation of single-photons.5354 Single-photon sources in the strong coupling regime can be expected to have very high extraction efficiencies and be time-bandwidth limited.55 Encouragingly single-photon emission has been reported recently for a dot in a strongly coupled pillar microcavity. 56 Another interesting recent development is the ability to locate a single quantum dot within the cavity, as this ensures the largest possible coupling and removes background emission, as well as other undesirable effects, due to other dots in the cavity. Above we discussed techniques to control the dot position on the growth surface. The other way is to position the cavity around the dot. One technique combines micro-photoluminescence spectroscopy to locate the dot position, with in-situ laser photolithography to pattern markers on the wafer surface.57 An alternative involves growing a vertical stack of dots so that their location can be revealed by scanning the wafer surface, 58 as shown in Fig.3. Recently this technique has allowed larger coupling energies for a single dot in a photonic bandgap defect cavity.48 Photon Indistinguishability Cavity effects are important for rendering different photons from the source indistinguishable, which is essential for many applications in quantum information. When two identical photons are incident simultaneously on the opposite input ports of a 50/50 beamsplitter, they will always exit via the same output port, 59 as shown schematically in Fig.4a. This occurs because of a destructive interference in the probability amplitude of the final state in which one photon exits through each output port. The amplitude of the case where both photons are reflected exactly cancels with that where both are transmitted, due to the π/2 phase change upon reflection, provided the two photons are entirely identical. Two-photon interference of two single-photons emitted successively from a quantum dot in a weakly-coupled pillar microcavity was first reported by the Stanford group.60 Fig. 4b shows a schematic of their experiment. Notice the reduction of the co-incidence count rate measured between detectors in either output port, when the two photons are injected simultaneously (Fig.4c). The dip does not extend completely to zero, indicating that the two photons sometime exit the beamsplitter in opposite ports. The measured reduction in co-incidence rate at zero delay of 69%, implies an overlap for the single-photon wavepackets of 0.81, after correcting for the imperfect single-photon visibility of the interferometer. Two-photon interference dips of 66% and 75% have been reported by Bennett et al61 and Vauroutsis et al. 62 Similar results have been obtained for a single dot in a photonic bandgap defect cavity.63 This two-photon interference visibility is limited by the finite coherence time of the photons emitted by the quantum dot,64 which renders them distinguishable. The depth of the dip in Fig.4c depends upon the ratio of radiative decay time to the coherence time of the dot, ie R=2τdecay/τcoh. When unity, the coherence time is limited by radiative decay and the source will display perfect 2-photon interference. The most successful approach thus far has been to extend τcoh by resonant optical excitation of the dot and reduce τdecay using the Purcell effect in a pillar microcavity, to values R~1.5. the future higher visibilities may be achieved with a larger Purcell enhancement, using a single dot cavity in the strong- coupling regime or with electrical gating described in the next section. A source of indistinguishable single-photons was used by Fattal et al to generate entanglement between post-selected pairs. 65 66 This involves simply rotating the polarisation of one of the photons incident on the final beamsplitter in Fig.4a by 90o. By post-selecting the results where the two photons arrive at the beamsplitter at the same time and where there is one photon in each output arm (labelled 1 and 2), the measured pairs should correspond to the Bell state ψ− = 1/√2 (¦H1 V2 > - ¦V1 H2 >) Eq.1 Note that only if the two photons are indistinguishable and thus the entanglement is only in the photon polarisation, are the two terms in Eq1 able to interfere. Analysis of the density matrix published by Fattal et al65 reveals a fidelity of the post-selected pairs to the state in Eq.1 of 0.69, beyond the classical limit of 0.5. This source of entangled pairs has an importance difference to that based on the biexciton cascade described below. Post-selection implies that the photons are destroyed when this scheme succeeds. This is a problem for some quantum information applications such as LOQC, but could be usefully applied to quantum key distribution.65 Single-Photon LEDs An early proposal for an electrical single-photon source by Kim et al67 was based upon etching a semiconductor heterostructure displaying Coulomb blockade. However, the light emission from this etched structure was too weak to allow the second-order correlation function to be studied. Recently encouraging progress has been made towards the realisation of a single-photon source based on quantising a lateral electrical injection current.6869 However the most successful approach so far has been to integrate self-assembled quantum dots into conventional p-i-n doped junctions. In the first report of electrically-driven single-photon emission by Yuan et al,70 the electroluminescence of a single dot was isolated by forming a micron-diameter emission aperture in the opaque top contact of the p-i-n diode. Fig.5a shows an improved emission aperture single-photon LED after Bennett et al, 71 which incorporates an optical cavity formed between a high reflectivity Bragg mirror and the semiconductor/air interface in the aperture. This structure forms a weak cavity, which enhances the measured collection efficiency 10-fold compared to devices without a cavity. 72 Single-photon pulses are generated by exciting the diode with a train of short voltage pulses. The second order correlation function g(2)(τ) of either the X or X2 electroluminescence (Fig.5c) shows the suppression of the zero delay peak indicative of single-photon emission.71 The finite rate of multi-photon pulses is due mostly to background emission from layers other than the dot, which is also seen for non-resonant optical excitation. Electrical contacts also allow the temporal characteristics of the single-photon source to be tailored. By applying a negative bias to the diode between the electrical injection pulses, Bennett et al73 reduced the jitter in the photon emission time <100ps. This allowed the repetition rate of the single-photon source to be increased to 1.07GHz (Fig.5d) while retaining good single- photon emission characteristics (Fig.5e). Electrical gating could provide a technique for producing time-bandwidth- limited single-photons from quantum dots. Another promising approach is to aperture the current flowing through the device.7475 This is achieved by growing a thin AlAs layer within the intrinsic region of the p-i-n junction and later exposing the mesa to wet oxidation in a furnace, converting the AlAs layer around the outer edge of the mesa to insulating Aluminium oxide. By careful control of the oxidation time, a µm-diameter conducting aperture can be formed within the insulating ring of AlOx. Such structures have the advantage of exciting just a single dot within the structure, thereby reducing the amount of background emission. The oxide annulus also confines the optical mode laterally within the structure, potentially allowing high photon extraction efficiency. Altering the nanostructure or materials that comprise the quantum dot allows considerable control over the emission wavelength and other characteristics. Most of the experimental work done so far has concentrated on small InAs quantum dots emitting around 900-950nm, as these have well understood optical properties and can be detected with low noise Si single-photon detectors. On the other hand the shallow confinement potentials of this system means they emit only at low temperatures. At shorter wavelengths optically-pumped single-photon emission has been demonstrated at ~350nm using GaN/AlGaN,76 500nm using CdSe/ZnSSe77 and 682nm InP/GaInP78 quantum dot. The former two systems have been shown to operate at 200K. It is very important for quantum communications to develop sources at longer wavelengths in the fibre optic transmission bands at 1.3 and 1.55µm. This may be achieved using InAs/GaAs heterostructures by depositing more InAs to form larger quantum dots. These larger dots offer deeper confinement potentials than those at 900nm and thus often display room temperature emission.79 Optically pumped single-photon emission at telecom wavelengths has been achieved using a number of techniques to prepare low densities of longer wavelength dots, including a bimodal growth mode in MBE to form low densities of large dots,80 ultra-low growth rate MBE81 and MOCVD.82 Recently, the first electrically-driven single-photon source at a telecom wavelength has been demonstrated.83 Generation of Entangled Photons By collecting both the X2 and X photons emitted by the biexciton cascade, a single quantum dot may also be used as a source of photon pairs. Polarisation correlation measurements on these pairs discovered that the two photons were classically-correlated with the same linear polarisation.848586 This occurs because the cascade can proceed via one of two intermediate exciton spin states, as described above and shown in Fig.2a, one of which couples to two H- and the other two V-polarised photons. The emission is thus a statistical mixture of |HX2HX> and |VX2VX>, although exciton spin scattering during the cascade (discussed below) ensures there are also some cross-polarised pairs. The spin splitting87,88 of the exciton state of the dot distinguishes the H and V polarised pairs and prevents the emission of entangled pairs predicted by Benson et al. 89 If this splitting could be removed, the H and V components would interfere in appropriately designed experiments. The emitted 2-photon state should then be written as a superposition of HH and VV, which can be recast in either the diagonal (spanned by D, A) or circular (σ+, σ-) polarisation bases, ie Φ+ = 1/√2 (¦HX2 HX > + ¦VX2 VX >) = 1/√2 (¦DX2 DX > + ¦ΑX2 ΑX >) = 1/√2 (¦σ+X2 σ X > + ¦σ X >) Eq.2. Equal weighting of the HH and VV terms assumes the source to be unpolarised, as indicated by experimental measurements. Eq.2 suggests that, for zero exciton spin splitting, the biexciton cascade generates entangled photon pairs, similar to those seen for atoms.90 Entanglement of the X or X2 photons was recently observed experimentally for the first time by Stevenson, Young and co-workers,9192 using two different schemes to cancel the exciton spin splitting. An alternative approach by Akopian et al, 93 using dots with finite exciton splitting, post-selects photons emitted in a narrow spectral band where the two polarisation lines overlap. The exciton spin splitting depends on the exciton emission energy, tending to zero for InAs dots emitting close to 1.4eV and then inverting for higher emission energy. 94 95 These correspond to shallow quantum dots for which the carrier wavefunctions extend into the barrier material reducing the electron-hole exchange. Zero splitting can be achieved by either careful control of the growth conditions to achieve dots emitting close to the desired energy, or by annealing samples emitting at lower energy.94 The exciton spin splitting may be continuously tuned by applying a magnetic field in the plane of the dot.96 It has been observed that the signatures of entanglement then appear only when the exciton splitting is close to zero.91 Other promising schemes to tune the exciton splitting are now emerging, including application of strain97 and electric field.9899 Figure 6a plots polarisation correlations reported by Young et al92 for a dot with zero exciton splitting (by control of the growth conditions). Pairs emitted in the same cascade (ie zero delay) shows a very striking positive correlation (co- polarisation) measuring in either, rectilinear or diagonal bases and anti-correlation (cross-polarisation) when measuring in circular basis. This is exactly the behaviour expected for the entangled state of Eq.2. In contrast, a dot with finite splitting shows polarisation correlation for the rectilinear basis only, with no correlation for diagonal or circular measurements, see Figure 6b. The strong correlations seen for all three bases in Fig.6a could not be produced by any classical light source or mixture of classical sources and is proof that the source generates entangled photons. The measured92 two-photon density matrix (Fig.6c) projects onto the expected 1/√2 (¦HX2 HX > + ¦VX2 VX >) state with fidelity (ie probability) 0.702 ± 0.022, exceeding the classical limit (0.5) by 9 standard deviations. Two processes contribute to the ‘wrongly’ correlated pairs which impair the fidelity of the entangled photon source. The first of these is due to background emission from layers in the sample other than the dot. This background emission, which is unpolarised and dilutes the entangled photons from the dot, limited the fidelity observed in the first report91 of triggered entangled photon pairs from a quantum dot and has been subsequently reduced with better sample design.92 The second mechanism, which is an intrinsic feature of the dot, is exciton spin scattering during the biexciton cascade. It is interesting that this process does not seem to depend strongly upon the exciton spin splitting. It may be reduced by suppressing the scattering using resonant excitation or alternatively using cavity effects to reduce the time required for the radiative cascade. Outlook The past several years have seen remarkable progress in quantum light generation using semiconductor devices. However, despite considerable progress many challenges still remain. The structural integrity of cavities must continue to improve, thereby enhancing quality factors. This, combined with the ability to reliably position single dots within the cavity, will further enhance photon collection efficiencies and the Rabi energy in the strong coupling regime. It is also important to realise all the benefits of these cavity effects in more practical electrically-driven sources. Meanwhile bandstructure engineering of the quantum dots will allow a wider range of wavelengths to be accessed for both single and entangled photon sources, as well as structures that can operate at higher temperatures. Techniques for fine tuning the characteristics of individual emitters will also be important. One of the most interesting aspects of semiconductor quantum optics is that we may be able to use quantum dots not only as quantum light emitters, but also as the logic and memory elements which are required in quantum information processing. Although LOQC is scalable theoretically, quantum computing with photons would be much easier with a useful single-photon non-linearity. Such non-linearity may be achieved with a quantum dot in a cavity in the strong coupling regime. Encouragingly strong coupling of a single quantum dot with various type of cavity has already been observed in the spectral domain. Eventually it may even be possible to integrate photon emission, logic, memory and detection elements into single semiconductor chips to form a photonic integrated circuit for quantum information processing. The author would like to thank Mark Stevenson, Robert Young, Anthony Bennett, Martin Ward and Andy Hudson for their useful comments during the preparation of the manuscript and the UK DTI “Optical Systems for Digital Age”, EPSRC and EC Future and Emerging Technologies programmes for supporting research on quantum light sources. TextBox : Photon Correlation Measurements The photon statistics of light can be studied via the second order correlation function, g(2)(τ), which describes the correlation between the intensity of the light field with that after a delay τ and is given by100 This function can be measured directly using the Hanbury-Brown and Twiss101 interferometer, comprising a 50/50 beamsplitter and two single-photon detectors, shown in the figure. For delays much less than the average time between detection events (ie for low intensities), the distribution in the delays between clicks in each of the two detectors is proportional to g(2)(τ). For a continuous light source with random emission times, such as an ideal laser or LED, g(2)(τ)=1. It shows there is no correlation in the emission time of any two photons from the source. A source for which g(2)(τ=0)>1 is described as 'bunched' since there is an enhanced probability of two photons being emitted within a short time interval. Photons emitted by quantum light sources are typically 'anti-bunched', (g(2)(τ=0)<1) and tend to be separated in time. In communication and computing systems, we are interested in pulsed light sources, for which the emission occurs at times defined by an external clock. In this case g(2)(τ) consists of a series of peaks separated by a clock period. For an ideal single-photon source, the peak at zero time delay is absent, g(2)(τ=0)=0; as the source cannot produce more than one photon per excitation period, clearly the two detectors cannot fire simultaneously. The figure shows g(2)(τ) recorded for resonant pulsed optical excitation of the X emission of a single quantum dot in a pillar microcavity. Notice the almost complete absence of the peak at zero delay: the definitive signature of a single- photon source. The weak peak seen at τ=0 demonstrates that the rate of two-photon emission is 50 times less than that of an ideal laser with the same average intensity. The bunching behaviour observed for the finite delay peaks is explained by intermittent trapping of a charge carrier in the dot.102 This trace was taken for quasi-resonant laser excitation of the dot which avoids creating carriers in the surrounding semiconductor. For higher energy laser excitation, the suppression in g(2)(0) is typically reduced indicating occasional 2-photon pulses due to emission from the layers surrounding the dot, but can be minimised with careful sample design. Figure textbox: (a) Schematic of the set-up used for photon correlation measurements, (b) second order correlation function of the exciton emission of a single dot in a pillar microcavity. Figure Captions Figure 1: Self assembled quantum dots (a) Image of a layer of InAs/GaAs self assembled quantum dots recorded in an Atomic Force Microscope (AFM). Each yellow blob corresponds to a dot with typical lateral diameters of 20-30nm and a height of 4-8nm. (b) AFM image23 of a layer of InAs quantum dots whose locations have been seeded by a matrix of nanometer sized pits patterned onto the wafer surface. Under optimal conditions up to 60% of the etch pits contain a single dot (Courtesy of P Atkinson & D A Ritchie, Cambridge). (c) Cross-sectional STM image of an InAs dot inside a GaAs device (Courtesy of P. Koenraad, Eindhoven). Figure 2: Optical spectrum of a quantum dot. (a) Schematic of the biexciton cascade of a quantum dot. (b) Typical photoluminescence spectrum of a single quantum dot showing sharp line emission due to the biexciton X2 and exciton X photon emitted by the cascade. The inset shows the polarisation splitting of the transitions originating from the spin splitting of the exciton level. Figure 3: SEM images of semiconductor cavities, including pillar microcavities (a)56 and (b), microdisk (c)51 and photonic bandgap defect cavities (d)47, (e) and (f).48 (Structures fabricated at Univ Wuerzburg (a), CNRS-LPN (UPR- 20), Marcoussis (b, c, e), Univ Cambridge (d), UCSB/ETHZ Zurich (f)) Figure 4: Two Photon Interference. (a) If the two photons are indistinguishable, the two outcomes resulting in one photon in either arm interfere destructively. This results in the two photons always exiting the beamsplitter together. (b) Schematic of an experiment using two photons emitted successively from a quantum dot, (c) experimental data showing suppression of the co-incidence rate in (b) when the delay between input photons is zero due to two-photon interference.60 (Courtesy of Y Yamamoto, Stanford Univ.) Figure 5: Electrically driven single-photon emission. (a) Schematic of a single-photon LED. (b) Electroluminescence spectra of the device. Notice the spectra are dominated by the exciton X and biexciton X2 lines, which have linear and quadratic dependence on drive current, respectively. Other weak lines are due to charged excitons. (c) second order correlation function recorded for the exciton (i) and biexciton (ii) emission lines, (d) time-resolved electroluminescence from a device operate with a 1.07GHz repetition rate, (e) measured (i) and modelled (ii) second order correlation function of the biexciton electroluminescence at 1.07GHz. (adapted from Refs. 71and 73) Figure 6: Generation of entangled photons by a quantum dot. (a) Degree of correlation measured for a dot with exciton polarisation splitting S=0 µeV in linear (i), diagonal (ii) and circular (iii) polarisation bases as a function of the delay between the X and X2 photons (in units of the repetition cycle). The correlation is defined as the rate of co-polarised pairs minus the rate of cross-polarised pairs divided by the total rate. Notice that the values at finite delay show no correlation, as expected for pairs emitted in different laser excitation cycles. More interesting are the peaks close to zero time delay, corresponding to X and X2 photon emitted from the same cascade. The presence of strong correlations for all three types of measurement for the dot with zero exciton splitting can only be explained if the X and X2 polarisations are entangled. (b) Degree of correlations measured for the dot in (a) subject to in-plane magnetic field so as to produce an exciton polarisation splitting of S=25 µeV. Notice that the correlation in diagonal and circular bases have vanished, indicating only classical correlations at finite splitting. (c) Two-photon density matrix of the device emission in (a). The strong off-diagonal terms appear due to entanglement. (adapted from Ref 92) References 1 Gisin, N., Ribordy, G., Tittel, W. & Zbinden, H. Quantum cryptography. Rev. Mod Physics 74, 145-195 (2001). 2 Dusek, M., Lutkenhaus, N. & Hendrych, M. Quantum Cryptography. Progress in Optics 49, Edt. E. Wolf (Elsevier 2006). 3 Waks, E., Inoue, K., Santori, C., Fattal, D., Vucković, J., Solomono, G. S. & Yamamoto, Y. Secure communication: Quantum cryptography with a photon turnstile. Nature 420, 762-762 (2002). 4 Bouwmeester, D., Pan, J. W., Mattle, K., Eibl, M., Weinfurter, H. & Zeilinger, A. Experimental quantum teleportation. Nature 390, 575–579 (1997). 5 Briegel, H.-J., Dür, W., Cirac, J. I. & Zoller, P. Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication. Phys. Rev. Lett. 81, 5932-5935 (1998). 6 Knill, E., Laflamme, R. & Milburn, G. J. A scheme for efficient quantum computation with linear optics. Nature 409, 46–52 (2001). 7 Kok, P., Munro, W. J., Nemoto, K., Ralph, T. C., Dowling, J. P. & Milburn, G. J. Linear optical quantum computing. Quant- ph/0512071 (2005). 8 Giovannetti, V., Lloyd, S. & Maccone, L. Quantum-enhanced measurements: Beating the standard quantum limit. Science 306, 1330-1336 (2004). 9 Brendel J, Gisin N, Tittel W and Zbinden H, Pulsed Energy-Time Entangled Twin-Photon Source for Quantum Communication Phys. Rev. Lett. 82, 2594 (1999). 10 Shih, Y. H. & Alley, C. O. New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion. Phys. Rev. Lett. 61, 2921–2924 (1988) 11 Ou, Z. Y. & Mandel, L. Violation of Bell’s inequality and classical probability in a two-photon correlation experiment. Phys. Rev. Lett. 61, 50–53 (1988). 12 Fasel, S., Alibart, O., Tanzilli, S., Baldi, P., Beveratos, A., Gisin, N. & Zbinden, H. High quality asynchronous heralded single- photon source at telecom wavelength. New J. Phys. 6, 163 (2004). 13 Edamatsu, K., Oohata, G., Shimizu, R. & Itoh, T. Generation of ultraviolet entangled photons in a semiconductor, Nature 431, 167–170 (2004). 14 Scarani V, Riedmatten H de, Marcikic I, Zbinden H, Gisin N, Eur. Phys. J D 32, 129-138 (2005). 15 Migdall, A., Branning, D. & Casteletto S. Tailoring single-photon and multiphoton probabilities of a single-photon on-demand source. Phys. Rev. A 66, 053805 (2002). 16 Pittman, T.B., Jacobs, B.C. & Franson, J.D. Single-photons on pseudodemand from stored parametric down-conversion. Phys. Rev. A 66, 042303 (2002). 17 Kimble, H. J., Dagenais M. & Mandel L. Photon Antibunching in Resonance Fluorescence. Phys. Rev. Lett. 39, 691-695 (1977). 18 Diedrich F. & Walther H. Nonclassical radiation of a single stored ion. Phys. Rev. Lett. 58, 203-206 (1987). 19 Michler P. et al, in Single Quantum Dots (Springer, Berlin 2003), p315. 20 Lounis B. and Orrit M., Single Photon Sources, Rep. Prog. Phys. 68, 1129 (2005). 21 Bimberg, D., Grundmann, M. & Ledentsov N. N. Quantum Dot Heterostructures (Wiley, Chichester, 1999). 22 Song, H. Z., Usuki, T., Hirose, S., Takemoto, K., Nakata, Y., Yokoyama, N. & Sakuma, Y. Site-controlled photoluminescence at telecommunication wavelength from InAs/InP quantum dots. Appl. Phys. Lett. 86, 113118 (2005). 23 Atkinson P. et al, Site control of InAs quantum dots using ex-situ electron-beam lithographic patterning of GaAs substrates, Jpn. J. Appl. Phys., 45, 2519-2521 (2006). 24 Landin, L., Miller, M. S., Pistol, M.-E., Pryor, C. E. & Samuelson, L. Optical studies of individual InAs quantum dots in GaAs: Few-particle effects. Science 280, 262-264 (1998). 25 Gammon, D., Snow, E. S., Shanabrook, B. V., Katzer, D. S. & Park, D. Fine structure splitting in the optical spectra of single GaAs quantum dots. Phys. Rev. Lett. 76, 3005 (1996). 26 Kulakovskii, V. D., Bacher, G., Weigand, R., Kümmell, T., Forchel, A., Borovitskaya, E., Leonardi, K. & Hommel, D. Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots. Phys. Rev. Lett. 82, 1780-1783 (1999). 27 Michler, P., Kiraz, A., Becher, C., Schoenfeld, W. V., Petroff, P. M., Zhang, L., Hu, E. & Imamoglu, A. A quantum dot single- photon turnstile device. Science 290, 2282-2285 (2000). 28 Santori, C., Pelton, M., Solomon, G., Dale, Y. & Yamamoto, Y. Triggered single-photons from a quantum dot. Phys. Rev. Lett 86, 1502-1505 (2001). 29 Zwiller, V., Blom, H., Jonsson, P., Panev, N., Jeppesen, S., Tsegaye, T., Goobar, E., Pistol, M.-E., Samuelson, L. & Björk, G. Single quantum dots emit single-photons at a time: Antibunching experiments. Appl. Phys. Lett. 78, 2476 (2001). 30 Thompson, R. M., Stevenson, R. M., Shields, A. J., Farrer, I., Lobo, C. J., Ritchie, D. A., Leadbeater, M. L. & Pepper, M. Single- photon emission from exciton complexes in individual quantum dots. Phys. Rev. B 64, 201302 (2001).. 31 Moreau, E., Robert, I., Manin, L., Thierry-Mieg, V., Gérard, J. M. & Abram, I, Quantum cascade of photons in semiconductor quantum dots. Phys. Rev. Lett. 87, 183601 (2001). 32 Regelman, D. V., Mizrahi, U., Gershoni, D., Enhrenfreund, E., Schoenfeld, W. V. & Petroff, P. M. Semiconductor quantum dot: A quantum light source of multicolor photons with tunable statistics. Phys. Rev. Lett. 87, 257401 (2001). 33 Kiraz, A., Falth, S., Becher, C., Gayral, B., Schoenfeld, W. V., Petroff, P. M., Zhang, L., Hu, E. & Imamoglu, A. Photon correlation spectroscopy of a single quantum dot. Phys. Rev. B 65, 161303 (2002). 34 Shields, A. J., Stevenson, R. M., Thompson, R., Yuan, Z. & Kardynal, B. Nano-Physics and Bio-Electronics (Elsevier, Amsterdam 2002). 35 Vahala, K. J. Optical microcavities. Nature 424, 839–846 (2003). 36 Barnes, W. L., Björk, G., Gérard, J. M., Jonsson, P., Wasey, J. A. E., Worthing, P. T. & Zwiller, V. Solid-state single-photon sources: light collection strategies. Euro. Phys. Journal D 18, 197 (2002). 37 Purcell E., Phys. Rev. 69, 681 (1946). 38 Moreau, E., Robert, I., Gérard, J. M., Abram, I., Manin, L. & Thierry-Mieg, V. Single-mode solid-state single-photon source based on isolated quantum dots in pillar microcavities. Appl. Phys. Lett. 79, 2865 (2001) 39 Pelton, M., Santori, C., Vucković, J., Zhang, B., Solomon, G. S., Plant, J. & Yamamoto, Y. Efficient source of single-photons: A single quantum dot in a micropost microcavity. Phys. Rev. Lett. 89 233602 (2002). 40 Vucković, J., Fattal, D., Santori, C., Solomon, G. S., Yamamoto, Y., Enhanced single-photon emission from a quantum dot in a micropost microcavity, Appl. Phys. Lett. 82, 3596 (2003). 41 Bennett, A. J., Unitt D., Atkinson P., Ritchie D. A. & Shields A. J., High performance single-photon sources from photo- lithographically defined pillar microcavities. Opt. Express 13, 50 (2005). 42 Yablonovitch, E., Inhibited spontaneous emission in solid state physics and electronics. Phys. Rev. Lett. 58, 2059-2062 (1987). 43 Song B-S, Noda S, Asano T and Akahane, Y., Ultra-high-Q photonic double heterostructure nanocavity. Nature Materials 4, 207- 210 (2005). 44 Notomi M et al, Appl. Phys. Lett. 88, 041112 (2006). 45 Kress, A., Hofbauer, F., Reinelt, N., Kaniber, M., Krenner, H. J., Meyer, R., Böhm, G. & Finley, J. J. Manipulation of the spontaneous emission dynamics of quantum dots in two-dimensional photonic crystals. Phys. Rev. B 71, 241304 (2005). 46 Englund, D., Fattal, D., Waks, E., Solomon, G., Zhang, B., Nakaoka, T., Arakawa, Y., Yamamoto, Y. & Vucković, J. Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal. Phys. Rev. Lett. 95, 013904 (2005). 47 Gevaux, D. G., Bennett, A. J., Stevenson, R. M., Shields, A. J., Atkinson, P., Griffiths, J., Anderson, D., Jones, G. A. C. & Ritchie, D. A. Enhancement and suppression of spontaneous emission by temperature tuning InAs quantum dots to photonic crystal cavities. Appl. Phys. Lett. 88, 131101 (2006). 48 Hennessy et al, quant-ph/0610034; to be published in Nature on 22 Feb doi:10.1038/nature05586 49 Reithmaier, J. P., Sek, G., Löffler, A., Hofmann, C., Kuhn, S., Reitzenstein, S., Keldysh, L. V., Kulakovskii, V. D., Reinecke, T. L. & Forchel, A. Strong coupling in a single quantum dot−semiconductor microcavity system. Nature 432, 197-200 (2004). 50 Yoshie, T., Scherer, A., Hendrickson, J., Khitrova, G., Gibbs, H. M., Rupper, G, Ell, C., Shchekin, O. B. & Deppeet, D. G. Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity. Nature 432, 200-203 (2004). 51 Peter, E., Senellart, P., Martrou, D., Lemaître, A., Hours, J., Gérard, J. M. & Bloch, J. Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity. Phys. Rev. Lett. 95, 067401 (2005). 52 LeThomas N., Woggon U., Schöps O., Artemyev M.V., Kazes M., Banin U., Nano Lett. 6, 557 (2006). 53 Kuhn A., Hennrich M., Rempe G., Determinsitic single photon source for distributed quantum networking, Phys. Rev. Lett. 89, 067901 (202) 54 McKeever J., Boca A., Boozer A.D., Miller R., Buck J.R., Kuzmich A., Kimble H.J., Determinsitic generation of single photons from one atom trapped in a cavity, Science 303, 1992 (2004). 55 Cui G and Raymer M.G., Quantum efficiency of single photon sources in cavity-QED strong-coupling regime, Optics Express 13, 9660 (2005). 56 Press, D., Goetzinger, S., Reitzenstein, S., Hofmann, C., Loeffler, A., Kamp, M., Forchel, A. & Yamamoto, Y. Photon antibunching from a single quantum dot-microcavity system in the strong coupling regime. Quant-ph/0609193 (2006). 57 Lee K. H. et al, Registration of single quantum dots using cryogenic laser photolithography, Appl. Phys. Lett. 88, 193106 (2006) 58 Badolato A. et al, Deterministic Coupling of single quantum dots to single nanocavity modes, Science 308, 1158 (2005). 59 Hong, C. K., Ou, Z. Y. & Mandel, L. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett. 59, 2044-2046 (1987). 60 Santori, C., Fattal, D., Vucković, J., Solomon, G. S. & Yamamoto, Y. Indistinguishable photons from a single-photon device. Nature 419, 594-597 (2002). 61 Bennett, A. J., Unitt, D., Atkinson, P., Ritchie, D. A. & Shields, A. J. Influence of exciton dynamics on the interference of two photons from a microcavity single-photon source. Opt. Express 13, 7772-7778 (2005). 62 Varoutsis, S., Laurent, S., Kramper, P., Lemaître, A., Sagnes, I., Robert-Philip, I. & Abram, I. Restoration of photon indistinguishability in the emission of a semiconductor quantum dot. Phys. Rev. B 72, 041303 (2005). 63 Laurent, S., Varoutsis, S., Le Gratier, L., Lemaître, A., Sagnes, I., Raineri, F., Levenson, A., Robert-Philip, I. & Abram, I. Indistinguishable single photons from a single quantum dot in two-dimensional photonic crystal cavity. Appl. Phys. Lett. 87, 163107 (2005). 64 Kammerer. C., Cassabois, G., Voisin, C., Perrin, M., Delalande, C., Roussignol, Ph. & Gérard, J. M. Interferometric correlation spectroscopy in single quantum dots. Appl. Phys. Lett. 81, 2737-2739 (2002).. 65 Fattal, D., Diamanti, E., Inoue, K. & Yamamoto, Y. Quantum teleportation with a quantum dot single-photon source. Phys. Rev. Lett. 92, 037904 (2004). 66 Fattal, D., Inoue, K., Vucković, J., Santori, C., Solomon, G. S. & Yamamoto, Y. Entanglement formation and violation of Bell’s inequality with a semiconductor single-photon source. Phys. Rev. Lett. 92, 037903 (2004). 67 Imamoglu, A. & Yamamoto, Y. Turnstile device for heralded single photons: Coulomb blockade of electron and hole tunneling in quantum confined p-i-n heterojunctions. Phys. Rev. Lett. 72, 210 (1994). 68 Cecchini, M., De Simoni, G., Piazza, V., Beltram, Berre, H. E. & Ritchie, D. A. Surface acoustic wave-driven planar light-emitting device. Appl. Phys. Lett. 85, 3020-3022 (2005). 69 Gell, J. R. et al, Surface acoustic wave driven luminescence from a lateral p-n junction. Appl. Phys. Lett. accepted (2006). 70Yuan, Z., Kardynal, B. E., Stevenson, R. M., Shields, A. J., Lobo, C. J., Cooper, K., Beattie, N. S., Ritchie, D. A. & Pepper, M. Electrically driven single-photon source. Science 295, 102–105 (2002). 71 Bennett, A. J., Unitt, D. C., See, P., Shields, A. J., Atkinson, P, Cooper, K. & Ritchie, D. A. A microcavity single-photon emitting diode. Appl. Phys. Lett. 86, 181102 (2005). 72 Abram, I., Robert, I. & Kuszelewicz, R. Spontaneous emission control in semiconductor microcavities with metallic or Bragg mirrors. IEEE. J. Of Quant. Elect. 34, 71-76 (1998) 73 Bennett, A. J., Unitt, D. C., See, P., Shields, A. J., Atkinson, P., Cooper, K. & Ritchie, D. A. Electrical control of the uncertainty in the time of single-photon emission events. Phys Rev B 72, 033316 (2005) 74 Ellis, D., Bennett, A. J., Shields, A. J., Atkinson, P. & Ritchie, D. A. Electrically addressing a single self-assembled quantum dot. Appl. Phys. Lett. 88, 133509 (2006). 75 Lochman, A., Stock, E., Schulz, O, Hopfer, F., Bimberg, D., Haisler, V. A., Toropov, A. I., Bakarov, A. K. & Kalagin, A. K. Electrically driven single quantum dot polarised single photon emitter. Electron. Lett. 42, 774-775 (2006). 76 Kako, S., Santori, C., Hoshino, K., Götzinger, S., Yamamoto, Y. & Arakawa, Y. A gallium nitride single-photon source operating at 200 K. Nature Materials 5, 887-892 (2006). 77 Sebald, K., Michler, P., Passow, T., Hommel, D., Bacher, G. & Forchel, A. Single-photon emission of CdSe quantum dots at temperatures up to 200 K. Appl. Phys. Lett. 81, 2920-2922 (2002). 78 Aichele, T., Zwiller, V. & Benson O. Visible single-photon generation from semiconductor quantum dots. New J. Phys. 6, 90 (2004). 79 Le Ru, E.C., Fack, J. & Murray, R. Temperature and excitation density dependence of the photoluminescence from annealed InAs/GaAs quantum dots. Phys. Rev. B 67, 245318 (2003). 80 Ward, M. B., Karimov, O. Z., Unitt, D. C., Yuan, Z. L., See, P., Gevaux, D. G., Shields, A. J., Atkinson, P. & Ritchie, D. A. On- demand single-photon source for 1.3 µm telecom fiber. Appl. Phys. Lett. 86, 201111 (2005). 81 Zinoni, C., Alloing, B., Monat, C., Zwiller, V., Li, L. H., Fiore, A., Lunghi, L., Gerardino, A., de Riedmatten, H., Zbinden, H. & Gisin, N. Time-resolved and antibunching experiments on single quantum dots at 1300nm. Appl. Phys. Lett. 88, 131102 (2006). 82 Miyazawa T., Takemoto, K., Sakuma, Y., Hirose, S., Usuki, T., Yokoyama, N., Miyazawa, T., Takatsu, M. & Arakawa, Y., Single-Photon Generation in the 1.55-um Optical-Fiber Band from an InAs/InP Quantum Dot, Jpn. J. Appl. Phys., vol 44, no 20, pp. L620-622 (2005) 83 Ward M B, Farrow T, See P, Yuan Z L, Karimov O Z, Bennett A J, Shields A J, Atkinson P, Cooper K, Ritchie D A,, Electrically driven telecommunication wavelength single-photon source Appl. Phys. Lett. 90, 063512 (2007) 84 Stevenson, R. M., Thompson, R. M., Shields, A. J., Farrer, I., Kardynal, B. E., Ritchie, D. A. & Pepper, M. Quantum dots as a photon source for passive quantum key encoding. Phys. Rev. B 66, 081302 (2002). 85 Santori, C., Fattal, D., Pelton, M., Solomon, G. S. & Yamamoto, Y. Polarization-correlated photon pairs from a single quantum dot. Phys. Rev. B 66, 045308 (2002). 86 Ulrich, S. M., Strauf, S., Michler, P., Bacher, G. & Forchel, A. Triggered polarization-correlated photon pairs from a single CdSe quantum dot. Appl. Phys. Lett. 83, 1848–1850 (2003). 87 van Kesteren, H. W., Cosman, E. C., van der Poel, W. A. J. A. & Foxon C. T. Fine structure of excitons in type-II GaAs/AlAs quantum wells. Phys. Rev. B 41, 5283–5292 (1990) 88 Blackwood, E., Snelling, M. J., Harley, R. T., Andrews, S. R. & Foxon C. T. B. Exchange interaction of excitons in GaAs Heterostructures. Phys. Rev. B 50, 14246–14254 (1994) 89 Benson, O., Santori, C., Pelton, M. & Yamamoto, Y. Regulated and entangled photons from a single quantum dot. Phys. Rev. Lett. 84, 2513-2516 (2000). 90 Aspect, A., Grangier, P. & Roger, G. Experimental tests of realistic local theories via Bell’s theorem. Phys. Rev. Lett. 47, 460-463 (1981). 91 Stevenson, R. M., Young, R. J., Atkinson, P., Cooper, K., Ritchie, D. A. & Shields, A. J. A semiconductor source of triggered entangled photon pairs. Nature 439, 179 (2006). 92 Young, R. J., Stevenson, R. M., Atkinson, P., Cooper, K., Ritchie, D. A. & Shields, A. J. Improved fidelity of triggered entangled photons from single quantum dots. New J. Phys. 8, 29 (2006). 93 Akopian, N., Lindner, N. H., Poem, E., Berlatzky, Y., Avron, J. & Gershoni, D. Entangled photon pairs from semiconductor quantum dots. Phys. Rev. Lett. 96, 130501 (2006). 94 Young, R. J., Stevenson, R. M., Shields, A. J., Atkinson, P., Cooper, K., Ritchie, D. A., Groom, K. M., Tartakovskii, A. I. & Skolnick, M. S. Inversion of exciton level splitting in quantum dots. Phys. Rev. B 72, 113305 (2005). 95 Seguin R., Schliwa A., Rodt S., Poetschke K., Pojl U.W. Bimberg D., Size-Dependent Fine-Structure Splitting in Self-Organized InAs/GaAs Quantum Dots, Phys. Rev. Lett. 95, 257402 (2005). 96 Stevenson, R. M., Young, R. J., See, P., Gevaux, D. G., Cooper, K., Atkinson, P., Farrer, I., Ritchie, D. A. & Shields, A. J. Magnetic-field-induced reduction of the exciton polarisation splitting in InAs quantum dots. Phys. Rev. B 73, 033306 (2006). 97 Seidl, S., Kroner, M., Högele, A. & Karrai, K. Effect of uniaxial stress on excitons in a self-assembled quantum dot. Appl. Phys. Lett. 88, 203113 (2006). 98 Geradot, B. D., Seidl, S., Dalgarno, P. A., Warburton, R. J., Granados, D., Garcia, J. M., Kowalik, K., Krebs, O., Karrai, K., Badolato, A. & Petroff, P. M. Manipulating exciton fine-structure in quantum dot with a lateral electric field. Cond-mat/0608711 (2006). 99 Kowalik, K., Lemaître, A., Laurent, S., Senellart, P., Voisin, P. & Gaj, J. A. Influence of an in-plane electric field on exciton fine structure in InAs-GaAs self-assembled quantum dots. Appl. Phys. Lett. 86, 041907 (2005). 100 Walls, D. F. & Milburn, G. J. Quantum Optics (Springer, Berlin, 1994). 101 Hanbury Brown, R. & Twiss, R. Q. A New Type of Interferometer for Use in Radio Astronomy. Phil. Mag. 45, 663 (1954). 102 Santori, C., Fattal, D., Vucković, J, Solomon, G. S., Waks, E. & Yamamoto, Y. Submicrosecond correlations in photoluminescence from InAs quantum dots. Phys. Rev. B 69, 205324 (2004). 500 nm 500 nm Fig. 1 1375 1380 1385 Detection polarisation: Vertical Horizontal Photon Energy (meV) 1378.0 1378.5 1380.0 1380.5 (b)(a) Fig. 2 ground state 500 nm 500nm (c)(a) (b) (e) (f) Fig. 3 (a) (b) Fig. 4 substrate/buffer n-ohmic contact InAs QD insulator Al p-ohmic contact emission n+ Bragg mirror cavity layer contact metal p+ GaAs Semicon/air interface 905 910 915 X X-X x100 0.11µA 12.0µA wavelength (nm) 95.1µ A X+(b) -40 -20 0 20 40 Time (ns) -40 -20 0 20 40 delay (ns) (ii) X (c) (i) -10 -5 5 10 delay (ns) (i) calculated (ii) measured time (ns) Fig. 5 -0.05 (c) Real Part Imaginary Part Errors (magnitude) S = 0µeV delay period (/12.5ns) (a)(i) (ii) (iii) -15 0 15 S = 25µeV (b)(i) (iii) -15 0 15 Fig. 6 detector beamsplitter detector device emission -40 -20 0 20 40 delay, τ [ns] (a) (b) Fig. textbox

0704.0404

To the origin of the difference of FSI phases in $B\to\pi\pi$ and $B\to\rho\rho$ decays

arXiv:0704.0404v1 [hep-ph] 3 Apr 2007 To the origin of the difference of FSI phases in B → ππ and B → ρρ decays A.B. Kaidalov∗ and M.I. Vysotsky† ITEP, Moscow, Russia Abstract The final state interactions (FSI) model in which soft rescattering of low mass intermediate states dominates is suggested. It explains why the strong interaction phases are large in the Bd → ππ channel and are considerably smaller in the Bd → ρρ one. Direct CP asym- metries of Bd → ππ decays which are determined by FSI phases are considered as well. 1 Introduction There are three reasons to study FSI in B decays: to predict (or explain) the pattern of branching ratios, to study strong interactions, and to forsee in what decays direct CPV will be large. In view of this necessity a model for FSI in B decays to two light mesons is suggested and explored in the present paper. The probabilities of three B → ππ and three B → ρρ decays are measured now with good accuracy. The C-averaged branching ratios of these decays are presented in Table 1 [1]. Let us look at the ratio of the charge averaged Bd decay probabilities to the charged and neutral mesons: Br(Bd → ρ+ρ−) Br(Bd → ρ0ρ0) ≈ 20 , Rπ ≡ Br(Bd → π+π−) Br(Bd → π0π0) ≈ 4 . (1) ∗[email protected] †[email protected] http://arxiv.org/abs/0704.0404v1 Table 1 Mode Br(10−6) Mode Br(10−6) Bd → π+π− 5.2± 0.2 Bd → ρ+ρ− 23.1± 3.3 Bd → π0π0 1.3± 0.2 Bd → ρ0ρ0 1.16± 0.46 Bu → π+π0 5.7± 0.4 Bu → ρ+ρ0 18.2± 3.0 C-averaged branching ratios of B → ππ and B → ρρ decays. The large difference of Rρ and Rπ is due to the difference of FSI phases in B → ρρ and B → ππ decays (see below). In Section 2 we will determine the differences of FSI phases of tree amplitudes which describe B → ρρ and B → ππ decays into the states with isospins zero and two from the data presented in Table 1. As a next step we will suggest a mechanism which produces such phases. Once this mechanism is defined it becomes possible to calculate FSI phases of decay amplitudes into states with a definite isospin (not only their differences). A central question is: what intermediate states produce FSI phases in B-meson decays into two light mesons. In the weak decay b → uū(dd̄)d in the rest frame of a heavy quark (which is B-meson rest frame as well) three fast light quarks are produced. Their energies are of the order of MB/3 and momenta are more or less isotropically oriented. The energy of the fourth (spectator) quark is of the order of ΛQCD. This four quark state transforms mainly into multi pi-meson final state with the average pion multiplicity about 9 (this number follows from the experimentally known charged particles multiplicity in e+e− annihilation at Ecm = 3GeV multiplied by 1.5 ∗ 1.5 in order to take neutral pions and third quark jet into account). The total branching ratio of such decays is about 10−2. However such meson state does not transform into the state composed from two light mesons moving into opposite directions with momenta MB/2. What meson state does transform into two light mesons can be understood from the inverse reaction of two light meson scattering at the center of mass energy equal to the mass of B-meson. The produced hadronic state consists of two jets of particles moving in opposite directions. Each jet should originate from a quark-antiquark pair produced in the weak decay of b-quark. The square of invariant mass of a jet which contains spectator quark does not exceed MBΛQCD and is much smaller than M B. The energy of this jet is determined by that of a companion quark and is about MB/2. That is why the square of invariant mass of the second jet also does not exceed MBΛQCD. So for B-decays the mass of a hadron cluster which transforms into light meson in the final state should not exceed 1.5 GeV. Following these arguments in the calculation of the imaginary parts of the decay amplitudes we will take into account only two (relatively light) particle intermediate states for which branching ratios of B-meson are maximal. In Section 3 we will calculate FSI phases of tree amplitudes describing B → ππ decays taking into account ρρ, ππ and πa1 intermediate states which by t(u)-channel exchanges are converted into ππ. We will find that large probability of B → ρ+ρ− decay explains about half of FSI phases of B → ππ decays. Relatively small probability of B → π+π− decay prevents generation of noticeable FSI phase of B → ρρ amplitudes through B → π+π− → ρρ chain. We will demonstrate that the strong interaction phase of the penguin amplitude is opposite to the result of quark loop calculation, which is very important for the value of a direct CPV asymmetry Cπ+π− ≡ C+− discussed in Section 4. Predictions for CPV asymmetries C00 and S00 will be presented in Section 4 as well and the value of the unitarity triangle angle α will be extracted from the experimental data on CPV asymmetry S+−. Subject of rare B decays is an object of intensive study nowadays and an interested reader can find extensive list of references in a recent paper [2]. 2 Phenomenology; |δπ0 − δπ2 | and |δ 0 − δ Let us present B → ππ decay amplitudes in the so-called “t-convention”, in which the penguin amplitude with the intermediate c-quark multiplied by ud + VcbV cd + VtbV td = 0 is subtracted from the decay amplitudes [3]: MB̄d→π+π− = |VubV ∗ud|m2Bfπf+(0) iδπ2 + + e−iγ iδπ0 + V ∗tdVtb eiβPei(δ +δ̃π0 ) , (2) MB̄d→π0π0 = |VubV ∗ud|m2Bfπf+(0) iδπ2 − − e−iγ 1√ iδπ0 − V ∗tdVtb eiβPei(δ +δ̃π0 ) , (3) MB̄u→π−π0 = |VubV ∗ud|m2Bfπf+(0) e−iγA2e , (4) where Vik are the elements of CKM matrix, γ and β are the unitarity triangle angles and we factor out the product m2Bfπf+(0) which appears when the decay amplitudes are calculated in the factorization approximation. A2 and A0 are the absolute values of the decay amplitudes into the states with I = 2 and 0, generated by operators O1 and O2 (tree amplitudes), while P is the absolute value of QCD penguin amplitude (generated by operators O3 − O6 of effective nonleptonic Hamiltonian which describes b quark decays into the states without charm and strange quarks). δπ0 , δ 2 and δ̃ 0 are FSI phases of these three amplitudes, and it is very important for what follows that all of them are different. It is easy to understand why δπ0 is different from δ 2 : strong interaction depends on the isospin and is different for I = 0 and I = 2. For example, there are definitely quark-antiquark resonances with I = 0, while exotic resonances with I = 2 should be made from at least four quarks and their existence is questionable. The reason why δπ0 differs from δ̃ 0 is more sub- tle. Let us consider the intermediate state made from two charged ρ-mesons which contributes to FSI phases: Bd → ρ+ρ− → ππ. ρ+ρ− intermediate state contribution to FSI phases can be large since Br(Bd → ρ+ρ−) is big. Both tree and penguin induced amplitudes get FSI phases through this chain. Its contribution to δπ0 is proportional to (BrBd → ρ+ρ−)T/(BrBd → π+π−)T ≈ (BrBd → ρ+ρ−)/(BrBd → π+π−) ≈ 2.1, while that to δ̃π0 is proportional to (BrBd → ρ+ρ−)P/(BrBd → π+π−)P . How can we determine the penguin contributions to the probabilities of Bd → ρ+ρ− and Bd → π+π−-decays? The most straightforward way suggested in literature is to extract them from the probabilities of Bu → K0∗ρ+ and Bu → K0π+ decays to which tree amplitudes almost do not contribute [4, 5]1: Br(Bd → ρ+ρ−)P = η2 + (1− ρ)2 ]2 τBd Br(K0∗ρ+) ≈ 0.34·10−6 , 1Contribution of tree amplitudes to these decays comes from the rescattering (Bu → K+π0)T , K +π0 → K0π+, and taking into account CKM suppression of the tree am- plitudes of B → Kπ(K∗ρ) decays relative to the penguin amplitudes we can cautiously estimate tree contribution as not more than 10% of penguin one . Br(Bd → π+π−)P = η2 + (1− ρ)2 ]2 τBd Br(K0π+) ≈ 0.59·10−6 , where fρ = 209 MeV and fK∗ = 218 MeV are the vector meson decay constants, λ = 0.23, η = 0.34 and ρ = 0.20 are the CKM matrix parameters in Wolfenstein parametrization [6], fK/fπ = 1.2 and the central values of Br(Bu → K0∗ρ+) = (9.2±1.5)·10−6 and Br(Bu → K0π+) = (23.1±1.0)·10−6 [1] were used. The accuracy of equations (5) and (6) depends on the accuracy of d ↔ s interchange symmetry (U -spin symmetry) of b → d(s) transition amplitudes described by QCD penguin, however when the ratio of (5) to (6) is calculated uncertainty factors partially cancel out and we obtain rather stable result: instead of being enchanced as in the case of the tree amplitude intermediate vector mesons contribution into penguin Bd → π+π− amplitude is suppressed, (δ̃π0 )ρρ ≈ 1/2.8(δπ0 )ρρ. Taking into account that fraction of longitudinally polarized vector mesons produced in Bu → K0∗ρ+ decays is about 50% we get additional suppression of (δ̃π0 )ρρ by factor Finally, phase δπP comes from the imaginary part of the penguin loop with c-quark propagating in it [8]. In order to calculate δπP let us consider corresponding quark diagram. The charm penguin contribution is given by the following expression: P = −Pc(k2) = ) + i 1− 4m , (7) where k is the sum of momenta of two quarks to which gluon radiated from penguin decays: k = p1 + p2. One of these quarks forms π-meson with the spectator quark, so neglecting spectator quark momentum in the rest frame of B-meson we have p1 = ( ). The second quark forms another π-meson with d̄-quark radiated from penguin: p2 = x( ) where 0 < x < 1 is the fraction of π+ momentum carried by u-quark. Substituting k2 = xm2b into (7) and integrating it with the asymptotic quark distribution function in π-meson ϕπ(x) = x(1−x) we obtain the value of δπP which depends on the ratio 4m2c/m b . In particular, for mb = 5.3 GeV and mc = 1.9 GeV (which correspond to the masses of physical states) we obtain δπP ≈ 10o, a small positive value. A nonperturbative calculation of δπP described in Section 3 demonstrates that the sign of δπP can be negative. Our next task is to determine the difference of FSI phases δπ0 − δπ2 (the large value of it is responsible for a relatively small value of Rπ). If we neglect the penguin contribution, then from (2) - (4) we get the following expression: cos(δπ0 − δπ2 ) = B+− − 2B00 + 23 B+− +B00 − 23 , (8) where Bik’s are the C-averaged branching ratios, while τ0/τ+ ≡ τ(Bd)/τ(Bu) = 0.92. Substituting the central values from Table 1 we get |δπ0 − δπ2 | = 48o. Penguin contributions to Bik do not interfere with tree ones because α = π − β − γ is almost equal to π/2. Taking P 2 terms into account with the help of (6) (subtracting 0.59 and 0.30 from the first and the second lines of Table 1 numbers describing B → ππ data correspondingly) we get: |δπ0 − δπ2 | = 37o ± 10o . (9) The accuracy of this 11o decrease of the absolute value of the phases difference is determined by the accuracy of (6) and is not high. In recent paper [2] the global fit of B → ππ and B → πK decay data was made. The tree amplitudes of B → ππ decays were designated in [2] by T for B → π+π− and by C for B → π0π0. According to [2] the difference of FSI phases between C and T equals δC = −58o ± 10o, |C| = 0.37 ± 0.05, |T | = 0.57 ± 0.05 in the units of 104 eV. The phase shift between the isospin amplitudes is determined by these quantities: tan(δ0 − δ2) = 3TC sin(−δC) 2T 2 + TC cos δC − C2 , (10) and substituting the numbers we obtain: δ0 − δ2 = 40o ± 7o , (11) the result very close to (9). However, the same d ↔ s interchange symmetry was used in [2] when relating B → ππ and B → Kπ decays. Fit [2] was made in the same “t-convention” which we use (see the statement at the end of page 3 of the paper [2]: “for simplicity, we will assume ... Ptc = Ptu”), therefore the obtained results can be directly compared with ours. Now let us consider B → ρρ decays. According to BABAR and BELLE results ρ mesons produced in B decays are almost entirely longitudinally polarized (fL(ρ+ρ−) = 0.98± 0.03[9], fL(ρ+ρ0) = 0.91± 0.4 [10], fL(ρ0ρ0) = 0.86 ± 0.12 [11]). For B decays into the longitudinally polarized ρ-mesons we can write formulas analogous to (2) - (4) and we can find FSI phases difference with the help of analog of (8). Substituting the central values of branching ratios of B → ρρ decays from Table 1 we obtain: |δρ0−δ 2 | = 21o. In order to subtract the penguin contribution with the help of (5) we should take into account that in Bu → K0∗ρ+ decays the fraction of the longitudinally polarized vector mesons equals approximately 50% [12], so we should subtract 0.17 · 10−6 in case of decay to ρ+ρ− and 0.08 · 10−6 for decay into ρ0ρ0. In this way we obtain: |δρ0 − δ 2 | = 20o+8 −20o , (12) and the factor 2 difference between (12) and (9) or (11) is responsible for the different patterns of B → ρρ and B → ππ decay probabilities. Let us emphasize that while |δρ0 − δ 2 | being only one standard deviation from zero can be very small this is not so for |δπ0 − δπ2 |. 3 Calculation of the FSI phases of B → ππ and B → ρρ decay amplitudes Among three amplitudes of B → ππ decays (2)–(4) only two are independent. We will calculate FSI phases of B → π+π0 and B → π+π− amplitudes and extract from them FSI phases of amplitudes with a definite isospin. Our task is to take into account the intermediate state contributions into FSI phases. As it was argued in Introduction we should consider only two particle intermediate states with positive G-parity to which B-mesons have relatively large decay probabilities. Alongside with ππ and ρρ there is only one such state: πa1. So we will consider ρρ intermediate state which transforms into ππ by π exchange in t-channel, πa1 intermediate state which transforms into ππ by ρ exchange in t-channel and will take into account the elastic channel B → ππ → ππ as well. This approach is analogous to the FSI consideration performed in paper [13]. However in [13] 2 → 2 scattering amplitudes were considered to be due to elementary particle exchanges in t-channel. For vector particles exchanges s-channel partial wave amplitudes behave as sJ−1 ∼ s0 and thus do not decrease with energy (decaying meson mass). However it is well known that the correct behavior is given by Regge theory: sαi(0)−1. For ρ-exchange αρ(0) ≈ 1/2 and the amplitude decrease with energy as 1/ s. This effect is very spectacular for B → DD → ππ chain with D∗(D∗2) exchange in t-channel: αD∗(0) ≈ −1 and reggeized D∗ meson exchange is damped as s−2 ≈ 10−3 in comparison with elementary D∗ exchange (see for example [14]). For π-exchange, which gives a dominant contribution to ρρ → ππ transition (see below), in the small t region the pion is close to mass shell and its reggeization is not important. We will use Feynman diagram approach to calculate FSI phases from the triangle diagram with the low mass intermediate states X and Y (see Figure 1). Integrating over loop momenta d4k we assume that integrals over masses of intermediate states X and Y decrease rapidly with increase of these masses. Then choosing z axis in the direction of momenta of the produced meson M1 we can transform the integral over k0 and kz into the integral over the invariant masses of clusters of intermediate particles X and Y dk0dkz = dsXdsY (13) and deform integration contours in such a way that only low mass interme- diate states contributions are taken into account while the contribution of heavy states being small is neglected. In this way we get: M Iππ = M XY (δπXδπY + iT XY→ππ) , (14) where M XY are the decay matrix elements without FSI interactions and T J=0XY→ππ is the J = 0 partial wave amplitude of the process XY → ππ (T J = (SJ − 1)/(2i)) which originates from the integral over d2k⊥. For real T (14) coincides with the application of the unitarity condition for the calculation of the imaginary part of M while for the imaginary T the corrections to the real part of M are generated. Let us calculate the imaginary parts of B → ππ decay amplitudes which originate from B → ρρ → ππ chain with the help of unitarity condition 2: ImM(B → ππ) = d cos θ M(ρρ → ππ)M∗(B → ρρ) , (15) where θ is the angle between ρ and π momenta. For small values of θ or t π-exchange in t-channel dominates and the calculation of Feynman diagram 2in this section the phases which originate from CKM matrix elements are omitted. Figure 1: Diagram which describes FSI in the decay of heavy meson MQq into two light mesons M1 and M2. X and Y are the clusters of particles with small invariant masses sX , sY ≤ MQΛQCD, k is 4-momentum of a virtual particle propagating in t-channel. for ρρ → ππ amplitude with the elementary virtual π-meson exchange can be trusted, as it was noted above. It was already stressed that ρ-mesons pro- duced in B-decays are almost entirely longitudinally polarized. That is why we will take into account only longitudinal polarization for the intermediate ρ-mesons and amplitudes of B-decays into ππ and ρLρL are simply related MB+→ρ+ρ0 = − MB+→π+π0 , MBd→ρ+ρ− = − MBd→π+π− . (16) For the amplitude of ρ+ρ0 → π0π+ transition we have: iM(ρ+ρ0 → π0π+) = −i g2ρππ (p1 − k1)2 −m2π +)(k2ρ 0) , (17) where p1, k1 and k2 are ρ +, π0 and π+ momenta. From the width of ρ-meson we get g2ρππ/16π = 2.85. For the longitudinally polarized ρ-mesons in their center of mass system we have: + = k2ρ 0 = − 1 (t−m2π)(1 + ) +m2ρ , (18) where t = (p1 − k1)2. Changing the integration variable in (15) to t with the help of dt = (1− 2 m )d cos θ and introducing formfactor exp(t/µ2) with 3relative negative sign of the amplitudes follows from the expressions for transition formfactors in the factorization approximation, see for example [15]. the parameter µ2 ∼ 1 GeV 2 we obtain: ImMB→π+π0 = + (m2ρ−m g2ρππdt 16πM2B ∗ 4m2ρ (t−m2π)(1 + + 2m2ρ(1 + t−m2π exp(t/µ2) MB→π+π0 . (19) For µ2 = 2m2ρ the contributions of the first two terms in square brackets cancel, while the third term gives: ImMB→π+π0 = − g2ρππ 3.1MB→π+π0 , (20) and from (4) we get: δπ2 (ρρ) = −4.9o . (21) Let us note that in the limit MB → ∞ the ratio Br(Bd → ρρ)/Br(Bd → ππ) grows as M2B, that is why FSI phase δ 2 (ρρ) (and δ 0 (ρρ)) diminishes as 1/MB. The analogous consideration of ρ+ρ− intermediate state leads to the pos- itive FSI phase of Bd → π+π− amplitude which is enhanced relatively to δπ2 (ρρ) according to (16): δπ+−(ρρ) = +5.7 o , (22) and for FSI phase of the amplitude with isospin zero in the linear approxi- mation we get: δπ0 (ρρ) = δ +−(ρρ) + δπ+−(ρρ)− δπ2 (ρρ) . (23) We are able to extract the ratio A2/A0 from that of C-averaged Br(Bd → π+π−), Br(Bd → π0π0) and Br(Bu → π+π0), subtracting penguin contribu- tion as we did deriving (9): B̃+− + B̃00 − 1 , (24) = 0.80± 0.09 , (25) and, finally: δπ0 (ρρ) = 15 o , δπ0 (ρρ)− δπ2 (ρρ) = 20o . (26) In this way we see that B → ρρ → ππ chain generates half of the experi- mentally observed FSI phase difference of B → ππ tree amplitudes. It is remarkable that FSI phases generated by B → ππ → ρρ chain are damped by Br(B → ρ+ρ−, ρ+ρ0)/Br(B → π+π, π+π0) ratios and are a few degrees: 2(ππ) = ∗ δπ2 (ρρ) = −1.4o , δ +−(ππ) = ∗ δπ+−(ρρ) = 1.2o , (A0/A2)ρρ = 1.1 , δ 0(ππ) = 2.9 o , δ 0(ππ)− δ 2(ππ) ≈ 4o . (27) Next we will take into account ππ intermediate state. From Regge analy- sis of ππ elastic scattering we know that good description of the experimental data is achieved when the exchanges of pomeron, ρ and f trajectories in t- channel are taken into account [16]. Pomeron exchange dominates in elastic ππ → ππ scattering at high energies. For αP (0) = 1 the corresponding amplitude T is purely imaginary and the phases of matrix elements do not change [3]. However taking into account that pomeron is ”supercritical”, αP (0) ≈ 1.1, we obtain the phase of the amplitude generated by pomeron exchange 4 which cancels the phases generated by ρ and f exchanges for I = 2. For I = 0 the sum of ρ and f exchanges produces the purely imag- inary amplitude T and the phase of the amplitude M is due to pomeron ”supercriticallity”: δπ0 (ππ) = 5.0 o , δπ2 (ππ) = 0 o . (28) In paper [3] the pomeron exchange amplitude was considered as purely imaginary. As a result though important for branching ratios phase difference δπ0 (ππ)−δπ2 (ππ) was the same (pomeron contribution being universal cancels in the difference of phases) it came mainly from δπ2 (ππ) negative value. In 4The amplitude of 2 → 2 process due to supercritical pomeron exchange is T ∼ (s/s0) αP (t)(1 + exp(−iπαP (t)))/(− sin(παP (t))) = (s/s0)(1+∆)(i + ∆π/2), where in the last expression t = 0 was substituted and αP (0) = 1 +∆ was used (∆ ≈ 0.1). this way result for the absolute value of direct CP-asymmetry Cπ+π− was underestimated, see below. Finally πa1 intermediate state should be accounted for. Large branching ratio of Bd → π±a∓1 -decay ( Br(Bd → π±a∓1 ) = (40 ± 4) ∗ 10−6) is partially compensated by small ρπa1 coupling constant (it is 1/3 of ρππ one). As a result the contributions of πa1 intermediate state (which transforms into ππ by ρ-trajectory exchange in t-channel) to FSI phases equal approximately that part of ππ intermediate state contributions which is due to ρ-trajectory exchange. Assuming that the sign of the πa1 intermediate state contribution into phases is the same as that of elastic channel we obtain: δπ0 (πa1) = 4 o , δπ2 (πa1) = −2o . (29) Summing the imaginary parts of the amplitudes which follow from (21), (26), (28) and (29) we finally obtain: δπ0 = 23 o , δπ2 = −7o , δπ0 − δπ2 = 30o , (30) and the accuracy of these numbers is not high, at the level of 50%. The analogous consideration of the real parts of the loop corrections to B → ππ decay amplitudes leads to the diminishing of the (real) tree am- plitudes by ≈ 30%, and we can explain the experimentally observed value δπ0 − δπ2 ≈ 40o in our model while for ρρ final state the analogous difference is about three times smaller, δ 0 − δ 2 ≈ 15o. Let us estimate the phase of the penguin amplitude δπP considering charmed mesons intermediate states: B → D̄D, D̄∗D, D̄D∗, D̄∗D∗ → ππ 5. In Regge model all these amplitudes are described at high energies by exchanges of D∗(D∗2)-trajectories. An intercept of these exchange-degenerate trajectories can be obtained using the method of [17] or from masses of D∗(2007) – 1− andD∗2(2460) – 2 + resonances, assuming linearity of these Regge-trajectories. Both methodes give αD∗(0) = −0.8÷−1 and the slope α′D∗ ≈ 0.5GeV −2. The amplitude of D+D− → π+π− reaction in the Regge model proposed in papers [18, 19] can be written in the following form: TDD̄→ππ(s, t) = − e−iπα(t)Γ(1− αD∗(t))(s/scd)αD∗(t) , (31) 5These amplitudes are considered as penguin due to the proper combination of CKM matrix elements. where Γ(x) is the gamma function. The t-dependence of Regge-residues is chosen in accord with the dual models and is tested for light (u,d,s) quarks [18]. According to [19] scd ≈ 2.2GeV 2. Note that the sign of the amplitude is fixed by the unitarity in the t- channel (close to the D∗-resonance). The constant g20 is determined by the width of the D∗ → Dπ decay: g20/(16π) = 6.6. Using (14), analog of (15), (31) and the branching ratio Br(B → DD̄) ≈ 2 · 10−4 [20] we obtain the imaginary part of P and comparing it with the contribution of P in B → π+π− decay probability (6) we get δπP ≈ −3.5o6. A smallness of the phase is due to the low intercept of D∗-trajectory. The sign of δP is negative - opposite to the positive sign which was obtained in perturbation theory (7). Since DD̄-decay channel constitutes only ≈ 10% of all two-body charm- anticharm decays of Bd-meson [20] taking these channels into account we can easily get δP ∼ −10o , (32) which may be very important for the interpretation of the experimental data on direct CP asymmetry C+− discussed in the next section. 4 CP asymmetries of Bd(B̄d) → ππ decays The CP asymmetries are given by : Cππ ≡ 1− |λππ|2 1 + |λππ|2 , Sππ ≡ 2Im(λππ) 1 + |λππ|2 , λππ ≡ e−2iβ MB̄→ππ MB→ππ , (33) where ππ is π+π− or π0π0. From (2) for direct CP asymmetry in Bd(B̄d) → π+π− decays we readily obtain: C+− = − sinα[ 2A0 sin(δ0 − δ̃0 − δP ) + A2 sin(δ2 − δ̃0 − δP )]/ cos(δ0 − δ2)− A0P̃ cosα cos(δ0 − δ̃0 − δP )− 6In integration over cos θ the region θ ≪ 1 dominates. In this region representation (31) is valid. − A2P̃√ cosα cos(δ2 − δ̃0 − δP ) + P̃ 2] , (34) where V ∗tdVtb P . (35) In order to make a numerical estimate we should know the ratios A0/A2 and P/A2. The first one is given by (25) while the second one can be extracted from the ratio Br(Bu → K0π+)/Br(Bu → π0π+) assuming d ↔ s invariance of the strong interactions: Br(Bu → K0π+) Br(Bu → π0π+) f 2KP 2|V ∗tsVtb|2 A22|V ∗udVub|2 , (36) = 0.092(0.009) . (37) The numerical values of A0 andA2 are given with good accuracy by factor- ization calculation, while P appears to be 2.5 times larger than factorization result [3]. In view of this the validity of factor fK in (36) which originates from factorization calculation of the penguin amplitude is questinable. If factorization of the penguin amplitudes is not assumed then the ratio fK/fπ in (36) should be replaced by unity. In this way we get 20% larger value of P/A2 in (37) and we will take this value of uncertainty as an estimate of the theoretical accuracy of the determination of P : = 0.21(0.04) , (38) Taking into account that unitarity triangle angle α ≈ 90o and angles δ̃0 and δP are of the order of few degrees from (34) we obtain: C+− ≈ −0.28[1.1 sin(δ0 − δ̃0 − δP ) + sin(δ2 − δ̃0 − δP )] ≈ ≈ −0.56 sin((δ0 + δ2)/2− δ̃0 − δP ) . (39) In order to determine the lower bound on the value of C+− let us suppose that δ0 = 37 o, δ2 = 0 o (we keep the difference δ0− δ2 = 37o, as it follows from the data on B → ππ decay probabilities (9)), and neglect small values of δ̃0 and δP : C+− > −0.18 . (40) Concerning experimental number it could well happen that finally it will be considerably below our bound. In this case the result of nonperturbative calculation of penguin phase will be confirmed. Substituting in (39) δ0 = 30o, δ2 = −7o and δP from (32) we obtain the following central value: C+− = −0.21 . (41) It is instructive to compare the obtained numbers with the value of C+− which follows from the asymmetry ACP (K +π−) if d ↔ s symmetry is sup- posed [21]: C+− = ACP (K Γ(B → K+π−) Γ(B → π+π−) sin(β + γ) sin(γ) = 1.2(−2)(−0.093± 0.015)19.8 sin 82o sin 60o 0.87 = −0.24± 0.04 . (42) Let us note that one factor fπ/fK in the last equation appears from the matrix element of the tree operator, the second one - from the matrix element of the penguin operator. If because of nonfactorization of penguin amplitudes we will omit the factor which appears from the penguin [5], then the numbers in the right-hand sides of (40, 41) and (42) will become 20% smaller. The experimental results obtained by Belle [22] and BABAR [23] are contradictory CBelle+− = −0.55(0.09) , CBABAR+− = −0.21(0.09), (43) Belle number being far below (40) and (41). For direct CP asymmetry in Bd(B̄d) → π0π0 decay from (3) we readily obtain: C00 = − P̃ sinα[A0 sin(δ0 − δ̃0 − δP )− 2A2 sin(δ2 − δ̃0 − δP )]/ A0A2 cos(δ0 − δ2)− A0P̃ cosα cos(δ0 − δ̃0 − δP ) + A2P̃ cosα cos(δ2 − δ̃0 − δP ) + P̃ 2] , (44) C00 ≈ −1.06[0.8 sin(δ0 − δ̃0 − δP )− 1.4 sin(δ2 − δ̃0 − δP )] ≈ −0.6 , (45) considerably smaller than C+−. This unusually large direct CPV (measured by |C00|) is intriguing task for future measurements since the present exper- imental error is too big: exper 00 = −0.36(0.32) . (46) Belle and BABAR agree now on the value of another CPV asymmetry measured in Bd(B̄d) → π+π− decays: Sexper+− = −0.62 ± 0.09 [22, 23]. From this measurement the value of unitarity triangle angle α can be extracted. Neglecting the penguin contribution we get: sin 2αT = S+− , (47) αT = 109o ± 3o . (48) Penguin shifts the value of α. The accurate formula looks like: S+− = [sin 2α( cos(δ0 − δ2))− A2P̃√ sinα cos(δ2 − δ̃0 − δP )− A0P̃ sinα cos(δ0 − δ̃0 − δP )]/ cos(δ0 − δ2)− A0P̃ cosα cos(δ0 − δ̃0 − δP )− A2P̃√ cosα cos(δ2 − δ̃0 − δP ) + P̃ 2] , (49) and since all the phase shifts are not big the values of cosines in (49) are rather stable relative to their variations. For numerical estimates we take δ0 = 30 o, δ2 = −7o and neglect δ̃0 and δP . In this way we get: (α)ππ = 88 o ± 40(exper)± 50(theor) , (50) where the first error comes from uncertainty in S exper +− while the second one comes from that in the value of penguin amplitude, (38). Relatively large theoretical uncertainty in the value of P̃ does not prevent to determine α with good precision. The relative smallness of penguin contribution to B → ρρ decay am- plitudes allow us to determine α with better theoretical accuracy from the experimental measurement of (S+−)ρρ just as it was done in [24]. With the help of (5) we obtain: )ρρ = 0.060(0.012) , (51) where the same 20% uncertainty in extracting penguin amplitude is supposed. Using the ratio (A0/A2)ρρ determined in (27) from the (49) neglecting strong phases (which are much smaller than in the case of B → ππ decays) and taking into account the recent experimental result (S exper +− )ρρ = −0.06± 0.18 [1] we obtain: (α)ρρ = 87 o ± 50(exper)± 10(theor) . (52) Let us point out that considerably larger theoretical error quoted in [4] follows from the larger theoretical uncertainty in the value of penguin ampli- tude assumed in that paper. Our results for α should be compared with the numbers which follow from the global fit of unitarity triangle [6, 7]: αCKMfitter = (99.0+4.0−9.4) o , αUTfit = (93± 4)o . (53) We conclude this section with the prediction for the value of CPV asym- metry S00: S00 = [sin 2α( 2A0A2 cos(δ0 − δ2)) + 2A2P̃√ sinα cos(δ2 − δ̃0 − δP )− A0P̃ sinα cos(δ0 − δ̃0 − δP )]/ 2A0A2 cos(δ0 − δ2)− A0P̃ cosα cos(δ0 − δ̃0 − δP ) + 2A2P̃√ cosα cos(δ2 − δ̃0 − δP ) + P̃ 2] = 0.70± 0.15 , (54) a large asymmetry with the sign opposite to that of S+−. 5 Conclusions FSI appeared to be very important in B → ππ decays. The description of these interactions presented in the paper allows to explain the experimentally observed difference of the ratios of decay proba- bilities to the neutral and charged modes in B → ππ and B → ρρ decays. Rather large absolute value of direct CP asymmetry C+− (if confirmed experimentally) will be a manifestation of the negative sign of penguin FSI phase in accord with nonperturbative calculation and opposite to perturba- tive result. We are grateful to L.V.Akopyan for checking formulas, Jose Ocariz for recommendation to include the result for angle α which follows from CP asymmetry (S+−)ρρ and M.B.Voloshin for useful discussion. This work was supported by Russian Agency of Atomic Energy; A.K. was partly supported by grants RFBR 06-02-17012, RFBR 06-02- 72041-MNTI, INTAS 05-103-7515 and state contract 02.445.11.7424; M.V. was partly supported by grants RFBR 05-02-17203 and NSh-5603.2006.2. References [1] HFAG, http://www.slac.stanford-edu/xorg/hfag. [2] C.-W. Chiang, Y.-F. Zhou, JHEP 0612 (2006) 027. [3] A.B. Kaidalov, M.I. Vysotsky, hep-ph/0603013, accepted in Yad. Fiz. [4] M. Beneke, M. Gronau, J. Rohrer, M. Spranger, Phys.Lett. B638 (2006) 68. [5] M. Gronau, J.L. Rosner, Phys. Lett. B595 (2004) 339. [6] CKM fitter, http://ckmfitter.in2p3.fr. [7] UTfit, http://utfit.roma1.infn.it. [8] M. Bander, D. Silverman and A. Soni, Phys. Rev. Lett. 43, (1979) 242; G.M. Gérard and W.-S. Hou, Phys. Rev. D43, (1991) 2909. [9] B. Aubert et al., BABAR Collaboration, hep-ex/0607098 (2006). [10] B. Aubert et al., BABAR Collaboration, Phys. Rev. Lett. 97 (2006) 261801. [11] B. Aubert et al., BABAR Collaboration, hep-ex/0607097 (2006). [12] J. Zhong et al. Belle Collaboration, Phys. Rev. Lett. 95 (2005) 141801; B. Aubert et al., BABAR Collaboration, Phys. Rev. Lett. 97 (2006) 201801. [13] H-Y. Cheng, C-K. Chua and A.Soni, Phys. Rev. D71 (2005) 014030. [14] A.Deandrea et al., Int. J. Mod. Phys. (2006) 4425. [15] R.Aleksan et al., Phys. Lett. B356 (1995) 95. [16] K.G.Boreskov, A.A.Grigoryan, A.B.Kaidalov, I.I.Levintov, Yad. Fiz. 27, (1978) 813. [17] A.B.Kaidalov, Zeit. fur Phys. C12, (1982) 63. [18] P.E.Volkovitsky, A.B.Kaidalov, Sov.J.Nucl.Phys. 35, (1982) 909. [19] K.G.Boreskov, A.B.Kaidalov, Sov.J.Nucl.Phys. 37, (1983) 100. [20] Review of Particle Physics, W.-M. Yao et al., Journal of Physics G 33, (2006) 1. [21] R.Fleischer, Phys. Lett. B459, (1999) 306. [22] H.Ishino, Belle, talk at ICHEP06, Moscow (2006). [23] B.Aubert et al, BABAR Collaboration, hep-ex/0703016 (2007). [24] M.I.Vysotsky, Yad. Fiz. 69, (2006) 703.

