Datasets:

bluuebunny

/

arxiv_dataset_by_year

like 2

0911.1675

on the equivalence of different approaches for generating multisoliton solutions of the kpii equation

the unexpectedly rich structure of the multisoliton solutions of the kpii equation has been explored by using different approaches, running from dressing method to twisting transformations and to the tau-function formulation. all these approaches proved to be useful in order to display different properties of these solutions and their related jost solutions. the aim of this paper is to establish the explicit formulae relating all these approaches. in addition some hidden invariance properties of these multisoliton solutions are discussed.

introduction the kadomtsev–petviashvili (kp) equation in its version called kpii (ut −6uux1 + ux1x1x1)x1 = −3ux2x2, (1.1) where u = u(x, t), x = (x1, x2) and subscripts x1, x2 and t denote partial derivatives, is a (2+1)-dimensional generalization of the celebrated korteweg–de vries (kdv) equa- tion. there are two inequivalent versions of the kp equations, corresponding to the choice for the sign in the rhs as in (1.1) and to the alternative choice, which is referred to as kpi equation. the kp equations, originally derived as a model for small-amplitude, long-wavelength, weakly two-dimensional waves in a weakly disper- sive medium [1], were known to be integrable since the beginning of the 1970s [2,3], and can be considered as prototypical (2+1)-dimensional integrable equations. the kpii equation is integrable via its association to the operator l(x, ∂x) = −∂x2 + ∂2 x1 −u(x), (1.2) which defines the well known equation of heat conduction, or heat equation for short. the spectral theory of the operator (1.2) was developed in [4–7] in the case of a real potential u(x) rapidly decaying at spatial infinity, which, however, is not the most interesting case, since the kpii equation was just proposed in [1] in order to deal with 1 two dimensional weak transverse perturbation of the one soliton solution of the kdv. in fact, kpii admits a one soliton solution of the form u(x, t) = −(a −b)2 2 sech2 (a −b) 2 x1 + (a2 −b2) 2 x2 −2(a3 −b3)t)  , (1.3) where a and b are real, arbitrary constants. multisoliton solutions have also been ob- tained by different methods, in [8] trough hirota method, in [9] and [10] via dressing, in [11] by the wronskian technique and in [12] by using darboux transformations. in addition, in contrast with the kpi equation, also non elastic and resonant scattering of solitons was described in [13–17]. the problem of finding the most general n-soliton solution and their interactions has recently attracted a great deal of attention. in [18] a sort of dressing method was applied to "superimpose" n-solitons to a generic, smooth and decaying background and to obtain the corresponding jost solutions. there, the two soliton case was studied in details, showing that solitons can interact inelasti- cally and that they can be created and annihilated. by using a finite dimensional version of the sato theory for the kp hierarchy [19], in a series of papers [20–24], it was shown that the general n-soliton solution can be written in term of τ-functions and their structure was studied in details, showing that they exhibit nontrivial spatial interaction patterns, resonances and web structures. a survey of these results is given in [25] and applications to shallow water waves in [25,26]. in [27] solutions correspond- ing to n solitons "superimposed" to a generic smooth decaying background and the corresponding jost solutions were constructed by means of twisting transformations. a spectral theory of the heat operator (1.2) that also includes multisolitons has to be built. in [28] the inverse scattering transform for a perturbed one-soliton potential was derived. in [29] the initial value problem for the kpii equation with data not decaying along a line was linearized. however, the case of n-solitons is still open. in solving the analogous problem for the nonstationary schr ̈ odinger operator, asso- ciated to the kpi equation, the extended resolvent approach was introduced [30]. accordingly, in order to solve the spectral problem for the heat operator, when the potential u(x) describes n solitons, one needs to find among different procedures used in deriving these potentials just that one that can be exploited in building the corre- sponding extended resolvent. therefore, one needs to explore these different available procedures and their interrelation, which is the goal of this paper. the paper is organized as follows. in section 2 we introduce basic notations and sketch some basic topics of the standard scattering problem of operator (1.2) for the case of decaying potential. in section 3 we review the multisoliton potentials con- structed in [18], solving a ∂-problem given by a rational transformation of generic spectral data. in section 4 we consider the multisoliton potentials obtained in [27] by means of twisting transformations and we study their connection with the potentials given in section 3. in section 5 we derive some alternative, more symmetric repre- sentations of the pure solitonic potentials, and we show that they coincide with the representation obtained in [20–24] by using the τ-function approach. in section 6 a symmetric representation for the jost solutions is also obtained, showing that they can be obtained by a miwa shift [31]. finally, we discuss the invariance properties of the multisoliton potentials with respect to transformations of the soliton parameters. 2 2 background theory the kpii can be expressed as compatibility condition of a lax pair, [l, t ] = 0, where l is the heat operator defined in (1.2) and t is given by t (x, ∂x, ∂t) = ∂t + 4∂3 x1 −6u∂x1 −3ux1 −3∂−1 x1 ux2. (2.1) since the heat operator is not self-dual, one has to consider at the same time its dual ld (x, ∂x) = ∂x2 + ∂2 x1 −u(x) and, then, introduce the jost solution φ(x, k) and the dual jost solution ψ(x, k) obeying equations l(x, ∂x)φ(x, k) = 0, ld(x, ∂x)ψ(x, k) = 0, (2.2) where k is an arbitrary complex variable, playing the role of a spectral parameter. it is convenient to normalize the jost and dual jost solutions multiplying them by an exponential function as follows χ(x, k) = ei k x1+k2 x2φ(x, k), ξ(x, k) = e−i k x1−k2 x2ψ(x, k). (2.3) these functions satisfy the differential equations (−∂x2 + ∂2 x1 −2i k ∂x1 −u(x))χ(x, k) = 0, (2.4a) (∂x2 + ∂2 x1 + 2i k ∂x1 −u(x))ξ(x, k) = 0, (2.4b) and are chosen to obey normalization conditions at k-infinity lim k→∞χ(x, k) = 1, lim k→∞ξ(x, k) = 1, (2.5) so that, by (2.4), they are related to the potential u(x) of the heat equation by means of the relations u(x) = −2i lim k→∞k ∂x1χ(x, k) = 2i lim k→∞k ∂x1ξ(x, k). (2.6) the reality of the potential u(x), that we always assume here, is equivalent to the conjugation properties χ(x, k) = χ(x, −k), ξ(x, k) = ξ(x, −k). (2.7) in the case of a potential u(x) ≡u0(x) rapidly decaying at spatial infinity, according to [4–7], the main tools in building the spectral theory of the operator (1.2), as in the one dimensional case, are the integral equations, whose solutions define the related normalized jost solution χ0(x, k), and ξ0(x, k), i.e., χ0(x, k) = 1 + z dx′ g0(x −x′, k)u0(x′)χ0(x′, k), (2.8a) ξ0(x, k) = 1 + z dx′ g0(x′ −x, k)u0(x′)ξ0(x′, k), (2.8b) where g0(x, k) = −sgn x2 2π z dα θ(α(α −2 kr)x2)eiαx1−α(α−2 k)x2, (2.9) 3 is the green's function of the bare heat operator, (−∂x2 + ∂2 x1)g0(x, k) = δ(x). thanks to (2.8), the functions χ0 and ξ0 have the following asymptotic behaviour on the x-plane lim x→∞χ0(x, k) = 1, lim x→∞ξ0(x, k) = 1. (2.10) however, if the potential u(x) does not decay at spatial infinity, as it is the case when line soliton solutions are considered, the integral equations (2.8) are ill-defined and one needs a more general approach. a spectral theory of the kpii equation that also includes solitons has been investigated using the resolvent approach. in this framework it was possible to develop the inverse scattering transform for a solution describing one soliton on a generic background [28], and to study the existence of the (extended) resolvent for (some) multisoliton solutions [27]. the general theory is, to some extent, still a work in progress. in this paper, however, we consider only different approaches to the construction of soliton solutions and the corresponding jost solutions. 3 multisoliton potentials via dressing method in [18] we used a sort of dressing method to construct a potential u(x) describing n solitons superimposed to a generic background potential u0(x). more precisely, we considered a potential u(x) with spectral data obtained by a rational transformation of the spectral data of a generic background u0(x) and we transformed the problem of finding this u(x) into a ∂-problem on the corresponding jost solutions, which was solved by a dressing procedure. the rational transformation was chosen to depend on n pairs of real distinct parameters aj, bj (j = 1, . . . , n) and the jost solutions φ(x, k) and ψ(x, k) were required to obey suitable analyticity properties, normalization like in (2.3), (2.5) and conjugation property (2.7). the transformed potential u(x) is given by the following formula u(x) = u0(x) −2∂2 x1 det[c + f(x)], (3.1) where c is a real n × n constant matrix and f(x) is an n × n matrix function of x with entries flj(x) = x1 z (bl−aj)∞ y2=x2 dy1 ψ0(y, ibl)φ0(y, iaj), l, j = 1, . . . , n, (3.2) where φ0(x, k) and ψ0(x, k) are the jost and dual jost solutions of equations (2.2) with potential u0(x). notice that the matrix elements of f(x) are given in terms of values of the so-called cauchy–baker–akhiezer function [32] at points aj and bl with j, l = 1, . . . , n. in [18] we have shown that the matrix c does not need to be either regular, or diagonal. we also mentioned that in order to obtain a real potential it is enough to consider a more general situation, with complex parameters aj and bj such that aj = ̄ aπa(j), bj = ̄ bπb(j), j = 1, . . . , n, (3.3) where the bar denotes the complex conjugate and πa, πb are some permutations of the indices (with a proper modification of the constraints on the constant matrix c). 4 however, this case is essentially more complicated to be investigated and we do not consider it here. if, a posteriori, the background u0 is taken to be identically zero, then the eigen- functions in (3.2) are pure exponential functions, i.e., φ0(x, iaj) = eaj(x), ψ0(x, ibl) = e−bl(x), where for the future convenience we introduced aj(x) = ajx1 + a2 jx2, bl(x) = blx1 + b2 l x2, j, l = 1, 2, . . . , n. (3.4) then (3.2) takes the form flj(x) = e−bl(x)λljeaj(x), (3.5) with λ the cauchy matrix λlj = 1 aj −bl . (3.6) notice that in [18] the time dependence was not specified, but this simply amounts to taking into account the time dependence of the original jost solutions φ0(x, k) and ψ0(x, k) fixed by the choice of the second lax operator (2.1). in the case of u0(x, t) ≡0 this means that we have to use aj(x, t) = ajx1 + a2 jx2 −4a3 jt, bl(x, t) = blx1 + b2 l x2 −4b3 l t, (3.7) instead of (3.4). in [18] we addressed the problem of regularity of the potentials, as well as of their asymptotic behaviour, only for the case n = 2, i.e., for 2-soliton potentials. in this particular case, we were able to formulate the conditions on the 2 × 2 matrix c that guarantee the regularity of the potential u(x), and we showed that at large distances in the x-plane the potential decays exponentially fast except along certain specific rays, where it has a solitonic one dimensional behaviour of the form (1.3). the classification of the 2-soliton potentials obtained in [18] was then successively obtained and generalized to the case of n-soliton in [24,25]. 4 multisoliton potentials via twisting transforma- tions 4.1 equivalence of the n-soliton potentials derived in [18] and in [27] in this section we consider pure soliton potentials of the heat equation obtained by twisting transformations. the details of the construction of the twisting operators and of the corresponding potentials can be found in [27]. the transformation of a generic (smooth, decaying at infinity) background potential u0(x) into a new potential u(x), which describes n-solitons "superimposed" to the background u0(x), is parameterized by two sets of real parameters {a1, . . . , ana} and {b1, . . . , bnb}, which we assume all distinct, and an na × nb real matrix c. in this case we allow na and nb to be not necessarily equal, but na, nb ≥1, (4.1) 5 and denote n = max{na, nb}. the pure n-soliton potential follows directly from the expressions derived in [27] by taking u0(x) ≡0. precisely, we have u(x) = −2∂2 x1 log τ1(x), (4.2) with τ1(x) = det(ena + c f(x)) ≡det(enb + f(x)c), (4.3) where we introduced the na × na (resp. nb × nb) identity matrix ena (resp. enb), the diagonal matrices ea(x) = diag{eaj(x)}na j=1, e−b(x) = diag{e−bl(x)}nb l=1, (4.4) see (3.4), and the nb × na matrix function f(x) = ∥flj(x)∥j=1,...,na l=1,...,nb , where nota- tion (3.5) was used, so that f(x) = e−b(x)λea(x), (4.5) with λ a nb × na constant matrix with elements given in (3.6). let us show now that the potentials given in (3.1) in terms of n × n matrices c and f coincide with those obtained by the twisting transformations and expressed by means of matrices c and f in (4.3). this is obvious when na = nb and matrices c and c are nonsingular. then, it is enough to put c = c−1 so that the determinants in (3.1) and (4.3) are equal up to an unessential constant factor. the generic situation is a bit more complicated. in order to make explicit the size of the matrices involved we use here notation an for a n × n matrix a. let us consider, for definiteness, the case na ≤nb ≡n. let cn denote the n × n matrix constructed by adding nb −na zero rows to the matrix c. taking into account that the product fc in the second equality in (4.3) is an n × n-matrix, it is clear that (f)ncn = fc, where fn = (e−b)nλn(ea)n (4.6) and where the parameters aj with j = na + 1, na + 2, . . . , n in (ea)n and in λn can be chosen arbitrarily, provided they are different from the parameters bl. it is well known that the cauchy matrix λn is invertible and that λ−1 n = gn e λnfn, where the matrix e λn is obtained from the matrix λn by renaming aj the bj and vice-versa and where gn and fn are two invertible diagonal matrices. thus, for the second determinant in (4.3) we have det(enb + fc) = det(en + fn cn) = det(e−bλeagf)n det (e−a)n e λn(eb)n + g−1 n cnf −1 n
. the factor det(e−bλeagf)n does not contribute to the potential, due to the deriva- tive in (4.2). moreover, thanks to (3.5) and (4.6), the n ×n matrix e−a(x)e λeb(x) coin- cides with the n ×n-matrix f(x) in (3.5), once parameters aj and bj are exchanged. therefore, relations (4.3) and (4.2) indeed generate the same potential provided cn = gncfn (4.7) and parameters aj are renamed bj and vice-versa. 6 remark 4.1 in the case of a nonzero background potential u0(x) one can prove that the two potentials obtained by the two approaches are equivalent, up to an additional term ∂2 x1 log det fn(x). remark 4.2 here and below we omitted the time dependence as it can be easily switched on by using (3.7) instead of (3.4). 4.2 regularity conditions for the potential in view of the discussion on the necessary and sufficient conditions required to guar- antee the regularity of the multisoliton solutions of the equations in the hierarchy of the kpii equation, we recover here, in an equivalent formulation, sufficient conditions for regularity of multisoliton potentials of the heat equation and, then, of multisoliton solutions of kpii, already given in [21,25]. let us consider again the second equality in (4.3). we have det(enb + fc) = nb x n=0 x 1≤l1<l2<***<ln≤nb (fc)  l1, l2, . . . , ln l1, l2, . . . , ln  . (4.8) here and in the following a  l1, l2, . . . , ln l1, l2, . . . , ln  denotes the minor of the matrix a obtained by selecting rows l1, l1, . . . , ln and columns j1, j2, . . . , jn. the principal minor of the product fc in (4.8) can be written by the binet–cauchy formula as (fc)  l1, l2, . . . , ln l1, l2, . . . , ln  = x 1≤j1<j2<***<jn≤na n y m=1 e−blm(x) ! n y m=1 eajm(x) ! × λ  l1, l2, . . . , ln j1, j2, . . . , jn  c  j1, j2, . . . , jn l1, l2, . . . , ln  . (4.9) taking into account that f is a nb × na-matrix and c a na × nb-matrix we get that all minors of fc with n > min na, nb are zero. recalling that the term corresponding to n = 0 in (4.8) is 1, i.e., greater than zero, we deduce that a sufficient condition for having a regular solution is that the real matrix c satisfies the following characterization conditions λ  l1, l2, . . . , ln j1, j2, . . . , jn  c  j1, j2, . . . , jn l1, l2, . . . , ln  ≥0, (4.10) for any 1 ≤n ≤min{na, nb} and all minors, i.e., any choice of 1 ≤j1 < j2 < * * * < jn ≤na and 1 ≤l1 < l2 < . . . ln ≤nb. in order to get a nontrivial potential under substitution of τ1 in (4.2) it is necessary then to impose the condition that at least one of inequalities in (4.10) is strict. now, since any submatrix of a cauchy matrix λ is itself a cauchy matrix, for an arbitrary minor of λ we have λ  l1, l2, . . . , lp j1, j2, . . . , jp  = y 1≤i<m≤p (aji −ajm) (blm −bli) p y i,m=1 (aji −bjm) . (4.11) 7 then, we deduce that there are orders of the parameters a's and b's for which all minors of the matrix λ are positive. for instance, if a1 < a2 < * * * < ana−1 < ana < b1 < b2 < * * * < bnb−1 < bnb, (4.12) the regularity conditions (4.10) become c  l1, l2, . . . , ln j1, j2, . . . , jn  ≥0, (4.13) i.e., all minors of the matrix c must be non negative. these matrices are called totally nonnegative matrices [34]. obviously one obtains the same result by using the first equality in (4.3) and the binet–cauchy formula enables to check directly that the two determinants in (4.3) are equal. if we consider the multi-time multisoliton solution of the entire hierarchy related to the kpii equation obtained by the miwa shift (see [31]), as reported in (6.22), we get again an expansion of the form (4.9), where the exponents are now independent and we conclude that the regularity condition (4.10) is also necessary. 5 equivalence with the τ-function representation 5.1 determinant representations for the potential and jost so- lutions in [27] we derived the transformed potential u(x) (see (4.2 and (4.3)) and the jost solutions of the direct and dual problems (2.2) related to this potential. the corre- sponding normalized jost solutions are expressed as χ(x, k) = 1 + na x j=1 nb x l,l′=1 eaj(x)cjl′(enb + f(x)c)−1 l′l e−bl(x) bl + i k ≡1 + na x j,j′=1 nb x l=1 eaj(x)(ena + c f(x))−1 jj′ cj′le−bl(x) bl + i k , (5.1a) ξ(x, k) = 1 −i na x j=1 nb x l,l′=1 eaj(x) aj + i kcjl(enb + f(x)c)−1 ll′ e−bl(x) ≡1 − na x j,j′=1 nb x l=1 eaj(x) aj + i k(ena + c f(x))−1 jj′ cj′le−bl(x). (5.1b) in analogy to (4.3), all relations are given in two equivalent forms. while it is enough to keep only one of them, we continue to consider both of them to highlight the symmetry property of the whole construction with respect to the numbers na and nb, which play the role of topological charges. in order to obtain a representation of (4.3) and (5.1) in terms of τ-functions we need to perform some simple algebraic operations. first, thanks to the standard identity for the determinant of a bordered matrix, we can rewrite (5.1) as χ(x, k) = τ1,χ(x, k) τ1(x) , ξ(x, k) = τ1,ξ(x, k) τ1(x) , (5.2) 8 where τ1 is given in (4.3) and τ1,χ(x, k) = det      enb + f(x)c −e−b∗(x) b∗+ i k na x j=1 eaj(x)cj∗ 1      (5.3a) ≡det   ena + c f(x) − nb x l=1 c∗l e−bl(x) bl + i k ea∗(x) 1   , (5.3b) τ1,ξ(x, k) = det    enb + f(x)c e−b∗(x) na x j=1 eaj(x) aj + i kcj∗ 1    (5.3c) ≡det      ena + c f(x) nb x l=1 c∗l e−bl(x) ea∗(x) a∗+ i k 1     . (5.3d) in the above formulas, in the row and column bordering matrices enb + f c and ena + c f we use the subscript ∗to denote an index running, respectively, from 1 to nb and from 1 to na. let us notice that the function τ1 in (4.3) can also be obtained as a limiting value, i.e., τ1(x) = lim k→∞τ1,χ(x, k) = lim k→∞τ1,ξ(x, k). (5.4) using (2.3) to go back from (5.2) and (5.3) to the jost solutions φ(x, k) and ψ(x, k), one can check that they have poles, respectively, at k = ibl (l = 1, . . . , nb) and k = iaj (j = 1, . . . , na) with residua φbl(x) = res k=ibl φ(x, k), ψaj(x) = res k=iaj ψ(x, k), (5.5) which obey the relations φbl(x) = −i na x j=1 φ(x, iaj)cjl, ψaj(x) = i nb x l=1 cjlψ(x, ibl). (5.6) notice that these equations, together with the requirement of analyticity and the normalization condition (2.10), allow one to reconstruct the normalized jost solutions in (5.2) and (5.3) and, then, via (2.6) the potential u(x) in (4.2) and (4.3). next, we rewrite (5.3) as τ1,χ(x, k) = nb y l=1 e−bl(x) bl + i k ! det    (b∗+ i k)[eb(x) + λea(x)c] −1∗ na x j=1 eaj(x)cj∗ 1    ≡   na y j=1 eaj(x)  det   e−a(x) + ce−b(x)λ − nb x l=1 c∗l e−bl(x) bl + i k 1∗ 1   , 9 τ1,ξ(x, k) = nb y l=1 e−bl(x) ! det    eb(x) + λea(x)c 1∗ na x j=1 eaj(x) aj + i kcj∗ 1    ≡   na y j=1 eaj(x) aj + i k  det   [e−a(x) + ce−b(x)λ](a∗+ i k) nb x l=1 c∗le−bl(x) 1∗ 1   . by elementary transformations of the matrices on the right-hand sides, we can reduce them to a form where rows and columns with all elements equal to 1 or −1 are transformed into rows and columns with all elements equal to 0 except 1 at the last place. in order to present the result of these transformations we introduce τ2,χ(x, k) = nb y l=1 ebl(x)(bl + i k) ! τ1,χ(x, k), (5.7a) τ ′ 2,χ(x, k) =   na y j=1 e−aj(x) aj + i k  τ1,χ(x, k), (5.7b) τ2,ξ(x, k) = nb y l=1 ebl(x) bl + i k ! τ1,ξ(x, k), (5.7c) τ ′ 2,ξ(x, k) = na y l=1 e−aj(x)(aj + i k) ! τ1,ξ(x, k). (5.7d) then, τ2,χ(x, k) = det  δll′(bl + i k)ebl(x) + na x j=1 aj + i k aj −bl eaj(x)cjl′   nb l,l′=1 , (5.8a) τ ′ 2,χ(x, k) = det δjj′ e−aj(x) aj + i k + nb x l=1 cjle−bl(x) (aj′ −bl)(bl + i k) !na j,j′=1 , (5.8b) τ2,ξ(x, k) = det  δll′ ebl(x) bl + i k + na x j=1 eaj(x)cjl′ (aj −bl)(aj + i k)   nb l,l′=1 , (5.8c) τ ′ 2,ξ(x, k) = det δjj′(aj + i k)e−aj(x) + nb x l=1 cjl(bl + i k) aj′ −bl e−bl(x) !na j,j′=1 . (5.8d) let us now introduce the limits τ2(x) = lim k→∞(i k)−nbτ2,χ(x, k) = lim k→∞(i k)nbτ2,ξ(x, k), τ ′ 2(x) = lim k→∞(i k)naτ ′ 2,χ(x, k) = lim k→∞(i k)−naτ ′ 2,ξ(x, k), 10 that, thanks to (5.8), have the following explicit expressions τ2(x) = det  δll′ebl(x) + na x j=1 eaj(x)cjl′ aj −bl   nb l,l′=1 , (5.9a) τ ′ 2(x) = det δjj′e−aj(x) + nb x l=1 cjle−bl(x) aj′ −bl !na j,j′=1 . (5.9b) by (5.4) we have τ2(x) = nb y l=1 ebl(x) ! τ1(x), τ′ 2(x) =   na y j=1 e−aj(x)  τ1(x). (5.10) both functions in (5.9) are equivalent, in the sense that they generate the same po- tential u(x) = −2∂2 x1 log τ2(x) = −2∂2 x1 log τ ′ 2(x). (5.11) for the functions χ(x, k) and ξ(x, k) (see (2.3)) we get by (5.2) and (5.7) χ(x, k) = nb y l=1 (bl + i k)−1 ! τ2,χ(x, k) τ2(x) ≡   na y j=1 (aj + i k)  τ ′ 2,χ(x, k) τ ′ 2(x) , (5.12a) ξ(x, k) = nb y l=1 (bl + i k) ! τ2,ξ(x, k) τ2(x) ≡   na y j=1 (aj + i k)−1  τ ′ 2,ξ(x, k) τ ′ 2(x) . (5.12b) notice that functions in (5.8) can be obtained from functions in (5.9) by performing the following substitutions τ2(x) →τ2,χ(x, k) replacing eaj →eaj(aj + i k), ebl →ebl(bl + i k) (5.13a) τ ′ 2(x) →τ ′ 2,χ(x, k) replacing eaj →eaj(aj + i k), ebl →ebl(bl + i k), , (5.13b) τ2(x) →τ2,ξ(x, k) replacing eaj → eaj aj + i k, ebl → ebl bl + i k, (5.13c) τ ′ 2(x) →τ ′ 2,ξ(x, k) replacing eaj → eaj aj + i k, ebl → ebl bl + i k. (5.13d) these rules enable to significantly shorten the list of formulas given below (see also remark 6.1). remark 5.1 notice that under the transformation x →−x, na ← →nb and a ← →b the function χ is transformed into ξ and viceversa. 5.2 symmetric representations for the potential and the com- parison with the τ-function approach here we prove that the expression (5.11) for the potential is equivalent to that one obtained by means of the τ-function approach in the series of papers quoted in the introduction and surveyed in [25]. 11 we already mentioned that the double representations for the potential and the jost solutions derived above highlight the symmetric role played by the parameters aj and bl. to better exploit this fact we introduce na + nb (real) parameters {κ1, . . . , κna+nb} = {a1, . . . , ana, b1 . . . , bnb}, (5.14) and, by analogy with (3.4), we introduce kn(x) = κnx1 + κ2 nx2, n = 1, . . . , na + nb, (5.15) so that kn(x) = an(x), n = 1, . . . , na, and kn(x) = bn−na(x), n = na +1, . . ., na + nb. according to (3.7) the time dependence is taken into account simply by adding a term −4κ3 nt to the rhs of (5.15). let d denote an na × nb real matrix with elements given in terms of the elements of the matrix c as djl = nb y l′=1 (aj −bl′)−1cjl nb y l′=1, l′̸=l (bl −bl′), j = 1, . . . , na, l = 1, . . . , nb. (5.16) let us also introduce the constant (na + nb) × nb- and na × (na + nb)-matrices d and d ′ with, respectively, the following block structures d =  d enb  , d ′ = (ena, −d) , (5.17) and the constant, diagonal, real (na + nb) × (na + nb)-matrix γ = diag    na+nb y n′=1,n′̸=n (κn −κn′)−1, n = 1, . . . , na + nb   . (5.18) let us prove now that with one more rescaling of τ2(x) and τ ′ 2(x): τ(x) =   y 1≤l<l′≤nb (bl′ −bl)  τ2(x), (5.19a) τ ′(x) =   y 1≤j<j′≤na (aj −aj′)−1     na y j=1 nb y l=1 (aj −bl)−1  τ ′ 2(x), (5.19b) we get instead of (5.9) τ(x) = det k ek(x) d