0704.0405

An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries

An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries The Annals of Applied Probability 2007, Vol. 17, No. 2, 741–779 DOI: 10.1214/105051606000000899 c© Institute of Mathematical Statistics, 2007 AN INVARIANCE PRINCIPLE FOR SEMIMARTINGALE REFLECTING BROWNIAN MOTIONS IN DOMAINS WITH PIECEWISE SMOOTH BOUNDARIES1 By W. Kang and R. J. Williams Carnegie Mellon University and University of California, San Diego Semimartingale reflecting Brownian motions (SRBMs) living in the closures of domains with piecewise smooth boundaries are of in- terest in applied probability because of their role as heavy traffic ap- proximations for some stochastic networks. In this paper, assuming certain conditions on the domains and directions of reflection, a per- turbation result, or invariance principle, for SRBMs is proved. This provides sufficient conditions for a process that satisfies the definition of an SRBM, except for small random perturbations in the defining conditions, to be close in distribution to an SRBM. A crucial ingredi- ent in the proof of this result is an oscillation inequality for solutions of a perturbed Skorokhod problem. We use the invariance principle to show weak existence of SRBMs under mild conditions. We also use the invariance principle, in conjunction with known uniqueness results for SRBMs, to give some sufficient conditions for validating approximations involving (i) SRBMs in convex polyhedrons with a constant reflection vector field on each face of the polyhedron, and (ii) SRBMs in bounded domains with piecewise smooth boundaries and possibly nonconstant reflection vector fields on the boundary surfaces. 1. Introduction. Semimartingale reflecting Brownian motions (SRBMs) living in the closures of domains with piecewise smooth boundaries are of interest in applied probability because of their role as heavy traffic diffusion approximations for some stochastic networks. The nonsmoothness of the boundary for such a domain, combined with discontinuities in the oblique directions of reflection at intersections of smooth boundary surfaces, present Received May 2006; revised November 2006. 1Supported in part by NSF Grants DMS-03-05272 and DMS-06-04537. AMS 2000 subject classifications. 60F17, 60J60, 60K25, 90B15, 93E03. Key words and phrases. Semimartingale reflecting Brownian motion, piecewise smooth domain, invariance principle, oscillation inequality, Skorokhod problem, stochastic net- works. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Probability, 2007, Vol. 17, No. 2, 741–779. This reprint differs from the original in pagination and typographic detail. http://arxiv.org/abs/0704.0405v1 http://www.imstat.org/aap/ http://dx.doi.org/10.1214/105051606000000899 http://www.imstat.org http://www.ams.org/msc/ http://www.imstat.org http://www.imstat.org/aap/ http://dx.doi.org/10.1214/105051606000000899 2 W. KANG AND R. J. WILLIAMS challenges in the development of a rigorous theory of existence, uniqueness and approximation for such SRBMs. When the state space is an orthant and the direction of reflection is con- stant on each boundary face, a necessary and sufficient condition for weak existence and uniqueness of an SRBM is known [14]. This condition involves a so-called completely-S condition on the matrix formed by the reflection directions. An invariance principle for such SRBMs was established in [15] and used in [16] to justify heavy traffic diffusion approximations for cer- tain open multiclass queueing networks. Loosely speaking, the invariance principle shows that, assuming uniqueness in law for the SRBM, a process satisfying the definition of the SRBM, except for small perturbations in the defining conditions, is close in distribution to the SRBM. For more general domains with piecewise smooth boundaries, some con- ditions for existence and uniqueness of SRBMs are known. In particular, for convex polyhedrons with a constant direction of reflection on each boundary face, necessary and sufficient conditions for weak existence and uniqueness of SRBMs are known for simple convex polyhedrons (where precisely d faces meet at each vertex in d-dimensions) and sufficient conditions are known for nonsimple convex polyhedrons, see [4]. For a bounded domain that can be represented as a finite intersection of domains, each of which has a C1- boundary and an associated uniformly Lipschitz continuous reflection vector field, sufficient conditions for strong existence and uniqueness were provided by Dupuis and Ishii [6]; in fact, these authors study stochastic differential equations with reflection which include SRBMs. Despite these existence and uniqueness results, a general invariance principle for SRBMs living in the closures of domains with piecewise smooth boundaries has not been proved to date. (We note that for the special case when the directions of reflection are normal, that is, perpendicular to the boundary, there are a number of perturbation results for reflecting Brownian motions. Our emphasis here is on treating a wide range of oblique reflection directions.) Motivated by its potential for use in approximating heavily loaded stochas- tic networks that are more general than open multiclass queueing networks, in this paper, we formulate and prove an invariance principle for SRBMs living in the closures of domains with piecewise smooth boundaries with possibly nonconstant directions of reflection on each of the smooth bound- ary surfaces. An application of the results of this paper to the analysis of an internet congestion control model can be found in [13]. An outline of the current paper is as follows. The definition of an SRBM and assumptions on the domains and direc- tions of reflection are given in Sections 2 and 3, respectively. Some sufficient conditions for these assumptions to hold are provided in Section 3. Section 4 is devoted to proving the main result of this paper, namely, the invari- ance principle. A key element for our proof of this result is an oscillation INVARIANCE PRINCIPLE FOR SRBMS 3 inequality for solutions of a perturbed Skorokhod problem; this inequality, which may be of independent interest, is proved in Section 4.1. In Section 5 we give some applications of the invariance principle. We prove weak ex- istence of SRBMs under the conditions specified in Section 3. We also use the invariance principle, in conjunction with known uniqueness results for SRBMs, to give sufficient conditions for validating approximations involving (i) SRBMs in convex polyhedrons with a constant reflection vector field on each face of the polyhedron, and (ii) SRBMs in bounded domains with piece- wise smooth boundaries and possibly nonconstant reflection vector fields on the boundary surfaces. Beyond its possible use in justifying SRBM approximations for stochastic networks, the invariance principle might be used to justify numerical ap- proximations to SRBMs. A further possible extension of the results stated here would involve an invariance principle for stochastic differential equa- tions with reflection. The oscillation inequality for the perturbed Skorokhod problem and associated criteria for C-tightness described in Sections 4.1 and 4.2 are likely to be useful for this. We have not developed such an ex- tension here as that would involve introduction of extra assumptions that would make the result less relevant for potential applications to stochastic networks. In particular, the approximating processes would involve stochas- tic integrals driven by a Brownian motion, whereas in stochastic network applications, the Brownian motion typically only appears in the limit. 1.1. Notation, terminology and preliminaries. Let N denote the set of all positive integers, that is, N = {1,2, . . .}, R denote the set of real numbers, which is also denoted by (−∞,∞), R+ denote the nonnegative half-line, which is also denoted by [0,∞). For x ∈ R, we write |x| for the absolute value of x, [x] for the largest integer less than or equal to x, x+ for the positive part of x. For any positive integer d, we let Rd denote d-dimensional Euclidean space, where any element in Rd is denoted by a column vector. Let ‖ · ‖ denote the Euclidean norm on Rd, that is, ‖x‖ = ( i=1 x 1/2 for x ∈Rd, and 〈·, ·〉 denote the inner product on Rd, that is, 〈x, y〉= i=1 xiyi, for x, y ∈ Rd. We note that for any x ∈ Rd, ‖x‖ ≤ i=1 |xi|. Let R + denote the positive orthant in Rd, that is, Rd+ = {x ∈ R d :xi ≥ 0,1 ≤ i ≤ d}. Let B(S) denote the Borel σ-algebra on S ⊂ Rd, that is, the collection formed by intersecting all Borel sets in Rd with S. Let dist(x,S) denote the distance between x ∈Rd and S ⊂Rd, that is, dist(x,S) = inf{‖x−y‖ :y ∈ S}, with the convention that dist(x,∅) =∞ for x ∈Rd. Let Ur(S) denote the closed set {x ∈Rd : dist(x,S)≤ r} for any r > 0 and S ⊂Rd, where if S =∅, Ur(S) =∅ for all r > 0. Let Br(x) denote the closed ball {y ∈R d :‖y− x‖ ≤ r} for any x ∈Rd and r > 0. For any set S ⊂Rd, we write S for the closure of S, So for the interior of S and ∂S = S \So. For a finite set S, |S| denotes the number 4 W. KANG AND R. J. WILLIAMS of elements in S. For any v ∈Rd, v′ denotes the transpose of v. Inequalities between vectors in Rd should be interpreted componentwise, that is, if u, v ∈ d, then u ≤ (<)v means that ui ≤ (<)vi for each i ∈ {1, . . . , d}. For any matrix A, let A′ denote the transpose of A. For any function x :R+ → R x(t−) denotes the left limit of x at t > 0 whenever x has a left limit at t; unless explicitly stated otherwise, x(0−)≡ 0, where 0 is the zero vector in Rd. For any function x :R+ → R d, we let ∆x(t) = x(t)− x(t−) for t ∈ R+ when x(t−) exists. We let 0 be the constant deterministic function x :R+ → R such that x(t) = 0 for all t ∈R+. A domain in Rd is an open connected subset of Rd. For each continuously differentiable function f defined on some nonempty domain S ⊂Rd, ∇f(x) is the gradient of f at x ∈ S. For each x ∈ Rd, a neighborhood Vx of x is a bounded domain in Rd that contains x. For any nonempty domain S ⊂Rd, we say that the boundary ∂S of S is C1, if for each x ∈ ∂S there exists a Euclidean coordinate system Cx for R d centered at x, an rx > 0, and a once continuously differentiable function ϕx :R d−1 →R such that ϕx(0) = 0 and S ∩Brx(x) = {z = (z1, . . . , zd) ′ in Cx : zd >ϕx(z1, . . . , zd−1)} ∩Brx(x). Then, for x ∈ ∂S, the inward unit normal to ∂S at z ∈ ∂S ∩Brx(x) is given in the coordinate system Cx by n(z) = (1 + ‖∇ϕx(z1, . . . , zd−1)‖ 2)1/2 (−∇ϕx(z1, . . . , zd−1) where ∇ϕx(z1, . . . , zd−1) = ( , . . . , ∂zd−1 )′(z1, . . . , zd−1). For any nonempty convex set S ⊂Rd, we call a vector n ∈Rd \{0} an inward unit normal vector to S at x ∈ ∂S if ‖n‖= 1 and 〈n, y − x〉 ≥ 0 for all y ∈ S. Note that such a vector need not be unique. All stochastic processes used in this paper will be assumed to have paths that are right continuous with finite left limits (abbreviated henceforth as r.c.l.l.). A process is called continuous if almost surely its sample paths are continuous. We denote by D([0,∞),Rd) the space of r.c.l.l. functions from [0,∞) into Rd and we endow this space with the usual Skorokhod J1-topology (cf. Chapter 3 of [7]). We denote by C([0,∞),R d) the space of continuous functions from [0,∞) into Rd. The Borel σ-algebra on either D([0,∞),Rd) or C([0,∞),Rd) will be denoted by Md. The abbreviation u.o.c. will stand for uniformly on compacts and will be used to indicate that a sequence of functions in D([0,∞),Rd) (or C([0,∞),Rd)) is converg- ing uniformly on compact time intervals to a limit in D([0,∞),Rd) (or C([0,∞),Rd)). Consider W 1,W 2, . . . ,W , each of which is a d-dimensional process (possibly defined on different probability spaces). The sequence {W n}∞n=1 is said to be tight if the probability measures induced by the W n on the measurable space (D([0,∞),Rd),Md) form a tight sequence, INVARIANCE PRINCIPLE FOR SRBMS 5 that is, they form a weakly relatively compact sequence in the space of probability measures on (D([0,∞),Rd),Md). The notation “W n ⇒W” will mean that, as n → ∞, the sequence of probability measures induced on (D([0,∞),Rd),Md) by {W n} converges weakly to the probability measure induced on the same space by W . We shall describe this in words by saying that W n converges weakly (or in distribution) to W as n → ∞. The se- quence of processes {W n}∞n=1 is called C-tight if it is tight, and if each weak limit point, obtained as a weak limit along a subsequence, almost surely has sample paths in C([0,∞),Rd). The following proposition provides a useful criterion for checking C-tightness. Proposition 1.1. Suppose that, for each n ∈N, W n is a d-dimensional process defined on the probability space (Ωn,Fn, Pn). The sequence {W n}∞n=1 is C-tight if and only if the following two conditions hold: (i) For each η > 0 and T ≥ 0, there exists a finite constant Mη,T > 0 such that lim inf 0≤t≤T ‖W n(t)‖ ≤Mη,T ≥ 1− η.(1) (ii) For each ε > 0, η > 0 and T > 0, there exists λ ∈ (0, T ) such that lim sup Pn{wT (W n, λ)≥ ε} ≤ η,(2) where for x ∈D([0,∞),Rd), wT (x,λ) = sup u,v∈[t,t+λ] ‖x(u)− x(v)‖ : 0≤ t < t+ λ≤ T Proof. See Proposition VI.3.26 in [12]. � A d-dimensional process W is said to be locally of bounded variation if all sample paths of W are of bounded variation on each finite time interval. For such a process W , we define V(W ) = {V(W )(t), t≥ 0} such that for each t≥ 0, V(W )(t) = ‖W (0)‖ + sup ‖W (ti)−W (ti−1)‖ : 0 = t0 < t1 < · · ·< tl = t, l≥ 1 A triple (Ω,F ,{Ft, t ≥ 0}) will be called a filtered space if Ω is a set, F is a σ-algebra of subsets of Ω, and {Ft, t≥ 0} is an increasing family of sub-σ-algebras of F , that is, a filtration. Henceforth, the filtration {Ft, t≥ 0} will be simply written as {Ft}. If P is a probability measure on (Ω,F), then 6 W. KANG AND R. J. WILLIAMS (Ω,F ,{Ft}, P ) is called a filtered probability space. A d-dimensional process X = {X(t), t ≥ 0} defined on (Ω,F , P ) is called {Ft}-adapted if for each t ≥ 0, X(t) :Ω→ Rd is measurable when Ω is endowed with the σ-algebra Given a filtered probability space (Ω,F ,{Ft}, P ), a vector µ ∈R d, a d× d symmetric, strictly positive definite matrix Γ, and a probability distribution ν on (Rd,B(Rd)), an {Ft}-Brownian motion with drift vector µ, covariance matrix Γ, and initial distribution ν, is a d-dimensional {Ft}-adapted process defined on (Ω,F ,{Ft}, P ) such that the following hold under P : (a) X is a d-dimensional Brownian motion whose sample paths are almost surely continuous and that has initial distribution ν, (b) {Xi(t)−Xi(0)− µit,Ft, t≥ 0} is a martingale for i= 1, . . . , d, and (c) {(Xi(t)−Xi(0)−µit)(Xj(t)−Xj(0)−µjt)−Γijt,Ft, t≥ 0} is a mar- tingale for i, j = 1, . . . , d. In this definition, the filtration {Ft} may be larger than the one generated by X ; however, for each t≥ 0, under P , the σ-algebra Ft is independent of the increments of X from t onward. The latter follows from the martingale properties of X . If ν = δx, the unit mass at x ∈ R d, we say that X starts from x. 2. Definition of an SRBM. Let G= i∈I Gi be a nonempty domain in d, where I is a nonempty finite index set and for each i ∈ I , Gi is a nonempty domain in Rd. For simplicity, we assume that I = {1,2, . . . , I} and then |I|= I. For each i ∈ I , let γi(·) be a vector valued function defined from Rd into Rd. Fix µ ∈Rd, Γ a d × d symmetric and strictly positive definite covariance matrix and ν a probability measure on (G,B(G)), where B(G) denotes the σ-algebra of Borel subsets of the closure G of G. Definition 2.1 (Semimartingale reflecting Brownian motion). A semi- martingale reflecting Brownian motion (abbreviated as SRBM) associated with the data (G,µ,Γ,{γi, i ∈ I}, ν) is an {Ft}-adapted, d-dimensional pro- cess W defined on some filtered probability space (Ω,F ,{Ft}, P ) such that: (i) P -a.s., W (t) =X(t) + (0,t] γ i(W (s))dYi(s) for all t≥ 0, (ii) P -a.s., W has continuous paths and W (t) ∈G for all t≥ 0, (iii) under P , X is a d-dimensional {Ft}-Brownian motion with drift vector µ, covariance matrix Γ and initial distribution ν, (iv) for each i ∈ I , Yi is an {Ft}-adapted, one-dimensional process such that P -a.s., (a) Yi(0) = 0, (b) Yi is continuous and nondecreasing, INVARIANCE PRINCIPLE FOR SRBMS 7 (c) Yi(t) = (0,t] 1{W (s)∈∂Gi∩∂G} dYi(s) for all t≥ 0. We shall often refer to Y = {Yi, i ∈ I} as the “pushing process” associated with the SRBM W . When ν = δx, we may alternatively say that W is an SRBM associated with the data (G,µ,Γ,{γi, i ∈ I}) that starts from x. We will call (W,X,Y ) satisfying Definition 2.1 an extended SRBM associated with the data (G,µ,Γ,{γi, i ∈ I}, ν). Loosely speaking, an SRBM behaves like a Brownian motion in the inte- rior of the domain G and it is confined to G by instantaneous “reflection” (or “pushing”) at the boundary, where the allowed directions of “reflection” at x ∈ ∂G are convex combinations of the vectors γi(x) for i such that x ∈ ∂Gi. Under the assumptions imposed on G and {γi, i ∈ I} in Sections 3.1 and 3.2 below, at each point on the boundary of G there is an allowed direction of reflection that can be used there which “points into the interior of G.” We end this section by introducing a related set-valued function I(·) and show a key property of it. Definition 2.2. For each x∈Rd, let I(x) = {i ∈ I :x ∈ ∂Gi}. The set-valued function I(·) has the following property called upper semi- continuity on ∂G. Lemma 2.1. For each x ∈ ∂G, there is an open neighborhood Vx of x in d such that I(y)⊂ I(x) for all y ∈ Vx.(4) Proof. We prove this lemma by contradiction. Suppose that the func- tion I(·) does not satisfy (4). Then there is a point x ∈ ∂G such that there is no open neighborhood Vx of x such that I(y)⊂ I(x) for all y ∈ Vx. Since the index set I is finite, there is an index k ∈ I \ I(x) and a sequence of points {yn} ⊂R d such that ‖yn−x‖< and k ∈ I(yn) for each n≥ 1. Hence yn ∈ ∂Gk for all n≥ 1. Since ∂Gk is closed and yn → x as n→∞, we con- clude that x ∈ ∂Gk. This implies that k ∈ I(x), which is a contradiction, as desired. � 3. Assumptions on the domain G and the reflection vector fields {γi}. 3.1. Assumptions on the domain G. We henceforth assume that the domain G satisfies assumptions (A1)–(A3) below. In the case when G is bounded, assumptions (A2)–(A3) follow from assumption (A1) (see Lem- mas A.1 and A.2 in the Appendix for details). If the domain G is a convex polyhedron satisfying assumption (A1), then assumptions (A2)–(A3) hold by Lemma A.3 in the Appendix. 8 W. KANG AND R. J. WILLIAMS (A1) G is a nonempty domain in Rd with representation Gi,(5) where for each i ∈ I , Gi is a nonempty domain, Gi 6=R d, and the boundary ∂Gi of Gi is C 1. For each i ∈ I , we let ni(·) be the unit normal vector field on ∂Gi that points into Gi. (A2) For each ε ∈ (0,1) there exists R(ε) > 0 such that for each i ∈ I , x ∈ ∂Gi ∩ ∂G and y ∈G satisfying ‖x− y‖<R(ε), we have 〈ni(x), y − x〉 ≥−ε‖x− y‖.(6) (A3) The function D : [0,∞)→ [0,∞] defined such that D(0) = 0 and D(r) = sup J 6=∅ (∂Gj ∩ ∂G) Ur(∂Gj ∩ ∂G) for r > 0, satisfies D(r)→ 0 as r→ 0.(8) Remark. Assumption (A2) is reminiscent of the uniform exterior cone condition (cf. [9], page 195). We say that a region G ⊂ Rd satisfies a uni- form exterior cone condition if for each x0 ∈ ∂G, there is a truncated closed right circular cone Vx0 , with nonempty interior and vertex x0, satisfying Vx0 ∩G= {x0}, and the truncated cones Vx0 are all congruent to some fixed truncated closed right circular cone V . By comparing assumption (A2) with the uniform exterior cone condition, we see that assumption (A2) implies the uniform exterior cone condition. On the other hand, under assumption (A1), assumption (A2) is implied by a family of uniform exterior cone condi- tions where for each ε ∈ (0,1), the axis of the truncated closed right circular cone at x ∈ ∂G is along the vector −ni(x) and all of the truncated closed right circular cones are congruent to a truncated closed right circular cone whose height and base radius are R(ε) and R(ε)( 1 − 1)1/2 respectively. Assumption (A2) holds automatically if G is convex. We also note that as- sumption (A2) is strictly weaker than the uniform exterior sphere condition. The definition of the uniform exterior sphere condition is similar to that for the uniform exterior cone condition where a closed ball with x0 on its bound- ary takes the place of the truncated closed right circular cone Vx0 . It can be checked that for the domain G = {(x, y) ∈ R2 :y < |x|α} with α ∈ (1,2), the uniform exterior sphere condition fails to hold, but assumption (A2) holds. In fact, at the point (0,0) ∈R2, there is no r > 0 and y ∈R2 such that Br(y)∩ ∂G= {(0,0)}. INVARIANCE PRINCIPLE FOR SRBMS 9 Remark. For the definition of D(·) in (A3), we adopt the convention that the supremum over an empty set is zero and dist(x,∅) =∞. Since ∂Gi∩ ∂G 6=∅ for at least one i ∈ I , the function D(·) satisfies limr→∞D(r) =∞. Furthermore, D(r1)≤D(r2) whenever r1, r2 ∈ [0,∞) and r1 ≤ r2. Assump- tion (A3) requires that for any nonempty subset J ⊂ I , the intersection of tubular neighborhoods of the boundaries {∂Gj ∩ ∂G : j ∈ J } given by the j∈J Ur(∂Gj ∩ ∂G) “converges” to the intersection of the boundaries given by the set j∈J (∂Gj ∩ ∂G) as r approaches 0. Property (8) need not always hold. For example, let G1 = {(x, y) ∈ R 2 :y < e−x 2/2, x ∈ R} and G2 = {(x, y) ∈ R 2 :y > 0, x ∈ R}. Then ∂G1 ∩ ∂G2 =∅. But for each r > 0, Ur(∂G1)∩Ur(∂G2) 6=∅. Hence D(r) =∞ for each r > 0. 3.2. Assumptions on the reflection vector fields {γi}. We henceforth as- sume that there are vector fields {γi(·), i ∈ I} satisfying assumptions (A4)– (A5) below. (A4) There is a constant L > 0 such that for each i ∈ I , γi(·) is a uni- formly Lipschitz continuous function from Rd into Rd with Lipschitz con- stant L and ‖γi(x)‖= 1 for each x ∈Rd. (A5) There is a constant a ∈ (0,1), and vector valued functions b(·) = (b1(·), . . . , bI(·)) and c(·) = (c1(·), . . . , cI(·)) from ∂G into R + such that for each x ∈ ∂G, i∈I(x) bi(x) = 1, j∈I(x) i∈I(x) bi(x)n i(x), γj(x) ≥ a,(9) i∈I(x) ci(x) = 1, j∈I(x) i∈I(x) ci(x)γ i(x), nj(x) ≥ a.(10) We note here for future use that by (A4), if we set ρ0 = , then for any x, y ∈ Rd satisfying ‖x − y‖ < ρ0, we have ‖γ i(x) − γi(y)‖ < a/4 for each i ∈ I . So for each 0< ρ < ρ0/4, by (9)–(10) and the normalization of b(·), c(·), γi(·), nj(·) for i, j ∈ I , we obtain j∈I(x) y∈B4ρ(x) i∈I(x) bi(x)n i(x), γj(y) ≥ a/2(11) j∈I(x) y∈B4ρ(x) i∈I(x) ci(x)γ i(y), nj(x) ≥ a/2.(12) 10 W. KANG AND R. J. WILLIAMS The use of B4ρ(x) here is related to the form in which this is used in Section Remark. Assumption (A4) is equivalent to (3.4) in [6] whenG is bounded. Property (10) means that, at each point x ∈ ∂G, there is a convex combi- nation γ(x) = i∈I(x) ci(x)γ i(x) of the vectors {γi(x), i ∈ I(x)} that can be used there such that γ(x) “points into” G. Property (9) is in a sense a dual condition to property (10), where the roles of γi and ni are reversed for i ∈ I(x). This property (9) is used in showing the oscillation inequality in Theorem 4.1 below. Assumption (A5) is an analogue of Assumption 1.1 in [4]. When G is bounded, (10) is similar to condition (3.6) in [6] (we as- sume some additional uniformity through the lack of dependence of a on It is straightforward to see using the triangle inequality that the following condition (A5)′ implies (A5). (A5)′ There is a ∈ (0,1) and vector valued functions b, c from ∂G into RI+ such that for each x ∈ ∂G, i∈I(x) bi(x) = 1, and for each i ∈ I(x), bi(x)〈n i(x), γi(x)〉 ≥ a+ j∈I(x)\{i} bj(x)|〈n j(x), γi(x)〉|,(13) i∈I(x) ci(x) = 1, and for each i ∈ I(x), ci(x)〈γ i(x), ni(x)〉 ≥ a+ j∈I(x)\{i} cj(x)|〈γ j(x), ni(x)〉|.(14) Condition (A5)′(ii) is similar to condition (3.8) in [6], although here we assume additional uniformity through the lack of dependence of a on x. As noted in [6], their condition (3.8) can be phrased in terms of a nonsingu- lar M-matrix requirement [2]. (This is sometimes also called a generalized Harrison–Reiman type of condition [10].) Since that nonsingular M-matrix property is invariant under transpose, and this property for the transpose corresponds to a local form of (A5)′(i), one might conjecture that there is an equivalence between the existence of a nonnegative vector valued function b such that (A5)′(i) holds for each x ∈ ∂G and the existence of a nonnegative vector valued function c such that (A5)′(ii) holds for each x ∈ ∂G. Indeed we have the following lemma. We have stated the two (equivalent) condi- tions (i) and (ii) in specifying (A5)′ to preserve a parallel with (A5) and since both properties can be useful in proofs. Furthermore, in light of the following lemma, verifying either condition suffices for both to hold. INVARIANCE PRINCIPLE FOR SRBMS 11 Lemma 3.1. There is a constant a ∈ (0,1) and a vector valued function b :∂G → RI+ such that (A5) ′(i) holds for each x ∈ ∂G if and only if there is a constant a ∈ (0,1) and a vector valued function c :∂G→ RI+ such that (A5)′(ii) holds for each x ∈ ∂G. Proof. We just prove the “if” part; the “only if” part can be proved in a similar manner. We suppose that there is a constant a ∈ (0,1) and a vector valued function c :∂G → RI+ such that (A5) ′(ii) holds for each x ∈ ∂G. For fixed x ∈ ∂G, consider the square matrix A(x) whose diagonal entries are given by the (positive) elements 〈ni(x), γi(x)〉 for i ∈ I(x) and whose off-diagonal entries are given by−|〈ni(x), γj(x)〉| for i ∈ I(x), j ∈ I(x), j 6= i. Let E be the square matrix having the same dimensions as A(x) and whose entries are all equal to one. By the theory of M-matrices (see [2], Chapter 6, especially condition (M35)), condition (ii) of (A5) ′ implies that A(x)− a E is a nonsingular M- matrix, that is, A(x)− a E has nonnegative diagonal entries and nonpositive off-diagonal entries and it can be written in the form s(x)I −B(x) where B(x) is a matrix with nonnegative entries and s(x)> 0 is a constant that is strictly larger than the spectral radius of B(x). Since the nonsingular M-matrix property is invariant under transpose (cf. (G21) in Chapter 6 of [2]), then A ′(x)− a E is also a nonsingular M-matrix. Hence, there is a vector b̃(x) = (b̃i(x) : i ∈ I(x)) with nonnegative entries such that (A′(x)− a E)b̃(x)> 0 (cf. (I27) in Chapter 6 of [2]). We can extend b̃(x) to an I-dimensional vector b(x) and normalize it so that i∈I(x) bi(x) = 1. Then (A5)′(i) holds with a in place of a. � 4. Invariance principle. In this section we state and prove an invariance principle for an SRBM living in the closure of a domain G with piecewise smooth boundary and having associated reflection fields {γi, i ∈ I}, where G, {γi, i ∈ I} satisfy assumptions (A1)–(A5) of Section 3. (These assump- tions hold throughout this section.) We shall first state a preliminary result called an oscillation inequality (see Theorem 4.1), then we use it to prove a tightness result (see Theorem 4.2). Finally, we establish the invariance principle (see Theorem 4.3). 4.1. Oscillation inequality. The following oscillation inequality is the key to the proof of the tightness result claimed in Theorem 4.2. In this subsec- tion, for any 0≤ t1 < t2 <∞ and any integer k ≥ 1, D([t1, t2],R k) denotes the set of functions w : [t1, t2]→R k that are right continuous on [t1, t2) and have finite left limits on (t1, t2]. For w ∈D([t1, t2],R Osc(w, [t1, t2]) = sup{‖w(t)−w(s)‖ : t1 ≤ s < t≤ t2},(15) Osc(w, [t1, t2)) = sup{‖w(t)−w(s)‖ : t1 ≤ s < t < t2}.(16) 12 W. KANG AND R. J. WILLIAMS Note that we do not explicitly indicate the dependence on k in the notation. Recall the constants a,L from assumptions (A4)–(A5), the functions R(·) from assumption (A2) and D(·) from (7). Let ρ0 = Theorem 4.1 (Oscillation inequality). There exists a nondecreasing func- tion Π: (0,∞)→ (0,∞] satisfying Π(u)→ 0 as u→ 0, such that Π depends only on the constants I, a and the function D(·), and such that whenever 0 < ρ <min{ R(a/4) }, 0 < δ < , 0 ≤ s < t < ∞, w,x ∈ D([s, t],Rd) and y ∈D([s, t],RI) satisfy: (i) w(u) ∈Bρ(x0)∩Uδ(G) for all u ∈ [s, t], for some x0 ∈G, (ii) w(u) = w(s) + x(u) − x(s) + (s,u] γ i(w(v))dyi(v) for all u ∈ [s, t], (iii) for each i ∈ I , (a) yi(s)≥ 0, (b) yi is nondecreasing and ∆yi(u)≤ δ for all u ∈ (s, t], (c) yi(u) = yi(s) + (s,u] 1{w(v)∈Uδ(∂Gi∩∂G)} dyi(v) for all u ∈ [s, t], (iv) D(Π(Osc(x, [s, t]) + δ))< then we have that the following hold: Osc(w, [s, t])≤Π(Osc(x, [s, t]) + δ),(17) Osc(y, [s, t])≤Π(Osc(x, [s, t]) + δ).(18) Proof. Let Π0(u) = u for all u > 0. Define Πm : (0,∞)→ (0,∞], m= 1, . . . , I, inductively such that Πm(u) = Πm−1(u) + (I+2)u+ (D(Πm−1(u) + (I+2)u) + 2u). Here the sum of any element of [0,∞) with ∞ is ∞ and D(∞) is defined to equal ∞. For each m= 0,1, . . . , I, the function Πm is nondecreasing and depends only on I, a and D(·). For each m= 1, . . . , I and u > 0, Πm−1(u)≤ Πm(u). By assumption (A3), we conclude (using an induction proof) that Πm(u)→ 0 as u→ 0, for m= 0,1, . . . , I. Let Π(·) = ΠI(·). Fix 0 < ρ < min{ R(a/4) }, 0 < δ < , 0 ≤ s < t < ∞. Suppose that w,x ∈D([s, t],Rd) and y ∈D([s, t],RI) satisfy (i)–(iv) in the statement of Theorem 4.1. For each nonempty interval [t1, t2]⊂ [s, t], let I[t1,t2] = {i ∈ I :w(u) ∈ Uδ(∂Gi ∩ ∂G) for some u ∈ [t1, t2]}, INVARIANCE PRINCIPLE FOR SRBMS 13 the indices of the boundary surfaces that w(·) comes close to in the time interval [t1, t2]. For each 0 ≤ m ≤ I, define Tm = {[t1, t2] ⊂ [s, t] : |I[t1,t2]| ≤ m}. Note that under the partial ordering of set inclusion, Tm increases with m. To prove the theorem, we will prove by induction that for each 0≤m≤ I and each interval [t1, t2] ∈ Tm, (17)–(18) hold with [t1, t2] in place of [s, t] and Πm(·) in place of Π(·). The result for m= I yields the theorem. Suppose that m= 0. Then T0 = {[t1, t2]⊂ [s, t] : |I[t1,t2]|= 0}. Fix an inter- val [t1, t2] ∈ T0. Since I[t1,t2] =∅ and (iii)(c) holds, the function y does not increase on the time interval (t1, t2], that is, yi(t2)− yi(t1) = 0 for all i ∈ I . Then, for t1 ≤ u < v ≤ t2, w(v)−w(u) = x(v)− x(u).(19) So in this case, Osc(w, [t1, t2]) = Osc(x, [t1, t2])≤Osc(x, [t1, t2]) + δ,(20) Osc(y, [t1, t2]) = 0≤Osc(x, [t1, t2]) + δ.(21) Thus, (17)–(18) hold with Π0(·) in place of Π(·) and [t1, t2] in place of [s, t] for each interval [t1, t2] ∈ T0. For the induction step, let 1 ≤ m ≤ I and suppose that (17)–(18) hold with Πm−1(·) in place of Π(·) and [t1, t2] in place of [s, t] for each interval [t1, t2] ∈ Tm−1. Now fix [t1, t2] ∈ Tm. If |I[t1,t2]| ≤ m − 1, then [t1, t2] ∈ Tm−1 and so by the induction assumption we have that (17)–(18) hold with [t1, t2] in place of [s, t] and Πm−1(·) [and hence Πm(·)] in place of Π(·). Thus, it suffices to consider [t1, t2] ⊂ [s, t] such that |I[t1,t2]| =m. For i /∈ I[t1,t2], by (iii)(c), yi(t2)− yi(t1) = 0, and so by (ii), for t1 ≤ u < v ≤ t2, we have w(v)−w(u) = x(v)− x(u) + i∈I[t1,t2] (u,v] γi(w(r))dyi(r).(22) Let Πm(u) = Πm−1(u) + (I + 2)u for all u > 0, and η = Osc(x, [t1, t2]) + δ. For any M ∈ (0,∞] and any nonempty set J ⊂ I , let FMJ = {z ∈R d : dist(z, ∂Gi ∩ ∂G)<M for all i ∈ J }. Note that FMJ = ∅ when there is an i ∈ J such that ∂Gi ∩ ∂G = ∅. Since Πm(·)≤Πm(·)≤Π(·), D(·) and Π(·) are nondecreasing, and Osc(x, [t1, t2])≤ Osc(x, [s, t]), we have by (iv) that D(Πm(η))≤D(Πm(η))≤D(Π(η))< .(23) Note that this implies Πm(η)<∞ since D(∞) =∞. We now consider two cases. 14 W. KANG AND R. J. WILLIAMS Case 1. Suppose that w(r) ∈ F Πm(η) I[t1,t2] for all r ∈ [t1, t2]. Fix u, v such that t1 ≤ u < v ≤ t2. Since we have that w(v) ∈ j∈I[t1,t2] Πm(η) (∂Gj ∩ ∂G), by the definition of D(·) and (23), there is z ∈ j∈I[t1,t2] (∂Gj ∩ ∂G) such ‖w(v)− z‖ ≤D(Πm(η))< .(24) For each r ∈ [t1, t2], by (i) we have that w(r) ∈ Uδ(G), and so there is z such that ‖w(r)− zr‖ ≤ 2δ. Hence by (i) and (24) we have ‖zr − z‖ ≤ ‖zr −w(r)‖+ ‖w(r)− x0‖+ ‖x0 −w(v)‖+ ‖w(v)− z‖ ≤ 2δ + ρ+ ρ+ ρ/2< 4ρ < R(a/4) ‖w(r)− z‖ ≤ ‖w(r)− x0‖+ ‖x0 −w(v)‖+ ‖w(v)− z‖ ≤ ρ+ ρ+ ρ/2< 4ρ. By (6) and (25) we have 〈nj(z), z − zr〉 ≤ ‖z − zr‖ for each j ∈ I(z) and r ∈ [t1, t2].(27) Note that I(z) ⊃ I[t1,t2]. Recalling the definition of b(·) from assumption (A5), on dotting the vector j∈I(z) bj(z)n j(z) with both sides of (22) and rearranging, we obtain i∈I[t1,t2] (u,v] j∈I(z) bj(z)n j(z), γi(w(r)) dyi(r) j∈I(z) bj(z)〈n j(z),w(v)−w(u)〉(28) j∈I(z) bj(z)〈n j(z), x(v)− x(u)〉. So by (11), (22), (24)–(28), and the fact that j∈I(z) bj(z) = 1, bj(z)≥ 0 for j ∈ I , we have i∈I[t1,t2] (yi(v)− yi(u)) INVARIANCE PRINCIPLE FOR SRBMS 15 i∈I[t1,t2] (u,v] j∈I(z) bj(z)n j(z), γi(w(r)) dyi(r) j∈I(z) bj(z)〈n j(z),w(v)− z〉+ j∈I(z) bj(z)〈n j(z), z − zu〉 j∈I(z) bj(z)〈n j(z), zu −w(u)〉 − j∈I(z) bj(z)〈n j(z), x(v)− x(u)〉 ≤D(Πm(η)) + ‖z − zu‖+ 2δ + ‖x(v)− x(u)‖ ≤D(Πm(η)) + 2δ + ‖x(v)− x(u)‖ (‖z −w(v)‖+ ‖w(v)−w(u)‖+ ‖w(u)− zu‖) ≤D(Πm(η)) + 2δ + ‖x(v)− x(u)‖ D(Πm(η)) + ‖x(v)− x(u)‖+ i∈I[t1,t2] (yi(v)− yi(u)) + 2δ {D(Πm(η)) + 2δ + ‖x(v)− x(u)‖}+ i∈I[t1,t2] (yi(v)− yi(u)). Hence i∈I[t1,t2] (yi(v)− yi(u))≤ {D(Πm(η)) + 2δ + ‖x(v)− x(u)‖} {D(Πm(η)) + 2η}. On multiplying through by 4 , we obtain i∈I[t1,t2] (yi(v)− yi(u))≤ {D(Πm(η)) + 2η} ≤Πm(η).(29) Hence, by (29) and the fact that for any x ∈Rd, ‖x‖ ≤ i=1 |xi|, we have Osc(y, [t1, t2])≤Πm(Osc(x, [t1, t2]) + δ),(30) and by (22), (29) and the definitions of Πm(·) and Πm(·), we have Osc(w, [t1, t2])≤Osc(x, [t1, t2]) + {D(Πm(η)) + 2η} ≤Πm(Osc(x, [t1, t2]) + δ), as desired. 16 W. KANG AND R. J. WILLIAMS Case 2. Suppose that there is t3 ∈ [t1, t2] such that w(t3) /∈ F Πm(η) I[t1,t2] Define σ = inf{u ∈ [t1, t2] :w(u) /∈ F Πm(η) I[t1,t2] }. Then σ ≤ t2. For each u ∈ [t1, σ), w(u) ∈ F Πm(η) I[t1,t2] and so by a similar analysis to that for Case 1, we obtain for each v ∈ [t1, σ), Osc(w, [t1, v])≤ η+ (D(Πm(η)) + 2η) Osc(y, [t1, v])≤ (D(Πm(η)) + 2η). By the right continuity of paths we have w(σ) /∈ F Πm(η) I[t1,t2] . Then there is an i ∈ I[t1,t2] such that dist(w(σ), ∂Gi ∩ ∂G) ≥ Πm(η), and it follows that w does not reach Uδ(∂Gi ∩ ∂G) during the interval [σ, t2]. To see this, let τ = inf{u ∈ [σ, t2] : dist(w(u), ∂Gi ∩ ∂G) ≤ δ} with the convention that the infimum of an empty set is ∞. If τ ≤ t2, then by the right continuity of w(·) and since Πm(η)> δ, we have τ > σ and dist(w(τ), ∂Gi∩∂G)≤ δ. Also, since |I[t1,t2]|=m, we have [σ,u] ∈ Tm−1 for each u ∈ [σ, τ). By the induction assumption and letting u→ τ , we have ‖w(τ−)−w(σ)‖ ≤Πm−1(η). By (ii), (iii)(b) and since ‖γi(·)‖= 1, we have ‖∆w(τ)‖ ≤ ‖∆x(τ)‖+ ∆yi(τ)≤Osc(x, [t1, t2]) + Iδ ≤ Iη. Then simple manipulations yield dist(w(σ), ∂Gi ∩ ∂G)≤ ‖w(σ)−w(τ−)‖+ ‖∆w(τ)‖+ dist(w(τ), ∂Gi ∩ ∂G) ≤Πm−1(η) + Iη+ δ <Πm(η). This contradicts the fact that dist(w(σ), ∂Gi∩∂G)≥Πm(η), and so confirms that w does not reach Uδ(∂Gi ∩ ∂G) in [σ, t2]. Thus we must have [σ, t2] ∈ Tm−1. Hence we have by the induction assumption that Osc(w, [t1, t2])≤ sup v∈[t1,σ) Osc(w, [t1, v]) + ‖∆w(σ)‖+Osc(w, [σ, t2]) ≤ η + (D(Πm(η)) + 2η) + Iη+Πm−1(η) ≤Πm(Osc(x, [t1, t2]) + δ) INVARIANCE PRINCIPLE FOR SRBMS 17 Osc(y, [t1, t2])≤ sup v∈[t1,σ) Osc(y, [t1, v]) + ‖∆y(σ)‖+Osc(y, [σ, t2]) (D(Πm(η)) + 2η) + Iη+Πm−1(η) ≤Πm(Osc(x, [t1, t2]) + δ). On combining all of the cases above, we have Osc(w, [t1, t2])≤Πm(Osc(x, [t1, t2]) + δ),(31) Osc(y, [t1, t2])≤Πm(Osc(x, [t1, t2]) + δ).(32) This completes the induction step. � Remark. The proof of the above theorem was inspired by the proof of Lemma 4.3 of [4]. Because of the condition (i) in Theorem 4.1, the oscilla- tion inequality given here is localized. Similar, but nonlocalized, oscillation inequalities were proved in [15] when G = Rd+ and in [3] for a sequence of convex polyhedrons; in these cases, the direction of reflection was constant on each boundary face. 4.2. C-tightness result. Throughout this subsection and the next, we suppose that the following assumption holds in addition to (A1)–(A5). Assumption 4.1. There is a sequence of strictly positive constants {δn}∞n=1 such that for each positive integer n, there are processes W n, W̃ n,Xn, αn having paths in D([0,∞),Rd) and processes Y n, Ỹ n, βn having paths in D([0,∞),RI) defined on some probability space (Ωn,Fn, Pn) such that: (i) Pn-a.s., W n = W̃ n + αn and W̃ n(t) ∈Uδn(G) for all t≥ 0, (ii) Pn-a.s., W n(t) =Xn(t)+ (0,t] γ i,n(W n(s−),W n(s))dY ni (s) for all t≥ 0, where for each i ∈ I , γi,n :Rd ×Rd → Rd is Borel measurable and ‖γi,n(y,x)‖= 1 for all x, y ∈Rd, (iii) Y n = Ỹ n+βn, where βn is locally of bounded variation and Pn-a.s., for each i ∈ I , (a) Ỹ ni (0) = 0, (b) Ỹ ni is nondecreasing and ∆Ỹ i (t)≤ δ n for all t > 0, (c) Ỹ ni (t) = (0,t] 1{W̃n(s)∈Uδn (∂Gi∩∂G)} dỸ ni (s), (iv) δn → 0 as n → ∞, and for each ε > 0, there is ηε > 0 and nε > 0 such that for each i ∈ I , ‖γi,n(y,x)− γi(x)‖< ε whenever ‖x− y‖< ηε and n≥ nε, 18 W. KANG AND R. J. WILLIAMS (v) αn → 0 and V(βn)→ 0 in probability, as n→∞, (vi) {Xn} is C-tight. Remark. A simple case in which (iv) above holds is where γi,n(y,x)≡ γi(y). In (v), V(βn) is the total variation process for βn (cf. Section 1.1). The following theorem will play an important role in the proof of the in- variance principle. It will be used to show that a sequence of processes sat- isfying suitably perturbed versions of the defining conditions for an SRBM [cf. (i)–(vi) above] is C-tight. Theorem 4.2 (C-tightness). Suppose that Assumption 4.1 holds. Define Zn = (W n,Xn, Y n) for each n. Then the sequence of processes {Zn}∞n=1 is C-tight. Remark. Note that C-tightness of {W n}, {Xn} and {Y n} implies C- tightness of {Zn} (see Chapter VI, Corollary 3.33 of [12] for details). Proof of Theorem 4.2. References here to (i)–(vi) are to the condi- tions in Assumption 4.1. Simple algebraic manipulations yield Pn-a.s., W̃ n(t) = X̃n(t) + (0,t] γi,n(W n(s−),W n(s))dỸ ni (s)(33) = X̃n(t) + Ṽ n(t) + (0,t] γi(W̃ n(s))dỸ ni (s),(34) where X̃n(t) =Xn(t) + −αn(t) + (0,t] γi,n(W n(s−),W n(s))dβni (s) Ṽ n(t) = (0,t] (γi,n(W n(s−),W n(s))− γi(W n(s)))dỸ ni (s) (0,t] (γi(W n(s))− γi(W̃ n(s)))dỸ ni (s). The hypotheses on αn, the total variation process V(βn) of βn, and the fact that ‖γi,n(y,x)‖ = 1 for all x, y ∈ Rd and each i ∈ I , imply that the process −αn(·) + (0,·] γi,n(W n(s−),W n(s))dβni (s) INVARIANCE PRINCIPLE FOR SRBMS 19 converges to 0 in probability as n→∞. Combining this with the fact that {Xn}∞n=1 is C-tight, we obtain that {X̃ n}∞n=1 is C-tight. Recall the positive nondecreasing function Π(·) from Theorem 4.1, and the constants a, L and functions R(·) and D(·) from assumptions (A1)–(A5) in Section 3. Recall also that ρ0 = Fix ρ, ε, η, T such that 0 < ρ <min{ R(a/4) }, ε > 0, η > 0 and T > 0. By assumption (A3), there is a constant r1 > 0 such that D(r)<min for all r ∈ (0, r1].(37) Since Π(u) → 0 as u → 0, there are constants 0 < r3 < r2 < min{r1, such that Π(r)< for all r ∈ (0, r3].(38) By (iv), there are 0< ε̃ <min{ } and n0 > 0 such that for all n≥ n0, ‖y−x‖<2ε̃ ‖γi,n(y,x)− γi(x)‖< .(39) By (iv)–(vi), and Proposition 1.1, there exist an integer n1 > n0, a con- stant M̃η,T > 0 and λ̃ ∈ (0, T ), such that for all n≥ n1, 0≤s≤T ‖X̃n(s)‖ ≤ M̃η,T ≥ 1− η/2,(40) Pn{wT (X̃ n, λ̃)≥ ε̃} ≤ η/4,(41) 0≤s≤T ‖αn(s)‖< 6ILr2 ≥ 1− η/4,(42) δn <min 8(1 + I) .(43) To prove C-tightness of {W̃ n} and {Ỹ n} (and hence of {W n}, {Y n}), by Proposition 1.1, it suffices to show that there exists a constant Nη,T > 0 such that for all n≥ n1, Pn{wT (W̃ n, λ̃)≥ ε} ≤ η,(44) Pn{wT (Ỹ n, λ̃)≥ ε} ≤ η,(45) 0≤s≤T ‖W̃ n(s)‖ ≤Nη,T ≥ 1− η,(46) 0≤s≤T ‖Ỹ n(s)‖ ≤Nη,T ≥ 1− η.(47) 20 W. KANG AND R. J. WILLIAMS For each n≥ 1, let Fn be a set in Fn such that Pn(Fn) = 1 and on Fn, properties (iii)(a)–(c) hold, (33)–(36) hold, and W̃ n(t) ∈ Uδn(G) for all t≥ 0. Fix a t such that 0≤ t < t+ λ̃≤ T . Let τn = inf{s≥ t :W̃ n(s) ∈ Uδn(∂Gi ∩ ∂G) for some i ∈ I}.(48) For each n≥ n1, let wT (X̃ n, λ̃)< ε̃, sup 0≤s≤T ‖αn(s)‖< 6ILr2 0≤s≤T ‖X̃n(s)‖ ≤ M̃η,T Then by (40)–(42) and the definition of Fn, P{Hn} ≥ 1− η.(50) Fix ωn ∈Hn. By the definition of wT (x,λ) in (3), we have that, r,s∈[t,t+λ̃] ‖X̃n(s,ωn)− X̃n(r,ωn)‖< ε̃.(51) Now there are two cases to consider for n ≥ n1 and u, v fixed such that t≤ u < v ≤ t+ λ̃. Case 1. ωn ∈ {τn > v}. In this case, by (iii)(c), Ỹ n(·, ωn) does not increase on the interval (u, v], that is, Ỹ ni (v,ω n)− Ỹ ni (u,ω n) = 0 for all i ∈ I . Then by (34) and (36), W̃ n(v,ωn)− W̃ n(u,ωn) = X̃n(v,ωn)− X̃n(u,ωn).(52) Hence, by (51), ‖W̃ n(v,ωn)− W̃ n(u,ωn)‖ ≤ sup r,s∈[t,t+λ̃] ‖X̃n(s,ωn)− X̃n(r,ωn)‖< ε̃ < ε/8, and we also have ‖Ỹ n(v,ωn)− Ỹ n(u,ωn)‖= 0< ε/2. Case 2. ωn ∈ {τn ≤ v}. Then there is an i ∈ I such that W̃ n(τn, ωn) ∈ Uδn(∂Gi ∩ ∂G), since the set Uδn(∂Gi ∩ ∂G) is closed and W̃ n(·, ωn) is right continuous. It follows that there is some x0 ∈ ∂G (which depends on ω such that W̃ n(τn, ωn) is in the closed ball Bδn(x0)⊂Bρ(x0). To apply the INVARIANCE PRINCIPLE FOR SRBMS 21 oscillation inequality in Theorem 4.1, we first prove the following: W̃ n(r,ωn) ∈Bρ(x0) for all r satisfying τ n ≤ r ≤ v.(53) For the proof of (53), let ξn = inf{r ≥ τn :W̃ n(r,ωn) /∈Bρ(x0)} ∧ v.(54) By the definition of ξn, W̃ n(r,ωn) ∈Bρ(x0) for each r ∈ [τ n, ξn). In order to apply the oscillation inequality in Theorem 4.1 on the time interval [τn, ξn), we show that D(Π(Osc(X̃n(·, ωn) + Ṽ n(·, ωn), [τn, ξn)) + δn))< .(55) For each r ∈ (0, T ], by (i)–(iii) and (33), (49), (43), we have that ‖W n(r−, ωn)−W n(r,ωn)‖ ≤ ‖W̃ n(r−, ωn)− W̃ n(r,ωn)‖+ ‖αn(r−, ωn)−αn(r,ωn)‖ ≤ ‖∆X̃n(r,ωn)‖+2 sup 0≤s≤T ‖αn(s)‖+ Iδn ≤ ε̃+ < 2ε̃. Hence by (39), for each r ∈ (0, T ], ‖γi,n(W n(r−, ωn),W n(r,ωn))− γi(W n(r,ωn))‖ ≤ .(56) By (36), (56), Assumption (A4), (i) and (49), we have that for any s1, s2 such that u≤ s1 < s2 ≤ v, ‖Ṽ n(s2, ω n)− Ṽ n(s1, ω (s1,s2] ‖γi,n(W n(r−, ωn),W n(r,ωn)) − γi(W n(r,ωn))‖dỸ ni (r,ω (s1,s2] ‖γi(W n(r,ωn))− γi(W̃ n(r,ωn))‖dỸ ni (r,ω (Ỹ ni (s2, ω n)− Ỹ ni (s1, ω (s1,s2] L‖W n(r,ωn)− W̃ n(r,ωn)‖dỸ ni (r,ω (Ỹ ni (s2, ω n)− Ỹ ni (s1, ω 6ILr2 (Ỹ ni (s2, ω n)− Ỹ ni (s1, ω 22 W. KANG AND R. J. WILLIAMS ‖Ỹ n(s2, ω n)− Ỹ n(s1, ω σn = inf{s≥ τn :Osc(Ỹ n(·, ωn), [τn, s))> r2}.(58) Note that Osc(Ỹ n(·, ωn), [τn, s)) as a function of s defined on (τn,∞) is left continuous with finite right limits and is nondecreasing. By the right continuity of Ỹ n, we know that Osc(Ỹ n(·, ωn), [τn, s))→ 0 as s ↓ τn. Thus, σn > τn, Osc(Ỹ n(·, ωn), [τn, σn))≤ r2 and on {σ n <∞}, Osc(Ỹ n(·, ωn), [τn, σn]) ≥ r2. By (57), (51), (43), the choice of ε, and since t≤ τ n ≤ ξn ≤ v ≤ t+ λ̃, we have Osc(X̃n(·, ωn) + Ṽ n(·, ωn), [τn, ξn ∧ σn)) + δn ≤Osc(X̃n(·, ωn), [τn, ξn ∧ σn)) +Osc(Ṽ n(·, ωn), [τn, ξn ∧ σn)) + δn ≤Osc(X̃n(·, ωn), [τn, ξn ∧ σn)) Osc(Ỹ n(·, ωn), [τn, ξn ∧ σn)) + δn ≤ ε̃+ r2 + δ n < r3. Then by (38) and the monotonicity of D(·), we have D(Π(Osc(X̃n(·, ωn) + Ṽ n(·, ωn), [τn, ξn ∧ σn)) + δn)) ≤D(r2)≤D(r1)< We claim that σn ≥ ξn.(61) To prove (61), we proceed by contradiction and suppose that σn < ξn. Then by (60), with x= X̃n(·, ωn) + Ṽ n(·, ωn) and δ = δn, condition (iv) of The- orem 4.1 holds with [s, t] = [τn, σn − 1/m] for all m sufficiently large. By applying Theorem 4.1 and letting m→∞, we obtain using (34), (38) and (59) that, Osc(Ỹ n(·, ωn), [τn, σn)) ≤Π(Osc(X̃n(·, ωn) + Ṽ n(·, ωn), [τn, ξn ∧ σn)) + δn)(62) ≤Π(r3)< INVARIANCE PRINCIPLE FOR SRBMS 23 By (62), (iii)(b) and (43), we obtain that Osc(Ỹ n(·, ωn), [τn, σn])≤ + Iδn < r2. This contradicts the fact that Osc(Ỹ n(·, ωn), [τn, σn]) ≥ r2 on {σ n < ∞}, and so (61) holds and (55) follows by (60). By applying Theorem 4.1 on [τn, ξn − 1/m] and then letting m→∞, we obtain using (61), (59) and (38), that Osc(W̃ n(·, ωn), [τn, ξn)) ≤Π(Osc(X̃n(·, ωn) + Ṽ n(·, ωn), [τn, ξn ∧ σn)) + δn) and similarly, Osc(Ỹ n(·, ωn), [τn, ξn))< .(63) Then we have ‖W̃ n(ξn−, ωn)− x0‖ ≤ ‖W̃ n(ξn−, ωn)− W̃ n(τn, ωn)‖+ ‖W̃ n(τn, ωn)− x0‖ + δn. Using hypotheses (ii), (iii)(b), and (33), (51), we obtain ‖W̃ n(ξn, ωn)− W̃ n(ξn−, ωn)‖ ≤ ‖X̃n(ξn, ωn)− X̃n(ξn−, ωn)‖ ‖γi,n(W n(ξn−, ωn),W n(ξn, ωn))‖ × (Ỹ ni (ξ n, ωn)− Ỹ ni (ξ n−, ωn)) ≤ ε̃+ Iδn. Hence ‖W̃ n(ξn, ωn)− x0‖ ≤ ‖W̃ n(ξn−, ωn)− x0‖ + ‖W̃ n(ξn, ωn)− W̃ n(ξn−, ωn)‖ + δn + ε̃+ Iδn ≤ ε̃+ (I+1)δn + < ρ/8 + ρ/8 + ρ/8< ρ/2. 24 W. KANG AND R. J. WILLIAMS It follows from this that ξn = v and (53) holds, as desired. Then, by (33), (51), (iii)(b), (iii)(c), (63) and (43), we have ‖W̃ n(v,ωn)− W̃ n(u,ωn)‖ ≤ sup r,s∈[u,v] ‖X̃n(s,ωn)− X̃n(r,ωn)‖+ (Ỹ ni (v,ω n)− Ỹ ni (u,ω ≤ ε̃+ (Ỹ ni (v,ω n)− Ỹ ni (u∨ τ n, ωn)) (Ỹ ni (u ∨ τ n, ωn)− Ỹ ni (u,ω n))(64) ≤ ε̃+ IOsc(Ỹ n(·, ωn), [u∨ τn, v)) + ∆Ỹ ni (v,ω n) + Iδn ≤ ε̃+ I + Iδn + Iδn < ‖Ỹ n(v,ωn)− Ỹ n(u,ωn)‖ ≤ (Ỹ ni (v,ω n)− Ỹ ni (u,ω (Ỹ ni (v,ω n)− Ỹ ni (u∨ τ n, ωn)) (Ỹ ni (u ∨ τ n, ωn)− Ỹ ni (u,ω Here we have used the fact that Ỹi does not increase on (u, τ n ∨ u) and can jump at most by δn at τn, by the definition of τn and (iii)(c). On combining the results from Case 1 and Case 2, we obtain that for each n≥ n1, u,v∈[t,t+λ̃] ‖W̃ n(v,ωn)− W̃ n(u,ωn)‖ : 0≤ t≤ t+ λ̃≤ T < ε(66) u,v∈[t,t+λ̃] ‖Ỹ n(v,ωn)− Ỹ n(u,ωn)‖ : 0≤ t≤ t+ λ̃≤ T < ε.(67) Hence since ωn ∈Hn was arbitrary, by (50), we have that (44) and (45) hold for all n≥ n1. Next we show that there is a constant Nη,T > 0 such that (46) and (47) hold for all n≥ n1. By (66)–(67) above, we have that for each n≥ n1, ω INVARIANCE PRINCIPLE FOR SRBMS 25 Hn, t such that 0≤ t < t+ λ̃≤ T and t≤ u < v ≤ t+ λ̃, ‖W̃ n(v,ωn)− W̃ n(u,ωn)‖< ε(68) ‖Ỹ n(v,ωn)− Ỹ n(u,ωn)‖< ε.(69) Then, for each 0≤ s≤ T , by (68), (69), (49) and (33), we have ‖W̃ n(s,ωn)‖ ≤ ‖W̃ n(s,ωn)− W̃ n(0, ωn)‖+ ‖W̃ n(0, ωn)‖ [T/λ̃]+1∑ ‖W̃ n(iλ̃∧ s,ωn)− W̃ n((i− 1)λ̃ ∧ s,ωn)‖+ ‖X̃n(0, ωn)‖ ≤ ([T/λ̃] + 1)ε+ M̃η,T ‖Ỹ n(s,ωn)‖ ≤ ‖Ỹ n(s,ωn)− Ỹ n(0, ωn)‖ [T/λ̃]+1∑ ‖Ỹ n(iλ̃ ∧ s,ωn)− Ỹ n((i− 1)λ̃∧ s,ωn)‖ ≤ ([T/λ̃] + 1)ε. Here [T/λ̃] is the greatest integer less than or equal to T/λ̃. Let Nη,T = ([T/λ̃] + 1)ε+ M̃η,T . Then we obtain that for n≥ n1 and ω n ∈Hn, 0≤s≤T ‖W̃ n(s,ωn)‖ ≤Nη,T(70) 0≤s≤T ‖Ỹ n(s,ωn)‖ ≤Nη,T .(71) Then by (50), we have that (46) and (47) hold for all n≥ n1. Finally by applying Proposition 1.1, we have the C-tightness of {W̃ n} and {Ỹ n}. It then follows that {(W̃ n,Xn, Ỹ n)}∞n=1 is C-tight. Since Z (W̃ n,Xn, Ỹ n) + (αn,0, βn) where αn,V(βn) → 0 in probability as n→∞, then {Zn}∞n=1 is also C-tight. � 4.3. Invariance principle for SRBMs. The main theorem of the paper is the following. Theorem 4.3 (Invariance principle for SRBMs). Suppose that Assump- tion 4.1 holds. Define Zn = (W n,Xn, Y n) for each n. Then the sequence of 26 W. KANG AND R. J. WILLIAMS processes {Zn}∞n=1 is C-tight and any (weak) limit point of this sequence is of the form Z = (W,X,Y ) where continuous d-dimensional processes W,X and a continuous I-dimensional process Y are defined on some probability space (Ω,F , P ) such that conditions (i), (ii) and (iv) of Definition 2.1 hold with Ft = σ{Z(s) : 0≤ s≤ t}, t≥ 0. If, in addition, the following conditions (vi)′ and (vii) hold, then any weak limit point of the sequence {Zn}∞n=1 is an extended SRBM associated with the data (G,µ,Γ,{γi, i ∈ I}, ν). If furthermore the following condition (viii) holds, then W n ⇒W as n→∞ where W is an SRBM associated with (G,µ,Γ,{γi, i ∈ I}, ν). (vi)′ {Xn} converges in distribution to a d-dimensional Brownian mo- tion with drift µ, covariance matrix Γ and initial distribution ν. (vii) For each (weak) limit point Z = (W,X,Y ) of {Zn}∞n=1, {X(t) − X(0)− µt, Ft, t≥ 0} is a martingale. (viii) If a process W satisfies the properties in Definition 2.1, the law of W is unique, that is, the law of an SRBM associated with the data (G,µ,Γ,{γi, i ∈ I}, ν) is unique. Remark. We note that (vi)′ implies that (vi) of Assumption 4.1 holds. Proof of Theorem 4.3. By Theorem 4.2, we have that the sequence {Zn}∞n=1 is C-tight. Let Z = (W,X,Y ) be a (weak) limit point of {Z n}∞n=1, that is, there is a subsequence {nk} of {n} such that Z nk ⇒Z as k→∞. It also follows that Z̃nk ≡ (W̃ nk ,Xnk , Ỹ nk)⇒Z as k→∞. By the C-tightness of {Zn}, we obtain that Z has continuous paths a.s. For the purpose of verifying that Z satisfies the listed properties in Definition 2.1, one may invoke the Skorokhod representation theorem to assume, without loss of generality, that Znk and Z̃nk converge u.o.c. to Z a.s. as k→∞ and V(βnk) converges u.o.c. to 0 a.s. as k → ∞. With this simplification, it is easily verified that the properties of {Znk} and {Z̃nk} imply that Z has properties (ii) and (iv)(a)–(b) of Definition 2.1. For the verification of property (i) of Definition 2.1, note that for each k, a.s. for each t≥ 0, W nk(t) =Xnk(t) + (0,t] γi,nk(W nk(s−),W nk(s))dβ i (s) (0,t] (γi,nk(W nk(s−),W nk(s))− γi(W nk(s)))dỸ i (s) (0,t] γi(W nk(s))dỸ i (s). The sum of the first two terms on the right-hand side of the above equality converges a.s. to X(t) as k → ∞. The third term on the right-hand side INVARIANCE PRINCIPLE FOR SRBMS 27 converges a.s. to 0 as k→∞, by property (iv) and the fact that a.s., s∈(0,t] ‖W nk(s)−W nk(s−)‖ ≤ sup s∈(0,t] ‖∆Xnk(s)‖+ I sup s∈(0,t] ‖∆Y nk(s)‖→ 0 as k→∞. It remains to show that for each i ∈ I and t≥ 0, a.s., (0,t] γi(W nk(s))dỸ i (s)→ (0,t] γi(W (s))dYi(s) as k→∞. This follows directly from Lemma A.4. For the verification of property (iv)(c) of Definition 2.1, it suffices to show that for each i ∈ I , m= 1,2, . . . , a.s. for each t≥ 0, Yi(t) = (0,t] fm(W (s))dYi(s),(72) where {fm} m=1 is a sequence of real valued continuous functions defined on Rd such that for each m, the range of fm is [0,1], fm(x) = 1 for x ∈ U1/m(∂Gi ∩ ∂G) and fm(x) = 0 for x /∈ U2/m(∂Gi ∩ ∂G). The existence of such a sequence of continuous functions {fm} m=1 can be shown using Urysohn’s lemma (cf. [8], page 122). Then (72) is a consequence of Lemma A.4, property (iii) of Ỹ i and the fact that δ nk → 0 as k→∞. Indeed, a.s., for each t≥ 0, Yi(t) = lim i (t) = lim (0,t] {W̃nk (s)∈U nk (∂Gi∩∂G)} i (s) = lim (0,t] fm(W̃ nk(s))dỸ i (s) (0,t] fm(W (s))dYi(s). Thus, Z satisfies properties (i), (ii) and (iv) of Definition 2.1 with Ft = σ{Z(s) : 0≤ s≤ t}, t≥ 0. Assuming properties (vi)′ and (vii) hold, Z satisfies (iii) of Definition 2.1. Then Z is an extended SRBM associated with the data (G,µ,Γ,{γi, i ∈ I}, ν). If in addition, property (viii) holds, then the law of W is unique. Since each weak limit W is an SRBM associated with the data (G,µ,Γ,{γi, i ∈ I}, ν) and the law of such an SRBM is unique, then by a standard argument, W n ⇒W as n→∞ where W is an SRBM associated with (G,µ,Γ,{γi, i ∈ I}, ν). � Some sufficient conditions for (vii) to hold are given in Proposition 4.2 of [15] for a simpler setting where G=Rd+. Two of those conditions generalize to our setting here and can be proved in the same manner as in [15]. For completeness, we state the ensuing result here. 28 W. KANG AND R. J. WILLIAMS Proposition 4.1. Suppose that Assumption 4.1 and (vi)′ of Theorem 4.3 hold. If, in addition, one of the following conditions (I)–(II) holds, then condition (vii) of Theorem 4.3 is satisfied, and any weak limit point of {Zn}∞n=1 is an extended SRBM associated with (G,µ,Γ,{γ i, i ∈ I}, ν). (I) For any triple of d-dimensional {Ft}-adapted processes (W,X,Y ) defined on some filtered probability space (Ω,F ,{Ft}, P ) and satisfying con- ditions (i), (ii) and (iv) of Definition 2.1 together with the condition that X, under P , is a d-dimensional Brownian motion with drift vector µ, co- variance matrix Γ and initial distribution ν, the pair (W,Y ) is adapted to the filtration generated by X and the P -null sets. (II) Xn = X̌n + εn1 , Y n = Y̌ n + εn2 , W n = W̌ n + εn3 , where ε 1 , ε 2 , ε 3 are processes converging to 0 in probability as n→∞, and: (a) {X̌n(t)− X̌n(0)}∞n=1 is uniformly integrable for each t≥ 0, (b) there is a sequence of constants {µn}∞n=1 in R d such that limn→∞µ n = µ, (c) for each n, {X̌n(t)− X̌n(0)−µnt, t≥ 0} is a Pn-martingale with respect to the filtration generated by (W̌ n, X̌n, Y̌ n). In the rest of this work, we focus on applications of the invariance prin- ciple and in particular on giving sufficient conditions for property (viii) of Theorem 4.3 to hold. 5. Applications of the invariance principle. In Section 5.1, we prove weak existence of SRBMs associated with data (G,µ,Γ,{γi, i ∈ I}, ν) satisfying (A1)–(A5) of Section 3. This is accomplished by constructing a sequence of approximations whose weak limit points are SRBMs. The invariance prin- ciple is used to prove the C-tightness of the approximations and that any weak limit point is an SRBM. In Sections 5.2 and 5.3, using known results on uniqueness in law for SRBMs, we illustrate the invariance principle for certain domains and directions of reflection. 5.1. Weak existence of SRBMs. Theorem 5.1. Suppose that assumptions (A1)–(A5) of Section 3 hold. Then there exists an SRBM associated with the data (G,µ,Γ,{γi, i ∈ I}, ν). Proof. We construct a sequence of approximations to an SRBM and use the invariance principle to establish weak convergence along a subse- quence to an SRBM. In the following we will use R(·) from assumption (A2), L > 0 from as- sumption (A4), a > 0 from assumption (A5), and ρ0 = . Fix ε > 0 and INVARIANCE PRINCIPLE FOR SRBMS 29 0 < ρ < min{ R(a/4) }. By assumption (A3), there is a constant r1 > 0 such that D(r)<min for all r ∈ (0, r1]. Recall the properties of Π(·) from Theorem 4.1. Since Π(u)→ 0 as u→ 0, there are constants 0< r3 < r2 <min{r1, } such that Π(r)< for all r ∈ (0, r3]. Fix ε̃ and δ such that 0< ε̃ <min{ 24ILr2 } and 0< 2δ <min{r3 8(1+I) We will construct a d-dimensional stochastic processW δ and an I-dimensional “pushing” process Y δ , such that W δ approximately satisfies the conditions defining an SRBM for the data (G,µ,Γ,{γi, i ∈ I}, ν) (cf. Assumption 4.1). The idea for this construction is to use a Brownian motion X with drift vector µ, covariance matrix Γ and initial distribution ν. Away from ∂G, the increments of W δ are determined by those of X . For any time t ≥ 0 such that W δ(t−) ∈ ∂G, we add an instantaneous jump to W δ(t−) to obtain W δ(t) ∈G. Here W δ(0−) =X(0). The size of the jump is such that W δ(t) is a strictly positive distance (depending on δ) from the boundary of G. The jump vector is obtained as a measurable function of W δ(t−). To ensure the measurability, each point x on ∂G is associated with a nearby point x̄, chosen in a measurable way from a fixed countable set of points in ∂G. The jump vector for x is one associated with x̄. We now specify the mapping x→ x̄ and the associated jump vector more precisely. By assumption (A5)(ii), for each x ∈ ∂G, there is c(x) ∈RI+ such that i∈I(x) ci(x) = 1 and min j∈I(x) i∈I(x) ci(x)γ i(x), nj(x) ≥ a.(73) By (73), Lemma 2.1 and the fact that ni(·) is continuous on ∂Gi for each i ∈ I , we have that for each x ∈ ∂G there is rx ∈ (0, δ) such that for each y ∈Brx(x)∩ ∂G, I(y)⊂ I(x)(74) j∈I(x) i∈I(x) ci(x)γ i(x), nj(y) .(75) It follows, using the C1 nature of ∂Gi and the fact that n i(y) is the inward unit normal to ∂Gi at y ∈ ∂G for each i ∈ I(y), that (by choosing rx even 30 W. KANG AND R. J. WILLIAMS smaller if necessary) for each x ∈ ∂G there is m(x)> 0 and rx ∈ (0, δ) such that for each y ∈Brx(x)∩ ∂G, (74)–(75) hold and y + λ i∈I(x) ci(x)γ i(x) ∈G for all λ ∈ (0,m(x)).(76) Let Borx(x) denote the interior of the closed ball Brx(x) for each x ∈ ∂G. The collection {Borx(x) :x ∈ ∂G} is an open cover of ∂G and it follows that there is a countable set {xk} such that ∂G⊂ kBrxk (xk) and {xk} ∩BN (0) is a finite set for each integer N ≥ 1. We can further choose the set {xk} to be minimal in the sense that for each strict subset C of {xk}, {Brx(x) :x ∈C} does not cover ∂G. Let Dk = (Brxk (xk) \ ( i=1 Brxi (xi)) ∩ ∂G for each k. Then Dk 6= ∅ for each k, {Dk} is a partition of ∂G, and for each x ∈ ∂G there is a unique index i(x) such that x ∈Di(x). For each x ∈R d, let x, if x /∈ ∂G, xi(x), if x ∈ ∂G. Note that for all x ∈Rd, ‖x− x̄‖< δ.(77) For each i ∈ I and x∈Rd, let γi,δ(x) = γi(x̄).(78) The mapping x → x̄ is Borel measurable on Rd and hence γi,δ is a Borel measurable function from Rd into Rd. We construct (W δ, Y δ) as follows. Let X defined on some filtered proba- bility space (Ω,F ,{Ft}, P ) be a d-dimensional {Ft}-Brownian motion with drift µ and covariance matrix Γ such that X is continuous surely and X(0) has distribution ν. Let τ1 = inf{t≥ 0 :X(t) ∈ ∂G} W δ(t) =X(t), Y δ(t) = 0 for 0≤ t < τ1. Note that W δ(τ1−) exists on {τ1 <∞} since X has continuous paths and in the case that τ1 = 0, W δ(0−)≡X(0). On {τ1 <∞}, define Y δi (τ1) = 0, i /∈ I(W δ(τ1−)), ci(W δ(τ1−)) m(W δ(τ1−)) , i ∈ I(W δ(τ1−)), W δ(τ1) =X(τ1) m(W δ(τ1−)) i∈I(W δ(τ1−)) ci(W δ(τ1−))γ i,δ(W δ(τ1−)) INVARIANCE PRINCIPLE FOR SRBMS 31 So W δ, Y δ have been defined on [0, τ1) and at τ1 on {τ1 <∞}, such that: (i) W δ(t) = X(t) + i∈I γ i,δ(W δ(0−))Y δi (0) + (0,t] γi,δ(W δ(s−))dY δi (s) for all t ∈ [0, τ1]∩ [0,∞), where W δ(0−) =X(0), (ii) W δ(t) ∈G for t ∈ [0, τ1]∩ [0,∞), (iii) for i ∈ I , (a) Y δi (0)≥ 0, (b) Y δi is nondecreasing on [0, τ1]∩ [0,∞), (c) Y δi (t) = Y i (0) + (0,t] 1{W δ(s)∈U2δ(∂Gi∩∂G)} dY i (s) for t ∈ [0, τ1] ∩ [0,∞), (iv) ‖∆Y δ(t)‖ ≡ ‖Y δ(t)−Y δ(t−)‖ ≤ δ for t ∈ [0, τ1]∩ [0,∞), where Y δ(0−)≡ Note that (iii)(c) above contains the expression W δ(s) ∈U2δ(∂Gi∩∂G). The reader may wonder why 2δ appears instead of δ. The reason is that at a jump time s of Y δi , W δ(s−) ∈ ∂Gi ∩ ∂G and so dist(W δ(s), ∂Gi ∩ ∂G)≤ ‖W δ(s)−W δ(s−)‖+ ‖W δ(s−)−W δ(s−)‖ ≤ δ + δ = 2δ. Proceeding by induction, we assume that for some n ≥ 2, τ1 ≤ · · · ≤ τn−1 have been defined, and W δ, Y δ have been defined on [0, τn−1) and at τn−1 on {τn−1 <∞}, such that (i)–(iv) above hold with τn−1 in place of τ1. Then we define τn =∞ on {τn−1 =∞}, and on {τn−1 <∞} we define τn = inf{t≥ τn−1 :W δ(τn−1) +X(t)−X(τn−1) ∈ ∂G}. For τn−1 ≤ t < τn, let Y δ(t) = Y δ(τn−1), W δ(t) =W δ(τn−1) +X(t)−X(τn−1), and on {τn <∞}, let Y δi (τn) = Y δi (τn−1), i /∈ I(W δ(τn−)), Y δi (τn−1) + ci(W δ(τn−)) m(W δ(τn−)) , i ∈ I(W δ(τn−)), W δ(τn) =W δ(τn−) m(W δ(τn−)) i∈I(W δ(τn−)) ci(W δ(τn−))γ i,δ(W δ(τn−)) 32 W. KANG AND R. J. WILLIAMS In this way, W δ, Y δ have been defined on [0, τn) and at τn on {τn <∞} such that (i)–(iv) hold with τn in place of τ1. By construction {τn} n=1 is a nondecreasing sequence of stopping times. Let τ = limn→∞ τn. On {τ =∞}, the construction of (W δ, Y δ) is complete. We now show that {τ < ∞} = ∅. In fact, if {τ < ∞} 6= ∅, let ω ∈ {τ < ∞}. The above construction gives (W δ(·, ω), Y δ(·, ω)) on the time interval [0, τ(ω)). For each t ∈ [0, τ(ω)), we have W δ(t,ω) =X(t,ω) + γi,δ(W δ(0−, ω))Y δi (0, ω) (0,t] γi,δ(W δ(s−, ω))dY δi (s,ω). SinceX is continuous on [0,∞), ‖γi,δ(x)‖= 1 for each x ∈Rd and i∈I Y i (0, ω)≤ δ, there are constants λ̃ ∈ (0, τ(ω)) and M̃ > 0 (depending on ω) such wτ(ω)(X(·, ω) + γ i,δ(W δ(0−, ω))Y δi (0, ω), λ̃)< ε̃(80) 0≤t≤τ(ω) ∥∥∥∥∥X(·, ω) + γi,δ(W δ(0−, ω))Y δi (0, ω) ∥∥∥∥∥≤ M̃,(81) where w (·, ·) is defined in (3). By the choice of ε̃, δ made at the beginning of this proof, (77)–(78) and the uniform Lipschitz property of the γi(·), i ∈ I , it follows that (39) and (43) hold with γi,δ(y) and 2δ in place of γi,n(y,x) and δn, respectively. Then by similar pathwise analysis to that used in Case 1 and 2 of the proof of Theorem 4.2, with W̃ n =W n =W δ, αn = 0, γi,n(y,x) = γi,δ(y) for each i ∈ I and x, y ∈ Rd, Xn = X + i∈I γ i,δ(W δ(0−))Y δi (0), Y n = Y δ , Ỹ n = Y δ − Y δ(0), βn = Y δ(0) and δn = 2δ, we obtain that (71) holds for any T < τ(ω) with ωn = ω, Nη,T = ([τ(ω)/λ̃] + 1)ε+ M̃ . It follows that supi∈I sups∈[0,τ(ω)) Y i (s,ω) is finite. By the nondecreasing property of Y δi (·, ω) on [0, τ(ω)) for each i ∈ I , Y i (τ(ω)−, ω) exists and is finite for each i ∈ I . Then by (79) and the continuity of X , we see that W δ(τ(ω)−, ω) exists and is finite. By the construction of Y δ and the fact that i∈I(x) ci(x) = 1 for all x∈ ∂G, we have that Y δi (τ(ω)−, ω) = m(W δ(τn(ω)−, ω)) ∧ δ.(82) Since τn(ω) ↑ τ(ω) as n → ∞ and W δ(τ(ω)−, ω) exists, it follows that {W δ(τn(ω)−, ω)} n=1 converges to W δ(τ(ω)−, ω) ∈ ∂G as n → ∞. Conse- quently, {W δ(τn(ω)−, ω)} n=1 is a bounded sequence in ∂G and so by the INVARIANCE PRINCIPLE FOR SRBMS 33 definition of the sets {Dk} which form a partition of ∂G, there is a finite set C such that {W δ(τn(ω)−, ω)} n=1 ⊂ Hence, m(W δ(τn(ω)−, ω))≤ inf m(xk)> 0,(83) and so the right-hand side of (82) is infinite. On the other hand, since supi∈I sups∈[0,τ(ω)) Y i (s,ω) is finite, the left-hand side of (82) is finite. This yields the desired contradiction and so {τ < ∞} = ∅ and we have con- structed (W δ, Y δ) on [0,∞). From the construction above, we can see that W δ and Y δ are well-defined stochastic processes with sample paths in D([0,∞),Rd) and D([0,∞),RI). They are adapted to the filtration generated by X and satisfy (i)–(iv) above with [0,∞) in place of [0, τ1]. Consider a sequence of sufficiently small δ’s, denoted by {δn}, such that δn ↓ 0 as n → ∞. For each δn, let (W δ , Y δ ) be the pair constructed as above for the same process X . By the above properties and the fact that for each i ∈ I and x, y ∈Rd, ‖γi,δ (y)− γi(x)‖ ≤ ‖γi(ȳ)− γi(x)‖ ≤ L‖ȳ− x‖ ≤ L(δn + ‖y − x‖), we obtain that Assumption 4.1 holds with W̃ n =W n =W δ , αn = 0, γi,n(y,x) = (y) for each i ∈ I and x, y ∈Rd, Xn =X+ i∈I γ i,δn(W δ (0−))Y δ i (0), Y n = Y δ , Ỹ n = Y δ (0), βn = Y δ (0) and 2δn in place of δn. By invok- ing the first part of Theorem 4.3, we obtain that {Zδ }∞n=1 = {(W δn ,Xδ )}∞n=1 is C-tight and any weak limit point Z of this sequence satisfies conditions (i), (ii) and (iv) of Definition 2.1 with Ft = σ{Z(s) : 0 ≤ s ≤ t}, t≥ 0. Note that condition (vi)′ of Theorem 4.3 holds trivially. Furthermore, = {Xδ (t)−Xδ (0)− µt, t≥ 0}= {X(t)−X(0)− µt, t≥ 0} is a mar- tingale with respect to the filtration generated by X . Since W δ , Y δ adapted to this filtration, it follows that M δ is a martingale with respect to the filtration generated by W δ , Y δ (which in fact is the same as that generated by X). For each t≥ 0, Xδ (t)−Xδ (0) =X(t)−X(0) and so trivially this forms a uniformly integrable sequence as n varies. It fol- lows from Proposition 4.1 that condition (vii) of Theorem 4.3 holds. Hence, any weak limit point of {Zδ }∞n=1 is an extended SRBM with the data (G,µ,Γ,{γi, i ∈ I}, ν). � 5.2. SRBMs in convex polyhedrons with constant reflection fields. Exis- tence and uniqueness in law for SRBMs living in convex polyhedrons with a constant reflection field on each boundary face has been studied by Dai and 34 W. KANG AND R. J. WILLIAMS Williams [4]. In this subsection, we state a consequence of our invariance principle using the results in [4] to establish uniqueness in law. In this case, G is defined in terms of I (I≥ 1) d-dimensional unit vectors {ni, i ∈ I} and an I-dimensional vector β = (β1, . . . , βI) ′ such that G≡ {x ∈Rd : 〈ni, x〉 ≥ βi for all i ∈ I}.(84) It is assumed that G is nonempty and that the set {(n1, β1), . . . , (n I, βI)} is minimal in the sense that no proper subset defines G. For each i ∈ I , let Fi denote the boundary face: {x ∈G : 〈ni, x〉= βi}. Then, n i is the inward unit normal to Fi. A constant vector field γ i of unit length specifies the direction of reflection associated with Fi. Definition 5.2. For each ∅ 6=K⊂ I , define FK = i∈KFi. Let F∅ =G. A set K⊂ I is maximal if K 6=∅, FK 6=∅ and FK 6= FK̄ for any K̄ ⊃ K such that K̄ 6=K. In [4], Dai and Williams introduced the following assumption. Assumption 5.1. For each maximal K⊂ I , (S.a) there is a positive linear combination n= i∈K bin i (bi > 0 ∀i ∈K) of the {ni, i ∈K} such that 〈n,γi〉> 0 for all i ∈K, (S.b) there is a positive linear combination γ = i∈K ciγ i (ci > 0 ∀i ∈K) of the {γi, i ∈K} such that 〈ni, γ〉> 0 for all i ∈K. Remark. For the given G and constant vector fields {γi, i ∈ I}, As- sumption 5.1 is equivalent to assumption (A5). Definition 5.3. The convex polyhedron G is simple if for each K⊂ I such that K 6=∅ and FK 6=∅, exactly |K| distinct faces contain FK. Remark. The polyhedron G is simple if and only if K is maximal for every K such that ∅ 6= K ⊂ I and FK 6= ∅. It is shown in [4] that when G is simple, (S.a) holds for all maximal K⊂ I if and only if (S.b) holds for all maximal K⊂ I. Dai and Williams [4] showed that Assumption 5.1 is sufficient for exis- tence and uniqueness in law of SRBMs living in G with the reflection fields {γi, i ∈ I} and fixed starting point. [They also showed that condition (S.b) holding for all maximal K⊂ I is necessary for existence of an SRBM starting from each point in G. Consequently, when G is simple, Assumption 5.1 is necessary and sufficient for existence of an SRBM starting from each point in G.] This yields the following consequence of our invariance principle. INVARIANCE PRINCIPLE FOR SRBMS 35 Theorem 5.4. Let G be a nonempty domain such that G is a convex polyhedron of the form (84) (with minimal description), and let {γi, i ∈ I} be a family of constant vector fields of unit length satisfying Assumption 5.1. Suppose that Assumption 4.1 and (vi)′, (vii) of Theorem 4.3 hold. Then W n ⇒W as n→∞ where W is an SRBM associated with (G,µ,Γ,{γi, i ∈ I}, ν). Proof. Clearly (A1) holds. Assumptions (A2)–(A3) hold by Lemma A.3. Since for each i ∈ I , γi(·) is a constant vector field of unit length, as- sumption (A4) holds trivially. Assumption (A5) is implied by Assumption 5.1. Hence by Theorem 4.3, the only thing that we have to check is condition (viii) of Theorem 4.3, that is, uniqueness in law for SRBMs in convex poly- hedrons with constant reflection fields of unit length. But this is proved in Theorem 1.3 of [4] for a fixed starting point in G and follows by a standard conditioning argument for the initial distribution ν. � 5.3. SRBMs in bounded domains with piecewise smooth boundaries. Dupuis and Ishii [6] have established sufficient conditions for the existence and path- wise uniqueness of reflecting diffusions living in the closures of bounded domains with piecewise smooth boundaries. In this subsection, we state a consequence of our invariance principle using the results in [6] to establish uniqueness in law. Theorem 5.5. Let G be a bounded domain and {γi, i ∈ I} be a family of reflection fields that satisfy assumptions (A1)–(A4) and (A5)′ in Section 3. We further assume that for each i ∈ I , γi(·) is once continuously differ- entiable with locally Lipschitz continuous first partial derivatives. Suppose that Assumption 4.1 and (vi)′, (vii) of Theorem 4.3 hold. Then W n ⇒W as n→∞ where W is an SRBM associated with (G,µ,Γ,{γi, i ∈ I}, ν). Remark. We remind the reader that in view of Lemma 3.1, to verify condition (A5)′, one only needs to show that (i) or (ii) holds for all x ∈ ∂G. However, as can be seen from the proof below, both forms of the condition can be useful. Proof of Theorem 5.5. This theorem follows from Theorem 4.3 and uniqueness in law for the associated SRBMs. The latter follows by a standard argument from the pathwise uniqueness established in Corollary 5.2 of [6] for their Case 2. The conditions required for that case are satisfied in particular because (A5)′(ii) implies condition (3.8) of [6]. That condition (3.8) readily implies condition (3.6) of [6]; and, by [5], under the additional smoothness assumptions imposed on the γi in the statement of our theorem, condition (3.8) also implies condition (3.7) in [6]. In addition, (A5)′(i) implies that 36 W. KANG AND R. J. WILLIAMS for each x ∈ ∂G, 〈γi(x), ni(x)〉> 0 for each i ∈ I(x), and furthermore, since (A5)′ implies (A5), we have by (A5)(i) that the origin does not belong to the convex hull of the {γi(x) : i ∈ I(x)}. � APPENDIX: AUXILIARY LEMMAS Lemma A.1. Suppose that G is bounded. If assumption (A1) holds, then assumption (A2) holds. Proof. To see this, suppose G is bounded and assumption (A1) holds. Fix ε ∈ (0,1). For each i ∈ I and z ∈ ∂Gi ∩ ∂G, by the C 1 property of ∂Gi, there is a neighborhood Vz of z and a constant R(ε, i, z) > 0 such that for all x ∈ Vz ∩ ∂Gi ∩ ∂G and y ∈Gi such that ‖x− y‖<R(ε, i, z), 〈ni(x), y − x〉 ≥−ε‖y − x‖.(85) Assumption (A2) then follows by a standard compactness argument. � Lemma A.2. Suppose that G is a nonempty bounded domain satisfying (5), where for each i ∈ I , Gi is a nonempty domain. Then assumption (A3) holds. Proof. We prove the lemma by contradiction. Suppose that assumption (A3) does not hold. Then, since there are only finite many J ⊂ I , J 6=∅, there is an ε > 0, a nonempty set J ⊂ I , a sequence {rn} ⊂ (0,∞) with rn → 0 as n→∞, a sequence {xn} ⊂R d such that for each n, xn ∈ j∈J Urn(∂Gj∩ ∂G) and dist(xn, j∈J (∂Gj ∩ ∂G)) > ε. But since G is bounded, {xn} is bounded and without loss of generality we may assume that xn → x as n→∞ for some x ∈Rd. It follows that x ∈ j∈J (∂Gj ∩ ∂G), since for each j ∈ J , dist(x,∂Gj ∩ ∂G)≤ ‖xn − x‖+dist(xn, ∂Gj ∩ ∂G)≤ ‖xn − x‖+ rn → 0 as n→∞. This is inconsistent with xn → x and dist(xn, j∈J (∂Gj ∩∂G))> Lemma A.3. Suppose (A1) holds where Gi = {x ∈R d : 〈ni, x〉> βi} for i ∈ I,(86) {ni, i ∈ I} is a finite collection of d-dimensional vectors of unit length, and for I= |I|, β = (β1, . . . , βI) ′ is an I-dimensional vector. (Thus, G is a convex polyhedron.) Assume that for each i ∈ I , ∂Gi ∩ ∂G 6=∅. Then assumptions (A2) and (A3) hold. INVARIANCE PRINCIPLE FOR SRBMS 37 Proof. Assumption (A2) holds automatically since G is convex. In or- der to show that assumption (A3) holds, we just need to show that for each J ⊂ I with J 6=∅, (∂Gj ∩ ∂G) Ur(∂Gj ∩ ∂G) → 0(87) as r→ 0. Fix J ⊂ I such that J 6=∅. Then j∈J (∂Gj ∩∂G) is the collection of all solutions x ∈Rd to the following system of linear inequalities: 〈ni, x〉 ≥ βi for all i ∈ I, 〈−ni, x〉 ≥ −βi for all i ∈ J . Suppose that j∈J (∂Gj ∩∂G) 6=∅, that is, (LS) has at least one solution. By a theorem of Hoffman [11], with supporting lemmas proved by Agmon [1], there is a constant C > 0 (depending only on {ni, i ∈ I} and not on β) such that for any x ∈Rd there exists a solution x0 ∈R d of (LS) with ‖x− x0‖ ≤C (βi − 〈n i, x〉)+ + (−βi − 〈−n i, x〉)+ .(88) For r > 0, any x ∈ j∈J Ur(∂Gj ∩ ∂G) satisfies the following: 〈ni, x〉 ≥ βi − r for all i ∈ I, (r-LS) 〈−ni, x〉 ≥ −βi − r for all i ∈ J . Then by (88), there is x0 ∈ j∈J (∂Gj ∩ ∂G) such that (∂Gj ∩ ∂G) ≤ ‖x− x0‖ ≤ 2C|I|r. It follows that (87) holds when j∈J (∂Gj ∩ ∂G) 6=∅. Now suppose that j∈J (∂Gj ∩∂G) =∅, that is, (LS) has no solution. We shall use an argument by contradiction to show that j∈J Ur(∂Gj ∩∂G) =∅ for all r sufficiently small. Suppose that this is not true. Then we have that⋂ j∈J Ur(∂Gj ∩ ∂G) 6= ∅ for all r ∈ (0,∞). As we have seen before, any j∈J Ur(∂Gj ∩ ∂G) is a solution to (r-LS). We now construct a Cauchy sequence. Let x1 ∈ j∈J U1/2(∂Gj ∩ ∂G). Then x1 is a solution to ( -LS). Since ( 1 -LS) has at least one solution, by the theorem of Hoffman [11] (using the fact that the constant C depends only on {ni, i ∈ I}), we conclude that there is a solution x2 to ( -LS) such that ‖x1−x2‖ ≤ , where C ′ = 2C|I|. Continuing in this manner, we can obtain a sequence {xn} n=1 such that for each n ≥ 1, ‖xn − xn+1‖ ≤ and xn+1 is a solution of ( -LS). The 38 W. KANG AND R. J. WILLIAMS sequence {xn} n=1 is Cauchy. Hence, there is an x ∗ ∈Rd such that xn → x as n→ ∞, and x∗ is a solution to (LS). This contradicts the supposition j∈J (∂Gj ∩ ∂G) =∅. Thus we have that j∈J Ur(∂Gj ∩ ∂G) =∅ for all r sufficiently small, and for such r, (∂Gj ∩ ∂G) Ur(∂Gj ∩ ∂G) by convention. Combining the above we see that for each J ⊂ I with J 6=∅, (87) holds and hence assumption (A3) holds. � Remark. In fact, under the assumptions of Lemma A.3, there is a con- stant C > 0 such that D(u) ≤ Cu for each u ≥ 0 and D(·) defined as in assumption (A3). Lemma A.4. Given T > 0, functions φ,{φn}∞n=1 in D([0,∞),R d), and χ,{χn}∞n=1 in D([0,∞),R), suppose that sup0≤s≤T ‖φ n(s)− φ(s)‖ → 0 and sup0≤s≤T |χ n(s)−χ(s)| → 0 as n→∞. Assume that χn is nondecreasing for each n. Then for any sequence of real valued continuous functions {fn}∞n=1 defined on Rd such that fn converges uniformly on each compact set to a continuous function f :Rd →R, we have (0,t] fn(φn(s))dχn(s)→ (0,t] f(φ(s))dχ(s) as n→∞,(89) uniformly for t ∈ [0, T ]. Proof. By replacing χn(·) and χ(·) by χn(·)− χn(0) and χ(·)− χ(0), respectively, we may assume that χn(0) = χ(0) = 0. It is straightforward to see by the uniform convergence of {χn} to χ on [0, T ] that χ inherits the nondecreasing property of the {χn}. By the triangle inequality, 0≤t≤T (0,t] fn(φn(s))dχn(s)− (0,t] f(φ(s))dχ(s) ≤ sup 0≤t≤T (0,t] (fn(φn(s))− f(φ(s)))dχn(s) ∣∣∣∣(90) + sup 0≤t≤T (0,t] f(φ(s))d(χn(s)− χ(s)) ∣∣∣∣. For the first term on the right-hand side of the above inequality, we have 0≤t≤T (0,t] (fn(φn(s))− f(φ(s)))dχn(s) ≤ sup 0≤s≤T |fn(φn(s))− f(φ(s))|χn(T ), INVARIANCE PRINCIPLE FOR SRBMS 39 where the right-hand side member above tends to zero as n → ∞ by the uniform convergence of φn to φ on [0, T ] (which implies uniform boundedness of {φn} on [0, T ]), the uniform convergence of fn to f on compact sets, the continuity of f , and the convergence of χn(T ) to χ(T ). For the second term, note that since f(φ(·)) ∈D([0,∞),R), by Theorem 3.5.6, Proposition 3.5.3 and Remark 3.5.4 of [7], there is a sequence of step functions {zk}∞k=1 of the zk(·) = zk(tki )1[tk )(·),(91) where 1 ≤ lk < ∞, 0 = t 1 < t 2 < · · · < t < ∞ and sup0≤s≤T |f(φ(s)) − zk(s)| → 0 as k→∞. Then 0≤t≤T (0,t] f(φ(s))d(χn(s)− χ(s)) ≤ sup 0≤t≤T (0,t] (f(φ(s))− zk(s))d(χn(s)− χ(s)) + sup 0≤t≤T (0,t] zk(s)d(χn(s)− χ(s)) ≤ sup 0≤s≤T |f(φ(s))− zk(s)|(χn(T ) + χ(T )) + sup 0≤t≤T |zk(tki )||(χ n − χ)((tki+1 ∧ t)−)− (χ n − χ)((tki ∧ t)−)|. For fixed k, the last term above can be made as small as we like for all n sufficiently large since χn → χ uniformly on [0, T ]. The desired result follows. Remark. The proof of Lemma A.4 is a modification of the proof of the related Lemma 2.4 in [4]. The difference in assumptions is that in [4] it is assumed that φn → φ in the J1-topology rather than uniformly on [0, T ], χn, χ ∈C([0,∞),R+) rather than χ n, χ ∈D([0,∞),R), and there is a single function f rather than a sequence {fn}. REFERENCES [1] Agmon, S. (1954). The relaxation method for linear inequalities. Canadian J. Math. 6 382–392. MR0062786 [2] Berman, A. and Plemmons, R. J. (1979). Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York. MR0544666 [3] Dai, J. G. and Dai, W. (1999). A heavy traffic limit theorem for a class of open queueing networks with finite buffers. Queueing Systems 32 5–40. MR1720547 http://www.ams.org/mathscinet-getitem?mr=0062786 http://www.ams.org/mathscinet-getitem?mr=0544666 http://www.ams.org/mathscinet-getitem?mr=1720547 40 W. KANG AND R. J. WILLIAMS [4] Dai, J. G. and Williams, R. J. (1995). Existence and uniqueness of semimartingale reflecting Brownian motion in convex polyhedrons. Theory Probab. Appl. 40 1– 40. MR1346729 [Correctional note (2006) 50 346–347 MR2222685.] [5] Dupuis, P. and Ishii, H. (1991). On oblique derivative problems for fully nonlinear second-order elliptic PDE’s on domains with corners. Hokkaido Math J. 20 135– 164. MR1096165 [6] Dupuis, P. and Ishii, H. (1993). SDEs with oblique reflection on nonsmooth do- mains. Ann. Probab. 21 554–580. MR1207237 [Correction note (2007) 6 pages, submitted.] [7] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York. MR0838085 [8] Folland, G. B. (1999). Real Analysis: Modern Techniques and Their Applications, 2nd ed. Wiley, New York. MR1681462 [9] Gilbarg, D. and Trudinger, N. S. (1977). Elliptic Partial Differential Equations of Second Order. Springer, Berlin. MR0473443 [10] Harrison, J. M. and Reiman, M. I. (1981). Reflected Brownian motion on an orthant. Ann. Probab. 9 302–308. MR0606992 [11] Hoffman, A. J. (1952). On approximate solutions of systems of linear inequalities. J. of Research of the National Bureau of Standards 49 263–265. MR0051275 [12] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, New York. MR0959133 [13] Kang, W., Kelly, F. P., Lee, N. H. and Williams, R. J. (2007). State space collapse and diffusion approximation for a network operating under a fair band- width sharing policy. Preprint. [14] Taylor, L. M. and Williams, R. J. (1993). Existence and uniqueness of semi- martingale reflecting Brownian motions in an orthant. Probab. Theory Related Fields 96 283–317. MR1231926 [15] Williams, R. J. (1998). An invariance principle for semimartingale reflecting Brow- nian motions in an orthant. Queueing Systems 30 5–25. MR1663755 [16] Williams, R. J. (1998). Diffusion approximations for open multiclass queueing net- works: Sufficient conditions involving state space collapse. Queueing Systems Theory Appl. 30 27–88. MR1663759 Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, Pennsylvania 15213 E-mail: [email protected] Department of Mathematics University of California at San Diego 9500 Gilman Drive La Jolla, California 92093 E-mail: [email protected] http://www.ams.org/mathscinet-getitem?mr=1346729 http://www.ams.org/mathscinet-getitem?mr=2222685 http://www.ams.org/mathscinet-getitem?mr=1096165 http://www.ams.org/mathscinet-getitem?mr=1207237 http://www.ams.org/mathscinet-getitem?mr=0838085 http://www.ams.org/mathscinet-getitem?mr=1681462 http://www.ams.org/mathscinet-getitem?mr=0473443 http://www.ams.org/mathscinet-getitem?mr=0606992 http://www.ams.org/mathscinet-getitem?mr=0051275 http://www.ams.org/mathscinet-getitem?mr=0959133 http://www.ams.org/mathscinet-getitem?mr=1231926 http://www.ams.org/mathscinet-getitem?mr=1663755 http://www.ams.org/mathscinet-getitem?mr=1663759 mailto:[email protected] mailto:[email protected] Introduction Notation, terminology and preliminaries Definition of an SRBM Assumptions on the domain G and the reflection vector fields {i} Assumptions on the domain G Assumptions on the reflection vector fields {i} Invariance principle Oscillation inequality C-tightness result Invariance principle for SRBMs Applications of the invariance principle Weak existence of SRBMs SRBMs in convex polyhedrons with constant reflection fields SRBMs in bounded domains with piecewise smooth boundaries Appendix: Auxiliary lemmas References Author's addresses

0704.0406

Finite Drude weight for 1D low temperature conductors

Finite Drude weight for 1D low temperature conductors Dariush Heidarian and Sandro Sorella Istituto Nazionale di Fisica della Materia (INFM)-Democritos, National Simulation Centre, and Scuola Internazionale Superiore di Studi Avanzati (SISSA), I-34014 Trieste, Italy We apply well established finite temperature QuantumMonte Carlo techniques to one dimensional Bose systems with soft and hardcore constraint, as well as to spinless fermion systems. We give clear and robust numerical evidence that, as expected, no superfluid density for Bosons or Meissner fraction for fermions. is possible at any non zero temperature in one dimensional interacting Bose or fermi lattice models, whereas a finite Drude weight is generally observed in gapless systems, in partial disagreement to previous expectations. PACS numbers: 74.25.Fy,71.27.+a,71.10.Fd I. INTRODUCTION In the last decades there have been a lot of numerical and theoretical works to understand the role of strong correlation in lattice model Hamiltonians.1,2,3,4,5,6,7 Re- cently this issue has acquired an increasing attention and remarkable importance, due to the recent advances in the realization of optical lattices. In these experiments ultra- cold atoms behave as boson particles trapped on particu- lar lattice sites, whereas the interaction and the hopping parameters can be tuned continuously. This important achievement has opened the possibility to verify directly the crucial role played by the electron correlation in very important model Hamiltonians defined on a lattice. An important example is the realization of a Mott insulating state in a system with strong on site repulsion8,9. More- over quite recently the possibility to include the Fermi statistics in optical lattices appears very promising and interesting.10 In 1D spinless fermion systems are equivalent to inter- acting Bose systems with hard-core constraint and are described by the same low energy theory -the Luttinger liquid theory-. Indeed this theory holds also for soft- core bosons, as shown in Ref.(7). Therefore, as far as the transport properties are concerned one should expect the same behavior both for fermions and bosons. On the other hand for lattice models, even in absence of disor- der, the current does not commute with the Hamiltonian, implying its possible decay at finite temperature due to the backscattering processes11. In this case the dynam- ical current-current correlation function also decays in time, leading to a current Fourier transform without δ function at zero energy, namely without a finite Drude weight within the linear response theory. Until few decades ago the absence of the Drude weight was the expected behavior of all interacting metals in lat- tice models or in real solids at finite temperature. How- ever a quite clear numerical evidence has been reported in Ref.12 that current should not decay in integrable 1D models, namely for Hamiltonians that can be solved by Bethe ansatz techniques in 1D. These models essentially possess some hidden conservation law, that was conjec- tured to forbid the current decay process.12,13 Later sev- eral groups have reproduced this surprising effect14,15, with a noticeable exception that a finite Drude weight at finite temperature was found also for non-integrable models.15 On the other hand, from purely theoretical grounds this issue is not settled yet: in Ref.11 it was ar- gued that backscattering processes can be effective also at finite temperature and in 1D non integrable models, whereas in Ref.16, it was proposed that also some par- ticular non integrable model could provide a conserved current. In this work we propose that the general behavior of 1D gapless systems is eventually characterized by a finite Drude weight at finite temperature, and we have found no exception in the models that we have studied. This conclusion is based on a careful and systematic numerical work on fairly generic one dimensional Bose and Fermi systems, that all show the same behavior, even though strong finite size effects are observed in the non integrable cases. In the following we investigate the behavior of the Drude weight in 1D systems in the thermodynamic limit and finite temperature. Model and Method : We have studied hardcore and softcore bosons in a 1D lattice with periodic boundary conditions. The Hamiltonian studied reads, iai+1 + h.c.) + ni(ni − 1) +V nini+1 +Wnini+2 − µni The sum is over all lattice sites i, a i/ai is the boson creation/annihilation operator at site i, henceforth ni is the particle number at site i andµ is the chemical poten- tial. t is the hopping amplitude which is set to one, U is the on-site repulsion, whereas V and W are the nearest and the next-nearest neighbor interactions, respectively. For hardcore bosons in the U → ∞ limit the Hamilto- nian can be mapped onto an S = 1/2 spin system with Szi = ni − 1/2 and S i = a i . In this work we present our results for the half filled case of hardcore and soft- core models. Most of our results have been obtained by Quantum Monte Carlo (QMC), using the stochastic se- ries expansion (SSE)5,17 with the directed loop update18. http://arxiv.org/abs/0704.0406v1 Superfluid density ρs (or spin stiffness in the equiva- lent spin model), is defined as the second derivative of the free energy with respect to a twist in the boundary conditions. In order to compute this quantity by QMC, it is convenient to apply linear response theory, relating this quantity to the current current response function Λ(q, iωn) = dτ exp(iωnτ)〈J(q, τ)J(−q, 0)〉/N , where J is the current operator and ωn is Matsubara frequency. Then the following expression for the superfluid density is obtained: ρs = 〈−K〉 − Λ(q = 0; iωn = 0) = 〈W 2〉 where 〈K〉 is the average kinetic energy per site, ωn = 2πn/β are the Matsubara frequencies andW is the wind- ing number. Similarly the Drude weight is obtained with the same expression but with a different order in the limit ω → 0 and q → 0, namely15,19,20 D = 〈−K〉 − ReΛ(q = 0, ω → 0). (3) In SSE one can obtain Λ very accurately in terms of Matsubara frequencies. Therefore analytic continuation of the data is required. In order to avoid difficulties of extrapolation to iωn → 0 at large temperatures, we have worked at relatively low temperatures (β ≥ 10). In principle, due to the different order of lim- its, the Drude weight and the superfluid density may be different when the following quantity remains fi- nite in the thermodynamic limit15: D − ρs = En=Em β exp(−βEn)|〈ψn|J |ψm〉| 2/L, where, J is the current operator, while En and |ψn〉 are the n th eigen- value and eigenstate of the many body system, respec- tively. The current operator can be written as J(q = 0) = ) where H+ al+1 and b is the bond index, corresponding to the site index l. The ensemble average of product of two local operatorsHσ1 andHσ2 (τ)Hσ1 (0)〉 = n,m=0 (τ − β)n(−τm) 〈ψk|H HmHσ1 |ψk〉 (4) where τ is the imaginary time, Z is partition function and the summation over n and m comes from Taylor- expansion of e(−β+τ)H and e−τH . Following Ref.17 the relation (4) can be simplified to (β − τ)ns−m−2τm (ns − 1)! (ns −m− 2)!m! N b1b2,σ1σ2m where ns is the length of sequence of the local operators and it changes in each QMC sampling. N b1b2,σ1σ2m is the number of times that two operatorsHσ1 and Hσ2 appear in this sequence with distance of m local operators, and 〈...〉W indicates an arithmetic average using configura- tions with relative weight W . In this work we introduce an efficient way to sample by SSE the current-current response function. To this end, we multiply expression (5) by eiωnτ and integrate over the imaginary time τ , we obtain: 1F1(m+ 1, ns; 2iπn)N b1b2σ1σ2 where 1F1(m+ 1, n; z) = (n− 1)! (n−m− 2)!m! dx exp(zx)xm(1− x)n−m−2 (7) is the confluent hypergeometric function. Therefore, the current-current correlation acquires contributions determined by length of operator string ns. All these contributions are stochastically sampled in an efficient way, and in each statistical measurement the correlation function Λ(q = 0, iωn) has the following estimator: σ1,σ2=± 1F 1(m+ 1, ns; 2iπn)N m (8) where Nσ1σ2m = b1,b2 N b1b2,σ1σ2m . Discussion: At zero temperature, for non degenerate ground state, the Drude weight and the superfluidity are the same. In a 1D system at any finite temperature ρs is expected to be zero in the thermodynamic limit, whereas the Drude weight can be non-zero. For hardcore and soft- core bosons in a 1D lattice, a systematic size scaling of the superfluid density ρs clearly shows that this quantity vanishes in the thermodynamic limit and for any finite temperature (see figures 1 and 2). Further, we find that, for a fixed set of parameters and at half filling, all super- fluidity data versus 1/L collapse to one curve whenever the x-axis is appropriately scaled with the temperature T (see figures 1 and 2). This analysis suggests the scal- ing form ρs(β, L) ≡ ρs(β/L). If one takes the order of limit T → 0 after L → ∞, superfluidity remains zero even at zero temperature. Notice that by taking first the limit T → 0 and then L → ∞ superfluidity has a finite value for the gapless phase, but this is not a signature of superfluidity, rather the occurrence of a finite zero tem- perature Drude weight. Though in 1D is not possible to have a finite superfluid density at any non zero tempera- ture, several authors have identified the finite zero tem- perature Drude weight with the superfluid density for a superfluid with vanishing critical temperature. We be- lieve that this identification is a bit confusing and there- fore we prefer to think about absence of superfluidity and superconductivity in 1D systems, as commonly reported in the textbooks. Fig. 3 shows the current-current correlation versus ωn in the metallic and insulating phases of an integrable FIG. 1: (color online) Superfluid stiffness for an integrable (a) and a non-integrable (b) model versus β/L. The system size L is ranging from 50 to 1200. FIG. 2: (color online) Superfluidity of the soft-core bosons versus scaled system size at half filling, the on-site interaction is U = 4 model (W = 0, U = ∞). The zero-frequency value is the superfluid density ρs and the limit ωn → 0 gives the Drude weight D. For W = 0 at zero temperature, there exists a critical value Vc/t = 2 below which the Drude weight is finite. In the first case (a) shown in Fig.(3) with V/t = 2 the Drude weight has a finite value at any finite temperature, which is consistent with the previous works12. In the insulating phase (case b) with V/t = 3, the superfluid density coincides with the Drude weight and they both tend to zero as the system size increases. In a non-integrable model such as hard-core bosons with nearest and next nearest neighborer interactions earlier works have suggested zero Drude weight as system size increases. With SSE we can go to very large system sizes and low temperatures and check the scaling depen- dence of the Drude weight. In Fig. 4 we have plotted current-current correlation versus Matsubara frequency for different L, and a fixed temperature T = 1/100. As shown in the same Figure (4) we have also found a finite Drude weight at finite T in the celebrated Bose-Hubbard model with softcore constraint and in several other mod- FIG. 3: (color online) (a) Current-current correlation for an integrable model in the metallic phase. The zero frequency data shows superfluidity while the extrapolation to n → 0 is the Drude weight. D remains finite with increasing L while ρs vanishes. (b) In the insulating phase D and ρs have the same value and both tend to zero by increasing L. FIG. 4: (color online) Response function vs. n for (a) hard- core bosons with V/t = 1.5, W/t = 1, T/t = 1/100 and (b) Bose-Hubbard model with softcore constraint and U/t = 2, µ/t = −0.4, T/t = 1/25. The system sizes ranges from L = 100 to L = 800. els (not shown). Although some evidence that few par- ticular non integrable models could have a finite Drude weight at finite temperature have been reported before, here we have found a very convincing evidence that this behavior should be generic for 1D gapless system regard- less from their integrability. We have supported this statement by state of the art numerical calculations ob- tained for very large system sizes and low temperature so that all possible extrapolations are perfectly under con- trol. In conclusion it turns out that, at low energy, all gap- less lattice models studied scale to the Luttinger liquid fixed point where the backscattering is a marginally irrel- evant coupling and the current is therefore conserved at the fixed point. This is therefore a peculiar and generic feature of 1D. Indeed in 2D systems, such as hardcore bosons with n.n. repulsion in a square and triangular lattice, we found no difference between ρs and D. Acknowledgments We thank M. Troyer for useful discussions. This work is partially supported by COFIN-2005 and CNR. 1 E. L. Pollock and D. M. Ceperley Phys. Rev. B 36, 8343 (1987). 2 G. G. Batrouni, R. T. Scalettar and G. T. Zimanyi Phys. Rev. Lett. 65, 1765 (1990), ibidem Phys. Rev. B 46, 9051 (1992). 3 L. I. Plimak, M. K. Olsen, and M. Fleischhlauer Phys. Rev. A 70, 013611 (2004). 4 S. Wessel, F. Alet, M. Troyer, and G. G. Batrouni, Phys. Rev. A 70, 053615 (2004). 5 A. W. Sandvik, Phys. Rev. B 56, 11678 (1997). 6 M. P. A. Fisher, P.B. Weichman, G. Grinstein and D. S. Fisher, Phys. Rev. B 40, 546 (1989). 7 see e.g. M. A. Cazalilla J. Phys. B 37, S1 (2004) and ref- erences therein. 8 M. Greiner et al. Nature (London) 415, 39 (2002). 9 M. Greiner et al. Nature (London) 426, 537 (2003). 10 see e.g. H. Moritz et al. Phys. Rev. Lett. 94, 210401 (2005) and references therein. 11 A. Rosch and N. Andrei Phys. Rev. Lett. 85, 1092 (2000). 12 X. Zotos and P. Prelovs̈ek Phys. Rev. B 53, 983 (1996). 13 H. Castella, X. Zotos and P. Prelovs̈ek Phys. Rev. Lett. 74, 972 (1995). 14 D. Poilblanc and et al., Europhys. Lett. 22, 537 (1993). 15 S. Kirchner, H. G. Evertz and W. Hanke Phys. Rev. B 59, 1825 (1999). 16 S. Fujimoto and N. Kawakami, Phys. Rev. Lett. 90, 197202 (2003); ibid S. Fujimoto and N. Kawakami Jour. Phys. A 31, 465 (1998). 17 A. W. Sandvik, J. Phys. A 25, 3667 (1992). 18 O. F. Syljuasen and A. W. Sandvik, Phys. Rev. E 66, 046701 (2002). 19 D. J. Scalapino, S. R. White and S. Zhang Phys. Rev. B 47, 7995 (1993). 20 In principle there is a subtle issue related to the ω → 0 limit, that should be employed for real frequencies. We assume here that the analytic continuation of the function Λ(iωn) is possible, as it is obvious on any finite cluster, and therefore this limit can be obtained by interpolation of Matsubara frequencies around ω = 0, namely at small enough temperatures.

0704.0407

Density dependent hadronic models and the relation between neutron stars and neutron skin thickness

arXiv:0704.0407v1 [nucl-th] 3 Apr 2007 Density dependent hadronic models and the relation between neutron stars and neutron skin thickness S.S. Avancini,1 J.R. Marinelli,1 D.P. Menezes,1 M.M.W. Moraes,1 and C. Providência2 Depto de F́ısica - CFM - Universidade Federal de Santa Catarina Florianópolis - SC - CP. 476 - CEP 88.040 - 900 - Brazil Centro de F́ısica Teórica - Dep. de F́ısica - Universidade de Coimbra - P-3004 - 516 - Coimbra - Portugal In the present work we investigate the main differences in the lead neutron skin thickness, binding energy, surface energy and density profiles obtained with two different density dependent hadron models. Our results are calculated within the Thomas-Fermi approximation with two different numerical prescriptions and compared with results obtained with a common parametrization of the non-linear Walecka model. The neutron skin thickness is a reflex of the equation of state properties. Hence, a direct correlation between the neutron skin thickness and the slope of the symmetry energy is found. We show that within the present approximations the asymmetry parameter for low momentum transfer polarized electron scattering is not sensitive to the model differences. PACS number(s): 21.65.+f,24.10.Jv,95.30.Tg,26.60.+c I. INTRODUCTION The relation between neutron star properties which are obtained from adequate equations of state (EoS) and the neutron skin thickness has long been a topic of investi- gation in the literature. The details of this relation and the important quantities to be discussed have been well established in [1], where it was shown that the difference between the neutron and the proton radii, the neutron skin thickness, is linearly correlated with the pressure of neutron matter at sub-nuclear densities. This is so be- cause the properties of neutron stars are obtained from appropriate EoS whose symmetry energy depends on the density and also controls the size of the neutron skin thickness in heavy and asymmetric nuclei, as 208 Pb, for instance. It is important to remember that the EoS in neutron stars is also very isospin asymmetric due to the β- equilibrium constraint. Hence, isospin asymmetry plays a major role in the un- derstanding of the density dependence of the symmetry energy and the consequences it may arise [2]. In [3, 4] it was shown that the models that yield smaller neutron skins in heavy nuclei tend to yield smaller neutron star radii due to a softer EoS. Neutron stars are believed to have a solid crust formed by nonuniform neutron rich matter in β-equilibrium above a liquid mantle. In the inner crust nuclei coex- ist with a gas of neutrons which have dripped out. The properties of this crust as, for instance, its thickness and pressure at the crust-core interface depend a lot on the density dependence of the EoS used to describe it [4, 5]. On the other hand, it is well known [6, 7] that the ex- istence of phase transitions from liquid to gas phases in asymmetric nuclear matter (ANM) is intrinsically related with the instability regions which are limited by the spin- odals. Instabilities in ANM described within relativistic mean field hadron models, both with constant and den- sity dependent couplings at zero and finite temperatures have already been investigated [7] and it was shown that the main differences occur at finite temperature and large isospin asymmetry close to the boundary of the insta- bility regions. In neutral neutron-proton-electron (npe) matter the electrons are also included. In a thermody- namical calculation the instabilities almost completely disappear due to the high electron Fermi energy [8]. However, in a dynamical calculation which includes the Coulomb interaction and allows for independent neutron, proton and electron fluctuations [9, 10], it is seen that the electron dynamics tends to restore the short wave- length instabilities although moderated by the high elec- tron Fermi energy. Moreover, it is also known that the liquid-gas phase transition in ANM can lead to an isospin distillation phe- nomenon, characterized by a larger proton fraction in the liquid phase than in the gas phase. This is due to the repulsive isovector channel of the nuclear interaction [11–13]. In a recent work the spinodal section and related quan- tities, as the neutron to proton density fluctuations re- sponsible for the distillation effect, has been studied within different relativistic models [8]. It was shown that the distillation effect within density dependent relativis- tic models decreases with density above a nuclear density of ∼ 0.02−0.03 fm−3, a result similar to the one obtained with the SLy230a parametrization of Skyrme interaction [14] and contrary to the results found with the more com- mon relativistic parametrizations with no density depen- dent coupling parameters. In the last case the distillation effect becomes always larger as the density increases. Also, the behavior of the symmetry energy obtained with density dependent models is closer to what one ob- tains with non-relativistic models than with other rel- ativistic models with constant couplings [7]. In an at- tempt to understand this behavior, a comparison be- tween the non-relativistic Skyrme effective force and rel- ativistic mean field models at subsaturation densities was performed [15]. It was shown that the relativistic mod- els could also be reduced to an energy density functional similar to the one describing the Skyrme interaction. http://arxiv.org/abs/0704.0407v1 There have already been some efforts in order to com- pare nuclear matter and finite nuclei properties obtained both with relativistic and non-relativistic models [16, 17] but there is no clear or obvious explanations for the differ- ences. At very low densities both, the relativistic and the non-relativistic approaches predict a non-homogeneous phase commonly named pasta phase, formed by a com- petition between the long-range Coulomb repulsion and the short-range nuclear attraction [18]. Based on the above arguments, it is very important that an accurate experimental measurement of the neu- tron skin thickness is achieved. This depends on a precise measurement of both the charge and the neutron radius. The charge radius is already known within a precision of one percent for most stable nuclei, using the well-known single-arm and non-polarized elastic electron scattering technique as well as the spectroscopy of muonic atoms [19] . For the neutron radius, our present knowledge has an uncertainty of about 0.2 fm [20]. However, using po- larized electron beams it is possible to obtain the neutron distribution in nuclei in a fairly model independent way, as first discussed in [21] and, as a consequence, to obtain the desired neutron radius. In fact, the Parity Radius Experiment (PREX) at the Jefferson Laboratory [22] is currently running to measure the 208Pb neutron radius with an accuracy of less than 0.05 fm, using polarized electron scattering. In the present work, we use two different hadronic models that incorporate density dependence in differ- ent ways. The first one, to which we refer next as the TW model is a density dependent hadronic model with the meson-to-nucleon couplings explicitly dependent of the density [23, 24]. In the following it is used to cal- culate the neutron skin thickness of 208Pb, which is a neutron-rich heavy nucleus. This model was chosen be- cause it is based on a microscopic calculation, fits well many nuclei properties and, as stated above, has shown to provide results which are different from the usual NL3 [25] and TM1 [26] parametrizations for the non-linear Walecka model (NLWM), having a richer density depen- dence of the symmetry energy than most of the rela- tivistic nuclear models. The original motivation for the development of this density dependent hadronic model [27, 28] was to reproduce results obtained with the rel- ativistic Dirac-Brueckner Hartree-Fock (DBHF) theory [29]. Later the DBHF calculations for nuclear matter were taken only as a guide for a suitable parametrization of the density dependence of the meson-nucleon coupling operators [24, 30]. Moreover, density dependent hadronic models can also be a useful tool in obtaining EoS for neutron stars even if hyperons are to be considered [32], which is not the case if NL3 or TM1 are used. Both, NL3 and TM1, can only be used if the EoS is restricted to ac- commodate neutrons, protons and the leptons necessary to enforce β-stability. Once hyperons are included, the nucleons acquire a negative effective mass above∼ 3−4ρ0 densities [33, 34], where ρ0 is the nuclear saturation den- sity. The second model, that we refer to as NLωρ model, includes non-linear σ − ρ and ω − ρ couplings [3, 35– 37] which allow to change the density dependence of the symmetry energy of the most common parametrizations of the NLWM that show essentially a linear behavior of the symmetry energy with density. However, the symme- try energy determines the behavior of isospin asymmetric matter and therefore is intrinsically related to the char- acteristics of the EoS that can describe neutron stars. Within this model the authors of [3] have shown that the neutron skin thickness of 208Pb was sensitive to the isovector channel of the nuclear interaction and there was a correlation between neutron skin thickness of nuclei and properties of neutron stars. For the sake of completeness, the results of the present work, whenever possible are compared with the results obtained with the NL3 parametrization of the NLWM, known to describe finite nuclei properties well. We perform two different numerical calculations to ob- tain the 208Pb properties: a Thomas-Fermi approxima- tion based on the liquid-gas phase transition developed in [38] and a Thomas-Fermi approximation based on a method proposed in [39], where a harmonic oscillator ba- sis is used. We restrict ourselves to the Thomas-Fermi approximation because, as we show in the Results sec- tion at the end of the paper, for the purpose of obtaining correct surface energy and neutron-skin thickness, it is almost as good as the solution of the Dirac equation. At this point it is worth mentioning that the scalar- isovector δ mesons, which play an important role in the isospin channel, could also be incorporated in our work as done in [7, 9, 40] but in order to make the compar- isons among different approximations as simple as possi- ble, they will be included in a future work. Finally, as we are interested in nuclei ground state properties, all calculations are performed at zero temperature. II. THE TW DENSITY DEPENDENT HADRONIC MODEL Next we describe the main quantities of the TWmodel, which has density dependent coupling parameters. The Lagrangian density reads: L = ψ̄ i∂µ − ΓvV µ − τ · bµ −e (1 + τi3) − (M − Γsφ) (∂µφ∂ µφ−m2sφ2)− m2vVµV Bµν ·Bµν + m2ρbµ ·bµ− µν (1) where φ, V µ, bµ and Aµ are the scalar-isoscalar, vector- isoscalar and vector-isovector meson fields and the pho- ton field respectively, Ωµν = ∂µVν − ∂νVµ , Bµν = ∂µbν − ∂νbµ − Γρ(bµ × bν), Fµν = ∂µAν − ∂νAµ and τp3 = 1, and τn3 = −1. The parameters of the model are: the nucleon mass M = 939 MeV, the masses of the mesons ms, mv, mρ, the electromagnetic coupling con- stant e = 4π/137 and the density dependent coupling constants Γs, Γv and Γρ, which are adjusted in order to reproduce some of the nuclear matter bulk properties shown in Table I, using the following parametrization: Γi(ρ) = Γi(ρsat)hi(x), x = ρ/ρsat, (2) hi(x) = ai 1 + bi(x+ di) 1 + ci(x + di)2 , i = s, v (3) hρ(x) = exp[−aρ(x− 1)], (4) with the values of the parameters mi, Γi(ρsat), ai, bi, ci and di, i = s, v, ρ given in [24]. This model does not include self-interaction terms for the meson fields (i.e. κ = 0, λ = 0 and ξ = 0 ) as in NL3 or TM1 parametriza- tions for the NLWM. The field equations of motion follow from the Euler- Lagrange equations. When they are obtained, some care has to be taken since the coupling operators depend on the baryon fields ψ̄ and ψ through the density. When the partial derivatives of L are performed relatively to the fields ψ̄ and ψ, they yield extra terms due to the func- tional dependence of the coupling operators. The new terms are absent in the usual Quantum Hadrodynamic (QHD, NLWM) models [25, 26, 31]. The equations of motion for the fields read: µ +m2φ)φ = Γsψ̄ψ, (5) µν +m2vV µ = Γvψ̄γ µψ, (6) µν +m2ρb ψ̄τγµψ, (7) ψ̄(1 + τ3)γ µψ, (8) [γµ(i∂ µ − Σµ)−M∗]ψ = 0 , (9) whereM∗ =M−Γsφ. Notice that in the equation of mo- tion for the baryon field ψ the vector self-energy consists of two terms, Σµ = Σ µ + Σ µ , where: Σ(0)µ = ΓωVµ + τ · bµ + (1 + τ3)Aµ, (10) ΣRµ = V νjν + bν · jν3 − where Σ µ is the usual vector self-energy, ρ̂uµ = jµ with u2 = 1 jν = ψ̄γνψ, j 3 = ψ̄τγ νψ and, as a result of the derivative of the Lagrangian with respect to ρ a new term appears, ΣRµ , which is called rearrangement self- energy and has been shown to play an essential rôle in the applications of the theory. This term guarantees the thermodynamical consistency and the energy-momentum conservation. For more detailed calculations, at zero and finite temperatures, please refer to [41]. In the static case there are no currents in the nucleus and the spatial vector components are zero. Therefore, the mesonic equations of motion become: ∇2φ = m2sφ− Γsρs, (12) ∇2V0 = m2vV0 − Γvρ, (13) ∇2b0 = m2ρb0 − ρ3, (14) ∇2A0 = −eρp, (15) where ρs =< ψ̄ψ > is the scalar density, ρ = ρp + ρn, ρ3 = ρp − ρn and ρp and ρn are the proton and neutron densities. A. Thomas-Fermi approximation We first define the functional Ω = E − µpBp − µnBn, (16) where E is the energy, µp (µn) is the proton (neutron) chemical potential and Bp (Bn) is the proton (neutron) number. Within the semi-classical Thomas-Fermi ap- proximation, the energy of the nuclear system with par- ticles described by the one-body phase-space distribution function f(r,p, t) at position r, instant t with momentum p is given by (2π)3 fi(r,p, t) p2 +M∗2 + Vi (∇φ)2 +m2sφ2 − (∇V0)2 −m2vV 20 −(∇b0)2 −m2ρb20 − (∇A0)2 where Vp = ΓvV0 + b0 + eA0 , Vn = ΓvV0 − γ = 2 refers to the spin multiplicity and the distribu- tion functions for protons and neutrons are fi = θ(k Fi(r) − p2), i = p, n . In this approach, the scalar, proton and neutron densities become: ρs(r) = i=p,n ∫ kFi(r) with ǫ = p2 +M∗2 and d3rρi, ρi(r) = k3Fi(r). From the above expressions we get for (16) (∇φ)2 − (∇V0)2 − (∇b0)2 − (∇A0)2 + Vef Vef = 2 −m2vV 20 −m2ρb20 − µpρp − µnρn i=p,n ∫ kFi dpp2ǫ+ΓvV0ρ+Γρ ρ3 + eA0ρp (18) Minimization of Ω with respect to kFi(r), i = p, n, gives rise to the following conditions k2Fp +M ∗2 − ΓvV0 − b0 − eA0 − ΣR0 k2Fn +M ∗2 − ΓvV0 + b0 − ΣR0 where the rearrangement term is ΣR0 = ρ V0 + − ∂ Γs ρs φ. From the above equations we obtain kFp = 0 and kFn = 0 or, for kFp or kFn different from zero, k2Fp +M ∗2 + ΓvV0 + b0 + eA0 +Σ 0 , (19) k2Fn +M ∗2 + ΓvV0 − b0 +Σ 0 . (20) The values of kFp and kFn are obtained inverting these two last equations. Such density dependences in the coupling parameters do not affect the energy functional but of course affect its derivative such as the pressure density and the chem- ical potentials. As already discussed in the literature [7–9, 32], the rearrangement term is crucial in obtaining different behaviors in physical properties related to the chemical potentials or to their derivatives with respect to the density, such as spinodal regions, as compared with the more common NL3 or TM1 parametrizations. III. NLωρ MODEL The Lagrangian density that incorporates the extra non-linear σ − ρ and ω − ρ couplings [3, 35–37] reads L = ψ̄ i∂µ − gvV µ − τ · bµ −e (1 + τi3) − (M − gsφ) (∂µφ∂ µφ−m2sφ2)− κφ3 − 1 λφ4 − 1 m2vVµV µ − 1 Bµν ·Bµν + m2ρbµ · bµ − +g2ρbµ · bµ[Λsg2sφ2 + Λvg2vVµV µ], (21) where Ωµν , Bµν and Fµν are defined after eq.(1). The parameters of the model are again the masses and the couplings, which are now constants, i.e., gs replaces Γs, gv replaces Γv and gρ replaces Γρ. Non-linear σ terms are also included. We have followed the prescription of [3], where the starting point was the NL3 parametrization and the gρ coupling was adjusted for each value of the coupling Λi studied in such a way that for kF = 1.15 fm−1 (not the saturation point) the symmetry energy is 25.68 MeV. In the present work we set Λs = 0 as in [37]. Notice that other possibilities for this model with σ − ρ and ω − ρ couplings have already been discussed in the literature as in [4], for instance. The mesonic equations of motion in the Thomas-Fermi approximation become ∇2φ = m2sφ− gsρs + φ3 (22) ∇2V0 = m2vV0 − gvρ+ 2Λvg2v V0 g2ρb20, (23) ∇2b0 = m2ρb0 − ρ3 + 2Λvg 0 , (24) ∇2A0 = −eρp, (25) and the expression for the energy reads ∫ kFi(r) (2π)3 p2 +M∗2 (∇φ)2 +m2sφ2 − (∇V0)2 −m2vV 20 −(∇b0)2 −m2ρb20 − (∇A0)2 +gvV0ρ+ ρ3b0 + eA0ρp φ4 − Λvg2vV 20 g2ρb20 . (26) All other expressions are very similar to the ones ob- tained from the TW model and can be read off from them bearing in mind that the density dependent cou- plings have to be replaced by the constant couplings. In particular the chemical potentials do not contain the re- arrangement term ΣR0 . IV. NUMERICAL RESULT VIA A NUCLEATION PROCESS At this point, eqs. (12-15) for the TW model and eqs. (22-25) for the NLωρmodel have to be solved numerically in a self-consistent way and hence, initial and boundary conditions for each equation are necessary. One of the methods we use here is based on a prescription given in [38], where these conditions are obtained from a situation of phase coexistence in a mean field approximation with classical meson fields and no electromagnetic interaction. The method is well explained in [38] and, as we are using different models here, just the main equations are written next. For the TW model, the equilibrium equations for ho- mogeneous matter for the fields are: m2sφ− Γs ρs = 0, (27) m2vV0 − Γv ρ = 0, (28) m2ρb0 − ρ3 = 0, (29) and for the energy and pressure density: E = 1 ∫ kFi p2 +M∗2 V 20 + b20, (30) ∫ kFi V 20 + +ρΣR0 . (31) For the NLωρ model, the equilibrium equations for ho- mogenous matter, energy density and pressure become: m2sφ− gsρs + φ3 = 0, (32) m2vV0 − gvρ+ 2Λvg2v V0 g2ρb20 = 0, (33) m2ρb0 − ρ3 + 2Λvg 0 = 0, (34) E = 1 ∫ kFi p2 +M∗2 2 −m2vV 20 −m2ρb20 + gvV0ρ+ φ4 − Λvg2vV 20 g2ρb20. (35) ∫ kFi V 20 + φ3 − λ φ4 + Λvg 0. (36) Based on the geometrical construction and Gibbs con- ditions for phase coexistence, i.e., the pressure and both chemical potentials are equal in both phases, we build the binodal section given in Fig. 1. Notice that we have defined the proton fraction of the system as . (37) The binodal section yields the boundary conditions which we need. For the same pressure, two points, with differ- ent proton fractions are found. For each of these points, the meson fields and the densities are well defined and used as the initial and boundary conditions in eqs. (12- 15), which are then solved. Once the meson fields are obtained, all the quantities that depend on them, as the energy, pressure densities, chemical potentials, baryonic densities, etc are also computed. The solution is a droplet with a certain proton fraction surrounded by a gas of neutrons. If stable nuclei are calculated, the gas vanishes because the energy of the system lies below the neutron drip line and the finite nuclei properties are easily calcu- lated. This is the general method, but the results depend strongly on the model used because of the reasons dis- cussed in Section VI. V. NUMERICAL RESULT WITHIN A HARMONIC OSCILLATOR BASIS Here a different prescription for solving the equations of motion and the thermodynamical quantities within the Thomas-Fermi approximation is used. According to [39], meson field equations of motion of the Klein-Gordon type with sources can be carried out by an expansion in a com- plete set of basis states. The harmonic oscillator func- tions with orbital angular momentum equal to zero are then chosen. The oscillator length is given by , b0 = , (38) where M is the nucleon mass and ω0 is the oscillator frequency. The meson fields and their corresponding in- homogeneous part can be expanded as Λ(r) = ΛnRn0(r), SΛ(r) = SΛnRn0(r), (39) where Λ(r) = φ(r), V0(r), b0(r) and Rnl(r) = l+1/2 n−1 (x 2)exp(−x2/2), (40) where x = r/b0 is the radius measured in units of the oscillator length, Nnl = 2(n− 1)!/(l + n− 1/2)! (41) is the normalization constant and Lmn (x 2) are the asso- ciated Laguerre polynomials. For the calculation of the meson fields l = 0 in the expressions given below. Once the ansatz given by eqs.(39) are substituted into eqs.(12- 14), a set of inhomogeneous equations is obtained: Hnn′Λn′ = SΛn (42) where Hnn′ = δnn′ b−2B (2(n− 1) + 3/2) +m +δnn′+1b n(n+ 1/2) + δn+1n′b n′(n′ + 1/2). vΛ =0.01 vΛ =0.025 0 0.1 0.2 0.3 0.4 0.5 FIG. 1: Binodal section for the NL3, TW and NLωρ parametrizations. Only the massive fields can be calculated with this method because the convergence of the Coulomb field, which has a long range, is very slow. The Green’s func- tion method is then chosen to describe the electromag- netic interaction: A0(r) = e r′2dr′ρp(r ′)Gc(r, r ′), (44) Gc(r, r 1/r for r > r′ 1/r′ for r′ > r. VI. RESULTS A. Parity Violating Electron Scattering and the Neutron Radius We start this section by defining the asymmetry for polarized electron scattering of a hadronic target as A = dσ+/dΩ− dσ−/dΩ dσ+/dΩ+ dσ−/dΩ , (46) where dσ±/dΩ is the differential cross section for initially polarized electrons with positive(+) and negative (−) he- licities. As the electromagnetic interaction is not sensi- tive to the above difference, the asymmetry becomes de- pendent of the weak interaction between the electron and the target. Moreover, we know from the Standard Model that the neutral Z-boson couples more strongly to the neutron than to the proton. Those reasonings were then used in [21] to first propose a clean way to determine the neutron distribution in nuclei. If we consider elastic scattering on an even-even target nucleus, the asymmetry can be written in the form: V + β ρn(q) ρp(q) ]. (47) In the above expression, G, α, a and β V are Standard Model coupling constants as defined in [21], q is the trans- ferred momentum by the electron to the nucleus and, ρn(p)(q) = d3r j0(qr)ρn(p)(r), (48) ρn(p)(r) being the neutron (proton) distribution in con- figuration space and j0 the spherical Bessel function of order zero. It is then clear that a small q measurement of the asymmetry gives the neutron radius of the distri- bution once the proton radius is well known. The proton and neutron mean-square radius are defined as R2i = d3rr2ρi(r) d3rρi(r) , i = p, n. (49) The neutron skin thickness is defined as θ = Rn −Rp. (50) In the PREX experiment mentioned in the Introduc- tion, the asymmetry is expected to be measured at q ≈ 0.4 fm−1 [22]. Also, because the target is a heavy nu- cleus (208Pb), the above results for the asymmetry should be reconsidered for a detailed comparison with the exper- iment, since they were obtained using a Plane Wave Born Approximation for the electron [43]. For our present pur- poses, eq. (47) is sufficient to illustrate the sensitivity to the different model parametrizations and is used next in the presentation of our numerical results. The surface energy per unit area of the droplets in the small surface thickness approximation, excluding the electromagnetic field, reads [38] . (51) However, as the electromagnetic interaction does not con- tribute to surface properties directly, we have kept the same definition for the surface energy. In Table II we show the neutron and proton radius, the neutron skin thickness, the binding energy and the sur- face energy obtained within the Thomas-Fermi approxi- mation and the two different numerical prescriptions de- scribed in the previous sections. All the results are sensi- tive to the numerical calculation although the analytical approximation is the same. When the nucleation method is performed, the neutron radius is systematically larger, what results in a thicker neutron skin. This is correlated with the fact that the surface energy is lower within the nucleation calculation than within the harmonic oscilla- tor method. Within the same numerical prescription, the neutron skin thickness is smaller with the TW model than with the NL3. As the coupling strength Λv increases in the NLωρ model, the results move from the original NL3 to the TW results for all quantities, except the pro- ton radius, which oscillates a little. We have also in- cluded the results obtained with the HS parametrization [44] because we have used this parametrization in order to compare the TF and the Dirac results for the cross sec- tions, as discussed in the following. As this parametriza- tion is known not to give as good results as the other parametrizations of the NLWM for finite nuclei, we do not comment on the results it provides. Notice that the experimental radius for the protons is obtained from the charge radius Rc and it is given by Rp = R2c − 0.64 in fm [39]. Our results can be compared with experimental and other theoretical results found in the literature. The proton radius, which is known to better than 0.001 fm is better described within the TW model. This quantity is practically independent of the ω− ρ interaction strength in the NLωρ model as far as the HO numerical prescrip- tion is used. The neutron radius, on the other had, is strongly model dependent with drastic consequences in the neutron skin thickness calculation. The experimen- tal values for θ are still very uncertain and all our re- sults fall inside the experimental confidence interval. We shall comment on possible restrictions to the neutron skin thickness in the next section. NL3 provides the best re- sults for the binding energy. In [25], the results shown for the proton and neutron radius are respectively 5.52 and 5.85 fm, yielding a skin of 0.33 fm, larger than ours. Notice, however, that in [25] the Dirac equation was explicity solved. In [4], the authors obtained a value of 0.21 fm for the neutron skin thickness and a binding energy of -7.89 MeV within a different parametrization of the NLωρ model. Again in this case the Dirac equation was solved. In Fig. 2 we show the difference between neutron and proton densities at the Pb surface for the models dis- cussed in the present work with the Thomas-Fermi ap- proximation solved in a harmonic oscillator basis. While the curves deviate a little in between 6.0 and 8.0 fm, at the very surface they are similar, but a small discrepancy, reflecting the differences in the neutron skin can be seen. In Fig. 3 we display again the difference between neu- tron and proton densities within both numerical calcula- tions of the TW and NL3 models. These two Thomas- Fermi calculations should have given more similar results. However the nucleation method predicts a very small sur- face energy for the NL3 parametrization, and therefore, a large radius. This may be related to the choice of the boundary conditions and a deeper comparison between the two methods will be pursued. Next we present our results for the asymmetry given by eq. (47) as a function of the transfered momentum. We begin with Fig.4 which displays the results for the HS parametrization of the Walecka model. The curve labeled no structure means the case where Zρn(r) = Nρp(r) and the other two curves are obtained within the TF approx- imation and the full solution of the Dirac equation in 6 7 8 9 0.000 0.005 0.010 0.015 0.020 0.025 NL ( v ) NL ( v ) r(fm) FIG. 2: Difference between neutron and proton densities ob- tained with the Thomas-Fermi approach solved in a harmonic oscillator basis for the models discussed in the present work. the Hartree approximation. At the momentum trans- fer values of recent experimental interest (around ≃ 0.4 fm−1), the curves are almost identical. A careful analy- sis of the same results in a different scale shows us that the asymmetry changes 12 and 11 percent respectively within the Dirac and TF approximations in comparison with the no structure case. Since it is the measurement of the asymmetry in this low momentum transfer region that will provide the accurate result for the neutron skin thickness, we have restricted our calculations to the TF approximation, as stated in the Introduction. In Fig. 5a we show the asymmetry obtained with the NL3 model for both numerical calculations in the TF ap- proximation, i.e, nucleation and HO expansion methods. In this case, the agreement is very satisfactory even for larger q-values, although the small numerical discrepan- cies is reflected in a ∼ 10 percent difference in the pre- dicted neutron skin thickness, as can be seen from Table II. Finally, in Fig. 5b our results for the NLωρ (using two different values for the ω− ρ coupling constant) and the TWmodels within the HO numerical prescription are shown. Again, at low momentum transfers, all curves co- incide. However, it should be noticed that even for two different model parametrizations which lead us to identi- cal neutron skin thicknesses, a measurement of the asym- metry in a higher q-region with a modest experimental precision, can distinguish between them. Also, we should expect that the asymmetry presents more structure in this high momentum transfer region if we solve the Dirac equation instead of using the TF approach, once the high q value region is much more sensitive to the central part of the neutron distribution, which is known to be flat in the TF approximation. These differences can be seen in Fig.4. NL3 nucl TW nucl NL3 HO TW HO r(fm) FIG. 3: Difference between neutron and proton densities ob- tained with the Thomas-Fermi approach solved with both nu- merical prescriptions for the TW model. VII. DIFFERENT EOS, DIFFERENT NEUTRON SKINS For the sake of completeness, at this point, we discuss some of the differences between the TW, the NLωρ mod- els and the NL3 parametrization of the NLWM. From Fig. 1 one can see that the largest possible pres- sure for a phase coexistence in the TW model is much lower, and appears at a lower proton fraction than the NL3 model. This gives rise to a thinner crust within the TW model, which may imply that the more exotic pasta shapes will not form [5]. The NLωρ model goes on a different direction, i.e., the pressure becomes higher than the one obtained with the NL3 as the Λv coupling is turned on. Although the nuclear matter properties fitted to parametrize the models are quite similar (see Table I), the way the EoS behaves when extrapolated to higher or lower densities can vary a lot from a density depen- dent hadron model to one of the parametrizations of the NLWM. Moreover, as seen from Table I, although the ef- fective mass at saturation density is lower with the TW than with the NL3, it can accommodate hyperons if an EoS for stellar matter is necessary, contrary to the usual 0.0 0.5 1.0 1.5 2.0 no structure HS-Dirac HS-TF A q(fm-1) 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 2.5x10-7 5.0x10-7 7.5x10-7 1.0x10-6 1.3x10-6 1.5x10-6 no structure HS-Dirac HS-TF q(fm-1) FIG. 4: Parametrization HS, comparison Thomas-Fermi-HO versus Dirac-HO NL3 parametrization [32–34]. TABLE I: Nuclear matter properties. NL3 NLωρ TW [25] [36] [24] Λv = 0.01 Λv = 0.02 Λv = 0.025 B/A (MeV) 16.3 16.3 16.3 16.3 16.3 ρ0 (fm −3) 0.148 0.148 0.148 0.148 0.153 K (MeV) 271 271 271 271 240 Esym. (MeV) 37.4 34.9 33.1 32.3 32.0 M∗/M 0.60 0.60 0.60 0.60 0.56 L (MeV) 118 88 68 61 55 Ksym (MeV) 100 -46 -53 -34 -124 Another quantity of interest in asymmetric nuclear matter is the nuclear bulk symmetry energy, shown in Table I for the saturation point. The differences in the symmetry energy at densities larger than the nuclear sat- uration density is still not well established, but has al- ready been extensively discussed in the literature even 0.0 0.5 1.0 1.5 2.0 q(fm-1) NL3 nucl NL3 HO no structure 0.0 0.5 1.0 1.5 2.0 q(fm-1) NL ( v=0.01) NL ( v=0.025) FIG. 5: Asymmetry obtained with a) NL3 with both numer- ical prescriptions and b)parametrizations NLωρ and TW for the TW model [7, 8, 16, 32]. Again, for the sake of completeness we reproduce these results here because the neutron skin thickness and the neutron star EoS are related by this quantity [1–4], which is usually defined as Esym = 12 ∂2E/ρ , with δ = −ρ3/ρ = 1 − 2yp. The symmetry energy can be analytically rewritten as Esym = ρ, (52) for the TW model and as Esym = ρ (53) with the effective ρ-meson mass defined as [3] = m2ρ + 2g for the NLωρ model. In both cases kFp = kF (1 + δ) 1/3, kFn = kF (1− δ)1/3, with kF = (1.5π 2ρ)1/3 and ǫF = k2F +M ∗2. In equa- tions (52) and (53) the second term dominates at large densities. It is seen that the non-linear ρ − ω terms introduce a non-linear density behavior in the symme- try energy of the NLWM parametrizations such as NL3 and TM1. In TW the non-linear density behavior en- ters through the density dependent coupling parameters. These non-linear density behavior is important because the linear behavior of NL3 and TM1 parametrizations predicts too high symmetry energy at densities of impor- tance for neutron star matter which has direct influence on the proton fraction dependence with density. From Fig. 6, it is easily seen that the symmetry energy ob- tained with the TW model behaves in a very different way, as compared with NL3. In [4] a relation between the symmetry energy and the nuclear binding energy is discussed : the harder the EoS, the more the symmetry (fm )−3ρ vΛ =0.01 vΛ =0.025 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 FIG. 6: Symmetry energy for the NL3, TW and NLωρmodels. energy rises with density. The density dependence dis- cussed in [4] is of the type introduced in [3, 36] through the inclusion of a σ− ρ and/or ω− ρ couplings and then, similar with the NLωρ model discussed here. One can observe that as the strength of the coupling increases, the symmetry energy gets closer to the TW curve. In fact, in [8] it was shown that once this kind of coupling is introduced with a reasonable strength, the symmetry energy at low densities tends to behave as the TW model. The symmetry energy can be expanded around the nu- clear saturation density and reads Esym(ρ) = Esym(ρ0) + ρ− ρ0 ρ− ρ0 where L and Ksym are respectively the slope and the curvature of the nuclear symmetry energy at ρ0 and they are calculated from L = 3ρ0 ∂Esym(ρ) |ρ=ρ0 Ksym = 9ρ20 ∂2Esym(ρ) |ρ=ρ0 . These two quantities can provide important information on the symmetry energy at both high and low densi- ties because they characterize the density dependence of the energy symmetry. In a recent work [49], the authors found a correlation between the slope of the symmetry energy and the neutron skin thickness. In their work 21 sets of the non-relativistic Skyrme potential were inves- tigated and only 4 of them were shown to have L values consistent with the values extracted from experimental isospin diffusion data from heavy ion collisions. In fact, the extracted value was L = 88 ± 25 MeV [50], which gives a very strong constraint on the density dependence of the nuclear symmetry energy and consequently on the EoS as well. A detailed analysis of Table I shows that, if this constraint is to be taken seriously, neither the NL3 nor the TW model satisfy it. Nevertheless, the NLωρ slope interpolates beautifully between the NL3 and TW slope values. Once again it is seen that the increase in Λv approximates the NL3 model values for the slope and energy symmetry to the TW values. Moreover, we have also tried to find a correlation between the θ values shown in Table II and L values displayed in Table I. We found that, as far as some numerical imprecision are consid- ered, larger values of L correspond to larger values of the neutron skin, as seen in Fig. 7. Let’s now go back to the problem of solving the dif- ferential equations within the nucleation numerical pre- scription. As we need boundary conditions arising from the liquid-gas phase coexistence in order to solve eqs. (12-15) for the TW model and eqs. (22-25) for the NLωρ model, the binodal sections are essential and the spinodal sections, which separate the regions of stable to unsta- ble matter are also of interest. If we had displayed the binodals in a ρp versus ρn plot, as it is done with the spinodals in Fig 8, we could see that the spinodals sur- faces lie inside the binodal sections and share the critical point corresponding to the highest pressure. In Fig. 8 the spinodals for the three different mod- els discussed in this work are shown. Once again, some of these results can also be found in the recent litera- ture [7, 8], but we include them here to make a direct link with the binodals. The instability of the ANM sys- tem is essentially determined by density fluctuations in the isoscalar channel. Although the spinodals are, by themselves, not relevant in calculations performed at the thermodynamical equilibrium, the isospin channel is very sensitive to the instabilities occurring below the nuclear saturation density. The spinodal is determined by the values of pressure, proton fraction and density for which the determinant of Fij = ∂ρi∂ρj , (56) where F is the free energy density, goes to zero. A de- tailed analysis of this quantity can be found in [8, 42]. From Fig. 8, it is seen that the instability region in the ρp/ρn plane, defined by the inner section of the spinodal curve is larger for the TW than for the NL3 model. The size of the instability region depends on the derivative of the chemical potentials with respect to the neutron and proton densities. At low densities different models exhibit different behaviors. The presence of the rearrangement term in the TW model also plays a decisive role. Even though a rela- tively large compensation exists between scalar and vec- tor mesons in the isoscalar channels within the rearrange- ment term at low densities, the spinodal region is defined by the derivative of the chemical potential and therefore of the rearrangement term. Next we examine the spinodals obtained with differ- ent coupling strengths for the NLωρ model. As seen in Fig. 8, there is almost no difference between the different curves. They all fall around the original NL3 curve but once again, they tend to the TW curve as the coupling strength increases. However, contrary to the TW model, it was shown in [9] that the direction of the instability in Λ =0.01 Λ =0.02 Λ =0.025v 0.16 0.18 0.22 0.24 50 60 70 80 90 100 110 120 L (MeV) FIG. 7: Correlation between the neutron skin θ and the slope of the symmetry energy L. NLωρ increases distillation as the density increases, and the larger the coupling Λv the larger the effect. Finally, to end this section, let’s make our points clear: we have used a simple mean field theory approach to ob- tain the boundary conditions for the equations of motion of the meson fields in the nucleation prescription. These boundary conditions depend on the model used and are intrinsically related with the liquid-gas phase transition which, in turn, can be well understood by studying the coexistence surfaces of the corresponding models. On the other hand, the neutron skin thickness shows a lin- ear correlation with the slope of the symmetry energy, as already pointed out in [49] for non-relativistic mod- els. Based on the different behaviors found with density dependent hadronic models and the NLWM, an obvious consequence is the fact that the neutron skin thickness depends on the choice of the model. VIII. CONCLUSIONS We have calculated the 208Pb neutron skin thickness with two different density dependent hadronic models, the TW and the NLωρ model, and one of the most used parametrizations of the NLWM, the NL3. The calcu- lations were done within the Thomas-Fermi approxima- tion, which gives quite accurate results for the asymme- try in the momentum transfer range of interest for the calculation of neutron skins. In implementing the nu- merical results two different prescriptions were used: the first one based on the nucleation process and the second one based on the harmonic oscillator basis method. We have seen that when the nucleation method is performed, the neutron radius is systematically larger, what results vΛ =0.01 vΛ =0.025 ρ (fm )−3 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08 FIG. 8: Spinodal section in terms of ρp versus ρn for the NL3, TW and NLωρ models. in a thicker neutron skin. This is a consequence of the fact that the surface energy is lower within the nucleation calculation than within the harmonic oscillator method. Within the same numerical prescription, the neutron skin thickness is smaller with the TW model than with the NL3. As the coupling strength Λv increases in the NLωρ model, the neutron skin thickness moves from the orig- inal NL3 towards the TW results. We have also found that although the neutron skin thickness is model depen- dent, the asymmetry at low momentum transfers (below 0.5 fm−1) is very similar for all models and all numerical prescriptions. As q increases, the asymmetry also be- comes model dependent. The density profiles obtained from the solution of the Dirac equation exhibits oscil- lations near the center of the nucleus, behavior which is not reproduced within the Thomas-Fermi approximation. This fact shows up in the asymmetry at large momentum transfers and therefore all the calculations should be re- produced by solving the Dirac equation. This calculation is already under investigation. It is worth mentioning that the neutron skin thickness has shown to give hints on the equations of state that are suitable to describe neutron stars. Moreover, in [49] a correlation between the slope of the symmetry energy and the neutron skin thickness was found for Skyrme- type models. We have observed that this correlation was also present in the density dependent models we have studied in the present work. ACKNOWLEDGMENTS This work was partially supported by CNPq(Brazil), CAPES(Brazil)/GRICES (Portugal) under project 100/03 and FEDER/FCT (Portugal) under the projects POCTI/FP/63419/2005 and POCTI/FP/63918/2005. [1] S. Typel and B.A. Brown, Phys. Rev. C 64, 027302 (2001). [2] A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys. Rep. 411, 325 (2005). [3] C.J. Horowitz and J.Piekarewicz, Phys. Rev. Lett. 86, 5647 (2001). [4] J.Piekarewicz, nucl-th/0607039. Proceedings of the ”In- ternational Conference on Current Problems in Nuclear Physics and Atomic Energy” (May 29 - June 3, 2006) Kyiv, UKRAINE. [5] F. Duchoin and Haensel, Phys. Lett. B 485, 107 (2000). [6] Ph. Chomaz, C. Colonna and J. Randrup, Phys. Rep. 389, 263 (2004). [7] S.S. Avancini, L. Brito, D. P. Menezes and C. Providência, Phys. Rev. C 70, 015203 (2004). [8] S.S. Avancini, L. Brito, Ph. Chomaz, D. P. Menezes and C. Providência, Phys. Rev. C 74, 024317 (2006). [9] C. Providência, L. Brito, S.S. Avancini, D. P. Menezes and Ph. Chomaz, Phys. Rev. C 73, 025805 (2006). [10] L. Brito, C. Providência, A.M.S. Santos, S.S. Avancini, D. P. Menezes and Ph. Chomaz. Phys. Rev. C (2006), C 74, 045801 (2006); C. Providência, L. Brito, A.M.S. Santos, D.P. Menezes and S.S. Avancini, Phys. Rev. C 74, 045802 (2006). [11] Ph. Chomaz and F. Gulminelli, Phys. Lett. B447, 221 (1999) 221; H. S. Xu, et al, Phys. Rev. Lett. 85, 716 (2000). [12] C. Ducoin, Ph. Chomaz and F. Gulminelli, Nucl. Phys. 771, 68 (2006). [13] Ph. Chomaz, Nucl. Phys. A 685, 274c (2001). [14] E. Chabanat, P. Bonche, P. Haensel, J. Meyer and R. Schaeffer, Nucl. Phys. A 627, 710 (1997). [15] C. Providência, D. P. Menezes, L. Brito and Ph. Chomaz, in preparation. [16] B.A. Li, C.M. Ko and W. Bauer, Inter. J. Mod. Phys. E 7, 147 (1998). [17] K. Pomorski, P. Ring, G.A. Lalazissis, A. Baran, Z. Lo- jewski, B. Nerlo-Pomorska, M. Warda, Nucl. Phys. A 624, 349 (1997). [18] D. G. Ravenhall, C. J. Pethick, and J. R. Wilson, Phys. Rev. Lett. 50, 2066 (1983); M. Hashimoto, H. Seki, and M. Yamada, Prog. Theor. Phys.71, 320 (1984). [19] H. de Vries, C.W. de Jager and C. de Vries, Atomic and Nuclear Data Tables 36, 495 (1987). [20] C.J. Horowitz, S.J. Pollock, P.A. Souder and R. Michaels, Phys. Rev. C 63, 025501 (2001). [21] T.W. Donnelly, J. Dubach and I. Sick, Nucl. Phys. A503 589 (1989). [22] K.A. Aniol et al. (HAPPEX) (2005), nucl- ex/0506010; ibidem, nucl-ex/0506011; R. Michaels, P.A. Souder and G.M. Urciuoli (2005), URL http://hallaweb.jlab.org/parity/prex. [23] H. Lenske and C. Fuchs, Phys. Lett. B 345, 355 (1995); C. Fuchs, H. Lenske and H.H. Wolter, Phys. Rev. C 52, 3043 (1995). [24] S. Typel and H. H. Wolter, Nucl. Phys. A656, 331 (1999). [25] G. A. Lalazissis, J. König and P. Ring, Phys. Rev. C 55, 540 (1997). [26] K. Sumiyoshi, H. Kuwabara, H. Toki, Nucl. Phys. A 581, 725 (1995). [27] C. Fuchs, H. Lenske and H.H. Wolter, Phys. Rev. C 52, 3043 (1995). [28] H. Lenske and C. Fuchs, Phys. Lett. B 345, 355 (1995). [29] B. ter Haar and R. Malfliet, Phys. Rep. 149, 207 (1987). [30] T. Niks̆ić, D. Vretenar, P. Finelli and P. Ring, Phys. Rev. C 66, 024303 (2002). [31] B. Serot and J.D. Walecka, Advances in Nuclear Physics 16, Plenum-Press, (1986) 1. [32] S.S. Avancini and D.P. Menezes, Phys. Rev. C 74, 015201 (2006). [33] D.P. Menezes and C. Providência, Phys. Rev. C 68, 035804 (2003); Braz. J. Phys. 34, 724 (2004). [34] A.M.S. Santos and D.P. Menezes, Phys. Rev. C 69, 045803 (2004). [35] C.J. Horowitz and J.Piekarewicz, Phys. Rev.C 64, 062802 (2001). [36] J.K. Bunta and S. Gmuca, Phys. Rev. C 68, 054318 (2003). [37] J.K. Bunta and S. Gmuca, Phys. Rev. C 70, 054309 (2004). [38] D.P. Menezes and C. Providência, Nucl. Phys. A 650, 283 (1999); D.P. Menezes and C. Providência, Phys. Rev. C 60, 024313 (1999); D.P. Menezes and C. Providência, Phys. Rev. C 64, 044306 (2001). [39] Y.K. Gambhir, P. Ring and A. Thimet, Ann. Phys. 198, 132 (1990). [40] T. Gaitanos, M. Di Toro, S. Typel, V. Baran, C. Fuchs, V. Greco and H. H. Wolter, Nucl. Phys. A 732, 24 (2004). [41] S.S. Avancini, M.E. Bracco, M. Chiapparini and D.P. Menezes, J. Phys. G 30, 27 (2004); S.S. Avancini, M.E. Bracco, M. Chiapparini and D.P. Menezes, Phys. Rev. C 67, 024301 (2003). [42] J. Margueron and P. Chomaz, Phys. Rev. C 67, 041602 (2003). [43] C.J. Horowitz, Phys. Rev. C57, 3430 (1998). [44] C.J. Horowitz and B.D. Serot, Nucl. Phys. A 368, 503 (1981). [45] G.Fricke, C. Bernhardt, K.Heilig, L.A. Schaller, L. Schellinberg, E.B. Shera,C.W. de Jager, At. Data Nucl. Data Tables 60 (1995)177. [46] G. Audi, A.H. Waptra, C. Thibault, Nucl. Phys. A 729, 337 (2003). [47] A. Krasznahorkay et a., Nucl. Phys. A 731, 224 (2004). [48] V.E. Starodubsky, N.M. Hintz,Phys. Rev. C49,2118(1994). [49] L. Chen, C.M. Ko and B. Li, nucl-th/0610057. [50] M.B. Tsang et al., Phys. Rev. Lett. 92, 062701 (2004). TABLE II: 208 Pb properties model approximation Rn Rp θ B/A σ (fm) (fm) (fm) MeV Mev/fm2 NL3 TF+nucleation 5.88 5.65 0.24 -7.77 0.76 NL3 TF+HO 5.79 5.57 0.22 -7.79 0.96 NLωρ, Λv = 0.01 TF+HO 5.77 5.57 0.20 -7.73 0.98 NLωρ, Λv = 0.02 TF+HO 5.75 5.57 0.17 -7.65 0.99 NLωρ, Λv = 0.025 TF+HO 5.74 5.58 0.16 -7.63 1.00 TW TF+nucleation 5.71 5.50 0.22 -6.42 1.08 TW TF+HO 5.68 5.52 0.16 -7.46 1.10 HS TF+HO 5.70 5.47 0.24 -6.10 1.37 exp.[45] 5.44 exp. [46] -7.87 exp. [47] 0.12± 0.07 exp. [48] 0.20± 0.04