y 1≤l<l′≤nb (bl −bl′) , τ′(x) = det d ′ e−k(x)γ k ′
y 1≤j<j′≤na (aj −aj′) , (5.20) where in analogy with (4.4) we introduced the diagonal (na +nb)×(na + nb)-matrix ek(x) = diag{ekn(x)}na+nb n=1 , (5.21) 12 and the new constant nb × (na + nb) and (na + nb) × na matrices k = ∥kln ∥, kln = nb y l′=1, l′̸=l (κn −bl′), l = 1, . . . , nb, n = 1, . . . , na + nb, (5.22) k ′ = ∥k′ nj ∥, k′ nj = na y j′=1, j′̸=j (aj′ −κn), n = 1, . . . , na + nb, j = 1, . . . , na. (5.23) notice that they have a block structure since kl,na+l′ and k ′ j′j (l, l′ = 1, . . . , nb, j, j′ = 1, . . . , na) are diagonal submatrices. we also emphasize that, since na, nb ≥1, as stated in (4.1), the constant matrices k, k ′, d and d ′ are not squared and, therefore, the determinants cannot be decomposed into products of determinants, which would imply u(x) ≡0. in order to prove (5.20), let us notice that k ek(x) d
nb l,l′=1 ≡   na+nb x n=1   nb y l′′=1, l′′̸=l (κn −bl′′)  ekn(x) dnl′   nb l,l′=1 = na x j=1 eaj(x)djl′ nb y l′′=1, l′′̸=l (aj −bl′′) + nb x l′′′=1 ebl′′′(x)δl′′′,l′ nb y l′′=1, l′′̸=l (bl′′′ −bl′′), where (5.17), (5.21) and (5.22) were used. all terms in the last sum vanish except for l′′′ = l. thus by (5.16) we get det k ek(x) d
nb l,l′=1 = (−1)nb(nb−1)/2   y 1≤l<l′≤nb (bl′ −bl)2  det  δll′′ebl(x) + na x j=1 eaj(x)cjl′ aj −bl   nb l,l′=1 . the determinant in the rhs coincides with the determinant in (5.9a), which proves the first equality in (5.20). the second one is reduced to (5.9b) by analogy, where also the explicit expression for matrix γ in (5.18) must be taken into account. by considering convenient linear operations on the rows and columns of the matri- ces in (5.20) one can get more symmetric expressions of τ(x) and τ ′(x) which involve only κ's. let us consider the first equality in (5.20). we can subtract the last row of matrix k from all the preceding rows. then the l-th (l > 1) row will read as (bl −bnb) qnb−1 l′=1, l′̸=l(κn −bl′). each factor bl −bnb for l = 1, . . . , nb can then be ex- tracted from the determinant and we repeat the same procedure with the last but one row, and up to the second one. the first row will then have all 1's, and the second row will have entries κl −b1 for l = 1, . . . , na + nb. then one can easily shift κl −b1 →κl by using the first row. a similar transformation can be used to transform the generic element of the subsequent rows into κj n. finally, instead of (5.20) one obtains τ(x) = det  v ek(x) d  , τ′(x) = det  d ′ e−k(x)γ v ′ , (5.24) 13 where by v and v ′ we denote the "incomplete vandermonde matrices," i.e., the nb × (na + nb)- and (na + nb) × na-matrices given as v =    1 . . . 1 . . . . . . κnb−1 1 . . . κnb−1 na+nb   , v ′ =    1 . . . κna−1 1 . . . . . . 1 . . . κna−1 na+nb   . (5.25) remark 5.2 in the expressions (5.20) all objects with the exception of kn(x) are invariant with respect to an overall shift of all parameters a's and b's (or, equivalently, all κ's) by the same constant, while in (5.25) this invariance is not obvious. in fact, following the same procedure used for transforming (5.20) into (5.25) one can get matrices v and v ′ constructed from powers of κn + z instead of κn, where z is totally arbitrary. the expressions for the potential are invariant with respect to rescaling (5.10) and (5.19) and, consequently, we have by (4.2) u(x) = −2∂2 x1 log τ(x) = −2∂2 x1 log τ ′(x). (5.26) this twofold expression for the potential follows also directly, as thanks to (5.10), (5.14), (5.15) and (5.19), τ(x) = (−1)nanb+na(na−1)/2 na+nb y n=1 ekn(x) ! v (κ1, . . . , κna+nb)τ ′(x), (5.27) where v denotes the vandermonde determinant v (κ1, . . . , κna+nb) = det    1 . . . 1 . . . . . . κna+nb−1 1 . . . κna+nb−1 na+nb    ≡ y 1≤m<n≤na+nb (κn −κm). (5.28) thus, we have shown the equivalence of representations (4.2) and (5.11) for the potential with those expressed as determinants of product of three and four matrices given in (5.24). these representations coincide with the representations obtained using the τ-function approach in [21] and studied in detail in [22–25]. notice that the special block form of the matrices d and d ′ in (5.17), in general, is not preserved, when in the determinants (5.24) we perform a renumbering of the parameters κn, for instance in order to have κ1 < κ2 < * * * < κna+nb. this problem will be studied in details in section 6.3. 5.3 explicit representation for the tau-functions in order to study the behaviour of the potential and jost solutions, which we plan to perform in a forthcoming publication, it is convenient to derive an explicit represen- tations for the determinants involved, see also [20–25]. these representations involve the maximal minors of matrices d and d ′ (see (5.17)) for which we use the simplified notations: d(n1, . . . , nnb) = d  n1, n2, . . . , nnb 1, 2, . . . , nb  , (5.29) 14 i.e., determinant of the nb × nb-matrix, that consists of {n1, . . . , nnb} rows of the matrix d and all its columns, and d ′(n1, . . . , nna) = d ′  1, 2, . . . , na n1, n2, . . . , nna  , (5.30) i.e., determinant of the na × na-matrix that consists of all rows of the matrix d ′ and {n1, . . . , nna} columns of this matrix. then, by using the binet–cauchy formula for the determinant of a product of matrices and notation (5.28), we can rewrite relations (5.24) in the form τ(x) = x 1≤n1<n2<***<nnb≤na+nb fn1,...,nnb nb y l=1 eknl(x), (5.31a) τ ′(x) = x 1≤n1<n2<***<nna≤na+nb f ′ n1,...,nna na y j=1 e−knj (x), (5.31b) where fn1,n2,...,nnb = v (κn1, . . . , κnnb) d(n1, . . . , nnb), (5.32a) f ′ n1,n2,...,nna = v (κn1, . . . , κnna) d ′(n1, . . . , nna) na y j=1 γnj. (5.32b) from (5.27) it follows that the coefficients f and f ′ in the two expansions (5.31b) and (5.31a) are related by the equation f ′ n1,...,nna = (−1)nanb+na(na−1)/2 fe n1,e n2,...,e nnb v (κ1, . . . , κna+nb), (5.33) where {n1, . . . , nna} and {e n1, e n2, . . . , e nnb} are two disjoint ordered subsets of the set of numbers running from 1 to na + nb. let us mention that in our construction the two equivalent representations for the τ-functions in (5.27) were obtained as a consequence of the two equalities in (2.6), while, of course, they can be proved directly by means of (5.31) and (5.33). see [24], where this property is called duality. remark 5.3 notice that, in agreement with (5.32) and (5.33), we have na y j=1 γnjv (κn1, . . . , κnna) = det π(−1)nanb+na(na−1)/2 v (κe n1, . . . , κe nnb) v (κ1, . . . , κna+nb). (5.34) and d′(n1, . . . , nna) = det πd(e n1, . . . , e nnb), (5.35) where π is the matrix performing the permutation from (κn1, . . . , κnna, κe n1, . . . , κe nnb) to (κ1, . . . , κna+nb). 15 6 jost solutions and invariance properties 6.1 properties of matrices d and d ′ matrices d and d ′ introduced in (5.17) obey rather interesting properties. as follows directly from the definition, they are orthogonal in the sense that d ′ d = 0, (6.1) where zero in the rhs is a na × nb-matrix. moreover, since the matrices d† d = enb + d†d, d ′ d ′† = ena + dd†, (6.2) where † denotes hermitian conjugation of matrices (in fact, transposition here), are invertible, the matrices (see [33]) d
(−1) = (enb + d†d)−1d†, (6.3) d ′
(−1) = d ′†(ena + dd†)−1, (6.4) are, respectively, the left inverse of the matrix d and the right inverse of the matrix d ′, i.e., d
(−1) d = enb, d ′d ′
(−1) = ena. (6.5) products of these matrices in the opposite order give the real self-adjoint (na + nb) × (na + nb)-matrices p = d d
(−1) = d(enb + d†d)−1d
†, (6.6) p ′ = d ′
(−1) d ′ = d ′
†(ena + dd†)−1 d ′, (6.7) which are orthogonal projectors, i.e., p 2 = p, (p ′)2 = p ′, pp ′ = 0 = p ′p, (6.8) and complementary in the sense that p + p ′ = ena+nb. (6.9) orthogonality of the projectors follows from (6.1) and the last equality from obvious relations of the kind (enb + d†d)−1d† = d†(ena + dd†)−1. 6.2 symmetric representations for the jost solutions in order to get a τ-representation for the jost solutions, we use (5.12) and notice that rescaling (5.19) does not modify the substitution rules given in (5.13). in terms of notations (5.14), (5.15) and (5.21) these rules read as τ(x) →τχ(x, k) replacing ek →ek(κ + i k), (6.10a) τ ′(x) →τ ′ χ(x, k) replacing ek →ek(κ + i k), (6.10b) τ(x) →τξ(x, k) replacing ek → ek κ + i k, (6.10c) 16 τ ′(x) →τ ′ ξ(x, k) replacing ek → ek κ + i k, (6.10d) where κ + i k denotes the diagonal (na + nb) × (na + nb)-matrix κ + i k = diag{κ1 + i k, . . . , κna+nb + i k} (6.11) and analogously for the matrix (κ+i k)−1. explicitly these replacements give (see (5.24)) τχ(x, k) = det  v ek(x)(κ + i k) d  , (6.12a) τ ′ χ(x, k) = det  d ′ e−k(x)(κ + i k)−1γ v ′ , (6.12b) τξ(x, k) = det  v ek(x)(κ + i k)−1 d  , (6.12c) τ ′ ξ(x, k) = det  d ′ e−k(x)(κ + i k)γ v ′ , (6.12d) thus, by (5.12) we have nb y l=1 (bl + i k) ! χ(x, k) = τχ(x, k) τ(x) ≡ na+nb y n=1 (κn + i k) ! τ ′ χ(x, k) τ ′(x) , (6.13a) nb y l=1 (bl + i k)−1 ! ξ(x, k) = τξ(x, k) τ(x) ≡ na+nb y n=1 (κn + i k)−1 ! τ ′ ξ(x, k) τ ′(x) . (6.13b) the jost solutions themselves are then given by (2.3) and in order to simplify their analyticity properties it is convenient to renormalize them in the following way φ(x, k) → φ(x, k) nb y l=1 (bl + i k) , ψ(x, k) → nb y l=1 (bl + i k) ! ψ(x, k). (6.14) then, to preserve relations given in (2.3) we also normalize χ(x, k) and ξ(x, k) ac- cordingly, so that instead of (6.13) we have χ(x, k) = τχ(x, k) τ(x) ≡ na+nb y n=1 (κn + i k) ! τ ′ χ(x, k) τ ′(x) , (6.15a) ξ(x, k) = τξ(x, k) τ(x) ≡ na+nb y n=1 (κn + i k)−1 ! τ ′ ξ(x, k) τ ′(x) . (6.15b) now χ(x, k) is a polynomial with respect to k of the order knb and ξ(x, k) is a mero- morphic function of k that becomes a polynomial of the order kna after multiplication by qna+nb n=1 (κn+i k). in other words, now φ(x, k) is an entire function of k and ψ(x, k) is meromorphic with poles at all points k = iκn, n = 1, . . . , na + nb. introducing the discrete values of φ(x, k) at these points as a (na + nb)-row φ(x, iκ) = {φ(x, iκ1), . . . , φ(x, iκna+nb)}, (6.16) 17 and the residuals of ψ(x, k) at these points ψκn(x) = res k=iκn ψ(x, k), (6.17) as a (na + nb)-column ψκ(x) = {ψκ1(x), . . . , ψκna+nb(x)}t, (6.18) we get thanks to (5.14), (5.16) and (5.17) that the relations (5.6) take the more symmetric form φ(x, iκ) d = 0, d ′ ψκ(x) = 0. (6.19) it is necessary to mention that after the renormalization (6.13) the asymptotic condi- tions (2.5) become lim k→∞(i k)−nbχ(x, k) = 1, lim k→∞(i k)nbξ(x, k) = 1, (6.20) and the relations (2.6) take the form u(x) = −2 lim k→∞(i k)−nb+1∂x1χ(x, k) = 2 lim k→∞(i k)nb+1∂x1ξ(x, k). (6.21) finally let us point out that by introducing an infinite set of times, and, precisely, replacing kn(x) with the formal series (see [31]) kn(t1, t2, . . .) = ∞ x j=1 κj ntj, (6.22) appropriate choices of the times tj's provide the multisoliton solutions of any non- linear evolution equation in the hierarchy related to the kpii equation, as well as the corresponding jost solutions. in particular the multisoliton solutions of the kpii equation are obtained by choosing t1 = x1, t2 = x2, t3 = −4t and all highest times equal to zero. remark 6.1 we showed in (6.10) that the jost solutions can be obtained by means of transformations, that are equivalent to the formal miwa shift used in the construc- tion of the baker–akhiezer functions in terms of the τ-functions. in fact, substitu- tions (6.10b) and (6.10a) can be obtained (up to an unessential factor i k) by con- sidering the multi-time τ-function obtained by replacing kn with the infinite formal series in (6.22) and, then, by shifting tj →tj −1/j(i/ k)j. similarly, the substitu- tions (6.10d) and (6.10c) are obtained (again up to an unessential factor −i/ k) by the shifts tj →tj + 1/j(i/ k)j. then, non formal jost solutions of the heat equation with a potential being a solution of kpii are derived by choosing t1 = x1, t2 = x2, t3 = −4t and all highest times equal to zero. 6.3 invariance properties of the multisoliton potential in some cases it is useful to rename the spectral parameters κn →e κn, for instance in such a way that the renamed parameters e κn are ordered as follows e κ1 < e κ2 < * * * < e κn . (6.23) 18 this permutation can be performed by means of a (na + nb) × (na + nb)-matrix π such that (e κ1, . . . , e κna+nb) = (κ1, . . . , κna+nb)π. (6.24) this matrix is unitary, π† = π−1, (6.25) and all rows and columns have one element equal to 1 and all other equal to 0. it is convenient to write this matrix in a block form like π =  π11 π12 π21 π22  , (6.26) where π11 is a na × na-matrix, π22 a nb × nb-matrix, π12 a na × nb-matrix and π21 a nb × na-matrix. then, representations (5.24) under the transformation (6.24) keep the same form if the matrices d and d ′ are transformed as follows d →e d = π d =  d1 d2  , d ′ →e d ′ = d ′ π† = (d ′ 1, −d ′ 2), (6.27) where, from (5.17), we have d1 = π11d + π12, d2 = π21d + π22, (6.28) d ′ 1 = π† 11 −dπ† 12, −d ′ 2 = π† 21 −dπ† 22. (6.29) while the size of blocks in (6.27) are the same as in (5.17), the block structure in general is different. nevertheless, thanks to the unitarity of the matrix π, relation (6.1) remains valid for the transformed matrices, which by (6.27) means that d ′ 1d1 = d ′ 2d2. (6.30) also relations (6.19) are preserved for the transformed quantities and transformed projectors e p and e p ′ can be built. let us mention that the special block structure of the matrices d and d ′ in (5.17) determines them uniquely in correspondence to a given potential. one can release this condition, without changing the potential, by multiplying the matrix d by any nonsingular nb × nb-matrix from the right and the matrix d ′ by any nonsingular na × na-matrix from the left. in fact, determinants of these matrices cancel out in (5.26) and (6.13). viceversa one can use this procedure for bringing e d and e d ′ back from the block structure (6.27) to the special block structure in (5.17). this can be done by using matrices (d2)−1 and (d ′ 1)−1, if they exist, to perform the following transformations e d →e d(d2)−1 =  d1(d2)−1 enb  , e d ′ →(d ′ 1)−1 e d ′ = (ena, −(d ′ 1)−1d ′ 2), (6.31) and, then, by noticing that thanks to (6.30) d1(d2)−1 = (d ′ 1)−1d ′ 2 = e d. (6.32) taking into account that both representations in (5.24) are equivalent, we deduce that matrices d2 and d ′ 1, if invertible, are simultaneously invertible. thus, we proved 19 that in the case of a permutation π such that the matrix d2 (or d ′ 1) is nonsingular, the permutation of κ's is equivalent to a transformation of the matrix d to e d, or, correspondingly, by (5.16), to a transformation of the matrix c. this is always the case when the matrix π has a diagonal block structure, i.e., when π12 = π21 = 0. in this situation both matrices d2 and d ′ 1 are invertible and one can use the above substitution. notice that in this case the permutation of κ's does not mix the original parameters a's with b's (see (5.14)). in general in making a permutation of κ's, say, reducing them to the order given in (6.23), the block structure (5.17) is lost and we can only say that the τ-functions are given by (5.24) and (6.12), where both matrices d and d ′ have at least two nonzero maximal minors. 7 concluding remarks here we described relations between different representations, existing in the literature, of the multisoliton potentials of the heat operator and we derived forms of these representations that will enable, in a forthcoming publication, a detailed study of the asymptotic behaviour of the potentials themselves and the corresponding jost solutions on the x-plane. we also presented various formulations of the conditions that guarantee the regularity of the potential. nevertheless, the essential problems of determining the necessary conditions of regularity of the multisoliton potential is left open. in this context, let us mention the special interest of the specific subclass of potentials satisfying strict inequalities in (4.10), which by (5.31a) is equivalent to requiring that all fn1, . . . ,nb are of the same sign (analogously, by (5.31b) that all f ′ n1, . . . ,na are of the same signs). these conditions identify fully resonant soliton solutions (cf. [20, 21]). when such conditions are imposed, all maximal minors of matrices d and d ′ are different from zero, as follows from (5.32a) and (5.32b) and, then, thanks to the invariance properties discussed above, we can always permute parameters κ's in any way, for instance as in (6.23), and at the same time deal with transformed matrices e d and e d ′, which have a special block structure as in (5.17). acknowledgments this work is supported in part by the grant rfbr # 08-01-00501, grant rfbr–ce # 09-01-92433, scientific schools 795.2008.1, by the program of ras "mathematical methods of the nonlinear dynamics," by infn and by consortium e.i.n.s.t.e.i.n. akp thanks department of physics of the university of salento (lecce) for kind hospitality. references [1] b. b. kadomtsev and v. i. petviashvili, "on the stability of solitary waves in weakly dispersive media," sov. phys. dokl. 192 (1970) 539–541 [2] v. s. dryuma, "analytic solution of the two-dimensional korteweg–de vries (kdv) equation," sov. jetp lett. 19 (1974) 387–388 20 [3] v. e. zakharov and a. b. shabat, "a scheme for integrating the non-linear equa- tions of mathematical physics by the method of the inverse scattering problem," func. anal. appl. 8 (1974) 226–235 [4] m. j. ablowitz, d. bar yacoov and a. s. fokas, "on the inverse scattering trans- form for the kadomtsev-petvishvili equation," stud. appl. math. 69 (1983) 135- 143 [5] v. g. lipovsky, "hamiltonian structure of the kadomtsev–petviashvili–ii equa- tion in the class of decreasing cauchy data," funkts. anal. prilog. 20 (1986) 35-45 [6] m. v. wickerhauser, "inverse scattering for the heat operator and evolutions in 2+1 variables," commun. math. phys. 108 (1987) 67-89 [7] p. g. grinevich and s. p. novikov, "two-dimensional inverse scattering problem at negative energy and generalized analytic functions. i. energy below a ground state," func. anal. appl. 22 (1988) 19–27 [8] j. satsuma, "n-soliton solution of the two-dimensional korteweg-de vries equa- tion," j. phys. soc. jap. 40 (1976) 286–290 [9] s. v. manakov, v. e. zakharov, l. a. bordag, a. r. its, and v. b. matveev, "two-dimensional solitons of the kadomtsev-petviashvili equation and their in- teraction," phys. lett. a 63(1977) 205–206 [10] b.g. konopelchenko, solitons in multidimensions: inverse spectral transform method, world scientific pu. co., singapore (1993) [11] n.c. freeman, j.j.c. nimmo, "soliton-solutions of the korteweg–de vries and kadomstev–petviashvili equations: the wronskian technique," phys lett. 95a (1983) 1–3 [12] v. b. matveev and m. a. salle, darboux transformations and solitons, springer, berlin (1991) [13] j.w. miles, "obliquely interacting solitary waves," j. fluid mech. 79 (1977) 157–169 and "resonantly interacting solitary waves," j. fluid mech. 79 (1977) 171–179 [14] v.e. zakharov, "shock waves propagated on solitons on the surface of a fluid," radiophysics and quantum electronics 29 (1986) 813-817 [15] m. boiti, f. pempinelli, a. k. pogrebkov and b. prinari, "study of some non- decaying solutions for the heat equation" in nonlinearity, integrability and all that. twenty years after needs'79, boiti m, martina l, pempinelli f, prinari b and soliani g eds, pp 42–50, world scientific pu. co., singapore (2000) [16] b. prinari, "on some nondecaying potentials and related jost solutions for the heat conduction equation," inverse problems 16(2000) 589–603 [17] e. medina, "an n soliton resonance solution for the kp equation: interaction with change of form and velocity," lett. math. phys. 62(2002) 91–99 21 [18] m. boiti, f. pempinelli, a. pogrebkov and b. prinari, "towards an inverse scat- tering theory for non decaying potentials of the heat equation," inverse problems 17 (2001) 937–957 [19] m. sato, "soliton equations as dynamical systems on an infinite dimensional grassmanian manifold," rims kokyuroku (kyoto university) 439(1981) 30-46 [20] g. biondini and y. kodama, "on a family of solutions of the kadomtsev– petviashvili equation which also satisfy the toda lattice hierarchy," j. phys. a: math. gen. 36 (2003) 10519–10536 [21] g. biondini and s. chakravarty, "soliton solutions for the kadomtsev-petviashvili ii equation," j. math. phys.47, 033514 (2006) 1–26 [22] g. biondini and s. chakravarty, "elastic and inelastic line-soliton solutions of the kadomtsev–petviashvili ii equation," math. comp. simul. 74 (2007) 237–250 [23] g. biondini, "elastic line-soliton interactions of the kadomtsev-petviashvili ii equation," phys. rev. lett. 99, 064103 (2007) 1–4 [24] s. chakravarty and y. kodama, "classification of the line-soliton solutions of kpii," j. phys. a. 41(2008) 275209 [25] s. chakravarty and y. kodama, "soliton solutions of the kp equation and ap- plication to shallow water waves," stud. app. math. 123(2009) 83-151 [26] g. biondini, k.-i. maruno, m. oikawa, h. tsuji, "soliton interactions of the kadomtsev–petviashvili equation and generation of large-amplitude water waves," stud. app. math. 122 (2009) 377-394 [27] m. boiti, f. pempinelli, a.k. pogrebkov and b. prinari, "building extended re- solvent of heat operator via twisting transformations," theor. math. phys. 159 (2009) 721-733 [28] m. boiti, f. pempinelli, a.k. pogrebkov and b. prinari, "inverse scattering theory of the heat equation for the perturbed 1-soliton potential," j. math. phys. 43 (2002) 1044-1062 [29] j. villarroel and m.j. ablowitz, "the cauchy problem for the kadomtsev- petviashili ii equation with nondecaying data along a line," stud. appl. math.109 (2002) 151-162, and "on the initial value problem for the kpii equation with data that does not decay along a line," nonlinearity17 (2004) 1843-1866 [30] m. boiti, f. pempinelli and a. k. pogrebkov, "scattering transform for the nonstationary schr ̈ odinger equation with a bidimensionally perturbed n-soliton potential," j. math. phys. 47 123510 (2006) 1-43 [31] t. miwa, m. jimbo and e. date, solitons: differential equations, symmetries and infinite dimensional algebra, cambridge university press, cambridge (2000) [32] p.g. grinevich, a.yu. orlov, "virasoro action on riemann surfaces, grassmani- ans, det ∂and segal–wilson τ-function," in "problems of modern quantum field theory," ed. a. a. belavin, a. v. klimyk and a. b. zamolodchikov, springer, new york (1989) 86–106 22 [33] f. r. gantmacher, th ́ eorie des matrices, editions jacques gabay (1990) [34] f.r. gantmacher and m.g. krein, oscillation matrices and kernels and small vi- brations of mechanical systems, ams, providence (2002), oszillationsmatrizen, oszillationskerne und kleine schwingungen mechanischer systeme, akademie- verlag, berlin (1960) 23

0911.1676

universal dynamical decoupling: two-qubit states and beyond

uhrig's dynamical decoupling pulse sequence has emerged as one universal and highly promising approach to decoherence suppression. so far both the theoretical and experimental studies have examined single-qubit decoherence only. this work extends uhrig's universal dynamical decoupling from one-qubit to two-qubit systems and even to general multi-level quantum systems. in particular, we show that by designing appropriate control hamiltonians for a two-qubit or a multi-level system, uhrig's pulse sequence can also preserve a generalized quantum coherence measure to the order of $1+o(t^{n+1})$, with only $n$ pulses. our results lead to a very useful scheme for efficiently locking two-qubit entangled states. future important applications of uhrig's pulse sequence in preserving the quantum coherence of multi-level quantum systems can also be anticipated.

introduction decoherence, i.e., the loss of quantum coherence due to system-environment coupling, is a major obstacle for a variety of fascinating quantum information tasks. even with the assistance of error corrections, decoherence must be suppressed below an acceptable level to realize a useful quantum operation. analogous to refocusing techniques in nuclear magnetic resonance (nmr) studies, the dy- namical decoupling (dd) approach to decoherence sup- pression has attracted tremendous interest. the central idea of dd is to use a control pulse sequence to effectively decouple a quantum system from its environment. during the past years several dd pulse sequences have been proposed. the so-called "bang-bang" control has proved to be very useful [1, 2, 3] with a variety of exten- sions. however, it is not optimized for a given period t of coherence preservation. the carr-purcell-meiboom- gill (cpmg) sequence from the nmr context can sup- press decoherence up to o(t 3) [4]. in an approach called "concatenated dynamical decoupling" [5, 6], the decoher- ence can be suppressed to the order of o(t n+1) with 2n pulses. remarkably, in considering a single qubit subject to decoherence without population relaxation, uhrig's (optimal) dynamical decoupling (udd) pulse se- quence proposed in 2007 can suppress decoherence up to o(t n+1) with only n pulses [4, 7, 8]. in a udd se- quence, the jth control pulse is applied at the time tj = t sin2( jπ 2n+2), j = 1, 2 * * * , n. (1) in most cases udd outperforms all other known dd con- trol sequences, a fact already confirmed in two beautiful experiments [9, 10, 11]. as a dramatic development in ∗[email protected] theory, yang and liu proved that udd is universal for suppressing single-qubit decoherence [12]. that is, for a single qubit coupled with an arbitrary bath, udd works regardless of how the qubit is coupled to its bath. given the universality of udd for suppression of single-qubit decoherence, it becomes urgent to examine whether udd is useful for preserving quantum coher- ence of two-qubit states. this extension is necessary and important because many quantum operations involve at least two qubits. conceptually there is also a big differ- ence between single-qubit coherence and two-qubit co- herence: preserving the latter often means the storage of quantum entanglement. furthermore, because quan- tum entanglement is a nonlocal property and cannot be affected by local operations, preserving quantum entan- glement between two qubits by a control pulse sequence will require the use of nonlocal control hamiltonians. in this work, by exploiting a central result in yang and liu's universality proof [12] for udd in single-qubit systems and by adopting a generalized coherence mea- sure for two-qubit states, we show that udd pulse se- quence does apply to two-qubit systems, at least for pre- serving one pre-determined type of quantum coherence. the associated control hamiltonian is also explicitly con- structed. this significant extension from single-qubit to two-qubit systems opens up an exciting avenue of dy- namical protection of quantum entanglement. indeed, it is now possible to efficiently lock a two-qubit system on a desired entangled state, without any knowledge of the bath. encouraged by our results for two-qubit systems, we then show that in general, the coherence of an arbi- trary m-level quantum system, which is characterized by our generalized coherence measure, can also be preserved by udd to the order of 1+o(t n+1) with only n pulses, irrespective of how this system is coupled with its envi- ronment. hence, in principle, an arbitrary (but known) quantum state of an m-qubit system with m = 2m levels can be locked by udd, provided that the required control 2 hamiltonian can be implemented experimentally. to es- tablish an interesting connection with a kicked multi-level system recently realized in a cold-atom laboratory [13], we also explicitly construct the udd control hamilto- nian for decoherence suppression in three-level quantum systems. this paper is organized as follows. in sec. ii, we first briefly outline an important result proved by yang and liu [12]; we then present our theory for udd in two-qubit systems, followed by an extension to multi-level quantum systems. in sec. iii, we present supporting results from some simple numerical experiments. section iv discusses the implications of our results and then concludes this paper. ii. udd theory for two-qubit and general multi-level systems a. on yang-liu's universality proof for single-qubit systems for our later use we first briefly describe one central result in yang and liu's work [12] for proving the uni- versality of the udd control sequence applied to single- qubit systems. let c and z be two time-independent hermitian operators. define two unitary operator u (n) ± as follows: u (n) ± (t ) = e−i[c±(−1)nz](t −tn) × e−i[c±(−1)(n−1)z](tn −tn−1) * * * × e−i[c∓z](t2−t1)e−i[c±z]t1. (2) yang and liu proved that for tj satisfying eq. (1), we must have  u (n) − † u (n) + = 1 + o(t n+1), (3) i.e., the product of  u (n) − † and u (n) + differs from unity only by the order of o(t n+1) for sufficiently small t . in the interaction representation, zi(t) ≡eictze−ict = ∞ x p=0 (it)p p! [c, [c, ...[c, z]]] | {z } p folds , (4) hence the above expression for u (n) ± can be rewritten in the following compact form u (n) ± (t ) = e−ict j h e−i
t 0 ±fn(t)zi(t)dti , (5) where t is the final time, j denotes the time-ordering operator, and fn(t) = (−1)j, for t ∈(tj, tj+1). (6) as an important observation, we note that though ref. [12] focused on single-qubit decoherence in a bath, eq. (3) was proved therein for arbitrary hermitian operators c and z. this motivated us to investigate under what conditions the unitary evolution operator of a controlled two-qubit system plus a bath can assume the same form as eq. (2). b. decoherence suppression in two-qubit systems quantum coherence is often characterized by the mag- nitude of the off-diagonal matrix elements of the system density operator after tracing over the bath. in single- qubit cases, the transverse polarization then measures the coherence and the longitudinal polarization measures the population difference. such a perspective is often helpful so long as its representation-dependent nature is well understood. in two-qubit systems or general multi- level systems, the concept of quantum coherence becomes more ambiguous because there are many off-diagonal ma- trix elements of the system density operator. clearly then, to have a general and convenient coherence measure will be important for extending decoherence suppression studies beyond single-qubit systems. here we define a generalized polarization operator to characterize a certain type of coherence. specifically, as- sociated with an arbitrary pure state |ψ⟩of our quantum system, we define the following polarization operator, p|ψ⟩≡2|ψ⟩⟨ψ| −i, (7) where i is the identity operator. this polarization oper- ator has the following properties: p2 |ψ⟩= i, p|ψ⟩|ψ⟩= |ψ⟩, p|ψ⟩|ψ⊥⟩= −|ψ⊥⟩, (8) where |ψ⊥⟩represents all other possible states of the sys- tem that are orthogonal to |ψ⟩. hence, if the expectation value of p|ψ⟩is unity, then the system must be on the state |ψ⟩. in this sense, the expectation value of p|ψ⟩ measures how much coherence of the |ψ⟩-type is con- tained in a given system. for example, in the single-qubit case, p|ψ⟩measures the longitudinal coherence if |ψ⟩is chosen as the spin-up state, but measures the transverse coherence along a certain direction if |ψ⟩is chosen as a superposition of spin-up and spin-down states. most im- portant of all, as seen in the following, the generalized polarization operator p|ψ⟩can directly give the required control hamiltonian in order to preserve the quantum coherence thus defined. we now consider a two-qubit system interacting with an arbitrary bath whose self-hamiltonian is given by he = c0. the qubits interact with the environment via the interaction hamiltonian hje = σj xcx,j + σj ycy,j + σj zcz,j for j = 1, 2, where σj x, σj y, and σj z are the stan- dard pauli matrices, and cα,j are bath operators. we 3 further assume that the qubit-qubit interaction is given by h12 = p k,l={x,y,z} cklσ1 kσ2 l , where the coefficients ckl may also depend on arbitrary bath operators. a gen- eral total hamiltonian describing a two-qubit system in a bath hence becomes h = he + h1e + h2e + h12 = c0 + σ1 xcx,1 + σ1 ycy,1 + σ1 zcz,1 + σ2 xcx,2 + σ2 ycy,2 + σ2 zcz,2 + σ1 xσ2 xcxx + σ1 xσ2 ycxy + σ1 xσ2 zcxz + σ1 yσ2 xcyx + σ1 yσ2 ycyy + σ1 yσ2 zcyz + σ1 zσ2 xczx + σ1 zσ2 yczy + σ1 zσ2 zczz. (9) for convenience each term in the above total hamiltonian is assumed to be time independent (this assumption will be lifted in the end). focusing on the two-qubit subspace, the above total hamiltonian is seen to consist of 16 linearly-independent terms that span a natural set of basis operators for all possible hermitian operators acting on the two-qubit sys- tem. this set of basis operators can be summarized as {xi}i=1,2,*** ,16 = {σk ⊗σl}, (10) where σk, σl ∈{i, σx, σy, σz}, with the orthogonality con- dition trace(xjxk) = 4δjk. but this choice of basis oper- ators is rather arbitrary. we find that this operator basis set should be changed to new ones to facilitate operator manipulations. in the following we examine the suppres- sion of two types of coherence, one is associated with non-entangled states and the other is associated with a bell state. 1. preserving coherence associated with non-entangled states let the four basis states of a two-qubit system be |0⟩= | ↑↑⟩, |1⟩= | ↑↓⟩, |2⟩= | ↓↑⟩, and |3⟩= | ↓↓⟩. the projector associated with each of the four basis states is given by |0⟩⟨0| = p0 = 1 4(1 + σ1 z)(1 + σ2 z), |1⟩⟨1| = p1 = 1 4(1 + σ1 z)(1 −σ2 z), |2⟩⟨2| = p2 = 1 4(1 −σ1 z)(1 + σ2 z), |3⟩⟨3| = p3 = 1 4(1 −σ1 z)(1 −σ2 z). (11) as a simple example, the quantum coherence to be pro- tected here is assumed to be p|0⟩= 2|0⟩⟨0| −i. we now switch to the following new set of 16 basis operators, y1 = p|0⟩= 2p0 −i = 1 2(−i + σ1 z + σ2 z + σ1 zσ2 z), y2 = p0 + p1 = 1 2(i + σ1 z), y3 = p0 −p1 + 2p2 = 1 2(i −σ1 z + 2σ2 z), y4 = p0 −p1 −p2 + 3p3 = 1 2(i −σ1 z −σ2 z + 3σ1 zσ2 z), y5 = |1⟩⟨3| + |3⟩⟨1| = 1 2(σ1 x −σ1 xσ2 z), y6 = −i|1⟩⟨3| + i|3⟩⟨1| = 1 2(σ1 y −σ1 yσ2 z), y7 = |2⟩⟨3| + |3⟩⟨2| = 1 2(σ2 x −σ1 zσ2 x), y8 = −i|2⟩⟨3| + i|3⟩⟨2| = 1 2(σ2 y −σ1 zσ2 y), y9 = |1⟩⟨2| + |2⟩⟨1| = 1 2(σ1 xσ2 x + σ1 yσ2 y), y10 = −i|1⟩⟨2| + i|2⟩⟨1| = 1 2(σ1 yσ2 x −σ1 xσ2 y), y11 = |0⟩⟨1| + |1⟩⟨0| = 1 2(σ2 x + σ1 zσ2 x), y12 = −i|0⟩⟨1| + i|1⟩⟨0| = 1 2(σ2 y + σ1 zσ2 y), y13 = |0⟩⟨2| + |2⟩⟨0| = 1 2(σ1 x + σ1 xσ2 z), y14 = −i|0⟩⟨2| + i|2⟩⟨0| = 1 2(σ1 y + σ1 yσ2 z), y15 = |0⟩⟨3| + |3⟩⟨0| = 1 2(σ1 xσ2 x −σ1 yσ2 y), y16 = −i|0⟩⟨3| + i|3⟩⟨0| = 1 2(σ1 xσ2 y + σ1 yσ2 x). (12) using this new set of basis operators for a two-qubit system, the total hamiltonian becomes a linear combi- nation of the yj (j = 1 −16) operators defined above, i.e., h = 16 x j=1 wjyj, (13) where wj are the expansion coefficients that can contain arbitrary bath operators. the above new set of basis operators have the following properties. first, the oper- ator y1 in this set are identical with p|0⟩and hence also satisfies the interesting properties described by eq. (8). second, [yj, y1] = 0, for j = 1, 2, * * * , 10; {yj, y1}+ = 0, for j = 11, 12, * * * , 16, (14) where [*] represents the commutator and {*}+ represents 4 an anti-commutator. third,   10 x i=1 aiyi, 16 x j=11 bjyj  = 16 x j=11 cjyj, 10 x i=1 aiyi !   10 x j=1 bjyj  = 10 x j=1 cjyj, 16 x i=11 aiyi !   16 x j=11 bjyj  = 10 x j=1 cjyj. (15) with these observations, we next split the total uncon- trolled hamiltonian into two terms, i.e., h = h0 + h′, where h0 = w1y1 + w2y2 + * * * + w10y10, (16) and h′ = w11y11 + * * * + w16y16. (17) evidently, we have the anti-commuting relation {y1, h′}+ = 0, (18) an important fact for our proof below. consider now the following control hamiltonian de- scribing a sequence of extended udd π-pulses hc = n x j=1 πδ(t −tj)y1 2 . (19) after the n control pulses, the unitary evolution operator for the whole system of the two qubits plus a bath is given by (ħ= 1 throughout) u(t ) = e−i[h0+h′](t −tn )(−iy1) × e−i[h0+h′](tn −tn−1)(−iy1) * * * × e−i[h0+h′](t3−t2)(−iy1) × e−i[h0+h′](t2−t1)(−iy1) × e−i[h0+h′]t1. (20) we can then take advantage of the anti-commuting rela- tion of eq. (18) to exchange the order between (−iy1) and the exponentials in the above equation, leading to u(t ) = (−iy1)ne−i[h0+(−1)n h′](t −tn) × e−i[h0+(−1)n−1h′](tn−tn−1) * * * × e−i[h0+h′](t3−t2) × e−i[h0−h′](t2−t1) × e−i[h0+h′]t1 = (−iy1)ne−ih0t j h e−i
t 0 fn (t)h′ i(t)dti ≡(−iy1)nu(n) + (t ). (21) here fn(t) is already defined in eq. (6), the second equality is obtained by using the interaction represen- tation, with h′ i(t) ≡eih0thie−ih0t, and the last line defines the operator u(n) + (t ). clearly, u(n) + is exactly in the form of u (n) + defined in eqs. (2) and (5), with h0 replacing c and h′ replacing z. this observation motivates us to define u(n) − (t ) ≡e−ih0t j h e−i
t 0 −fn(t)h′ i(t)dti , (22) which is completely in parallel with u (n) − defined in eq. (5). as such, eq. (3) directly leads to  u(n) − † u(n) + = 1 + o(t n+1). (23) with eq. (23) obtained we can now evaluate the co- herence measure. in particular, for an arbitrary initial state given by the density operator ρi, the expectation value of p|0⟩at time t is given by trace{u(t )ρiu †(t )p|0⟩} = trace{(−iy1)nu(n) + ρi  u(n) + † (iy1)np|0⟩} = trace{(−iy1)nu(n) + ρip|0⟩  u(n) − † (iy1)n} = trace{  u(n) − † u(n) + ρip|0⟩} = trace{ρip|0⟩}  1 + o(t n+1)  , (24) where we have used p|0⟩= y1, y 2 1 = i, and the anti- commuting relation between p|0⟩and h′. equation (24) clearly demonstrates that, as a result of the udd se- quence of n pulses, the expectation value of p|0⟩is pre- served to the order of 1 + o(t n+1), for an arbitrary initial state. if the initial state is set to be |0⟩, i.e., trace{ρip|0⟩} = 1, then the expectation value of p|0⟩ remains to be 1 + o(t n+1) at time t , indicating that the udd sequence has locked the system on the state |0⟩= | ↑↑⟩. in our proof of the udd applicability in preserving the coherence p|ψ⟩associated with a non-entangled state, the first important step is to construct the control operator y1 = p|ψ⟩and then the control hamiltonian hc. as is clear from eq. (8), each application of the control op- erator y1 = p|0⟩leaves the state |0⟩intact but induces a negative sign for all other two-qubit states. it is in- teresting to compare the control operator y1 with what can be intuitively expected from early single-qubit udd results. suppose that the two qubits are unrelated at all, then in order to suppress the spin flipping of the first qubit (second qubit), we need a control operator σ1 z (σ2 z). as such, an intuitive single-qubit-based control hamilto- nian would be hc,single = π 2 n x j=1 δ(t −tj)(σ1 z + σ2 z). (25) 5 this intuitive control hamiltonian differs from eq. (19), hinting an importance difference between two-qubit and single-qubit cases. indeed, here the qubit-qubit inter- action or the system-environment coupling may directly cause a double-flipping error | ↑↑⟩→| ↓↓⟩, which can- not be suppressed by hc,single. the second key step is to split the hamiltonian h into two parts h0 and h′, with the former commuting with y1 and the latter anti- commuting with y1. once these two steps are achieved, the remaining part of our proof becomes straightforward by exploiting eq. (23). these understandings suggest that it should be equally possible to preserve the coher- ence associated with entangled two-qubit states. 