0704.0409

On the over-barrier reflection in quantum mechanics with multiple degrees of freedom

CERN-PH-TH/2007-065 On the over-barrier reflection in quantum mechanics with multiple degrees of freedom D.G. Levkova1, A.G. Panina,b2, S.M. Sibiryakovc,a3 aInstitute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary prospect 7a, Moscow 117312, Russia. bMoscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudny 141700, Moscow Region, Russia. cTheory Group, Physics Department, CERN, CH-1211 Geneva 23, Switzerland. Abstract We present an analytic example of two dimensional quantum mechanical system, where the exponential suppression of the probability of over–barrier reflection changes non-monotonically with energy. The suppression is minimal at certain “optimal” ener- gies where reflection occurs with exponentially larger probability than at other energies. 1 Introduction Tunneling and over–barrier reflection are the characteristic non–perturbative phenomena in quantum mechanics. They typically occur with exponentially small probabilities, P ∝ e−F/~ , (1) where F is the suppression exponent; still, the above phenomena are indispensable in under- standing a wide variety of physical situations, from the generation of baryon number asym- metry in the early Universe [1] to chemical reactions [2] and atom ionization processes [3]. During the last decades extensive investigations of tunneling processes in systems with many degrees of freedom have been performed [2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. These studies [email protected] [email protected] [email protected], [email protected] http://arxiv.org/abs/0704.0409v2 revealed a rich variety of features of multidimensional tunneling which are in striking contrast to the properties of one–dimensional tunneling and over–barrier reflection. In particular, the following phenomenon has been observed: the probability of tunneling may depend non- monotonically on the total energy of the system and exhibit resonance-like peaks. One can envisage three physically different mechanisms of this phenomenon. The first mechanism, present already in one-dimensional case, is tunneling via creation of a metastable state. In this case the tunneling probability at the maximum of the resonance is exponentially higher than at other energies. On the other hand, the resonance width ∆E is exponentially suppressed; so, after averaging with an energy distribution of a finite width the effect of the resonance is washed out in the semiclassical limit ~ → 0. The second possible mechanism of non-monotonic behavior of P(E) is quantum interference [7, 13] (see also [14]). In this case the peak value of the tunneling probability is only by a factor of order one higher than the average value, while the width of the resonance scales as ∆E ∝ ~. Again, the resonances become indiscernible in the semiclassical limit. In both these cases the resonances can be attributed to the subleading semiclassical corrections, i.e. non-monotonic behavior of the pre-exponential factor omitted in Eq. (1). The third possibility is that the suppression exponent F (E) is non-monotonic. In this case the existence of the “resonances” is the leading semiclassical effect: the optimal tunneling probability at the maximum of the resonance is exponentially higher than the probability at other energies. At the same time the resonance width scales as4 ∆E ∝ ~. This last possibility of “optimal tunneling” is definitely of interest; yet, it did not receive much attention in literature. We are aware of only a few works mentioning non–monotonic dependence of the suppression exponent on energy [15, 16, 14]. It is worthwhile studying this phenomenon in detail; this can provide a new insight into the dynamics of multidimensional tunneling. In this paper we consider the process of over–barrier reflection in a simple model with two degrees of freedom. Our setup is interesting in two respects. First, the model under study is essentially non–linear and the variables cannot be separated; still, over–barrier reflections in this model can be described analytically within the semiclassical framework. Thus, this model can serve as an analytic laboratory for the study of multidimensional tunneling. Sec- ond, the suppression exponent F of the reflection process behaves non–monotonically as the 4This follows from the representation P(E) ∝ exp F (Eo) F ′′(Eo)(E − Eo)2 of the tunneling probability in the vicinity of the maximum. total energy E changes. We demonstrate that the function F (E) possesses a number of local minima E = Eo, where reflection is optimal. We stress that the process we study is exponentially preferable at “optimal” energies as compared to other energies. Our model describes the motion of a quantum particle in the two dimensional harmonic waveguide (see Refs. [8, 10, 14] for similar models). The Hamiltonian is w2(x, y) , where x, y are the Cartesian coordinates and m is the mass of the particle. The function U = mω2w2/2 represents the waveguide potential in two dimensions: a particle with small energy is bound to move along the line w(x, y) ≈ 0. We do not introduce a potential barrier across the waveguide and consider the case when the line w = 0 stretches all the way from x → −∞ to x → +∞. We also assume that the function w(x, y) is linear in the initial asymptotic region, w(x, y) → y as x→ −∞ . In the present paper we consider two particular cases of the function w(x, y) describing waveguides with one and two sharp turns5, see Fig. 1. The motion of the particle at x → −∞ is a superposition of free translatory motion in x direction and oscillations of frequency ω along y coordinate; the state of such a particle is fully characterized by two quantum numbers, the total energy E and y–oscillator excitation number N . The particle sent into the waveguide from the asymptotic region x→ −∞ with given E, N may either continue to move towards x → +∞, or reflect back into the region x→ −∞. We are interested in the probability P(E,N) of reflection. Let us discuss reflections at the classical level. [Note that the classical counterpart of N is the energy of transverse oscillations.] Consider first the waveguide with one sharp turn (Fig. 1a). One observes that the outcome of the classical evolution, i.e. whether or not the particle reflects from the turn, depends not only on the total energy E, but on other dynamical quantities as well. In particular, the direction of the momentum of the particle in the vicinity of the turn (point C on the graph) is important. This means that the entire dynamics in the waveguide should be taken into account in order to determine the possibility of classical reflection. This is in sharp contrast with the situation in one–dimensional case, where reflection from the potential barrier (or transition through it) is ensured by the value of the conserved energy of the particle. 5The explicit expressions for the waveguide functions w(x, y) will be presented in the subsequent sections. Figure 1: The equipotential contour U = E for the waveguides with (a) one and (b) two sharp turns. An example of classical trajectory is shown in the case (b). Now, consider the waveguide with two turns. The model is characterized by the angles of the turns and the distance L between them (see Fig. 1b). Suppose the particle starts moving classically from x → −∞ with N = 0 along the valley w = 0. Then, the transverse oscillations get excited only after the particle crosses the first turn, point C ′ on the plot, so that at the time of arrival to the second turn (point C) approximately ωτ/2π oscillations are made, where τ ∼ L m/2E is the time of motion between the two turns. The state of the particle (coordinates and momenta) at which it comes across the second turn depends periodically on the phase of transverse oscillations ωτ . Hence, one expects that the regime of motion of the classical particle can change from transmission to reflection and back as the energy grows (τ decreases); the energies where it happens can be roughly estimated as mω2L2 2(2πn)2 . (2) We will see that this is indeed the case for the waveguides with certain angles of the turns. At some values of E, N the reflection process cannot proceed classically. Then, at the quantum mechanical level its probability is exponentially suppressed, F (E,N) > 0. It is natural to call such a process “over–barrier reflection”6. The central quantity to be studied below is the suppression exponent F (E,N) of this process. The above discussion suggests that F (E,N), being determined by the entire dynamics in the waveguide, may be a highly non–trivial function. For the particular case of the waveguide with alternating regimes of 6By this term we want to emphasize that the process is classically forbidden. Recall, however, that there is no actual potential barrier across the waveguide in our setup. classical reflections and transmissions F should oscillate: F = 0 at the energies where the classical reflections are allowed, and F > 0 at the energies where the reflections are classically forbidden. One can expect that the similar oscillatory behavior of the suppression exponent persists for other two–turn models as well. Now, instead of reaching zero, F may possess a number of local positive minima implying that the reflection at the “optimal” energies is still a tunneling process. Let us emphasize the difference of the “optimal tunneling” from quantum interference and resonance phenomena in our two–turn model. The interference of the de Broglie waves reflected from the two turns can, in principle, lead to oscillations in the reflection proba- bility P(E). One can estimate the positions of the interference peaks by equating the De Broglie wavelength of the particle to an integer fraction of the distance between the turns, 2mE ∼ L/n. This yields the energies of the interference peaks, Eintn ∼ (2πn)2~2 This formula is completely different from Eq. (2) for the peaks due to “optimal tunneling”. In particular, the distance between the adjacent inteference peaks, ∆Eint ∼ scales proportional to ~. Thus, these peaks should be averaged over in the semiclassical limit. Besides, the amplitude of the interference peaks is at most of order one and does not affect the suppression exponent. Indeed, the exponential increase of the scattering amplitude can arise due to quantum interference only in the presence of a resonant state with exponentially long life–time. This state should be supported somewhere in between the turns and should be classically stable. In Sec. 4.2 we show that such states are absent in our system. One concludes that the peak–like structure of the probability P(E) of “optimal tunneling” is caused by completely different physical reasons as compared to the case of resonance scattering in quantum theory. It is worth noting that the phenomenon of “optimal tunneling” has an important imple- mentation in field theory. Recently it was argued [17] (see also Ref. [16]) that the probability of tunneling induced by particle collisions [18, 19] reaches its maximum at a certain “op- timal” energy and stays constant7 at higher energies. This result, if generic, provides the 7As opposed to the quantum mechanical case, the tunneling probability does not decrease at energies higher than the “optimal” one. This is due to the possibility, specific to the field theoretical setup, to emit the excess of energy into a few hard particles, so that tunneling effectively occurs at the “optimal” energy. answer to the long–standing question [20] about the high–energy behavior of the probability of collision–induced nonperturbative transitions in field theory. The quantum mechanical model presented here supports the generic nature of the phenomenon of “optimal tunnel- ing”; the simplicity of our model enables one to get an intuitive insight into the nature of this phenomenon. The paper is organized as follows. In Sec. 2 we review the semiclassical method of complex trajectories, which is exploited in the rest of the paper. Reflections in the waveguides with one and two turns are considered in Secs. 3 and 4 respectively. We discuss our results in Sec. 5. In appendix we analyze the validity of some assumptions made in the main body of the paper. 2 The semiclassical method We start by describing the semiclassical method8 of complex trajectories which will be used in the study of over–barrier reflections. We concentrate on the derivation of the formula for the suppression exponent F (E,N) (see Refs. [2, 8, 9] for the details of the method and Ref. [19] for the field theory formulation). In what follows we use the system of units ~ = m = ω = 1 , where the Hamiltonian takes the form, p2x + p y + w 2(x, y) . (3) One starts with the amplitude of reflection into the state with definite coordinates xf < 0 , yf , A = 〈xf , yf |e−iĤ(tf−ti)|E, N〉 . (4) Here |E, N〉 is the initial state of the particle moving in the asymptotic region xi → −∞ with fixed translatory momentum p0 = 2(E −N) and the oscillator excitation number N . Semiclassically, 〈xi, yi|E, N〉 = eip0xi cos ′)dy′ + π/4 , (5) 8Note that the method has been confirmed by the explicit comparison with the exact quantum mechanical results in Refs. [8, 9, 14]; specifically, the recent check [14] deals with the case when the dependence of the suppression exponent on energy is not monotonic. where xi, yi denote initial coordinates, 2N − y′2 , (6) and we omitted the pre-exponential factor which is irrelevant for our purposes. Using Eq. (5), one rewrites the amplitude (4) as a path integral, dxidyi [dx][dy] xf , yf xi, yi eiS+ip0xi cos ′)dy′ + π/4 , (7) where S is the classical action of the model (3). In the semiclassical case the integral (7) is dominated by the (generically complex) saddle point. Note that, as we continue the integrand in Eq. (7) into the plane of complex coor- dinates, one of the exponents constituting the initial oscillator wave function grows, while the other becomes negligibly small. Within the validity of our approximation, we omit the decaying exponent by writing ′)dy′ + π/4 → exp ′)dy′ , (8) with the standard choice9 of the branch of the square root in Eq. (6). One proceeds by finding the saddle point for the integral (7) with the substitution (8). Extremization with respect to x(t), y(t) leads to the classical equations of motion, ẍ = −wwx , ÿ = −wwy . (9) Differentiating with respect to xi ≡ x(ti), yi ≡ y(ti), one obtains, ẋi = p0 = 2(E −N) , ẏi = py(yi) = 2N − y2i . The latter equations are equivalent to fixing the total energy E and initial oscillator energy N of the complex trajectory, ẋ2i +N , (10a) ẏ2i + y . (10b) 9 The correct branch is fixed by drawing a cut between the oscillator turning points y = ± 2N , and choosing Im py > 0 at y ∈ R, y > 2N , see, e.g., Refs. [21]. Substituting the saddle–point configuration10 into Eq. (7), one obtains the amplitude of the process with exponential accuracy, A ∝ eiS+iB(xi, yi) , where the term B(xi, yi) = p0xi + ′)dy′ (11) is the initial–state contribution. For the inclusive reflection probability one writes, dxfdyf |A|2 ∝ dxfdyf e iS−iS∗+iB−iB∗ . The integral over the final states can also be evaluated by the saddle point technique; ex- tremization with respect to xf ≡ x(tf ), yf ≡ y(tf) fixes the boundary conditions in the asymptotic future, Im ẋf = Im xf = 0 , Im ẏf = Im yf = 0 . (12) In this way one obtains the expression (1) for the reflection probability, where the suppression exponent F is given by the value of the functional F (E, N) = 2 ImS + 2 ImB(xi, yi) evaluated on the saddle–point configuration — a complex trajectory satisfying the boundary value problem (9), (10), (12). The contribution B(xi, yi) of the initial state is simplified after one uses the asymptotic form of the solution at t→ −∞ (xi → −∞), x = p0t + x0 , y = ae −it + āeit . (13) Equations (10) guarantee that the quantities p0 = 2(E −N) and 2aā = N are real, since E, N ∈ R. Therefore, one may introduce two real parameters T , θ as follows, 2 Im x0 = −p0T , ā = a∗eT+θ . (14) One finds for the initial term (11), 2 Im B(xi, yi) = Im 2p0xi − 2Narccos(yi/ 2N) + yi 2N − y2i = −p20T −N(T + θ) + Im(yiẏi) , 10For simplicity we assume that the saddle–point configuration is unique. Otherwise, one should take the saddle point corresponding to the weakest exponential suppression. and thus F = 2 Im S̃ −ET −Nθ , (15) where S̃ is the classical action of the system (3) integrated by parts, S̃ = −1 xẍ+ yÿ + w2(x, y) . (16) Let us comment on the physical meaning of the parameters T , θ. Consider two trajectories which are solutions to the boundary value problem (9), (10), (12) at neighbouring values of E, N . The differential of the quantity 2 Im S̃ as one deforms one trajectory into the other is d (2 Im S̃) = d Im(2S + xiẋi + yiẏi) = Im(xidẋi − ẋidxi + yidẏi − ẏidyi) = EdT +Ndθ , where in the last equality we used the asymptotic form (13), (14) of the solution. Then, from Eq. (15) one finds, dF (E,N) = −TdE − θdN . (17) Thus, the parameters T and θ are (up to sign) the derivatives of the suppression exponent with respect to energy E and initial oscillator excitation number N respectively. Our final remark is that the boundary value problem (9), (10), (12) is invariant with respect to the trivial time translation symmetry, t→ t+ δt , δt ∈ R , (18) which can be fixed in any convenient way. 3 The model with one turn To warm up, we consider the simplest model, where the waveguide has one sharp turn, w = y θ(−x+ y tg β) + cos β (x sin β + y cos β) θ(x− y tg β) . (19) Here θ(x) is the step function. It is convenient to use the rotated coordinate system, cos β − sin β sin β cos β The waveguide function takes the form, w = η cos β − ξ sin β θ(−ξ) . (20) Figure 2: The equipotential contour w2(x, y) = 2N for the waveguide (20) and the trajectory of the critical solution with energy N/ cos2 β. The equipotential contour w2(ξ, η) = const is shown in Fig. 2. One observes that the motion of the particle in two regions, ξ < 0 and ξ > 0, decomposes into the translatory motion and oscillations in the coordinates x, y and ξ, η respectively (see. Eqs. (19) and (20)); the frequency of η–oscillations in the latter case is cos β. Due to the presence of the step function, the first derivatives of the potential (20) are discontinuous11 at ξ = 0. Strictly speaking, the semiclassical method is not applicable in this situation [21]. Thus, the formula (20) should be regarded as an approximation to some waveguide function with smooth turn. Generically the width of the smoothened turn is characterized by a parameter b; the sharp–turn approximation (20) corresponds to b → 0. An example of smoothening is provided by the following substitution in Eq. (20), θ(ξ) → θb(ξ) = 1 + e−ξ/b . (21) The semiclassical description can be used as long as the de Broglie wavelength of the particle is small compared to the linear size of the potential12, 1/ E ≪ b. We conclude that the sharp–turn and semiclassical approximations are valid simultaneously for smooth waveguides 1 ≫ b≫ 1/ E . (22) 11Note that the potential itself is continuous. 12Another semiclassical condition is that the energy is sufficient to excite a lot of oscillator levels, E ≫ 1. It is satisfied provided Eq. (22) holds. An important property of the model (20) is invariance of the classical equations of motion (9) under the rescaling of the coordinates, x→ Λx , y → Λy . (23) Using the transformation (23), one may express a solution x(t), y(t) with energy E in terms of the “normalized” one, x = x̃ E , y = ỹ where the solution x̃(t), ỹ(t) has unit energy; its initial oscillator excitation number is ν = N/E . The suppression exponent (15) takes the form, F (E, N) = Efβ(ν) , (24) where fβ(ν) is the exponent for the “normalized” solution. Substituting the expression (24) into Eq. (17), one obtains, fβ(ν) = −T − θν . (25) We will exploit Eq. (25) in the end of this section. Now, we proceed to finding the “normal- ized” trajectories. At certain initial data ν > νcr the particle can reflect from the turn classically, so that fβ(ν > νcr) = 0 . Let us find the value of νcr. In the region ξ < 0 the classical solution takes the form, x(t) = p0t+ x0 , (26a) y(t) = A0 sin(t + ϕ) . (26b) Having crossed the line ξ = 0 (line AB in Fig. 2), the classical particle can never return back into the region ξ < 0. Indeed, in this case it moves at ξ > 0 with constant momentum pξ > 0. Thus, the particle can reflect classically only if its trajectory touches the line ξ = 0. The potential of our model has ill–defined derivatives at ξ = 0, and the fate of the particle moving along the line AB depends on the particular choice of the smoothening of the potential. In appendix we consider the motion of the classical particle in the case when nonzero smoothening of width b is switched on. For a class of smoothenings we show that in the small vicinity (δξ ∼ b) of any trajectory touching the line ξ = 0 there exists some “smoothened” trajectory, which reflects classically from the turn. Consequently, below we associate the trajectories touching the line ξ = 0 with the classical reflected solutions. One notices that the inclination of the trajectory (26) is bounded from above therefore, the classical trajectory of the particle can touch the line ξ = 0, that is, y/x = ctg β only at A0/p0 ≥ ctg β . (27) From Eqs. (27), (26), (10) one extracts the condition for the particle to reflect classically from the turn, ν ≥ νcr = cos2 β . (28) The critical classical solution at ν = νcr touches the line ξ = 0 at η = 0 (point C in Fig. 2), where its trajectory xcr(t) = 2t sin β , (29) ycr(t) = 2 sin t cos β . has the largest inclination. We now turn to the classically forbidden reflections at ν < νcr, which are described by the boundary value problem (9), (10), (12). One makes the following important observation. The waveguide function (20) has the form of two analytic functions glued together at ξ = 0. Hence, the equations of motion (9) can be continued analytically to the complex values of coordinates in two different ways, starting from the regions ξ < 0 and ξ > 0 respectively. In this way one obtains two complex solutions, ξ−(t), η−(t) and ξ+(t), η+(t). These solutions and their first derivatives should be matched at some moment of time t1, ξ(t1) = 0. [Note that the matching time t1 does not need to be real.] Below we conventionally refer to these solutions as the ones belonging to the regions ξ < 0 and ξ > 0. By the same reasoning as above we find that once the particle arrives into the region ξ > 0, it never reflects back to ξ < 0, unless pξ = 0. So, in the region ξ > 0 one writes, ξ+(t) = 0 , (30a) η+(t) = cos β sin(t cos β + ϕη) , (30b) where the “normalization” condition E = 1 has been used explicitly. Due to the conditions in the asymptotic future, Eqs. (12), the parameter ϕη is real. We use the translational invariance (18) to set ϕη = 0. Note that we again associate the trajectory going along the line ξ = 0 with the reflected one. The physical picture of over–barrier reflection that comes to mind matches with the new mechanism of multidimensional tunneling proposed recently in Refs. [9, 11]. The process proceeds in two steps. The first step, which is exponentially suppressed, is formation of the periodic classical orbit (30) oscillating along the line ξ = 0. This orbit is unstable. At the second step of the process the unstable orbit decays classically forming a trajectory going back to x → −∞ at t → +∞. Clearly, the second step does not affect the suppression exponent of the whole process, and we do not consider it explicitly. In what follows we concentrate on the determination of the tunneling trajectory describing the first step of the process. One should find the solution at ξ < 0 and impose the boundary conditions (10). Note, however, that the energy of our solution is fixed already. As for the initial oscillator excitation number ν, it does not change during the evolution in the region ξ < 0. Thus, one may fix it at the matching time t = t1. One writes, (ẏ2 + y2) = cos2 β + sin2 β sin2(t1 cos β) . This complex equation allows one to express t1 as sin(t1 cos β) = −i νcr − ν sin β , (31) where the choice of the sign is dictated by the condition in footnote 9. It is convenient to introduce notation t1 = iT1, T1 ∈ R. In order to find the suppression exponent fβ(ν), one needs to evaluate the parameters T (ν), θ(ν). At ξ < 0 the solution has the form, x−(t) = p0(t− iT/2) + x′0 , (32a) y−(t) = ae −it + a∗eT+θ+it , (32b) where the definitions (13), (14) have been taken into account explicitly, so that p0, x 0 ∈ R. One evaluates p0, x 0, a, T , θ by matching the coordinates x±, y± and their first derivatives νcr0.20.150.10.050 Figure 3: The suppression exponent fβ(ν) for the waveguide (20); β = π/3. ẋ±, ẏ± at t = iT1; this yields x′0 = 0 , p0 = 2(1− ν) , a = i 1− ν/ cos2 β T + θ cos2 β − ν sin β The last two equations, together with Eq.(25), define the function fβ(ν), fβ(ν) = cos β arcsh νcr − ν sin β − ν cos β arcsh νcr − ν sin β (νcr − ν)(1− ν) this finction is plotted in Fig. 3. One observes that at ν → νcr the quantities T1, T, θ, fβ tend to zero, and the complex trajectory tends to the classically allowed critical solution, cf. Eqs. (29), 2 sin β , a→ i√ cos β . At ν = 0 one has, fβ(0) = −2 + cos β arcth (cos β) . (33) To summarize, we obtained the suppression exponent for the reflection of a particle in the simplest waveguide with one sharp turn. Figure 4: The equipotential contour w2(x, y) = 2N ′ for the waveguide (35) and the trajectory of the critical solution with energy N ′/ cos2 β > EB. The matching points C, C ′ are shown by the thick black dots. 4 The model with two turns 4.1 Introducing the system In the model of the previous section the suppression exponent was proportional to energy because of the coordinate rescaling symmetry (23). Now, we are going to demonstrate that small violation of this symmetry results in highly non–trivial graph for F (E). One introduces a second turn into the waveguide, see Fig. 4. We want to consider this turn as a small perturbation, so, we assume its angle α to be smaller than β. It is convenient to introduce two additional coordinate systems, x′, y′ and ξ, η, bound to the central and rightmost parts of the waveguide respectively. They are related to the original coordinate system x, y as follows, cosα sinα − sinα cosα cos β − sin β sin β cos β x′ − L Note that the origin of the coordinate system ξ, η is shifted by the distance L. The waveguide function is w = θ(−x′)θ(−ξ)y + θ(−ξ)θ(x′)y′ cosα + θ(ξ)η cosα cos β ; (35) it consists of three pieces glued together continuously at x′ = 0 and ξ = 0 (lines A′B′ and AB in Fig. 4 respectively). At t→ −∞ the particle comes flying from the asymptotic region x′ < 0, where w = y. In the intermediate region x′ > 0, ξ < 0 the particle moves in the x′ direction oscillating along the y′ coordinate with the frequency cosα. Finally, in the region ξ > 0 its motion is free in the coordinates ξ, η; the frequency of η–oscillations is cosα cos β. The model (35) no longer possesses the symmetry (23): rescaling of coordinates changes the length L of the central part of the waveguide. In what follows it is convenient to work in terms of the rescaled dynamical variables, x̃ = x/L , ỹ = y/L . In new terms the parameter L disappears from the classical equations of motion, entering the theory through the overall coefficient L2 in front of the action. The initial–state quantum numbers are also proportional to L2, E = L2Ẽ , N = L2Ñ . (36) Thus, the conditions (22) for the validity of the semiclassical approximation are satisfied in the limit L→ ∞ , Ẽ, Ñ = fixed . The suppression exponent takes the form F (E,N) = L2F̃ (Ẽ, Ñ) . (37) To simplify notations, we omit tildes over the rescaled quantities in the rest of this sec- tion. Rescaling back to the physical units can be easily performed in the final formulae by implementing Eqs. (36), (37). 4.2 Classical evolution Let us begin this subsection by demonstrating that there are no stable classical solutions localized in the region between the turns. This is important for the determination of the tunneling probability, since such stable solutions could lead to exponential resonances in the tunneling amplitude. The argument proceeds as follows. Any trajectory which is localized in the intermediate region should reflect from the line AB infinitely many times. Each reflection involves touching the unstable orbit living at the line AB. This implies that the trajectory itself is unstable. We proceed by determining the region of initial data E, N , which correspond to the classical reflections. [For brevity we will refer to this region as the “classically allowed region”, as opposed to the “classically forbidden region” where reflections occur only at the quantum mechanical level. We stress that these are the regions in the plane of quantum numbers E, N .] Let us search for the critical classical solutions which correspond to the smallest initial oscillator number N = Ncr(E) at given energy E. As in the previous section, one finds that the particle must get stuck at the line13 AB for some time in order to reflect back. Let us first make an assumption inspired by the study of the one-turn model that the critical solutions touch the line AB at their maximum inclination point (point C in Fig. 4). We will see shortly that this is true only at energies above a certain value EB, see Eq. (50). Still, the analysis based on the above assumption enables one to catch the qualitative features of the critical line N = Ncr(E). Besides, the analysis is considerably simplified in this case; we postpone the accurate study until the end of this subsection. Keeping in mind the above remarks, one writes for the solution in the intermediate region, x′cr(t) = t 2E sin β + 1 , (38a) y′cr(t) = cos β sin(t cosα) . (38b) Before entering the intermediate region, the particle crosses the line A′B′ (point C ′ in Fig. 4). The initial oscillator number N is most conveniently calculated at the moment t = t0 ≡ − 2E sin β of crossing. Using the relations (34) one obtains, ẋcr(t0) = sin β cosα− cos β sinα cos cosα√ 2E sin β , (39) and thus Ncr(E) = E − ẋ2cr(t0) = E − E sin β cosα− cos β sinα cos cosα√ 2E sin β , E > EB . 0.05 0.15 0.90.80.70.60.50.40.30.10 Figure 5: The boundary N = Ncr(E) of the classically allowed region at E > EB for the waveguide model (35); β = π/3, α = π/30. The region of the classically allowed initial data lies above this boundary. The empty circles correspond to the energies E = En, where the curve N = Ncr(E) touches its lower envelope N = E cos 2(β + α). As an example, we show in Fig. 5 the region of the classically allowed initial data for β = π/3, α = π/30. One observes that the function Ncr(E) oscillates between two linear envelopes, E cos2(β + α) and E cos2(β − α); the period of oscillations decreases as E → 0. Moreover, the curve Ncr(E) has a number of minima at the points E = E n . This means that the energies E = Ecrn are optimal for reflection: in the vicinity of any point E = E n , N = Ncr(E n ) reflections become exponentially suppressed independently of whether the energy gets increased or decreased. This feature is particularly pronounced in the case α+β = π/2, when the lower envelope coincides with the line N = 0. Then, the classical reflections (i.e. reflections with the probability of order 1) at N = 0 are possible only in the vicinities of the points 8π2(n− 1/2)2 This is the case we used in Introduction to illustrate the effect. The minima E = Ecrn exist at other values of the parameters as well. For instance, let 13We do not consider reflections from the line A′B′. They disappear at larger values of N than reflections from the line AB if α is small enough. us find the positions of these minima in the case α ≪ 1. One differentiates Eq. (40) with respect to energy and obtains, Ecrn = En π(n− 1/2) arcsin ctg β 2πα(n− 1/2) +O(α2) , (41) where 8π2(n− 1/2)2 sin2 β are the points where the curve N = Ncr(E) touches its lower envelope. The argument of arcsine in Eq. (41) should be smaller than one, so, the minima Ecrn exist only at large enough n ≥ n0 ≡ ctg β + 1 , (43) where [·] stands for the integer part. Let us make several comments. First, note that n0 ∼ O(1/α), consequently, all the optimal points Ecrn lie in the region of small energies E ∼ 1/n20 ∼ O(α2). Second, as we pointed out before, the formula (40) for the function Ncr(E) holds at E > EB. Comparing the expressions (42), (43) and (50), one observes that En0 > EB if tg β > 1. So, there does exist a range of energies where the non-monotonic behavior of the function Ncr(E) can be inferred from the formula (40). In fact, the conclusion about the existence of the local minima of Ncr(E), as well as the expressions (41), (42), (43) determining their positions, remain valid also at E < EB. This follows from the rigorous analysis of the boundary of the classically allowed region to which we turn now. The reader who is more interested in the tunneling processes may skip this part and proceed directly to subsection 4.3. Now, we do not appeal to the Ansatz (38). Instead, we start with the general solution in the intermediate region, x′ = p′0(t− t0) , (44a) y′ = A′0 sin [(t− t0) cosα+ ϕ′] . (44b) It is convenient to parametrize it by the total energy E = p′20 /2 + cos 2 αA′20 /2 and the “inclination” γ defined by the relation p′0/A 0 = tg γ cosα . Expressions (44) take the following form, 2E (t− t0) sin γ , (45a) cos γ sin [(t− t0) cosα + ϕ′] . (45b) The constants t0 and ϕ ′ are fixed by demanding the trajectory (45) to reflect classically from the second turn, i.e. touch the line ξ = 0 at t = 0, (x′ − 1) cos β − y′ sin β = 0 , = ctg β . These conditions imply, t0 = − 2E sin γ tg2 β tg2 γ − 1 , (46a) ϕ′ = − cosα√ 2E sin γ tg2 β tg2 γ − 1− arccos . (46b) One sees that the classical reflections are possible only at γ ∈ [0; β]; the boundary value γ = β reproduces the solution (38). In order to find Ncr(E), one should minimize the value of the incoming oscillator exci- tation number with respect to γ at fixed E. At t = t0, when the particle crosses the first turn, p0 ≡ ẋ(t0) = 2E(cosα sin γ − sinα cos γ cosϕ′) . (47) Since N = E − p20/2, one can maximize the value of the translatory momentum p0 instead of minimizing N(γ). Formula (39) represents the value γ = β lying at the boundary of the accessible γ–domain; this value should be compared to p0(γ) taken at local maxima. Let us consider the case α ≪ 1. At large enough energies, E ∼ 1, Eq. (47) is dominated by the first term, which grows with γ, so that the maximum of p0(γ) is indeed achieved at γ = β. At small energies, however, the second term in Eq. (47) becomes essential because of the quickly oscillating cosϕ′ multiplier: the frequency of cosϕ′ oscillations grows as E → 0, and at E ∼ α2, in spite of the small magnitude proportional to sinα, the second term produces the sequence of local maxima of the function p0(γ). One expects the parameters of the trajectory at small α not to be very different from the ones at α = 0 (the latter case was considered in Sec. 3). So, we write, γ = β − δγ , where 0 < δγ ≪ 1. Expanding the expressions (46), (47) and taking into account that E ∼ α2 one obtains, ϕ′ = − 2E sin β (1 + δγ ctg β) , (48a) 2E(sin β − δγ cos β − α cos β cosϕ′) . (48b) Now, the local maxima of the initial translatory momentum can be obtained explicitly by differentiating Eqs. (48) with respect to δγ. One finds the sequence of them, δγn = −tg β + sin2 β cos β 2πn− π − arcsin 2E sin2 β α cos β . (49) Only the maxima with δγn > 0 should be taken into account. The local maxima exist when E ≤ EB ≡ α2 cos2 β 2 sin4 β . (50) Substituting Eq. (49) into the expressions (48), one evaluates the values of p0 at the local maxima, p0,n(E) =2 2E sin β − 2E sin2 β 2πn− π − arcsin 2E sin2 β α cos β 2E cos β 1− 2E sin α2 cos2 β The graphs Nn(E) = E − p20,n(E)/2 are shown in Fig. 6 for the case β = π/3, α = π/30. Each graph is plotted for the energy range E > EAn restricted by the condition δγn > 0. They are presented together with the curve given by the formula (40). By definition, the critical solution corresponds to the lowest of these graphs. Clearly, for each “local” curve representing the n-th local minimum of N(γ) there is a range of energies EAn < E < EBn where it lies lower than the “global” curve (40). This means that the parameter γ of the critical solution changes discontinuously across the points E = EBn . Correspondingly, the curve Ncr(E) has a break at these points. On the other hand, the function Ncr(E) is smooth at the points An as the “local” graphs end up exactly at δγ = 0, where the parameters of the n-th “local” solution coincide with the ones of the “global” solution. To summarize, we have observed that the boundary of the classically allowed region is given by a collection of many branches of classical solutions, each branch being relevant in its own energy interval. We will see that a similar branch structure is present in the complex trajectories describing over–barrier reflections in the classically forbidden region of E, N . 4.3 Classically forbidden reflections In this subsection we demonstrate that the suppression exponent F (E, N) viewed as a function of energy at fixed N exhibits oscillations deep inside the classically forbidden region 0.02 0.04 0.06 0 0.05 0.1 0.15 0.2 Figure 6: The graphs Nn(E) corresponding to the local minima of the function N(γ) (dashed lines) plotted together with the “global” curve, Eq. (40) (solid line); β = π/3, α = π/30. The critical curve N = Ncr(E) is obtained by taking the minimum among all the graphs. of initial data. This result comes without surprise if one takes into account the non-monotonic behavior of the boundary Ncr(E) of the classically allowed region. Indeed, the curve N = Ncr(E) coincides with the line F (E,N) = 0. One has, = −∂EF N=Ncr(E) so that (Ecrn , N n ) = 0 . We conclude that the points E = Ecrn are the local minima of the function F (E) at fixed N = N crn . It is natural to expect that such local minima of F (E) exist at other values of N as well. To illustrate this fact explicitly, we study the complex trajectories, solutions to Eqs. (9), (10), (12). Following the tactics of the previous section, we find solutions in three separate regions: initial region x′ < 0, final region ξ > 0, and the intermediate region x′ > 0, ξ < 0. These solutions, together with their first derivatives, should be glued at t = t0, when the complex trajectory crosses the line x′ = 0, and at t = t1, when ξ = 0. Besides, we are looking for the tunneling solution which ends up oscillating along the line AB, see Fig. 4. As discussed in Sec. 3 this assumes existence of the second step of the process: classical decay of the unstable orbit living at ξ = 0; the latter decay is described by a real trajectory14 going to x → −∞ at t→ +∞. The solution in the final region ξ > 0 is (cf. Eqs. (30)), ξ+(t) = 0 , (51a) η+(t) = cosα cos β sin(t cosα cos β) , (51b) where we used the time translation invariance (18) to fix the final oscillator phase ϕη = 0. In the intermediate region x′ > 0, ξ < 0 one writes, x′(t) = p′0t + x 0 , (52a) y′(t) = a′e−it cosα + ā′eit cosα . (52b) Note that the final solution (51) does not contain free parameters; thus, the matching of x′, ẋ′, y′, ẏ′ at t = t1 enables one to express all the parameters in Eqs. (52) in terms of one complex variable t1, p′0 = 2E sin β cos φ1 , (53a) x′0 = 1 + [sin φ1 − φ1 cos φ1] , (53b) eiφ1/ cos β [sinφ1 + i cos β cosφ1] , (53c) ā′ = e−iφ1/ cos β [sinφ1 − i cos β cos φ1] , (53d) where we introduced φ1 = t1 cosα cos β. As the energy of the solution has been fixed already, the only remaining initial condition involves initial oscillator excitation number at x′ < 0, see Eqs. (10). It is convenient to impose this condition at the matching point t = t0. One recalls the definition of the matching time t0, p′0t0 + x 0 = 0 , 14One wonders why this trajectory does not reflect from the turn A′B′ on its way back. This concern is removed by the observation that the trajectory produced in the decay of the unstable orbit is not unique: in appendix we show that the decay can occur at any point of the segment AC giving rise to a whole bunch of potential decay trajectories. Most of these trajectories pass through the turn A′B′ without reflection. which, after taking into account the expressions (53a), (53b), leads to the following equation, cosα√ 2E sin β sinφ1 cos β − cosφ1∆φ = 0 , (54) where ∆φ = cosα(t1 − t0). At t = t0 one has, ẋ(t0) = p 0 cosα− ẏ′(t0) sinα = 2(E −N) , and thus √ = ctgα sin β cosφ1 − sin φ1 sin∆φ− cos β cosφ1 cos∆φ . (55) As before, ν = N/E. Two complex equations (54), (55) determine the matching times t0, t1, and, consequently, the complex trajectory. Although these equations cannot be solved explicitly, they can be simplified in the case α ≪ 1, which we consider from now on. For concreteness, we study reflections at N = 0. It is important to keep in mind that in the region of interest E ∼ Ecrn ∼ O(α2); thus, one should regard all the momenta p and oscillator amplitudes a, ā, as the quantities of order O(α). At the same time, for the distances along the waveguide one has x ∼ O(1), so that the real parts of time intervals may be parametrically large, Re t ∼ x/p ∼ O(1/α). Further on, it will be convenient to work in terms of real variables, so, we represent φ1 and ∆φ as φ1 = cosα cos β(τ1 + iT1) , ∆φ = cosα(τ + i∆T ) . Note that τ and ∆T are the real and imaginary parts of the time interval t1 − t0 which the particle spends in the intermediate region. Now, equation (54) enables one to express 2E sin βch(T1 cos β) +O(α) , (56) τ1 = − τ cos β cos β −∆T cth(T1 cos β) +O(α3) . (57) Note that τ1 ∼ O(α), τ ∼ O(1/α). Then, the real part of Eq. (55) implies that ch(T1 cos β) = sin β 1 + α ctgβ cos τe∆T +O(α2) . (58) While deriving this formula we imposed T1 < 0 which follows from the requirement that in the limit α → 0 equation (31) should be recovered; besides, we assumed e∆T ∼ O(1). Substituting Eq. (58) into Eq. (56) and the imaginary part of Eq. (55), we obtain the final set of equations, 2E = α ctgβ cos τe∆T +O(α2) , (59a) (1 + ∆T )e−∆T = α ctgβτ sin τ +O(α) . (59b) These two nonlinear equations, still, cannot be solved explicitly. Nevertheless, one can get a pretty accurate idea about the structure of their solutions. Before proceeding to the analysis of the above equations, let us derive a convenient expression for the suppression exponent F0(E) ≡ F (E,N = 0). Note that on general grounds one expects to obtain an expression of the form, F0(E) = E(fβ(0) +O(α)) , where fβ(0) is given by Eq. (33). We are interested in the O(α) correction in this expression, so, one must be careful to keep track of the subleading terms during the derivation. Making use of the equations of motion, one obtains for the incomplete action (16) of the system, 2 Im S̃ = Im p′0 = 2E sin β Im(cosφ1) . Substitution of Eqs. (56), (57), (58) into this formula yields 2 Im S̃ = 2E −1−∆T − α ctg β cos τe∆T cos2 β + 2∆T +O(α2) For the parameter T one has (see Eqs. (14)), T = − 2 Im x0 2 Im(x(t0)− p0t0) = 2(T1 −∆T ) + sinα Im y′(t0) , (60) where in the last equality we used Eqs. (34) and x′(t0) = 0. The quantity Im y ′(t0) is evaluated by using Eqs. (52b), (53) and (58); one finds, Im y′(t0) = − ctg β cos τe∆T +O(α) Substituting everything into the formula (15), we obtain, F0(E) = E fβ(0)− 4α ctg β cos τ ∆T e∆T +O(α2) . (61) This expression implies that determination of the O(α) correction to the suppression expo- nent involves finding τ , ∆T with O(1)–accuracy. This is precisely the level of accuracy of Eqs. (59). Below we will also need the following formulae, which can be easily obtained by using T = −F. and Eq. (60), = fβ(0) + 2(∆T + 1) +O(α) , (62) 2(∆T + 1 +O(α)) . (63) Note that, though the suppression exponent differs from that in the one–turn case only by O(α) correction, its derivative gets modified in the zeroth order in α. Now, we are ready to analyze Eqs. (59). One begins by solving Eq. (59b) graphically, see Fig. 7. The important property of this equation is as follows. One notices that the l.h.s. of Eq. (59b) is always smaller than 1, the maximum being achieved at ∆T = 0. Therefore, the solutions to this equation are confined to the bands τ sin τ < This corresponds to τ ∈ [0; 2π(n1 − 1) + δτn1 ] or τ ∈ [2πn− π − δτn; 2πn + δτn] , n ≥ n1 (64) where δτn = arcsin 2πα(n− 1/2) +O(α) , + 1 , (65) with [·] in the last formula standing for the integer part. The forbidden bands, where τ sin τ > tgβ/α, are marked in Fig. 7 by yellow shading. The property (64) introduces a topological classification of the solutions τ , ∆T to Eqs. (59). Namely, these solutions fall into a set of continuous branches: the “local” branches τn(E), ∆Tn(E) living inside the strips τ ∈ [2πn − π − δτn; 2πn + δτn], n ≥ n1, and the “global” branch τg(E), ∆Tg(E) inhabiting the very first band τ ∈ [0; 2π(n1 − 1) + δτn1 ]. As follows from the definition of τ , the topological number n counts the number of y′–oscillations during the evolution in the intermediate region. Let us consider the “global” branch. From Eqs. (59) one has, τg → 2π(n1 − 1) +O(α lnα) , ∆Tg → ln(tg β/α) , E → 0 , τg → 0 , ∆Tg → −1 , E → +∞ . 10π 9π 8π 7π 6π 5π4π 3π 2π π 0 g 4 5 10π 9π 8π 7π 6π 5π4π 3π 2π π 0 g 4 5 Figure 7: Curves representing solutions to Eq. (59b); β = π/3, α = π/30. By inspection of Fig. 7 one can work out the qualitative behavior of the functions τg(E), ∆Tg(E). Alternatively, these functions can be found numerically. They are plotted in Fig. 8 for the case β = π/3, α = π/30 (the curves marked with “g”). One observes that at high enough energies the function ∆Tg(E) exhibits oscillations around the line ∆T = −1. According to the formula (63) this means that the function F0(E)/E is non-monotonic, it attains local minima at the points E ′n = 8π2(n− 1/2)2 1 + 2αe−1ctgβ +O(α2) . (66) Moreover, if n ≥ n′0 ≡ fβ(0) exp fβ(0) + 1 (67) there exist Eon = E n(1 + O(α)), such that ∆T (E n) = −1 − fβ(0)/2. Then, according to Eq. (62) the points Eon are the “optimal” energies corresponding to the local minima of the suppression exponent F0(E). At low energies the function ∆Tg(E) ceases to oscillate and becomes large and positive. According to Eq. (62) this means that the suppression exponent F0,g(E) of the “global” solution becomes negative at low energies15, see Fig. 9. This is a clear signal that the 15It is worth mentioning that Eqs. (59) and the expression (61) for the suppression exponent become 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 E’4E’5 Figure 8: Several first branches of solutions to Eqs. (59): “global” branch (“g”) and two “local” branches (“4”, “5”); β = π/3, α = π/30. 0.05 0.15 0.25 0 0.2 0.4 0.6 0.02 0.04 0.120.110.100.09 Figure 9: The suppression exponent F0(E) for the “global” and first “local” (n = 4) branches; β = π/3, α = π/30. The vicinity of intersection of the graphs is enlarged in the upper right corner. “global” solution becomes unphysical at these energies and its contribution to the reflection probability should be discarded: negative suppression exponent contradicts the unitarity requirement16, P < 1. One is forced to conclude that at low energies reflection is described by the “local” solutions. Let us study them in detail. For the n-th branch one obtains, τn → 2πn+O(α lnα) , ∆Tn → ln(tg β/α) , E → 0 , τn → 2πn− π , ∆Tg → +∞ , E → +∞ . From Fig. 7 one learns that the n-th solution passes through the points ∆Tn = −1 , τ = 2πn or τ = 2πn− π . (68) inapplicable at large ∆T : the assumption e∆T ∼ O(1) which was used in the derivation of these equations gets violated. Nevertheless, by analyzing the full equations (54), (55) one can show that dF0,g/dE = −Tg is large and positive at E → 0. This is sufficient for concluding that F0,g(E) is negative in the low–energy domain. 16Another indication that the “global” solution is unphysical at small E is that the function τg(E) is bounded from above. Indeed, τ is the time interval the particle spends in the intermediate part of the waveguide, one expects it to tend to infinity as E → 0 for a physically relevant solution. Thus, each curve ∆Tn(E) has one sharp dip, its minimum is smaller than −1, see Fig. 8. As in the case with the “global” branch, the points (68) represent the extrema of the functions F0,n(E)/E; the positions of the local minima are again given by Eq. (66). Making use of Eq. (61), we find that the suppressions F0,n(E) of the “local” branches are large and positive at high energies. Hence, these solutions give subdominant contributions to the reflection probability at such E as compared to the “global” solution. As energy decreases, F0,n(E) also decreases, then makes one oscillation and drops to negative values at small E. The latter property means that each “local” branch becomes unphysical at small enough energies. The suppression exponent of the first “local” branch (corresponding to n = 4 in the case β = π/3, α = π/30) is presented in Fig. 9. An alert reader may have already guessed that we have met here the typical Stokes phenomenon [21]. In fact, the Stokes phenomenon is specific to the situations where some integral (e.g., the path integral (7) in our case) is evaluated by the saddle–point method. Essentially, it means the following: as one gradually changes the parameters of the integral in question, a given saddle point may become non–contributing after the values of these parameters cross a certain curve drawn in the parameter space, the Stokes line. Since the result of the computation should be continuous, this phenomenon occurs only for subdomi- nant saddle points (saddle–point trajectories in our case). Unfortunately, apart from several heuristic conjectures [21, 12], sometimes rather suggestive [13], there is presently no general method of dealing with the Stokes phenomenon in the semiclassical calculations. However, in the situation encountered above it suffices to use the simplest logic lying at the heart of all other approaches17. When gathering the final result for the suppression exponent, we follow two guidelines. First, it is clear that, as energy decreases, each branch becomes unphysical before F0,n(E) crosses zero. On the other hand, at high energies one should pick up the branch corresponding to the smallest value of the suppression exponent. Looking at Fig. 9, one notes that the curves F0,g(E), F0,4(E) have two intersections, A and B. At E > EB one chooses the “global” branch. In the region EA < E < EB we switch to the first “local” branch, because in this region F0,4(E) < F0,g(E). Naively, at E = EA one should jump back to the “global” branch; however, in order to preserve unitarity at small energies, we suppose that somewhere in between the points B and A the “global” branch becomes non–contributing, so that one should stay at the “local” branch at E < EA. Similarly, the adjacent “local” branches have 17The simplification in the present case is related to the fact that we concentrate on the dominant semi- classical contribution, leaving aside the subdominant ones. two intersections; as the energy decreases, we switch from n-th branch to n + 1-th at the first intersection, and stay there until the intersection with the n+2-th branch. Overall, one obtains the graph for the suppression exponent plotted in Fig. 10. The suppression exponent 0.02 0.04 0.06 0.08 0 0.05 0.1 0.15 0.2 0.25 Figure 10: The final result for the suppression exponent F0(E) in the region of small energies; β = π/3, α = π/30. The points where different branches merge are shown with thick black dots. oscillates between two linear envelopes, F = E(fβ(0) ± 4e−1α ctg β); oscillations pile up in the region of low energies. The reflection process is optimal in the vicinities of the minima of the function F0(E). 5 Discussion By considering a class of two–dimensional waveguide models, we have demonstrated explicitly that the probability of over–barrier reflection can be non–monotonic function of energy. The origin of the effect lies in the classical dynamics: the parameters of the complex trajectory describing over–barrier reflection change quasi-periodically as the energy gets decreased. This results in the oscillatory behavior of the suppression exponent. Reflection occurs with exponentially larger probability in the vicinities of “optimal” energies (local minima of the suppression exponent) while being highly suppressed in between. Our results are obtained for a fairly specific class of waveguides, namely, the ones with very sharp turns. However, the qualitative features observed in this paper should be valid for quite general waveguide models: a classical particle with high energy feels any large–scale turn of the waveguide as a sharp one18; if two turns are separated by a long interval of free motion, one arrives to the model (35). We remark that the phenomenon of optimal tunneling has been observed also in numerical investigation of a smooth waveguide, see Ref. [14]. The branch structure of solutions observed in the region of small energies is interesting from the mathematical point of view. We have shown that there exists an infinite sequence of complex trajectories marked by the topological number n. Each branch produces physically consistent result for the suppression exponent in some energy interval; outside of this interval the n-th branch would correspond either to highly suppressed transitions (high energies) or to violation of unitarity (low energies). We collected the final graph for the suppression exponent basing on the empirical considerations, which hardly may be acknowledged as satisfactory. Our study clearly shows that the method of complex trajectories should be equipped with a convenient rule to pick up the physical trajectory among the discrete set of solutions to the boundary value problem (9), (10), (12) (in other words, the method to deal with the Stokes phenomenon). Presently, such a rule is absent. We note that the described physical phenomenon of optimal tunneling is present inde- pendently of the way the branches of solutions are glued together. The result at relatively high energies is given by the “global” branch, which displays a large number of local minima if n′0 > n1, see Eqs. (67), (65). This is the case for the illustrative example considered throughout this paper, see Fig. 9. As a final remark, we point out some open issues. We have calculated the suppression exponent of reflection using the sharp–turn approximation. It would be instructive to extend our analysis by finding corrections due to the finite turn widths. The motivation is twofold. First, the analysis performed in appendix implies existence of a rich variety of distinctive semiclassical solutions contributing almost equally into the reflection probability. This fea- ture might be a manifestation of chaos [7] which is present in our system but hidden by the sharp–turn approximation. [Note that chaos is inherent in a very similar waveguide model 18More precisely, one should compare the width b of the turn to the quantity 2π , where p0 is the translatory momentum of the particle and ω stands for the frequency of transverse oscillations; if b≪ 2πp0 one is in the class of models with sharp turns. with smooth potential, see Ref. [14].] Clearly, the structure of solutions in the vicinities of the turns is worth further investigation. Second, it was proposed recently in Refs. [9, 11] that the process of dynamical tunneling in quantum systems with multiple degrees of freedom (including field theoretical models, see Refs. [19]) can proceed differently from the ordinary case of one–dimensional tunnel- ing. Namely, classically unstable state can be created during the process; this state decays subsequently into the final asymptotic region. The analysis performed in the present paper naturally conforms with this tunneling mechanism: all our complex trajectories are matched with the unstable orbit living at the turn. Still, the sharp–turn approximation does not allow to distinguish between the truly unstable trajectories staying at the turn forever and those which reflect from the turn in a finite time. To decide whether the tunneling mechanism of Refs. [9, 11] is indeed realized in our model one needs to go beyond the sharp–turn approx- imation. Then, the candidate for the “mediator” unstable state is the “excited sphaleron”, the solution considered in the appendix. Presumably, in our model one can answer analyti- cally to the question of whether or not the “excited sphaleron” acts as an intermediate state of the tunneling process. This study is quite beyond the scope of the present paper and we leave it for future investigations. Acknowledgments. We are indebted to F.L. Bezrukov and V.A. Rubakov for the en- couraging interest and helpful suggestions. This work is supported in part by the Russian Foundation for Basic Research, grant 05-02-17363-a; Grants of the President of Russian Federation NS-7293.2006.2 (government contract 02.445.11.7370), MK-2563.2006.2 (D.L.), MK-2205.2005.2 (S.S.); Grants of the Russian Science Support Foundation (D.L. and S.S.); the personal fellowship of the “Dynasty” foundation (awarded by the Scientific board of ICFPM) (A.P.) and INTAS grant YS 03-55-2362 (D.L.). D.L. is grateful to Universite Libre de Bruxelles and EPFL (Lausanne) for hospitality during his visits. A Classical motion near the turn In this appendix we analyze the motion of the particle near the sharp turn of the waveguide (20) at nonzero smoothening of the turn, see, e.g., Eq. (21). We suppose that in the small vicinity of the turn the function w(ξ, η) can be represented in the form w(ξ, η) = cos β (η − bv(ξ/b)) , (69) where v(ψ) does not depend explicitly on b. Moreover, we consider the case when v(ψ) has a maximum19, v′(ψ0) = 0 . (70) Due to the property (70) one immediately obtains the exact periodic solution to the equations of motion (9), which we call “excited sphaleron” [9], ξsp = bψ0 , ηsp = Aη sin(t cos β + ϕη) + bv(ψ0) . (71) We are going to show that this solution is unstable: a small perturbation above it grows with time and the particle flies away to either end of the waveguide. In particular, there are solutions that describe the decay of the sphaleron to ξ → −∞ both at t → ±∞. Clearly, such solutions correspond to reflections from the turn. In the vicinity of the sphaleron the trajectory of the particle can be represented in the form, ξ = bψ(t) , η = ηsp(t) + bρ(t) , (72) where ψ, ρ ∼ O(1). Writing down the classical equations of motion (9) in the leading order in b, one obtains, Aη sin(2s)v ′(ψ) , (73) + 4ρ = 4[v(ψ)− v(ψ0)] , (74) where s = (t cos β +ϕη)/2. It is worth noting that the right hand side of Eqs. (73), (74) are of different order in b. We will see that due to this difference ρ = 0 in the leading order in b. Let us first consider the linear perturbations above the excited sphaleron, ψ = ψ0 + δψ , δψ ≪ 1 . Equation (73) can be linearized with respect to δψ leading to the Mathieu equation δψ + 2q sin(2s)δψ = 0 , with canonical parameter q = −2v′′0Aη/b > 0. As q ∼ O(1/b) ≫ 1, one can apply the WKB formula, A cosW dW/ds , (75) 19For the smoothening (21), the properties (69), (70) hold with v(ψ) = ψtgβ , ψ0 ≈ 1.28. where |A| ≪ 1, and sin(2s′) . Note that we have chosen the solution symmetric with respect to time reflections, δψ(π/2− s) = δψ(s) . (76) At s ∈ [0; π/2] the exponent W is real and the particle gets stuck at ψ ≈ ψ0, oscillating around this point with high frequency dW/ds ∼ O(b−1/2). At s < 0 the solution (75) grows exponentially, meaning that the particle flies away from the excited sphaleron, δψ(s < 0) = A cos(W (0)− π/4) |dW/ds| e|W (s)−W (0)| . In what follows, we choose A cos(W (0)− π/4) < 0, so that δψ < 0 at s < 0. Let us denote by s1 < 0 the point where δψ becomes formally equal to −1, A cos(W (0)− π/4) |dW/ds| e|W (s1)−W (0)| = −1 . In what follows we suppose that s1 ∼ O(1), hence, A is exponentially small. Then, in the vicinity of this point, |s− s1| ≪ 1, one has, δψ = − exp −2q sin(2s1)(s1 − s) = − exp 4v′′0Aη sin(2s1) (s1 − s)√ . (77) We notice that δψ evolves from exponentially small values to δψ ∼ O(1) during the charac- teristic time |s− s1| ∼ O( When δψ ∼ O(1) the linear approximation breaks down and one has to solve the nonlinear equation (73). Using s = s1 +O( b) one writes Aη sin(2s1)v ′(ψ) . (78) This equation permits to draw a useful analogy with one–dimensional particle moving in the effective potential Veff (ψ) = −4b−1Aη sin(2s1)v(ψ) (see Fig. 11). This auxiliary particle starts in the region near the maximum of the potential at (s − s1)/ b → +∞ with energy E ≈ Vmax and rolls down toward ψ → −∞ at (s−s1)/ b→ −∞. In this limit v(ψ) → ψ tg β and the solution takes the form ψ = C1 + C2(s− s1) + 2b−1Aη sin(2s1) tg β (s− s1)2 . Vmax Figure 11: The effective potential for Eq. (78). Note that the coefficients C1, C2 here are not independent: they are determined by the parameter s1 through matching of the solution with Eq. (77) at (s− s1)/ b→ +∞. We do not need their explicit form, however. Let us argue that the function ρ remains small during the whole evolution of the particle in the vicinity of the sphaleron. Indeed, in the linear regime one has δψ ≪ 1 and the r.h.s. of Eq. (74) is small. So, ρ does not get excited. On the other hand, the nonlinear evolution of ψ proceeds in a short time interval ∆s = O( b); so, again, ρ is suppressed by some power of b. The trajectory (72) found in the vicinity of the sphaleron should be matched at 1 ≫ |s− s1| ≫ with the free solution in the asymptotic region ξ < 0, see Eqs. (26). It is straightforward to check that matching can be performed up to the second order in (t− t1), which is consistent with our approximations. In this way one determines the free asymptotic solution which, up to corrections of order O(b), coincides with the sinusoid coming from ξ → −∞ at t → −∞ and touching the line ξ = 0 at t = t1. Now we recall that, by construction, the obtained solution is symmetric with respect to time reflections, ξ(s) = ξ(π/2− s) , η(s) = η(π/2− s) . This means that it satisfies ξ → −∞ at t → ±∞. This solution describes reflection of the particle from the turn. The reasoning presented in this appendix puts considerations of the main body of this paper on the firm ground: we have found the “smoothened” solutions which reflect classically from the turn, and in the limit b→ 0 coincide with the free solutions of Sec. 3 touching the line ξ = 0. It is worth mentioning that, apart from the reflected solution we have found, in the vicinity of any trajectory touching the line ξ = 0 there exists a rich variety of qualitatively different motions. First of all, one may successfully search for solutions which are odd with respect to time reflections (Eq. (76) with minus sign). Such solutions, though close to the reflected ones at t < 0, describe transmissions of the particle through the sharp turn into the asymptotic region ξ → +∞. Relaxing the time reflection symmetry, one can find solutions leaving the vicinity of the turn at any point η < 0, which is different, in general, from the starting point η = η(s1). Yet another types of solutions are obtained in the case when the amplitude A of δψ–oscillations at s ∈ [0; π/2] is so small that δψ does not reach the values of order one during the time period s ∈ [−π/2; 0]. If the particle is still in the vicinity of the point ψ0 at s = −π/2, it remains for sure in this vicinity at s ∈ [−π; −π/2], because the r.h.s. of Eq. (73) is positive again. In this way one obtains solutions, which spend two, three, etc. sphaleron periods at ψ ≈ ψ0 before escaping into the asymptotic regions ψ → ±∞. In the leading order in b all these solutions correspond to the identical initial state, and (in the case of classically forbidden transitions) to the same value of the suppression exponent. However, an accurate study of the dynamics in the vicinity of the the sphaleron is generically required to obtain the correct value of the suppression exponent in the case b ∼ 1, cf. Ref. [14]. References [1] V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B 155, 36 (1985). [2] W. Miller and T. George, J. Chem. Phys. 56, 5668 (1972); 57, 2458 (1972); W. H. Miller, Adv. Chem. Phys. 25, 69 (1974). [3] A. M. Perelomov, V. S. Popov and M. V. Terent’ev, ZHETF 51, 309 (1966); V. S. Popov, V. Kuznetsov and A. M. Perelomov, ZHETF 53, 331 (1967). [4] M. Davis and E. Heller, J.Chem.Phys. 75, 246 (1981). [5] M. Wilkinson, Physica 21D, 341 (1986); S. Takada and H. Nakamura, J. Chem. Phys. 100, 98 (1994); S. Takada, P. Walker and M. Wilkinson, Phys. Rev. A 52, 3546 (1995); S. Takada, J. Chem. Phys. 104, 3742 (1996). [6] W. Miller, J. Phys. Chem. A 105, 2942 (2001). [7] O. Bohigas, D. Boose, R.Egydio de Carvalho, V. Marvulle, Nucl. Phys. A560, 197 (1993); S. Tomsovic, D. Ullmo, Phys. Rev. E 50, 145 (1994); A. Shudo, K.S. Ikeda, Phys. Rev. Lett. 74, 682 (1994); Physica D 115, 234 (1998). E. Doron, S.D. Frischat, Phys. Rev. Lett. 75, 3661 (1995); Phys.Rev. E 57, 1421 (1998); S.C. Creagh, N.D. Whelan, Phys. Rev. Lett. 77, 4975 (1996). [8] G. F. Bonini, A. G. Cohen, C. Rebbi and V. A. Rubakov, Phys. Rev. D 60, 076004 (1999) [arXiv:hep-ph/9901226]. [9] F. Bezrukov and D. Levkov, J. Exp. Theor. Phys. 98, 820 (2004) [Zh. Eksp. Teor. Fiz. 125, 938 (2004)] [arXiv:quant-ph/0312144]; “ Transmission through a potential bar- rier in quantum mechanics of multiple degrees of freedom: Complex way to the top”, arXiv:quant-ph/0301022. [10] C.S. Drew, S.C. Creagh, R.H. Tew, Phys. Rev. A 72, 062501 (2005). [11] K. Takahashi, K.S. Ikeda, Europhys. Lett. 71 (2), 193 (2005); Phys. Rev. Lett. 97, 240403 (2006). [12] S. Adachi, Ann.Phys. 195, 45 (1989). [13] A. Shudo, K.S. Ikeda, Phys.Rev.Lett. 76, 4151 (1996). [14] D. G. Levkov, A. G. Panin and S. M. Sibiryakov, “Complex trajectories in chaotic dynamical tunneling,” arXiv:nlin.cd/0701063. [15] A. S. Ioselevich and E. I. Rashba, Sov. Phys. JETP 64, 1137 (1986) [Zh. Eksp. Theor. Fiz. 91, 1917 (1986)]. [16] M. B. Voloshin, Phys. Rev. D 49, 2014 (1994). [17] D. G. Levkov and S. M. Sibiryakov, Phys. Rev. D 71, 025001 (2005) [arXiv:hep-th/0410198]; JETP Lett. 81, 53 (2005) [Pisma Zh. Eksp. Teor. Fiz. 81, 60 (2005)] [arXiv:hep-th/0412253]. http://arxiv.org/abs/hep-ph/9901226 http://arxiv.org/abs/quant-ph/0312144 http://arxiv.org/abs/quant-ph/0301022 http://arxiv.org/abs/nlin/0701063 http://arxiv.org/abs/hep-th/0410198 http://arxiv.org/abs/hep-th/0412253 [18] M. P. Mattis, Phys. Rept. 214, 159 (1992); P. G. Tinyakov, Int. J. Mod. Phys. A 8, 1823 (1993); V. A. Rubakov and M. E. Shaposhnikov, Usp. Fiz. Nauk 166, 493 (1996) [Phys. Usp. 39, 461 (1996)], [arXiv:hep-ph/9603208]. [19] V. A. Rubakov, D. T. Son and P. G. Tinyakov, Phys. Lett. B 287, 342 (1992); A. N. Kuznetsov and P. G. Tinyakov, Phys. Lett. B 406, 76 (1997) [arXiv:hep-ph/9704242]; F. Bezrukov, D. Levkov, C. Rebbi, V. A. Rubakov and P. Tinyakov, Phys. Rev. D 68, 036005 (2003) [arXiv:hep-ph/0304180]. [20] A. Ringwald, Nucl. Phys. B 330, 1 (1990). O. Espinosa, Nucl. Phys. B 343, 310 (1990). [21] P. V. Elyutin, V. D. Krivchenkov “Nonrelativistic Quantum Mechanics in problems”, Moscow, Fizmatlit (2001). M. V. Berry, K. E. Mount, Rep. Prog. Phys. 35, 315 (1972). http://arxiv.org/abs/hep-ph/9603208 http://arxiv.org/abs/hep-ph/9704242 http://arxiv.org/abs/hep-ph/0304180 Introduction The semiclassical method The model with one turn The model with two turns Introducing the system Classical evolution Classically forbidden reflections Discussion Classical motion near the turn