2. preserving coherence associated with entangled states consider a different coherence property as defined by our generalized polarization operator p|ψ⟩, with |ψ⟩ taken as a bell state | ̃ 0⟩= 1 √ 2 [| ↑↓⟩+ | ↓↑⟩]. (26) the other three orthogonal basis states for the two-qubit system are now denoted as | ̃ 1⟩, | ̃ 2⟩, | ̃ 3⟩. for example, they can be assumed to be | ̃ 1⟩= 1 √ 2[| ↑↑⟩+ | ↓↓⟩], | ̃ 2⟩= 1 √ 2[| ↑↑⟩−| ↓↓⟩], and | ̃ 3⟩= 1 √ 2[| ↑↓⟩−| ↓↑⟩]. to preserve such a new type of coherence, we follow our early procedure to first construct a control operator ̃ y1 and then a new set of basis operators. in particular, we require ̃ y1 = p| ̃ 0⟩= 2| ̃ 0⟩⟨ ̃ 0| −i = 1 2(−i + σ1 xσ2 x + σ1 yσ2 y −σ1 zσ2 z). (27) we then construct other 9 basis operators that all com- mute with ̃ y1, e.g., ̃ y2 = 1 2(i + σ1 xσ2 x), ̃ y3 = 1 2(i −σ1 xσ2 x + 2σ1 yσ2 y), ̃ y4 = 1 2(i −σ1 xσ2 x −σ1 yσ2 y −3σ1 zσ2 z), ̃ y5 = 1 2(σ1 zσ2 x −σ1 xσ2 z), ̃ y6 = 1 2(σ2 y −σ1 y), ̃ y7 = 1 2(σ2 x −σ1 x), ̃ y8 = −1 2(σ1 yσ2 z −σ1 zσ2 y), ̃ y9 = 1 2(σ1 z + σ2 z), ̃ y10 = −1 2(σ1 xσ2 y + σ1 yσ2 x). (28) the remaining 6 linearly independent basis operators are found to be anti-commuting with ̃ y1. they can be writ- ten as ̃ y11 = 1 2(σ1 x + σ2 x), ̃ y12 = −1 2(σ1 yσ2 z + σ1 zσ2 y), ̃ y13 = 1 2(σ1 xσ2 z + σ1 zσ2 x), ̃ y14 = −1 2(σ1 y + σ2 y), ̃ y15 = 1 2(σ1 z −σ2 z), ̃ y16 = 1 2(σ1 xσ2 y −σ1 yσ2 x). (29) the total hamiltonian can now be rewritten as h = ̃ h0 + ̃ h′, in which ̃ h0 = ̃ w1 ̃ y1 + ̃ w2 ̃ y2 + * * * + ̃ w10 ̃ y10 (30) and ̃ h′ = ̃ w11 ̃ y11 + * * * + ̃ w16 ̃ y16. (31) it is then evident that if we apply the following control hamiltonian, i.e., ̃ hc = n x j=1 πδ(t −tj) ̃ y1 2 = n x j=1 π 4 δ(t −tj)(−i + σ1 xσ2 x + σ1 yσ2 y −σ1 zσ2 z), (32) the time evolution operator of the controlled total system becomes entirely parallel to eqs. (20) and (21) (with an arbitrary operator o replaced by ̃ o). hence, using the n control pulse described by eq. (32), the quantum coherence defined by the expectation value of p| ̃ 0⟩can be preserved up to 1 + o(t n+1), for an arbitrary initial state. if the initial state is already the bell state | ̃ 0⟩(i.e., coincides with the |ψ⟩that defines our coherence measure p|ψ⟩), then our udd control sequence locks the system on this bell state with a fidelity 1 + o(t n+1), no matter how the system is coupled to its environment. the constant term in the control hamiltonian ̃ hc can be dropped because it only induces an overall phase of the evolving state. all other terms in ̃ hc represent two-body and hence nonlocal control. this confirms our initial ex- pectation that suppressing the decoherence of entangled two-qubit states is more involving than in single-qubit cases. we have also considered the preservation of another bell state 1 √ 2[| ↑↓⟩−| ↓↑⟩]. following the same procedure outlined above, one finds that the required udd control hamiltonian should be given by ̃ hc = − n x j=1 π 4 δ(t −tj)(i + σ1 xσ2 x + σ1 yσ2 y + σ1 zσ2 z), (33) 6 which is a pulsed heisenberg interaction hamiltonian. such an isotropic control hamiltonian is consistent with the fact the singlet bell state defining our quantum co- herence measure is also isotropic. c. udd in m-level systems our early consideration for two-qubit systems sug- gests a general strategy for establishing udd in an ar- bitrary m-level system. let |0⟩, |1⟩, * * * , |m −1⟩be the m orthogonal basis states for an m-level system. their associated projectors are defined as pj ≡|j⟩⟨j|, with j = 0, 1, * * * , m −1. without loss of generality we con- sider the quantum coherence to be preserved is of the |0⟩- type, as characterized by p|0⟩= 2|0⟩⟨0| −i. as learned from sec. ii-b, the important control operator is then v1 = p|0⟩= 2p0 −i, (34) with v 2 1 = i. a udd sequence of this control operator can be achieved by the following control hamiltonian ̃ hc = n x j=1 πδ(t −tj)v1 2 . (35) in the m-dimensional hilbert space, there are totally m 2 linearly independent hermitian operators. we now divide the m 2 operators into two groups, one commutes with v1 and the other anti-commutes with v1. specifi- cally, the following m −1 operators v2 = p0 + p1, v3 = p0 −p1 + 2p2, * * * vm = p0 −p1 −... −pm−2 + (m −1)pm−1 (36) evidently commutes with v1. in addition, other (m −2)(m −1) basis operators, denoted vm+1, vm+2, * * * , vm+(m−2)(m−1), also commute with v1. this is the case because we can construct the following 1 2(m −2)(m −1) basis operators |k⟩⟨l| + |l⟩⟨k| (37) with 0 < k < m and k < l < m. the other 1 2(m −2)(m −1) basis operators that commute with v1 are constructed as −i|k⟩⟨l| + i|l⟩⟨k|, (38) also with 0 < k < m and k < l < m. all the remaining 2(m −1) basis operators are found to anti-commute with v1. specifically, they can be written as vm+(m−1)(m−2)+2l−1 = |0⟩⟨l| + |l⟩⟨0|; vm+(m−1)(m−2)+2l = −i|0⟩⟨l| + i|l⟩⟨0|, (39) where 1 ≤l ≤m −1. the total hamiltonian for an uncontrolled m-level sys- tem interacting with a bath can now be written as hm = h0 + h′, h0 = m2−2m+2 x j=1 wjvj, h′ = m2 x j=m2−2m+3 wjvj, (40) where wj are the expansion coefficients that may contain arbitrary bath operators. with the udd control sequence described in eq. (35) tuned on, the unitary evolution operator can be easily investigated using [v1, h0] = 0 and {v1, h′}+ = 0. in- deed, it takes exactly the same form (with y1 →v1) as in eq. (21). we can then conclude that, the quantum coherence property p|ψ⟩associated with an arbitrarily pre-selected state |ψ⟩in an m-level system can be pre- served with a fidelity 1 + o(t n+1), with only n pulses. for an m-qubit system, m = 2m. in such a multi-qubit case, our result here indicates the following: if the initial state of an m-qubit system is known, then by (i) setting |ψ⟩the same as this initial state, and then (ii) setting p|ψ⟩as the control operator, the known initial state will be efficiently locked by udd. certainly, realizing the re- quired control hamiltonian for a multi-qubit system may be experimentally challenging. recently, a multi-level system subject to pulsed exter- nal fields is experimentally realized in a cold-atom lab- oratory [13]. to motivate possible experiments of udd using an analogous setup, in the following we consider the case of m = 3 in detail. to gain more insights into the control operator v1, here we use angular momentum operators in the j = 1 subspace to express all the nine basis operators. specifically, using the eigenstates of the jz operator as our representation, we have jx = 1 √ 2   1 1 1 1  , jy = 1 √ 2   −i i −i i  , jz =   1 0 −1  . (41) as an example, we use the state (1, 0, 0)t to define our coherence measure. the associated control operator v1 is then found to be v1 = jz + j2 z −i. (42) interestingly, this control operator involves a nonlinear function of the angular momentum operator jz. this requirement can be experimentally fulfilled, because re- alizing such kind of operators in a pulsed fashion is one 7 main achievement of ref. [13], where a "kicked-top" sys- tem is realized for the first time. the two different con- texts, i.e., udd by instantaneous pulses and the delta- kicked top model for understanding quantum-classical correspondence and quantum chaos [13, 14, 15], can thus be connected to each other. for the sake of completeness, we also present below those operators that commute with v1, namely, v2 = i + 1 2jz −1 2j2 z, v3 = −i −1 2jz + 5 2j2 z, v4 = −1 √ 2(j+jz + jzj−), v5 = i √ 2(j+jz −jzj−), (43) where j± = jx ± ijy; and those operators that anti- commute with v1, namely, v6 = 1 √ 2 (jzj+ + j−jz), v7 = i √ 2(j−jz −jzj+), v8 = 1 2(j2 + + j2 −), v9 = i 2(j2 −−j2 +). (44) some linear combinations of these operators will be re- quired to construct the control hamiltonian to preserve the coherence associated with other states. iii. simple numerical experiments to further confirm the udd control sequences we ex- plicitly constructed above, we have performed some sim- ple numerical experiments. we first consider a model of a two-spin system coupled to a bath of three spins. the total hamiltonian in dimensionless units is hence given by h = 5 x m=3 x j={x,y,z} bj,mσm j + 5 x n=1 x k={x,y,z} 5 x m>n x j={x,y,z} cjkσm j σn k + hc, (45) where the first two spins constitute the two-qubit system in the absence of any external field, hc represents the udd control hamiltonian, and the coefficients bj,m and cjk take randomly chosen values in [0, 1] in dimensionless units. in addition, to be more realistic, we replace the instantaneous δ(t −tj) function in our control hamil- tonians by a gaussian pulse, i.e., (1/c√π)e−[(t−tj)2/c2], 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.8 0.85 0.9 0.95 1 1.05 t f(t) without pulses extended udd, n=8 intuitive pulse, n=8 fig. 1: (color online) expectation value of the coherence measure p|ψ⟩, denoted f(t), as a function of time in dimen- sionless units, with |ψ⟩being the non-entangled state | ↑↑⟩ of a two-qubit system. the bath responsible for the deco- herence is modeled by a three-spin system detailed in the text. the bottom curve is without any control and the de- coherence is significant. the middle curve is calculated from a control hamiltonian intuitively based on two independent qubits. the top solid curve represents significant decoherence suppression due to our two-qubit udd control hamiltonian described by eq. (19). with c = t/100 unless specified otherwise. further, we set t = 0.1, because this scale is comparable to the de- coherence time scale. figure 1 depicts the time dependence of the expecta- tion value of the coherence measure p|ψ⟩, denoted f(t), with |ψ⟩being the non-entangled state | ↑↑⟩of the two- qubit system. the initial state of the system is also taken as the non-entangled state | ↑↑⟩. as is evident from the uncontrolled case (bottom curve) , the decoherence time scale without any decoherence suppression is of the or- der 0.1 in dimensionless units. turning on the two-qubit udd control sequence described by eq. (19) for n = 8, the decoherence (top solid curve) is seen to be greatly suppressed. we have also examined the decoherence sup- pression using a udd sequence based on the single-qubit- based intuitive control hamiltonian hc,single described by eq. (25). as shown in fig. 1, hc,single can only produce unsatisfactory decoherence suppression. similar results are obtained in fig. 2, where we aim to preserve the coherence measure p|ψ⟩associated with the bell state defined in eq. (26). apparently, with the assistance of our two-qubit udd control sequence, the system is seen to be locked on the bell state with a fi- delity close to unity at all times. figure 2 also presents 8 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 t f(t) without pulses extended udd, n=8 intuitive pulse, n=8 fig. 2: (color online) same as in fig. 1, but for p|ψ⟩associ- ated with a bell state defined in eq. (26). the smooth dashed curve represents significant decoherence without control. the drastically oscillating dashed curve is calculated from an intu- itive single-qubit-based control hamiltonian, showing strong population transfer from the initial state to other two-qubit states. the top solid curve represents signficant decoherence suppression due to our two-qubit udd control sequence in eq. (32). the parallel result if the control hamiltonian is given by hc,single shown in eq. (25). the drastic oscillation of f(t) in this case indicates that strong population oscilla- tion occurs, thereby demonstrating again the difference between single-qubit decoherence suppression and two- qubit decoherence suppression. using the same initial state as in fig. 2, fig. 3 de- picts d ≡ 1 2t
t 0 ||ρ(t) −ρi||dt, i.e., the time-averaged distance between the actual time-evolving density matrix from that of a completely locked bell state, for c = t/100 and c = t/1000, with different number of udd pulses. it is seen that, at least for the number of udd pulses con- sidered here, c = t/100 = 1/1000 (about one hundredth of the decoherence time scale) already suffices to preserve a bell state. that is, there seems to be no need to use much shorter pulses such as c = t/1000 = 1/10000, be- cause the case of c = t/1000 (dashed line) in fig. 3 shows little improvement as compared with the case of c = t/100 (solid line). this should be of practical in- terest for experimental studies of two-qubit decoherence suppression. finally, we show in fig. 4 the decoherence suppression of a three-level quantum system, with the control opera- tor given by eq. (42). here the bath is modeled by other four three-level subsystems, and the total hamiltonian is 0 2 4 6 8 10 12 14 16 18 20 10 −2 10 −1 n d c=1/10000 c=1/1000 fig. 3: (color online) the time-averaged distance d between the actual density matrix from that of a completely locked bell state, for c = t/100 and c = t/1000, versus the number of udd pulses. the initial state is the same as in fig. 2. chosen as h = 5 x m=2 x α={x,y,z} bj,mjα,m + 5 x n=1 x α={x,y,z} 5 x m>n x β={x,y,z} cαβjα,mjβ,n + hc, (46) where jα,m represents the jx, jy, or jz operator associated with the mth three-level subsystem, with the first being the central system and the other four being the bath. the coupling coefficients are again randomly chosen from [0, 1] with dimensionless units. the results are analogous to those seen in fig. 1 and fig. 2, confirming the general applicability of our udd control sequence in multi-level quantum systems. note also that even for the n = 2 case (middle curve in fig. 4), decoherence suppression already shows up clearly. the results here may motivate experimental udd studies using systems analogous to the kicked-top system realized in ref. [13]. iv. discussion and conclusion so far we have assumed that the system-bath cou- pling, the bath self-hamiltonian, and the system hamil- tonian in the absence of the control sequence are all time- independent. this assumption can be easily lifted. in- deed, as shown in a recent study by pasini and uhrig for 9 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 t f(t) n=2 n=10 without pulses fig. 4: (color online) expectation value of the coherence measure p|ψ⟩, denoted f(t), as a function of time in dimen- sionless units, with |ψ⟩being one basis state of a three-level system. the central system is coupled with a bath modeled by other four three-level subsystems. the bottom curve rep- resents significant decoherence without decoherence control. the top two curves represent decoherence suppression based on the control operator constructed in eq. (42), for n = 2 and n = 10. single-qubit systems [16], the udd result holds even af- ter introducing a smooth time dependence to these terms. the proof in ref. [16] is also based on yang and liu's work [12]. a similar proof can be done for our extension here. take the two-qubit case with the control operator y1 as an example. if h0 and h′ are time-dependent, then the unitary evolution operator in eq. (20) is changed to u(t ) = (−iy1)nj h e−i
t tn [h0+(−1)nh′]dti × j  e −i
tn tn−1[h0+(−1)n−1h′]dt  * * * × j h e−i
t3 t2 [h0+h′]dti × j h e−i
t 2 t 1 [h0−h′]dti × j h e−i
t1 0 [h0+h′]dti = (−iy1)nj h e−i
t 0 h0dti j h e−i
t 0 fn (t)h′ i(t)dti , (47) with h′ i(t) = j h ei
t 0 h0dti h′j h e−i
t 0 h0dti . (48) because the term j h e−i
t 0 h0dti in eq. (47) does not affect the expectation value of our coherence measure, the final expression for the coherence measure is essentially the same as before and is hence again given by its initial value multiplied by 1 + o(t n+1). our construction of the udd control sequence is based on a pre-determined coherence measure p|ψ⟩that charac- terizes a certain type of quantum coherence. this implies that our two-qubit udd relies on which type of deco- herence we wish to suppress. indeed, this is a feature shared by uhrig's work [7] and the yang-liu universal- ity proof [12] for single-qubit systems (i.e., suppressing either transverse decoherence or longitudinal population relaxation). can we also efficiently suppress decoherence of different types at the same time, or can we simulta- neously preserve the quantum coherence associated with entangled states as well as non-entangled states? this is a significant issue because the ultimate goal of decoher- ence suppression is to suppress the decoherence of a com- pletely unknown state and hence to preserve the quan- tum coherence of any type at the same time. fortunately, for single-qubit cases: (i) there are already good insights into the difference between decoherence suppression for a known state and decoherence suppression for an unknown state [17, 18] (with un-optimized dd schemes); and (ii) a very recent study [19] showed that suppressing the longi- tudinal decoherence and the transverse decoherence of a single qubit at the same time in a "near-optimal" fashion is possible, by arranging different control hamiltonians in a nested loop structure. inspired by these studies, we are now working on an extended scheme to achieve efficient decoherence suppression in two-qubit systems, such that two or even more types of coherence properties can be preserved. thanks to our explicit construction of the udd control sequence for non-entangled and entangled states, some interesting progress towards this more am- bitious goal is being made. for example, we anticipate that it is possible to preserve two types of quantum co- herence of a two-qubit state at the same time, if we have some partial knowledge of the initial state. it is well known that decoherence effects on two-qubit entanglement can be much different from that on single- qubit states. one current important topic is the so-called "entanglement sudden death" [20], i.e., how two-qubit entanglement can completely disappear within a finite duration. since the efficient preservation of two-qubit entangled states by udd is already demonstrated here, it becomes certain that the dynamics of entanglement death can be strongly affected by applying just very few control pulses. in this sense, our results on two-qubit systems are not only of great experimental interest to quantum entanglement storage, but also of fundamental interest to understanding some aspects of entanglement dynamics in an environment. to conclude, based on a generalized polarization opera- tor as a coherence measure, we have shown that udd also applies to two-qubit systems and even to arbitrary multi- level quantum systems. the associated control fidelity is 10 still given by 1 + o(t n+1) if n instantaneous control pulses are applied. this extension is completely general because no assumption on the environment is made. we have also explicitly constructed the control hamiltonian for a few examples, including a two-qubit system and a three-level system. our results are expected to advance both theoretical and experimental studies of decoherence control. v. acknowledgments this work was initiated by an "sps" project in the faculty of science, national university of singapore. we thank chee kong lee and tzyh haur yang for discus- sions. j.g. is supported by the nus start-up fund (grant no. r-144-050-193-101/133) and the nus "yia" (grant no. r-144-000-195-101), both from the national univer- sity of singapore. [1] l. viola and s. lloyd, phys. rev. a 58, 2733 (1998). [2] l. viola, e. knill, and s. lloyd, phys. rev. lett. 82, 2417 (1999). [3] j. j. l. morton et al., nature physics 2, 40 (2006); j. j. l. morton et al., nature 455, 1085 (2008). [4] g. s. uhrig, new journal of physics 10, 083024 (2008). [5] k. khodjasteh and d.a. lidar, phys. rev. lett. 95, 180501 (2005). [6] k. khodjasteh and d. a. lidar, phys. rev. a 75, 062310 (2007). [7] g. s. uhrig, phys. rev. lett. 98, 100504 (2007). [8] b. lee, w. m. witzel, s. das sarma, phys. rev. lett. 100, 160505 (2008). [9] m. j. biercuk, h. uys, a. p. vandevender, n. shiga, w. m. itano, and j. j. bolinger, nature 458, 996 (2009). [10] m. j. biercuk, h. uys, a. p. vandevender, n. shiga, w. m. itano, and j. j. bolinger, phys. rev. a 79, 062324 (2009). [11] j. f. du, x. rong, n. zhao, y. wang, j. h. yang, and r. b. liu, nature 461, 1265 (2009). [12] w. yang and r. b. liu, phys. rev. lett. 101, 180403 (2008). [13] s. chaudhury, a. smith, b. e. andersdon, s. ghose, and p. s. jessen, nature 461, 768 (2009). [14] f. haake, quantum signatures of chaos 2nd ed. (springer-verlag, berlin, 1999). [15] see, for example, j. wang and j. b. gong, phys. rev. lett. 102, 244102 (2009) for an extensive list of the kicked-top model literature and its relevance to several areas. [16] s. pasini and g. s. uhrig, arxiv:0910.0417 (2009). [17] w. x. zhang, n. p. konstantinidis, v. v. dobrovitski, b. n. harmon, l. f. santos, and l. viola, phys. rev. b 77, 125336 (2008). [18] w. x. zhang, v. v. dobrovitski, l. f. santos, l. viola, and b.n. harmon, phys. rev. b 75, 201302 (r) (2007). [19] j. r. west, b. h. fong, and d. a. lidar, arxiv:0908.4490v2 (2009). [20] t. yu and j. h. eberly, science 323, 598 (2009).

0911.1677

logical primes, metavariables and satisfiability

for formulas f of propositional calculus i introduce a "metavariable" mf and show how it can be used to define an algorithm for testing satisfiability. mf is a formula which is true/false under all possible truth assignments iff f is satisfiable/unsatisfiable. in this sense mf is a metavariable with the "meaning" 'f is sat'. for constructing mf a group of transformations of the basic variables ai is used which corresponds to 'flipping" literals to their negation. the whole procedure corresponds to branching algorithms where a formula is split with respect to the truth values of its variables, one by one. each branching step corresponds to an approximation to the metatheorem which doubles the chance to find a satisfying truth assignment but also doubles the length of the formulas to be tested, in principle. simplifications arise by additional length reductions. i also discuss the notion of "logical primes" and show that each formula can be written as a uniquely defined product of such prime factors. satisfying truth assignments can be found by determining the "missing" primes in the factorization of a formula.

introduction introductions to the problem of satisfiability can be found in textbooks and reviews, some of them available in the net (see e.g. [1],[2]). one of the unsolved questions of the field is whether satisfiability can be determined in polynomial time ("p=np ?"). other questions center around efficient techniques to determine satisfying assignments (see [3,4] for new approaches), and to identify classes of "hard" problems which inherently seem to consume large computing time. i believe that some insight into the difficulties can be gained by using algebraic tools. i have outlined some of them in a previous note [5]. in particular the notion of 'logical primes' and the group of flipping transformations appear helpful in analyzing formulas and deriving general theorems. i will recall these notions and some consequences in sections i , ii and iii. then i will introduce the metaformula and a related quantity, the parityformula, which encodes whether f has an even or an odd number of satisfying solutions in section iv. in sections v and vi algorithms with encreasing effectiveness in determining satisfiability are introduced. i. definitions we consider a finite algebra v with two operations + and x, and denote by 1 and 0 their neutral elements, respectively, i.e. (1) ax1=a, a+0=a additionally, the operations are associative and commutative, and the distributive law (2) ax(b+c)=axb + axc is assumed to hold in v. two more properties are required, namely: 3 (3) a+a=0 (4) axa=a it is clear from these definitions that v may be identified with the boolean algebra of propositional calculus, where "x" corresponds to the logical "and" and "+" to the logical "xor" (exclusice or). to each element of v we introduce its "negation" by (5) ~a := a+1 from (2), (3) and (4) it is clear that ~axa = 0 as is appropriate for a negation. ii. consequences. as a first consequence of equ.s (1) - (5) we can state the following theorem: (ti) dim(v) = |v| = 2n for some natural number n i.e. the number of elements of v is necessarily a power of 2. this is not surprising, of course, if one has the close resemblence of v to propositional calculus in mind. but here it is to be deduced solely from the algebraic properties. all proofs are given in the appendix. in order to formulate a second consequence it is necessary to introduce the notion of "logical primes". we define: 4 (di) p ε v is a (logical) prime, iff for any a ε v pxa=0 implies a=0 or a=~p. if not clear by definition, the name "prime" will become clear by the following theorems (tii) there are exactly ld|v|=n many primes in v. and: (tiii) each element of v has a unique decomposition into primes: (6) a = πjpj where the product refers to the x-operation, and jειa, and ιa=ιb iff a=b this property can be formulated alternatively with the negated primes ~pj via (7) a = σj ~pj with jε cιa (cιa is the complement of ιa in {0,1,..., n-1} ) the neutral elements 0 and 1 are special cases. 1 is expressed as the empty product according to (6), whereas the sum extends over all primes. for 0 the sum- representation is empty, but the product extends over all possible primes. a property which is extremely helpful in calculations is (8) ~pjx~pk = ~pk δjk (δjk = 1 iff j=k, 0 otherwise) which with the aid of (5) can be written pjxpk = pj + ~pk = ~pj + pk for k=j note, that no use has been made of the correspondence of {v,+,x, 0 , 1 } to propositional calculus, up to now. we can even proceed further and define the analogue of truth assignments. consider the set of maps t:v {0,1} . we call t 5 "allowed" iff there is a relationship between the image of a "sum" or a "product" and the image of the single summands or factors. in formula: (9) t(a+b) = f(t(a),t(b)) and t(axb) = g(t(a),t(b)) with some functions f and g and all a,bεv. these relations suffice to show theorem iv (tiv) there are exactly n different allowed maps tj , and they fulfill: (10) tj(~pk) = δjk given functions f and g of (9) one can also use (10) as a definition and extend tj to all elements of v via (7). in one last step we assume n=2n for some natural number n. then (tv) n distinct elements ak ( different from 0, 1) can be found, such that (11) ~ps = (πjsjaj)( πk(1-sk)~ak) where s= σr2r-1sr is the binary representation of s. in words: each element of v can be written as a "sum" of "products" of all ak and ~ak. e.g. for n=3 one has p2=a2x~a1x~a3 as one of the eight primes.the ak are not necessarily unique. e.g., for n=3, given ak, the set a1,a3, a1x~a2+~a1xa2 will serve the same purpose (with a different numbering convention in (11)). iii. propositional calculus. propositional calculus (pc) consists of infinitely many formulas which can be constructed from basic variables ak with logical functions (like "and", "or" and 6 negation). even for a finite set of n basic variables bn={a1,a2,...an} there are infinitely many formulas arizing from combinations of the basic variables. these formulas can be grouped into classes of logically equivalent formulas. that is, formulas f and f' belong to the same class iff their values under any truth assignment t:bn {0,1} are the same. members of different classes are logically inequivalent, i.e. there is at least one truth assignment for which their values differ. this finite set of classes for fixed n can be identified with the algebra v of the foregoing section. neutral elements of the operations x and +, 1 and 0, are interpreted as complete truth and complete unsatisfiability. in order to see how operations + and x correspond to logical operations "and" and "or" we define a new operation v in v via (12) a v b = a + b + axb with this definition the defining relations (1) - (5) can be reformulated in terms of v and x, and the algebraic structure of a boolean algebra for formulas becomes obvious. v is the logical "or", x the logical "and". relation (12) reduces logical considerations to simple algebraic manipulations in which + and x can be used as in multiplication and addition of numbers. additionally the simplifying relations a+a=0 = ~axa and axa=a, a+~a=1 hold. consider for illustration the so called "resolution" method. it states that avb and ~avc imply bvc. a "calculational" proof of this statement might run as follows. (from now on we skip the x-symbol for multiplication) we make use of the fact that in pc the implication a b is identical to ~a v b : 7 (avb)(~avc) bvc = ~((avb)(~avc))vbvc = (1+(a+~ab)(~a+ca))+b+c+bc+(b+c+bc)( 1+(a+~ab)(~a+ca)) = 1 +ac+~ab + +(b+c+bc)(a+~ab)(~a+ca) = 1 +ac+~ab+abc+~ab+~bac = 1 +ac(1 +b+~b) = 1 in other words: the implication is a tautology ( true under all truth assignments) as claimed. tiii and tv tell us that each formula f of pc has a unique decomposition into a "sum" of "products" of its independent variables ak. because of (8) and (12) the sum in (7) may be written as a "v"-sum. thus (7) takes the form of a disjunctive normal form (dnf) and it can as well be transformed into a conjunctive normal form (cnf) as given by (6). for the neutral element 0 one has (13) 0 = (a1va2v...van)x(~a1v...an)x...x(~a1v...~an) with all possible primes. according to (6) each formula f has a similar representation, but with some prime factors missing. from the primes present one can immediately read off the truth assignments for which f evaluates to 0, thus the missing factors give the truth assignments for which f is satisfiable. note, however, that each factor in the prime representation of a formula involves all ak . so one way of determining satisfying assignments or test a formula for satisfiability consists of transforming a given cnf representation of the formula to its standard form (6). this can be done e.g. by "blowing up" each factor until all ak are present. e.g. avbv~c = (avbv~cvd)(avbv~cv~d) from 3 to 4 variables. since each new factor has to be treated in the same way, until n is reached, this is a o(2n) - 8 process in principle, which makes the difficulty in finding a polynomial time algorithm for testing satisfiability understandable. also from (7) with (10) and (8) it follows that the satisfying assignments of a formula f= σj ~pj are given by the negated primes which do not show up in the cnf representation. in particular, the number of satisfying assignments is equal to the number of summands in this equation. furthermore, they can be read off immediately, since, according to (10) ts(f) = 1 iff the corresponding ~ps shows up in the sum. also the ts must coincide with the 2n possible truth assignments t :bn {0,1}. one may choose the numbering such that the values of ts on bn are given by the binary representation s= σr2r-1ts(ar). as a last example for the usefulness of the algebraic approach we consider the number of satisfying assignments of a formula f of pc , #(f) and show that this number does not change if some (or all) of the variables ak are "flipped", i.e. substituted by their negation and vice versa: (14) #(f(a1,...,an)) = #(f(a1,...~ai,...~aj,...)) to prove this "conservation of satisfiability" we consider a group of transformations {r0,...rn-1} which negate the ak according to the following definition: rs negates all ar (and ~ar likewise) for which sr in the binary representation of s is non zero. in formula, for any truth assignment tj (15) tj(rs(ar)) = (1-sr) tj(ar) + sr(1- tj(ar)) and s= σr2r-1sr. 9 it is easy to see that the rs form a group with r0 = id , and each rs induces a permutation πs of of the ~pj which is actually a transposition given by (15') πs(j) = s + j - 2σr2r-1srjr =: (s,j) thus rs simply permutes the primes pk and therefore in the representation of f in (6) or (7) their number is not changed. the fact may also be stated as (16) tj(rs(f)) = t (s,j) (f) and therefore #(f)= σjtj(f) = σjtj(rs(f)) = #(rs(f)) which proves (14). one may also conclude from (16) that for satisfiable f each rs(f) is satisfiable. more precise: if tj(f) =1 for some j, then for any k there is a flipping operation rs such that tk(rs(f))=1, namely s=(k,j). likewise, for any rs one can find a tk such that tk(rs(f))=1. on the other hand, if f is not satisfiable, none of the rs(f) can be satisfiable, otherwise one would have tk(rs(f))=1 for some k and thus tj(f) =1 for some j, contrary to the assumption that f is not sat. iv. the metaformula. for any formula f of n variables we write f(a1,...,an) and define the metaformula by "adding" with respect to the or-operation all "flipped" versions of f: (17) mf(a1,...,an) = r0(f) v r1(f) v ... v rn-1(f) 10 where n= 2n. from the considerations at the end of the foregoing section it is immediately clear that mf is not satisfiable if f is not, and that mf is a tautology if f is satisfiable: (18) mf = 1 iff fε sat; mf = 0 iff fε sat thus, considered as a logical variable itself, mf represents the satisfiability of f. mf can only take the two values 0 and 1 depending on whether f is sat or not. that is why i call mf a metatheorem or metaformula. very similarly one may introduce a "parityformula" pf in substituting the or-operation in the definition (17) by the exclusive xor. analogously to (18) one can show that pf = 0 iff pf has an even number of satisfying truth assignments, pf = 1 iff pf has un odd number of satisfying assignments. v. sat algorithm. we now turn to the question how mf can be utilized to formulate sat algorithms. since either mf = 1 or mf = 0 it is sufficient to test one single truth assignment in order to determine whether f is sat or not. thus the satisfiability of f can be determined in linear time in the length of mf. nothing is gained so far, however, since the length of mf is of order n times the length of f. thus, instead of testing all n tj on f to determine its satisfiability in the metatheorem approach one first constructs an order-n variant of f and checks it with a single tj. simplifications may arise, however in the process of constructing mf. 11 note first that mf can be constructed in n steps. for this purpose consider the shift operator (19) d(k)(f) = f v rq(f) with q=2k-1 and k=1,...,n note that all n operators rs can be generated by the n operators rq with q=2k-1 and k=1,...,n. e.g. r29 = r16r8r4r1. furthermore it is easy to see that rq flips the variable ak and therefore d(k) is independent of ak, and (19) may be rewritten as (20) d(k)(f) = f(a1,..., ak= 1, ...,an) v f(a1,..., ak= 0, ..., an). in terms of shift operators mf may be rewritten as (21) mf = dn(dn-1(...d2(d1(f)...) = :d(n)...d(1)(f) we can now consider systematic approximations on mf. namely the series of lth order approximations (22) mf (l) = d(l)...d(1)(f) = d(l)(mf (l-1)) ; mf (0) = f. from this definition we may write mf (l) as (23) mf (l) = f v r1(f) v ... v rq-1(f) with q=2l. we will show next that a properly chosen truth assignment for testing the l-th approximation can give a wealth of information. check tq with q=2l on mf (l). let us assume that tq(mf (l)) = 1. then one of the ri(f) is true under that truth assignment, therefore there is a truth assignment which satisfies f, thus f is satisfiable. if on the other hand tq(mf (l)) = 0, then we may conclude the following: tq(ri(f)) = 0 for all i 12 ε {1, 2,..., q}. therefore also t(i,q)(f) = 0 according to (16). in conclusion, see (15') for the definition of (i,q): (24) if tq(mf (l)) = 0 then f is not satisfied by truth assignments tk for k ε {q, q+1,..., 2q-1} (q= 2l) an effective check for satisfiability of f may therefore run as follows: checksat [f,n] set s=1 1 set f = d(s)(f) if ts(f)=1 then stop and return "f is sat" s=s+1 if s=n then stop and return "f is not sat" goto 1 checksat determines satisfiability in n steps each of which is linear in the length of the formula. in each step the number of excluded truth assignments is doubled, as well as the chance to find a satisfying assignment if there is one. however, the formulas to be checked become longer and longer in each step, therefore it remains an order-n process in principle. a look at (20) reveals that the procedure corresponds to a successive elimination of variables. further optimizations require length reductions in the formulas mf (l) which arise in the approximation process. vi. length reduction. in this section we assume f to be given in conjunctive normal form (cnf): 13 (25) f = c1c2...cm with m clauses of the form (26) c = l v r where l is a literal corresponding to one of the variables ak or its negation, and r may itself be written in the form (26) and so forth until r is a literal. in the process of eliminating variables described in the foregoing section the following well known rules can help to reduce formulas in length. (a) (lvr)(~lvr) = r (b) (lvr)(~lvr') = lr' v ~lr (27) (c) l(~lvr) = lr (d) (lvr)(lvr') = l v ~lrr' (e) l(lvr) = l in a cnf-formula one encounters terms of the form (lvr1)...(lvrs)(~lvs1)...(~lvst) which may be rewritten by the aid of (27): (28) f(l):= (lvr1)...(lvrs)(~lvs1)...(~lvst) = ls1s2...st v ~lr1r2...rs note that the cnf form on the l.h.s. is split into a disjunction of two cnf-formulas on the r.h.s.. the variable l does not show up in the r an s by definition. if we eliminate the variable l, which is exactly what happens when the shift operator d is applied, one reads off the r.h.s.: (29) f(l) v f(~l) = s1s2...st v r1r2...rs 14 a disjunction of cnf-forms independent of l. in any practical application of the approximation algorithm outlined in the foregoing section to a cnf-formula g one might proceed as follows: collect all clauses with variable a1 and ~a1. call the remaining factor gr. then one has (in the notation of (28) with l=a1) (30) d(1)(g) = g v r1(g) = g(a1,...) v g(~a1,...) = (s1s2...st v r1r2...rs)gr where neither the r and s nor gr depend on a1. the collection procedure is polynomial and the resulting formula is not longer than the original one in terms of symbols. but it is not a cnf formula anymore. if one wants to repeat the process and apply the same rules, one has to split (30) into two cnf formulas and apply the procedure to each. now in effect the formula length has doubled (nearly) and one encounters the exponential behaviour typical of np-problems. simplifications might arise from the s and r factors, however. all of them are shorter than the clauses one started with because they do not contain a1 anymore. if an s or r is reduced to a single varible l, the application of (27b) can eliminate several clauses in one stroke. from this consideration it becomes clear that an effective algorithm will involve a clever choice of consecutive variables. conclusion. two new formal tools to deal with propositional calculus and the problem of satisfiability were discussed; namely the notion of logical primes [5 ] and the 15 metaformula. it was shown that each equivalence class of boolean formulas has a unique representation as a product of logical primes. therefore the satisfiability of a formula can be formulated as a problem of prime factorization. the notion of the metavariable or metaformula enables one to formulate well known procedures for determining satisfiability in a systematic manner. a simple program was formulated which checks for sat in n (number of basic variables) linear steps. nonetheless the procedure cannot do the job in polynomial time because the length of the formula to be checked in each step basically doubles. steps to optimize the procedure by proper length reductions were indicated. appendix the proofs for theorems (ti) to (tv) are straightforward and only basic ideas will be sketched here. proof of ti: for n=1 v consists only of the trivial elements 0 and 1 . thus we assume |v|>2. for some nontrivial s define ks={a|axs=0 }. obviously ~s and 0 ε ks. analogously for k~s. it is easy to show that ks and k~s are subgroups of v with respect to + , and both have only 0 in common. thus each a ε v has a unique decomposition a=u+v where u ε ks and v ε k~s . let | ks |=ns, and | k~s |= n~s. next we count elements which do not belong to ks or k~s. define: eks(u0) = {u0+v| v ε k~s\ 0} with u0 ε ks. | eks(u0) | = n~s-1 from the definition. next one shows that eks(a) and eks(b) have no elements in common unless a=b. thus |v|= ns-1+ n~s +|σu eks(u) |= ns-1+ n~s +( ns-1)|eks(u)|= ( ns-1)(1+ n~s-1)+ n~s 16 = ns n~s . since both ks and k~s are subfields of v (with neutral elements ~s and s with respect to x) one can apply the same line of argument to each of them until one reaches the trivial field v0={ 0, 1} which has |v0|=2. thus both ns and n~s , and therefore |v| is a power of 2. next the proof of (tii) can proceed via induction over n=ld(|v|). again one considers the subfields ks and k~s of a v with | v|= 2n+1 and their sets of primes pj and qj which exist by assumption. then one shows that all pj + s are primes in v, and qj + ~s dto. furthermore one can show that no two of these primes of v or their negations coincide, and, secondly, that any possible prime of v is necessarily one of them. thus the pj + s and qj + ~s constitute the set of primes of v, and their number is by assumption ld(ns)+ld(n~s) = n+1. the fact that different negated pk are orthogonal, equ. (8), is proven as follows: for i=j pjx~pi ε kpi by definition of k. but since pi is prime, kpi = { 0,~pi}. thus either pjx~pi = 0 which implies (because also pj is prime) that ~pi is either 0 or equal to ~pj both in contradiction to assumptions, therefore : or pjx~pi = ~pi . which is equivalent to the claim. along the same line of thought - considering ks and k~s for s=some prime element of v - it can be proven that each element of v has a unique decomposition into primes, equ. (7) or (6). proof of (tiv). 17 first note that both functions f(x,y) and g(x,y) in equ. (9) can take values 0 or 1 only, and they are symmetric because of the commutativity of the operations x and +. then from (1) and (9) setting t(a)=0 or 1 respectively one gets 0 = g(0,t(1 )) = g(t(1 ),0) and 1 = g(1,t(1 )) = g(t(1 ),1) and t(0) = g(1, t(0)) = g(0, t(0 )) from ax 0= 0 . if one chooses t(0) = 0 then t(1) = 0 leads to a contradiction, as well as setting both values equal to 1. one is left with the choice (a) t(0 ) = 0 and t(1) = 1 (b) t(0 ) = 1 and t(1) = 0 we adopt choice (a) in the following. as a consequence 0=g(0,1) = g(1,0) = g(0,0) and 1 = g(1,1) and, from (1) for + 0=f(0,0)=f(1,1) and 1=f(1,0)=f(0,1) . let t be fixed. because of (8): 0=g(t(~p),t(~q)) for different p,q. thus either t(~p)=t(~q)=0 or the two assignments have different value. if t(~pk)=0 for all k, one gets a contradiction to 1=σk~pk and 0=f(0,0). thus at least for one k t(~pk)=1. but then for all other j t(~pj)=0 because of 0=g(0,1) and the orthogonality relation (8). thus for each t there is exactly one ~pk with truth assignment 1, and all other ~p giving 0. now consider two different maps t, t' with t(~pk)=1 and t'(~pl)=1. then k and l must be different, otherwise the two maps would coincide. repeating this argument with a third t'' and so on leads to the conclusion that there are exactly as many allowed maps as there are primes. we can label the maps as we would like to, so the most natural choice is equ. (10). 18 as for theorem v, the easiest way to prove the existence of n=ld(n) ak is to construct them from the uniquely defined primes: ar = σi σsσl ~pi δ(i,s+2kl) where δ is the kronecker δ and the s and l sums run from 2k-1 to 2k-1 and from 0 to 2n-k-1 respectively. constructing them inductively is more instructive because one encounters choices which lead to different sets of ak. the seemingly complicated formula above is obsolete once one uses the binary representation of all quantities which is given by the bijection f tn-1(f) ...ti(f)...t0(f) for any f. in particular the ai take the simple form: a1 = ....1010101010101010 a2 = ....1100110011001100 a3 = ....1111000011110000 and so on. references. [1] welzl, emo: boolean satisfiability – combinatorics and algorithms. lecture script, http://www.inf.ethz.ch/~emo/smallpieces/sat.ps [2] cook, s. a., and mitchell, d. g.: finding hard instances of the satisfiability problem: a survey. imacs series in discrete mathematics and theoretical computer science, vol 5, 1997. [3] schuh, b. r.:testing satisfiability in polynomial time, submitted for publication in international journal of unconventional computing [4] schuh, b. r.: mean value satisfiability, to be published [5] schuh, b. r.: algebraic properties of propositional calculus, arxiv:0906.2133v1