0704.0411

Molecular circuits based on graphene nano-ribbon junctions

Microsoft Word - APL-Ribbon_junctions_20070328-Text_with_Figures-ARXIV.doc Molecular circuits based on graphene nano-ribbon junctions Zhiping Xu† Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China Graphene nano-ribbons junctions based electronic devices are proposed in this Letter. Non- equilibrium Green’s function calculations show that nano-ribbon junctions tailored from single layer graphene with different edge shape and width can act as metal/semiconductor junctions and quantum dots can be implemented. In virtue of the possibilities of patterning monolayer graphene down to atomic precision, these structures, quite different from the previously reported two-dimensional bulk graphene or carbon nanotube devices, are expected to be used as the building blocks of the future nano-electronics. Keyword: graphene nano-ribbon, electronic transport, metal/semiconductor junction, quantum dot † Email: [email protected] Nano-electronics, or molecular electronics have been proposed as the alternative to silicon in future technical applications1 and have attracted great interests recently. In virtue of their unique structures and various functions, these nanostructures possess intriguing electromagnetic, mechanical and optical features. Especially, carbon based nanostructure, such as fullerene, graphene and carbon nanotubes are the most interesting structures because of their rich variety of excellent physical properties. For instance, anomalous quantum hall effects (QHE) and massless Dirac electronic behavior have been discovered in the graphene systems2, 3, and these discovery has sparked lots of investigations on this unique two-dimensional material. Tailored from monolayer graphene, graphene ribbon (GNR) with finite width has been shown to hold unusual electronic properties4, depending on their edge shape and width. In more details, ribbons with zigzag edges (ZGNRs) possess spin-polarized peculiar edges states and spin-polarized electronic state provides half-metallicity under transverse electric field and has great potential in the application as spintronics5. In contrast, the armchair edged ribbons (AGNR) can be either metallic or semiconducting depending on their width6, AGNR with width Na (named as NaAGNR in the conventional nomenclature) has been shown to be metallic only if Na = 3k + 2 and semiconducting otherwise, where k is an integer. From the experimental point of view, the fascinating feature of the ribbons is that the graphene material can be easily patterned using standard micro- or nano-electronics lithography methods. Unlike the carbon nanotubes or other low-dimensional nanostructures, the GNRs with intricate sub-micrometer structures can now be fabricated7, 8, 9, and it is believed that a combination of standard lithographic and chemical methods will help to pattern the graphene with atomic precision down to the molecular level. The high mobility μ = 2.7 m2/V.s, large elastic mean free path le = 600nm and phase coherence lengths lφ= 1.1 μm observed7 in the epitaxial graphene patterned suggest the use of pure GNR structures as the building blocks of the nanoscale confined and coherent electronic circuits. To realize the components such as field transistors9 and coulomb blockade devices, experimentally controllable metal/semiconductor junctions and quantum dots will be essential. As proposed by Chico et al.10, 11, these can be achieved by jointing different carbon nanotubes. However, the fabrication and control of the nanostructure of graphene ribbons are much more convenient than introducing pentagon-heptagon defects in carbon nanotubes as discussed, therefore it is interesting to investigate the possibilities of ribbon junction based nano-circuits. To this end, we have proposed several kinds of the GNRs based electronic devices in this Letter. We show that, by controlling the tailoring process of GNRs with different edge shape and width, the metal/semiconductor junctions and quantum dots can be easily implemented experimentally. To validate this, electronic transport calculation using the non-equilibrium Green’s function method have been carried out following Landauer’s approach12. The electronic structure of the graphene lattice is described using the nearest-neighbour π-orbital tight-binding model and the hopping parameter Vppπ = 2.75 eV is used. This simple topological model gives quantitative results comparing with the LDA results except for the gap opening at small width as the consequence of the length changing of σ bonds6. By solving the Green’s function, the conductance was finally calculated as G = G0Tr[ΓLGRΓRGA] and the density of state is expressed as D = –ImTr[GR]/π11, where G0 = 2e2/h is the unit quanta of conductance including the spin degeneracy, GR(A) is the retarded (advanced) Green’s function of the conductor and ΓL(R) is the spectral density describing the coupling between the left (right) lead and the conductor. In our model, the leads are represented using semi-infinite graphene ribbons attached to the conductor region, with the same shape and width. First of all we investigate straight metal/semiconducting junction 11AGNR/10AGNR. The structure of the junction is considered by simply patching two different straight ribbons together, leave a width mismatching at the interface. The result shown in Fig. 1 indicates a gap Eg = 0.93 eV near the Fermi energy and the imperfection at the interface induces a deviation of conductance from the step-like curve of the perfect ribbon. However, the van Hove singularities which are the characteristics of 1D system remain. To examine the detailed electronic structure of the junction, a spatial-resolved localized density of states (LDOS) analysis is helpful. We have grouped the atoms into slices according to their distance from the interface. Each 4.26 Å long slice (a unit cell of the perfect AGNR) in the 10AGNR, 11AGNR and interface part contain 20, 22 and 21 atoms respectively. The LDOS averaged at different slices are plotted in Fig. 1. From the semiconducting 10AGNR side we find the LDOS is distorted near the interface and gap state appears through the contact with metallic 11AGNR. However at slices far from the interface, at slice 3 for example, the perfect semiconduting behavior is mostly recovered. At the scattering interface the van Hove singularities have been smoothed and 1D metallic structure gradually emerges as the distance from interface increases from the 11AGNR side. The arising of gap state near the interface characterizes the metal- semiconductor junction and suggests the possibilities of building Schottky devices. Furthermore, L-shape GNR junctions with different orientations can be constructed. For instance, the LDOS of 8ZGNR/15AGNR junction with a π/6 joint is analysized in Fig. 2. As expected, the edge state of the 8ZGNR spreads into the semiconducting 15AGNR side. Because of the ZGNR possess spin-polarized structure, so this half-metal/semiconducting junction inspires interests in the spin-transport devices. Beside of the metal/semiconductor junction, the semiconductor/semiconductor junctions have also been investigated and defect states in gap appear at the interface. Moreover in the ZGNR/ZGNR junctions, zero-conductance dips13 near Fermi energy have been observed, caused by the complete backward scattering. The metal-semiconductor junction also suggests quantum dot devices through combing two of them together. We now consider the junction 12AGNR/11AGNR/12AGNR. In this structure a central metallic ribbon is sandwiched by two semiconductor barriers where quantized states can be formed. Our calculation results depicted in Fig. 3 show two sharp DOS peaks inside the gap of semiconducting 12AGNR containing 7 unit cells, with energy E1,2 = 0.2025 and -0.2025 eV. As seen from the spatial-resolved LDOS at E = 0.2025, the bounded state is localized inside the 11AGNR region. The structure of the quantum levels can be further tuned by changing the length of 11AGNR. From our calculation, as it changes from 1 to 8 unit cells, the energy spacing between the nearest peaks around Fermi energy, i.e. ΔE = E1-E2, gradually decreases from 0.785 eV to 0.385 eV and their DOS becomes higher and sharper. We have also observed quantized edge states within the 10AGNR/7ZGNR/10AGNR junctions through introducing two π/6 joints. The results are shown in Fig. 3 where we can found 7 LDOS peaks inside the zero-conductance gap. The quantized states with E = -0.3525, -0.1625, - 0.05, 0, 0.05, 0.1625 and 0.3525 correspond to different LDOS patterns (see Fig. 4 for E = 0.3525). The higher the energy, the more nodes of the bounded standing wave have. The electron wave quantized pattern depends on the structure of the central region. In conclusion, we have proposed nano-electronic circuits based on graphene nano-ribbon junctions. Through tailoring GNRs into junctions of different edge shape and width, we can implement metal/semiconductor junctions and quantum dots in principle. In virtue of the possibility of molecular level patterning based on lithography and chemical methods, these devices are expected to be fabricated easier in comparison with other structures such as the single molecule or carbon nanotubes junctions, and are expected to find great applications in the large-scale integrated nano-circuits in future. The work is supported by the National Science Foundation of China through Grants 10172051, 10252001, and 10332020 and the Hong Kong Research Grant Council (NSFC/RGC N HKU 764/05 and HKU 7012/04P). ZX also thanks Prof. Wenhui Duan, Dr. Tao Zhou and Dr. Haiyun Qian from the Department of Physics in Tsinghua University for their help on the calculation. 1N. J. Tao, Nature Nanotechnology 1 173 (2006). 2K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos and A. A. Firsov, Nature 438, 197 (2005). 3Y. Zhang, Y. Tan, H. L. Stormer and P. Kim, Nature 438, 201 (2005). 4Y. Kobayashi, K. Fukui, T. Enoki, K. Kusakabe and Y. Kaburagi, Phys. Rev. B 71, 193406 (2005). 5Y. Son, M. L. Cohen and S. G. Louie, Nature 444, 347 (2006). 6Y. Son, M. L. Cohen and S. G. Louie, Phys. Rev. Lett. 97, 216803 (2006). 7C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First and W. A. de Heer, Science 312, 119 (2006). 8S. Liu, F. Zhou, A. Jin, H. Yang, Y. Ma, H. Li, C. Gu, L. Lu, B. Jiang, Q. Zheng, S. Wang and L. Peng, Acta Physica Sinica 54, 4251 (2005). 9Z. Chen, Y. Lin, M. Rooks and P. Avouris, http://arvix.org/abs/cond-mat/0701599, (2007). 10L. Chico, V. H. Crespi, L. X. Benedict, S. G. Louie and M. L. Cohen, Phys. Rev. Lett. 76, 971 (1996). 11L. Chico, M. P. Lopez Sancho and M. C. Munoz, Phys. Rev. Lett. 81, 1278 (1998). 12J. Lu, J. Wu, W. Duan, F. Liu, B. Zhu and B. Gu, Phys. Rev. Lett. 90, 156601 (2003). 13K. Wakabayashi, Phys. Rev. B 64, 125428 (2001). FIG. 1. The metal/semiconducting junction 11AGNR/10AGNR: (Top) Conductance and DOS of the whole system. (Bottom) LDOS at slices near the interface. Slice n (n = 1, 2 and 3) represents the n-th nearest slice to the interface and the vertical scale of DOS is 0.2. FIG. 2. Spatial-resolved LDOS in metal/semiconducting junction 8ZGNR/15AGNR, the vertical scale of DOS is 0.2. FIG. 3. Quantum dot structure based on 12AGNR/11AGNR/12AGNR junction: (Top) Conductance and DOS at low bias, where two isolated sharp peaks appear inside the gap; (Bottom) Spatial- resolved LDOS at E = 0.2025 eV, the grey dot represents the ionic site and the radius of circle around it corresponds to the value of LDOS. FIG. 4. Quantum dot structure based on 10AGNR/7ZGNR/10AGNR junction: (Top) Conductance and DOS; (Bottom) Spatial-resolved LDOS at E = 0.3525 eV. Figure 1 Figure 2 Figure 3 Figure 4

0704.0412

Unit groups of integral finite group rings with no noncyclic abelian finite subgroups

UNIT GROUPS OF INTEGRAL FINITE GROUP RINGS WITH NO NONCYCLIC ABELIAN FINITE SUBGROUPS MARTIN HERTWECK Abstract. It is shown that in the units of augmentation one of an integral group ring ZG of a finite group G, a noncyclic subgroup of order p2, for some odd prime p, exists only if such a subgroup exists in G. The corresponding statement for p = 2 holds by the Brauer–Suzuki theorem, as recently observed by W. Kimmerle. 1. Introduction Is a finite subgroup H of units in the integral group ring ZG of a finite group G necessarily isomorphic to a subgroup of G? Of course, torsion coming from the coefficient ring should be excluded, that is, only finite subgroups H in V(ZG), the group of units of augmentation one in ZG, will be considered. The question was raised by Higman in his thesis (1940), where he gave an affirmative answer when G is metabelian nilpotent or the affine group over a prime field; cf. Sandling (1981). In the survey of Sandling (1984) it is included as Problem 5.4, and noted that an affirmative answer for metabelian G was finally given by Roggenkamp (1981); but see also Cliff, Sehgal and Weiss (1981), and Marciniak and Sehgal (2003) for a more recent result, giving a generalization based on a theorem of Weiss (1988). These results are really about certain ‘large’ torsion-free normal subgroups of V(ZG). For a more complete discussion, see Chapter 4 in Sehgal’s book (1993). As a sort of converse, one may fix a finite group H and look for groups G for which H embeds into V(ZG), again hoping for the best, but little is known in this respect. What is known is that if a cyclic group H of prime power order embeds into some unit group V(ZG), then H also embeds into G (due to an observation of Cohn and Livingstone (1965); see also Zassenhaus (1974)), and only recently in Hertweck (2007b) it was shown that the restriction on the order can be removed if in addition G is assumed to be solvable. In this spirit, Marciniak, at a satellite conference of the ICM 2006, asked whether a group G necessarily has a subgroup isomorphic to Klein’s four group provided this is the case for V(ZG). Kimmerle immediately observed that this is implied by the Brauer–Suzuki theorem (rendered in Kimmerle (2006)), see Section 2. Our complementary result is as follows. Theorem A. Let G be a finite group. Suppose that V(ZG) has a noncyclic abelian subgroup of order p2, for some odd prime p. Then the same is true for G (i.e., Sylow p-subgroups of G are not cyclic). Date: October 30, 2018. 2000 Mathematics Subject Classification. Primary 16S34, 16U60; Secondary 20C05. Key words and phrases. integral group ring, torsion unit, partial augmentation. http://arxiv.org/abs/0704.0412v1 2 MARTIN HERTWECK It is easy to verify that a finite p-group with no noncyclic abelian subgroup is either cyclic or a (generalized) quaternion group, see Theorem 4.10 in Gorenstein (1968). It comes to mind that the theory of cyclic blocks might be used in the proof, but it is pretty simple and makes only use of a fact about vanishing of partial augmentations of torsion units, established in Hertweck (2006, 2007a). We remark that both results (whether p is even or odd) for a solvable group G are covered by Theorem 5.1 in Dokuchaev and Juriaans (1996). Note that a group G whose Sylow 2-subgroups are cyclic has a normal 2-comple- ment, by Burnside’s well known criterion, see Theorem 4.3 in Gorenstein (1968). We obtain the following corollary. Corollary 1. Let G be a finite group having cyclic Sylow p-subgroups for some prime p. Then any finite p-subgroup of V(ZG) is isomorphic to a subgroup of G. Finally, we remark that, as with other results in this field, the theorem can be formulated for more general coefficient rings than Z, notably for the semilocalization of Z at the prime divisors of the order of G. Unfortunately, it is definitely wrong for p-adic coefficient rings. 2. Kimmerle’s observation Coming back to the initial question, we mention that in the hope for further positive results, it is natural to impose restrictions on the prime divisors of the finite subgroup H , i.e., to consider only π-groups H for some set π of primes (a singleton {p}, to begin with), as has been done before in work on the stronger Zassenhaus conjecture (ZC3), cf. Dokuchaev and Juriaans (1996). It is well known that then, one can assume that Oπ′(G), the largest normal π ′-subgroup of G, is trivial, for H has an isomorphic image under the natural map ZG → ZG/Oπ′(G), see the remark after Theorem 2.2 in Dokuchaev and Juriaans (1996). This derives from the vanishing of certain partial augmentations of the elements of H . Recall that for a group ring element u = g∈G agg (all ag in Z), its partial augmentation with respect to an element x of G, or rather its conjugacy class xG in G, is the sum g∈xG ag; we will denote it by εx(u). The result of Cohn and Livingstone mentioned in the introduction really says that if an element h of H is of prime power order, then there exists an element x in G of the same order such that εx(h) 6= 0. Note that εz(u) = az for an element z in the center of G. An old yet fundamental result from Berman (1955) and Higman (1940) asserts that if εz(h) 6= 0 for an element h in H and some z in the center of G, then h = z. Coming to Marciniak’s question, suppose that G has no subgroups isomorphic to Klein’s four group. For our purpose, we can assume that O2′(G) = 1 and that Sylow 2-subgroups of G are not cyclic. Thus Sylow 2-subgroups of G are (generalized) quaternion, and by the Brauer–Suzuki theorem, from Brauer and Suzuki (1959), G contains a unique involution z. For an involution u in V(ZG), the Cohn–Livingstone result gives εz(u) 6= 0, and therefore u = z by the Berman–Higman result, answering Marciniak’s question in the affirmative. Theorem B (Kimmerle). Let G be a finite group. Suppose that V(ZG) has a subgroup isomorphic to Klein’s four group. Then the same is true for G. We do not know of a proof avoiding the use of the Brauer–Suzuki theorem. Suppose that Sylow 2-subgroups of G are quaternion groups. Then the theorem implies that finite 2-subgroups of V(ZG) are cyclic or quaternion groups. Taking UNIT GROUPS WITH NO NONCYCLIC ABELIAN FINITE SUBGROUPS 3 into account the structure of the quaternion groups, and the Cohn–Livingstone result, one obtains the following corollary. Corollary 2. Let G be a finite group whose Sylow 2-subgroups are quaternion groups (ordinary or generalized). Then any finite 2-subgroup of V(ZG) is isomor- phic to a subgroup of G. 3. Proof of Theorem A The partial augmentations of a torsion unit in V(ZG) encode its character values in a way establishing a connection to group elements which respects a divisibility relation between orders. We will make use of a lemma which is an easy consequence of this fact. Lemma 3. Let u be a torsion unit in V(ZG) of, say, order n. Let s be a natural integer coprime to n, so that st ≡ 1 mod n for another natural integer t. Then for all x in G whose order divide n, we have εx(u s) = εxt(u). Proof. Let ζ be a primitive n-th complex root of unity, and let σ be the Galois auto- morphism ofQ(ζ) sending ζ to ζs. Let x1, . . . , xk be representatives of the conjugacy classes of G whose elements have order dividing n. Note that then xt1, . . . , x k is an- other system of representatives. By Theorem 2.3 in Hertweck (2007a), εx(u) 6= 0 is possible only for elements x whose order divide n. Thus for any ordinary irreducible character χ of G, we have εxi(u s)χ(xi) = χ(u s) = χ(u)σ = εxi(u)χ(xi) εxi(u)χ(x i ) = (u)χ(xi). Since the character table of G, stripped off from any additional information, is an invertible matrix, it follows that εxi(u s) = εxt (u) for all indices i, which proves the lemma. � Corollary 4. Let u be a torsion unit in V(ZG) of, say, order n. Then for any x in G whose order divides n, s∈(Z/nZ)× s∈(Z/nZ)× εxs(u). Corollary 5. Suppose that for a prime divisor p of the order of G, all elements of order p in G are conjugate to a power of some fixed element x. Let u be a torsion unit in V(ZG) of order p. Then i=1 u i and i=1 x i have the same partial augmentations. Proof. Let k be the number of conjugacy classes of elements of order p in G. By Corollary 4 and Theorem 2.3 in Hertweck (2007a), εxi(u) = yG : y∈〈x〉 εy(u) = Applying again Theorem 2.3 from Hertweck (2007a), the corollary follows. � 4 MARTIN HERTWECK We will apply this by means of the following formula relating ranks of an idem- potent to arithmetical properties of the group. Corollary 6. Suppose that for a prime divisor p of the order of G, all elements of order p in G are conjugate to a power of some fixed element x. Suppose further that V(ZG) contains an elementary abelian subgroup U of order p2. Then for any ordinary character χ of G, (1) χ χ(1) + (p+ 1) χ(xi) We now turn to the proof of Theorem A. Suppose that G has a cyclic Sylow p-subgroup P (p = 2 is allowed). Let x be an element of order p in P , and set N = NG(〈x〉). Suppose further that V(ZG) contains an elementary abelian subgroup U of order p2. Let χ be the character of G which is induced from the principal irreducible character of P . Then the rank in (1) is (|G : P |+ |N : P |(p2 − 1)). If χ is a character of G which is induced from a faithful irreducible character of P , the rank in (1) is (|G : P | − |N : P |(p+ 1)). The difference of these ranks is |N : P |(p2 + p)/p2, which is not an integer. This contradiction proves the theorem. In view of Corollaries 1 and 2, one may be tempted to investigate the analogous problem for groups with dihedral Sylow 2-subgroups. These groups were classified by Gorenstein and Walter, and listed, for example, on p. 462 in Gorenstein (1968). To indicate what can be done by now, we end with an example. Note that the order of a finite subgroup of V(ZG) divides the order of G, see Lemma 37.3 in Sehgal (1993); a fact which, surprisingly enough from today’s point of view, is in this generality not recorded in Higman’s thesis. Example 7. For the alternating group A7, any finite 2-subgroup of V(ZA7) is isomorphic to a subgroup of A7. Proof. Sylow 2-subgroups of A7 are dihedral of order 8. Let x be an element of order 4 in A7. Then x G and (x2)G are the only conjugacy classes of elements of order 4 and 2, respectively. There is an (irreducible) character χ of A7 of degree 6 which is afforded by a deleted permutation representation. We have χ(x) = 0 and χ(x2) = 2. Let U be a finite 2-subgroup of V(ZA7). If U is of order 2, then U is rationally conjugate to a subgroup of A7 by Corollary 3.5 in Hertweck (2006). If U is of order 4, the Luthar–Passi method as described in Hertweck (2007a) is not sufficient to guarantee rational conjugacy to a subgroup of A7: for a unit u of order 4 in V(ZA7) one cannot exclude the possibility of having (εx2(u), εx(u)) = (2,−1) when χ(u) = 4. In this case, also χ(u−1) = 4. Anyway, U is isomorphic to a subgroup of A7, and the same is true if U is a Klein’s four group. Suppose that U is abelian of order 8. By the Cohn–Livingstone result, U is not cyclic. Set e = 1 u∈U u. Since e is an idempotent, χ(u) is a rational integer. If U is elementary abelian, then χ(e) = 1 (χ(1) + 7χ(x2)) = 20 , which is impossible. UNIT GROUPS WITH NO NONCYCLIC ABELIAN FINITE SUBGROUPS 5 Thus U contains 3 elements of order 2 and 4 elements of order 4. Trying out all possibilities shows that again χ(e) is not a rational integer. It remains to consider the case when U is the quaternion group. Let u be an element of order 4 in U . Since χ(u2) = χ(x2), the restriction of the character χ to U is the sum of four linear characters and the one of degree two. But this is impossible since χ is afforded by a rational representation, while the character of degree two of the quaternion group comes from the block of the rational quaternion algebra (whence the name of the group). � References Berman, S. D. (1955). On the equation xm = 1 in an integral group ring. Ukrain. Mat. Ž. 7:253–261. Brauer, R., Suzuki, M. (1959). On finite groups of even order whose 2-Sylow group is a quaternion group. Proc. Nat. Acad. Sci. U.S.A. 45:1757–1759. Cliff, G. H., Sehgal, S. K., Weiss, A. R. (1981). Units of integral group rings of metabel- ian groups. J. Algebra 73(1):167–185. Cohn, J. A., Livingstone, D. (1965). On the structure of group algebras. I. Canad. J. Math. 17:583–593. Dokuchaev, M. A., Juriaans, S. O. (1996). Finite subgroups in integral group rings. Can- ad. J. Math. 48(6):1170–1179. Gorenstein, D. (1968). Finite groups. New York: Harper & Row. Hertweck, M. (2006). On the torsion units of some integral group rings. Algebra Colloq. 13(2):329–348. Hertweck, M. (2007a). Partial augmentations and Brauer character values of torsion units in group rings. Comm. Algebra, to appear (e-print arXiv:math.RA/0612429v2). Hertweck, M. (2007b). The orders of torsion units in integral group rings of finite solvable groups. Comm. Algebra, to appear (e-print arXiv:math.RT/0703541). Higman, G. (1940). Units in group rings. Ph.D. thesis. University of Oxford (Balliol College). Kimmerle, W. (2006). Arithmetical properties of finite groups. Talk delivered at the Math Colloquium of the Vrije Universiteit Brussel. Marciniak, Z., Sehgal, S. K. (2003). The unit group of 1 + ∆(G)∆(A) is torsion-free. J. Group Theory 6(2):223–228. Roggenkamp, K. W. (1981). Units in integral metabelian grouprings. I. Jackson’s unit theorem revisited. Quart. J. Math. Oxford Ser. (2) 32:209–224. Sandling, R. (1981). Graham Higman’s thesis “Units in group rings”. In: Integral rep- resentations and applications (Oberwolfach, 1980). Lecture Notes in Math. Vol. 882. Berlin: Springer, pp. 93–116. Sandling, R. (1984). The isomorphism problem for group rings: a survey. In: Orders and their applications (Oberwolfach, 1984). Lecture Notes in Math. Vol. 1142. Berlin: Springer, pp. 256–288. Sehgal, S. K. (1993). Units in integral group rings. Pitman Monographs and Surveys in Pure and Applied Mathematics Vol. 69. Harlow: Longman Scientific & Technical. Weiss, A. (1988). Rigidity of p-adic p-torsion. Ann. of Math. (2) 127(2):317–332. Zassenhaus, H. (1974). On the torsion units of finite group rings. In: Studies in mathe- matics (in honor of A. Almeida Costa). Lisbon: Instituto de Alta Cultura, pp. 119–126. Universität Stuttgart, Fachbereich Mathematik, IGT, Pfaffenwaldring 57, 70550 Stuttgart, Germany E-mail address: [email protected] arXiv:math.RA/0612429v2 arXiv:math.RT/0703541 1. Introduction 2. Kimmerle's observation 3. Proof of Theorem A References

0704.0413

Exotic Hadron in Pole-dominated QCD Sum Rules

Exotic Hadron in Pole-dominated QCD Sum Rules Toru Kojo 1,∗), Daisuke Jido, 2 and Arata Hayashigaki 3 1 Department of Physics, Kyoto University, Kyoto 606-8502, Japan 2 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606–8502, Japan 3Institut für Theoretische Physik, J.W. Goethe Universität, D-60438 Frankfurt am Main, Germany We study pentaquark Θ+(I = 0, J = 1/2) in the QCD sum rules emphasizing that we can not extract any properties of the pentaquark outside of the Borel window. To find the appropriate Borel window, we prepare a favorable set up of the correlators and carry out the operator product expansion up to dimension 15 within factorization hypothesis. Our procedures reduce the unwanted high energy contaminations and enhance the low energy correlation. In the Borel window, independent analyses for the chiral-even and odd sum rules give the consistent values of the Θ+ mass, 1.68±0.22 GeV, and the residue. The parity is found to be positive. §1. Introduction The experimental announcement for the discovery of the pentaquark Θ+(1540)1) triggered tremendous amount of theoretical and experimental works on the exotic states. Although the existence of such exotic states is still not so obvious, the exotics provide a good opportunity to get the deeper insight of the hadron structures and their connection to QCD. One of approaches from QCD to exotics is the QCD sum rule (QSR),2) which relates informations of QCD to the hadronic properties through the correlator analysis for the interpolating fields of hadrons. The Borel transformed sum rules with the simplest pole + continuum parametrization are given as (i = 0, 1 correspond to the chiral even and odd part, respectively) (ope) i (−Q 2) = λ2i e −m2/M2 + ds e−s/M (ope) i (s), (1 where the relation ±mλ20 = λ21 holds due to the spinor structure and the relative sign of the residues λ2i represents with the parity of the resonance state. Using these sum rules, we can derive the approximated expressions of the mass and residue as a function of M and sth. To extract properties of the low energy excitations with good accuracy, we need to treat sum rules in the appropriateM2-region, i.e., Borel window, where the low energy correlation is large enough compared to the contaminations from high energy components which have no relations with properties of low-lying resonances. The setting the Borel window is the most important step in QSR and, only within this window, we can evaluate the physical quantities. In the exotic cases, as reported in Ref. 3), it is extremely difficult to find the appropriate Borel window in contrast to the usual meson and baryon cases. ∗) e-mail address: [email protected] typeset using PTPTEX.cls 〈Ver.0.9〉 http://arxiv.org/abs/0704.0413v1 2 Toru Kojo, Daisuke Jido and Arata Hayashigaki This is because the OPE convergence is very slow and the unwanted high energy components dominate the spectral integral. In addition, we often encounter the artificial stability of the physical quantities against M2-variation. This is the case that physical quantities depend too strongly on the threshold parameter sth and not on the low energy correlations which we want to extract. To attack these serious problems common to the exotics, we propose a new approach and apply it to the Θ+, assuming its quantum number as I = 0, J = 1/2, as one example of the exotics.4) §2. OPE and favorable set up of the correlators To find the Borel window, it is necessary to increase low energy informations in the spectral function efficiently and, at the same time, reduce high energy contami- nations. For these purposes, we take the following treatments. Inclusion of the higher dimension terms of OPE is particularly important be- cause they strongly reflect the low energy dynamics. For example, in the case of the sum rules for ρ and A1 mesons, the dim.0 and 4 terms are the same due to the chiral symmetry realized in the high energy, and the splitting of the masses is explained only after the inclusion of dim.6 terms, 〈q̄q〉2, which appear due to the chiral sym- metry breaking. From these observations, we perform the OPE calculation up to dim.15 within factorization hypothesis both for taking into account the low energy correlations and for the confirmation of good OPE convergence. As later shown, simple inclusion of the low energy correlations through the higher dimension terms is found to be not sufficient to find the Borel window because high energy contaminations are too large in the QSR for the exotics. To reduce the high energy contaminations, we take the difference between correlators for two interpolating fields with similar structure but different chirality each other, i.e., d4x eiq·x ∣T [P (x)P̄ (0) − t S(x)S̄(0)] ds e−s/M Im[ΠP0 (s)− tΠS0 (s)] q̂ + Im[ΠP1 (s)− tΠS1 (s)] , (2.1) where Π0, Π1 correspond to the chiral even and odd part respectively, and P = ǫabcǫdef ǫcfg{uTaCdb}{uTd Cγµγ5de}γµCsTg , (2.2) S = ǫabcǫdef ǫcfg{uTaCγ5db}{uTd Cγµγ5de}γµCsTg . (2.3) Here the only difference in these interpolating fields is that the first diquark structures have the opposite chirality. Let us first explain in the case of the chiral even part. Since they show the same behavior in high energy due to the chiral symmetry, after the subtraction of two correlators (t = 1 case), the irrelevant high energy contributions are expected to be canceled out in the similar way as the Weinberg sum rules.5) In terms of OPE, this cancellation corresponds to the cancellation of the lower dimension terms. It is not a priori evident whether the low energy correlations remain enough even after the subtraction because the low energy contribution could also cancel out. Our Borel Exotic Hadron in Pole-dominated QCD Sum Rules 3 analysis, however, reveals that, in the case of t = 1, the large low energy correlation remains enough even after the subtraction. As a result, we can find the Borel window in the relatively large M2-region. On the other hand, for the chiral odd part, the subtraction procedure corre- sponding to t = 1 case does not lead the cancellation of the high energy components because chiral odd part is constructed of the chiral symmetry breaking terms. How- ever, in the case of t = 1, the OPE convergence is found to be very good and then we can find the Borel window in the small M2-region where high energy contaminations are suppressed due to the Borel factor e−s/M in the integral of the spectral function. §3. Borel analysis for mass and residue Here we explain our criterion for the Borel window. The lower bound of the Borel window is given so that the highest-dimensional terms in the truncated OPE are less than 10% of its whole OPE, while the upper bound is determined by the region where the absolute value of the pole contribution is larger than the absolute value of the integrated spectral function in the region s ≥ sth. Note that the 50% pole contribution in our criterion is extremely large in comparison with any prior pentaquark sum rules, where the pole contributions are not more than 20%.3) To recognize the problems in the case of QSR for exotics, let us see Fig.1 for M2-dependence of the mass in the cases of t = −1, 0, 10 corresponding to PP̄ +SS̄, PP̄ , SS̄ cases respectively. The threshold value is fixed to typical value sth = 2.2 GeV. In these cases, we fail to find stabilities of the mass in the M2-region lower than the upper bound of the Borel window. The stabilities above the upper bound are simply artifacts which often appear in QSR. Fig.1 shows that typical mass of PP̄ case is much smaller than that of SS̄, and then we can expect that the low energy correlation of PP̄ is much larger than that of SS̄. This observation leads that even after the subtraction PP̄ − SS̄ (t = 1 case), the low energy correlation can remain enough. Now we see the case of around t = 1. We tune the value of t around t = 1 to get the widest Borel window. As expected, for even part (t = 0.9), the high energy SS (t = 10) PP (t = 0) PP (t = 0) SS (t = ‐1) SS (t = ‐1) SS (t = 10) Fig. 1. The behavior of the mass as a function of M2 for t = −1, 0, 10. The left arrows represent the upper bound of the Borel window. In the smaller M2-region than the upper bound, we can not find stable region of the mass. The stabilities above the upper bound are simply artifacts which often appear in QSR. 4 Toru Kojo, Daisuke Jido and Arata Hayashigaki even odd (t=1.1)(t=0.9) Fig. 2. The behavior of the mass as a function of M2. The left (right) arrows represent the upper (lower) bound of the Borel window. We succeed to find the Borel window and stabilities of mass. contaminations are canceled out due to chiral symmetry and we find the wide Borel window in the relatively large M2-region. On the other hand, for odd part (t = 1.1), thanks to the good OPE convergence, we also find the wide Borel window in the small M2-region. The threshold values are taken to make the physical quantities most stable in the Borel window. The best stability is achieved with sth = 2.2 GeV (even) and 2.1 GeV (odd), giving mΘ+ = 1.64 GeV (even) and 1.72 GeV (odd) respectively. The values of the residue are also obtained from the chiral even and odd sum rules as λ20 = (3.0±0.1)× 10−9 GeV12 and λ21/mΘ+ = (3.4 ± 0.2) × 10−9 GeV12. It is remarkable that these numbers are quite similar with the close t. This implies that our analysis investigates consistently the same state in the two independent sum rules. Note that from the relative sign of the residues, we assign positive parity to the observed Θ+ state. In conclusion, we perform the Borel analysis for Θ+ with special setup of the correlators in order to find the Borel window. Within uncertainties of the condensate value, independent analyses for the chiral-even and odd sum rules give the consistent values of the Θ+ mass, 1.68± 0.22 GeV, and the residue. The parity is found to be positive. Acknowledgements We thank Profs. M. Oka, A. Hosaka and S.H. Lee for useful discussions about QSR for the exotics during the YKIS2006 on ”New Frontiers on QCD” held at the Yukawa Institute for Theoretical Physics. This work is supported in part by the Grant for Scientific Research (No.18042001) and by Grant-in-Aid for the 21st Century COE ”Center for Diversity and Universality in Physics” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. References 1) T. Nakano et al., Phys. Rev. Lett. 91 (2003), 012002. 2) M.A. Shifman, A.I. Vainshtein, and V.I. Zakharow, Nucl. Phys. B 147 (1979), 385. 3) R.D. Matheus and S. Narison, Nucl. Phys. Proc. Suppl. 152 (2006), 236. 4) T. Kojo, A. Hayashigaki, and D. Jido, Phys. Rev. C 74 (2006), 045206 5) S. Weinberg, Phys. Rev. Lett. 18 (1967), 507. Introduction OPE and favorable set up of the correlators Borel analysis for mass and residue

0704.0414

Leaky modes of a left-handed slab

Leaky modes of a left-handed slab A. Moreau LASMEA, UMR CNRS 6602, Université Blaise Pascal, 24 avenue des Landais, 63177 Aubière, France. D. Felbacq GES UMR CNRS 5650, Université de Montpellier II, Bat. 21, CC074, Place E. Bataillon, 34095 Montpellier Cedex 05, France. Using complex plane analysis we show that left-handed slab may support either leaky slab waves, which are backward because of negative refraction, or leaky surface waves, which are backward or forward depending on the propagation direction of the surface wave itself. Moreover, there is a general connection between the reflection coefficient of the left-handed slab and the one of the corresponding right-handed slab (with opposite permittivity and permeability) so that leaky slab modes are excited for the same angle of incidence of the impinging beam for both structures. Many negative giant lateral shifts can be explained by the excitation of these leaky modes. Keywords: Left-handed materials, leaky modes, complex plane analysis 1 Introduction Left-handed materials [1] have long been considered a theoretical oddity. Since it has been demonstrated that they could be produced using metamaterials [2], they have attracted much attention. The basic physics of left-handed materials (LHM) is truly exotic and has been completely ignored until recently, it renews the physics of lamellar structures to the extend that a bare slab of LHM exhibits many surprising properties : it can for instance support unusual guided modes [3,4] or behave as a perfect lens [5]. In this paper, we study the exotic properties of the different types of leaky waves supported by a left-handed slab. Given the importance of the left-handed slab for both fundamental and applied works, there is obviously a need for a clear understanding of these properties. We particularly show that two types of leaky waves are supported by such a structure (i) leaky slab waves which are always backward due to negative refraction and (ii) leaky surface waves which do not exist for a right-handed slab and which can be backward or forward. The excitation of these modes leads to positive or negative giant lateral shifts, the latter being rather exotic [6]. http://arxiv.org/abs/0704.0414v3 2 Leaky modes and giant lateral shifts A leaky mode [6] is a solution of the wave equation which verifies the relation dispersion of a structure but with a propagative solution above and (or) under the structure. Whereas a guided mode has a real propagation constant, the propagation constant of a leaky mode is complex because the energy of the waves leaks out of the structure and the waves is attenuated. A leaky wave is thus a complex solution of the dispersion relation and a complex plane analysis is thus particularly relevant for a thorough analysis of its properties. Let us underline that a leaky mode may be either forward, which is common, or backward, leading to a propagation constant which has a positive (respectively negative) imaginary part. Let us consider a slab characterized by ε2 and µ2 surrounded by right-handed media with ε1 and µ1 (resp. ε3 and µ3) above (resp. under) the slab as shown figure 1. The values we have chosen for ε2 and µ2 are arbitrary but realistic [7] so that this structure could be realized using split-ring resonators and wires. ε µ3 3 ε µ1 1 ε µ2 2 Figure 1: The LHM slab of thickness h surrounded by right-handed media. We may assume that ε1 µ1 ≥ ε3 µ3 with no loss of generality. The relation dispersion of such a structure can be written r21 r23 = exp(−2iγ2 h) (1) where γi = εi µi k 0 − α2, k0 = ωc = and rij = κi−κj κi+κj with κi = in TE polarization (or κi = in TM polarization). Since ε1 µ1 ≥ ε3 µ3 and we are concerned with leaky waves, we will only consider values of α such that α < ε1 µ1 k0, which means that the solution will always be propagative at least in medium 1. Let us now consider the reflection coefficient of a plane wave exp(i(αx+γ z−ω t)) coming from upwards with an angle of incidence θ so that α = n k0 sin θ. Its reflection coefficient can be written r23 exp(2iγ2 h)− r21 1− r21 r23 exp(2iγ2 h) using the above definitions. It is obvious that when the relation dispersion is verified, then the reflection coefficient presents a pole. A leaky mode thus corresponds to a pole of the reflection coefficient. A zero, located on the other side of the real axis, corresponds to each pole. As we will see in the following, a zone where the phase of r quickly varies lies between a pole and its corresponding zero. This zone crosses the real axis, so that the presence of a pole is responsible for a swift variation of the phase on the real axis. When considering the reflection of a gaussian beam on a structure whose reflection co- efficient has a modulus equal to one (so that it can be written r = eiφ), the lateral displacement of the reflected beam’s barycenter along the interface is given by the well known formula δ = −dφ . (3) This lateral displacement is the sign that a leaky wave has been excited by the incident beam. The reflected beam then has two components : the part which is reflected by the first interface of the structure (whose barycenter is not particularly displaced) and the leaky wave itself [6]. The reflected beam is heavily distorted by the leaky wave and presents an exponentially decreasing tail so that its barycenter is largely displaced : this is the so-called giant lateral shift. This effect is sometimes called a giant Goos-Hänchen effect, but in this case the shift is due to the excitation of a leaky mode [6] and not, as in the real Goos-Hänchen effect [8,9], to the total reflection. 3 The left-handed slab With left-handed materials, though, negative lateral shifts seem to be much more common [10–14] than once expected [6]. Here we will consider the case of a left-handed slab (i.e. if ε2 < 0 and µ2 < 0) and explain why the leaky modes supported by such a structure are usually backward. Our explanations will be supported by a complex plane analysis of the leaky modes. Here the expression (2) of the reflection coefficient remains perfectly valid. We will now distinguish two cases : the case when the solution is propagative in the left-handed medium and the case when the solution is evanescent in region 2. 3.1 Leaky slab modes When the field is propagative in the left-handed slab, large negative lateral shifts have been reported but not interpreted [13]. These shifts are due to the excitation of leaky slab modes or Perot-Fabry resonances of the slab at non normal incidence. Such leaky modes have already been studied for a right-handed slab [15] and they can be considered as constructive interferences of the multiple beams which are produced by reflections on the interfaces of the slab. In the case of a left-handed slab, since the first beam undergoes a negative refraction as shown figure 2 these constructive interferences will logically generate a backward leaky mode. We may thus conclude that the existence of such a backward leaky mode is linked to negative refraction. Figure 2: Modulus of the field for a thick left-handed slab with ǫ1 = ǫ3 = µ1 = µ3 = 1, ǫ2 = −3,µ2 = −1 and h = 60 λ using a gaussian incident beam with a waist of 20 λ and an incidence angle of θ = 45. This argument is not a proof, though : unexpected lateral shifts have been reported when the beams interfere destructively [16]. But if the leaky modes are backward, then the corresponding solutions of the dispersion relation and the poles of the reflection coefficient should have a negative imaginary part. This is what is shown figure 3. Figure 3: The phase of the reflection coefficient in a part of the complex plan [0, n1 k0] + i[−k0 ]. Each black point represents a pole and each white point a zero. The cut line is clearly visible here. The rapid variation of the phase which is due to each pole is obvious. Two types of leaky slab waves should be distinguished (i) L2 waves which are leaky in both the upper and the lower media and (ii) L1 waves which are leaky only in the upper medium and evanescent in the lower one. The latter correspond to the poles located under the cut line. Using complex plane analysis we will now try to show that all the solutions of the disper- sion relation 1 are located in the lower part of the complex plane, meaning that all the leaky modes are backward. When the relation dispersion is satisfied, then the following condition holds : |r23 r21| = e2 γ h. (4) As demonstrated in the annex, |rij > 1 whenever one of the media is left-handed. Since medium 2 is left-handed then the condition h > 1 (5) should be satisfied, which is possible for γ′′2 > 0 and therefore for α ′′ < 0 (see the annex for details). The fact that rij > 1 is thus the main reason why the poles of r are under the axis and why the leaky slab modes are backward. We must underline the fact that our demonstration is valid only for the first Riemann sheet : our proof cannot exclude that there may be some poles on the other Riemann sheet above the real axis, corresponding to forward L1 leaky slab waves when ε1 µ1 > ε3 µ3. But we could not find any. 3.2 Leaky surface modes Let us now consider the situation in which the field is evanescent in the left- handed medium. Then γ2 is purely imaginary on the real axis. Since e 2 γ′′ h tends towards infinity when h → +∞ then relation (4) can be verified only if r23 has a pole (r21 cannot have one since the field is always propagative in the upper medium). This means that the structure may support a leaky mode only if the interface between medium 2 and 3 can support a guided mode. It is now well-known that such an interface actually supports a surface mode [17,18] which can, depending on media 2 and 3, be backward (resp. forward) corresponding to a pole under the real axis (resp. above the real axis but on the other Riemann sheet). The leaky wave always has the same propagation direction as the surface mode, whatever the thickness of the slab, as shown figure 4. In the case of a forward leaky wave, only the zero belongs to the first Riemann sheet, just under the real axis. The pole shown figure 4 belongs to the other Riemann sheet. 0.05 0.15 0.25 0.35 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 −0.45 −0.35 −0.25 −0.15 −0.05 1.7 1.75 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 Figure 4: Location of the poles in the α complex plane for different values of h with ε1 = 9, µ1 = µ3 = ε3 = 1 and (a) ε2 = −0.5 and µ2 = −1.5, showing a forward surface mode and (b) ε2 = −5 and µ2 = −0.5, showing a backward surface mode. Figure 5 finally shows the excitation of a backward leaky surface wave by a gaussian beam. The chosen values of µ2 may be obtained with simple split ring resonators [19] for instance. Figure 5: Modulus of the field for a left-handed slab with ǫ1 = 9,ǫ3 = µ1 = µ3 = 1, ǫ2 = −0.5,µ2 = −1.5 and h = 0.6 λ using a gaussian incident beam with a waist of 20 λ and an incidence angle of θ = 21.496. The pole corresponding to the leaky mode is located at αp = (1.0993 + 0.001267i) k0. 4 Fundamental property Let us a consider a structure with left-handed materials. We will call corresponding right-handed structure the structure obtained by replacing any left-handed medium by a medium with opposite permittivity and permeability, without changing the geometrical parameters. In this section, we will concentrate on the link between the reflection coefficient of a left-handed slab and the one of its corresponding right-handed structure. Let us consider the interface between a right-handed medium labelled i and a left-handed medium j. The reflection coefficient of such an interface is rij . We will now define r+ij the reflection coefficient of an interface between medium i and right-handed medium characterized by |εj| and |muj|. It is not difficult to see, from the expression of rij that . (6) This allows to understand why the Goos-Hänchen shift of an interface between a right- and a left-handed medium is the opposite of the corresponding right-handed structure [11] since the phases of both structures are opposite on the real axis. The reflection coefficient r can now be written e2iγ2 h 1− e2iγ2 h r+23 e −2iγ2 h − r+21 1− r+21 r+23 e−2iγ2 h Since except when z is on the cut line, then γ(z∗) = γ(z)∗ and hence r+ij(z) ∗) so that r(z)∗ = ∗) e2iγ2(z ∗)h − r+21(z∗) 1− r+21(z∗) r+23(z∗) e2iγ2(z , (10) which can simply be written r(z)∗ = r+(z∗), (11) where r+ is the coefficient reflection of the corresponding right-handed slab. Note that this relation does not hold on the cut line, but that it holds for the two Riemann sheets. This means that the poles of the left-handed slab and the poles of the corresponding right-handed slab are symmetrical with respect to the real axis. This means that L2 waves can be excited for the same incidence angle for both structures. This is not the case for L1 modes : the function r on the real axis is continuous with the lower part of the first Riemann sheet whatever the situation and the poles which are above the cut line thus have no effect on the real axis. As an example, we have computed the field in TE polarization inside and around the slab when it is illuminated with a gaussian beam for the left-handed slab and its corresponding right-handed structure. The results are shown figures 6 and 7. Figure 6: Modulus of the field for a symmetrical slab with ǫ1 = ǫ3 = 9, µ1 = µ3 = 1, ǫ2 = 1.5, µ2 = 1 and h = 1.3 λ using a gaussian incident beam with a waist of 20 λ and an incidence angle of θ = 22.78. Figure 7: Modulus of the field for a symmetrical slab with ǫ1 = ǫ3 = 9, µ1 = µ3 = 1, ǫ2 = −1.5, µ2 = −1. and h = 1.3 λ using a gaussian incident beam with a waist of 20 λ and an incidence angle of θ = 22.78. The pole corresponding to the leaky mode is located at αp = (1.16823− 0.01125i) k0 5 The grounded left-handed slab The grounded left-handed slab is a much more simple structure for (i) there is no need to distinguish two different types of leaky slab modes and (ii) the structure can not support any leaky surface mode. All the leaky modes are then slab modes and are found for α < n2 k0. The reflection coefficient of the grounded slab is given by (2) with r23 = −1 for the TE polarization and r23 = 1 for the TM polarization so that the relation dispersion gives |r12| = e2 γ h. (12) Since |r12| > 1 then all the solutions of the dispersion relation are located in the lower part of the complex plane so that they are all backward. It is then easy to show that the relation r+(z)∗ = r(z∗) still holds. As a consequence, the leaky modes of a grounded left-handed slab and of its corresponding right-handed structure can be excited for the same angle of incidence of the impinging beam. 6 Conclusion In this paper, we have thoroughly studied the leaky modes of a left-handed slab for realistic values of the permittivity and permeability of the left-handed medium [7,19,20] which can be obtained using structures like split-ring resonators. Our results can be summarized as follows. Left-handed slab may support two types of leaky modes : • Leaky slab modes, which are always backward because of the negative refraction phenomenon. When the transmission is not null, leaky modes of the left-handed slab and of its corresponding right-handed structure are excited for the same angle of incidence. • Leaky surface modes, which may be backward or forward depending on the propa- gation direction of the surface wave itself. This work could help to interpret many giant lateral shifts as excitations of exotic leaky waves [12, 13, 16]. Since the existence of backward slab waves is linked to the property of negative refraction, and since these leaky waves constitute a signature of a left-handed slab behavior we think that they could be used to characterize the left-handedness of metamaterial or photonic crystal structures far better than other methods [21]. Acknowledgments This work has been supported by the French National Agency for Research (ANR), project 030/POEM. The authors would like to thank Alexandru Cabuz and Kevin Vynck for their help. Annex In this annex, we will clearly define the choice we have made for the definition of the complex square root and prove that for z on the first Riemann sheet (but not on the cut line) we have |rij(z)| > 1 when media i and j are not both right-handed. Since the square root can be continued on the complex plane, r and rij can be continued as well. We have chosen to take 2 with z = r eiθ and θ ∈]−π, π], as a definition of the square root. This means that we have placed the cut line on the negative part of the real axis and if x is a positive real, −x = i x. This defines the square root on the entire complex plane, to which we refer as the first Riemann sheet. When we write that z is on the second Riemann sheet, it will mean that we have taken the opposite of defined above. With this choice, we have (i) ℜ( z) ≥ 0 (ii) for z on both sheets but not on the cut line (iii) if ℑ(z) < 0, ℑ( z) < 0 and if ℑ(z) > 0, ℑ( z) > 0 (iv) the function γ(z) = ǫ µ k20 − z2 has a cut line on the real axis (on ] − ∞,−n k0] ∪ [n k0,+∞] more precisely) and the function γ on the real axis is continuous with the part of the complex plane which is under the cut line : when z passes through the cut line from the first Riemann sheet (coming from the lower part of the plane) to the second Riemann sheet, γ(z) is continuous. When a function which can be written using γ(z) presents a pole, it must be found either (i) for z on the first Riemann sheet and under the real axis (we will say that the pole itself is on the first Riemann sheet in this case) or (ii) for z on the second Riemann sheet but above the real axis. We have rij = κi − κj κi + κj . (13) The modulus of rij reads as |rij|2 = (κi − κj) (κ∗i − κ∗j ) (κi + κj) (κ i + κ |κi|2 + |κj|2 − 2 (κ′i κ′j + κ′′i κ′′j ) |κi|2 + |κj|2 + 2 (κ′i κ′j + κ′′i κ′′j ) , (15) where κ = κ′ + i κ′′. Let us define x and y the real and imaginary part of z = x + i y on the first Riemann sheet. Let us assume that x > 0. We have n2 k20 − z2 = n2 k20 − x2 + y2 − 2 i x y. (17) If y > 0, then x y > 0 and thus ℑ(n2 k20 − z2) < 0 so that finally ℑ(γ) < 0. If y < 0, then x y < 0 so that ℑ(γ) > 0. Since γ(−z) = γ(z) the result will hold for x < 0 too and for x = 0, γ(z) is real and positive so that the result obviously holds. So the imaginary part of γ(z) is positive (resp. negative) when the imaginary part of z is negative (resp. positive). For any right-handed medium, κ has the same property than γ. For a left-handed medium, since κ = γ or κ = γ depending on the polarization, the imaginary part of κ has the sign of ℑ(z). Since i and j are not both right-handed, then κ′′i and κ′′j have not the same sign and the product κ′′i κ j is always negative. Since ℜ( z) > 0 for all z on the first Riemann sheet then κ′i κ j is always negative too. Finally, since κ′i κ j + κ j < 0, we have |rij| > 1 for all z except on the real axis. Please note that rij is not, in the particular case of a left-handed medium, the reflection coefficient on the interface [22]. REFERENCES [1] V. Veselago, “The Electrodynamics of substances with simultaneously negative values of ǫ and µ”, Usp. Fiz. Nauk. 92 517 (1967). [2] R. Shelby, D. Smith, and S. Shultz, “Experimental Verification of a Negative Index of Refraction”, Science 292 77 (2001). [3] I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Guided modes in negative-refractive- index waveguides”, Phys. Rev. E 67 057602 (2003). [4] P. Tichit, A. Moreau, and G. Granet, “Localization of light in a lamellar structure with left-handed medium : the Light Wheel”, Opt. Expr. 15 14961–14966 (2007). [5] J. Pendry, D. Schuring, and D. Smith, “Controlling Electromagnetic Fields”, Science 312 1780 (2006). [6] T. Tamir and H. Bertoni, “Lateral displacement of optical beams at multilayered and periodic structures”, J. Opt. Soc. Am. 61 1397 (1971). [7] D. R. Smith, S. Schultz, P. Makos, and C. M. Soukoulis, “Determination of effec- tive permittivity and permeability of metamaterials from reflection and transmission coefficients”, Phys. Rev. B 65 195104 (2002). [8] F. Goos and H. Haenchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion”, Ann. Phys. 1 333 (1947). [9] D. Felbacq, A. Moreau, and R. Smaali, “Goos-Haenchen effect in the gaps of photonic crystals”, Opt. Lett. 28 1633 (2003). [10] A. Lakhtakia, “On plane wave remittances and Goos-Haenchen shifts of planar slabs with negative real permittivity and permeability”, Electromagnetics 23 71 (2003). [11] P. Berman, “Goos-Haenchen shift in negatively refractive media”, Phys. Rev. E 66 067603 (2002). [12] I. Shadrivov, A. A. Zharov, and Y. Kivshar, “Giant Goos-Haenchen effect at the reflection from left-handed materials”, Appl. Phys. Lett. 83 2713 (2002). [13] L. Wang and S. Zhu, “Large negative lateral shifts from the Kretschman-Raether configuration with left-handed materials”, Appl. Phys. Lett. 87 221102 (2005). [14] A. Moreau and D. Felbacq, “Comment on ’Large negative lateral shifts from the Kretschman-Raether configuration with left-handed materials”’, Appl. Phys. Lett. 90 066102 (2007). [15] F. Pillon, H. Gilles, S. Girard, M. Laroche, R. Kaiser, and A. Gazibegovic, “Goos- Haenchen and Imbert-Fedorov shifts for leaky guided modes”, J. Opt. Soc. Am. B 22 1290 (2005). [16] X. Chen and C. Li, “Lateral shift of the transmitted light beam through a left-handed slab”, Phys. Rev. E 69 066617 (2004). [17] R. Ruppin, “Surface polaritons of a left-handed medium”, Phys. Lett. A 277 61 (2000). [18] I. Shadrivov, A. Sukhorukov, Y. Kishvar, A. Zharov, A. Boardman, and P. Egan, “Non-linear surface waves in left-handed materials”, Phys. Rev. E 69 016617 (2004). [19] C. Soukoulis, T. Koschny, J. Zhou, M. Kafesak, and E. Economou, “Magnetic re- sponse of split ring resonators at terahertz frequencies”, Phys. Stat. Sol. B 244 1181–1187 (2007). [20] S. O’Brien and J. B. Pendry, “Photonic band-gap effect and magnetic activity in dielectric composites”, J. Phys. : Condens. Matter 14 4035–4044 (2002). [21] J. Kong, B. Wu, and Y. Zhang, “Lateral displacement of a Gaussian beam reflected from a grounded slab with negative permittivity and permeability”, Appl. Phys. Lett. 80 2084 (2002). [22] D. Felbacq and A. Moreau, “Direct evidence of negative refraction at media with negative ε and µ”, J. Opt. A : Pure and Appl. Opt. 5 L9 (2003). Introduction Leaky modes and giant lateral shifts The left-handed slab Leaky slab modes Leaky surface modes Fundamental property The grounded left-handed slab Conclusion