0911.1678

industrial-strength formally certified sat solving

boolean satisfiability (sat) solvers are now routinely used in the verification of large industrial problems. however, their application in safety-critical domains such as the railways, avionics, and automotive industries requires some form of assurance for the results, as the solvers can (and sometimes do) have bugs. unfortunately, the complexity of modern, highly optimized sat solvers renders impractical the development of direct formal proofs of their correctness. this paper presents an alternative approach where an untrusted, industrial-strength, sat solver is plugged into a trusted, formally certified, sat proof checker to provide industrial-strength certified sat solving. the key novelties and characteristics of our approach are (i) that the checker is automatically extracted from the formal development, (ii), that the combined system can be used as a standalone executable program independent of any supporting theorem prover, and (iii) that the checker certifies any sat solver respecting the agreed format for satisfiability and unsatisfiability claims. the core of the system is a certified checker for unsatisfiability claims that is formally designed and verified in coq. we present its formal design and outline the correctness proofs. the actual standalone checker is automatically extracted from the the coq development. an evaluation of the certified checker on a representative set of industrial benchmarks from the sat race competition shows that, albeit it is slower than uncertified sat checkers, it is significantly faster than certified checkers implemented on top of an interactive theorem prover.

introduction advances in boolean satisfiability sat technology have made it possible for sat solvers to be routinely used in the verification of large industrial problems, including safety-critical domains that require a high degree of assurance such as the railways, avionics, and automotive industries [9,15]. however, the use of sat solvers in such do- mains requires some form of assurance for the results. this assurance can be provided in two different ways. first, the solver can be proven correct once and for all. however, this approach had limited success. for example, lescuyer et al. [11] formally designed and verified a sat solver using the coq proof-assistant [2], but without any of the techniques and optimizations used in modern solvers. reasoning about these optimizations makes the arxiv:0911.1678v2 [cs.lo] 17 dec 2009 sat solver - industrial strength - large & complex - un-trusted (ad-hoc) - proof generating industrial - strength certifed sat solver certified sat checker - standalone executable - small & clear - trusted (formal) - proof checking cnf proof cnf yes/no fig. 1: high-level view of an industrial-strength formally certified sat solver. formal correctness proofs exceedingly hard. this was shown by the work of mari ́ c [12], who verified the algorithm used in the argo-sat solver but restricted the verification to the pseudo-code level, and in particular, did not verify the actual solver itself. in addition, the formal verification has to be repeated for every new sat solver (or even a new version of a solver), or else the user is locked into using the specific verified solver. alternatively, a proof checker can be used to validate each individual outcome of the solver independently; this requires the solver to produce a proof trace that is viewed as a certificate justifying the outcome of the solver. this approach was used to design several checkers such as tts, booleforce, picosat and zchaff [16]. however, these checkers are typically implemented by the developers of the sat solvers whose output they are meant to check, which can lead to bugs being masked, and none of them was formally designed or verified, which means that they provide only limited assurance. the problems of both approaches can be circumvented if the checker rather than the solver is proven correct, once and for all. this is substantially simpler than proving the solver correct, because the checker is comparatively small and straightforward. it does not lead to a system lock-in, because the checker can work for all solvers that can produce proof traces (certificates) in the agreed format. this approach was followed by weber and amjad [20] in their formal development of a proof checker for zchaff and minisat proof traces. their core idea is to replay the derivation encoded in the proof trace inside lcf-style interactive theorem provers such as hol 4, isabelle, and hol light. since the design and implementation of these provers is based on a small trusted kernel of inference rules, assurance is very high. however this comes at the cost of usability: their checker can run only inside the supporting prover, and not as a standalone tool. moreover, performance bottlenecks become prominent when the size of the problems increases. here, we follow the same general idea of a formally certified proof checker, but depart considerably from weber and amjad in how we design and implement our solu- tion. we describe an approach where one can plug an untrusted, industrial-strength sat solver into a formally certified sat proof checker to provide an industrial-strength cer- tified sat solver. we designed, formalized and verified the sat proof checker for both satisfiable and unsatisfiable problems. in this paper, we focus on the more interesting aspect of checking unsatisfiable claims; satisfiable certificates are significantly easier to formally certify. our certified checker shruti is formally designed and verified using the higher-order logic based proof assistant coq [2], but we never use coq as a checker; instead we automatically extract an ocaml program from the formal development that is compiled to a standalone executable that is used independently of coq. in this regard, our approach prevents the user to be locked-in to a specific proof assistant, something that was not possible with weber and amjad's approach. a high-level architectural view of our approach is shown in figure 1. since it combines certification and ease-of-use, it enables the use of certified checkers as regular components in a sat-based verification work flow. 2 propositional satisfiability 2.1 satisfiability solving given a propositional formula, the goal of satisfiability solving is to determine whether there is an assignment of the boolean truth values (i.e., true, false) to the variables in the formula such that the formula evaluates to true. if such an assignment exists, the given formula is said to be satisfiable or sat, otherwise the formula is said to be unsat- isfiable or unsat. many problems of practical interest in system verification involved proving unsatisfiability, one concrete example being bounded model checking [4]. for efficiency purposes, sat solvers represent the propositional formulas in con- junctive normal form (cnf), where the entire formula is a conjunction of clauses. each clause itself denotes a disjunction of literals, which are simply (boolean) variables or negated variables. an efficient cnf representation uses non-zero integers to represent literals. a positive literal is represented by a positive integer, whilst a negated one is denoted by a negative integer. as an example, the (unsatisfiable) formula (a ∧b) ∨(¬a ∧b) ∨(a ∧¬b) ∨(¬a ∧¬b) over two propositional variables a and b can thus be represented as 1 2 0 -1 2 0 1 -2 0 -1 -2 0 the zeroes are delimiters that separate the clauses from each other. sat solvers take a boolean formula, for example represented in the dimacs no- tation used here, and produce a sat/unsat claim. a proof-generating sat solver produces additional evidence (or certificates) to support its claims. for a sat claim, the certificate simply consists of an assignment. it is usually trivial to check whether that assignment-and thus the original sat claim-is correct. one simply substitutes the boolean values given by the assignment in the formula and then evaluates the over- all formula, checking that it indeed is true. for unsat claims, the evidence is more complicated, and the solvers need to return a resolution proof trace as certificate. un- surprisingly, checking these unsat certificates is more complicated as well. 2.2 proof checking when the solver claims a given problem is unsat, we can independently re-play the proofs produced by the solver, to check that the solver's output is correct. for a given problem, if we can follow the resolution inferences given in the proof trace to derive an empty clause, then we know that the proof trace correctly denotes an unsat instance for the problem, and we can conclude that the given problem is indeed unsat. a proof trace consists of the original clauses used during resolution and the interme- diate resolvents obtained by resolving the original input clauses. the part of the proof trace that specifies how the input clauses have been resolved in sequence to derive a conflict is organized as chains. these chains are often referred to as the regular input resolution proofs, or the trivial proofs [1,3]. we call the input clauses in a chain its an- tecedents and its final resolvent simply its resolvent. designing an efficient checking methodology relies to some extent on the representation of the proof trace produced by the solvers. representing proof chains as a trivial resolution proof is a key con- straint [3]. the trivial resolution proof requires that the clauses generated to form the proof should be aligned in such a manner in the trace so that whenever a pair of clauses is used for resolution, at most one complementary pair of literals is deleted. another important constraint that is needed for efficiency reasons is that at least one pair of complementary literals gets deleted whenever any two clauses are used in the trace for resolution. this is needed because we want to avoid search during checking, and if the proof trace respects these two criteria we can design a checking algorithm that can val- idate the proofs in a single pass using an algorithm which is linear in the size of the input clauses. 2.3 picosat proof representation most proof-generating sat solvers [3,7,23] preserve these two criterions. we carried out our first set of experiments with picosat [3]. picosat ranked as one of the best solvers in the industrial category of the sat competitions 2007 and 2009, and in the sat race 2008. moreover, picosat's proof representation is in ascii format. it was reported by weber and amjad [22] that some other solvers such as minisat generate proofs in a more compact, but binary format. another advantage of picosat's proof trace format besides using an ascii format is its simplicity. it does not record the information about pivot literals as some of the other solvers such as zchaff. it is however straightforward to develop translators for formats used by other sat solvers [19]. as a proof of concept, we developed a translator from zchaff's proof format to picosat's proof format, which can be used for validating unsatisfiability proofs obtained with zchaff. a picosat proof trace consists of rows representing the input clauses, followed by rows encoding the proof chains. each row representing a chain consists of an asterisk (*) as place-holder for the chain's resolvent,3 followed by the identifiers of the clauses involved in the chain. each chain row thus contains at least two clause identifiers, and 3 this is generated by picosat; there is another option of generating proof traces from picosat where instead of the asterisk the actual resolvents are generated delimited by a single zero from the rest of the chain. denotes an application of one or more of the resolution inference rule, describing a trivial resolution derivation. each row also starts with a non-zero positive integer de- noting the identifier for that row's (input or resolvent) clause. in an actual trace there are additional zeroes as delimiters at the end of each row, but we remove these before we start proof checking. for the unsat formula shown in the previous section, the corresponding proof trace generated from picosat looks as follows: 1 1 2 2 -1 2 3 1 -2 4 -1 -2 5 * 3 1 6 * 4 2 5 the first four rows denote the input clauses from the original problem (see above) that are used in the resolution, with their identifiers referring to the original clause numbering, whereas rows 5 and 6 represent the proof chains. in row 5, the clauses with identifiers 3 and 1 are resolved using a single resolution rule, whilst in row 6 first the original clauses with identifier 4 and 2 are resolved and then the resulting clause is resolved against the clause denoted by identifier 5 (i.e., the resolvent from the previous chain), in total using two resolution steps. picosat by default creates a compacted form of proof traces, where the antecedents for the derived clauses are not ordered properly within the chain. this means that there are instances in the chain where we resolve a pair of adjacent clauses and no literal is deleted. this violates one of the constraints we explained above, thus we cannot deduce an existence of an empty clause for this trace unless we re-order the antecedents in the chain. picosat comes with an uncertified proof checker called tracecheck that can not only check the outcome of picosat but also corrects the mis-alignment of traces. the outcome of the alignment process is an extended proof trace and these then become the input to the certified checker that we design. 3 the shruti certified proof checker our approach to efficient formally certified sat solving relies on the use of a certified checker program that has been designed and implemented formally, but can be used in practice, independently of any formal development environment. rather than verifying a separately developed checker we follow a correct-by-construction approach in which we formally design and mechanically verify a checker using a proof assistant to achieve the highest level of confidence, and then use program extraction to obtain a standalone executable. our checker shruti takes an input cnf file which contains the original problem description and a proof trace file and checks whether the two together denote an (i) unsat instance, and (ii) that they are "legitimate". our focus here is on (i) where we check that each step of the trace is correctly applying the resolution inference rule. however, in order to gain full assurance about the unsat claim, we also need to check that all input clauses used in the resolution in the trace are taken from the original problem, i.e., that the proof trace is legitimate. shruti provides this as an option, but as far as we are aware most checkers do not do this check. our formal development follows the lcf style [17], and in particular, only uses definitional extensions, i.e., new theorems can only be derived by applying previously derived inference rules. axiomatic extensions though possible are prohibited, since one can assume the existence of a theorem without a proof. thus, we never use it in our own work. we use the the coq proof assistant [2] as a development tool. coq is based on the calculus of inductive constructions and encapsulates the concepts of typed higher- order logic. it uses the notion of proofs as types, and allows constructive proofs and use of dependent types. it has been successfully used in the design and implementation of large scale certification of software such as in the compcert [10] project. for our development of shruti, we first formalize in coq the definitions of res- olution and its auxiliary functions and then prove inside coq that these definitions are correct, i.e., satisfy the properties expected of resolution. once the coq formalization is complete, ocaml code is extracted from it through the extraction api included in the coq proof assistant. the extracted ocaml code expects its input in data structures such as tables and lists. these data structures are built by some glue code that also handles file i/o and pre-processes the read proof traces (e.g., removes the zeroes used as sepa- rators for the clauses). the glue code wraps around the extracted checker and the result is then compiled to a native machine code executable that can be run independently of the proof-assistant coq. 3.1 formalization in coq in this section we present the formalization of shruti. its core logic is formalized as a shallow embedding in coq. in a shallow embedding we identify the object data types (types used for shruti) with the types of the meta-language, which in our case happens to be the coq datatypes. since most checkers read the clause representation on integers (e.g., using the di- macs notation) we incorporate integers as first class elements in our formalization, so we do not have to map literals to booleans. thus, inside coq, we denote literals by integers, and clauses by lists of integers. antecedents (denoting the input clauses) in a proof chain are represented by integers and a proof chain itself by a list of integers. a resolution proof is represented internally in our implementation by a table consisting of (key,binding) pairs. the key for this table is the identifier obtained from the proof chains read from the input proof trace file. the binding of this table is the actual re- solvent obtained by resolving the clauses specified (in the proof trace). when the input proof trace is read, the identifier corresponding to the first row of the proof chain be- comes the starting point for resolution checking. once the resolvent is calculated for this the process is repeated for all the remaining rows of the proof chain, until we reach the end of the trace input. if the identifier for the last row of the proof chain denotes an empty resolvent, we conclude that the given problem and its trace represents an unsat instance. we use the usual notation for quantifiers (∀, ∃) and logical connectives (∧, ∨, ¬) but distinguish implication over propositions (⊃) and over types (→) for presentation clar- ity, though inside coq they are exactly the same. the notation ⇒is used during pattern matching (using match −with) as in other functional languages. for type annotation we use :, and for the cons operation on lists we use ::. the empty list is denoted by nil. the set of integers is denoted by z, the type of polymorphic list by list and the list of integers by list z. list containment is represented by ∈and its negation by / ∈. the function abs computes the absolute value of an integer. we use the keyword definition to present our function definitions. it is akin to defining functions in coq. main data structures that we used in the coq formalization are lists, and finite maps (hashtables with integer keys and polymorphic bindings). we define our resolution function (▷ ◁) with the help of two auxiliary functions union and auxunion. both functions compute the union of two clauses represented as integer lists, but differ in their behavior when they encounter complementary literals: whereas union removes both literals and then calls auxunion to process the remainder of the lists; auxunion copies both the literals into the output and thus produces a tauto- logical clause. ideally, if the sat solver is sound and the proof trace reflects the sound outcome, then for any pair of clauses that are resolved, there will be only one pair of complementary literals and we do not need two functions. however in reality, a solver or its proof trace can have bugs and it can create instances of clauses in the trace with multiple complementary pair of literals. hence, we employ the two auxiliary functions to ensure that the resolution function deals with this in a sound way. we will later explain in more detail the functionality of the auxiliary functions but both functions expect the input clauses to respect three well-formedness criteria: there should be no duplicates in the clauses (nodup); there should be no complementary pair of literals within any clause (nocomppair), and the clauses should be sorted by absolute value (sorted). the predicate wf encapsulates these properties. definition wf c = nocomppair c ∧nodup c ∧sorted c the assumptions that there are no duplicates and no complementary pair of literal within a clause are essentially the constraints imposed on input clauses when the reso- lution function is applied in practice. sorting is enforced by us to keep the complexity of our algorithm linear. the union function takes a pair of sorted (by absolute value) lists of integers, and an accumulator list, and computes the resolvent by doing a pointwise comparison on input literals. definition union (c1 c2 : list z)(acc : list z) = match c1, c2 with | nil, c2 ⇒app (rev acc) c2 | c1, nil ⇒app (rev acc) c1 | x :: xs, y :: ys ⇒if (x + y = 0) then auxunion xs ys acc else if (abs x < abs y) then union xs (y :: ys)(x :: acc) else if (abs y < abs x) then union (x :: xs) ys (y :: acc) else union xs ys (x :: acc) we already pointed out above that this function and the auxiliary union function - auxunion (shown below) that it employs are different in behaviour if the literals being compared are complementary. however, when the literals are non-complementary, if they are equal, only one copy is put in the resolvent whilst when they are unequal both are kept in the resolvent. once a single run of any of the clauses is finished, the other clause is merged with the accumulator. actual sorting in our case is done by simply reversing the accumulator (since all elements are in descending order). definition auxunion (c1 c2 : list z)(acc : list z) = match c1, c2 with | nil, c2 ⇒app (rev acc) c2 | c1, nil ⇒app (rev acc) c1 | x :: xs, y :: ys ⇒if (abs x < abs y) then auxunion xs (y :: ys) (x :: acc) else if (abs y < abs x) then auxunion (x :: xs) ys (y :: acc) else if x=y then auxunion xs ys (x :: acc) else auxunion xs ys (x :: y :: acc) we can now show the actual resolution function denoted by ▷ ◁below. it makes use of the union function. definition c1 ▷ ◁c2 = (union c1 c2 nil) given a problem representation in cnf form and a proof trace that respects the well- formedness criterion and is a trivial resolution proof for the given problem, shruti will deduce the empty clause and thus validate the solver's unsat claim. conversely, whenever shruti validates a claim, the problem is indeed unsat-shruti will never deduce an empty clause for a sat instance and will thus never give a false posi- tive. if shruti cannot deduce the empty clause, it invalidates the claim. this situation can be caused by three different reasons. first, counter to the claim the problem is sat. second, the problem may be unsat but the resolution proof may not represent this because it may have bugs. third, the traces either do not respect the well-formedness criteria, or do not represent a trivial resolution proof. in this case both the problem and the proof may represent an unsat instance but our checker cannot validate it as such. in this respect our checker is incomplete. 3.2 soundness of the resolution function here we formalize the soundness criteria for our checker and present the soundness theorem stating that the definition of our resolution function is sound. we need to prove that the resolvent of a given pair of clauses is logically entailed by the two clauses. thus at a high-level we need to prove that: ∀c1 c2 c3 * (c3 = c1 ▷ ◁c2) ⊃{c1, c2} | = c3 where | = denotes the logical entailment. we can use the deduction theorem ∀a b c * {a, b} | = c ≡(a ∧b ⊃c) to re-state what we intuitively would like to prove: ∀c1 c2 * (c1 ∧c2) ⊃(c1 ▷ ◁c2) however instead of proving the above theorem directly we prove its contraposition: ∀c1c2 * ¬(c1 ▷ ◁c2) ⊃¬(c1 ∧c2) in our formalization a clause is denoted by a list of non-zero integers, and a con- junction of clauses is denoted by a list of clauses. we now present the definition of the logical disjunction and conjunction functions that operate on the integer list represen- tation of clauses. we do this with the help of an interpretation function that maps an integer to a boolean value. the function eval shown below maps a list of integers to a boolean value by using the logical disjunction ∨and an interpretation i of the type z →bool. definition eval nil i = false eval (x :: xs) i = eval x i ∨(eval xs i) we now define what it means to perform a conjunction over a list of clauses. the function and shown below takes a list of clauses and an interpretation i (with type z →bool) and returns a boolean which denotes the conjunction of all the clauses in the list. definition and nil i = true and (x :: xs) i = (eval x i) ∧(and xs i) the interpretations that we are interested in are the logical interpretations which means that if we apply an interpretation i on a negative integer the value returned is the logical negation of the value returned when the same interpretation is applied on the positive integer. definition logical i = ∀(x : z) * i(−x) = ¬(i x) thus we can now state the precise statement of the soundness theorem that we proved for our checker as: theorem 1. soundness theorem ∀c1c2 * ∀i * logical i ⊃¬(eval (c1 ▷ ◁c2) i) ⊃¬(and [c1, c2] i) proof. the proof begins by structural induction on c1 and c2. the first three sub-goals are easily proven by term rewriting and simplification by unfolding the definitions of ▷ ◁, eval and and. the last sub-goal is proven by doing a case split on if-then-else and then using a combination of induction hypothesis and generating conflict among some of the assumptions. a detailed transcription of the coq proof is available from http://sites.google.com/site/certifiedsat/. 3.3 correctness of implementation further to ensure that the formalization of our checker is correct we check that the union function is implemented correctly. we check that it preserves the following properties of the trivial resolution function. these properties are: 1. a pair of complementary literals is deleted in the resolvent obtained from resolving a given pair of clauses (theorem 2). 2. all non-complementary pair of literals that are unequal are retained in the resolvent (theorem 3). 3. for a given pair of clauses, if there are no duplicate literals within each clause, then for a literal that exists in both the clauses of the pair, only one copy of the literal is retained in the resolvent (theorem 4). we have proven these properties in coq. the actual proof consists of proving several small and big lemmas - in total about 4000 lines of proof script in coq (see the proofs online). the general strategy is to use structural induction on clauses c1 and c2. for each theorem, this results in four main goals, three of which are proven by contradiction since for all elements l, l/ ∈nil. for the remaining goal a case-split is done on if-then-else, thereby producing sub-goals, some of whom are proven from induction hypotheses, and some from conflicting assumptions arising from the case-split. theorem 2. a pair of complementary literals is deleted. ∀c1 c2 * wf c1 ⊃wf c2 ⊃uniquecomppair c1 c2 ⊃ ∀l1 l2 * (l1 ∈c1) ⊃(l2 ∈c2) ⊃(l1 + l2 = 0) ⊃ (l1 / ∈(c1 ▷ ◁c2)) ∧(l2 / ∈(c1 ▷ ◁c2)) note that to ensure that only a single pair of complementary literals is deleted we need to assume that there is a unique complementary pair (uniquecomppair). the above theorem will not hold for the case with multiple complementary pairs. for the following theorem we need to assert in the assumption that for any literal in one clause there exists no literal in the other clause such that the sum of two literals is 0. this is defined by the predicate nocomplit. theorem 3. all non-complementary, unequal literals are retained. ∀c1 c2 * wf c1 ⊃wf c2 ⊃ ∀l1 l2 * (l1 ∈c1) ⊃(l2 ∈c2) ⊃ (nocomplit l1 c2) ⊃(nocomplit l2 c1) ⊃ (l1 ̸= l2) ⊃(l1 ∈(c1 ▷ ◁c2)) ∧(l2 ∈(c1 ▷ ◁c2)) theorem 4. only one copy of equal literals is retained (factoring). ∀c1 c2 * wf c1 ⊃wf c2 ⊃ ∀l1 l2 * (l1 ∈c1) ⊃(l2 ∈c2) ⊃(l1 = l2) ⊃ ((l1 ∈(c1 ▷ ◁c2)) ∧(count l1 (c1 ▷ ◁c2) = 1)) in order to check the resolution steps for each row, one has to collect the actual clauses corresponding to their identifiers and this is done by the findclause function. the function findclause takes a list of clause identifiers (dlst), an accumulator (acc) to collect the list of clauses, and requires as input a table that has the information about all the input clauses (ctbl). if a clause id is processed, then its resolvent is fetched from the resolvent table (rtbl), else obtained from ctbl. if there is no entry for a given id in the resolvent table and in the clause table, an error is signalled. this error denotes the fact that there was an input/output problem with the proof trace file due to which some input clauses in the proof trace could not be accessed properly. this could have happened either because the proof trace was ill-formed accidentally or wilfully tampered with. the function that uses the ▷ ◁function recursively on a list of input clause chain is called chainresolution and it simply folds the ▷ ◁function from left to right for every row in the proof part of the proof trace file. definition chainresolution lst = match (lst : list (list z)) with | nil ⇒nil | (x :: xs) ⇒list.fold left (▷ ◁) xs x the function findandresolve is our last function defined in coq world for unsat check- ing and provides a wrapper on other functions. once the input clause file and proof trace files are opened and read into different tables, findandresolve starts the checking pro- cess by first obtaining the clause ids from the proof part of the proof trace file, and then invoking findclause to collect all the clauses for each row in the proof part of the proof trace file. once all the clauses are obtained the function chainresolution is called and applied on the list of clauses row by row. for each row the resolvent is stored in a sep- arate table. the checker then simply checks if the last row has an empty clause, and if there is one, it agrees with the sat solver and says yes, the problem is unsat, else no. if the proof trace part of the trace file contains nothing (ill-formed) then there would be no entry for an identifier in the trace table (ttbl), and this is signalled by en error state consisting of a list with a single zero. since zeros are otherwise prohibited to be a legal part of the cnf problem description, we use them to signal error states. similarly, if the clause and trace table both are empty, then a list with two zeros is output as an error. 3.4 program extraction we extract the ocaml code by using the built-in extraction api in coq. at the time of extraction we mapped several coq datatypes and data structures to equivalent ocaml ones. for optimization we made the following replacements: 1. coq booleans by ocaml booleans. 2. coq integers (z) by ocaml int. 3. coq lists by ocaml lists. 4. coq finite map by ocaml's finite map. 5. the combination of app and rev on lists in the function union, and auxunion was replaced by the tail-recursive list.rev append in ocaml. replacing coq zs with ocaml integers gave a performance boost by a factor of 7- 10. making minor adjustments by replacing the coq finite maps by ocaml ones and using tail recursive functions gave a further 20% improvement. an important conse- quence of our extraction is that only some datatypes, and data structures get mapped to ocaml's; the key logical functionality is unmodified. the decisions for making changes in data types and data structures are a standard procedure in any extraction process using coq [2]. table 1: comparison of our results with hol 4 and tracecheck. number of resolutions (inferences) shown for hol 4 is the number that hol 4 calculated from the proof trace obtained from running zverify - the uncertified checker (for zchaff) that amjad used to obtain the proof trace. shruti resolution count is obtained from the proof trace generated by the uncertified checker tracecheck. in terms of inferences/second, we are 1.5 to 32 times faster than amjad's hol 4 checker, whilst a factor 2.5 slower than tracecheck. all times shown are total times for all the three checkers.the symbol z? denotes that zchaff timed out after an hour. no. benchmark hol 4 shruti tracecheck resolutions time inf/sec resolutions time inf/s time inf/s 1. een-tip-uns-numsv-t5.b 89136 4.61 19335 122816 0.86 142809 0.36 341155 2. een-pico-prop01-75 205807 5.70 36106 246430 1.67 147562 0.48 513395 3. een-pico-prop05-50 1804983 58.41 30901 2804173 20.76 135075 8.11 345767 4. hoons-vbmc-lucky7 3460518 59.65 58013 4359478 35.18 123919 12.95 336639 5. ibm-2002-26r-k45 1448 24.76 58 1105 0.004 276250 0.04 27625 6. ibm-2004-26-k25 1020 11.78 86 1132 0.004 283000 0.04 28300 7. ibm-2004-3 02 1-k95 69454 5.03 13807 114794 0.71 161681 0.35 327982 8. ibm-2004-6 02 3-k100 111415 7.04 15825 126873 0.9 140970 0.40 317182 9. ibm-2002-07r-k100 141501 2.82 50177 255159 1.62 157505 0.54 472516 10. ibm-2004-1 11-k25 534002 13.88 38472 255544 1.77 144375 0.75 340725 11. ibm-2004-2 14-k45 988995 31.16 31739 701430 5.42 129415 1.85 379151 12. ibm-2004-2 02 1-k100 1589429 24.17 65760 1009393 7.42 136036 3.02 334236 13. ibm-2004-3 11-k60 z? z? - 13982558 133.05 105092 59.27 235912 14. manol-pipe-g6bi 82890 2.12 39099 245222 1.59 154227 0.50 490444 15. manol-pipe-c9nidw s 700084 26.79 26132 265931 1.81 146923 0.54 492464 16. manol-pipe-c10id s 36682 11.23 3266 395897 2.60 152268 0.82 482801 17. manol-pipe-c10nidw s z? z? - 458042 3.06 149686 1.21 381701 18. manol-pipe-g7nidw 325509 8.82 36905 788790 5.40 146072 1.98 398378 19. manol-pipe-c9 198446 3.15 62998 863749 6.29 137320 2.50 345499 20. manol-pipe-f6bi 104401 5.07 20591 1058871 7.89 134204 2.97 356522 21. manol-pipe-c7b i 806583 13.76 58617 4666001 38.03 122692 15.54 300257 22. manol-pipe-c7b 824716 14.31 57632 4901713 42.31 115852 18 272317 23. manol-pipe-g10id 775605 23.21 33416 6092862 50.82 119891 21.08 289035 24. manol-pipe-g10b 2719959 52.90 51416 7827637 64.69 121002 26.85 291532 25. manol-pipe-f7idw 956072 35.17 27184 7665865 68.14 112501 30.74 249377 26. manol-pipe-g10bidw 4107275 125.82 32644 14776611 134.92 109521 68.13 216888 4 experimental results we evaluated our certified checker shruti on a set of benchmarks from the sat races of 2006 and 2008 and the sat competition of 2007. we present our results on a sample of the sat race benchmarks in table 1. the results for shruti shown in the table are for validating proof traces obtained from the picosat solver. our experiments were carried out on a server running red hat on a dual-core 3 ghz, intel xeon cpu with 28gb memory. the hol 4 and isabelle based checkers [22] were evaluated on the sat race benchmarks shown in the table [21]. isabelle reported segmentation faults on most of the problems, whilst hol 4's results are summarized along with our's in table 1. hol 4 was run on an amd dual-core 3.2 ghz processor running ubuntu with 4gb of memory. we also compare our timings with that obtained from the uncertified checker tracecheck. since the size of the proof traces obtained from zchaff is substantially different than the size of the traces obtained from tracecheck on most problems, we decided to compare the speed of our checker with hol 4 and tracecheck in terms of resolutions (inferences) solved per second. we observe that in terms of inferences/sec we are 1.5 to 32 times faster than hol 4 and 2.5 times slower than tracecheck. times shown for all the three checkers in the table are the total times including time spent on actual resolution checking, file i/o and garbage collection. amjad reported that the version of the checker he has used on these benchmarks is much faster than the one published in [22]. as a proof of concept we also validated the proof traces from zchaff by translating them to picosat's trace format. the performance of shruti in terms of inf/sec on the translated proof traces (from zchaff to picosat) was similar to the performance of shruti when it checked picosat's traces obtained directly from the picosat solver – something that is to be expected. 4.1 discussion the coq formalization consisted of 8 main function definitions amounting to nearly 160 lines of code, and 4 main theorems shown in the paper and 4 more that are about maps (not shown here due to space). overall the proof in coq was nearly 4000 lines consisting of the proofs of several big and small lemmas that were essential to prove the 4 main theorems. the extracted ocaml code was approximately 2446 lines, and the ocaml glue code was 324 lines. we found that there is no implementation of the array data type which meant that we had to use the type of list. since lists are defined inductively, it is easier to do reasoning with them, although implementing very fast and efficient functions on these is impossible. in a recent development related to coq, there has been an emergence of a tool called ynot [14] that can deal with arrays, pointers and file related i/o in a hoare type theory. future work in certification using coq should definitely investigate the relevance and use of this. we noticed that the ocaml compiler's native code compilation does produce effi- cient binaries but the default settings for automatic garbage collection were not useful. we observed that if we do not tune the runtime environment settings of ocaml by set- ting the values of ocmalrunparam, as soon as the input proof traces had more than a million inferences, garbage collection would kick in so severely that it will end up consuming (and thereby delaying the overall computation) as much as 60% of the total time. by setting the initial size of major heap to a large value such as 2 gb and making the garbage collection less eager, we noticed that the computation times of our checker got reduced by upto a factor of 7 on proof traces with over 1 million inferences. 5 related work recent work on checking the result of sat solvers can be traced to the work of zhang & malik [23] and goldberg & novikov [8], with additional insights provided in recent work [1,19]. besides weber and amjad, others who have advocated the use of a checker include bulwahn et al [5] who experimented with the idea of doing reflective theorem proving in isabelle and suggested that it can be used for designing a sat checker. in a recent paper [12], mari ́ c presented a formalization of sat solving algorithms in isabelle that are used in modern day sat solvers. an important difference is that whereas we have formalized a sat checker and extracted an executable code from the formalization itself, mari ́ c formalizes a sat solver (at the abstract level of state machines) and then implements the verified algorithm in the sat solver off-line. an alternative line of work involves the formal development of sat solvers. ex- amples include the work of smith & westfold [18] and the work of lescuyer and con- chon [11]. lescuyer and conchon have formalized a simplified sat solver in coq and extracted an executable. however, the performance results have not been reported on any industrial benchmarks. this is because they have not formalized several of the key techniques used in modern sat solvers. the work of of smith & westfold involves the formal synthesis of a sat solver from a high level description. albeit ambitious, the preliminary version of the sat solver does not include the most effective techniques used in modern sat solvers. there has been a recent surge in the area of certifying smt solvers. m. moskal recently provided an efficient certification technique for smt solvers [13] using term- rewriting systems. the soundness of the proof checker is guaranteed through a formal- ization using inference rules provided in a term-rewriting formalism. l. de moura and n. bjørner [6] presented the proof and model generating features of the state-of-the-art smt solver z3. 6 conclusion in this paper we presented a methodology for performing efficient yet formally certified sat solving. the key feature of our approach is that we did a one-off formal design and reasoning of the checker using coq proof-assistant and extracted an ocaml program which was used as a standalone executable to check the outcome of industrial-strength sat solvers such as picosat and zchaff. our certified checker can be plugged in with any proof generating sat solver with previously agreed certificates for satisfiable and unsatisfiable problems. on one hand our checker provides much higher assurance as compared to uncertified checkers such as tracecheck and on the other it enhances us- ability and performance when compared to the certified checkers implemented in hol 4 and isabelle. in this regard our approach provides an arguably optimal middle ground between the two extremes. we are investigating on optimizing the performance aspects of our checker even further so that the slight difference in overall performance between uncertified checkers and us can be further minimized. acknowledgements. we thank h. herbelin, y. bertot, p. letouzey, and many more people on the coq mailing list who helped us with coq questions. we also thank t. weber and h. amjad for answering our questions on their work and also carrying out industrial benchmark evaluation on their checker. a. p. landells helped out with server issues. this work was partially funded by epsrc grant ep/e012973/1, and eu grants ict/217069 and ist/033709. references 1. p. beame, h. a. kautz, and a. sabharwal. towards understanding and harnessing the poten- tial of clause learning. j. artif. intell. res. (jair), 22:319–351, 2004. 2. y. bertot and p. castéran. interactive theorem proving and program development. coq'art: the calculus of inductive constructions, 2004. 3. a. biere. picosat essentials. journal on satisfiability, boolean modeling and computation, 4:75–97, 2008. 4. a. biere, a. cimatti, e. clarke, o. strichman, and y. zhu. advances in computers, chapter bounded model checking. academic press, 2003. 5. l. bulwahn, a. krauss, f. haftmann, l. erkök, and j. matthews. imperative functional programming with isabelle/hol. in theorem proving in higher order logics, pages 134– 149, 2008. 6. l. m. de moura and n. bjørner. proofs and refutations, and z3. in proceedings of the lpar 2008 workshops, 2008. 7. n. een and n. sorensson. an extensible sat-solver. in e. giunchiglia and a. tacchella, editors, sat, volume 2919 of lecture notes in computer science, pages 502–518. springer, 2003. 8. e. i. goldberg and y. novikov. verification of proofs of unsatisfiability for cnf formulas. in design, automation and test in europe conference, pages 10886–10891, march 2003. 9. j. hammarberg and s. nadjm-tehrani. formal verification of fault tolerance in safety-critical reconfigurable modules. international journal on software tools for technology transfer (sttt), 7(3):268–279, 2005. 10. x. leroy and s. blazy. formal verification of a c-like memory model and its uses for verifying program transformations. journal of automated reasoning, jan 2008. 11. s. lescuyer and s. conchon. a reflexive formalization of a sat solver in coq. in emerging trends of tphols, 2008. 12. f. mari ́ c. formalization and implementation of modern sat solvers. journal of automated reasoning, 43(1):81–119, june 2009. 13. m. moskal. rocket-fast proof checking for smt solvers. in tools and algorithms for the construction and analysis of systems, pages 486–500, 2008. 14. a. nanevski, g. morrisett, a. shinnar, p. govereau, and l. birkedal. ynot: dependent types for imperative programs. in international conference on functional programming, pages 229–240, 2008. 15. m. penicka. formal approach to railway applications. in formal methods and hybrid real- time systems, pages 504–520, 2007. 16. sat competition. http://www.satcompetition.org/. 17. d. s. scott. a type-theoretical alternative to iswim, cuch, owhy. theor. comput. sci., 121(1-2):411–440, 1993. 18. d. r. smith and s. j. westfold. synthesis of propositional satisfiability solvers. technical report, kestrel institute, april 2008. 19. a. van gelder. verifying propositional unsatisfiability: pitfalls to avoid. in theory and applications of satisfiability testing, pages 328–333, may 2007. 20. t. weber. efficiently checking propositional resolution proofs in isabelle/hol. 6th interna- tional workshop on the implementation of logics, 2009. 21. t. weber and h. amjad. private communication. 22. t. weber and h. amjad. efficiently checking propositional refutations in hol theorem provers. journal of applied logic, 7(1):26–40, 2009. 23. l. zhang and s. malik. validating sat solvers using an independent resolution-based checker: practical implementations and other applications. in design, automation and test in europe conference, pages 10880–10885, march 2003.

0911.1679

xor gate response in a mesoscopic ring with embedded quantum dots

we address xor gate response in a mesoscopic ring threaded by a magnetic flux $\phi$. the ring, composed of identical quantum dots, is symmetrically attached to two semi-infinite one-dimensional metallic electrodes and two gate voltages, viz, $v_a$ and $v_b$, are applied, respectively, in each arm of the ring which are treated as the two inputs of the xor gate. the calculations are based on the tight-binding model and the green's function method, which numerically compute the conductance-energy and current-voltage characteristics as functions of the ring-electrodes coupling strengths, magnetic flux and gate voltages. quite interestingly it is observed that, for $\phi=\phi_0/2$ ($\phi_0=ch/e$, the elementary flux-quantum) a high output current (1) (in the logical sense) appears if one, and only one, of the inputs to the gate is high (1), while if both inputs are low (0) or both are high (1), a low output current (0) appears. it clearly demonstrates the xor behavior and this aspect may be utilized in designing the electronic logic gate.