0704.0415

Coulomb blockade of field emission from nanoscale conductors

untitled Coulomb blockade of field emission from nanoscale conductors O. E. Raichev* Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Prospekt Nauki 45, 03028, Kiev, Ukraine �Received 9 February 2006� Theoretical description of the field emission of electrons from nanoscale objects weakly coupled to the cathode is presented. It is shown that the field-emission current increases in a steplike fashion due to single- electron charging which leads to abrupt changes of the effective electric field responsible for the field emission. A detailed consideration of the current-voltage characteristics is carried out for a nanocluster modeled by a metallic spherical particle in the close vicinity of the cathode and for a cylindrical silicon nanowire grown on the cathode surface. PACS number�s�: 79.70.�q, 73.23.Hk, 73.40.Gk I. INTRODUCTION The discrete nature of electric charge reveals itself in the transport of electrons through small conductors �nanopar- ticles or other nanoscale objects� weakly coupled to the source and drain electrodes �current-carrying leads� owing to the Coulomb blockade effect. Numerous manifestations of the charge quantization in transport properties, the most fa- miliar of them are the Coulomb blockade oscillations of the electric current as a function of the gate voltage and the Coulomb staircase in the current-voltage characteristics, have attracted considerable attention in the past years.1 Since the fundamentals of the transport theory in the Coulomb blockade regime have been established,2–4 the Coulomb blockade-based physics has been applied to various issues of electron transport in mesoscopic systems, and the field of its applications expands in line with the advances in nanotech- nology. Usually, the influence of the Coulomb blockade on the current in two-terminal devices is considered under assump- tion that the coupling between the nanoscale object and the leads is not sensitive to the number of electrons N determin- ing the object charge eN. This corresponds to the introduc- tion of ohmic �or nearly ohmic� effective resistances describ- ing this coupling. Though this assumption often works well, it can be violated, for example, in nanomechanical systems,5–7 where charging of the object gives rise to its displacement towards one of the leads thereby changing its tunnel coupling to both leads. In this paper we study a situ- ation when the sensitivity of the tunnel coupling to the num- ber of electrons does not require a mechanical displacement and is determined by the nature of tunneling. This implies a device layout and conditions similar to those used in the recent experiments on field emission of electrons from me- tallic nanoclusters,8–10 silicon nanowires11–15 and nanocones,16,17 and carbon nanotubes �see, for example, Refs. 11 and 18–26�, when small �nanoscale� objects are formed on the source electrode �cathode�, the latter is then negatively biased with respect to the drain electrode �anode� in vacuum. The current between the electrodes flows owing to the field emission of electrons from nanoscale objects, because the electric field F at the tips of the objects is higher than in the other places of the device. The field-emission current is described by the Fowler-Nordheim formula27 I = ASF2 exp�− F �, F = 4�2m 3�e�� W3/2, �1� where m is the free electron mass, W is the work function of the emitting material, S is the effective emitting area, and A is a constant expressed through the work function and Fermi energy �F of the emitting material �e�3��F/W 4�2 � ��F + W� . �2� The effective field F, which describes the tunnel coupling between the nanoscale object and the anode, depends on the object charge, which is induced by the applied voltage V =V1−V2, where V1 and V2 are the cathode and anode poten- tials, respectively. Under conditions of Coulomb blockade, i.e., when the electric connection between the cathode and the object is weak and the charging energy of the object considerably exceeds the temperature T, the continuous variation of the voltage V leads to discrete changes of the object charge in units of e, and, consequently, to correspond- ing discrete changes of the field F. Therefore, one may in- troduce the field FN, which is a function of the discrete num- ber N and continuous variable V. Next, if the current in the device is limited by the field emission, the single-electron tunneling processes become important. This means that, at a fixed voltage V, the object stays mostly in the states with N and N−1 electrons, the number N is determined by the volt- age. In the N-electron state, no electrons can come to the object from the cathode until an electron leaves the object by tunneling through the barrier, see Fig. 1�a�. Then the object appears in the N−1-electron state and returns to the N-electron state before the next Fowler-Nordheim tunneling event takes place. The field-emission current in these condi- tions is given by Eq. �1� with F=FN and can be denoted as IN. If the bias eV increases, the state with N+1 electrons becomes more favorable, and the current changes in a step- like fashion from IN to IN+1. This leads to staircaselike current-voltage characteristics, which may look similar to the usual Coulomb staircases.28–30 However, since the sensitivity of the tunneling to the number of electrons is involved, the staircaselike current-voltage characteristics can exist under rather peculiar conditions, when the source-drain bias is or- ders of magnitude larger than the charging energy. The rest of the paper is devoted to quantitative studies based on the physical idea outlined above. In Sec. II we give the basic equations and calculate the current in the simplest case of an idealized emitter shown in Fig. 1�b�. In Sec. III we calculate the current from a nanocluster modeled by a spheri- cal particle on the metallic cathode surface and from a semi- conductor wire �nanowhisker� grown perpendicular to the cathode surface. The discussion and concluding remarks are given in the last section. II. GENERAL CONSIDERATION We consider the case of classical �or metallic� Coulomb blockade, when the electron energy level separation in the nanoscale object can be neglected in comparison to both temperature and charging energy. Since the object is assumed to be weakly coupled to the cathode, we study the sequential tunneling process and not the coherent one. It is convenient to investigate the electron transport by applying the kinetic equation2 �Master equation� for the distribution function PN describing the probability for the object to be in the state with N electrons. Assuming that the electric connection be- tween the cathode and the object is characterized by the con- ductance G, this equation is written as = QN+1 − QN, �3� where 1 − exp�− �EN/T� �PN − PN−1 exp�− �EN/T� + PNIN/�e� . �4� Here �EN= �e 2 /C��N−1/2−C2V /e is the difference in Cou- lomb energies for the objects with N and N−1 electrons, C is the total capacitance, and C2 is the capacitance of the object with respect to the anode �the capacitance with respect to the cathode is given by C1=C−C2�. The first term in expression �4� has the usual form2 and corresponds to the current be- tween the object and the cathode. It is written as a difference of the contributions describing the departure of an electron from the object in the N-electron state and arrival of an elec- tron at the object in the N−1-electron state. The second term corresponds to the field-emission current from the object in the N-electron state. Since no electrons come to the object from the anode, this term does not contain a contribution describing arrival of electrons. In the stationary case, Eq. �3� is reduced to the form QN=const, where the constant can be chosen equal to zero. After determining PN from the equation QN=0 with the use of the normalization condition NPN=1, the total current is given by PNIN. �5� Under the condition GT� �e � IN, which means that the object is in thermal equilibrium with the cathode, the stationary solution of Eq. �3� is written as PN=Z −1 exp�−EN /T�, where EN= �e 2 /2C��N−C2V /e 2 is the Coulomb energy, and Z N exp�−EN /T� is the partition function. The current in this case is determined by the expression J = Z−1 IN exp�− EN/T� . �6� Let us apply the solution �6� to the idealized model of emitter, Fig. 1�b�, when the emission takes place from a spherical nanoparticle of radius R, placed at a distance d from the cathode. The distance between the cathode and an- ode is L. The connection c-p denotes a low-transparent con- tact �for example, tunnel barrier� between the particle and the cathode, which does not contribute to the field-emission properties and electrostatics of the device. Assuming d�R, we have C=R, C2=Rd /L, and neglect the charge polariza- tion of the particle because this polarization is small in com- parison to the total charge eN induced by the applied voltage. The number of electrons is estimated as N�C2V /e =RdF0 / �e�, where F0=−V /L is the applied electric field. The effective field for the nanoparticle with N electrons is FN = �e �N /R2, and the partial currents IN in these conditions are given by IN = AS�eN/R 2�2exp�− FR2/�e�N� , �7� where the emitting area S, in the idealized model considered here, can be approximated by the total surface area of the nanoparticle, S=4�R2. In Fig. 2 we plot the current-voltage characteristics of the idealized emitter, calculated according to Eqs. �6� and �7�, where A is given by Eq. �2� with W =5.1 eV and �F=5.5 eV �taken for Au�, and the geometrical parameters are chosen as R=5 nm and d=0.5 �m. The char- acteristics look like staircases with flat regions �plateaus� be- tween the steps, which are visible even at room temperature. It is possible to estimate the relative heights of the steps by calculating the ratio of the currents IN and IN−1 emitted from the nanoparticle with N and N−1 electrons � exp� FR2 �e�N�N − 1� � . �8� In spite of the fact that the charged nanoparticle typically contains a large number of electrons, N 100, one can al- FIG. 1. �a� The mechanism of single-electron tunneling in the Fowler-Nordheim regime. �b� Schematic representation of the ide- alized emitter. ways find a regime when the ratio IN / IN−1 is not small in comparison to unity. This necessarily implies a weak Fowler- Nordheim tunneling, when F /F=FR2 / �e �N�1. In the calculations described above, the applicability of the Fowler-Nordheim formula requires R�W / �e �F, which is rewritten as R�e2N /W, or, according to N�RdF0 / �e�, as �e �F0�W /d, independent of the nanoparticle radius. This condition is satisfied at high enough applied voltages. If �e �F0=eV /L�W /d, the approximation of a triangular poten- tial barrier is not quite good, and one should consider the tunneling through the barrier described by the potential en- ergy W−e2N�1/R−1/r� at r�R, where r is the distance from the center of the spherical nanoparticle; the tunneling through the potential barrier of this form is described in Ref. 31. Even under the condition �e �F0�W /d, which is satisfied in the calculations shown in Fig. 2, the relative change of the current per one step, IN / IN−1−1, appears to be significant, because the exponent FR2 / �e �N�N−1� in Eq. �8� is estimated as c�W / �e �F0d� 2, where the dimensionless constant c =4/3�2me4 /�2W is noticeably larger than unity. If the current is high enough, the field emission cannot remain the bottleneck for the electron transfer from the cath- ode to the anode, and a finite resistance G−1 becomes essen- tial. The nanoparticle in these conditions is no longer in equi- librium with the cathode. This means that the distribution PN is established kinetically, and several states with different charges coexist at a fixed voltage �see the inset in Fig. 2�. As a consequence, the Coulomb blockade features are washed out. This case requires a numerical solution of the equation QN=0. The corresponding current-voltage characteristics of the idealized emitter calculated by using the RC time C /G =100 ps are also shown in Fig. 2. The degradation of the current steps appears to be stronger with increasing voltage, because the current increases and the nanoparticle-cathode link becomes more important. The shape of the steps in this case resembles the usual Coulomb staircase. III. MORE COMPLEX EXAMPLES After demonstrating the possibility of the Coulomb- blockade staircase of the field emission on a model example, it is worth to consider more complex cases. Indeed, the model example discussed above has certain disadvantages. First of all, it is hardly possible to connect a particle placed far from the cathode surface by a link �c-p in Fig. 1�b� which does not contribute to the electrostatic properties of the device. Second, the model of uniform charging is insuf- ficient: the charge polarization of the nanoscale object ap- pears to be important and should be always taken into ac- count, see below in this section. Therefore, the model shown in Fig. 1�b� is suitable only for the purposes of illustration of the basic physics described by Eqs. �3�–�6�. To have a closer approach to reality, we point out that the nanoscale objects investigated in the above-cited experiments on field emission can be roughly divided into two classes: the objects whose dimensions in all directions are comparable �nanoclusters or nanoparticles�, and the objects whose length in the direction of the applied field is much larger than their transverse size �nanowires or nanowhiskers�. The following consideration is carried out for the cases of nanoclusters and nanowires of the simplest geometries, when the electric fields FN and the ca- pacitances C and C2 can be determined consistently by solv- ing corresponding electrostatic problems. The current is cal- culated according to Eq. �6�, under the assumption that the objects are in equilibrium with the cathode. A. Field emission from nanoclusters Below we consider the field emission from a nanocluster modeled by a spherical metallic particle of radius R depos- ited on the flat cathode surface. To provide a finite capaci- tance C, one should assume a finite separation d-R between the particle and the metallic cathode plate �for instance, one can imagine that the particle resides on an oxidized surface�, see the inset to Fig. 3. Besides, this assumption provides electrical isolation of the particle from the cathode, which is FIG. 2. Current from the idealized emitter as a function of the applied field F0=−V /L for the case of small C /G �nanoparticle in thermal equilibrium with the cathode, upper curves� and for the case of C /G=100 ps �lower curves�, at the temperatures T=77 K �solid� and T=293 K �dashed�. The inset shows the distribution function PN at F0=5 10 5 V/cm for the second case. FIG. 3. Charge density per unit length in z direction for a spheri- cal metallic nanocluster placed at the distance 0.1 R from the me- tallic cathode. Here 0=F0R /2. The inset shows the geometry of the problem and the directions of the field emission �arrows�. a necessary condition for the Coulomb blockade. The elec- trostatics of the plane-sphere system is known, and the field and charge distributions in this case can be found in the form of rapidly converging infinite series arising from the poten- tials of image point charges and point dipoles.32 Such a con- sideration allows one to present the distribution of the elec- trostatic potential energy near the particle in the approximate U�r,�� � W + e��F0R�1 + �cos � − 1� − ��eN − C2V /C� r − R , �9� where r and � are the radial and azimuthal coordinates of the spherical coordinate system with the origin at the center of the particle, and �, , and � are the dimensionless constants of the order of unity, which are to be determined from nu- merical calculations. Such calculations also give us the ca- pacitances C and C2. 33 Note that if the charge quantization is neglected �so that N=C2V /e when the particle is in equilib- rium with the cathode�, � is identified with the field enhance- ment factor conventionally used in the physics of field emis- sion. The expression �9� provides an excellent description of the electrostatic potential at r−R�R /2 and at small �. It allows one to take into account deviations of the potential energy from the linear form W− �e �F�r−R� and, therefore, to find corrections to the Fowler-Nordheim tunneling exponent. Neglecting such corrections in the prefactor, we obtain the following expression for the partial currents IN = ASFN 2 exp�− F �� �e�FNR ��x� = � x2�x − 1��2 − arctan �x − 1� − x� , �10� where A is given by Eq. �2�, the dimensionless function ��x� describes the corrections to the tunneling exponent, and the effective emitting area S=2�R2�FN 2 / F�F0� �2�R2�FN / F� is reduced due to the angular dependence of the radial field described by Eq. �9�. The field FN is given by FN = �F0 + � �e�N − C̃2F0 , �11� where the quantity C̃2=C2L does not depend on the distance L between the cathode and anode. Note that, since we always assume that L is much larger than any dimension of the nanoscale object, the capacitance C2 is always proportional to 1 /L, and it is more convenient to replace C2 �V� by C̃2F0. This substitution also allows us to represent the Coulomb energy standing in Eq. �6� as �N − C̃2F0/�e� 2. �12� Further calculations are done for the separation d−R =0.1R, when C=2.16R, C̃2=1.74R 2, �=4.32, �=1.22, and =0.66. Figure 3 shows the distribution of negative charges on the surface of the spherical particle staying in equilibrium with the cathode for this case ��e �N= C̃2F0 is assumed�. The distribution of the radial field F�z� at the surface of the par- ticle is given by the same dependence, F�z� /F0= �z� / The field-emission current from the nanocluster described above has been calculated according to Eqs. �6� and �10�– �12� at R=5 nm. The results of the calculations shown in Fig. 4 demonstrate the staircaselike behavior caused by the Cou- lomb blockade. However, in contrast to the staircases shown in Fig. 2, the current continues to increase between the steps. This occurs because of electrostatic polarization of the nano- particle. According to Eq. �11�, when the particle charge is constant, the increase in the applied field F0 leads to an in- crease in the effective field FN because the factor � −�C̃2 /CR is positive. For the chosen particle radius, the steps of the current are clearly visible at liquid nitrogen tem- perature but poorly visible at room temperature. Neverthe- less, the Coulomb blockade features at room temperature be- come quite distinct in the plots of the derivative of the current, as shown in the inset to Fig. 4. B. Field emission from nanowires Let us consider the field emission from a small semicon- ductor wire modelled by a cylinder of radius R and length d, which ends with a hemispherical tip of the same radius, see the inset to Fig. 5. The cathode substrate upon which the wire is grown is assumed to be a metal �or a heavily doped semiconductor� so that one can use the method of image charges instead of solving the electrostatic problem in the whole space. The electric isolation of the wire from the cath- ode in this case takes place in a natural way, because a Schottky barrier is formed between the wire and the metallic cathode �in the case of semiconducting cathode there can be a heterobarrier or an interband p-n barrier�. In other words, FIG. 4. Current from the spherical nanocluster of radius R =5 nm as a function of the applied field F0=−V /L at T=4.2 K �solid� and 77 K �dashed�. The inset shows the derivative of the current at T=293 K. the wire region adjacent to the cathode becomes depleted of electrons and positively charged because of the presence of donors �we assume that the wire is uniformly doped with bulk donor density nD�. When a bias eV is applied between the cathode and anode, the wire acquires a con- siderable negative charge because of tunneling or thermi- onic emission of electrons from the cathode through the barrier. When the field emission from the wire of nan- oscale radius becomes essential, the density of induced negative charges per unit length of the wire appears to be much larger than the equilibrium charge density =�R2 �e �nD even if nD is of the order of 10 18 cm−3. For this reason, one can use the “metallic” approximation assuming that the charges in the wire are placed mostly on its surface. This means that the electron density distribution n�� ,z�, which depends on the radial coordinate � of the cy- lindrical coordinate system connected with the wire, is given by n�� ,z�= �2�� �e � �−1���−R� �z�+nD for z�d and n�� ,z� = �2�� �e � �−1���−�R2− �z−d�2 �z�+nD for d�z�d+R, where �z� is the density of negative charges on the surface per unit length. Since this approximation is based on the assumption that the screening length is small in comparison with the wire radius, it works better for wider wires. For silicon wires, whose field-emission properties are currently the subject of investigations,11–15 the metallic approximation remains suitable even for the radius of several nanometers, because, owing to the large effective masses and six-valley degeneracy, the density of electron states in n-Si appears to be high enough to provide the Thomas-Fermi screening length less than one nanometer for Fermi energies �F �0.01 eV. The metallic approximation, of course, fails to describe the region of the wire in the close vicinity of the cathode, where the depletion occurs. Nevertheless, since this region is a small part of the whole wire, see the charge dis- tribution in Fig. 5, its presence cannot considerably modify the parameters calculated as described below. According to the discussion given here, we search for the charge distribution �z� satisfying the integral equation U�z� = U0 − �e�F0z + � dz�K�z,z�� �z�� , �13� where U�z� is the potential energy counted from the Fermi level in the cathode material, U0 is the barrier height, and K�z ,z�� is the potential of interaction between the electrons in the points z and z� of the wire surface in the presence of the cathode plate, see the Appendix. Equation �13� is accom- panied with additional requirements: U�z�=0 at z�z0 and �z�=− D at z�z0, where z0 is the depletion edge coordi- nate, which is to be found self-consistently. The first of these requirements corresponds to a full screening of the bare po- tential U0− �e �F0z by the induced charges of the wire, while the second one models the presence of the positive charges in the depletion region z�z0. Once the distribution �z� is found, the total charge of the wire, −�0 d+Rdz �z�, as well as the distribution of electric field around the wire, can be cal- culated. To find the capacitance C and describe modification of the effective field under single-electron charging, one may calculate the variation of the total charge and the field at the tip �at z=d+R� with respect to a small variation of U0. Equa- tion �13� is solved numerically by using the method of itera- tions. The dependence of the effective field FN on F0 and N can be represented in the form similar to Eq. �11� FN = ��F0�F0 + � �e��N + B� − C̃2F0 C�F0�R , �14� while the Coulomb energy is written as 2C�F0� �N + B − C̃2F0/�e� 2. �15� These equations take into account a finite �though weak� dependence of the capacitance C and field enhancement fac- tor � on the applied field F0. The dependence of the param- eters C̃2 and � on F0 appears to be much weaker and can be neglected. The positive dimensionless constant B reflects the fact that the average number of induced charges is smaller than C̃2F0 / �e�. These features appear because the system un- der consideration is not entirely metallic and contains a depletion region whose length changes with F0. The numerical calculations leading to the results pre- sented below are done for U0=0.7 eV, which approximately corresponds to the Schottky barrier height for n-Si in contact with Al.34 The chosen donor density is nD=2 10 18 cm−3. The parameters standing in Eqs. �14� and �15�, however, are not sensitive to nD, except for the capacitance C, which changes within 10% when nD varies from 10 18 cm−3 to 2 1018 cm−3. Figure 5 shows the charge density distribu- tion for the wire of radius R=5 nm and length d=0.1 �m at F0=10 6 and 2 106 V/cm. The charge density shows a nearly linear growth through the main part of the wire and a sharp enhancement at the hemispherical tip from which the field emission occurs. The dependence of the field enhancement factor and capacitance on the applied electric field is shown in Fig. 6, and the other calcu- lated parameters are C̃2=2.44 dR, �=0.414, and B=12.14. FIG. 5. Charge density per unit length for the cylindrical wire whose geometry is shown in the inset �see parameters in the text�. The plots of the field-emission current calculated with the use of the parameters listed here are given in Fig. 7. The calculations are done according to Eqs. �6�, �14�, and �15�, and the Fowler-Nordheim formula for the partial current, IN=ASFN 2 exp�−F /FN�. Since the calculated radial electric field in the region of the tip weakly depends on z �in contrast to the case of the nanocluster studied above� and sharply decreases in the region of transition to the cylindrical part of the wire, the effective emitting area S is estimated by the total area of the hemispherical tip, S=2�R2. The work func- tion is taken for silicon, W=4.2 eV. Next, the Fermi energy standing in the expression for A, see Eq. �2�, is estimated from the equation �F��e �FinrTF, where Fin��F0 /� is the field inside the semiconductor near the end of the tip, rTF is the Thomas-Fermi screening length, and � is the dielectric constant of the semiconductor. Such an estimate, carried out for n-Si, leads to �F�0.1 eV at F0�2 10 6 V/cm. The pic- ture of Coulomb staircase shown in Fig. 7 is basically the same as that in Fig. 4. Again, the increase of the current with the applied field is determined by the increase of the effec- tive field �14� due to both single-electron charging �steps� and charge polarization under a constant charge �regions be- tween the steps�. The main difference is that the interval of the applied field needed for addition of one electron to the wire is considerably reduced, owing to the larger capacitance C2, and appears to be of about 1.2 V/�m �further reduction of this interval takes place with the increase of the wire length, see below�. Next, since the capacitance C increases considerably in comparison to the case of nanocluster of the same radius, the Coulomb blockade features at room tem- perature are poorly visible even in the derivative plot, see the inset. Nevertheless, these features remain pronounced at T =77 K. With the increase of the wire length d, the parameters entering Eqs. �14� and �15� are modified as shown in Fig. 8. The field enhancement factor and the capacitances increase nearly in a linear way, while the parameter �, which charac- terizes relative contribution of charging to the effective field, slightly decreases �for comparison, the idealized emitter con- sidered in the previous section is described by the parameters �=d /R, �=1, C=R, and C̃2=dR, where d is the distance from the cathode to the emitting sphere�. The increase of the total capacitance C makes it difficult to observe the Coulomb staircase in long wires. For example, at d=1 �m one should have temperatures considerably lower than 77 K. The inter- val of the applied field corresponding to the addition of one electron is inversely proportional to C̃2. This interval de- creases very fast with the increase of d and becomes equal to 2.5 102 V/cm at d=1 �m. FIG. 6. Field dependence of the enhancement factor and capaci- tance for the cylindrical wire with R=5 nm and d=0.1 �m. FIG. 7. Current from the cylindrical wire of radius R=5 nm and length 0.1 �m as a function of the applied field F0=−V /L at T =4.2 K �solid� and 77 K �dashed�. The inset shows the derivative of the current at T=293 K. FIG. 8. Dependence of the parameters �, �, C, and C̃2 on the length of the wire for R=5 nm and F0=5 10 5 V/cm. IV. CONCLUSIONS The key point of the presented theoretical study is the possibility of noticeable modification of the effective electric field causing the field emission from a nanoscale conductor by addition of just one electron to this conductor. Formally, this modification is described by introducing the effective field FN, which determines the partial current IN, and by evaluating the dependence of this field on the bias applied between the cathode and anode, see Eqs. �11� and �14�. As a result of this effect, the current-voltage characteristics of the field emission show steps in the Coulomb blockade regime. In other words, the steplike current-voltage characteristics related to single-electron charging �Coulomb staircases� may exist even under the conditions of field-emission experi- ments, when the applied bias is orders of magnitude larger than the charging energy. The steps on the current-voltage characteristics can be visible at 77 K in the case of field emission from nanoclusters and nanowires of 10 nm diam- eter and submicron length. In the regions between the steps, where the total charge of the nanoscale object is constant, the current increases with the increase of the applied bias owing to charge polarization. The staircases described in this work are similar to the usual Coulomb staircases obtained in the transport through small metallic islands28–30 or quantum dots �see Ref. 35 for review� with strong asymmetry in the barriers. In both cases, each step of the current is associated with addition of an electron to the nanoscale object, and the applied source-drain voltage drops mostly across the low-transparent barrier �the barrier between the object and the drain�. Therefore, the pe- riodicity of the steps in both cases is determined by the object-drain capacitance C2. However, the steps in the sec- ond case are formed due to shifts of effective �N-dependent� electrochemical potential of the object with respect to elec- trochemical potentials of the source and drain. For this rea- son, the usual Coulomb staircase shows well-defined steps when C2 is greater than the object-source capacitance C1, while in the opposite situation, C1�C2, the steps are sup- pressed and the current-voltage characteristic approaches to a linear dependence.29,30 In contrast, in the case described in this work the steps are formed due to changes in the prob- ability of Fowler-Nordheim tunneling from the object to the drain �anode�. That is why the steps are clearly visible under the condition C1�C2, which is imposed by the field- emission layout considered in this paper. To summarize, the sensitivity of the field emission to the number of electrons in the nanoscale object makes it possible to obtain the Coulomb staircases under the conditions when such staircases cannot be observed in the transport through small metallic islands or quantum dots. The quantitative consideration has been applied here to some simple models of the nanoscale objects, whose electro- static properties necessary for description of the field en- hancement and charging have been determined consistently. Consequently, the number of geometrical parameters charac- terizing the objects has been minimized. For example, the nanowire has been characterized only by its length d and radius R. In reality, the geometrical structure of objects is more complicated. For example, their tips may contain sharp regions which provide a more efficient field emission. In fact, high field-emission currents from nanoscale objects are typically observed at the applied fields of the order of 105 V/cm, which requires the field enhancement factors much larger than those calculated in this paper. On the other hand, the presence of sharp tips cannot strongly modify the capacitances of the objects. The general picture of the single- electron tunneling under the field-emission regime also re- mains valid. For possible application to experiments, the field enhancement due to charging can be described by equa- tions of the kind of Eqs. �11� and �14�, where � and � should be considered as parameters to be determined experimen- tally. At the present time, there is no experimental evidence of the Coulomb staircase phenomenon under the field emission. Though the current-voltage characteristics sometimes show steplike features, see, for example, Ref. 11, these features are not regular and, most probably, should be attributed to insta- bilities of the emission process and burning out of the emit- ting material. There are numerous reasons which make ob- servation of the phenomena considered in this paper difficult. First of all, in most cases the nanoscale objects on the cath- ode surface form dense arrays. This means that the field emission takes place from a macroscopic number of objects which are electrostatically coupled. The charging and field- emission properties appear to be considerably different36 from those of individual objects. The Coulomb blockade phenomena in this case should be dramatically suppressed by the size dispersion of the objects and by the effects of mutual screening. Investigation of field emission from individual ob- jects is possible in the cases of metallic nanoclusters8–10 and carbon nanotubes.26 However, there exists the problem of electric isolation of these objects from the cathode, which is one of the necessary conditions for Coulomb blockade. No special attempts to achieve such an isolation in the field- emission experiments have been undertaken so far, except for the nanomechanical system investigated in Ref. 7, where the electron emission from an isolated Au island to a submicron-sized electrode has been observed. Most of the experiments on field emission are carried out at room tem- perature, though existing experimental techniques also allow measurements at liquid nitrogen temperature. This means that the Coulomb blockade phenomena can be observed only for small-sized objects whose capacitances are low enough �see the results of Sec. III�. Besides, the interval of the ap- plied field corresponding to addition of one electron strongly decreases in the case of emission from long nanowires, which requires high resolution with respect to field. In sum- mary, a search for the Coulomb blockade features in the field-emission current would require a special planning of experiment. The author hopes that the presented theoretical study will stimulate experimental investigations in this direc- tion. ACKNOWLEDGMENTS The author is grateful to A. I. Klimovskaya for stimulat- ing discussions. APPENDIX: KERNEL OF EQUATION (13) If z�d and z��d, K�z ,z��=K0�z ,z��−K0�−z ,z��, where K0�z,z�� = � ��z − z��2 + 2R2�1 − cos �� . �A1� If z�d and z��d, K�z ,z��=K0�z ,z��−K0�−z ,z��, where K0�z,z�� = � ��d − z�2 + 2R�d − z�cos �� + 2R2�1 − sin �� cos �� . �A2� If z�d and z��d, K�z ,z��=K0�z ,z��−K0�z ,−z��, where K0�z,z�� = � ��d − z��2 + 2R�d − z��cos � + 2R2�1 − sin � cos �� . �A3� Finally, if z�d and z��d, K�z,z�� = � � �e��2R2�1 − cos � cos �� − sin � sin �� cos ���� �4d2 + 4dR�cos � + cos ��� + 2R2�1 + cos � cos �� − sin � sin �� cos ��� . �A4� In Eqs. �A2�–�A4�, cos �= �z−d� /R and cos ��= �z�−d� /R, so � and �� are the azimuthal angles. The integrals are taken over the polar angle �. The function K�z ,z�� is also representable in the form of full elliptic integrals. *Electronic address: [email protected] 1 Single Charge Tunneling, edited by H. Grabert and M. H. De- voret, NATO ASI Series B 294 �Plenum Press, New York, 1992�. 2 I. O. Kulik and R. I. Shekhter, Zh. Eksp. Teor. Fiz. 68, 623 �1975� �Sov. Phys. JETP 41, 308 �1975� . 3 D. V. Averin and K. K. Likharev, J. Low Temp. Phys. 62, 345 �1986�; and in Mesoscopic Phenomena in Solids, edited by B. L. Altshuler, P. A. Lee, and R. A. Webb �Elsevier, Amsterdam, 1991�. 4 C. W. J. Beenakker, Phys. Rev. B 44, 1646 �1991�. 5 L. Y. Gorelik, A. Isacsson, M. V. Voinova, B. Kasemo, R. I. Shekhter, and M. Jonson, Phys. Rev. Lett. 80, 4526 �1998�. 6 A. Erbe, C. Weiss, W. Zwerger, and R. H. Blick, Phys. Rev. Lett. 87, 096106 �2001�. 7 D. V. Scheible, C. Weiss, J. P. Kotthaus, and R. H. Blick, Phys. Rev. Lett. 93, 186801 �2004�. 8 M. E. Lin, R. P. Andres, and R. Reifenberger, Phys. Rev. Lett. 67, 477 �1991�. 9 M. E. Lin, R. Reifenberger, and R. P. Andres, Phys. Rev. B 46, 15490 �1992�. 10 M. E. Lin, R. Reifenberger, A. Ramachandra, and R. P. Andres, Phys. Rev. B 46, 15498 �1992�. 11 C. S. Chang, S. Chattopadhyay, L. C. Chen, K. H. Chen, C. W. Chen, Y. F. Chen, R. Collazo, and Z. Sitar, Phys. Rev. B 68, 125322 �2003�. 12 S. Johnson, A. Markwitz, M. Rudolphi, H. Baumann, S. P. Oei, K. B. K. Teo, and W. I. Milne, Appl. Phys. Lett. 85, 3277 �2004�. 13 N. N. Kulkarni, J. Bae, C.-K. Shih, S. K. Stanley, S. S. Coffee, and J. G. Ekerdt, Appl. Phys. Lett. 87, 213115 �2005�. 14 J. C. She, K. Zhao, S. Z. Deng, J. Chen, and N. S. Xu, Appl. Phys. Lett. 87, 052105 �2005�. 15 J. C. She, S. Z. Deng, N. S. Xu, R. H. Yao, and J. Chen, Appl. Phys. Lett. 88, 013112 �2006�. 16 Q. Wang, J. J. Li, Y. J. Ma, Z. L. Wang, P. Xu, C. Y. Shi, B. G. Quan, S. L. Yue, and C. Z. Gu, Nanotechnology 16, 2919 �2005�. 17 Y. L. Chueh, L. J. Chou, S. L. Cheng, J. H. He, W. W. Wu, and L. J. Chen, Appl. Phys. Lett. 86, 133112 �2005�. 18 A. G. Rinzler, J. H. Hafner, P. Nikolaev, L. Lou, S. G. Kim, D. Tomanek, P. Nordlander, D. T. Colbert, and R. E. Smalley, Sci- ence 269, 1550 �1995�. 19 P. G. Collins and A. Zettl, Appl. Phys. Lett. 69, 1969 �1996�. 20 Q. H. Wang, A. A. Setlur, J. M. Lauerhaas, J. Y. Dai, E. W. Seelig, and R. P. H. Chang, Appl. Phys. Lett. 72, 2912 �1998�. 21 S. Fan, M. G. Chapline, N. R. Franklin, T. W. Tombler, A. M. Cassell, and H. Dai, Science 283, 512 �1999�. 22 R. H. Baughman, A. A. Zakhidov, and W. A. de Heer, Science 297, 787 �2002�. 23 S. H. Jo, Y. Tu, Z. P. Huang, D. L. Carnahan, J. Y. Huang, D. Z. Wang, and Z. F. Ren, Appl. Phys. Lett. 84, 413 �2004�. 24 M. Mauger, V. T. Binh, A. Levesque, and D. Guillot, Appl. Phys. Lett. 85, 305 �2004�. 25 N. de Jonge, M. Allioux, M. Doytcheva, M. Kaiser, K. B. K. Teo, R. G. Lacerda, and W. I. Milne, Appl. Phys. Lett. 85, 1607 �2004�. 26 Z. Xu, X. D. Bai, E. G. Wang, and Z. L. Wang, Appl. Phys. Lett. 87, 163106 �2005�. 27 R. H. Fowler and L. W. Nordheim, Proc. R. Soc. London, Ser. A 119, 173 �1928�. 28 J. B. Barner and S. T. Ruggiero, Phys. Rev. Lett. 59, 807 �1987�. 29 K. Mullen, E. Ben-Jacob, R. C. Jaklevic, and Z. Schuss, Phys. Rev. B 37, 98 �1988�. 30 R. Wilkins, E. Ben-Jacob, and R. C. Jaklevic, Phys. Rev. Lett. 63, 801 �1989�. 31 L. D. Landau and E. M. Lifshitz, Quantum Mechanics �Perga- mon, Oxford, 1977�. 32 W. R. Smythe, Static and Dynamic Electricity �McGraw-Hill, New York, 1968�. 33 If d=R, the parameters are obtained analytically: �=7��3� /2 �4.21, =93��5� /56��3�−3/4, �=�2 /8, and C2= �� 2 /6�R2 /L, but C is equal to infinity �C diverges in a logarithmic way as the separation d-R goes to zero�. 34 Metal-Semiconductor Interfaces, edited by A. Hiraki �IOS Press, Amsterdam, 1995�. 35 L. P. Kouwenhoven, C. M. Marcus, P. L. McEuen, S. Tarucha, R. M. Westervelt, and N. S. Wingreen, Electron Transport in Quan- tum Dots, in Proceedings of the Advanced Study Institute on Mesoscopic Electron Transport, edited by L. L. Sohn, L. P. Kou- wenhoven, and G. Schön �Kluwer, 1997�. 36 T. A. Sedrakyan, E. G. Mishchenko, and M. E. Raikh, cond-mat/ 0504042 �unpublished�.

0704.0416

Origamis with non congruence Veech groups

Origamis with non congruence Veech groups Gabriela Schmithüsen In this article we give an introduction to origamis (often also called square-tiled surfaces) and their Veech groups. As main theorem we prove that in each genus there exist origamis, whose Veech groups are non congruence subgroups of SL2(Z). The basic idea of an origami is to obtain a topological surface from a few combina- torial data by gluing finitely many Euclidean unit squares according to specified rules. These surfaces come with a natural translation structure. One assigns in general to a translation surface a subgroup of GL2(R) called the Veech group. In the case of surfaces defined by origamis, the Veech groups are finite index sub- groups of SL2(Z). These groups are the objects we study in this article. One motivation to be interested in Veech groups is their relation to Teichmüller disks and Teichmüller curves, see e.g. the article [H 06] of F. Herrlich in the same volume: A translation surface of genus g defines in a geometric way an embedding of the upper half plane into the Teichmüller space Tg of closed Rie- mann surfaces of genus g. The image is called Teichmüller disk. Its projection to the moduli space Mg is sometimes a complex algebraic curve, called Teichmüller curve. More precisely this happens, if and only if the Veech group is a lattice in SL2(R). In this case the algebraic curve can be determined from the Veech group up to birationality. It is hard to determine the Veech group for a general translation surface. How- ever, if the translation surface comes from an origami there is a special approach to this problem. It is based on the idea of describing origamis by finite index sub- groups of F2, the free group in two generators. This leads to a characterization of origami Veech groups as the images in SL2(Z) of certain subgroups of Aut(F2), the automorphism group of F2. Using this approach we will calculate Veech groups of two origamis explicitly. They turn out to be non congruence groups. Starting from these examples we obtain infinite sequences of origamis all of whose Veech groups are non congruence groups. This leads to the following theorem. Theorem 1. Each moduli space Mg (g ≥ 2) contains an origami curve whose Veech group is a non congruence group. In Section 1 we introduce origamis and present different equivalent ways to de- scribe them. In Section 2 we give a glance on the mathematical context. We describe, how an origami defines a family of translation surfaces and explain roughly , how one obtains a Teichmüller curve in moduli space starting from http://arxiv.org/abs/0704.0416v1 an origami. We introduce Veech groups and shortly point out their relation to Teichmüller curves. In Section 3 we turn to Veech groups of origamis and present a characterization of them in terms of automorphisms of the free group F2 in two generators. We use this characterization to calculate two examples explicitly. Finally, in Section 4 we show that these two examples produce Veech groups that are non congruence groups and give a method to construct out of them infinite sequences of Veech groups that are again non congruence groups. The first part (Section 1 - Section 3) of this article is meant to give a handy in- troduction to origamis and an overview on some of our results about their Veech groups. In the second part we state and prove Theorem 1 based on the results in the PhD thesis [S 05] of the author. For a broader introduction and overview on origamis and Teichmüller curves as well as for references to the larger context, we refer the the reader e.g. to [HeSc 06], [S 04] and [S 05]. Acknowledgments: I would like to thank Frank Herrlich for his support in respect of the content and for his proof reading, Stefan Kühnlein for helpful dis- cussions and suggestions especially on non congruence groups and the organizers of the conference for giving me the opportunity to contribute to these proceedings. This work was partially supported by a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD). 1 ORIGAMIS 3 1 Origamis There are several ways to define origamis. We start with the somehow playful description that we have learned from [Lo 05], where also the name origami was introduced: An origami is obtained by gluing the edges of finitely many copies Q1, . . . , Qd of the Euclidean square Q via translations according to the following rules: • Each left edge shall be identified to a right edge and vice versa. • Similarly, each upper edge shall be identified to a lower one. • The arising closed surface X shall be connected. We only study what is called oriented origamis in [Lo 05] and call them just origamis. Example 1.1. a) The simplest example is the origami that is made from only one square. There is precisely one possibility to glue its edges according to the rules. One obtains a torus E. We call this origami the trivial origami O0. Figure 1: The trivial origami. Opposite edges are glued. Observe that the four vertices of the square are all identified and become one point on the closed surface E. We call this point ∞. b) We now consider an origami made from four squares, see Figure 2. Some identifications of the edges are already done in the picture. For all other edges those having same labels are glued. The origami is called L(2, 3) for obvious reasons. 2 3 4 a b c Figure 2: The origami L(2, 3). Opposite edges are glued. 1 ORIGAMIS 4 Observe that in this case the vertices labeled with • and the vertices labeled with ◦ are respectively identified and become two points on the closed surface X. By calculating the Euler characteristic one obtains, that the genus of the surface X is 2. c) Finally, we consider an example with five squares, see Figure 3. Here, edges with same labels are identified. For the unlabeled edges, those which are opposite to each other are glued. We call the origami D. 1 2 3 • • • Figure 3: The origami D. Edges with the same label and unlabeled edges that are opposite are glued. In this case, we obtain the three identification classes ◦, ⋆ and • for the vertices. The genus of the closed surface X is again 2. Origamis as coverings of a torus Observe, that the trivial origami O0 from Example 1.1 a) is universal in the following sense: If X is the closed surface that arises from an arbitrary origami O and E the torus that arises from O0, then we have a natural map X → E by mapping each of the unit squares of the origami O that form the surface X to the one unit square of O0 that forms the torus E. This map is a covering that is unramified except over the one point ∞ ∈ E. Conversely, given a closed surface X together with such a covering p : X → E, we obtain a decomposition of X into squares by cutting X along the preimages of the edges of the one square of O0 that forms E. This motivates the following definition of origamis. Definition 1.2. An origami O of genus g and degree d is a covering p : X → E of degree d from a closed, oriented (topological) surface X of genus g to the torus E that is ramified over at most one marked point ∞ ∈ E. Remark that we have fixed here one torus E and one point ∞ ∈ E. In particular we may furthermore fix a point M 6= ∞ on E and a set of standard generators of the fundamental group π1(E,M) that do not pass through ∞. That way we obtain a fixed isomorphism ∗) ∼= F2, (1) 1 ORIGAMIS 5 where E∗ = E−∞ and F2 = F2(x, y) is the free group in two generators x and y. Describing E by gluing the edges of the unit square via translations, we choose M to be the midpoint of the unit square and the standard generators to be the horizontal and the vertical simply closed curve through M , see Figure 4. Figure 4: Generators of π1(E Example 1.3. In Example 1.1, in a) the covering is the identity id : E → E. In b) we have a covering p : X → E of degree 4 that is ramified in the two points labeled by • and ◦. Recall that the genus of X is 2. In c) we have a covering p : X → E of degree 5 ramified in the two points labeled by • and ⋆. Observe that though the point on X labeled by ◦ is a preimage of ∞, the covering is not ramified in this point. The genus of X is again 2. Definition 1.4. We say that two origamis O1 = (p1 : X1 → E) and O2 = (p2 : X2 → E) are equivalent, if there is a homeomorphism ϕ : X1 → X2 with p1 = p2 ◦ ϕ. Description by a pair of permutations An origami O = p : (X → E) of degree d defines (up to conjugation in Sd) • a homomorphism m : F2 = F2(x, y) → Sd or equivalently • a pair of permutations (σa, σb) in Sd as follows: Let M1, . . . , Md be the preimages of the point M (defined as above) under p. Furthermore, let m : π1(E ∗,M) → Sym(M1, . . . ,Md) be the monodromy map defined by p, i.e. for the closed path c ∈ π1(E ∗,M) the point Mi is mapped to Mj by m(c) if and only if the lift of the curve c to X via p, that starts in Mi, ends in Mj. Choosing an isomorphism Sym(M1, . . . ,Md) ∼= Sd and using the isomorphism ∗) ∼= F2 fixed in (1) makes m into a homomorphism from F2 to Sd. We set σa = m(x) and σb = m(y). Observe that this homomorphism depends on the chosen isomorphism to Sd and on the choice of the origami in its equivalence class only up to conjugation in Sd. Therefore we consider two homomorphisms m1 : F2 → Sd and m2 : F2 → Sd to be equivalent, if they are conjugated by an element in Sd. Similarly we call two pairs (σa, σb) and (σ b) in Sd equivalent, if they are simultaneously conjugated, i.e. there is some s ∈ Sd such that σa = sσ −1 and σb = sσ 1 ORIGAMIS 6 Example 1.5. In Example 1.1 we obtain for the origami L(2, 3) in b) the mon- odromy homomorphism m : F2 → S4, x 7→ (2 3 4) and y 7→ (2 1), and thus σa = (2 3 4) and σb = (2 1). For the origami D in c) we similarly obtain the permutations σa = (1 2 3) and σb = (1 4 5)(2 3). Description as finite index subgroups of F2 Origamis can be equivalently described as finite index subgroups of F2, the free group in two generators, as stated in the following remark. The characterization of the Veech groups of origamis is mainly based on this observation. Remark 1.6. We have a one-to-one correspondence: origamis up to equivalence ↔ finite index subgroups of F2 up to conjugacy. More precisely, this correspondence is given as follows: Let O = (p : X → E) be an origami. Define E∗ = E − {∞} and X∗ = X − p−1(∞). Thus we may restrict p to the unramified covering p : X∗ → E∗. This defines an embedding of the corresponding fundamental groups: U = π1(X ∗) ⊆ π1(E ∗) ∼= F2 Again we use the fixed isomorphism in (1), see also Figure 4. Changing the origami in its equivalence class leads to a conjugation of U with an element in F2. The index of the subgroup of F2 is the degree d of the covering p. Conversely, given a finite index subgroup U of F2 we retrieve the origami in the following way: Let v : Ẽ∗ → E∗ be a universal covering of E∗. By the theorem of the universal covering, π1(E ∗) is isomorphic to Deck(Ẽ∗/E∗), the group of deck transformations of Ẽ∗/E∗. Furthermore, the finite index subgroup U of Deck(Ẽ∗/E∗) corresponds to an unramified covering p : X∗ → E∗ of finite degree. This can be extended to a covering X → E, where X is a closed surface. Example 1.7. In Example 1.1, we obtain the following subgroups of F2: In a), X∗ is the once punctured torus itself and U = F2. In b), X∗ is a genus 2 surface with 2 punctures. Thus U = π1(X ∗) is a free group of rank 5. Keeping in mind that we use the identification π1(E ∗) ∼= F2 = F2(x, y) described in Figure 4, one can read off from the picture in Figure 2 that U = < x3, xyx−1, x2yx−2, yxy−1, y2 > In c), X∗ is a genus 2 surface with three punctures. Thus U is a free group of rank 6. More precisely, we read off the picture in Figure 3, that U = < x3, xyx−2, x2yx−1, yxy−1, y2xy−2, y3 > 2 TRANSLATION STRUCTURES AND VEECH GROUPS 7 Description as a finite graph Finitely, sometimes it is convenient to describe an origami O = (p : X → E) as a finite, oriented labeled graph: Namely, let U be the finite index subgroup of F2 (unique up to conjugation) that corresponds to O as described in the last paragraph. Then we represent the origami by the Cayley-Graph of U ⊆ F2: The vertices of the graph are the coset representatives. They are labeled with a representative of the coset. The edges are labeled with x and y. For each vertex (with label w ∈ F2) there is an x-edge from it to the vertex that belongs to the coset of wx. And similarly there is a y-edge to the vertex that belongs to the coset wy. Example 1.8. The following figure shows the Cayley-graph for the origami L(2, 3) from Example 1.1: ?>=<89:;ȳ GFED@ABCīd x // ?>=<89:;x̄ x // �� GFED@ABC Figure 5: Graph for O = L(2, 3). 2 Translation structures and Veech groups Translation structures Recall that an atlas on a surface is called translation atlas, if all transition maps are translations. An origamiO = (p : X → E) naturally defines an SL2(R)-family of translation structures µA (A ∈ SL2(R)) on X ∗ = X − p−1(∞) as follows: • As first step, observe that each A ∈ SL2(R) naturally defines a translation structure ηA on the torus E itself by identifying it with C/ΛA, where and ΛA is the lattice < > in C (2) • Then define the translation structure µA on X ∗ by lifting ηA via p, i.e. µA = p Using the first description of an origami that we gave by gluing squares, we obtain the translation structure µI (where I is the identity matrix), if we identify the squares with the Euclidean unit square in C. We obtain µA for a general matrix 2 TRANSLATION STRUCTURES AND VEECH GROUPS 8 A ∈ SL2(R) from this by identifying the squares with the parallelogram spanned by the two vectors Thus the SL2(R)-variations of the translation structure µI can be thought of as affine shearing of the unit squares, see Figure 6. Figure 6: Sheared translation structure for the origami L(2, 3). From an origami to a Teichmüller curve in the moduli space By the SL2(R)-family of translation structures, the origami O = (p : X → E) defines a specific complex algebraic curve called Teichmüller curve in the moduli space Mg of closed Riemann surfaces of genus g. We state this construction here only briefly as motivation and refer e.g. to the overview article [HeSc 06] for a detailed description and links to references. A particular nice configuration of such Teichmüller curves is described in [H 06] in this volume. The Teichmüller curve in Mg is obtained from the origami in the following way: • The translation structure µA described in the previous paragraph is in par- ticular a complex structure on the surface X∗ which can be extended to the closed surface X . The Riemann surface (X, µA) together with the identity map id : X → X as marking then defines a point in the Teichmüller space Tg. Thus we obtain the map: ι̂ : SL2(R) → Tg, A 7→ [(X, µA), id]. • If A ∈ SO2(R), then the affine map z 7→ A ·z is holomorphic. Thus the map ι̂ factors through SO2(R). Furthermore using that SL2(R) modulo SO2(R) is isomorphic to the upper half plane H, one obtains a map ι : H ∼= SO2(R)\SL2(R) → Tg In fact, this map is an embedding that is in the same time holomorphic and isometric. A map with this property is called Teichmüller embedding and its image ∆ in Teichmüller space is called a Teichmüller disk or a geodesic disk. 2 TRANSLATION STRUCTURES AND VEECH GROUPS 9 • Finally, one may compose the map ι with the projection to the moduli space Mg. The image of ∆ in Mg is a complex algebraic curve. A curve in Mg that arises like this as the image of a Teichmüller disk is called Teichmüller curve. Note: More generally, one obtains a Teichmüller disk ∆ in a similar way starting from an arbitrary translation surface (or a bit more general: from a flat surface). However, the image of such a disk ∆ in moduli space is not always a complex algebraic curve; in fact its Zariski closure tends to be of higher dimension. It is an interesting question how to decide whether a translation surface leads to a Teichmüller curve. One possible answer to this is given by the Veech group which we introduce in the following paragraph. Veech groups Let X∗ be a connected surface and µ a translation structure on it. One assigns to it a subgroup of GL2(R) called Veech group as described in the following: We con- sider the group Aff+(X∗, µ) of all orientation preserving affine diffeomorphisms, i.e. orientation preserving diffeomorphisms that are locally affine maps of the plane C, see Figure 7. Here – and throughout the whole article – we identify C with R2 by the map z 7→ (Re(z) , Im(z))t. Thus an affine diffeomorphism f can be written in terms of local charts as f : z = (Re(z), Im(z))t 7→ A · (Re(z), Im(z))t+ z0 with A ∈ GL2(R) and z0 ∈ C. Observe that A does not depend on the chart, since µ is a translation structure. Thus one obtains a well defined map D : Aff+(X∗, µ) → GL2(R), f 7→ A called Derivative map. Definition 2.1. The Veech group Γ(X∗, µ) of the translation surface (X∗, µ) is the image of the derivative map D: Γ(X∗, µ) = D(Aff+(X∗, µ)) z 7→ Az + z0 Figure 7: An affine diffeomorphism of a translation surface 3 VEECH GROUPS OF ORIGAMIS 10 Example 2.2. Let (X∗, µ) be C/ΛI with the natural translation structure. Here I is the identity matrix and ΛI is the corresponding lattice as defined in (2). An affine diffeomorphisms of C/ΛI lifts to an affine diffeomorphism of C respect- ing the lattice. Conversely, each such diffeomorphism descends to C/ΛI. Thus, we have in this case Γ(X∗, µ) = SL2(Z). Veech groups and Teichmüller curves As indicated in the paragraph about Teichmüller curves, the Veech group “knows” whether a translation surface defines a Teichmüller curve in moduli space or not. More precisely, one has the following statement: Fact: Let X be a surface of genus g and X∗ = X−{P1, . . . , Pn} for finitely many points P1, . . . , Pn on X . Furthermore let µ be a translation structure on X Then (X∗, µ) defines a Teichmüller curve C if and only if the Veech group Γ(X∗, µ) is a lattice in SL2(R). In this case, the curve C is (antiholomorphic) birational to H/Γ(X∗, µ). We describe the relation to Teichmüller curves here just as motivation and in order to give a glance at the general frame. We have therefore resumed theorems contributed by several authors condensed in what is here called “fact”. A good access to it can be found e.g. in [EG 97] or [Z 06]. A broader overview on Veech groups of translation surfaces is given e.g. in [HuSc 01] and in [Le 02]. Teichmüller disks, Teichmüller curves and Veech groups have intensively been studied by numerous authors, starting from Thurston [T 88] and Veech himself [V 89]. We refer to [S 04] and [HeSc 06] for more comprehensive overviews on references. 3 Veech groups of origamis Let O = p : (X → E) be an origami. We have seen in Section 2 that O defines an SL2(R)-family of translation structures µA (A ∈ SL2(R)) on X ∗ = X − p−1(∞). The corresponding Veech groups are not very different. In fact, they are all conjugated to each other. More precisely, we have: Γ(X∗, µA) = AΓ(X ∗, µI)A Thus, we may restrict to the case where A = I which justifies the following definition. Definition 3.1. The Veech group Γ(O) of the origamiO is defined to be Γ(X∗, µI). From Example 2.2 it follows that the Veech group of the trivial origami O0 (de- fined in Example 1.1) is SL2(Z). For a general origami one can show that Γ(O) 3 VEECH GROUPS OF ORIGAMIS 11 is a finite index subgroup of SL2(Z). In fact, also the converse is true as it was shown by Gutkin and Judge in [GJ 00]: A Veech group is a finite index subgroup of SL2(Z) if and only if it comes from an origami. From this it follows in particular by the Fact presented in Section 2 on page 10 that an origami always defines a Teichmüller curve in the moduli space. Characterization of origami Veech groups Recall from Section 1 that an origami O corresponds (up to equivalence) to a finite index subgroup U of F2 = F2(x, y), the free group in two generators (up to conjugation). This description enables us to give a characterization of its Veech group entirely in terms of F2 and its automorphisms. For this we need the following two ingredients: • Let β̂ : Aut(F2) → Out(F2) ∼= GL2(Z) be the natural projection. The fact that we only consider orientation preserving diffeomorphisms applies to only taking automorphisms of Aut(F2) that are mapped to elements in SL2(Z). We denote Aut +(F2) = β̂ −1(SL2(Z)) and restrict to the map β̂ : Aut+(F2) → SL2(Z). • Let Stab(U) = {γ ∈ Aut+(F2)|γ(U) = U} Using these ingredients, it was shown in [S 04] that Veech groups of origamis can be described as stated in the following theorem. Theorem 2 (Proposition 1 in [S 04]). For the Veech group Γ(O) of the origami O holds: Γ(O) = β̂(Stab(U)) Let us make two comments on this description: One consequence is, that one obtains an algorithm that can calculate the Veech group of an arbitrary origami explicitly. This algorithm is described in detail in [S 04]. As an other consequence, we have now a characterization of all origami Veech groups as stated in the following corollary. Corollary 3.2. A finite index subgroup of SL2(Z) occurs as origami Veech group if and only if it is the image of the stabilizing group Stab(U) ⊆ Aut+(F2) for some finite index subgroup U in F2. Thus the question, which finite index subgroups of SL2(Z) are Veech groups be- comes roughly speaking the same as the question which subgroups of Aut+(F2) are such stabilizing groups. So far, there is no general answer known. 3 VEECH GROUPS OF ORIGAMIS 12 In [S 05] it was shown that many congruence subgroups of SL2(Z) are Veech groups. Recall that a congruence group of level n is a subgroup of SL2(Z) that is the full preimage of some subgroup of SL2(Z/nZ) under the natural homomor- phism SL2(Z) → SL2(Z/nZ) and n shall be minimal with this property. For prime level congruence groups the following statement is shown in [S 05, Theorem 4] Theorem 3. Let p be prime. All congruence groups Γ of level p are Veech groups except possibly p ∈ {2, 3, 5, 7, 11} and Γ has index p in SL2(Z). This result is generalized to a statement for arbitrary n in [S 05, Theorem 5] Presenting the Veech group Γ and the quotient H/Γ for an origami As mentioned above, using Theorem 2 the Veech group of an origami can be calculated explicitly. The Veech groups are described as subgroups of SL2(Z) by generators and coset representatives. We use for the notation that SL2(Z) is generated by S and T , with and T = Recall furthermore from the discussion on Veech groups and Teichmüller curves in Section 2 on page 10 that for a Veech group Γ we are in particular interested in the quotient H/Γ, since this quotient is birational to the corresponding Te- ichmüller curve. Here Γ acts as Fuchsian group on the upper half plane H, which is endowed with the Poincaré metric. Since an origami Veech group Γ is a finite index subgroup of SL2(Z), the quotient H/Γ comes with a natural triangulation. More precisely, we choose the funda- mental domain for the action of SL2(Z) on H that is the geodesic pseudo-triangle ∆ with vertices P = −1 i, Q = 1 i and P∞ = ∞. Figure 8: Fundamental domain of SL2(Z). The surface H/SL2(Z) is obtained by identifying the vertical edges P∞ and Q∞ via T and the edge PQ with itself (with fixed point i) via S. For an arbitrary subgroup Γ of SL2(Z) of finite index we obtain a fundamental domain as a union of translates of the triangle ∆: for each coset A we take the 3 VEECH GROUPS OF ORIGAMIS 13 triangle A(∆), where A is a representative of the coset. The identification of the edges is given by the elements in Γ. Gluing the edges gives the quotient surface H/Γ, filling in the cusps leads to a closed Riemann surface endowed with a triangulation. We draw stylized pictures of the fundamental domains that indicate the triangles (see Figure 9 and 10). The triangles are labeled with a coset representative, edges that are identified are labeled with the same letter and vertices that are identified with the same number. Vertices that come from cusps (i.e. points at ∞) are marked with •. In particular, one can read off from these stylized pictures the genus and the number of cusps of the quotient surface H/Γ. Two examples: the origami L(2,3) and the origami D The origami L(2,3): In [S 04, Example 3.5] the Veech group is calculated as follows: Γ(L(2, 3)) = < More precisely, one obtains the generators presented as products of S and T as well as a list of coset representatives. • List of generators: = T 3, = TST 2ST−1T−1, = TSTST−1S, = T 2STST−1S−1T−2, • List of representatives: I, T, S, T 2, TS, ST, T 2S, TST, T 2ST Hence, Γ(L(2, 3)) is a subgroup of index 9 in SL2(Z). The stylized picture of the quotient H/Γ(L(2, 3)) is determined in [S 04, Example 3.6] and is shown here in Figure 9. 3 VEECH GROUPS OF ORIGAMIS 14 TTSTT Figure 9: Fundamental domain of Γ(L(2, 3)). From this one can read off that the genus of the quotient H/Γ(L(2, 3)) is 0 and that it has 3 cusps, namely the vertices labeled by 1,4 and 5. It follows in par- ticular that the corresponding Teichmüller curve has genus 0. The origami D: The Veech group of the origami D is calculated in [S 05, Section 7.3.2]. It has index 24 in SL2(Z) and the following generators: = −I, A1 = = T 3, = ST 6S−1, A3 = −7 16 = (T 2S)T 4(T 2S)−1. = (TS)T 4(TS)−1, A5 = −20 11 = (TST 2S)T 5(TST 2S)−1, −18 −5 = (ST 3S)T 2(ST 3S)−1, The following is a system of cosets representatives: I , T , S , T 2 , TS , ST , T 2S , TST , ST 2 , STS , T 2ST , TST 2 , ST 5 , ST 3 , T 2S , TST 3 , TST 2S , ST 4 , ST 3S , TST 2ST−1 , TST 2ST−2 , TST 2ST−3 ; TST 2ST−4 , ST 3ST 4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 15 The corresponding origami curve C(D) has genus 0. It is shown with its natu- ral triangulation in Figure 10. It has six cusps, namely C1, C2, C3, C4, C5 and C6. TT TTS TTSTT STTSTT TSTTS TSTTST−1 3ST ST Figure 10: The origami curve to D. 4 Veech groups that are non congruence groups Theorem 3 implies that there are many congruence groups which are Veech groups. How about non congruence groups? In this section we will see that the Veech groups for the two examples, the origami L(2, 3) and the origami D, studied in the last paragraph are both non congruence groups. Furthermore, we give a construction that produces for both of them an infinite sequence of origamis whose Veech group is a non congruence group. We use this in order to prove our main theorem. 4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 16 An other generalization of the example L(2, 3) was given by Hubert and Lelièvre in [HL 05], where they show for certain “L-shaped” origamis or square-tiled sur- faces, how they are called there, that their Veech groups are non congruence groups. These surfaces are all of genus 2, hence it follows that there are infinitely many origamis of genus 2 whose Veech group is a non congruence group. Recall that a group is a congruence group, whose level is a divisor of n, if and only if it contains the principal congruence group Γ(n) = { mod n} = kernel(proj : SL2(Z) → SL2(Z/nZ)) In [S 04, Proposition 3.8] it was shown using a proof of Stefan Kühnlein that the Veech group of L(2, 3) is a non congruence group. The basic tool for this is the general level that is defined for any subgroup Γ of SL2(Z) as follows: For each cusp we define its amplitude to be the smallest natural number n such that there is an element of Γ conjugated in SL2(Z) to the matrix which fixes the cusp. Observe that this is equal to the number of triangles around the vertex that represents the cusp in our stylized picture of the quotient surface (see Figures 9 and 10). The general level of Γ is the least common multiple of the amplitudes of all its cusps. A theorem of Wohlfahrt [W 64, Theorem 2] states that the level and the general level of a congruence group coincide. The amplitude of the three cusps of H/Γ(L(2, 3)) labeled with 1, 4 and 5 in Fig- ure 9 is 3, 2 and 4 respectively. Hence, the general level of Γ(L(2, 3)) is 12. Then it is shown in the proof that Γ(L(2, 3)) does not contain Γ(12) which gives the contradiction. The same method can be used in order to show that Γ(D) is a non congruence group. We here carry out the proof for it. Observe from Figure 10 that the six cusps C1, . . . , C6 have the amplitude 3, 6, 4, 4, 5 and 2, respectively. Thus the general level is 60. Proposition 4.1. The Veech group Γ(D) is a non congruence group. Proof. Suppose that Γ = Γ(D) is a congruence group. Since the general level of Γ is 60, we have by the theorem of Wohlfahrt mentioned above, that Γ(60) is a subgroup of Γ. We will use the following facts, which can be checked e.g. in Figure 10: ∈ Γ, A6 = −18 −5 ∈ Γ and T = 4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 17 In order to verify this in Figure 10, use that A1 = T 3 and A6 = S −1T 2S−1T−1S−1TS−1T−3S−1. We will find an element in Γ whose projection to SL2(Z/60Z) is equal to that of T , which gives us the desired contradiction. Recall that SL2(Z/60Z) ∼= SL2(Z/4Z)× SL2(Z/3Z)× SL2(Z/5Z). We identify in the following these two groups. Furthermore we denote by p4, p3, p5 and p60 the projection from SL2(Z) to SL2(Z/4Z), SL2(Z/3Z), SL2(Z/5Z) and SL2(Z/60Z), respectively. Then p60 = p4 × p3 × p5. We have p60(A1) = ( ) and p60(A6) = ( The order of p4(A1) in SL2(Z/4Z) is 4, the order of p3(A1) in SL2(Z/3Z) is 1 and the order of p5(A1) in SL2(Z/5Z) is 5. We also say: The order of p60(A1) is (4, 1, 5). Since 7 ≡ 3 mod 4 and 7 ≡ 2 mod 5 we have p60(A 1) = ( ) (4) Furthermore: p60(A 6) = ( and with the same notation as above p60(A 6) has the order (1, 3, 5). Thus p60(A 6 ) = ( ). (5) From (4) and (5) it follows that p60(A 6 · A 1) = ( ) = p60( ) = p60(T ) But A206 · A 1 ∈ Γ and T /∈ Γ, thus Γ(60) = ker(p60) cannot be contained in Γ. Therefore, Γ cannot be a congruence group of level 60. Contradiction! 4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 18 Sequences of origamis with non congruence Veech groups Starting from the origamis L(2, 3) and D we will define respectively a sequence On, such that for each n ∈ N the Veech group Γ(On) again is a non congruence group. The basic idea is to “copy and paste”: we will cut the origami along a segment, take n copies of it and glue them along the cuts. In Figure 11 we show the origami On for L(2, 3): 1 3 4 5 7 8 . . . 4n-7 4n-5 4n-4 4n-3 4n-1 4n Figure 11: n copies of L(2, 3). Opposite edges are glued. Using the description of an origami by a pair of permutations from Section 1, On is given as: σa = (1 3 4 5 7 8 9 11 12 . . . 4n−3 4n−1 4n), σb = (1 2)(5 6) . . . (4n−3 4n−2). Observe that the genus of On is n + 1 and it has 2n cusps: n of order 3 (all n marked by • in Figure 11), and n of order 1 (all n marked by ◦ in Figure 11). Finally, we want to present the origami On by the finite index subgroup Hn = ∗) of F2, that corresponds to On by Remark 1.6. Recall from Example 1.7 that for O1 = L(2, 3), we obtain the free group of rank 5: U = H1 =< g1 = x 3, g2 = xyx −1, g3 = x 2yx−2, g4 = yxy −1, g5 = y 2 >= F5. The group Hn is obtained as as follows: Hn = < g 1 , g 1 gj g 1 ∈ F5 | i ∈ {0, . . . , n− 1} and j ∈ {2, . . . , 5} > In Figure 12, we show the origami Dn: 1 2 3 6 7 8 . . . . . . . . . 5n-4 5n-3 5n-2 b1 a2 b2 an • • • * • • • • • • Figure 12: n copies of D. Edges with the same label or unlabeled opposite edges are glued. 4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 19 The pair of permutations describing Dn is: σa = (1 2 3 6 7 8 . . . 5n− 4 5n− 3 5n− 2), σb = (1 4 5)(6 9 10) . . . (5n− 4 5n− 1 5n)(2 3)(7 8) . . . (5n− 3 5n− 2) The genus of Dn is 2n and it has n+ 2 cusps: 2 of order 2n (marked as • and ⋆) and n of order 1 (all n marked by ◦). Again, we present On by the corresponding finite index subgroup Hn of F2. We have from Example 1.7 that U = H1 = F6, the free group of rank 6: U =< g′1 = x 3, g′2 = xyx −2, g′3 = x 2yx−1, g′4 = yxy −1, g′5 = y 2xy−2, g′6 = y 3 > = F6 And similarly as above, we obtain: Hn = < g 1 , g 1 ∈ F6 | i ∈ {0, . . . , n− 1} and j ∈ {2, . . . , 6} > We will see in the following that for both sequences all Veech groups Γ(On) are non congruence groups. More precisely, we will show: Proposition 4.2. For both sequences On the following holds: • Γ(On) ⊆ Γ(O1), which is for both sequences a non congruence group. • More generally one has: n divides m ⇒ Γ(Om) ⊆ Γ(On). • Different origamis in one sequence have different Veech groups, i.e.: Γ(On) 6= Γ(Om) for n 6= m. To prove this, let us detect that we are in the following more general setting. Setting A: • Let U be a finite index subgroup of F2. Then U is a free group of rank k for some k ≥ 2, i.e. U = < g1, . . . , gk > = Fk • Let α : Fk → Z be the projection w 7→ ♯g1w where ♯g1w is the number of g1 in the word w = w(g1, . . . , gk) with g counted as −1. • Let Hn be the kernel of pn ◦ α, where pn : Z → Z/nZ is the natural projection, i.e. Hn = < g 1 , g 1 gj g 1 ∈ Fk | i ∈ {0, . . . , n− 1} and j ∈ {2, . . . , k} > . 4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 20 • Finally, let H0 be the kernel of α, i.e.: Hn = << g2, . . . , gk >>U , is the normal subgroup in U generated by g2, . . . , gk. Observe that we are in this setting with U = π1(X ∗) =< x3, xyx−1, x2yx−2, yxy−1, y2 > for the origami L(2, 3) and U = π1(X ∗) =< x3, xyx−2, x2yx−1, yxy−1, y2xy−2, y3 > for the origami D. In order to prove the properties in Proposition 4.2, we will need that U fulfills the following a bit technical condition: Property B: Let U =< g1, . . . , gk > (k ≥ 2) be as above a finite index subgroup of F2 of rank k and {wi}i∈I a system of coset representatives with w1 = id. Suppose that U has the following property: ∀ j ∈ I − {1} : wj << g2, . . . , gk >>U w j 6⊆ U. One can check by hand that for both origamis, L(2, 3) and D, this property is fulfilled. In this setting we obtain the following conclusions. Proposition 4.3. Let n ∈ N ∪ {0}. Let U be a finite index subgroup of F2 fulfilling property B. With the notations from Setting A, we have: a) The normalizer of Hn in F2 is equal to U : NormF2(Hn) = U b) Stab (Hn) ⊆ StabAut+(F2) c) Recall that U = Fk, the free group in k generators. Let βn : Aut(Fk) → GLk(Z/nZ) be the natural projection. Then Stab (Hn) is equal to β−1n ({A = (ai,j)1≤i,j≤k ∈ GLk(Z/nZ)| a1,2 = . . . = a1,k = 0}) ∩ G. Here we use the notation Z/(0Z) = Z thus β0 is the natural projection Aut(Fk) → GLk(Z). Proof. By definition Hn is normal in U , i.e. U ⊆ NormF2(Hn). Let now w be an element of F2\U . Hence, w = wj ·u for some j ∈ I−{1}, u ∈ U . By Property B, there exists some h0 ∈ << g2, . . . , gk >>U = H0, such that wjh0w j 6∈ U . Therefore we have w(u −1h0u)w −1 6∈ U . But u−1h0u ∈ H0 ⊆ Hn, since H0 is normal in U . This shows that w 6∈ NormF2(Hn). 4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 21 This follows from a), since for a subgroup H of F2 in general holds: (H) ⊆ Stab (NormF2(H)), see e.g. [S 06, Remark 3.1]. Define M = {A = (ai,j)1≤i,j≤k ∈ GLk(Z/nZ)| a1,2 = . . . = a1,k = 0}. Let γ ∈ G. We have to show that γ(Hn) = Hn if and only if βn(γ) ∈ M . Let furthermore pkn : Fk → (Z/nZ) k be the natural projection. Consider the following commutative diagram: Hn = p n(Hn) ⊆ (Z/nZ) βn(γ) // (Z/nZ)k ⊇ pkn(Hn) = Hn Since pkn is surjective and Hn is the full preimage of Hn = p n(Hn), it follows that γ(Hn) = Hn if and only if βn(γ)(Hn) = Hn. Observe finally that: Hn = {(0, x2, . . . , xk) ∈ (Z/nZ) k} and StabGLk(Z/nZ)(Hn) = { A = (ai,j)1≤i,j≤k ∈ GLk(Z/nZ)| (y1, . . . , yk) = A · (0, x2, . . . , xk) ⇒ y1 = 0 } = {A = (ai,j) ∈ GLk(Z/nZ)| a1,2 = . . . = a1,k = 0} Theorem 2 suggests the following notation. Definition 4.4. Let U be a subgroup of F2. With β̂ : Aut+(F2) → SL2(Z) as in Theorem 2, we define Γ(U) = β̂(Stab and call Γ(U) the Veech group of U . We now obtain from Proposition 4.3 the following conclusions. Corollary 4.5. Suppose that we are in the same situation as in Proposition 4.3, in particular that U is a finite index subgroup of F2 fulfilling property B. Then we have for all n ∈ N: a) Stab (H0) ⊆ StabAut+(F2) (Hn) and Γ(H0) ⊆ Γ(Hn). 4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 22 b) If m ∈ N with n|m, then: (Hm) ⊆ StabAut+(F2) (Hn) and Γ(Hm) ⊆ Γ(Hn). (H0) = (Hn) and Γ(H0) = Γ(Hn) Proof. a) and b): Let γ ∈ G. By Proposition 4.3 we have that ∀n ∈ N : γ ∈ Stab (Hn) ⇔ βn(γ) = A = (ai,j) with a1,2 ≡ . . . ≡ a1,k ≡ 0 mod n and γ ∈ Stab (H0) ⇔ β0(γ) = A = (ai,j) with a1,2 = . . . = a1,k = 0. Thus we have for all n ∈ N and for all m ∈ N with n|m, that (H0) ⊆ StabAut+(F2)(Hm) ⊆ StabAut+(F2)(Hn). We have in particular by the definition of the Veech group of a subgroup of F2: Γ(H0) ⊆ Γ(Hm) ⊆ Γ(Hn). ⊆ follows from a). ⊇ follows from Remark [S 06, Remark 3.1]. We now return to the language of origamis: Let O be an origami, U the corre- sponding subgroup of F2. Define for U the subgroups Hn (n ∈ N) as in Setting A and let On be the origamis corresponding to the groups Hn. By Corollary 4.5 and Theorem 2 we obtain immediately the following result. Proposition 4.6. If U has the Property B, then ∀n ∈ N : Γ(On) ⊆ Γ(O) and ∀n,m ∈ N : n|m ⇒ Γ(Om) ⊆ Γ(On). In particular, if Γ(O) is a non congruence group, each Γ(On) is a non congruence group. Thus in this case, we obtain infinitely many origamis whose Veech group is a non congruence group. In order to conclude Proposition 4.2, it is now just left to prove the last item. But this follows , since we have (see [S 05]) for both sequences On, the one coming from the origami L(2, 3) and the one coming from the origami D, that ∈ Γ(On) ⇔ 3n divides s. (6) 4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 23 This finishes the proof of Proposition 4.2. Furthermore, Theorem 1 follows from Proposition 4.2. Remark: From Corollary 4.5 and (6) it follows that Γ(H0) has infinite index in SL2(Z). Furthermore it is non trivial, since it contains for L(2, 3) respectively B3 = for D. REFERENCES 24 References [EG 97] C.J. Earle, F.P. Gardiner: Teichmüller disks and Veech’s F -structures. American Mathematical Society. Contemporary Mathematics 201, 1997 (p. 165–189). [GJ 00] E. Gutkin, C. Judge: Affine mappings of translation surfaces. Duke Mathematical Journal 103 No. 2, 2000 (p. 191–212). [HeSc 06] F. Herrlich, G. Schmithüsen: On the boundary of Teichmüller disks in Teichmüller and in Schottky space. To appear in Handbook of Teichmüller theory. Ed. A. Papadopoulos, European Mathematical Society, 2006. [H 06] F. Herrlich: A comb of origami curves in M3. Proceedings of Symposium on Transformation Groups, Yokohama, November 2006. [HL 05] P. Hubert, S. Lelièvre: Noncongruence subgroups in H(2). International Mathematics Research Notices 2005, No.1 , 2005 (p. 47–64). [HuSc 01] P. Hubert, T. Schmidt: Invariants of translation surfaces. Annales de l’Institut Fourier 51 No. 2, 2001 (p. 461–495). [Le 02] S. Lelièvre: Veech surfaces associated with rational billiards. Preprint, 2002. arXiv:math.GT/0205249. [Lo 05] P. Lochak: On arithmetic curves in the moduli space of curves. J. Inst. Math. Jussieu 4, No. 3, 2005 (p. 443–508). [S 04] G. Schmithüsen: An algorithm for finding the Veech group of an origami. Experimental Mathematics 13 No. 4, 2004 (p. 459–472). [S 05] G. Schmithüsen: Veech Groups of Origamis. Dissertation (PhD thesis), Karlsruhe 2005. Elektronisches Volltextarchiv EVA Universität Karlsruhe. http://www.ubka.uni-karlsruhe.de/eva/ [S 06] Schmithüsen,G.: Examples for Veech groups of origamis. In: The Geome- try of Riemann Surfaces and Abelian Varieties. Contemp. Math. 397, 2006 (p. 193–206). [T 88] W. Thurston: On the geometry and dynamics of diffeomorphisms of sur- faces. Bulletin (New Series) of the American Mathematical Society 19 No. 2, 1988 (p. 417–431). [V 89] W.A. Veech: Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Inventiones Mathematicae 97 No.3 1989, (p. 553–583). REFERENCES 25 [W 64] K. Wohlfahrt: An Extension of F. Klein’s Level Concept. Illinois Journal of Mathematics 8, 1964 (p. 529–535). [Z 06] A. Zorich: Flat Surfaces in collection ”Frontiers in Number Theory, Physics and Geometry. Volume 1: On random matrices, zeta functions and dynamical systems”, Ed. P. Cartier, B. Julia, P. Moussa, P. Vanhove, Springer- Verlag, 2006 (p. 439–586). Origamis Translation structures and Veech groups Veech groups of origamis Veech groups that are non congruence groups

0704.0417

Thermodynamic Stability - A note on a footnote in Ruelle's book

UWThPh-2007-09 Thermodynamic Stability – A note on a footnote in Ruelle’s book Bernhard Baumgartner1 Institut für Theoretische Physik, Universität Wien Boltzmanngasse 5, A-1090 Vienna, Austria April 3, 2007 Abstract Thermodynamic stable interaction pair potentials which are not of the form “positive function + real continuous function of positive type” are presented in dimension one. Construction of such a potential in dimension two is sketched. These constructions use only elementary calculations. The mathematical background is discussed separately. PACS numbers: 05.20.-y, 02.20.-a, 02.40.Ft Keywords: thermodynamic stability, convex cone 1 Introduction In Ruelle’s book [R69] on statistical mechanics, in section 3.2 concerning one species of classical particles in Rν , you can read: 1 PROPOSITION. If the pair potential Φ can be written in the form Φ = Φ1 + Φ2 (1) where Φ1 is positive, and Φ2 is a real continuous function of positive type, then Φ is stable. [email protected] http://arxiv.org/abs/0704.0417v1 TDS BB April 3, 2007 2 “Positive” is meant here and throughout this paper as nowhere negative, “stable” means ∃E0 ∈ R such that ∀N, ∀{x1...xN} ⊂ Rν : U(x1 · · ·xN) ≧ N · E0, (2) where U(x1 · · ·xN ) = i 6=j Φ(xj − xi). (3) This proposition is accompanied by the 2 FOOTNOTE. It seems to be an open problem to construct a stable potential which is not of the form (1). We solve this problem in dimension 1, considering particles either in Z or in R, giving a detailed proof. In dimension 2 the problem can also be solved, but we give only a sketch of the ideas. 1 To make it simple, we consider only pair potentials which are bounded con- tinuous functions and state the stability property as 3 DEFINITION. A bounded continuous real valued function V on Rν is stable, E(ρ) := ρ(x)V (x− y)ρ(y)dνx dνy ≥ 0 (4) for every positive finite measure ρ(x)dνx on Rν. A bounded real valued function V on Zν is stable, if E(ρ) := ρ(~m)V (~m− ~n)ρ(~n) ≥ 0 (5) for every positive bounded function ρ(~m) on Zν . The stability property used in Ruelle’s Theorem is an immediate consequence. With ρ = i=1 δ(xi − xj) put into equation (4) one gets U(x1 · · ·xN ) = E(ρ)−N · V (0) ≥ −N · V (0). The main result of our considerations is stated as 4 THEOREM. Each of the following functions is a stable pair potential, but not a sum of a positive and a real valued positive definite function. 1. The function V : Z → R, defined as V (0) = V (2) = V (−2) = 1, V (1) = V (−1) = −1, (6) V (n) = 0 ∀n with |n| ≥ 3, 2. The function W : R → R, defined as W (x) = V (n)f(n− x+ y)f(y)dy, (7) with f a positive continuous function (−1 ) → R and V as defined in (6). 1 Construction in higher dimensions is still an open problem. TDS BB April 3, 2007 3 2 Properties of the interaction potentials Proof. Of part (1) of 4 Theorem. Denote the distribution of particles on the chain by the “density” ρ, a function Z → Z+. The interaction energy U becomes smaller, when the system is cut into non-interacting pieces: If ρ(n) ≥ ρ(n+1) divide the chain, cutting between n+1 and n+ 2. Moving the pieces apart, one looses the energy 2[ρ(n)− ρ(n + 1)]ρ(n+ 2) + 2ρ(n+ 1)ρ(n + 3) ≥ 0. The symmetric procedure of cutting between n − 2 and n − 1 lowers the energy if ρ(n− 1) ≤ ρ(n). Now there remains a set of pieces of no more than three lattice points, with densities like 0 ≤ ρ(n− 1) ≤ ρ(n) ≥ ρ(n + 1) ≥ 0. Including the “self-energies” N · V (0) one gets for each piece, centered around n, E = ρ(n−1)2+ρ(n)2+ρ(n+1)2+2[ρ(n−1)ρ(n+1)−ρ(n−1)ρ(n)−ρ(n)ρ(n+1)] = [ρ(n− 1)− ρ(n) + ρ(n+ 1)]2 ≥ 0. Proving the stability of V . If V were the sum of a positive and a positive definite function, it would give V (n)µ(n) ≥ 0, (8) for each µ being both positive and positive definite. Now consider µ(5ν) = 1, µ(5ν ± 1) = , µ(5ν ± 2) = 0, (9) which is obviously positive. Positive definiteness is seen by using Bochner’s the- orem [RN55] and calculating the Fourier-Transform, with α ∈ (−π,+π]: µ̂(α) = µ(n)e−inα 5 δ(α) + ),+δ(α+ > 0. (10) But it does not give a positive value in (8): V (n)µ(n) = 2− 5 < 0. TDS BB April 3, 2007 4 The appearance of the numbers 5 and 5 may seem mysterious. Demystifying is the next section, where we present the “origin” of these V and µ. In this section we develop further use of these functions in R and in R2. Proof. Of part (2) of 4 Theorem. For N particles at x1 . . . xN consider the measure ρ(x) = δ(x− xj). (11) Adding the self-energies N ·W (0), we study ρ(x)W (x− y)ρ(y) dxdy V (n) ρf (x+ n)ρf (x) dx, (12) with ρf(x) := f(x − y)ρ(y) dy. Splitting the integral in (12) into pieces of intervals with unit length and defining ρf,x(m) = ρf(x+m) gives V (n)ρf (x+m+ n)ρf(x+m) ρf,x(p)V (p−m)ρf,x(m) ≥ 0, by part (1) of the theorem. So the potential W is stable. Now consider the distribution µD(x) = µ(m)δ(x−m), (13) using the sequence µ defined in (9). This distribution is positive and positive definite, as can be seen at its Fourier transform, which is (up to a factor) the same as in (10), now with µ̂D(α+2π) = µ̂D(α) periodically extended to all α ∈ R. This µD is used to show that the potential is not a sum of positive and positive definite functions: W (x)µD(x)dx V (n) ∫ + 1 dx δ(x−m)f(n− x+ y)f(y) V (n)µ(n) · f 2(y)dy < 0. (14) In the last step the final support of f is essential. TDS BB April 3, 2007 5 Construction of a stable pair potential in R2 being a function of the particle distances only may be done in the following way: • Use W (x) defined in (7), now with an f supported on (−1 ), convolute it twice with the distribution h(x) = e−ǫ|n|δ(x− 5n) : W1(x) = h(x− y)W (y − z)h(z)dy dz. • Take the mean value (times 2π) of all rotated versions: Wr(~x) = 1rW1(|~x|). • Smoothen out Wr with a positive continuous function g(r) with support on [0, 1 W2(~x) = g(|~x− ~y|) W1(|~y − ~z|) |~y − ~z| g(|~z|)d2y d2z. That the stability is not destroyed by the double convolution with h follows from a consideration as it is used in the equation (12). Written in a formal way: 〈ρ|W1 |ρ〉 = 〈ρ| h ∗W ∗ h− |ρ〉 = 〈ρ ∗ h|W |ρ ∗ h〉. Considering only smooth densities ρ(~x) one may take W1(x1)δ(x2) as a stable distribution in R2: 〈ρ|W1 · δ |ρ〉dim=2 = 〈ρy|W1 |ρy〉dim=1 d y ≥ 0. Now rotating the axes and taking the mean value does not destroy the stability. Once more a double convolution is done, now with g in order to get W2 as a bounded continuous potential acting in R2. 〈ρ|W2 |ρ〉 = 〈ρ| g ∗W ∗ g− |ρ〉 = 〈ρ ∗ g|W |ρ ∗ g〉 ≥ 0. Smoothing by convolution with g enables to consider again sets of particles rep- resented by delta-functions in ρ. To disprove the possibility of splitting W2 into a sum of a positive and a positive definite function one may use the µD of equ. (13) embedded into R µD(x, y) = µD(x)δ(y). Due to the smoothing of Wr by g and due to its decrease given by the decrease of h, the integral W2µD is finite: W2µD(x) dx = W2(0) + 2W2(1)µ(1) + 2 · W2(|5ν + n|)µ(n) TDS BB April 3, 2007 6 The bounded support of f and g is needed here as it was in equ. (14). The exponential decrease implies W2(|5ν + n|) = const. · e−5ǫ ν V (n) · 1 +O( The “const.” factor involves the integrals over f 2 and g2, the error term O( 1 gives the difference between e−5ǫ ν/5ν and e−ǫ (5ν+n)/(5ν + n). The summations over ν and n give ≈ 2 · const. · V (n)µ(n) · log(1/ǫ) +O e−5ǫ ν The first part is negative and increases without limit when ǫ → 0, while the other term remains finite. So W2 with small ǫ can not be a sum of positive and positive definite functions. 3 Mathematical background Only in applying Proposition 1 in statistical mechanics the Thermodynamic Limit is considered, not yet in the investigations of “stability”. Moreover, in the re- formulation in 3 Definition there is no mentioning of “particles”. What is used of properties of space are: A distance relation between points and an invariant measure. This allows for a more general version of the definition, concerning functions on groups. We keep the notation we used above: x and y are elements of the group, their “group product” is x+ y, the “inverse” of x is −x. 5 DEFINITION. Consider a bounded continuous real valued function V on a locally compact abelian group G which has the Haar measure dx. V is stable, if 〈ρ|V |ρ〉 := ρ(x)V (x− y)ρ(y)dx dy ≥ 0 (15) for every finite positive Borel measure ρ(x)dx. Stable functions can be added, multiplied by positive numbers, and limits may be formed. So they form a closed convex cone, which we call STB. This cone STB contains POS, the cone of positive functions, also PDF, the cone of positive definite functions and sums thereof. STB ⊃ POS + PDF (16) An investigation of the relations between these cones may proceed via investi- gation of the dual cones (see [V64, R62, G03]). The dual cones are subsets of V ′, the space of finite Borel measures µ(x)dx, which is the dual space to V, the TDS BB April 3, 2007 7 Banach space of bounded continuous functions. The dual cone to POS is POS′, the set of finite positive Borel measures, dual to PDF is PDF′, the set of finite positive definite Borel measures. The cone STB′ is given as the closure of the cone of convex combinations of “correlation measures” µ(x) = ρ(x)ρ(y + x)dy, (17) i.e. convolutions of finite positive Borel measures ρ(x)dx with their reflected ver- sion ρ(−x)dx. These correlation measures are both positive and positive definite: STB′ ⊂ POS′ ∩ PDF′ (18) Now the question of equality or inequality in this relation is related to the central problem which is our concern in this investigation, the question of equality or inequality in (16). If the closed cone POS′∩PDF′ contains an element µ which is not in the closed cone STB′, then, by definition of “dual cone”, there exists an element V ∈ STB such that V µ < 0, incompatible with a decompostion V = f + g, f ∈ POS, g ∈ PDF. For the groups Z2, Z3, Z4 there is equality in the equations (16) and (18), but not for Z5. 6 PROPOSITION. The intersection of POS′ ∩ PDF′ with the plane {(µ(−2) . . . µ(2))|µ(0) = 1} is completely characterized by its extremal points (0, 0, 1, 0, 0), (0, γ, 1, γ, 0), (γ, 0, 1, 0, γ), (1, 1, 1, 1, 1), with γ = ( 5 − 1)/2 = 1/(2| cos 4π/5|). Proof. By using Bochner’s theorem and analyzing the Fourier transform µ̂(k) = µ(n)e−2π k n/5. (19) On the other hand there is a bound for STB′ which cuts off a triangular subset of this convex quadrangle: 7 LEMMA. Each element of STB′ obeys the inequality µ(1) ≤ µ(n)/4. (20) Proof. STB′ is defined by its extremal rays, formed as correlation measures of positive densities. µ ∈ STB′, µ extremal ⇔ ∃ρ ≥ 0, µ(n) = ρ(m)ρ(m + n). TDS BB April 3, 2007 8 Assume, w.l.o.g., that ρ(−1) ≥ ρ(−2). then µ(1) = [ρ(−1) + ρ(1)] · [ρ(−2) + ρ(0) + ρ(2)]− [ρ(−1)− ρ(−2)] ρ(2)− ρ(−2)ρ(1) − x)(s + x) ≤ s Here s = m ρ(m), x = [ρ(−2) + ρ(0) + ρ(2)− ρ(−1)− ρ(1)] /2. Observe n µ(n) = s Remark: Also µ(2) obeys this inequality and µ(−1) = µ(1), µ(−2) = µ(2). Closer inspection reveals moreover two rounded edges of STB′. Now the extremal point with µ(n) as in equation (9) with ν = 0 is outside this boundary. And V (n) as in equation (6) is an element of STB, but outside of POS+PDF. 4 Conclusion For pair potentials which are bounded continuous functions the property of being “stable” can be reformulated without mention of particles. In this way it can be studied for abstract abelian groups. At the heart of the present investigation is the observation of a function V in Z5 which is stable, but indecomposable into a sum of positive and positive definite functions. This function V can also be used on Z. With some smoothing it can be used on R, and in damped periodically extended, rotationally symmetrized and again smoothed form on R2. Of course it is possible find sets of other examples nearby. So V (−1) = V (1) in Theorem 4 could be a little bit higher than −1. Only at −( 5+1)/4 ≈ −0.8 does it become decomposable. The construction of a rotationally invariant example for dimension two is not so simple. A nicer one, or one for higher dimension, is not yet known. References [R69] D. Ruelle: Statistical Mechanics: Rigorous Results (W. A. Benjamin, inc., New York) 1969. [RN55] F. Riesz and B. Sz. Nagy: Functional Analysis (Ungar, New York) 1955 [V64] Frederick A. Valentine: Convex sets (McGraw-Hill, NY (McGraw-Hill se- ries in higher mathematics)) 1964 [R62] W. Rudin: Fourier Analysis on Groups (Interscience, New York) 1962 [G03] “Convex cones and their faces” Chapter 3 in: H. Glöckner: Positive Def- inite Functions on Infinite-Dimensional Convex Cones; Memoirs AMS, 166, Number 789, 2003 Introduction Properties of the interaction potentials Mathematical background Conclusion

0704.0418

Entanglement entropy at infinite randomness fixed points in higher dimensions

Entanglement entropy at infinite randomness fixed points in higher dimensions Yu-Cheng Lin1, Ferenc Iglói2,3 and Heiko Rieger1 Theoretische Physik, Universität des Saarlandes, 66041 Saarbrücken, Germany Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary Institute of Theoretical Physics, Szeged University, H-6720 Szeged, Hungary (Dated: November 3, 2018) The entanglement entropy of the two-dimensional random transverse Ising model is studied with a numerical implementation of the strong disorder renormalization group. The asymptotic behavior of the entropy per surface area diverges at, and only at, the quantum phase transition that is governed by an infinite randomness fixed point. Here we identify a double-logarithmic multiplicative correction to the area law for the entanglement entropy. This contrasts with the pure area law valid at the infinite randomness fixed point in the diluted transverse Ising model in higher dimensions. PACS numbers: Valid PACS appear here Extensive studies have been devoted recently to under- stand ground state entanglement in quantum many-body systems [1]. In particular, the behavior of various entan- glement measures at/near quantum phase transitions has been of special interest. One of the widely used entan- glement measures is the von Neumann entropy, which quantifies entanglement of a pure quantum state in a bi- partite system. Critical ground states in one dimension (1D) are known to have entanglement entropy that di- verges logarithmically in the subsystem size with a uni- versal coefficient determined by the central charge of the associated conformal field theory [2]. Away from the crit- ical point, the entanglement entropy saturates to a finite value, which is related to the finite correlation length. In higher dimensions, the scaling behavior of the entan- glement entropy is far less clear. A standard expectation is that non-critical entanglement entropy scales as the area of the boundary between the subsystems, known as the ”area law” [3, 4]. This area-relationship is known to be violated for gapless fermionic systems [5] in which a logarithmic multiplicative correction is found. One might suspect that whether the area law holds or not depends on whether the correlation length is finite or diverges. However, it has turned out that the situation is more complex: numerical findings [7] and a recent analytical study [8] have shown that the area law holds even for critical bosonic systems, despite a divergent correlation length. This indicates that the length scale associated with entanglement may differ from the correlation length. Another ongoing research activity for entanglement in higher spatial dimensions is to understand topological contributions to the entanglement entropy [9]. The nature of quantum phase transitions with quenched randomness is in many systems quite different from the pure case. For instance, in a class of systems the critical behavior is governed by a so-called infinite- randomness fixed point (IRFP), at which the energy scale ǫ and the length scale L are related as: ln ǫ ∼ Lψ with 0 < ψ < 1. In these systems the off-critical regions are also gapless and the excitation energies in these so- called Griffiths phases scale as ǫ ∼ L−z with a nonuni- versal dynamical exponent z <∞. Even so, certain ran- dom critical points in 1D are shown to have logarith- mic divergences of entanglement entropy with universal coefficients, as in the pure case; these include infinite- randomness fixed points in the random-singlet universal- ity class [12, 13, 14, 15, 16] and a class of aperiodic singlet phases [17]. In this paper we consider the random quantum Ising model in two dimensions (2D), and examine the disorder- averaged entanglement entropy. The critical behavior of this system is governed by an IRFP [10, 11] implying that the disorder strength grows without limit as the system is coarse grained in the renormalization group (RG) sense. In our study, the ground state of the system and the entanglement entropy are numerically calculated using a strong-disorder RG method [18, 19], which yields asymp- totically exact results at an IRFP. To our knowledge this is the first study of entanglement in higher dimensional interacting quantum systems with disorder. The random transverse Ising model is defined by the Hamiltonian H = − 〈i,j〉 i . (1) Here the {σαi } are spin-1/2 Pauli matrices at site i of an L × L square lattice with periodic boundary conditions. The nearest neighbor bonds Jij(≥ 0) are independent random variables, while the transverse fields hi(≥ 0) are random or constant. For a given realization of random- ness we consider a square block A of linear size ℓ, and calculate the entanglement between A and the rest of the system B, which is quantified by the von Neumann entropy of the reduced density matrix for either subsys- tems: S = −Tr(ρA log2 ρA) = −Tr(ρB log2 ρB). (2) The basic idea of the strong disorder RG (SDRG) ap- proach is as follows [18, 19]: The ground state of the sys- tem is calculated by successively eliminating the largest http://arxiv.org/abs/0704.0418v2 FIG. 1: (color online). An example of typical ground state in the random quantum Ising model (a) in 1D, and (b) in 2D; it contains a collection of spin clusters of various sizes, which are formed and decimated during the RG. The entanglement of a block (shaded area) is give by the number of decimated clusters (indicated by red loops) that connect the block with the rest of the system. local terms in the Hamiltonian and by generating a new effective Hamiltonian in the frame of the perturbation theory. If the strongest bond is Jij , the two spins at i and j are combined into a ferromagnetic cluster with an effec- tive transverse field h̃(ij) = . If, on the other hand, the largest term is the field hi, the spin at i is decimated and an effective bond is generated between its neighbor- ing sites, say j and k, with strength J̃jk = JijJik . After decimating all degrees of freedom, we obtain the ground state of the system, consisting of a collection of indepen- dent ferromagnetic clusters of various sizes; each cluster of n spins is frozen in an entangled state of the form: (| ↑↑ · · · ↑︸ ︷︷ ︸ n times 〉+ | ↓↓ · · · ↓︸ ︷︷ ︸ n times 〉). (3) In this representation, the entanglement entropy of a block is given by the number of clusters that connect sites inside to sites outside the block [Fig. 1]. We note that correlations between remote sites also contribute to the entropy due to long-range effective bonds generated under renormalization. In 1D the RG calculation can be carried out analyti- cally and the disorder-averaged entropy Sℓ of a segment of length ℓ has been obtained as Sℓ = log2 ℓ [12]. In higher dimensions the RG method can only be imple- mented numerically. The major complication in this case is that the model is not self-dual and thus the location of the critical point is not exactly known. To locate the crit- -1.5 -1 -0.5 -1.5 -1 -0.5 0 -0.15 -0.1 -0.05 0 L = 16 L = 32 L = 64 -1.5 -1 -0.5 -1.5 -1 -0.5 0 (a) (b) (d) (e) =1.175 h =1.175 =1.175 =1.18 h0=1.17 PSfrag replacements ln ˜h ln ˜h ln ˜J ln ˜J ln ˜h /L0.55 FIG. 2: (color online). The distribution of the last deci- mated effective log-fields lneh∞, and the distribution of the last decimated effective log-bonds ln eJ∞ in the RG calcula- tions. At h0 = 1.175, the distributions, shown in (a) and (b), get broader with increasing system sizes, indicating the RG flow towards infinite randomness, i.e. the system is critical. A scaling plot of the data in (a) using energy-length scaling lneh∞ ∼ L ψ with ψ = 0.55 is presented in (c). The solid line is just a guide to the eye. The subfigures (d) and (e) show the log-field distribution at h0 = 1.18 and the log-bond dis- tribution at h0 = 1.17, respectively; the distributions show a power-law decaying tail in the low energy region, which is clear evidence that the system is in the Griffiths phases. ical point, we can make use of the fact that the excitation energy of the system has the scaling behavior ln ǫ ∼ Lψ at criticality, while it follows ǫ ∼ L−z in the off-critical regions. In the numerical implementation of the SDRG method, the low-energy excitations of a given sample can be identified with the effective transverse field h̃∞ of the last decimated spin cluster, or with the effective coupling J̃∞ of the last decimated cluster-pair. In our implementation we set for convenience the transverse fields to be a constant h0 and the random bond variables were taken from a rectangular distribu- tion centered at J = 1 with a width ∆ = 0.5. The critical point was approached by varying the single control pa- rameter h0. Although this initial disorder appears to be = 1.170 = 1.175 = 1.180 = 1.185 = 1.190 L = 16 L = 24 L = 32 L = 40 L = 64 = 1.175 PSfrag replacements ln ℓℓ FIG. 3: (color online). Left panel: The disorder averaged block entropy per surface unit Sℓ/ℓ vs. the linear size of the block ℓ for a system size L = 64 for various values of h0. We observe that the entropy for ℓ = L/2 reaches its maximum at the critical point hc = 1.175 (cf. Fig 2). Right panel: The block entropy per surface area vs. ln ℓ on a log-scale for different system sizes L at the critical point. The data show a straight line (guided by the dashed line), corresponding to the scaling obeying the area law with a double-logarithmic correction, as given in Eq. (4). weak, the renormalized field and bond distributions be- come extremely broad even on a logarithmic scale [Fig. 2] at the critical point h0 = hc = 1.175. This indicates the RG flow towards infinite randomness. Slightly away from the critical point, both in the disordered Griffiths-phase with h0 = 1.18 and in the ordered Griffiths-phase with h0 = 1.17, the distributions have a finite width and obey quantum-Griffiths scaling h∞ ∼ L−z. At the critical point one has IRFP scaling lnh∞ ∼ Lψ and we estimate the scaling exponent as ψ = 0.55, quite close to the value ψ = 0.5 for the 1D case [18]. Now we consider the entanglement entropy near the in- finite randomness critical point. To obtain the disorder- averaged entanglement entropy Sℓ of a square block of size ℓ, we averaged the entropies over blocks in different positions of the whole system for a given disorder real- ization and then averaged over a few thousand samples. In Fig. 3 we show the entropy per surface unit Sℓ/ℓ = sℓ for different values of h0. This average entropy density is found to be saturated outside the critical point, which corresponds to the area law. At the critical point sℓ in- creases monotonously with ℓ, and the numerical data are consistent with a log-log dependence: Sℓ ∼ ℓ log2 log2 ℓ (4) as illustrated in Fig. 3. In this way we have identified an alternative route to locate the infinite randomness criti- cal point: it is given by the field h0 for which the average block entropy at ℓ = L/2 is maximal. Indeed the nu- merical results in Fig. 3 predict the same value of hc as obtained from the scaling of the gaps. We note that the same quantity, the position of the maxima of the average entropy, can be used for the random quantum Ising chain to locate finite-size transition points [21]. The log-log size dependence of the average entropy in Eq.(4) at criticality is completely new; it differs from the scaling behavior observed in 2D pure systems, like the area law, Sℓ ∼ ℓ, for critical bosonic systems [7, 8], or a logarithmic multiplicative correction to the area law, Sℓ ∼ ℓ log2 ℓ, as found in free fermions [5, 6, 7, 8]. This double-logarithmic correction can be understood via a SDRG argument: In the 1D case a characteristic length scale r at a given RG step is identified with the aver- age length of the effective bonds, i.e. the average size of the effective clusters. At the scale r(< ℓ) the frac- tion of the total number of spins, nr, that have not been decimated is given by nr ∼ 1/r [18]; these active (i.e., undecimated) spins have a finite probability to form a cluster across the boundary of the block (a segment ℓ in the 1D case) and thus to give contributions to the en- tanglement entropy. Repeating the renormalization until the scale r ∼ ℓ, the contributions to the entropy are summed up: Sℓ ∼ dr nr ∼ ln ℓ, leading to the log- arithmic dependence of the 1D model [12]. For the 2D case with the same type of RG transformation with a length scale r < ℓ, the fraction of active spins in the renormalized surface layer of the block is nr ∼ ℓ/r. Here we have to consider the situation in which some of these active surface spins would form clusters within the sur- face layer and thus contribute zero entanglement entropy; the number of the active spins that are already engaged in clusters on the surface at RG scale r is proportional to ln r, as known from the 1D case, and only O(1) of the active surface spins would form clusters connecting the block with the rest of the system. Consequently, the entropy contribution in 2D can be estimated as: dr nr/ ln r ∼ ℓ ln ln ℓ, i.e. a double-logarithmic ℓ- dependence, as reflected by the numerical data in Fig. 3. Based on the SDRG argument described above, the double-logarithmic correction to the area law appears to be applicable for a broad class of critical points in 2D with infinite randomness. For instance, the critical points of quantum Ising spin glasses are believed to belong to the same universality class as ferromagnets since the frustra- tion becomes irrelevant under RG transformation, and the same type of cluster formations as observed in our numerics for the ferromagnet is expected to be generated during the action of the RG. The entanglement entropy at the IRFP is completely determined by the cluster ge- ometries occurring during the SDRG. Another type of IRFP in higher dimensions occurs in the bond-diluted quantum Ising ferromagnet: The Hamiltonian is again given by (1), but now Jij = 0 with probability p and Jij = J > 0 with probability 1− p. At percolation threshold p = pc there is a quantum critical line along small nonzero transverse fields, which is con- p = 0.49 = 0.50 p = 0.52 1 10 10010 L = 128 L = 256 L = 512 = 0.5 PSfrag replacements FIG. 4: (color online). The entropy per surface area Sℓ/ℓ = sℓ vs. ℓ near the percolation threshold pc = 0.5 for the 2D bond-diluted Ising model at small transverse fields for L = 512. The curves converge to finite values for ℓ → ∞, corresponding to the area law. The inset shows sℓ − s∞ as a function of ℓ. s∞ is estimated from sL/2 at L = 512. The dashed line corresponds to ℓ−1. trolled by the classical percolation fixed point, and the energy scaling across this transition line obeys ln ǫ ∼ Lψ, implying an IRFP [20]. The ground state of the system is given by a set of ordered clusters in the same geom- etry as in the classical percolation model – only nearest neighboring sites are combined into a cluster. In this cluster structure, the block entropy, determined by the number of the clusters connecting the block and the rest of the system, is bounded by the area of the block, i.e. Sℓ ∼ ℓd−1 with d being the dimensionality of the sys- tem. To examine this, we determined the entanglement entropy by analyzing the cluster geometry of the bond- diluted transverse Ising model. Fig. 4 shows our results for the square lattice, which follow a pure area-law with an additive constant: Sℓ = aℓ+ b+O(1/ℓ). To summarize, we have found that the entanglement properties at quantum phase transitions of disordered systems in dimensions larger than one can behave quite differently. Generalizing our arguments for the 2D case, we expect for the random bond transverse Ising systems a multiplicative d-fold logarithmic correction to the area law in d dimensions at the critical point, whereas for diluted Ising model at small transverse fields the area law will hold in any dimension d > 1 at the percolation threshold. Although both critical points are described by infinite randomness fixed points, the structure of the strongly coupled clusters in both cases is fundamentally different, reflecting the different degrees of quantum me- chanical entanglement in the ground state of the two sys- tems. This behavior appears to be in contrast to one- dimensional systems governed by IRFPs [12]. Other disordered quantum systems in higher dimen- sions might also display interesting entanglement prop- erties: For instance, the numerical SDRG has also been applied to higher dimensional random Heisenberg anti- ferromagnets which do not display an IRFP [22]. The ground states involve both singlet spins and clusters with larger moments; therefore, we expect the correction to the area law to be weaker than a multiplicative loga- rithm and different from the valence bond entanglement entropy in the Néel Phase [23]. Useful discussions with Cécile Monthus are gratefully acknowledged. This work has been supported by the Na- tional Office of Research and Technology under Grant No. ASEP1111, by a German-Hungarian exchange pro- gram (DAAD-MÖB), by the Hungarian National Re- search Fund under grant No OTKA TO48721, K62588, MO45596. [1] For a review see: L. Amico et al., quant-ph/0703044. [2] P. Calabrese and J. Cardy, J. Stat. Mech. Theor. Exp. 2004, P06002 (2004). [3] M. Srednicki, Phys. Rev. Lett. 71, 666 (1993). [4] M.B. Plenio et al., Phys. Rev. Lett. 94, 060503 (2005); M. Cramer et al., Phys. Rev. A 73, 012309 (2006); M. Cramer and J. Eisert, New J. Phys 8 71 (2006). [5] M.M. Wolf, Phys. Rev. Lett. 96, 010404 (2006); D. Gioev and I. Klich, Phys. Rev. Lett. 96, 100503 (2006). [6] W. Li et al, Phys. Rev. B 74, 073103 (2006). [7] T. Barthel, M.-C. Chung, and U. Schollwöck, Phys. Rev. A 74, 022329 (2006). [8] M. Cramer, J. Eisert, and M.B. Plenio, Phys. Rev. Lett. 98, 220603 (2007). [9] A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006); M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006); E. Fradkin and J.E. Moore, Phys. Rev. Lett. 97, 050404 (2006). [10] C. Pich et al., Phys. Rev. Lett. 81, 5916 (1998). [11] O. Motrunich et al., Phys. Rev. B 61, 1160 (2000); Y.- C. Lin et al., Prog. Theor. Phys. Suppl. 138, 479 (2000). [12] G. Refael and J.E. Moore, Phys. Rev. Lett. 93, 260602 (2004). [13] G. Refael and J.E. Moore, Phys. Rev. B 76, 024419 (2007). [14] R. Santachiara, J. Stat. Mech. Theor. Exp. 2006, L06002 (2006). [15] N.E. Bonesteel and K. Yang, cond-mat/0612503. [16] A. Saguia et al., Phys. Rev. A 75, 052329 (2007). [17] F. Iglói, R. Juhász, and Z. Zimborás, Europhys. Lett. 79, 37001 (2007); R. Juhász and Z. Zimborás, J. Stat. Mech. Theor. Exp. 2007, P04004 (2007). [18] D.S. Fisher, Phys. Rev. B 50, 3799 (1994). D.S. Fisher, Phys. Rev. B 51, 6411 (1995). [19] F. Iglói and C. Monthus, Phys. Rep. 412, 277 (2005). [20] T. Senthil and S. Sachdev, Phys. Rev. Lett. 77, 5292 (1996). [21] F. Iglói, Y.-C. Lin, H. Rieger, and C. Monthus, Phys. Rev. B 76, 064421 (2007). [22] Y.-C. Lin et al, Phys. Rev. B 68, 024424 (2003); Y.- C. Lin et al, Phys. Rev. B 74, 024427 (2006). [23] F. Alet et al, cond-mat/0703027. http://arxiv.org/abs/quant-ph/0703044 http://arxiv.org/abs/cond-mat/0612503 http://arxiv.org/abs/cond-mat/0703027

0704.0419

Ultrasound attenuation of superfluid $^{3}$He in aerogel

Ultrasound Attenuation of Superfluid 3He in Aerogel H.C. Choi, N. Masuhara, B.H. Moon, P. Bhupathi, M.W. Meisel, and Y. Lee∗ Microkelvin Laboratory, Department of Physics, University of Florida, Gainesville, FL 32611-8440, USA N. Mulders Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA S. Higashitani, M. Miura, and K. Nagai Faculty of IAS, Hiroshima University, Kagamiyama 1-7-1, Higashi-Hiroshima 739-8521, Japan (Dated: October 27, 2018) We have performed longitudinal ultrasound (9.5 MHz) attenuation measurements in the B-phase of superfluid 3He in 98% porosity aerogel down to the zero temperature limit for a wide range of pressures at zero magnetic field. The absolute attenuation was determined by direct transmission of sound pulses. Compared to the bulk fluid, our results revealed a drastically different behavior in attenuation, which is consistent with theoretical accounts with gapless excitations and a collision drag effect. Liquid 3He has attracted intense interest for many decades in the field of low temperature physics [1]. In its normal state, liquid 3He has served as a paradigm for a Fermi liquid whose nature transcends 3He physics. The superfluid phases of 3He exhibit exotic and intrigu- ing features associated with the broken symmetries in the condensate, having an unconventional structure of the or- der parameter with spin triplet p-wave pairing. Liquid 3He is arguably the most well-understood system mainly because of its extreme intrinsic pureness at low temper- atures. Therefore, it has provided important insights in understanding other unconventional superconductors such as the high temperature superconductors, the heavy fermion superconductors, and in particular the more re- cently discovered Sr2RuO4, which is also thought to have the p-wave symmetry [2]. However, the same virtue has hampered the effort in pursuing answers to an important overarching question: how does the nature of a quantum condensate (spin triplet p-wave superfluid in this case) respond to increasing impurity or disorder? Observation of superfluid transitions in liquid 3He im- pregnated in high porosity aerogel in 1995 [3, 4] opened a novel path to introducing static disorder in liquid 3He. Aerogel possesses a unique structure, whose topology is at the antipode of widely studied porous media such as Vycor glass and metallic sinters. Due to its open struc- ture, there are no well-defined pores in aerogel and conse- quently, the liquid is in the proximity to the bulk. Ninety eight percent porosity aerogel, which has been used in most of the studies including this work, offers a corre- lated network of strand-like aggregates of SiO2 molecules whose structure can be characterized by the geometrical mean free path (ℓ ≃ 100 - 200 nm), the diameter of strand (r ≈ 3 nm), and the average inter-strand distance (d ≃ 25 - 40 nm). The coherence length of pure superfluid 3He, ξ0, which varies from 20 nm (34 bar) to 80 nm (0 bar), is at least an order of magnitude larger than the strand diameter but is comparable to ℓ and d. As a result, the scattering off the aerogel strand would have a significant influence on the superfluid. It is now well established that the superfluid transition temperature is significantly depressed from that of the bulk, and the effect of pair- breaking is progressively magnified at lower pressures, leading to the possibility of a quantum phase transition at Pc ≈ 6 bars [5]. To date, three distinct superfluid phases have been experimentally identified, namely the A-like, B-like, and A1-like phases [4, 6, 7, 8, 9]. The B-like phase and the A1-like phase in aerogel show striking sim- ilarity to their counterparts in the bulk superfluid [9, 10]. Detailed NMR studies [7, 8, 10] suggest that the aerogel B-phase has the same order parameter structure as the bulk B-phase. The aerogel A1-phase only appears in the presence of magnetic field as is the case in the bulk [9]. However, the aerogel A-phase exhibits quite a different behavior from the bulk A-phase (e.g. in NMR frequency shift and superfluid density), although the overwhelming experimental evidence suggests that it is an equal spin pairing state. Various interpretations or novel proposi- tions on the possible order parameter structure have been suggested for this phase [11, 12, 13]. Nuclear magnetic resonance and ultrasound spec- troscopy have been used in concert to investigate the mi- croscopic structure of the superfluid phases [1, 14]. These two experimental methods encompass complementary in- formation on the orbital (ultrasound) and spin (NMR) structure of the Cooper pairs. Rich spectra of order parameter collective modes in bulk superfluids, which are the fingerprints of specific broken symmetries in the system, have been mapped by ultrasound spectroscopic techniques [14]. In 2000, Nomura et al. [15] performed ul- trasound attenuation measurements on 98% aerogel us- ing a 16.5 MHz cw acoustic impedance technique. Their work was limited to a single pressure at 16 bars and down to 0.6 mK. Although their technique was not adequate in determining absolute attenuation, they managed to ex- tract the absolute sound attenuation after making auxil- http://arxiv.org/abs/0704.0419v1 Time (µs) FIG. 1: Acoustic response from the receiver vs. time at 34 bars for select temperatures ranging from 0.3 mK to 2.5 mK. The aerogel superfluid transition is marked by a small arrow. iary assumptions. A Bayreuth group [16] performed ab- solute sound attenuation measurements in aerogel (97% porosity) using a direct sound transmission technique at 10 MHz. They experienced poor transducer response, and observed self-heating and no depression in the aero- gel superfluid transition. We conducted high frequency sound transmission experiments in 98% porosity aero- gel, covering the whole phase diagram of the superfluid phases in aerogel, from 8 to 34 bars and from the transi- tion temperatures to as low as 200 µK. In this experiment, two matched LiNbO3 longitudi- nal sound transducers with the fundamental resonance at 9.5 MHz were used as a transmitter and a receiver. The 6.3 mm diameter transducers were separated by a Ma- cor spacer maintaining a 3.05 (± 0.02) mm sound path between the transducers where the aerogel sample was grown in situ. This scheme ensures the best contact be- tween the transducer surface and the aerogel, which is crucial for clean sound transmission at the boundaries. A 1 µs pulse was generated by the transmitter and de- tected by the receiver. Temperature was determined by a melting pressure thermometer (MPT) for T ≥ 1 mK and a Pt NMR thermometer for T ≤ 1 mK which was calibrated against the MPT. No non-linear response or self-heating was observed at the excitation level used in this work. All the data presented here, except for 8 bars, were taken while warming with a typical warming rate of 3 µK/min. A detailed description on the experimen- tal cell and experimental techniques can be found else- where [17, 18]. The temporal responses of the receiver taken at 34 bars are shown in Fig. 1 for select temperatures ranging from 0.3 to 2.5 mK. The primary response, which starts to rise around 8 µs, shows a rather broad response due to ring- ing of the high Q transducer (Q ∼ 103). The step-like structure of the receiver signal is caused by the slight mis- match in the spectra of the transducers [18]. Below the aerogel superfluid transition (marked around 2.1 mK by an arrow in Fig. 1) the primary response starts to grow and the trailing echoes emerge from the background, as the sound attenuation decreases in the superfluid. No change in the receiver signal was observed at the bulk superfluid transition. The multiple echoes follow a bona fide exponential decay in time. Absolute sound attenua- tion was obtained in the following manner [19]. First, the relative attenuation at each temperature was calculated using the area under the primary response curve by inte- grating the signal from the rising edge to a fixed point in time (23 µs point). The absolute attenuation at 0.4 mK and 29 bars, obtained using the primary signal and the echoes, was used as a reference point in converting the relative attenuation into the absolute attenuation. Due to a drastic mismatch in the acoustic impedance at the the transducer-aerogel/3He boundary, the signal absorb- tion at the surface of transducers was ignored [19]. The possible background contributions to attenuation from the quasi-particle scattering off the cavity wall [20] and the non-parallel alignment of the two transducers are es- timated to be negligible. The absolute attenuations on warming for several pres- sures are plotted as a function of temperature in Fig. 2(a). The superfluid transition is marked by the smooth drop in attenuation. Our aerogel superfluid transition tem- peratures are in excellent agreement with the previously reported values for all pressures [5, 21]. At 9.5 MHz in the bulk B-phase, a strong attenuation peak appears right below the superfluid transition. This peak is the result of the combined contributions from pair-breaking and cou- pling to the order parameter collective modes. Above the polycritical pressure, the B to A transition on warming is registered as a sharp step in attenuation. In aero- gel, none of these features exist. However, we did ob- serve a sharp step in attenuation on cooling for P > 14 bars, which implies the existence of the supercooled A- phase [19]. We were able to identify a rather smooth B to A transition on warming for 29 and 34 bars within ≈ 150 µK below the superfluid transition. This observation is consistent with the previous results obtained using a transverse acoustic impedance technique [13]. Therefore, most of the attenuation data presented here are in the aerogel B-phase. In the bulk B-phase with a clean gap, the attenuation follows α ∝ e−∆(T )/kBT below the atten- uation peak, practically reaching zero attenuation below T/Tc ≈ 0.6, due to thermally activated quasi-particles, where ∆(T ) is the temperature dependent gap and kB is the Boltzmann constant. In contrast, the attenuation in aerogel decreases rather slowly with temperature and remains high even at T/Tc ≈ 0.2. Furthermore, a pe- culiar shoulder feature appears at T/Tc ≈ 0.6 for higher pressures. This feature weakens gradually and eventually disappears at lower pressures, Fig. 2(a). Sound propagation for higher harmonics up to 96 MHz was measured for several temperatures and pressures, but no evidence of sound propagation was found above 30 MHz even at 0.3 mK, where the lowest attenuation is expected. Below about 10 mK, the scattering process is dominated by the temperature independent impurity scattering off the aerogel, and at 9.5 MHz, ωτi ∼ 0.1 for all pressures where τi = ℓ/vf (see below for ℓ). There- fore, the sound mode should remain in the hydrodynamic limit. This claim is bolstered by the observation of the strong frequency dependence in attenuation and the ab- sence of a temperature dependence in the normal fluid attenuation [15]. The coupling between the normal com- ponent of the superfluid 3He and the mass of the elas- tic aerogel modifies the conventional two-fluid hydrody- namic equations [22, 23]. This consideration leads to two (slow and fast) longitudinal sound modes with different sound speeds, cs = ca ρsρa/ρ, and cf = c1 1+ρaρs/ρnρ 1+ρa/ρn Here, cf(s) represents the speed of the fast (slow) mode, ρn(s) is the normal fluid (superfluid) density (ρ = ρn+ρs), ρa is the aerogel density, c1 is the speed of hydrodynamic sound in 3He, and finally ca is the sound speed of the bare aerogel. From the time of flight measurements, we found the sound speed in aerogel consistently lower (by ≈ 20%) than c1 for all pressures studied and in good agreement with the values obtained using the expression above [24]. Detailed analysis of sound velocity for various pressures will be presented in a separate publication. Low mass density and the compliant nature of aero- gel necessitate the consideration of effective momentum transfer upon quasi-particle scattering off the aerogel, which generates dragged motion of aerogel. Ichikawa et al. [25] incorporated the collision drag effect in calculat- ing the dispersion relation in the normal fluid. Their model offered a successful explanation for the experimen- tal results of the Northwestern group [15]. Recently, Hi- gashitani et al. [26, 27] extended this model to study the longitudinal sound (fast mode) propagation in superfluid 3He/aerogel within the framework of the two-fluid model. The drag effect can be described phenomenologically by a frictional force, ~Fd = (~vn −~va), introducing an addi- tional relaxation time τf , where ~vn(a) is the normal fluid component (aerogel) velocity. This effect is of particu- lar importance when ωτi < 1, and the total attenuation (Eq. (130) of ref. [27]) is ω2/2cf 1 + ρaρs/ρnρ ρ2aτf/ρρn 1 + ρa/ρn 4η/3ρc21 1 + ρaρs/ρnρ ), (1) where η is the shear viscosity of liquid 3He. The first term (αf ) arises from the frictional damping caused by the aerogel motion relative to the normal fluid compo- nent, and the second term (αv) from the conventional hydrodynamic sound damping associated with the viscos- ity. This expression allows us to extract ℓ in this system from our absolute attenuation at the transition temper- ature, αc. The inset of Fig. 3 shows our results of αc for 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 10 bar 14 bar 20 bar 25 bar 33 bar 34 bar T / T 10 bar 12 bar 14 bar 21 bar 25 bar 34 bar T (mK) FIG. 2: (a) Absolute attenuation for various pressures vs. temperature (color in on-line version). Thin solid lines are the results of a quadratic fit to the low temperature part (T/Tc <∼ 0.4) of the data at each pressure. (b) Normalized sound attenuation vs. normalized temperature. The results of theoretical calculation (solid lines, color in on-line version) are plotted along with the experimental results at 34 bars for comparison. various pressures. The solid lines are the result of calcu- lation using Eq. (1) for three different mean free paths, ℓ = 100, 120, and 140 nm. As can be seen, ℓ = 120 nm produces an excellent fit to our data for the whole pres- sure range, which is in good agreement with the val- ues obtained from the thermal conductivity (90 nm) [28] and spin diffusion (130 nm) [29] measurements. With the knowledge of the mean free path, one can calculate the full temperature dependence of sound attenuation in the superfluid phase. The results of the calculation (in the unitary limit) following the prescription described in ref. [27] are displayed in Fig. 2(b) along with the experi- mental results at 34 bars. The calculation reproduces all the important features observed in our measurements. In particular, the conspicuous shoulder structure appearing near T/Tc ≈ 0.6 at 33 bars softens at lower pressures and is completely absorbed in an almost linear temperature dependence below 20 bars. This behavior is the charac- teristic of αf [27]. A fast decrease in ρn right below Tc produces the bump in αf , and αf → 0 as T → 0. On the other hand, αv decreases monotonically and reaches a finite value due to non-zero ρn and the impurity states 0 5 10 15 20 25 30 35 0 10 20 30 P (bar) 100 nm 120 nm 140 nm P (bar) FIG. 3: Normalized zero temperature attenuation vs. pres- sure. The dashed line is a guide for eye. Inset: Pressure dependence of sound attenuation at Tc. The solid lines (color on-line) are the results of theoretical fit for ℓ = 100, 120, and 140 nm (see text). induced inside the gap as T → 0. The quantitative agree- ment between the theory and experiment, however, is not yet satisfactory. The calculation utilizes the isotropic ho- mogeneous scattering model (IHSM) [30], which tends to overestimate ∆(T ) and ρs compared to the experimen- tally determined values [3, 23]. As shown in ref. [31], the inhomogeneity gives rise to the reduction of the aver- age value of the order parameter and consequently yields larger η and ρn, which in turn increases α0 but decreases the frictional contribution. It is also expected that the non s-wave scattering components make non-trivial con- tributions to the viscous and frictional relaxation times in a direction that improves the quantitative agreement. Theoretical calculations based on the IHSM [27, 32] predict that the impurity states would completely fill the gap, leading to a gapless superfluid when τiTc < 1 for the B-phase in the unitary limit. We estimate 0.3 < τiTc < 1 for 10 < P < 34 bars with ℓ = 120 nm. The normalized zero temperature attenuation (α0/αc) obtained by ex- trapolating the low temperature part of the attenuation (solid lines in Fig. 2(a)) is plotted in Fig. 3, where α0/αc increases as the sample pressure is reduced and seems to approach unity near Pc ≈ 6 bars. Since the viscosity ratio is directly related to the density of states at zero energy through η(0)/η(Tc) = n(0) z, z = {2,4} for the {Born, unitary} limit where n(0) is the normalized den- sity of states at zero energy [27], the finite α0/αc is strong evidence of a finite n(0). The gapless behavior has been experimentally suggested by recent thermal conductivity (for P ≤ 10 bars) [28] and heat capacity (for 11 ≤ P ≤ 29 bars) [33] measurements. The pressure dependence of α0/αc is in qualitative agreement with the combined re- sults of Fisher et al. and Choi et al. Although all of these experimental techniques (including ours) are lim- ited to probe the impurity states near the Fermi level, the behavior is consistent with the theoretical predictions with gapless excitations. Unlike the thermodynamic and transport measurements, the high frequency ultrasound measurement has a potential to unveil a larger portion of the impurity states profile from the frequency depen- dence. We acknowledge support from an Alfred P. Sloan Re- search Fellowship (YL), NSF grants DMR-0239483 (YL), DMR-0305371 (MWM), and a Grant-in-Aid for Scientific Research on Priority Areas (No. 17071009) from MEXT of Japan (SH and KN). We would like to thank J.-H. Park for his technical assistance, and Jim Sauls, Peter Wölfle, and Bill Halperin for useful discussions. ∗ [email protected] [1] D. Vollhardt and P. Wölfle, The Superfluid Phases of Helium Three, (Taylor and Francis, London, 1990). [2] A.P. Mackenzie and Y. Maeno, Rev. Mod. Phys. 75, 657 (2003). [3] J.V. Porto and J.M. Parpia, Phys. Rev. Lett. 74, 4667 (1995). [4] D. T. Sprague et al., Phys. Rev. Lett. 75, 661 (1995). [5] K. Matsumoto et al., Phys. Rev. Lett. 79, 253 (1997). [6] D.T. Sprague et al., Phys. Rev. Lett. 77, 4568 (1996). [7] H. Alles et al., Phys. Rev. Lett. 83, 1367 (1999). [8] B.I. Barker et al., Phys. Rev. Lett. 85, 2140 (2000). [9] H.C. Choi et al ., Phys. Rev. Lett. 93, 145302 (2004). [10] V.V. Dmitriev et al., JETP Lett. 76, 312 (2002); V.V. Dmitriev et al., Physica B 329, 324 (2003). [11] G.E. Volovik, JETP Lett. 63, 301 (1996). [12] I.A. Fomin, J. Low Temp. Phys. 134, 769 (2004). [13] C. L. Vicente et al., Phys. Rev. B 72, 094519 (2005). [14] W.P. Halperin and E. Varoquaux, in Helium Three, ed. by W.P. Halperin and L.P. Pitaevski (Elsevier, Amster- dam, 1990). [15] R. Nomura et al., Phys. Rev. Lett. 85, 4325 (2000). [16] L. Hristakos, Ph.D. thesis, University of Bayreuth (2001). [17] H.C. Choi et al., will appear in J. Low Temp. Phys. [18] H.C. Choi, Ph.D. thesis, University of Florida (2007). [19] Y. Lee et al., will appear in J. Low Temp. Phys. [20] G. Eska et al., Phys. Rev. B 27, 5534 (1983). [21] G. Gervais et al., Phys. Rev. B 66, 054528 (2002). [22] M.J. McKenna, T. Slawecki, and J.D. Maynard, Phys. Rev. Lett. 66, 1878 (1991). [23] A. Golov et al. Phys. Rev. Lett. 82, 3492 (1999). [24] For example, c = 350 (± 10) m/s at 34 bars from our measurement, and cf = 370 m/s. [25] T. Ichikawa et al., J. Phys. Soc. Jpn. 70, 3483 (2001). [26] M. Miura et al., J. Low Temp. Phys. 134, 843 (2004). [27] S. Higashitani et al., Phys. Rev. B 71, 134508 (2005). [28] S. N. Fisher et al., Phys. Rev. Lett. 91, 105303 (2003). [29] J.A. Sauls et al., Phys. Rev. B 72, 024507 (2005). [30] E. V. Thuneberg et al., Phys. Rev. Lett. 80, 2861 (1998). [31] R. Hänninen and E.V. Thuneberg, Phys. Rev. B 67, 214507 (2003). [32] P. Sharma and J.A. Sauls, Physica B 329-333, 313 (2003). [33] H. Choi et al., Phys. Rev. Lett. 93, 145301 (2004). mailto:[email protected]

0704.0420

The Hourglass - Consequences of Pure Hamiltonian Evolution of a Radiating System

The Hourglass—Consequences of Pure Hamiltonian Evolution of a Radiating System Donald McCartor ABSTRACT Hourglass is the name given here to a formal isolated quantum system that can radiate. Starting from a time when it defines the system it represents clearly and no radiation is present, it is given straightforward Hamiltonian evolution. The question of what significance hourglasses have is raised, and this question is proposed to be more consequential than the measurement problem. 1 Hourglasses 2 Physics without true histories 3 But histories are sometimes good 4 Phlogiston and oxygen 5 A closer look at quantum engineering 6 Conclusion But I want to know the particular go of it – the plea of James Clerk Maxwell as a young child concerning, among many things, the bell-wires that ring the bells that summon servants. [Mahon] 1 Hourglasses Suppose that theory develops in such a way that quantum fields can be handled like nonrelativistic quantum mechanics. Then if we are interested in something, perhaps gooseberry bushes, we can model one as we would conceive it to be at some instant and then follow its development through time. And not only the atoms and molecules would be modeled, but also the radiation. This is a scheme for the imagination. The gooseberry bush, though not isolated, would grow within a suitable environment that would be an isolated system, complete in itself. We do learn well from isolated systems, both real ones in the laboratory and those envisaged in our theoretical musings. We will provide the bush with air, earth, and water. And there can be life-giving sunlight shining on it. As for the light that had been reflected or emitted from the bush before the present time, we will leave that out. Such light goes off and away, so it could only matter as information about what the http://arxiv.org/abs/0704.0420v1 bush had been doing. We will take the bush just as it is now. The gooseberry bush is then developed forward in time. Lagrange or Hamilton would have recognized what we are doing, for we are doing physics the classical way. We have an initial condition and we are finding out what will happen next. As we move toward the future, light shoots out from the bush, as we expect. But, disconcertingly, the bush starts to lose definition. Its parts lose their precise places. Within a few weeks it is a scarcely recognizable mess. Let’s go back in time, then. This is terrifyingly worse. The bush has been the subject of a vast conspiracy. Light has been streaming in on it from the entire universe. The bush swallows it up. Then at the present time this suddenly all stops. Time symmetry of the Hamiltonian makes it happen like that. This is the hourglass. It is really more like a cone, with the light streaming in before the set-up time forming one nappe and the light streaming out after it the other. But hourglass is a more colorful name. What to do? We will try to bring quantum mechanics to the rescue. We will make what is conventionally called a measurement, but cautiously. A place is chosen well outside the gooseberry bush, and a time chosen that is later than when we set the state of the bush up. A check is made of whether there are at this place and time any photons coming from the direction of the bush. By doing things this way, we won’t disturb the bush at all, and we don’t care if we disturb the escaping light. We get from this, of course, a probability distribution over various possibilities for photons at this place and time. Encouraged by this small success, we choose another place and time and do the same. And this is what is nice: the two measurements are compatible. Thus we get correlations between them, too. Emboldened by this opportunity, we do millions of them, which all formally combine into a single measurement with a single set of possible results. Each possible result of the single, combined measurement is a combination of results of all the individual measurements of light made at the various times and places. Thus each combined result constitutes a kind of movie of the gooseberry bush. What will the most probable of these results be like? This is the problem of the hourglass. To begin with, however, it may be that there is no hourglass. The deepest quantum theory might not provide a system with a state and its evolution. Or if it does, it could still be objected that the Hamiltonian evolution should not have been allowed to run on unchecked. There should have been many quantum jumps. By leaving them out, quantum mechanics has been misused, and what results is no matter. But Lagrange and Hamilton and would have been best pleased if these objections did not hold. And surely we would then hope to see in each of the most probable results something like a movie of a bush producing gooseberries: physics working right. The bushes in these movies would look much alike at the start but then gradually differ, as chance has it. We would learn something about how gooseberry bushes grow gooseberries! Certainly Lagrange and Hamilton would have thought the problem of the hourglass a leading one, if they had known of quantum mechanics. Indeed, every physicist might like to take a stab at guessing its solution, just to orient themselves in their science. Does the hourglass fail, and if so, where and why? Or if it does produce movies true to our world, but not from a developing quantum state that might be the true history of a gooseberry bush, rather from a “history” that does at one time represent a gooseberry bush well, but soon is unlike anything that ever did exist, then how can this be? 2 Physics without true histories Here is what I think about it. But before we go into that, see if you don’t agree that the hourglass question has gravity, and this regardless of the ideas that I or anyone might have for its answer. Now my guess is that quantum mechanics will give us movies of ripening gooseberries, produced by hourglasses through the means described or some- thing rather like that. And I think that to understand hourglasses, not to solve the measurement problem, is the central question for the understanding of quantum mechanics. For the measurement problem begs a question, which makes it futile. It assumes that we learn from physics simply because physics describes well those things that exist. Like this example from classical physics. There exist in a gas a multitude of zipping molecules. At any given moment, each particle has its particular position and momentum, and over time this forms their history. Physics has told us what a gas is—precisely what exists there. This is what lets us learn about gases. Undoubtedly this is how Boltzmann saw it. But when we look at the statistical mechanics he produced, and even more at that of Gibbs, a person will acquire deep qualms about this view- point. Boltzmann’s analysis of the collision of molecules seems like straightfor- ward common sense. He is looking at what they are likely to do. But when Loschmidt’s reversibility objection is brought forward, the lucidity vanishes. Gibbs’s more abstract statistical mechanics made the problem even starker. Gibbs found beautiful mathematical form in Boltzmann’s (and Maxwell’s) work, which he generalized. He held that thermodynamic systems should be represented as being in states that have the form of certain probability dis- tributions over classical states. Gibbs could not well understand what these probabilities were about, but he saw that his theory was good nevertheless. To keep this lack of clear comprehension from poisoning work with the theory, he devised a work-around. The axioms of probability theory are reflected in the axioms of finite set theory. One can effectively solve problems of probability by thinking about finite sets. So Gibbs suggested that we simply think about these probabilities in terms of sets. The word he used was ensembles. Gibbs described his intent in these words: “The application of this prin- ciple is not limited to cases in which there is a formal and explicit reference to an ensemble of systems. Yet the conception of such an ensemble may serve to give precision to notions of probability. It is in fact customary in the discus- sion of probabilities to describe anything which is imperfectly known as some- thing taken at random from a great number of things which are completely described.”[Gibbs] But physicists have never been able to accept gracefully that they don’t understand the elements of their science. So they have been moved to think that they do understand Gibbs’s probabilities somehow, and this has led to two missteps. One has been to regard the probabilities in Gibbs’s theory as being the result of our ignorance of the detailed state of the system we are considering. But when a probability distribution is useful, this is a very great step up in order from chaos. Ignorance cannot create order. If water always boils at the same temperature, it is not our fault. Rather than being so explained, for it is not, Gibbs’s theory shows that there is something deeply wrong with classi- cal mechanics. Classical statistical mechanics is not really a form of classical mechanics. It is quantum mechanics being born. The following words of Gibbs seem to show that Gibbs himself took the view just scotched. “The states of the bodies which we handle are certainly not known to us exactly. What we know about a body can generally be described most accurately and most simply by saying that it is one taken at random from a great number (ensemble) of bodies which are completely described.”[Gibbs] The impression that I get, though, is that Gibbs is cautiously hedging. He is not saying plainly, as he might have, that a body we handle will be in some completely described state, so that if we describe it with an ensemble, the probabilities in the ensemble simply represent our partial ignorance about that state. He does say plainly that his method seems to work. The other misstep has come about because quantum theory is a mirror of Gibbs’s statistical mechanics in the sense that it is based on what are prob- abilities in form (in other words, sets of non-negative real numbers that add up to one) and we don’t know what they mean in general. It is true that we can make good sense of them as real probabilities in various special cases. For instance, when quantum mechanics is applied to the Stern-Gerlach experiment, to see the detector react is like seeing a coin tossed. But in the general case no such kind of experience is directly implied by these probability forms. There are, for example, canonical distributions in quantum mechanics too, and we don’t ever expect to see a detector pick a pure state out of a hot cup of coffee. We then sometimes think about these formal probabilities in terms of ensembles, just as Gibbs did, and for the same reason. Where the formal probabilities are highest and the members of the ensemble most numerous, there the greatest significance will lie, whatever it may be. This is fine. But quite often physicists say that ensembles (that is to say, Gibbs’s work-around) provide the means to understand quantum theory. This is clearly wrong. But to get back to the measurement problem. As you well know, but for explicitness I will say it anyway, to see a problem in measurement is to suppose that quantum mechanics can describe the equipment in the lab as it exists at the start of an experiment, but when the representation is continued, the equipment becomes entangled with the microscopic systems it is examining and gets smeared. Then quantum mechanics has stopped describing what we know exists in the lab and needs to be corrected so that it will continue to describe what exists. But it isn’t so that quantum mechanics, if it is to show us some pre- dictability in nature, must provide us directly with histories of the existence of things, as by a developing wave packet. As evidence, I offer the hourglass. 3 But histories are sometimes good If physics does not work simply because it describes what exists, and if, rather, the way of the hourglass is right, then a corollary is that how we learn about nature necessarily becomes more indirect. We are given such information as radiation provides about something, not directly told what exists there. And for the purpose of inferring useful rules of nature’s behavior, what we deal with are imagined situations that we think typical of what we want to learn about, not faithful descriptions of actual things. No real radiating system is like an hourglass, except momentarily near the hourglass’s neck. But quantum engineering may temper the truth of that judgment just a bit. For there is also an engineering use of quantum mechanics where, somewhat as classical mechanics does it, for a time we can use a wave packet to represent the development of an actual situation we are dealing with. But this is rather more special, for we must take care to set things up so that this will work. The vacuum must be excellent, etc. Isolation is important. A simple example of quantum engineering is an ion that alternately blinks for a spell and remains dark for a spell while sitting in an ion trap that is irradi- ated by lasers. You can picture the ion well enough by thinking of Schrödinger evolution of a wave packet with occasional quantum jumps interspersed. You might then be tempted to think that everything can be handled effectively in the same way, at least in principle. We have just not been clever enough to find New York City’s wave packet and its measurement collapses. This is trouble. The worst of it is that you will be led to ignore hourglasses and what they imply, since clearly hourglasses cannot represent the history of things in the same manner that you have advantageously represented the history of the blinking ion. On the other hand, imagine that decades ago physicists had taken hour- glasses to their hearts, as well I think they might have. Then they could have been tempted to look upon representing an ion in a trap by Schrödinger evolu- tion of a wave packet with quantum jumps as ‘following the wrong philosophy’ (by trying to represent the actual histories of things with wave packets), and might have disdained to do so. There is a lesson here. Don’t take your philo- sophical ideas too seriously, we’re not good enough for that. I believe, though, that from hourglasses you would be able to infer that Schrödinger evolution with jumps is a simple and effective (not perfect) way to regard a blinking ion in a trap. The hourglasses would then be in this sense the more fundamental theory. 4 Phlogiston and oxygen But what is a quantum jump? Here is where I think the community of physi- cists has been careless in the use of words, perhaps mixed with real misun- derstanding. Two principles of quantum physics have been formulated. The first principle (promoted by Dirac and von Neumann) is that when a measure- ment is made on a system, an immediately following measurement will give the same result. Therefore, right after any measurement the system must be in the eigenstate corresponding to the value found. The second principle is that if the probabilities of the possible results of all the measurements that may be made on a system are defined, then there will be a (unique) quantum state that the system may be said to be in that will yield these probabilities. Add to this that sometimes two measurements may be made on a system without interfering with each other. Then when one of the two measurements has a certain result this will define a conditional probability for any result of the other measurement (simply divide the probability that both results occur by the probability that this result of the first measurement occurs). According to the second principle, then, there will be a quantum state that yields these probabilities (for the possible results of any measurement that may be made without interfering with, or suffering interference from, a given measurement that has had a certain result). Please notice that the argument above assumes that the set of all the measurements compatible with a given measurement effectively constitutes ‘all the measurements that may be made on a system’ as needed by the second principle. Now consider a system A in the state α. It is composed of two subsystems, B and C, in reduced states β and γ respectively. A measurement is made on subsystem B and it has a result. By the first principle, there is a quantum state β′ that will yield the probabilities of the possible results of any immedi- ately following measurement that might be made on subsystem B. And by the second principle, there is a quantum state γ′ that will yield the probabilities of the possible results of any compatible measurement made on subsystem C. For the supplanting in one’s considerations of β by β′ there is the historical name ‘collapse of the wave packet’. For the supplanting in one’s considerations of α by γ′ most physicists use the same phrase (or any of its several synonyms). It would easier to think about these things if different names were used for the two. ‘Collapse of the wave packet’ might be retained for the first and, say, ‘conditioning of the wave packet’ adopted for the second. This is all the more important because the first principle is an out and out mistake by Dirac and von Neumann, whereas the second is an inalienable part of quantum mechanics. To those two mathematically minded, and so logically minded, people, the dignity of quantum mechanics required that there be measurements, so that quantum mechanics might be real physics. And since quantum mechanics did not say that a system had to have, before the measurement, the value found in the measurement, the dignity of measurement required that it at least have that value afterward, or what sort of measurement was this anyway? Tacked on to this was the fact that so distressed Schrödinger: wave pack- ets spread interminably. If a developing wave packet were to represent the history of a system, which they assumed to be necessary, then the spreading had to be checked, and an occasional quantum jump such as their measurement theory presupposed might do that. And experiment lent some support. Above all, if an electron went splat somewhere on a screen, which they regarded as a measurement by the experi- menter of the electron’s position, then conservation of charge suggested strongly that the electron could be found subsequently thereabout. This was the origin of the phrase ‘collapse of the wave packet’. Too, the famous Stern-Gerlach experiment allows a following measurement of spin, which will give the same result as the first if the first measurement’s detection has been delicate enough. But the idea of a quickly following measurement is just not well-defined in general. And there are cases where the principle must prove false under any reasonable definition of a following measurement. For example, a particle might lose most of its energy in those collisions that measured its energy. Or if the momentum of a charged particle were measured by the curvature of its path in a magnetic field, the particle might end up going in the wrong direction, although this is, to be sure, correctable. Those events called “measurements” are what they are, and if they fall short of truly being measurements of properties, so be it! If the first principle is an error, then that leaves us with only one principle, the second, and people might then be inclined to continue to use the traditional phrase ‘collapse of the wave packet’, but now meaning the replacements the second principle defines. This would result in the transfer, in the course of history, of the meaning of the phrase from the first principle to the second. I think that this would have the same unhappy effect as if Lavoisier, not wishing to burden the world with a neologism, had instead given to the word phlogiston a new sense. 5 A closer look at quantum engineering The second principle has a very different flavor from the first. For it leads to conditional probabilities, and these lend themselves to imaginative thinking. In this mind-set you are free to take up points of view according to what you wish to learn. The first principle, however, leads to probabilities that are thought to be the properties of real events, such as an actual toss of a coin. You are now in a reality mind-set. That probability is as much a part of the coin toss as is the silver of the coin, and you must deal with it. You have no choice. But I don’t mean to say that this is an absolute difference between the two principles. Rather, they tend to lead us into these respective modes of thought, and vice versa. Bearing this in mind, let us look at the hourglass and quantum engineering. First consider the hourglass that represents a gooseberry bush. By choos- ing one among the more probable of the results of the course of observation of light, we will select what is in effect a likely movie of such a bush. We can look at the movie, and the marvelous algorithms of our brains will construct an idea of a gooseberry bush and follow it through its history. We have gotten something good out of this, and we have made no use of the conditional proba- bilities offered by the second principle at all. However, if we are not limited to one movie then we can use conditional probabilities as they are normally used, to explore various interesting possibilities while taking into account how likely they are when we are supplied with certain information. Notice that we have been thinking imaginatively. No one would suppose that we have directly grasped the reality of a gooseberry bush in our garden in this way, particularly because real gooseberry bushes do not start to exist at a special time. Now consider quantum engineering. By means of careful construction of the equipment a clearly defined situation can be set up where the power of wave packets to give understanding will be enhanced. On the other hand, here there can be significant entanglement. The power of our minds to achieve understanding through their everyday methods will be set at nought. Then for quantum engineering, a history formed by wave packet develop- ment with occasional saltations may be a quite good route to understanding. We would take up this idea of what exists simply because it is good enough to help us with the job at hand. And for this case, where we find it fit to think that we are dealing with an actual system that is an evolving wave packet, and with saltations that we regard as actual events, but in a way so different from that intended by Dirac and von Neumann, then perhaps a third term, say, ‘change of the wave packet’, would be appropriate for the saltations. These changes of the wave packet would differ from collapses of the wave packet because, although they would be thought of as real events just as col- lapses have been, they would be derived from conditioning of wave packets, in the following manner. When a system sends out radiation (or anything else) that will not return, in one way you can consider the system of interest to be the whole, including the radiation, and in another way you can consider it to be the reduced system that does not include the radiation. Upon observation of the radiation you will derive from the result and from the wave packet of the whole system a wave packet for the system less the radiation, and this we have called conditioning of the wave packet. But if before the observation your interest had been focussed on the system less the radiation, and thus on its reduced wave packet, then you will have gone from one wave packet to another wave packet for the system less the radiation. And since you are reckoning these wave packets as being portions of the system’s history, this looks like a quantum jump. This is what is meant by a change of the wave packet. There is no need to define any such change of the wave packet precisely, of course. No more is there need to suppose that it can be defined with precision. 6 Conclusion If hourglasses cannot be true histories, how can it be that we can learn from them? What lets them tell us how gooseberry bushes grow, when they are only momentarily like a gooseberry bush? I haven’t said a word about this yet. First of all, there is an assumption hidden behind this puzzlement of ours. The assumption is that we have no reason to be perplexed that we can learn from things that can be true histories. For if it did not seem so perfectly natural to us that we learn from true histories, then it would not appear unnatural to learn from what clearly cannot be a true history. But I think this assumption of ours is thoughtless, and I will try to explain why. We make judgments about when we are better informed and when less so. The ideas we hold true when thought to be better informed are compared with those that we held when not so well informed. In this way, through the device of taking the ideas that we presently have most confidence in as trustworthy, we try to gather how successfully our ideas tend to stack up against reality. It is not quite so simple, however, since we know from sad experience that the ideas we now trust may fail us. But we have the conviction, or hope, that if such happens we can land on our feet again. We will search for still better ideas until we find something that works. We are apt to give to this situation a logical cast. Namely, by positing that there is a best of all possible ideas in whose direction we are headed. This posit can be helpful. It can give us greater confidence in our search for better ideas. If we guess that this best idea will have a certain form, and we guess well, it can guide our search. But there is no necessity for this posit; all we really know is what was said above. Another thing we like to do is to find where things are and when. Our vision, touch, and hearing do this automatically all the time, and we often give them some conscious help, say by turning the head. When we are a teenager it is likely to occur to us that there must be a best of all possible such ideas, a complete map of where everything is, and has been, and perhaps will be too. A further thought may cross one’s mind. Maybe this is all that our world is. For instance, if one person likes another, this should show up in that person’s actions, which the map will completely define. Maybe the liking simply is those actions. Now I will propose some physics, the red dust theory. According to this theory the world is made up of an exceedingly large number of very fine specks of a scarlet dust. Because of its ruddiness, the dust is extremely beautiful, if only we could see it, but we will not be concerned with that. The red dust theory differs from most physics in that the flight of the particles does not have to satisfy a differential equation, it is merely continuous. The interpretation of the theory is quite simple. Where we find things there will be a crowd of these specks, and where we find vacancy they will be much sparser. But can our world be as this theory says? Surely it can. There will be among its solutions one that maps the entire history of our universe with extraordinary precision. The collisions of galaxies, the evolution of whales, the experiments in laboratories, all will be there and rightly shown. Now you may think that the red dust theory is hopelessly bad physics and should be ignored. It may be hopelessly bad, but it should not be ignored. It is a benchmark. If another physics theory is proposed, is it better than the red dust theory, and if so, just why? This is especially pertinent if the other theory intends, as does the red dust theory, to give a precise description of all that exists. Bohmian quantum mechanics is an example. But what I intend to put up against the benchmark is classical mechanics. Everyone will agree that classical mechanics is far better than the red dust theory. You can do things with classical mechanics; you can’t do anything with the red dust theory. For instance, you can pull a pendulum to the side and let it go. It will swing. Classical mechanics can give you the history of that swing ahead of time. The red dust theory has so many solutions compatible with the way things are at the start that it won’t tell you anything useful about how things will go. Our experience with classical mechanics is that it is practical, but why is this so? The most natural idea is that the world must at bottom be clas- sical mechanical. Since we understood the pendulum by assigning a classical mechanical state to it and evolving the state, there must then be an evolving classical mechanical state that the whole world is in, and that would explain why classical mechanics is so useful. When we look at the history of our universe, however, and particularly at the evolution of life over billions of years, and when we consider the resources that it is likely that classical mechanics has to offer in its solutions, it doesn’t really seem possible that there is any classical mechanical history that would match our universe’s history, no matter how exquisitely the initial conditions are chosen. For the more detailed structures of the classical representation must in time dissolve into lasting chaos, and I would think rather quickly. Still, this does depend on a point I don’t actually know the answer to. For in order to make the universe behave as you wish, that is to say, give a good account of continents rifting and hummingbirds feeding, it might be that to obtain each additional second of the desired history it is always sufficient to correctly calculate another, say, thousand decimal places for the positions and momenta of the molecules in the initial state. Or to the contrary, the first thousand decimal places might give you one second, the next thousand only a further half second, then a fourth of a second, and so on. Yet even if I am wrong in this, we would just go from Scylla to Charybdis. For in that case classical mechanics must be like the red dust theory, where, from our point of view, anything is possible, or too close to anything. In either case the classical solution set would imply no structure such as we experience in life. No sculpted dunes, no ants carting morsels, no shower of hail would pop out of it. Nor can one imagine any reason why the solution set would show a preference for depicting creatures learning classical mechanics, or if so doing benefiting by it. In short, there is a total disconnect between the fact that classical mechanics is useful and the hypothesis that the universe as a whole is a classical mechanical system. That leaves us with an unsolved mystery: why does classical mechanics work for us? And classical mechanics is the archetype of the kind of physics where we learn from what can be true histories of things. To my mind, the hourglass with observation of its emitted light is deeply conservative physics. It makes quantum mechanics as seamless a continuation of the physics of the previous centuries as is at all possible. This is because of the mathematical form of the hourglass, which is a continuous development from initial conditions, as well as the form of the observations, which impinge as little as can be. And when this leads to our being given movies rather than direct histories, then I am surprised (and amused) by this, but accept it for the sake of the qualities mentioned, which I consider to be virtues that promise. Nature is teaching us another lesson. Bohr’s old quantum theory was based on quantum jumps, and I think this was a wonderful piece of exploration in the dark. When Heisenberg’s new quantum mechanics came along, quantum jumps were kept. The jumps would allow direct histories to be retained as the foundation of our physics, though at the expense of the continuous Hamiltonian evolution of the wave packets (and at the expense of clear definition, for no one has ever been able to specify just when and where and what the quantum jumps are). Like Schrödinger, I am jarred by this. If we are given the choice of preserving philosophical principle or mathematical form, I think we should prefer mathematical form. Isn’t this what Copernicus did? A final thought: If learning from the movies provided by hourglasses is how we do physics, then to know why quantum mechanics works would be to know why all the inferences we might make from the movies will fit together with sufficient coherence. But to know this would require that we know all the things we might ever think of. It’s hopeless. Though we might nibble at the problem, by showing that the hourglasses have some needed characteristics. So I think the hourglasses will leave us with an essentially unfathomable mystery. References Gibbs, J. Willard [1981]: Elementary Principles in Statistical Mechanics , Wood- bridge, CT: Ox Bow Press, p. 17 and p. 163. Mahon, Basil [2003]: The Man Who Changed Everything, Chichester, UK: John Wiley & Sons Ltd. The hourglasses suggest that von Neumann’s measurement theory should be recast for imaginative use rather than for the description of actual situations. This gives one extra freedom in setting it up, and it can then work more effectively. An outline is here: McCartor, Donald [2004]: ‘Quantum Thought Experiments Can Define Na- ture’, Concepts of Physics, Vol. I, no. 1, pp. 105–150 and quant-ph 0702192. [email protected]