introduction in the present age of nanoscience and technology quantum confined model systems are used exten- sively in electronic as well as spintronic engineering since these simple looking systems are the funda- mental building blocks of designing nano devices. a mesoscopic normal metal ring is one such promising example of quantum confined systems. here we will explore the electron transport through a mesoscopic ring, composed of identical quantum dots and at- tached to two external electrodes, the so-called electrode-ring-electrode bridge, and show how such a simple geometric model can be used to design a logic gate. the theoretical description of elec- tron transport in a bridge system has been followed based on the pioneering work of aviram and rat- ner [1]. later, many excellent experiments [2, 3, 4] have been done in several bridge systems to un- derstand the basic mechanisms underlying the elec- tron transport. though in literature many theo- retical [5, 6, 7, 8, 9, 10, 11, 12, 13, 14] as well as experimental papers [2, 3, 4] on electron transport are available, yet lot of controversies are still present between the theory and experiment, and the com- plete knowledge of the conduction mechanism in this scale is not very well established even today. the electronic transport in the ring significantly de- pends on the ring-to-electrodes interface structure. by changing the geometry one can tune the trans- mission probability of an electron across the ring. this is solely due to the quantum interference ef- fect among the electronic waves traversing through different arms of the ring. furthermore, the elec- tron transport through the ring can be modulated in other way by tuning the magnetic flux, the so- called aharonov-bohm (ab) flux, that threads the ring. the ab flux threading the ring can change the phases of the wave functions propagating along the different arms of the ring leading to constructive or destructive interferences, and accordingly the transmission amplitude changes [15, 16, 17, 18, 19]. beside these factors, ring-to-electrodes coupling is another important issue that controls the electron transport in a meaningful way [19]. all these are the key factors which regulate the electron trans- mission in the electrode-ring-electrode bridge sys- tem and these effects have to be taken into account properly to reveal the transport mechanisms. the aim of the present paper is to describe the xor gate response in a mesoscopic ring threaded by a magnetic flux φ. the ring is contacted sym- metrically to the electrodes, and the two arms of the ring are subjected to two gate voltages va and vb, respectively (see fig. 1) those are treated as the two inputs of the xor gate. here we adopt a simple tight-binding model to describe the sys- tem and all the calculations are performed numer- ically. we address the xor behavior by studying the conductance-energy and current-voltage charac- teristics as functions of the ring-electrodes coupling strengths, magnetic flux and gate voltages. our study reveals that for a particular value of the mag- netic flux, φ = φ0/2, a high output current (1) (in the logical sense) is available if one, and only one, of the inputs to the gate is high (1), while if both the inputs are low (0) or both are high (1), a low output current (0) is available. this phenomenon clearly demonstrates the xor behavior which may be uti- lized in manufacturing the electronic logic gate. to the best of our knowledge the xor gate response in such a simple system has not been described earlier in the literature. the paper is organized as follow. following the introduction (section 1), in section 2, we present the model and the theoretical formulations for our calculations. section 3 discusses the significant re- sults, and finally, we summarize our results in sec- tion 4. 2 model and the synopsis of the theoretical background we begin by referring to fig. 1. a mesoscopic ring, composed of identical quantum dots (filled red cir- cles) and threaded by a magnetic flux φ, is attached symmetrically to two semi-infinite one-dimensional metallic electrodes. the ring is placed between two gate electrodes, viz, gate-a and gate-b. these gate electrodes are ideally isolated from the ring and can be regarded as two parallel plates of a capacitor. in our present scheme we assume that the gate volt- ages each operate on the dots nearest to the plates only. while, in complicated geometric models, the effect must be taken into account for the other dots, though the effect becomes too small. the dots a and b in the two arms of the ring are subjected to the gate voltages va and vb, respectively, and these are treated as the two inputs of the xor gate. the actual scheme of connections with the batteries for the operation of the xor gate is clearly presented in the figure (fig. 1), where the source and the gate voltages are applied with respect to the drain. based on the landauer conductance formula [20, 21] we determine the conductance (g) of the ring. 2 at very low temperature and bias voltage it can be expressed in the form, g = 2e2 h t (1) where t gives the transmission probability of an electron through the ring. this (t ) can be repre- sented in terms of the green's function of the ring and its coupling to the two electrodes by the rela- tion [20, 21], t = tr [γsgr rγdga r] (2) where gr r and ga r are respectively the retarded and advanced green's functions of the ring including the effects of the electrodes. the parameters γs and γd describe the coupling of the ring to the source and φ a b source drain gate−a gate−b figure 1: (color online). the scheme of con- nections with the batteries for the operation of the xor gate. a mesoscopic ring with embed- ded quantum dots (filled red circles) is attached to two semi-infinite one-dimensional metallic elec- trodes, viz, source and drain. the gate voltages va and vb, those are variable, are applied in the dots a and b of the two arms, respectively. the source and the gate voltages are applied with respect to the drain. drain, respectively. for the full system i.e., the ring, source and drain, the green's function is defined as, g = (e −h)−1 (3) where e is the injecting energy of the source elec- tron. to evaluate this green's function, the inver- sion of an infinite matrix is needed since the full system consists of the finite ring and the two semi- infinite electrodes. however, the entire system can be partitioned into sub-matrices corresponding to the individual sub-systems and the green's func- tion for the ring can be effectively written as, gr = (e −hr −σs −σd)−1 (4) where hr is the hamiltonian of the ring that can be expressed within the non-interacting picture like, hr = x i (ǫi0 + vaδia + vbδib) c† ici + x <ij> t  c† icjeiθ + c† jcie−iθ (5) in this hamiltonian ǫi0's are the on-site energies for all the sites (filled red circles) i, except the sites i = a and b where the gate voltages va and vb are applied, those are variable. these gate voltages can be incorporated through the site energies as ex- pressed in the above hamiltonian. c† i (ci) is the cre- ation (annihilation) operator of an electron at the site i and t is the nearest-neighbor hopping integral. θ = 2πφ/nφ0 is the phase factor due to the flux φ, where n represents the total number of sites/dots in the ring. similar kind of tight-binding hamiltonian is also used, except the phase factor θ, to describe the semi-infinite one-dimensional perfect electrodes where the hamiltonian is parametrized by constant on-site potential ǫ0 and nearest-neighbor hopping integral t0. the ring is coupled to the electrodes by the parameters τs and τd, where they (coupling pa- rameters) correspond to the coupling strengths with the source and drain, respectively. the parameters σs and σd in eq. (4) represent the self-energies due to the coupling of the ring to the source and drain, respectively, where all the informations of this cou- pling are included into these self-energies. to evaluate the current (i), passing through the ring, as a function of the applied bias voltage (v ) we use the relation [20], i(v ) = e π ̄ h ef +ev/2 z ef −ev/2 t (e) de (6) where ef is the equilibrium fermi energy. here we make a realistic assumption that the entire voltage is dropped across the ring-electrode interfaces, and it is examined that under such an assumption the i-v characteristics do not change their qualitative features. in this presentation, all the results are computed only at absolute zero temperature. these results 3 are also valid even for some finite (low) tempera- tures, since the broadening of the energy levels of the ring due to its coupling with the electrodes be- comes much larger than that of the thermal broad- ening [20]. on the other hand, at high tempera- ture limit, all these phenomena completely disap- pear. this is due to the fact that the phase coher- ence length decreases significantly with the rise of temperature where the contribution comes mainly from the scattering on phonons, and accordingly, the quantum interference effect vanishes. for the sake of simplicity, we take the unit c = e = h = 1 in our present calculations. 3 results and discussion to illustrate the results, let us first mention the values of the different parameters used for the nu- merical calculations. in the ring, the on-site energy ǫi0 is taken as 0 for all the sites i, except the sites i = a and b where the site energies are taken as va and vb, respectively, and the nearest-neighbor -8 0 8 e 0 1 2 ghel hcl -8 0 8 e -1 0 1 ghel hdl -8 0 8 e -1 0 1 ghel hal -8 0 8 e 0 1 2 ghel hbl figure 2: (color online). conductance g as a func- tion of the energy e for a mesoscopic ring with n = 8 and φ = 0.5 in the limit of weak-coupling. (a) va = vb = 0, (b) va = 2 and vb = 0, (c) va = 0 and vb = 2 and (d) va = vb = 2. hopping strength t is set to 3. on the other hand, for the side attached electrodes the on-site energy (ǫ0) and the nearest-neighbor hopping strength (t0) are fixed to 0 and 4, respectively. the fermi en- ergy ef is set to 0. to narrate the coupling effect, throughout the study we focus our results for the two limiting cases depending on the strength of the coupling of the ring to the source and drain. case i: the weak-coupling limit. it is described by the condition τs(d) << t. for this regime we choose τs = τd = 0.5. case ii: the strong-coupling limit. this is specified by the condition τs(d) ∼t. in this particular regime, we set the values of the pa- rameters as τs = τd = 2.5. the key controlling parameter for all these calculations is the magnetic flux φ which is set to φ0/2 i.e., 0.5 in our chosen unit c = e = h = 1. in fig. 2 we present the conductance-energy (g- e) characteristics for a mesoscopic ring with n = 8 in the limit of weak-coupling, where (a), (b), (c) and (d) correspond to the results for the different gate voltages. when both the two inputs va and vb are identical to zero i.e., both the inputs are low, the conductance g becomes exactly zero (fig. 2(a)) -8 0 8 e 0 1 2 ghel hcl -8 0 8 e -1 0 1 ghel hdl -8 0 8 e -1 0 1 ghel hal -8 0 8 e 0 1 2 ghel hbl figure 3: (color online). conductance g as a func- tion of the energy e for a mesoscopic ring with n = 8 and φ = 0.5 in the limit of strong-coupling. (a) va = vb = 0, (b) va = 2 and vb = 0, (c) va = 0 and vb = 2 and (d) va = vb = 2. for all energies. this reveals that the electron can- not conduct through the ring. similar response is also observed when both the two inputs are high i.e., va = vb = 2, and in this case also the ring does not allow to pass an electron from the source to the drain (fig. 2(d)). on the other hand, for the cases where any one of the two inputs is high and other is low i.e., either va = 2 and vb = 0 (fig. 2(b)) or va = 0 and vb = 2 (fig. 2(c)), the conduc- tance exhibits fine resonant peaks for some partic- ular energies. thus for both these two cases the 4 electron conduction takes place across the ring. at the resonances where the conductance approaches the value 2, the transmission probability t goes to unity, since the relation g = 2t follows from the landauer conductance formula (see eq. 1 with e = h = 1). all these resonant peaks are associ- ated with the energy eigenvalues of the ring, and accordingly, we can say that the conductance spec- trum manifests itself the electronic structure of the -8 0 8 v -.6 .6 hcl 0 i -8 0 8 v -.6 .6 hdl 0 i -8 0 8 v -.6 .6 hal 0 i -8 0 8 v -.6 .6 hbl 0 i figure 4: (color online). current i as a function of the bias voltage v for a mesoscopic ring with n = 8 and φ = 0.5 in the limit of weak-coupling. (a) va = vb = 0, (b) va = 2 and vb = 0, (c) va = 0 and vb = 2 and (d) va = vb = 2. ring. thus more resonant peaks can be obtained for larger rings corresponding to their energy eigen- values. now we justify the dependences of the gate voltages on the electron transport for these four dif- ferent cases. the probability amplitude of getting an electron across the ring depends on the quantum interference of the electronic waves passing through the upper and lower arms of the ring. for the sym- metrically connected ring i.e., when the two arms of the ring are identical with each other, the proba- bility amplitude is exactly zero (t = 0) for the flux φ = φ0/2. this is due to the result of the quantum interference among the two waves in the two arms of the ring, which can be obtained in a very sim- ple mathematical calculation. thus for the cases when both the two inputs (va and vb) are either low or high, the transmission probability drops to zero. on the other hand, for the other two cases the symmetry of the two arms of the ring is broken by applying the gate voltage either in the atom a or b, and therefore, the non-zero value of the transmis- sion probability is achieved which reveals the elec- tron conduction across the ring. thus we can pre- dict that the electron conduction takes place across the ring if one, and only one, of the inputs to the gate is high, while if both the inputs are low or table 1: xor gate behavior in the limit of weak- coupling. the current i is computed at the bias voltage 6.02. input-i (va) input-ii (vb) current (i) 0 0 0 2 0 0.378 0 2 0.378 2 2 0 both are high the conduction is no longer possible. this feature clearly demonstrates the xor behav- ior. with these characteristics, we get additional one feature when the coupling strength of the ring to the electrodes increases from the low regime to high one. in the limit of strong ring-to-electrodes coupling, all these resonances get substantial widths compared to the weak-coupling limit. the results are shown in fig. 3, where all the other parame- ters are identical to those in fig. 2. the contri- bution for the broadening of the resonant peaks in this strong-coupling limit appears from the imagi- nary parts of the self-energies σs and σd, respec- tively [20]. hence by tuning the coupling strength, we can get the electron transmission across the ring for the wider range of energies and it provides an important signature in the study of current-voltage (i-v ) characteristics. all these features of electron transfer become much more clearly visible by studying the i-v char- acteristics. the current passing through the ring is computed from the integration procedure of the transmission function t as prescribed in eq. 6. the transmission function varies exactly similar to that of the conductance spectrum, differ only in magni- tude by the factor 2 since the relation g = 2t holds from the landauer conductance formula eq. 1. as illustrative examples, in fig. 4 we show the current- voltage characteristics for a mesoscopic ring with n = 8 in the limit of weak-coupling. for the cases when both the two inputs are identical with each other, either low (fig. 4(a)) or high (fig. 4(d)), the current is zero for the entire bias voltages. this be- havior is clearly understood from the conductance 5 spectra, figs. 2(a) and (d), since the current is com- puted from the integration procedure of the trans- mission function t . for the two other cases where only one of the two inputs is high and other is low, a high output current is obtained which are clearly described in figs. 4(b) and (c). from these figures it is observed that the current exhibits staircase-like structure with fine steps as a function of the applied bias voltage. this is due to the existence of the sharp resonant peaks in the conductance spectrum in the weak-coupling limit, since the current is com- puted by the integration method of the transmission -8 0 8 v -4 0 hcl 4 i -8 0 8 v -4 0 hdl 4 i -8 0 8 v -4 0 hal 4 i -8 0 8 v -4 0 hbl 4 i figure 5: (color online). current i as a function of the bias voltage v for a mesoscopic ring with n = 8 and φ = 0.5 in the limit of strong-coupling. (a) va = vb = 0, (b) va = 2 and vb = 0, (c) va = 0 and vb = 2 and (d) va = vb = 2. function t . with the increase of the bias voltage v , the electrochemical potentials on the electrodes are shifted gradually, and finally cross one of the quantized energy levels of the ring. accordingly, a current channel is opened up which provides a jump in the i-v characteristic curve. addition to these behaviors, it is also important to note that the non-zero value of the current appears beyond a finite value of v , so-called the threshold voltage (vth). this vth can be controlled by tuning the size (n) of the ring. from these i-v characteristics the behavior of the xor gate response is clearly visi- ble. to make it much clear, in table 1, we present a quantitative estimate of the typical current am- plitude, computed at the bias voltage v = 6.02, in this weak-coupling limit. it shows that i = 0.378 only when any one of the two inputs is high and other is low, while for the other cases when either va = vb = 0 or va = vb = 2, i gets the value 0. in the same analogy, as above, here we also discuss the i-v characteristics for the strong-coupling limit. in this limit, the current varies almost continuously with the applied bias voltage and achieves much larger amplitude than the weak-coupling case as presented in fig. 5. the reason is that, in the limit of strong-coupling all the energy levels get broad- ened which provide larger current in the integration procedure of the transmission function t . thus by tuning the strength of the ring-to-electrodes cou- pling, we can achieve very large current amplitude table 2: xor gate behavior in the limit of strong- coupling. the current i is computed at the bias voltage 6.02. input-i (va) input-ii (vb) current (i) 0 0 0 2 0 2.846 0 2 2.846 2 2 0 from the very low one for the same bias voltage v . all the other properties i.e., the dependences of the gate voltages on the i-v characteristics are exactly similar to those as given in fig. 4. in this strong- coupling limit we also make a quantitative study for the typical current amplitude, given in table 2, where the current amplitude is determined at the same bias voltage (v = 6.02) as earlier. the re- sponse of the output current is exactly similar to that as given in table 1. here the non-zero value of the current gets the value 2.846 which is much larger compared to the weak-coupling case which shows the value 0.378. from these results we can clearly manifest that a mesoscopic ring exhibits the xor gate response. 4 concluding remarks to summarize, we have addressed xor gate re- sponse in a mesoscopic metallic ring threaded by a magnetic flux φ. the ring, composed of identi- cal quantum dots, is attached symmetrically to the source and drain. the upper and lower arms of the ring are subjected to the gate voltages va and vb, respectively those are taken as the two inputs of the xor gate. a simple tight-binding model is used to describe the full system and all the calcu- 6 lations are done in the green's function formalism. we have numerically computed the conductance- energy and current-voltage characteristics as func- tions of the ring-electrodes coupling strengths, mag- netic flux and gate voltages. very interestingly we have noticed that, for the half flux-quantum value of φ (φ = φ0/2), a high output current (1) (in the logical sense) appears if one, and only one, of the inputs to the gate is high (1). on the other hand, if both the inputs are low (0) or both are high (1), a low output current (0) appears. it clearly mani- fests the xor gate behavior and this aspect may be utilized in designing a tailor made electronic logic gate. in view of the potential application of this xor gate as a circuit element in an integrated cir- cuit, we would like to point out that care should be taken during the application of the magnetic field in the ring such that the other circuit elements of the integrated circuit are not effected by this field. in this presentation, we have calculated all the results by ignoring the effects of the temperature, electron-electron correlation, disorder, etc. due to these factors, any scattering process that appears in the arms of the ring would have influence on electronic phases, and, in consequences can disturb the quantum interference effects. here we have as- sumed that, in our sample all these effects are too small, and accordingly, we have neglected all these effects in our present study. the importance of this article is mainly con- cerned with (i) the simplicity of the geometry and (ii) the smallness of the size. to the best of our knowledge the xor gate response in such a sim- ple low-dimensional system has not been addressed earlier in the literature. references [1] a. aviram, m. ratner, chem. phys. lett. 29 (1974) 277. [2] t. dadosh, y. gordin, r. krahne, i. khivrich, d. mahalu, v. frydman, j. sperling, a. ya- coby, i. bar-joseph, nature 436 (2005) 677. [3] j. chen, m. a. reed, a. m. rawlett, j. m. tour, science 286 (1999) 1550. [4] m. a. reed, c. zhou, c. j. muller, t. p. bur- gin, j. m. tour, science 278 (1997) 252. [5] p. a. orellana, m. l. ladron de guevara, m. pacheco, a. latge, phys. rev. b 68 (2003) 195321. [6] p. a. orellana, f. dominguez-adame, i. gomez, m. l. ladron de guevara, phys. rev. b 67 (2003) 085321. [7] a. nitzan, annu. rev. phys. chem. 52 (2001) 681. [8] a. nitzan, m. a. ratner, science 300 (2003) 1384. [9] d. m. newns, phys. rev. 178 (1969) 1123. [10] v. mujica, m. kemp, m. a. ratner, j. chem. phys. 101 (1994) 6849. [11] v. mujica, m. kemp, a. e. roitberg, m. a. ratner, j. chem. phys. 104 (1996) 7296. [12] k. walczak, phys. stat. sol. (b) 241 (2004) 2555. [13] k. walczak, arxiv:0309666. [14] w. y. cui, s. z. wu, g. jin, x. zhao, y. q. ma, eur. phys. j. b. 59 (2007) 47. [15] r. baer, d. neuhauser, j. am. chem. soc. 124 (2002) 4200. [16] d. walter, d. neuhauser, r. baer, chem. phys. 299 (2004) 139. [17] k. tagami, l. wang, m. tsukada, nano lett. 4 (2004) 209. [18] k. walczak, cent. eur. j. chem. 2 (2004) 524. [19] r. baer, d. neuhauser, chem. phys. 281 (2002) 353. [20] s. datta, electronic transport in mesoscopic systems, cambridge university press, cam- bridge (1997). [21] m. b. nardelli, phys. rev. b 60 (1999) 7828. 7

0911.1680

the heavy quark free-energy at t<tc in ads/qcd

starting with the modified ads/qcd metric developed in ref.[1] we use the nambu-goto action to obtain the free energy of a quark-antiquark pair at t<tc, for which we show that the effective string tension goes to zero at tc=154mev.

introduction as shown in ref. [2] the free energy of a static, in- finitely massive quark-antiquark pair f is given by e−βf = l(⃗ rq)l†(⃗ rq)  , (1.1) where l(⃗ r) is the wilson-line l(⃗ r) = 1 n tr t exp  i β z 0 dτ ˆ a0(⃗ r, τ)  , (1.2) β = 1/t is the inverse temperature, ˆ a0 is the gluon field in the fundamental representation and τ denotes imag- inary time. according to the holographic dictionary [3] the right hand side of equation (1.1) is equal to the string partition function on the ads5 space with the integration contours on the boundary of ads5. in saddle-point ap- proximation: e−βf = l(⃗ rq)l†(rq)  ≈e−sng , (1.3) where sng is the nambu-goto action sng = 1 2πl2 s z d2ξ p dethab , (1.4) with the induced worldsheet metric hab = gμν ∂xμ ∂ξa ∂xν ∂ξb . (1.5) as described in reference [4], due to the symmetry of the problem, we may set up a cylindrical coordinate system in five-dimensional euclidean space. then the five coor- dinates are: ∗electronic address: [email protected] †electronic address: [email protected] ‡electronic address: [email protected] §electronic address: [email protected] q q x −d 2 b d 2 b τ r fig. 1: wilson loops in euclidean time with periodicity β = 2πr. • t - time • z - the bulk coordinate (extra 5th dimension) • x, r, φ - three spatial coordinates we parameterize an element of the surface connecting the two wilson loops with dξ1 = rdφ and dξ2 = dx, set t = 0 and reinterpret dτ ≡rdφ as euclidean time. we use the modified metric gμν developed in reference [1] effectively for nc = 3 and nf = 4. this metric has been further analysed as a possible solution of 5-d gravity in ref. [5] ds2 eucl = h(z)l2 z2 (r2dφ2 + dr2 + dx2 + dz2) , (1.6) h(z) = log(ǫ) log ((λz)2 + ǫ) , (1.7) λ = l−1 = 264mev , (1.8) ǫ = l2 s l2 = 0.48 . (1.9) the nambu-goto action determines the string surface with ls as string length: 2 sng = 1 2πl2 s 2π z 0 dφ d/2 z −d/2 dxl2h(z) z2 r p 1 + (z′)2 + (r′)2 = 1 ǫ d/2 z −d/2 dxh(z) z2 r p 1 + (z′)2 + (r′)2 (1.10) =: 1 ǫ d/2 z −d/2 dx l[z(x), z′(x), r(x), r′(x)] , (1.11) the quark and antiquark are separated by a distance d along the x-axis. the two polyakov loops are approx- imated by wilson loops of radius r = β/2π. l is the lagrangian density and the prime (′) denotes the deriva- tive with respect to x. the configuration in shown in figure 1. ii. euler-lagrange equations since the lagrangian in (1.11) does not depend on x explicitly, it is invariant under variations x →x + δx. it follows from noether's theorem that there exists a con- served quantity k given by k = h(z) * r z2 1 p 1 + (z′)2 + (r′)2 . (2.1) the euler-lagrange equations corresponding to the nambu-goto action (1.10) can be simplified with eq. (2.1) (see ref. [4]): r′′ −h2(z) * r k2z4 = 0 , z′′ −h(z) * r2 * (z∂zh(z) −2h(z)) k2z5 = 0 . (2.2) the boundary conditions are r (±d/2) = r = β 2π = 1 2πt , z (±d/2) = 0 . (2.3) iii. numerical solutions analysis of symmetry (cf. fig. 1) gives for the first derivatives: r′(0) = 0 , z′(0) = 0 . (3.1) but the conditions eqs. (2.3) and (3.1) are not given at the same point. it is more convenient to find conditions for the functions and their derivatives at the same point. analysis of eq. (2.2) shows that r′ and z′ must diverge near the boundary x →± d 2. in order to obtain some stable numerical solution, we have studied the behavior of r(x) and z(x) near the boundary, cf. ref. [4]. nu- merically, a small cutoffv is applied, then r(−d/2 + v), z(−d/2+v), r′(−d/2+v) and z′(−d/2+v) can be calcu- lated asymptotically and used as initial conditions. we do not prescribe the value of d. for a fixed value of r, we give an arbitrary value to k > 0, then calculate r, z and consequently the nambu-goto action sng for this value of k. by changing the value of k we obtain the nambu- goto action as a function of k, denoted by sng(k). we can express d as a function of k, and determine the dis- tance d associated with the nambu-goto action sng(k). in principle, the constructed numerical solutions r′(x) and z′(x) can vanish at different points due to the small difference between our asymptotic solution and the real solution. we adjust the initial conditions at the point v keeping the value of k fixed in such a way that r′(0) and z′(0) vanish at the same point. the nambu-goto action is given by sng = 2 ǫ z 0 −d/2 dxh(z) * r z2 q 1 + (z′)2 + (r′)2 = 2 ǫ z −d/2+v −d/2 dxh (za) * ra z2 a q 1 + (z′ a)2 + (r′ a)2 + 2 ǫ z 0 −d/2+v dxh (zn) * rn z2 n q 1 + (z′ n)2 + (r′ n)2, (3.2) where the subscript "a" denotes "asymptotic solution" near x = −d/2, while "n" means "numerical solution". in the last expression, the first integral is divergent at x = −d/2. but, as we have the explicit form of the 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 t[fm−1] d[fm] fig. 2: the euler-lagrange equations (2.2) have only so- lutions for temperature t and quark-antiquark separation d within the shaded area. 3 0.1 0.2 0.3 0.4 −5 −4 −3 −2 −1 0 d[fm] sng,reg b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c + + + + + + + + + + + + + + + + + + + ++ + + + + + ++ + + + + + + + ++ + + + + + + + + + ++ + + + + + + + + + + + + + + + * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * ** * * * * * * * * * * * * • - t = 98mev ◦- t = 118mev +- t = 138mev ⋆- t = 154mev fig. 3: regularized nambu-goto action as a function of quark-antiquark separation d for different temperatures t. asymptotic solutions ra(x) and za(x), we can expand the first integrand into power series near x = −d/2, and re- move the divergent terms. to compensate this removal, we should add the antiderivative of the divergent terms at x = −d/2 + v. this way, we obtain the regularized value of sng. integrating the euler-lagrange equations (2.2) for a wide range of initial values suggests that solutions exist only for a specific range of temperature t and quark- antiquark separation d (fig. 2). figure 3 defines the regularized nambu-goto action sng,reg for several temperatures t as a function of quark- antiquark separation d. a fit to the numerical calcula- tions gives sfit ng,reg = −0.48 t d + d −7.46 fm + 5.84 t fm2  . (3.3) iv. thermodynamic quantities using sfit ng,reg it is easy to calculate the free energy of the q ̄ q-system as a function of the q ̄ q separation: f = t * sfit ng,reg = −0.48 d + d −7.46 fm t + 5.84 fm2  . (4.1) fig. 4 shows the free energy. one can recognize the flat- tening of f when increasing temperature t for large di- tances d. the term linear in d yields the effective string tension: σeffective = −7.46 fm t + 5.84 fm2 . (4.2) the entropy writes as s = −∂f ∂t = 7.46 fm d , (4.3) 0.1 0.2 0.3 0.4 −5 −4 −3 −2 −1 0 d[fm] f[fm−1] b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c b c + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + ++ + + + + + + + ++ + + + + + + + + + ++ + + + + + + + + + + + + + + + * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * ** * * * * * * * * * * * * • - t = 98mev ◦- t = 118mev +- t = 138mev ⋆- t = 154mev fig. 4: free energy as a function of quark-antiquark separa- tion d for different temperatures t. and does not depend on temperature. such an entropy is well known for strong coupling qcd on a 3 dimensional lattice, where s = l*log (2d −1), with d = 3 and lis the length of the random path in lattice units connecting the quark and antiquark [6]. inner energy and string tension also do not depend on temperature and are given by e = f + t s = −0.48 d + 5.84 fm2 d , (4.4) and σ = 5.84 fm2 = σeffective(t = 0) , (4.5) respectively. v. confinement and phase transition the free energy eq. (4.1) contains a linear term and a coulomb-like term, which is absent in strong cou- pling lattice qcd. the linear term provides confinement: when it becomes zero there will be no confinement. from eq. (4.2) we can estimate the critical temperature for the confinement/deconfinement phase transition, which is roughly tc = 154 mev. this value is in agreement with lattice qcd results, i.e. t lattice c (nc = 3, nf = 3) = 155 ± 10 mev [7], but one must admit that we only solve a pure gluon theory without dynamical quarks, where, however, the input t = 0 potential has been fitted for nf = 4. vi. conclusion we have shown that the modified metric of ads/qcd proposed in ref. [4] can also be applied to the heavy quark potential. since we are not using the black hole metric this theory is restricted to the t < tc regime. we 4 0 0.2 0.4 0.6 0.8 0 2 4 6 tc t[fm−1] σeffective[fm−2] fig. 5: effective string tension as a function of temperature. the phase transition happens when the effective string tension vanishes. found that the modified ads5-metric produces confine- ment and the short distance coulombic behavior in this region. in previous works on loop-loop correlators [8, 9], these two features had to be added by hand, whereas here they follow from one action. we also can determine tc by demanding that the effective string tension vanishes. because of the singularity in the metric at zir = 1 λ √1 −ǫ/ ≈0.54fm the euler-lagrange equations (2.2) have only solution for a very limited range of boundary condition. in particular for large t they yield solutions only for very small qq separations making the fit (3.3) more hypothetical. [1] h. j. pirner and b. galow, phys. lett. b679, 51 (2009), 0903.2701. [2] l. d. mclerran and b. svetitsky, phys. rev. d24, 450 (1981). [3] e. witten, adv. theor. math. phys. 2, 253 (1998), hep- th/9802150. [4] j. nian and h. j. pirner (2009), 0908.1330. [5] b. galow, e. megias, j. nian, and h. j. pirner (2009), 0911.0627. [6] h. meyer-ortmanns, rev. mod. phys. 68, 473 (1996), hep-lat/9608098. [7] k. yagi, t. hatsuda, and y. miake, camb. monogr. part. phys. nucl. phys. cosmol. 23, 1 (2005). [8] a. i. shoshi, f. d. steffen, and h. j. pirner, nucl. phys. a709, 131 (2002), hep-ph/0202012. [9] a. i. shoshi, f. d. steffen, h. g. dosch, and h. j. pirner, phys. rev. d68, 074004 (2003), hep-ph/0211287.