0704.0421

The Sigma-D Relation for Planetary Nebulae: Preliminary Analysis

Serb. Astron. J. } 174 (2007), 73 - 76 Preliminary report THE Σ − D RELATION FOR PLANETARY NEBULAE: PRELIMINARY ANALYSIS D. Urošević1, B. Vukotić2, B. Arbutina1,2 and D. Ilić1 1Department of Astronomy, Faculty of Mathematics, University of Belgrade Studentski trg 16, 11000 Belgrade, Serbia 2Astronomical Observatory, Volgina 7, 11160 Belgrade 74, Serbia (Received: February 22, 2007; Accepted: March 30, 2007) SUMMARY: An analysis of the relation between radio surface brightness and diameter, so-called Σ−D relation, for planetary nebulae (PNe) is presented: i) the theoretical Σ − D relation for the evolution of bremsstrahlung surface brightness is derived; ii) contrary to the results obtained earlier for the Galactic supernova remnant (SNR) samples, our results show that the updated sample of Galactic PNe does not severely suffer from volume selection effect - Malmquist bias (same as for the extragalactic SNR samples) and; iii) we conclude that the empirical Σ −D relation for PNe derived in this paper is not useful for valid determination of distances for all observed PNe with unknown distances. Key words. planetary nebulae: general – Radio continuum: ISM – Methods: ana- lytical – Methods: statistical 1. INTRODUCTION The relation between radio surface bright- nesses and diameters of supernova remnants (SNRs), the so-called Σ−D relation, has been subject of the extensive discussions in the last more than fourty years. Due to improvements of the observational techniques (radio-interferometers), the several hun- dreds planetary nebulae (PNe) were resolved in the last two decades at radio frequencies, but the Σ−D relation for PNe was not discussed until now. By using radio data, some statistical methods were es- tablished in order to determine distances to PNe. The main method was related to the correlation be- tween radius of PNe and brightness temperature – R − Tb relation (Van de Steene and Zijlstra 1995, Zhang 1995, Phillips 2002). The different samples of Galactic PNe with known distances were defined in these papers. All the obtained empirical R − Tb relations were used for determination of distances to PNe for which the independent distances (in order of R− Tb dependence) were not obtained earlier. The samples of Galactic PNe are better for statistical analysis than the samples of Galactic SNRs. The selection effects should be smaller in the case of PN samples. However, the selection ef- fects surely influence the Galactic PN samples and the statistical determination of distances to Galactic PNe has to be highly uncertain. The main objectives of this paper are the fol- lowing: i) to derive a simple form of the theoretical Σ−D relation for PNe by analyzing the evolution of radio bremsstrahlung surface brightness, ii) to discuss whether the updated sample of radio PNe is affected by the selection effects, and, iii) to check whether the Σ−D relation is valid for determination of distances to PNe. http://arxiv.org/abs/0704.0421v3 D. UROŠEVIĆ et al. 2. ANALYSIS AND RESULTS 2.1. Theoretical Σ−D relation for PNe The thermal bremsstrahlung mechanism is re- sponsible for radiation of HII regions at radio wave- lengths. The bremsstrahlung volume emissivity εν of a PN can be shown to be (Rohlfs and Wilson 1996): εν [ergs s −1 cm−3 Hz−1] ∝ n2T−1/2, (1) where n is the volume density and T is the thermo- dynamic temperature of interstellar medium (ISM). The surface brightness can be expressed as: Σν ∝ ενD, (2) where D is the diameter of PN. Combining Eqs. (1) and (2), we obtain: Σν ∝ n D. (3) Our next step is to express dependance of n and T on D. For a constant velocity mass flow the den- sity distribution is ̺ = Ṁ 4πr2v , i.e. n ∝ D−x, where x = 2. Moreover, for the isothermal envelope with a power-law electron density distribution there is re- lationship between the shape of the density distri- bution and the power-law index of the radio contin- uum spectra (see Gruenwald and Aleman 2007, and references therein). Supposing that n ∝ D−2 and T=const. (HII regions are approximately isothermal at T ∼ 104 K), we obtain the simplest form of the theoretical Σ−D relation for PNe: Σν ∝ D −3. (4) This is a standard power-law form of the Σ − D relation which can be written in general form as Σ = AD−β , that is the same as in the case of SNRs. It is possible that x in density distribution is slightly higher, x & 2, and that the temperature is not strictly constant throughout the nebula. We can expect to see temperature gradients in PNe arising from radiation hardening. More energetic photons will travel further and when they are absorbed by the PN they will impart greater kinetic energy to the ions thereby producing a higher temperature. Using the numerical model results given by Evans and Dopita (1985), we calculate the dependence between logTe and logD and find the low slope (≈ 0.1). Therefore, this only slightly changes the slope of the theoretical Σ−D relation. The value β = 3 is then a theoretical lower limit, and the Σ − D relation could only be steeper, as one can see from Eq. (3). 2.2. The empirical Σ−D relation for PNe The most important prerequisite for deriving a proper empirical Σ − D relation is defining of a representative sample of PNe. The distances to the calibrators have to be determined by accurate meth- ods, e.g. trigonometric or spectroscopic parallaxes of central stars in PNe, or by a method that uses the expansion of nebulae. On the other hand, all sam- ples suffer from the severe selection effects that arise from limitation in sensitivity and resolution, but the most severe selection effect for the Galactic samples of PNe is Malmquist bias; i.e. intrinsically bright PNe are favored because they are sampled from a larger spatial volume compared to any given flux lim- ited survey. The result is a bias against low surface brightness nebulae such as highly evolved old PNe. In this paper we use the updated sample of PNe at the distances less than 0.7 kpc collected by Phillips (2002). The influence of Malmquist bias in this sam- ple is limited because of the limitation in distances to PNe. In addition, we assume that the distances are accurately determined for this sample of relatively close PNe. The empirical Σ − D relation at 5 GHz for 44 calibrators with distances less than 0.7 kpc (Phillips 2002) has the form: Σ56Hz = 2.33 +0.88 −0.64 · 10 −2.07±0.19 . (5) The parameters A and β are calculated by least- squares fitting procedure with correlation coefficient −0.86. The corresponding Σν −D diagram is shown in Fig. 1. 0.01 0.1 1 10 D [pc] Fig. 1. The Σ−D diagram at 5 GHz for 44 Galac- tic PNe with distances less than 0.7 kpc. The form of Eq. (5) is very close to the so- called trivial Σ−D form with β = 2 (for details see Arbutina et al. 2004). The additional test in order to estimate the validity of Eq. (5) pertains to the possi- ble dependence between the luminosity and diameter of PNe. The Lν−D diagram is shown in Fig. 2. The scatter in Lν − D plane shows that the correlation between Lν and D is poor (correlation coefficient = - 0.06) and therefore the physical dependence between L and D could not be confirmed by this statistical procedure. THE Σ−D RELATION FOR PLANETARY NEBULAE 0.01 0.1 1 10 D [pc] Fig. 2. The L −D plot at 5 GHz for 44 Galactic PNe with distances less than 0.7 kpc. 3. DISCUSSION The theoretical Σν −D relation (Eq. (4)) for PNe, derived in this paper, describes a trend of de- creasing radio surface brightness with increasing di- ameter of an object. The radiation mechanism used in this simple derivation is thermal bremsstrahlung. This is the basic process of production of the radio radiation in HII regions. The theoretically derived slope (β = 3) is steeper than the slope from the em- pirical relation given by Eq. (5). This discrepancy can be explained by the low quality of the sample of Galactic PNe or by the assumptions used in deriva- tion of theoretical relation. Due to small variation in power-law density distribution with x & 2 (Gru- enwald and Aleman 2007, and references therein) and approximately constant temperature of expand- ing envelope of PNe, theoretical slope can be slightly steeper than in Eq. (4). Therefore, we conclude that the theoretical relation has the correct form, but our empirical relation is under influence of bi- ases that could make the slope shallower. On the other hand, there are some attempts to show that evolution of PNe are not linear in log-log scales (e.g. Phillips 2004). These different dependences cannot be derived from the thermal bremsstrahlung radia- tion formula (Eq. (1)). A very interesting feature regarding the em- pirical relation for Galactic PNe (Eq. (5)) is that the slope is approximately equal to the slope of triv- ial Σ − D relation. Therefore, we conclude that Malmquist bias is not so severe as in cases of Galac- tic SNR samples. This slope (β ≈ 2) was obtained for the extragalactic samples of SNRs (except M82 sample) where Malmquist bias is small, because all the SNRs are at the approximately same distance (see Urošević 2002, Urošević et al. 2005). The large scatter in Lν − D plane (Fig. 2) suggests that the slope in Eq. (5) does not have real and valid physical interpretation. It is a kind of luminosity-diameter scattering artefact which pro- duces the trivial Σ ∝ D−2 form. Therefore, the rela- tion defined by Eq. (5) is not precise enough for de- termination of valid distances to Galactic PNe. This is due to the different biases: the limitations in sen- sitivity and resolution of radio surveys, the source confusion, Malmquist bias (in mild form), mixture of different types of PNe in the same sample, and insufficient precision in determining the distances to the 44 calibrators. 4. SUMMARY The main results of this paper may be sum- marized as follows: i) The theoretical Σν −D relation for the radio evolution of thermal bremsstrahlung surface brightness of PNe in form of Σν ∝ D −3 is de- rived. ii) Our results show that the updated sample of Galactic PNe does not severely suffer from vol- ume selection effect - Malmquist bias (same as in cases of the extragalactic SNR samples). This is opposite to results obtained earlier for the Galactic SNR samples. iii) Due to analysis of the Lν −D dependence, we conclude that the Σν −D relation for Galactic PNe is not useful for reliable determination of distances for all observed PNe with unknown distances. The above observation leads to the more gen- eral comment that PNe may have very different ini- tial conditions leading to independent evolutionary paths. These paths could follow the same theoreti- cal Σ−D curve but with varying intercepts, leading to the scatter such as the one found in this paper. Acknowledgements – The authors would like to thank the referee Prof. Nebojsa Duric for valuable com- ments which have improved this paper. This research has been supported by the Ministry of Science and Environmental Protection of the Republic of Serbia (Projects: No 146002, No 146003, No 146012, No 146016). D. UROŠEVIĆ et al. REFERENCES Arbutina, B., Urošević, D., Stanković, M. and Tešić, Lj.: 2004, Mon. Not. R. Astron. Soc., 350, Evans, I.N. and Dopita, M.A.: 1985, Astrophys. J. Suppl. Series, 58, 125 Gruenwald, R. and Aleman, A.: 2007, Astron. As- trophys., 461, 1019. Phillips, J.P.: 2002, Astrophys. J. Suppl. Series, 139, 199. Phillips, J.P.: 2004, Mon. Not. R. Astron. Soc., 353, 589. Rohlfs, K. and Wilson, T.L.: 1996, Tools of Radio Astronomy, Springer Urošević, D.: 2002, Serb. Astron. J., 165, 27 Urošević, D., Pannuti, T. G., Duric, N., Theodorou, A.: 2005, Astron. Astrophys., 435, 437. Van de Steene, G.C. and Zijlstra, A.A.: 1995, As- tron. Astrophys., 293, 541. Zhang, C.Y.: 1995, Astrophys. J. Suppl. Series, 98, Σ − D RELACIJA ZA PLANETARNE MAGLINE: PRELIMINARNA ANALIZA D. Urošević1, B. Vukotić2, B. Arbutina1,2 and D. Ilić1 1Department of Astronomy, Faculty of Mathematics, University of Belgrade Studentski trg 16, 11000 Belgrade, Serbia 2Astronomical Observatory, Volgina 7, 11160 Belgrade 74, Serbia UDK 524.37–77–54 Prethodno saopxtenje Prikazana je analiza tzv. Σ − D re- lacije izme�u povrxinskog sjaja na radio- frekvencijama i dijametra planetarnih maglina (PM): i) izvedena je teorijska Σ −D relacija za evoluciju povrxinskog sjaja stvorenog zakoqnim zraqenjem; ii) suprotno rezultatima dobijenim ranije za uzorke saqinjene od Galaktiqkih ostataka super- novih, naxi rezultati pokazuju da najnovije formirani uzorak Galaktiqkih PM ne trpi veliki uticaj zbog zapreminskog selekcionog efekta, tzv. Malmkvistovog selekcionog efekta (isto vaжi za vangalaktiqke uzorake ostataka supernovih); i iii) zakljuqujemo da empirijska Σ − D relacija za PM izvedena u ovom radu nije upotrebljiva za pouzdana odre�ivanja daljina do svih posmatranih PM sa nepoznatim daljinama.

0704.0422

Polarization conversion in a silica microsphere

Polarization conversion in a silica microsphere Pablo Bianucci, Chris Fietz, John W. Robertson, Gennady Shvets, and Chih-Kang Shih∗ Physics Department, The University of Texas at Austin, Austin, Texas 78712 (Dated: May 22nd, 2007) Abstract We experimentally demonstrate controlled polarization-selective phenomena in a whispering gallery mode resonator. We observed efficient (≈ 75%) polarization conversion of light in a silica microsphere coupled to a tapered optical fiber with proper optimization of the polarization of the propagating light. A simple model treating the microsphere as a ring resonator provides a good fit to the observed behavior. In the past few years, microresonators have received a lot of attention1. Whis- pering gallery mode (WGM) resonators2, such as microspheres,3 microtoroids4 and microrings5 have been the object of inten- sive research, both in their fundamental prop- erties (such as quality factors, non-linear effects6,7 and coupling to quantum systems8 among many) and applications that include lasers9,10, chemical11 and biological12 sens- ing and photonic devices13. Microsphere res- onators, particularly when coupled to a ta- pered optical fiber14,15, are very useful to characterize these properties and test new ideas due to their high Q-factors and ease of fabrication. Recent reports have shown a further step, taking into account the difference between modes with different polarizations in micro- spheres. In particular, changes in the output polarization after coupling into the resonator have been observed16 and transverse electric (TE) and transverse magnetic (TM) modes have been discriminated17. Polarization conversion has been observed in microrings5 and explained as a resonant enhancement of polarization coupling caused by waveguide bending. However, the mode structure of microspheres makes it possible to completely decouple the polarizations and still obtain conversion. In this article, we re- port on the observation of efficient, controlled polarization conversion by using a silica mi- crosphere resonator coupled to a tapered op- tical fiber. We demonstrate that highly ef- ficient polarization conversion (75% for our particular case, higher for better optimized conditions) is enabled by a specific orienta- tion between the incoming light polarization and fiber-resonator displacement. Specifi- http://arxiv.org/abs/0704.0422v2 cally, for a horizontally stacked, strongly cou- pled, fiber and resonator combination, a 45◦ incident polarization results in the largest conversion. The conversion results in a strong dip of the transmitted light with the original polarization and a strong spike in the orthogonally polarized transmission. We fabricated the tapered fiber using the “flame brush” technique18. This technique involves mechanically stretching the optical fiber while scanning a flame (oxy-hydrogen in our case) over the region to be tapered. Due to constraints in the maximum pulling length, the fiber tapers are not completely adiabatic, but typical losses are never larger than 50%. SEM studies of the tapers reveal a characteristic diameter close to 1 µm. The microsphere was fabricated using a CO2 laser to stretch and melt an optical fiber tip19. In this way it is easy to obtain spheres with di- ameters ranging from 10 µm to 200 µm. For this particular experiment the sphere diame- ter was measured using an optical microscope to be 52 µm (corresponding to an estimated free spectral range of 1.2 THz). We mounted the microsphere on a piezo- electric scanner which allowed us to finely position the sphere over a range of a few micrometers, and the stretched fiber taper on a piezoelectric stick-slip walker permitting both coarse and fine positioning of the fiber taper next to the sphere. Both sphere and PD PC2 PC1 Tapered Fiber Laser PR FC Microsphere 10 um FIG. 1: Experimental setup schematic. PR is a polarization rotator, FC a fiber coupler, PC1 and 2 are fiber polarization controllers, P a po- larizer and PD is an amplified photodiode. Inset: Image of a sphere near a tapered fiber. taper were then situated inside a compact, closed chamber. We used an external cav- ity tunable diode laser purchased from New Focus as the excitation source, centered at a wavelength near 927.85 nm. The polariza- tion rotator set the polarization of the laser which was then coupled into the optical fiber using a free-space coupler. A polarizer and an amplified photodiode at the fiber output were used to analyze the transmitted light. Space constraints in the chamber and lim- itations on the arrangement of the optical fiber caused bending of the fiber in differ- ent locations and subsequent scrambling of the input polarization. As a way to compen- sate for these changes in the polarization, we used two polarization controllers. The first one preceded the fiber taper, compensating for polarization changes up to the position of the microsphere. The second controller was placed after the fiber taper to ensure the lin- earity of the output polarization. Figure 1 shows a schematic of this experimental setup. We used the following procedure to mea- sure the degree of polarization conversion. First, the incoming polarization was selected by using the polarization rotator. Then we adjusted the first polarization controller to ensure the polarization at the fiber taper was linear and matched to one set of modes (“x- polarized”). The next step was to uncouple the taper from the sphere and make sure the output polarization was linear (we achieved this by turning the detection polarizer to its position for minimum transmission and then minimizing this transmission further with the second polarization controller). This orienta- tion of the detection polarizer is the one we call “orthogonal”. Rotating the polarizer 90 degrees (the “parallel” orientation) resulted in maximum transmission, with a contrast of about 95%, confirming the linear polar- ization of the output light. Finally, we posi- tioned the sphere and the tapered fiber try- ing to optimize the coupling, while measur- ing transmission spectra for both orientations of the detection polarizer. We repeated the procedure for two other incoming polariza- tions: one matched to the other set of sphere modes (“y-polarized”) and another at 45◦ be- tween the x- and y- polarization axis (“xy- polarized”). Figure 2 shows the resulting transmission -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 Frequency shift (GHz) Input Detection polariz. polariz. FIG. 2: Transmission spectra for different in- put polarizations. The resonant frequencies cor- respond to modes with l ≈ 496. The x- and y-polarizations are orthogonal and correspond to the polarization eigenmodes of the resonator. The xy-polarization is oriented at 45 degrees from both x and y. The dark traces correspond to the detection polarizer parallel to the input polarization and the light traces correspond to a crossed detection polarization. spectra for the different configurations. The cases for both the x- and y- polarized light show the same behavior: a set of transmission dips whenever the laser frequency hit a whis- pering gallery resonance when the detection polarization is parallel and no signal when it is perpendicular. The xy-polarized case is more interesting: the parallel detection po- larization shows dips for both sets of modes, while the orthogonal one shows transmission peaks at the whispering gallery resonances. At the highest peak, more than 70% of the incident light had its polarization converted. Most of the observed polarization conver- sion can be understood by using a simple ring resonator model for the whispering gallery modes. In this model, the transmission of polarized light through the resonator is given by14,20 τ(φ) = r − aeiφ 1− raeiφ , (1) where r is the field coupling coefficient be- tween the resonator and the waveguide, a is the attenuation due to the resonator intrinsic losses and φ = 2π(ν−ν0)tRT is the phase shift imposed by the resonator (ν and ν0 are the in- coming light frequency and the resonant fre- quency respectively, while tRT is the round- trip time in the resonator). The model is scalar, but we can include the polarization by simply assuming that modes with orthogonal polarizations are independent and neglecting cross-polarized couplings (using an analysis similar to that by Little and Chu21). In this way we obtain the same expression, with pos- sibly different parameters, for the transmis- sion of both polarizations. In our particu- lar case of whispering gallery modes in mi- crospheres, we can safely assume that modes with different polarizations are not degener- ate, so one of the polarizations will be unaf- fected by the presence of a resonance. This differs from the case of microrings5, where the conversion depends on coupling between TE and TM modes. The essence of the effect lies in the differ- ent resonator response for each polarization. For a strongly coupled fiber and microsphere, |τ | ≈ 1, but the phase shift ψ = arg(τ) is changed by ∆ψ = π as the frequency is sweeped across the resonance. Because the orthogonal polarization is transmitted unal- tered, the transmitted polarization rotates by as much as 90◦ for the initial xy-polarization. When the fiber and the resonator are horizon- tally stacked, the effect is maximized when the incident polarization is at 45◦ degrees with respect to the horizontal plane. Conversion efficiencies of up to 25% can be achieved if one of the polarizations is crit- ically coupled to the ring, i.e. is completely absorbed in+the resonator. Achieving higher efficiencies requires increasing the resonator- waveguide coupling to obtain a significant po- larization dependent phase shift which will change the final polarization state into one closer to the desired one. We can look in more detail at the data by concentrating into a pair of modes show- ing good conversion, now accounting for laser frequency drift between scans using a Fabry- Perot interferometer as a reference. This de- tailed spectrum can be seen in in Fig. 3. The resonance on the right side of Fig. 3, near a shift of 31 GHz, shows a polarization conver- sion of about 60%. The left-side resonance shows a conversion near 75%. The higher ef- ficiency is due to the leftmost mode being 26 27 28 29 30 31 Frequency shift (GHz) 30.5 31 26 26.5 27 27.5 28 FIG. 3: Detailed view of two modes showing polarization conversion. The dashed lines are fits using equations of the form of equation 1. The fit parameters for the leftmost features are a = 0.99997, r = 0.99977. The corresponding ones for the rightmost feature are a = 0.99999, r = 0.99993. more strongly coupled (displaying a broader feature) to the tapered fiber than the right- most one. Consistent with theoretical predic- tions, in both cases one of the polarizations is over-coupled to the ring. The lack of a shift in the center frequency of the features also indicates that each pair of peak and dip corresponds to a single resonant mode. This phenomenon could be useful for po- larization control in photonic devices, such as narrowband polarization-dependent filter- ing or switching, as shown in Fig. 4 or even arbitrary polarization manipulation. We have observed efficient polarization conversion on a microsphere resonator cou- PBSPBS a) b) FIG. 4: Schematic of a resonator working as wavelength-selective polarization switch. a) Two signals with different wavelengths+ (green and blue) and orthogonal polarizations pass un- changed through the waveguide and the un- coupled resonator. A polarization beamsplitter then routes the signals to different paths. b) The polarization of the resonant signal (blue) is converted by the coupled resonator, and both signals are sent through the same path. The resonator-waveguide coupling can be changed in different ways, including mechanical or optical22 means. pled to a tapered optical fiber and used a simple theoretical model to understand the phenomenon. The model does not involve di- rect coupling of the orthogonal polarizations, but rather a polarization-selective phase shift induced by the resonator. This effect should be common to all whispering gallery mode resonators and could be useful for polariza- tion control in photonic devices. Acknowledgments This work was supported by NSF-NIRT (DMR-0210383), the Texas Advanced Tech- nology program, and the W.M. Keck Foun- dation. G.S. and C.F. acknowledge support from ARO MURI grant no. W911NF-04-01- 0203. ∗ Electronic address: [email protected] 1 K. J. Vahala, Nature (London) 424, 839 (2003). 2 A. B. Matsko and V. S. Ilchenko, IEEE J. Sel. Top. Quantum Electron. 12, 3 (2006). 3 M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, Opt. Lett. 21, 453 (1996). 4 D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, Nature (London) 421, 925 (2003). 5 A. Melloni, F. Morichetti, and M. Martinelli, Opt. Lett. 29, 2785 (2004). 6 A. E. Fomin, M. L. Gorodetsky, I. S. Gru- dinin, and V. S. Ilchenko, J. Opt. Soc. Am. B 22, 459 (2005). 7 T. Carmon, H. Rokhsari, L. Yang, T. Kip- penberg, and K. J. Vahala, Phys. Rev. Lett. 94, 223902 (2005). 8 Y.-S. Park, A. K. Cook, and H. Wang, Nano. Lett. 6, 2075 (2006). 9 M. Cai and K. Vahala, Opt. Lett. 26, 884 (2001). 10 S. I. Shopova, G. Farca, A. T. Rosenberger, W. M. Wickramanayake, and N. A. Kotov, Appl. Phys. Lett. 85, 6101 (2004). 11 A. M. Armani and K. J. Vahala, Opt. Lett. 31, 1896 (2006). 12 S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, Opt. Lett. 28, 272 (2003). 13 F. Michelotti, A. Driessen, and M. Bertolotti, eds., Microresonators as building blocks for VLSI photonics, vol. 709 of AIP Conference Proceedings (American Institute of Physics, Melville, New York, 2003). 14 M. Cai, O. Painter, and K. J. Vahala, Phys. Rev. Lett. 85, 74 (2000). 15 J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, Opt. Lett. 22, 1129 (1997). 16 G. Guan and F. Vollmer, Appl. Phys. Lett. 86, 121115 (2005). 17 H. Konishi, H. Fujiwara, S. Takeuchi, and K. Sasaki, Appl. Phys. Lett. 89, 121107 (2006). 18 T. A. Birks and Y. W. Li, J. Lightwave Tech- nol. 10, 432 (1992). 19 D. S. Weiss, V. Sandoghar, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, Opt. Lett. 20, 1835 (1995). 20 D. D. Smith, H. Chang, and K. A. Fuller, J. mailto:[email protected] Opt. Soc. Am. B 20, 1967 (2003). 21 B. E. Little and S. T. Chu, IEEE Photon. Technol. Lett. 12, 401 (2000). 22 V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, Nature (London) 431, 1081 (2004).

0704.0423

Limits on WIMP-nucleon interactions with CsI(Tl) crystal detectors

Limits on the WIMP-nucleon interactions with CsI(Tl) crystal detectors H.S. Lee,1 H.C. Bhang,1 J.H. Choi,1 H. Dao,7 I.S. Hahn,4 M.J. Hwang,5 S.W. Jung,2 W.G. Kang,3 D.W. Kim,1 H.J. Kim,2 S.C. Kim,1 S.K. Kim,1, ∗ Y.D. Kim,3 J.W. Kwak,1, † Y.J. Kwon,5 J. Lee,1, ‡ J.H. Lee,1 J.I. Lee,3 M.J. Lee,1 S.J. Lee,1 J. Li,7 X. Li,7 Y.J. Li,7 S.S. Myung,1 S. Ryu,1 J.H. So,2 Q. Yue,7 and J.J. Zhu7 (KIMS Collaboration) DMRC and Department of Physics and Astronomy, Seoul National University, Seoul, Korea Department of Physics, Kyungpook National University, Daegu, Korea Department of Physics, Sejong University, Seoul, Korea Department of Science Education, Ewha Womans University, Seoul, Korea Department of Physics, Younsei University, Seoul, Korea Department of Engineering Physics, Tsinghua Universuty, Beijing, China Department of Engineering Physics, Tsinghua University, Beijing, China (Dated: November 4, 2018) The Korea Invisible Mass Search (KIMS) experiment presents new limits on the WIMP-nucleon cross section using data from an exposure of 3409 kg·d taken with low-background CsI(Tl) crystals at Yangyang Underground Laboratory. The most stringent limit on the spin-dependent interaction for a pure proton case is obtained. The DAMA signal region for both spin-independent and spin- dependent interactions for the WIMP masses greater than 20 GeV/c2 is excluded by the single experiment with crystal scintillators. PACS numbers: 95.35.+d, 14.80.Ly The existence of dark matter has been widely sup- ported by many astronomical observations on vari- ous scales [1][2][3]. Weakly interacting massive parti- cles (WIMPs) are a good candidate for dark matter well motivated by cosmology and supersymmetric models [4]. The Korea Invisible Mass Search (KIMS) experiment has developed low-background CsI(Tl) crystals to detect the signals from the elastic scattering of WIMP off the nu- cleus [5][6][7]. Both 133Cs and 127I are sensitive to the spin-independent (SI) and spin-dependent (SD) interac- tions of WIMPs. Recently, the role of CsI in the direct search for SD WIMP for pure proton coupling has been pointed out [8]. It is worth noting that 127I is the dom- inant target for the SI interactions in the DAMA exper- iment. The pulse shape discrimination (PSD) technique allows us to statistically separate nuclear recoil (NR) sig- nals of WIMP interactions from the electron recoil (ER) signals due to the gamma ray background [9][10]. The KIMS experiment is located at the Yangyang Un- deground Laboratory (Y2L) at a depth of 700 m under an earth overburden. Details of the KIMS experiment and the first limit with 237 kg·d exposure data can be found in the previous publication [11]. Four low-background CsI(Tl) crystals are installed in the Y2L and operated at a temperature of T = 0◦C. Throughout the exposure period, the temperature of the detector was kept sta- ble to within ±0.1◦C. Green-enhanced photomultiplier tubes (PMTs) are mounted at both ends of each crystal. The signals from the PMTs are amplified and recorded by a 500 MHz FADC. Each event is recorded for a pe- riod of 32 µs. Both PMTs on each crystal must have at least two photoelectrons within a 2 µs window to form an event trigger. We obtained 3409 kg·d WIMP search data TABLE I: Crystals used in this analysis and amount of data for each crystal Crystal mass (kg) data (kg·days) S0501A 8.7 1147 S0501B 8.7 1030 B0510A 8.7 616 B0510B 8.7 616 Total 34.8 3409 with four crystals, as shown in Table I. The energy is cali- brated using 59.5 keV gamma rays from an 241Am source. For calibration of the mean time, a variable used for the PSD, NR events are obtained with small crystals ( 3 cm × 3 cm × 3 cm ) using an Am-Be neutron source. Compton scattering events taken with the WIMP search crystals using the 137Cs source are used to determine the mean time distribution of the gamma background. Compton scattering events are also taken with the small crystals to verify that the mean time ditributions for both the test crystals and the WIMP search crystals are the same. In order to understand the nature of the PMT background, a dominant background at low energies, acrylic boxes are mounted on the same PMTs used for the crystals. The data obtained using this setup is used to develop the cuts for the rejection of PMT background. Since the decay time of the scintillation light in the CsI(Tl) crystal is rather long, photoelectrons are well separated at low energies and thereby enabling recon- struction of each photoelectron. The time distribution of photoelectrons in an event is fitted to a double exponen- http://arxiv.org/abs/0704.0423v2 sec)µMean Time ( FIG. 1: (color online). MT distribution of NR events (open squares), ER events (open circles) and WIMP search data (filled triangles) of S0501A crystal in the 5-6 keV range. Fitted PDF functions are overlayed. χ2/DOF =0.8 and 1.3 with DOF=38 and 35 for NR and ER events respectively. tial function given by f(t) = −(t− t0) −(t− t0) where τf and τs are decay time constants of fast and slow components, respectively, R is ratio between two components, and t0 is the time of the first photoelectron in the event. The mean time (MT ) of each event is then calculated using these quantities as t · f(t)dt/ f(t)dt. With this method, an improvement in PSD is achieved over the previous analysis where we used a simple math- ematical mean [11]. In order to reject the PMT back- ground, we applied cuts to the fit variable, τf . The ratio between the maximum log likelihood value of the dou- ble exponential fit and that of the single exponential fit is also used to reject the PMT background, since PMT background events tend to be shaped as single exponen- tial decay. To reject the background that originates from the radioactivity of the PMT, the asymmetry between the signals from two PMTs is applied. Finally events in which signals are recorded in more than one crystal are rejected. The event selection efficiency was estimated by applying the same analysis cuts to the neutron and gamma calibration samples. The efficiency depends on the measured energy and ranges from 30% at 3 keV to 60% above 5 keV. The estimation of the NR event rate is performed in each 1 keV bin from 3 to 11 keV for each crystal. TheMT distributions of NR events and ER events are compared with the WIMP search data in Fig. 1 for the 5-6 keV energy range. The probability density functions (PDF) for the ER and NR events are obtained by fitting these distributions. An unbinned maximum likelihood fit is Electron Equivalent Energy (keV) 3 4 5 6 7 8 9 10 11 FIG. 2: (color online). Extracted NR event rates of the S0501A (open circles), S0501B (filled circles), B0510A (filled squares), and B05010B (filled triangles) crystals and only sta- tistical errors (1σ) are shown. The points are shifted with respect to each other on the x-axis to avoid overlapping. performed with the log(MT ) distribution of the WIMP search data using the likelihood function, × exp{−(NNR,i +NER,i)} [NNR,iPDFNR,i(xk) +NER,iPDFER,i(xk)], where the index i denotes the i-th energy bin; n = NNR,i +NER,i is the total number of events; NNR,i and NER,i are the numbers of NR and ER events, respec- tively; PDFNR,i and PDFER,i are PDFs of NR and ER events, respectively; and xk = log(MT ) for each event. The NR event rates obtained for each bin and for each crystal after efficiency correction are shown in Fig. 2. The extracted NR event rates are consistent with a null observation of the WIMP signal. In order to obtain the expected measured energy spec- trum of a WIMP signal including instrumental effects, a Monte Carlo (MC) simulation with GEANT4 [12] is used. A recoil energy spectrum is generated for each WIMP mass with the differential cross section, form fac- tor, and quenching factor, as described in Ref. [13]. The spin-dependent form factor for 133Cs calculated by Toiva- nen [14] is used, while for 127I, Ressell and Dean’s cal- culation [15] is used. The photons generated with the fitted decay function described above are propagated to the PMT and digitized in the same manner as in the ex- periment. Subsequently, the photoelectrons within given time windows are counted to check the trigger condition and to calculate energy. In this manner, the trigger ef- ficieny and energy resolution is accounted for in the ex- pected energy spectrum. The trigger efficiency is found to be higher than 99% above 3 keV. The simulation is verified with the energy spectrum obtained using 59.5 keV gamma rays from 241Am. The peak position and TABLE II: Spin expectation values for 133Cs and 127I Isotope J < Sp > < Sn > Reference 133Cs 7/2 -0.370 0.003 [16] 127I 5/2 0.309 0.075 [15] width of the distribution are very well reproduced for each crystal as described in Ref [11]. The total WIMP rate, R, for each WIMP mass is ob- tained by fitting the measured energy spectrum to the simulated one. The 90% confidence level (CL) limit on R is calculated by the Feldman-Cousins’s approach in the case of Gaussian with a boundary at the origin [17] and then converted to the WIMP-nucleus cross section, σW−A. Subsequently, the limits on WIMP-nucleon cross section is obtained from Ref. [13][18] as follows: σW−n = σW−A where µn,A are the reduced masses of the WIMP-nucleon and WIMP-target nucleus of mass number A. CA/Cn = A2 for SI interactions and CA/Cn = 4/3{ap < Sp > +an < Sn >}2(J + 1)/J for SD interactions. Here ap, an are WIMP-proton and WIMP-neutron SD couplings respectively. The spin expectation values used for this analysis are shown in Table II. Following the “model- independent” framework [18], we report the allowed re- gion in two cases for SD interaction: one for an = 0, and the other for ap = 0. We express the WIMP-nucleon cross section as follows: σSIW−n = σW−A σSDW−n,p = σW−A µ2n,p (J + 1) < Sn,p >2 where we indicate pure proton (p, an = 0) and pure neutron (n, ap = 0) coupling for SD interaction. We also present the allowed region in the ap − an plane with the following relation [18]: where GF is the Fermi coupling constant. The uncertainty in the MT distribution results in the uncertainty of the NR event rate. The limited statistics of the calibration data and different crystals used for the neutron calibration and WIMP search data are the ma- jor sources of this uncertainty. The former is investigated by varying the fitted parameters in PDF function within errors. The lattter is estimated by changing the mean of MT by the difference between the crystals. The system- atic uncertainties from these two souces are combined in quadrature resulting in 20-30% of statistical uncertain- ties depending on the energy bins. In addition, there WIMP Mass (GeV) 210 310 410 DAMA region FIG. 3: (color online). Exclusion plot for the SD interaction in the case of pure proton coupling (an = 0) at the 90% confidence level WIMP Mass (GeV) 210 310 410 DAMA region FIG. 4: (color online). Exclusion plot for for the SD interac- tion in the case of pure neutron coupling (ap = 0) at the 90% confidence level are uncertainties in the MC estimation of the expected event rates due to the uncertainties in the quenching fac- tors and the difference of energy resolution between the MC simulation and the data. The systematic error from the MC simulation is estimated to be 13.3% of the limits. These systematic errors are combined with the statistical error in quadrature in the presented results. The limits on the SD interactions are shown in Fig. 3 and 4 in the cases of pure proton coupling and pure neu- tron coupling, respectively. We also show the results ob- tained from CDMS [19], NAIAD [20], SIMPLE [21], and -6 -4 -2 0 2 4 6 FIG. 5: (color online). Allowed region (90% confidence level) in ap − an plane by KIMS data (inside the solid line contour) for 50 GeV WIMP mass. Results of CDMS [19](dotted line) and NAIAD [20](dot-dashed line) are also shown. WIMP Mass (GeV) 210 310 410 FIG. 6: (color online). Exclusion plot for the SI interactions at the 90% confidence level. PICASSO [22]. The DAMA signal region is taken from Ref [23]. Our limit provides the lowest bound on the SD interactions in the case of pure proton coupling for a WIMP mass greater than 30 GeV/c2. The allowed region in the ap − an plane for the WIMP mass of 50 GeV/c2 is also shown in Fig. 5 together with the limits from CDMS and NAIAD. The limit for the SI interactions is shown in Fig. 6 together with the results of CDMS [24], EDEL- WEISS [25], CRESST [26], ZEPLIN I [27], and the 3σ signal region of DAMA (1-4) [28]. Although there are several experiments that reject the DAMA signal region, this is the first time that it is ruled out by a crystal de- tector containing 127I, which is the dominant nucleus for the SI interactions in the NaI(Tl) crystal. In summary, we report new limits on the WIMP- nucleon cross section with CsI(Tl) crystal detectors using 3409 kg·d exposure data. The DAMA signal regions for both SI and SD interactions are excluded for the WIMP masses higher than 20 GeV/c2 by the single experiment. The most stringent limit on the SD interaction in the case of purely WIMP-proton coupling is obtained. The authors thank Dr. J. Toivanen and M. Korte- lainen for the calculation of the SD form factor as well as for the useful discussions. This work is supported by the Creative Research Initiative Program of the Korea Science and Engineering Foundation. We are grateful to the Korea Middland Power Co. Ltd. and the staff mem- bers of the YangYang Pumped Storage Power Plant for providing us the underground laboratory space. ∗ [email protected] † Current address: National Cancer Center, Ilsan, Korea ‡ Current address: Department of Physics, Ewha Womans University, Seoul, Korea [1] K. G. Begeman, A. H. Broeils, and R. H. Sanders, Mon. Not. Roy. Astron. Soc. 249, 523 (1991). [2] D. N. Spergel et al., Astrophys. J. Suppl. 148, 175 (2003). [3] M. Tegmark et al., Phys. Rev. D 69, 103501 (2004). [4] G. Jungman, M. Kamionkowski, and K. Griest, Phys. Rep. 267, 195 (1996). [5] H. S. Lee et al., Nucl. Instr. Meth. A 571, 644 (2007). [6] Y. D. Kim et al., J. Korean. Phys. Soc. 40, 520 (2002). [7] Y. D. Kim et al., Nucl. Instr. Meth. A 552, 456 (2005). [8] T. A. Girard and F. Giuliani, Phys. Rev. D 75, 043512 (2007). [9] H. J. Kim et al., Nucl. Instr. Meth. A 457, 471 (2001). [10] H. Park et al., Nucl. Instr. Meth. A 491, 460 (2002). [11] H. S. Lee et al., Phys. Lett. B 633, 201 (2006). [12] S. Agostinelli et al., Nucl. Instr. Meth. A 506, 250 (2003). [13] J. D. Lewin and P. F. Smith, Astropart. Phys. 6, 87 (1996). [14] J. Toivanen and M. Kortelainen (2006), private commu- nication. [15] M. T. Ressell and D. J. Dean, Phys. Rev. C 56, 535 (1997). [16] F. Iachello, L.M.Krauss, and G. Maino, Phys. Lett. B 254, 220 (1991). [17] G. J. Feldman and R. D. Cousins, Phys. Rev. D 57, 3873 (1998). [18] D. R. Tovey et al., Phys. Lett. B 488, 17 (2000). [19] D. S. Akerib et al., Phys. Rev. D 73, 011102 (2006). [20] G. J. Alner et al., Phys. Lett. B 624, 186 (2005). [21] T. A. Girard et al., Phys. Lett. B 621, 233 (2005). [22] M. Barnabe-Heider et al., Phys. Lett. B 624, 186 (2005). [23] C. Savage, P. Gondolo, and K. Freese, Phys. Rev. D 70, 123513 (2004). [24] D. S. Akerib et al., Phys. Rev. Lett. 96, 011302 (2006). [25] V. Sanglard et al., Phys. Rev. D 71, 122002 (2005). [26] G. Angloher et al., Astropart. Phys. 23, 325 (2005). [27] G. J. Alner et al., Astropart. Phys. 23, 444 (2005). [28] R. Bernabei et al., Phys. Lett. B 480, 23 (2000); R. Bern- abei et al., Riv. Nuovo. Cim. 26, 1 (2003). mailto:[email protected]

0704.0424

Stopping effects in U+U collisions with a beam energy of 520 MeV/nucleon

Stopping effects in U+U collisions with a beam energy of 520 MeV/nucleon Xiao-Feng Luo,1, ∗ Xin Dong,1 Ming Shao,1 Ke-Jun Wu,2 Cheng Li,1 Hong-Fang Chen,1 and Hu-Shan Xu3 University of Science and Technology of China, Hefei, Anhui 230026, China Institute of Particle Physics, Hua-Zhong Normal University, Wuhan, Hubei 430079, China Institute of Modern Physics, Chinese Academy of Sciences, LanZhou, Gansu 730000, China (Dated: November 4, 2018) A Relativistic Transport Model (ART1.0) is applied to simulate the stopping effects in tip-tip and body-body U+U collisions, at a beam kinetic energy of 520 MeV/nucleon. Our simulation results have demonstrated that both central collisions of the two extreme orientations can achieve full stopping, and also form a bulk of hot, dense nuclear matter with a sufficiently large volume and long duration, due to the largely deformed uranium nuclei. The nucleon sideward flow in the tip-tip collisions is nearly 3 times larger than that in body-body ones at normalized impact parameter b/bmax < 0.5, and that the body-body central collisions have a largest negative nucleon elliptic flow v2 = −12% in contrast to zero in tip-tip ones. Thus the extreme circumstance and the novel experimental observables in tip-tip and body-body collisions can provide a good condition and sensitive probe to study the nuclear EoS, respectively. The Cooling Storage Ring (CSR) External Target Facility (ETF) to be built at Lanzhou, China, delivering the uranium beam up to 520 MeV/nucleon is expected to make significant contribution to explore the nuclear equation of state (EoS). PACS numbers: 24.10.Lx,25.75.Ld,25.75.Nq,24.85.+p I. INTRODUCTION In recent years, the ultra-relativistic high energy heavy ion collisions performed at SPS/CERN and RHIC/BNL sNN ∼ 10 − 200 GeV) focus on high temperature and low baryon density region in nuclear matter phase diagram [1] to search a new form of matter with par- tonic degree of freedom-the quark-gluon plasma (QGP) [2, 3, 4, 5]. However, no dramatic changes of experimen- tal observables, such as jet-quenching, elliptic flow and strangeness enhancement, have been observed yet [6]. On the other hand, the heavy ion collisions performed at the BEVALAC/LBNL and SIS/GSI [7, 8] in last two decades were used to produce hot and compressed nuclear mat- ter to learn more about the nuclear equation of state (EoS) [13, 14] at high baryon density and low temper- ature region of the phase diagram. Although we have made great efforts to study the nuclear EoS, theoreti- cally and experimentally, a solid conclusion can hardly be made. Then, it is still worthwhile to systematically study on the collision dynamics as well as the EoS observ- ables. Recently, for more understanding of the nuclear matter phase diagram and EoS at high net-baryon den- sity region, it is proposed to collide uranium on uranium target at External Target Facility (ETF) of Cooling Stor- age Ring (CSR) at Lanzhou, China with a beam kinetic energy of 520 MeV/nucleon. [10]. Uranium is the largest deformed stable nucleus, and has approximately an ellipsoid shape with the long and short semi-axis given by Rl = R0(1 + 2δ/3) and Rs = ∗contact author: [email protected] FIG. 1: (Color online) (a) body-body collisions (b) tip-tip collisions R0(1 − δ/3), respectively, where R0 = 7 fm is the effec- tive spherical radius and δ = 0.27 is the deformation pa- rameter [9]. Consequently, one has Rl/Rs = 1.3. In our simulation, we consider two extreme orientations: the so- called tip-tip and body-body patterns with the long and short axes of two nuclei are aligned to the beam direc- tion, respectively [12], see Fig. 1 for illustration. The two types of orientations can be identified in random ori- entations of U+U collisions by making proper cutoffs in experimental data, such as the particle multiplicities, el- liptic flow and so on [10, 11]. With the two extreme collision orientations, some novel stopping effects which are believed responsible for some significant experimental observables, such as particle production, collective mo- tion as well as attainable central densities, can be ob- tained. Due to the large deformation of the uranium nuclei [11, 12] , it is expected that the tip-tip collisions can form a higher densities nuclear matter with longer duration than in body-body or the spherical nuclei colli- sions, which is considered to be a powerful tool to study http://arxiv.org/abs/0704.0424v2 the nuclear matter phase transition at high baryon den- sity [12], and the body-body central collisions may reveal a largest out-of-plane elliptic flow (negative v2) at high densities, which can be a sensitive probe to extract the early EoS of the hot, dense nuclear matter [12, 17]. The novel experimental observables can be effectively utilized to study the possible nuclear matter phase transition and the nuclear EoS [12, 13, 14, 15, 16, 17, 18, 19, 20]. For comparing with tip-tip and body-body collisions, a type of gedanken ”sphere-sphere” collisions without deforma- tions of uranium nuclei are also included in the simula- tion. The ART1.0 model [21, 22] derived from Boltzmann- Uehling-Uhlenbeck (BUU) model [23] has a better treat- ment of mean field and Pauli-Blocking effects [23] than cascade models [24]. The fragments production mech- anism and partonic degree of freedom are not present in the ART1.0 model. A soft EoS with compressibility coefficient K = 200 MeV is used throughout the simu- lation and the beam kinetic energy of uranium nuclei is set to 520 MeV/nucleon if not specifically indicated. In the next section, we discuss about the stopping power ratio and selection of impact parameter b. In Sec. 3, the evolution of baryon and energy densities as well as thermalization of central collision systems are studied. In Sec. 4, some experimental observables, such as nucleon sideward flow and elliptic flow are also investigated. We summarize our results in Sec. 5. II. STOPPING POWER OF TIP-TIP AND BODY-BODY COLLISIONS Large stopping power can lead to remarkable pressure gradient in the compressed dense matter. It is generally also considered to be responsible for transverse collec- tive motion [25], the maximum attainable baryon and energy densities as well as thermalization of collision sys- tems. Thus, the study of the stopping power in U+U collisions may provide important information for under- standing the nuclear EoS and collision dynamics. A. Selection of impact parameter The nuclear stopping power and geometric effects in U+U collisions rely strongly on the impact parameter b. Considering the conceptual design of the CSR-ETF detector [10], two methods are invoked here to estimate the impact parameter. The first one is the multiplicity of forward neutrons with polar angle θ < 20o in the lab frame which can be covered by a forward neutron wall. The other method is to make use of the parameter Erat [26], which is the ratio of the total transverse kinetic en- ergy to the total longitudinal one. The particles are also required to be within θ < 20o in the lab frame, while the two qualities are calculated within the center of mass 0 0.2 0.4 0.6 0.8 1 tip-tip body-body 0 0.2 0.4 0.6 0.8 10 maxb/b FIG. 2: Upper: Forward neutron multiplicity and Lower: Erat, as a function of normalized impact parameter b/bmax in both tip-tip and body-body collisions. system (c.m.s.). Erat = Ezi (1) The normalized impact parameter b/bmax is used to rep- resent centralities of tip-tip and body-body collisions and the bmax of the two cases are quite different from each other. As shown in Fig. 2, with either method, obvious linear dependence of the normalized impact parameter are demonstrated in both tip-tip and body-body near central collisions. Then, the two methods can be com- bined to determine the impact parameter to identify the most central collision events in both tip-tip and body- body collisions. B. Stopping power ratio definition and evolution It is difficult to obtain a universally accepted estimate of the nuclear stopping power in heavy ion collisions due to a proliferation of definitions of the concept [27]. The stopping power ratio R [28] is employed to measure the degree of stopping and defined as: |Ptj |/ |Pzj | (2) , the total nucleon transverse momentum |Ptj | divided by the total absolute value of nucleon longitudinal momen- tum |Pzj | in the c.m.s.. The ratio is wildly used to de- scribe the degree of thermalization and nuclear stopping by low and intermediate energies heavy ion collisions. It’s a multi-particle observable on an event-by-event ba- sis, which for an isotropic distribution is unity. Fig. 3 shows the time and normalized impact param- eter dependence of the stopping ratio R for three con- ditions: tip-tip, body-body and sphere-sphere collisions. 0 10 20 30 40 =0max(a)b/b tip-tip body-body sphere-sphere 0.2 0.4 0.6 0.8 1 (b)Minibias tip-tip body-body sphere-sphere t(fm/c) maxb/b FIG. 3: (Color online) (a)The time evolution of the stopping ratio R in tip-tip, body-body and sphere-sphere central colli- sions, and (b) the stopping ratio R as a function of b/bmax in minimum biased collisions. When the ratio R reaches the value of 1, full stopping of the collision system is considered to be achieved, and the momenta is also isotropy, which are not sufficient but nec- essary for thermal equilibrium of collision systems [28]. For R > 1, it can be explained by preponderance of mo- mentum flow perpendicular to the beam direction [29]. It is shown that all of the three conditions can achieve full stopping when the stopping ratio R=1, the correspond- ing time for body-body and tip-tip central collisions are about 15 fm/c and 25 fm/c, respectively. Larger stopping ratio and faster evolution to full stopping are observed for body-body central collisions than tip-tip and sphere- sphere ones at the early stage, which may indicate a more violent colliding process for body-body central collisions due to the sizable initial transverse overlap region. Al- though the stopping ratio of tip-tip central collisions is lowest than the other two cases at the early time, it raises sharply later and even exceeding one. So, it means that longer reaction and passage time can be obtained in tip- tip central collisions than body-body and sphere-sphere ones, which may indicate the nucleons in tip-tip colli- sions can undergo more binary collisions to reach higher transverse momentum. In Fig. 3(b), the R of the three conditions are gradu- ally decrease with the increase of the normalized impact parameter. When b/bmax < 0.5, the ratio is always larger for tip-tip collisions than the other two cases, while for b/bmax > 0.5 all of the three conditions almost have the same stopping power ratio. III. BARYON, ENERGY DENSITY AND THERMAL EQUILIBRIUM Considering the discrepancy of stopping power be- tween tip-tip and body-body collisions, it is interesting 4 (a)Baryon Density tip-tip body-body Au-Au 0 10 20 30 40 50 (b)Energy Density tip-tip body-body Au-Au t(fm/c) FIG. 4: The evolution of (a) baryon and (b) energy densities in tip-tip, body-body and Au+Au central collisions. to study further about the baryon and energy densities evolution in both cases. As the full stopping and de- formation effects in U+U collisions, it is believed higher local baryon and energy densities system with long du- ration can be created, which is considered to be a signif- icant condition to study the nuclear EoS at high bayonic density region. A. The evolution of baryon and energy densities The evolution of baryon and energy densities in the central zone of tip-tip and body-body as well as Au+Au central collisions are illustrated in Fig. 4. In Fig. 4, it is observed the maximum attainable baryon and energy densities for both tip-tip and body- body central collisions are about 3.2 ρ0 and 0.8 GeV/fm respectively, while the Au+Au one are about 2.6 ρ0 and 0.6 GeV/fm3. Both the baryon and energy densities in U+U collisions are higher than the Au+Au one. Once a baryon density threshold of ρ > 2.5 ρ0 is required, the corresponding duration in tip-tip central collisions ∼ 20 fm/c (from ∼ 8 fm/c to ∼ 28 fm/c) is longer than ∼ 10 fm/c ( from ∼ 8 fm/c to ∼ 18 fm/c ) of body-body one, which is as predicted. But the peak densities have no significant discrepancy between the two cases unlike those at the energy region of the Alternating Gradient Synchrotron (AGS) [12], which may be attribute to the full stopping at the CSR energy. B. Thermalization of the U+U collision systems As mentioned before, the stopping ratio R = 1 is a necessary but not sufficient condition for thermal equi- librium of the collision system. In order to approach a thermal equilibrium, a long duration of reaction is needed for nucleons to undergo sufficient binary collisions. As 0 10 20 30 40 50 tip-tip body-body Au-Au 0 10 20 30 40 50 tip-tip body-body Au-Au t(fm/c) FIG. 5: The evolution of (a) volume with high density (ρ > 2.5ρ0) in tip-tip, body-body and Au+Au central collisions, and (b) the scaled mean kinetic energy 2 < Ek >, within a sphere of radius 2fm around the system mass center. shown in Fig. 4(a), obvious long duration has been ob- tained in both tip-tip and body-body central collisions. It is therefore possible thermal equilibrium at the time of freeze-out can be achieved. The Fig. 5(a) is the evolution of the volume with the high baryon density(ρ > 2.5 ρ0) for tip-tip, body-body and Au+Au central collisions, respectively. Both tip- tip and body-body central collisions have larger volumes than Au+Au one at the same beam kinetic energy 520 MeV/nucleon. Although the maximum volume attain- able for body-body central collisions(∼ 220 fm3) is about two times larger than tip-tip one(∼ 120 fm3), the peak volume of tip-tip central collisions lasts a much longer time of ∼ 10 fm/c (from ∼ 15 fm/c to ∼ 25 fm/c) and much more stable than body-body one. To estimate the temperature at the freeze-out time, the scaled mean ki- netic energy of all hadrons in a sphere of radius 2fm around the system mass center is calculated as 2 < Ek > [22], which is utilized to reflect the thermalization tem- perature T of the collision system approximately. As illustrated in Fig. 5 (b), both tip-tip and body-body central collisions show a flat region about 75 MeV and the corresponding time range are about 10 fm/c to 28 fm/c and 10 fm/c to 18 fm/c, respectively. Considering the time range of the flat region in Fig. 5 (b) associat- ing with the corresponding range in Fig. 5 (a) and also looking back to Fig. 4, we obtain a large volume of hot, dense nuclear matter in both tip-tip and body-body cen- tral collisions. Consequently, the extreme circumstance of sufficiently high temperature and density for a signif- icant large volume and long duration [12, 22] has been formed in tip-tip and body-body central collisions, which can provide a good opportunity to study the nuclear EoS as well as particles in medium properties, especially for tip-tip case. The time of freeze out should be cautiously determined (a)tip-tip π+∆+*N 0 10 20 30 40 50 60 80 (b)Body-Body π+∆+*N t(fm/c) FIG. 6: Evolution of the multiplicity of the free pion, N∗ + ∆, N∗ +∆+ π in (a)tip-tip ,and (b)body-body central colli- sions. for estimating the thermalization temperature of colli- sion system. In Fig. 6 the multiplicity evolution of free pion which are not bounded in baryon resonances and pion still bounded inside the excited baryon resonances (∆, N∗) (unborn pion) are displayed. At the Lanzhou CSR energy region (520 MeV/nucleon), the production and destruction of the ∆ resonances are mainly through NN ⇋ N∆ and ∆ → Nπ reactions in which the ∆ decay rate is always higher than that of the formation of this resonance and the production of pion is predominated by the decay of the ∆ resonances (∆ → Nπ) [30]. The total multiplicity of pion, ∆ and N∗ approaches a saturated level after a period of evolution, indicating the freeze- out time about t=28 fm/c and t=18 fm/c for tip-tip and body-body central collisions, respectively. The larger maximum attainable total multiplicity of pion, ∆ andN∗ and freeze out earlier indicates a existent of faster evo- lution and more violently reaction process for the body- body central collisions than tip-tip case consisting with the discussing before. The corresponding temperature about 75 MeV at freeze-out time can be extracted from the Fig. 5 (b), for both tip-tip and body-body central collisions. To further confirm this estimation, both the energy spectrum of the nucleon and negative-charged pion are studied within the polar angle range of 900±100 in the c.m.s.. The thermo- dynamic model [31] predicts that the energy spectra will be represented by a temperature T which characterizes a Maxwell-Boltzmann gas PEdEdΩ = const× e−Ekin/T (3) , where P and E are the particle momentum and total energy in the c.m.s.. Both the energy spectra and the Boltzmann fit results are shown in Fig. 7. The inverse slope (e.g. temperature T ) of the nucleons in tip-tip and body-body central collisions are about 73 MeV and 70 0 0.2 0.4 0.6 0.8 310 tip-tip body-body (a)Nucleon 0 0.2 0.4 0.6 0.8 -π(b) (GeV)kinE FIG. 7: (a) Nucleon, and (b) negative-charged pion energy spectrum at 900 ± 100 in the c.m.s. together with a Maxwell- Boltzmann fit for both tip-tip and body-body central colli- sions. The nucleon fit temperature for tip-tip and body-body are about 73 MeV and 70 MeV, respectively and that of pion are about 56 MeV and 52 MeV, respectively. MeV, respectively, which are in good agreement with the temperature extracted from the Fig. 5(b) at the freeze out time. The spectra of negative-charged pion show a different lower temperature than that of nucleon which may be explained by considering an equilibrated N and ∆ system at thermal freeze out and taking into account the kinematics of ∆ decay [32]. The nucleon temperature closely reflects the freeze-out temperature of tip-tip and body-body central collisions. In conclusion, thermalization (or near thermalization) of the collision system corresponding a freeze-out tem- perature about 75 MeV is likely to be achieved in both tip-tip and body-body central collisions. However, it’s also possible that the collision system is still in a non- equilibrium transport process on its path towards kinetic equilibration [30]. IV. THE COLLECTIVE FLOW OF U+U COLLISIONS Stopping of nuclei in heavy ion collision can lead to pressure gradient along different directions, result- ing in collective motion as spectators bounce-off [34] and participants squeeze-out effects [35]. Since last two decades, at Bevalac/LBNL and SIS/GSI energies the so-called ”collective flow” analysis has been estab- lished [15, 34, 35, 36, 37] to study the collective mo- tion of the products in heavy ion collisions. The collec- tive flow resulting from bounce-off and squeeze-out ef- fects, which can be explained well by the hydrodynamics model [34, 38], and also be in good agreement with the experimental data has been observed [39, 40]. Because of the large deformation of the uranium nuclei, a novel col- -1 -0.5 0 0.5 1 Soft:tip-tip Soft:body-body Cascade:tip-tip Cascade:body-body (a)b/b -1 -0.5 0 0.5 1 (b)b/b FIG. 8: The mean transverse momentum per nucleon pro- jected into the reaction plane, < px/A >, as a function of c.m.s. normalized rapidity is illustrated for tip-tip and body-body collisions. With normalized impact parameter cutoff:(a)b/bmax <= 0.5 (b)b/bmax > 0.5. lective motion is expected [12], to be used to extract the medium properties and nuclear matter EoS information. [15, 16, 17, 18, 19, 20]. To perform flow analysis, it is necessary to construct a imaginary reaction plane defined by direction of the beam (z) and the impact parameter vector b [43, 45, 46]. In our simulation, the x− z plane is just defined as the reaction plane with the beam direction along z positive direction and the impact parameter vector b along x positive direc- tion. In last two decades, there are mainly two methods to study the collective flow at the low and intermediate energies. One is the sphericity method [28, 34, 41, 42] which yields the flow angle relative to the beam axis of the major axis of the best-fit kinetic energy ellipsoid, and the other is to employ the mean transverse momentum per nucleon projected into the reaction plane, < px/A >, to perform nucleon sideward flow analysis [43, 44] which reflects the spectator bounce-off effects in the reaction plane. In recent years, it is usual to use an anisotropic transverse flow analysis method. With a Fourier expan- sion [47, 48] of the particle azimuthal angle φ distribu- tion with respect to the reaction plane, different har- monic coefficients can be extracted, among which the first harmonic coefficient v1, called directed flow (simi- lar to sideward flow) and the second harmonic coefficient v2, called elliptic flow are mostly interested. The ellip- tic flow reflects the anisotropy of emission particles in the plane perpendicular to the reaction plane while the directed flow describes the anisotropy in reaction plane. The Fourier expansion can be expressed as ∼ 1 + 2vncos(nφ) (4) Fig. 8 shows nucleon sideward flow, < px/A >, for both tip-tip and body-body minimum biased collisions (a)Nucleon Flow Parameter tip-tip body-body Au-Au(500MeV/A) 0 0.2 0.4 0.6 0.8 1 10 2(b)Nucleon v tip-tip body-body Au-Au(500MeV/A) maxb/b FIG. 9: (a)The nucleon flow parameter F and (b)the c.m.s. mid-rapidity ( −0.5 < y0 < 0.5 ), nucleon elliptic flow v2 of three collision conditions as a function of normalized impact parameter b/bmax with soft EoS. as a function of normalized rapidity, y(0) = Ycm/ycm, in which Ycm represents the particle rapidity in c.m.s. and ycm is the rapidity of the system mass center. To ex- tract the nuclear EoS information and also demonstrate the discrepancies of the nucleon sideward flow in tip-tip and body-body collisions, the cascade events [49], which neglect the mean field and pauli blocking effects are em- ployed here to compare with the soft EoS case. In Fig. 8(a),(b), with a soft EoS, it is noted that either tip-tip or body-body collisions show a spectator bounce-off ef- fect revealing an obvious ”S” shape [15, 49] at the mid- rapidity region of −0.5 < y0 < 0.5, while the cascade one appear a almost vanishing nucleon sideward flow. It can be understand by the nucleon sideward flow is related to the mean field, which is mainly responsible for the pressure gradient of the stopping nuclei, while the mean field has a strong dependence of the nuclear EoS. There- fore, the nucleon sideward flow is thought to be a good indirect probe to extract the nuclear EoS information, especially tip-tip case for its largely remarkable sideward flow. A cutoff on normalized impact parameter is also applied to explore the impact parameter dependence of nucleon sideward flow. As shown in Fig. 8(b), when b/bmax > 0.5 the curves of soft EoS and cascade are al- most superposed with each other, while for b/bmax < 0.5 large discrepancy is observed. The situation is quite sim- ilar to Fig. 3 (b), almost the same stopping power for b/bmax > 0.5 and large discrepancy for b/bmax < 0.5 in tip-tip and body-body minimum biased collisions, which means there exists a correlation between nuclear stopping power and sideward flow [33]. The normalized impact parameter dependence of the collective flow of nucleon is further studied, by analyzing the ”flow parameter” F [49] and also elliptic flow v2 for both tip-tip and body-body as well as Au+Au minimum biased collisions. The flow parameter F is a customarily used quality to describe the nucleon sideward flow quan- titatively defined as d < px/A > dy(0) y(0)=0 the slope of the mean transverse momentum per nucleon projected into the reaction plane at y(0) = 0. In Fig. 9(a), with b/bmax > 0.5, the nucleon flow pa- rameter F of tip-tip and body-body collisions are with similar value. This similarity, along with the almost same stopping ratioR in Fig. 3(b), indicates a existence of sim- ilar pressure gradient effects on nucleon sideward flow in the two collision orientations. While for b/bmax < 0.5, the flow parameter F of tip-tip collisions is nearly 3 times larger than that of body-body case. Even the sideward flow of Au+Au collisions is larger than the body-body one. It is further confirmed the tip-tip nucleon sideward flow is a more sensitive probe to extract the information of nuclear EoS than that of body-body one. The promi- nence high of the nucleon sideward flow in tip-tip colli- sions may be resulted from the stronger pressure gradient between the participants and spectators in the reaction plane than body-body one, due to the largely deformed nuclei. The normalized impact parameter dependent of nu- cleon elliptic flow v2 at the mid-rapidity region ( −0.5 < y0 < 0.5 ) is displayed in Fig. 9(b). A significant neg- ative elliptic flow v2 at this energy region is consistent with the excitation function of the elliptic flow studied before [50]. An largest negative v2 about −12% in body- body central collisions is observed which reflects the large geometric and squeeze-out effects in the collisions. While for tip-tip and Au+Au ones the maximum negative v2 are obtained at mid-centrality. Since both high baryon, en- ergy densities and large elliptic flow effects, which reflects an early EoS of the hot dense compression nuclear matter [17], are available in body-body central collisions. Thus the body-body nucleon elliptic flow can also be taken as a sensitive probe of nuclear EoS. The novel behaviors of nucleon collective flow in tip-tip and body-body col- lisions are mainly attributed to the large deformation of the uranium nuclei. V. SUMMARY In summary, the CSR-ETF at Lanzhou provide a good opportunity to systematically study the nuclear EoS at the high net-baryon density region of nuclear matter phase diagram. Due to the novel stopping effects in largely deformed U+U collisions, the simulation based on ART1.0 demonstrates that full stopping can be achieved and also a bulk of hot, high densities nuclear matter with large volume and long duration have been formed in both tip-tip and body-body collisions. Large nucleon sideward flow in tip-tip collisions and the significant negative nu- cleon elliptic flow in body-body central collisions can pro- vide a sensitive probe to extract nuclear EoS information. Thus the extreme circumstance and the novel collective flow in both tip-tip and body-body collisions can provide a good condition and sensitive probe to study the nu- clear EoS, respectively. More experimental observables of U+U collision dynamics should be further studied, due to the geometric effects. VI. ACKNOWLEDGEMENT This work is supported by National Natural Sci- ence Foundation of China (10575101,10675111) and the CAS/SAFEA International Partnership Program for Creative Research Teams under the grant number of CXTD-J2005-1. We wish to thank Bao-an Li, Feng Liu, Qun Wang, Zhi-Gang Xiao and Nu Xu for their valuable comments and suggestions. [1] M. A. Stephanov, Int. J. Mod. Phys. A20,4387 (2005); [2] C. Lourenco et al, Nuclear Physics A698,13-22 (2002); [3] N. Xu et al, Nucl. Phys. A751,109-126 (2005) [4] J. Adams et al, Nucl. Phys. A757,102-183 (2005); [5] K. Adcox et al, Nucl. Phys. A757,184-283 (2005); [6] P. Jacobs, X. N. Wang, Prog. Part. Nucl. Phys. 54, 433- 534(2005) [7] E. K. Hyde, Phys. Scr. 10 30-35 (1974) ; [8] C. Höhne, Nucl. Phys. A749,141c-149c (2005); [9] A. Bohr and B. Mottelson, Nuclear Structure 2,133 (1975); [10] Z. G. Xiao, talk presented at QM2006,ShangHai; [11] E. V. Shuryak, Phys. Rev. C 61,034905 (2000); [12] Bao-An Li, Phys. Rev. C 61,021903(R) (2000); [13] P. Danielewicz, nucl-th/0512009; [14] P. Danielewicz et al, Science 298,1592-1596 (2002); [15] K. G. R. Doss et al, Phys. Rev. Lett. 57,302 (1986); [16] J. J. Molitoris et al, Nucl. Phys. A447,13c (1985); [17] P. Danielewicz et al, Phy. Rev. Lett. 81,2438 (1998); [18] J. Hofmann et al, Phys. Rev. Lett. 36,88 (1976); [19] H. Stöocker and W. Greiner, Phys. Rep. 137,277 (1986); [20] H. Stöocker et al, Z. Phys. A 290,297 (1979); [21] Bao-An Li and C. M. Ko, Phys. Rev. C 52 ,2037 (1995); [22] Bao-An Li et al, Inter. Jour. Mod. Phys. E10,267 (2001); [23] G. F. Bertsch and S. D. Gupta, Phys.Rep. 160,189 (1988); [24] J. Cugnon, Phys. Rev. C 22,1885 (1980) [25] A. Andronic et al, Eur. Phys. J. A30,1 (2006); [26] B. Hong et al, Phys. Rev. C 66,034901 (2002); [27] S. P. Sorensen et al, CONF-9109221-3: DE92009006 (1991); [28] H. Ströbele et al, Phys. Rev. C 27,1349 (1983); [29] R. E. Renfordt and D. Schall et al, Phys. Rev. Lett. 53,763 (1984); [30] Bao-An Li and W. Bauer, Phys. Rev. C 44,450 (1991); [31] J. Gosset et al, Phys. Rev.C 16,629-657 (1977); [32] R. Brockmann et al, Phys. Rev. Lett. 53,2012 (1984); [33] W. Reisdorf et al, Phys. Rev. Lett. 92,232301 (2004); [34] H. A. Gustafsson et al, Phys. Rev. Lett. 52,1590 (1984); [35] H. H. Gutbrod et al, Phys. Rev. C 42,640 (1990); [36] R. E. Renfordt et al, Phys. Rev. Lett. 53,763 (1984); [37] H. Stöcker, J. A. Maruhn, W. Greiner, Phys. Rev. Lett. 44,725 (1980); [38] H. Stöcker et al, Phys. Rev. Lett. 47,1807 (1981); [39] W. Scheid et al, Phys. Rev. Lett. 32,741 (1974); [40] G. Buchwald et al, Phys. Rev. Lett. 52,1594 (1984); [41] J. Cugnon et al, Phys. Lett. B 109,167 (1982); [42] M. Gyulassy et al, Phys. Lett. B 110,185 (1982); [43] P. Danielewicz and and G. Odyniec, Phys. Lett. B 157,146 (1985); [44] H. A. Gustafsson et al, Z. Phys. A321,389 (1983); [45] J. Y. Ollitrault, Phys. Rev. D 48,1132 (1993); [46] R. J. M. Snellings et al, STAR Note 388 (1999) (arXiv: nucl-ex/9904003); [47] A. M. Poskanzer and S. A. Voloshin, Phys. Rev. C 58,1671 (1998); [48] S. A. Voloshin and Y. Zhang, Z. Phys. C 70,665-672 (1994); [49] F. Rami et al, Nucl. Phys. A646,367-384 (1999); [50] J. Y. Ollitrault, Nucl. Phys. A638,195-206 (1998); http://arxiv.org/abs/nucl-th/0512009 http://arxiv.org/abs/nucl-ex/9904003