0911.1681

applications of uwb technology

recent advances in wideband impulse technology, low power communication along with unlicensed band have enabled ultra wide band (uwb) as a leading technology for future wireless applications. this paper outlines the applications of emerging uwb technology in a private and commercial sector. we further talk about uwb technology for a wireless body area network (wban).

introduction there have been tremendous research efforts to apply ultra wide band (uwb) technology to the military and government sectors. some of them are already accomplished and some are intended for future. these applications are mainly categorized into three parts: communications and sensors, position location and tracking, and radar. this paper presents a brief discussion on the aforementioned applications. it is divided into five sections. section 2 outlines the application of uwb technology in communication and sensors. section 3 presents discussion on position location and tracking. in section 4, we talk about radar. section 5 presents the application of uwb in a wireless body area network (wban). the final section presents conclusion. ii. communications and sensors a. low data rate the power spectral density (psd) of uwb signals is extremely low, which enables uwb system to operate in the same spectrum with narrowband technology without causing undue interference. the solution on the market for today's indoor application is infrared or ultrasonic approaches. the line-of-sight propagation in infrared technology cannot be guaranteed all the time. it is also affected by shadows and light-related interferences. ultra sonic approach propagates with confined penetration. uwb technology is less affected by shadows and allows the transmission through objects. the innovative communication method of uwb at low data rate gives numerous benefits to government and private sectors. for instance, the wireless connection of computer peripherals such as mouse, monitor, keyboard, joystick and printer can utilize uwb technology. uwb allows the operation of multiples devices without interference at the same time in the same space. it can be used as a communication link in a sensor network. it can also create a security bubble around a specific area to ensure security. it is the best candidate to support a variety of wban applications. a network of uwb sensors such as electrocardiogram (ecg), oxygen saturation sensor (spo2) and electromyography (emg) can be used to develop a proactive and a smart healthcare system. this can benefit the patient in chronic condition and provides long term health monitoring. in uwb system, the transmitter is often kept simpler and most of the complexity is shifted towards receiver, which permits extremely low energy consumption and thus extends battery life. designing rake-receiver for low power devices is a com- plicated issue. energy detection receivers are the best approach to build simple receivers [1]. a rake receiver for on-body network is presented in [2], which shows that 1 or 2 fingers is sufficient to collect 50% to 80% of maximum energy for links with a distance of 15cm, independent of its placement on the body. different energy management schemes may also assist in extending the battery life . positioning with previ- ously unattained precision, tracking, and distance measuring techniques, as well as accommodating high node densities due to the large operating bandwidth are also possible [3]. global positioning system (gps) system is often available in low date rate applications and requires new solutions. the reduction in protocol overhead can decrease the energy consumption of the complex gps transceivers and extends the battery life. even though, a low data rate application using alternative phy concepts is currently discussed in ieee 802.15.4a [4] but tremendous research efforts are required to bring those systems to real world applications. b. high data rate the unique applications of uwb systems in different scenarios have initially drawn much attention, since many applications of uwb spans around existing market needs for high data rata applications. demand for high density multimedia applications is increasing, which needs innovative methods to better utilize the available bandwidth. uwb system has the property to fill the available bandwidth as demand increases. the problem of designing receiver and robustness against jamming are main challenges for high-rate applications [5]. the large high-resolution video screens can benefit from uwb. these devices stream video content wirelessly from video source to a wall-mounted screen. various high data rata applications include internet access and multimedia services, wireless peripheral interfaces and location based services. arxiv:0911.1681v3 [cs.ni] 2 apr 2015 table i contents and requirements for home networking and computing [9] service data rate (mbps) real time feature digital video 32 yes dvd, tv 2-16 yes audio 1.5 yes pc 32 no internet >10 no other <1 no regardless of the environment, very high data rate applications (>1 gbit/sec) have to be provided. the use of very large bandwidth at lower spectral efficiency has designated uwb system as a suitable candidate for high internet access and multimedia applications. the conventional narrowband system with high spectral efficiency may not be suitable for low cost and low power devices such as pda or other handheld devices. standardized wireless interconnection is highly desirable to replace cables and propriety plugs [6].the interconnectivity of various numbers of devices such as laptops and mobile phones is increasingly important for battery-powered devices. c. home network applications home network application is a crucial factor to make per- vasive home network environment. the wireless connectivity of different home electronic systems removes wiring clutter in living room. this is particularly important when we consider the bit rate needed for high definition television that is in excess of 30 mbps over a distance of at least few meters [6]. in ieee 1394, an attempt has been made to integrate entertainment, consumer electronics and computing within a home environment. it provides isochronous mode where data delivery is guaranteed with constant transmission speed. it is important for real time applications such as video broadcasts. the required data rates and services for different devices are given in table i. ieee 1394 also provides asynchronous mode where data delivery is guaranteed but no guarantee is made about the time of arrival of the data [7]. the isochronous data can be transferred using uwb technology. a new method which allows ieee 1394 equipment to transfer an isochronous data using a uwb wireless communication network is pre- sented in [8]. a connection management protocol (cmp) and ieee 1394 over uwb bridge module can exchange isochronous data through ieee 1394 over uwb network. iii. position location and tracking position location and tracking have wide range of benefits such as locating patient in case of critical condition, hikers injured in remote area, tracking cars, and managing a variety of goods in a big shopping mall. for active rf tracking and positioning applications, the short-pulse uwb techniques offer distinct advantages in precision time-of-flight measurement, multipath immunity for leading edge detection, and low prime power requirements for extended-operation rf identification (rfid) tags [10]. the reason of supporting human-space intervention is to identify the persons and the objects the user aims at, and identifying the target task of the user. knowing where a person is, we can figure out near to what or who this person is and finally make a hypothesis what the user is aiming at [11]. this human-space intervention could improve quality of life when used in a wban. in a wban, a number of intelligent sensors are used to gather patient's data and forwards it to a pda which is further forwarded to a remote server. in case of critical condition such as arrhythmic disturbances, the correct identification of patient's location could assist medical experts in treatment. iv. radar a short-pulse uwb techniques have several radar applica- tions such as higher range measurement accuracy and range resolution, enhanced target recognition, increased immunity to co-located radar transmissions, increased detection probability for certain classes of targets and ability to detect very slowly moving or stationary targets [10]. uwb is a leading technol- ogy candidate for micro air vehicles (mav) applications [12]. the nature of creating millions of ultra-wideband pulses per second has the capability of high penetration in a wide range of materials such as building materials, concrete block, plastic and wood. v. ultra wideband technology in wban a wban consists of miniaturised, low power, and non- invasive/invasive wireless biosensors, which are seamlessly placed on or implanted in human body in order to provide a smart and adaptable healthcare system. each tiny biosensor is capable of processing its own task and communicates with a network coordinator or a pda. the network coordinator sends patient's information to a remote server for diagnosis and pre- scription. a wban requires the resolution of many technical issues and challenges such as interoperability, qos, scalability, design of low power rf data paths, privacy and security, low power communication protocol, information infrastructure and data integrity of the patient's medical records. the average power consumption of a radio interface in a wban must be reduced below 100μw. moreover, a wban is a one-hop star topology where power budget of the miniaturised sensor nodes is limited while network coordinator has enough power budget. in addition, most of the complexity is shifted to the network coordinator due to its capability of having abundant power budget. the emerging uwb technology promises to satisfy the average power consumption requirement of the radio interface (100μw), which cannot be achieved by using narrowband radio communication, and increases the operating period of sensors. in the uwb system, considerable complexity on the receiver side enables the development of ultra-low-power and low-complex uwb transmitters for uplink communication, thereby making uwb a perfect candidate for a wban. the difficulty in detecting noise-like behavior and robustness of uwb signals offer high security and reliability for medical applications [5]. fig. 1. block diagram of pulse generator [16] existing technological growth has facilitated research in promoting uwb technology for a wban. the influence of human body on uwb channel is investigated in [13] and results about the path loss and delay spread have been reported. the behavior of uwb antenna in a wban has also been presented in [14]. moreover, a uwb antenna for a wban operating in close vicinity to a biological tissue is proposed in [15]. this antenna can be used for wban applications between 3 ghz and 6 ghz bands. a low complex uwb transmitter being presented in [16] adapts a pulse-based uwb scheme where a strong duty cycle is produced by restricting the operation of the transmitter to pulse transmission. this pulse-based uwb scheme allows the system to operate in burst mode with minimal duty cycle and thus, reducing the baseline power consumption. a low power uwb transmitter for a wban requires the calibration of psd inside the federal communication commission (fcc) mask for indoor application. the calibration process is a challenging task due to the discrepancies between higher and lower frequencies. two calibration circuits are used to calibrate the spectrum inside fcc mask and to calibrate the bandwidth. a pulse generator presented in fig. 1 activates triangular pulse generator and a ring oscillator simultaneously. during pulse transmission, the ring oscillator is activated by a gating circuit, thus avoiding extra power consumption. a triangular pulse is obtained at output when the triangular signal is multiplied with the output carrier produced by the oscillator. finite-difference time-domain (fdtd) is used to model uwb prorogation around the human body where the path loss depends on the distance and increases with a large fading vari- ance. the narrowband implementation is compared in terms of power consumption using a wban channel model. simulation results showed that the path loss near the body was higher than the path loss in free space. moreover, it was concluded that the performance of a uwb transmitter for a wban is better for best channel while for average channels narrow band implementation is a good solution [16].a considerable research efforts are required both at algorithmic and circuit level to make uwb a key technology for wban applications. vi. conclusions the uwb is a leading technology for wireless applications including numerous wban applications. in this paper, we discussed the current and future applications of emerging uwb technology in a private and commercial sector. we believe that the uwb technology can easily satisfy the energy consumption requirements of a wban. our future work includes the investigation of uwb technology for a non invasive wban. references [1] federal communications commission (fcc) fcc noi: rules regarding ultra-wideband transmission systems, et docket no. 98-153, sept. 1, 1998. [2] thomas zasowski, frank althaus, mathias stager, a. wittneben, and g. troster, uwb for non-invasive wireless body area net- works: channel measurements and results, ieee conference on ultra wideband systems and technologies, uwbst 2003, reston, virginia, usa, nov. 2003. [3] b allen, "ultra wideband wireless sensor networks", iee uwb symposium, june 2004. [4] dec. 2007, http://www.ieee802.org/15/pub/tg4a.html [5] b. allen, t. brown, k. schwieger, e. zimmermann, w. q. malik, d. j. edwards, l. ouvry, and i. oppermann, ultra wideband: appli- cations, technology and future perspectives, in proc. int. workshop convergent tech. oulu, finland, june 2005. [6] b allen, m ghavami, a armogida, a.h aghvami, the holy grail of wire replacement, iee communications engineer, oct/nov 2003. [7] jan. 2008, http://www.vxm.com/21r.49.html [8] s.h park, s.h lee, isochronous data transfer between av devices using pseudo cmp protocol in ieee 1394 over uwb network, ieice trans. commun., vol.e90-b, no.12 december 2007. [9] ben allen, white paper ultra wideband: technology and future perspectives v3.0, march 2005. [10] r.j. fontana, recent system applications of short-pulse ultra- wideband (uwb) technology, ieee microwave theory and tech., vol. 52, no. 9, september 2004. [11] t. manesis and n. avouris, survey of position location techniques in mobile systems, mobilehci, salzburg, september 2005. [12] r. j. fontana, e. a. richley et.al, an ultra wideband radar for micro air vehicle applications, reprinted from 2002 ieee conference on ultra wideband systems and technologies, may 2002. [13] yue ping zhang; qiang li, "performance of uwb impulse radio with planar monopoles over on-human-body propagation chan- nel for wireless body area networks," antennas and propagation, ieee transactions on , vol.55, no.10, pp.2907-2914, oct. 2007 [14] yazdandoost, k.y.; kohno, r., "uwb antenna for wireless body area network," microwave conference, 2006. apmc 2006. asia- pacific , vol., no., pp.1647-1652, 12-15 dec. 2006 [15] m. klemm, i.z. kovacs, et.al, comparison of directional and omni- directional uwb antennas for wireless body area network appli- cations, 18th international conference on applied electromagnetics and communications, pg 1-4, icecom 2005. [16] j. ryckaert, c. desset, a. fort, m. badaroglu, v. de heyn, p. wambacq, g. van der plas, s. donnay, b. van poucke, b. gyselinckx, "ultra-wide-band transmitter for low-power wireless body area networks: design and evaluation," ieee transactions on circuits and systems i: regular papers, vol.52, no.12, pp. 2515- 2525, dec. 2005

0911.1683

luminescent ions in silica-based optical fibers

we present some of our research activities dedicated to doped silica-based optical fibers, aiming at understanding the spectral properties of luminescent ions, such as rare-earth and transition metal elements. the influence of the local environment on dopants is extensively studied: energy transfer mechanisms between rare-earth ions, control of the valence state of chromium ions, effect of the local phonon energy on thulium ions emission efficiency, and broadening of erbium ions emission induced by oxide nanoparticles. knowledge of these effects is essential for photonics applications.

introduction during the last two decades, the development of sophisticated optical systems and devices based on fiber optics have benefited from the development of very performant optical fiber components. in particular, optical fibers doped with 'active' elements such as rare-earth (re) ions have allowed the extremely fast development of optical telecommunications [i,ii], lasers [iii] industries and the development of temperature sensors [iv]. the most frequently used re ions (nd3+, er3+, yb3+, tm3+) have applications in three main spectral windows: around 1, 1.5 and 2μm in fiber lasers and sensors based on absorption/fluorescence and around 1.5 μm for telecommunications and temperature sensors. re-doped fibers are either doped with one element (e.g. er3+ in line amplifiers for long haul telecommunications) or two elements (e.g. yb3+ and er3+ in booster amplifiers or powerful 1.5 μm lasers). in the second case, the non- radiative energy transfer mechanism from donor to acceptor is implemented to benefit from the good pump absorption capacity of the donor (e.g. yb3+ around 0.98 μm) and from the good stimulated emission efficiency of the acceptor (e.g. er3+ around 1.5 μm). all the developed applications of amplifying optical fibers are the result of long and careful optimization of the material properties, particularly in terms of dopant incorporation in the glass matrix, transparency and quantum efficiency. the exploited re-doped fibers are made of a choice of glasses: silica is the most widely used, sometimes as the result of some compromises. alternative glasses, including low maximum phonon energy (mpe) ones, are also used because they provide better quantum efficiency or emission bandwidth to some re ions particular optical transition. the icon example is the tm3+-doped fiber amplifier (tdfa) for telecommunications in the s-band (1.48-1.53 μm) [v], for which low mpe glasses have been developed: oxides [vi,vii], fluorides [viii], chalcogenides [ix]... however, these glasses have some drawbacks not acceptable at a commercial point of view: high fabrication costs, low reliability, difficult connection to silica components and, in the case of fiber lasers, low optical damage threshold and resistance to heat. to our knowledge, silica glass is the only material able to meet most of applications requirements, and therefore the choice of vitreous silica for the active fiber material is of critical importance. however a pure silica tdfa would suffer strong non-radiative de-excitation (nrd) caused by multiphonon coupling from tm3+ ions to the matrix. successful insulation of tm3+-ions from matrix vibrations by appropriate ion-site 'engineering' would allow the development of a practical silica-based tdfa. other dopants have recently been proposed to explore amplification over new wavelength ranges. bi-doped glasses with optical gain [x] and fiber lasers operating around 1100-1200 nm have been developed [xi,xii], although the identification of the emitting center is still not clear, and optimization of the efficiency is not yet achieved. transition metal (tm) ions of the ti-cu series would also have interesting applications as broad band amplifiers, super-fluorescent or tunable laser sources, because they have in principle ten-fold spectrally larger and stronger emission cross-sections than re ions. however, important nrd strongly reduces the emission quantum efficiency in silica. bi- and tm-doped fibers optical properties are extremely sensitive to the glass composition and/or structure to a very local scale. as for tm3+ ions, practical applications based on silica would be possible when the 'ion site engineering' will be performed in a systematic approach. this approach is proposed via 'encapsulation' of dopants inside glassy or crystalline nanoparticles (np) embedded in the fiber glass, like reported for oxyfluoride fibers [xiii] and multicomponent silicate fibers [xiv]. in np-doped-silica fibers, silica would act as support giving optical and mechanical properties to the fiber, whereas the dopant spectroscopic properties would be controlled by the np nature. the np density, mean diameter and diameter distribution must be optimized for transparency [xv]. in this context, our group has made contributions in various aspects introduced above. our motivations are both fundamental and application oriented. first, the selected dopants act as probes of the local matrix environment, via their spectroscopic variations versus ligand field intensity, site structure, phonon energy, statistical proximity to other dopants,... the studies are always dedicated to problems or limitation in applications, such as for erbium-doped fiber amplifier (edfa) and tdfa, or high temperature sensors. it is also important to use a commercially derived fabrication technique, here the modified chemical vapor deposition (mcvd), to assess the potential of active fiber components for further development. the aim of this paper is reviewing our contributions to improving the spectroscopic properties of some re and tm ions doped into silica. the article is organized as follows: section 0 describes the mcvd fabrication method of preform and fiber samples, and the common characterization techniques used in all studies. section 0 chapitre: is devoted to the study of energy transfers in erbium ion (er3+) and ytterbium:erbium (yb3+:er3+) heavily (co)doped fibers and the applications to fiber temperature sensors, whereas section 0 summarizes our original investigations on chromium (cr3+ and cr4+) in silica-based fibers. in section 0 chapitre:, we report on the spectroscopic investigations of thulium- (tm3+) doped fibers versus the material composition, including phonon interactions and non-radiative relaxations. in section 0 chapitre: are reported our recent discoveries in re-doped dielectric nanoparticles, grown by phase separation. experimental preforms and fibers fabrication all the fibers investigated in this article were drawn from preforms prepared by the modified chemical vapor deposition (mcvd) technique [xvi] at laboratoire de physique de la matière condensée (nice). in this process, chemicals (such as o2, sicl4) are mixed inside a glass tube that is rotating on a lathe. due to the flame of a burner moving along the tube, they react and extremely fine silica particles are deposited on the inner side of the tube. these soot are transformed into a glass layer (thickness is about few μm) when the burner is passing over. the cladding layers are deposited inside the substrate tube, followed by the core layers. germanium and phosphorus can be incorporated directly through the mcvd process. they are added to raise the refractive index. moreover, this last element is also added as a melting agent, decreasing the melting temperature of the glass. all the other elements (rare-earths, transition metals, aluminium, ...) are incorporated through the solution doping technique [xvii]. the last core layer is deposited at lower temperature than the preceding cladding layers, so that they are not fully sintered and left porous. then the substrate tube is filled with an alcoholic solution of salts and allowed to impregnate the porous layers. after 1-2 hours, the solution is removed, the porous layer is dried and sintered. when the deposition is complete, the tube is collapsed at 2000°c into a preform. in our case, the typical length of the preform is about 30 cm and the diameter is 10 mm. the preform is then put into a furnace for drawing into fiber. the preform tip is heated to about 2000°c. as the glass softens, a thin drop falls by gravity and pulls a thin glass fiber. the diameter of the fiber is adjusted by varying the capstan speed. a uv-curable polymer is used to coat the fiber. material characterizations refractive index profiles (rip) of the preforms were measured using a york technology refractive index profiler (p101), while the rips of the optical fibers were determined using a york technology refractive index profiler (s14). the oxide core compositions of the samples were deduced from measurement of the rip in the preform, knowing the correspondence between index rising and alo3/2, geo2, po5/2 concentration in silica glass from the literature [xviii,xix]. the composition was also directly measured on some preforms using electron probe microanalysis technique in order to compare results. a good agreement was found. the concentration of these elements is generally around few mol%. luminescent ions concentrations are too low to be measured through the rip. they were measured through absorption spectra. for example, tm3+ ion concentration has been deduced from the 785 nm (3h6=>3h4) absorption peak measured in fibers and using absorption cross-section reported in [xx]: abs(785 nm) = 8.7x10-25 m2. energy transfers in er3+ and yb3+:er3+ heavily doped silica fibers the non-radiative energy transfer processes are well-known phenomena that influence the optical properties of doped-materials. the first theoretical basis appears in the 50's with the förster-dexter's model [xxi,xxii] that treats this process as the result of dipole-dipole and multipole-multipole interactions. two energy transfer processes are described in fig. 1. when pumping a co-doped material some ions are promoted in one of their excited level. if some ions are close to each other, their wave-functions interpenetrate and the energy stored in the excited level of the donor ions is non-radiatively transferred to a resonant level of the acceptor ions. this process was turned to good account in yb3+:er3+ co-doped silica fibers for high power fiber amplifier [xxiii] and laser [xxiv] applications : it takes advantage of the strong absorption cross- section of yb3+ at 980 nm and of the high efficiency of the energy transfer. in the case of high doping levels for both species, another non-radiative energy transfer process can take place and allows exciting a higher level of the acceptor ion : it is the double energy transfer process (det) described by auzel [xxv]. this process was first used to convert infrared light from led to visible emission or to detect weak infrared signals with photomultipliers [xxvi,xxvii]. double energy transfer in er3+-doped fibers the clustering effect in er3+-doped silica fibers is now a well-accepted phenomenon, and its detrimental influence on the 1550-nm gain transition of such fibers is well established [xxviii]. for simplicity, modeling of clusters has consisted of considering that a fraction of the dopants were organized in ion pairs [xxix], in which an immediate energy transfer leads to an instantaneous relaxation of one excited ion. this model is in very good agreement with the experimental results obtained for saturable absorption and for gain measurements at low er3+ doping levels as in fiber amplifiers. at higher doping levels, ainslie et al. [xxx] showed that, in addition to the ions dispersed in the host, regions in which concentrations of rare-earth exceeding 40 wt% - called clusters – appear : in such a material the ion-pair model cannot be applied. we have developed a cluster model [xxxi] that differs from the ion-pair one by the fact that we consider that each ion of a cluster can efficiently transfer its energy to any of the other ions of the same cluster. when n ions of a cluster are excited, a succession of (n-1) fast relaxations by energy transfer leads to a situation in which all the ions of a cluster but one are de-excited. this model permits the determination of the proportion of the dopants organized in clusters and the transfer rate. in order to validate the model we realized a pump-absorption- versus-pump-power experiment with two fiber samples, er-1 and er-2, doped with 100 ppm and 2,500 ppm of er3+, respectively (fig. 2). this shows that the non-saturable absorption (nsa) grows dramatically with the er3+ concentration. we have attributed this behaviour to the presence of clusters containing a significant percentage of the dopants and in which efficient energy transfers allow these ions to relax rapidly after the absorption of a first pump photon. double energy transfer in highly yb3+:er3+-co-doped fibers the green fluorescence of er3+-doped optical fiber is a well-known phenomenon in 800 nm-pumped erbium-doped fibers. this emission results from the excited state absorption phenomenon and is characteristic of the emission from the 2h11/2 and 4s3/2-levels and consequently can be observed with any pumping scheme leading to the population of these levels. we have studied how these levels can be excited by det [xxxii] and a schematic energy diagram is shown in fig. 3. at low rare-earth concentrations, the large inter-ionic distances permit efficient single energy transfers, but the second energy transfer is very inefficient. for applications in which the green fluorescence is desirable, this second energy transfer must be enhanced. for that the rare- earth concentration must be as high as possible to reduce the distance between neighboring ions. in this case, a second phase, referred to as clusters, can appear in which the rare-earth ions concentration is particularly high. in order to quantify the fraction of active ions into clusters, we have studied the yb3+ and er3+ fluorescence dynamics in a highly co-doped fiber ([yb]=[er]=2,500 ppm) : the 1040 nm-fluorescence decay represents the population decay from the 2f5/2-metastable level of yb3+, and that of the green-fluorescence represents the evolution of the 2h11/2 and 4s3/2-populations of er3+. our setup allows simultaneous measurements of the counter-propagative visible emission and the lateral infrared emission. the experimental curves show two typical decays. fitted with our rate equations model [xxxii], they revealed that roughly 50% of both ions are organized in clusters in the co-doped fiber. this high percentage must be associated with very high yb-er transfer rates (3x106 s-1), one order of magnitude superior to the er3+:4i11/2 intermediate level relaxation rate (3.7x105 s-1): er3+ ions placed in their short lived 4i11/2 state have a higher probability to be excited to the 4f7/2 upper state than to relax spontaneously. the strong percentage of ions organized in clusters and the very high transfer rates are at the origin of the very good up-conversion efficiency. thermalization effects between excited levels in doped fibers: temperature sensor based on fluorescence of er3+ though the rare-earth ions are never in thermodynamical equilibrium because of the metastability of some levels, it has been demonstrated that the populations of the 2h11/2 and 4s3/2 levels responsible for the green emission in er3+-doped fibers are in quasi-thermal equilibrium. this effect has been observed for the first time in fluoride glass fibers [xxxiii] and can be attributed to the relatively long lifetime of these levels (400 μs) in that host. in silica, in spite of the two orders of magnitude shorter lifetime, a fast thermal coupling between both levels has been proposed [xxxiv] and confirmed experimentally [xxxv] (fig. 4). indeed these levels can be considered to be in quasi-thermal equilibrium, because of the small energy gap between them, about 800 cm-1, compared to the high energy gap between them and the nearest lower level, about 3000 cm-1. in this case, the lifetime of these levels is sufficient (1 μs) to allow populating the upper level from the lower one by phonon induced transitions. therefore r, the ratio of the intensities coming from both levels, can be written as: r  i 2h 11/2   i 4s 3/2    2h 11/2   4s 3/2   e 2h 11/2   e 4s 3/2   expe/kt   (1) where  is the frequency, e the emission cross section, k the boltzmann constant, e the energy gap between the two levels and t the temperature in degrees kelvin. in fig. 4 we show that the experimental data can be fitted by a function in agreement with equation (1). this is another example of an energy transfer process, this one being assisted by phonons. in order to take advantage of the high efficiency of the det in highly co-doped yb-er doped fiber and of the thermalisation effect between the higher levels involved in the green fluorescence in this kind of fiber, we have developed a new temperature sensor, unsensitive for strain. the dynamic obtained was 11 db in fig. 4 over the shown temperature range, leading to a mean rate of change of the green intensity ratio of approximately 0.016 db/k at 300 k. several temperature cycles have been carried out and we have observed a good repeatability. as for the stability, no modifications have been observed on the two intensities when the fiber was heated during several hours at temperatures up to 600°c. due to the strong absorption of the doped fiber in the signal wavelength range - the green emissions corresponding to transitions downto the fundamental level - and to the 15 db/km intrinsic absorption of the transparent fiber in the same wavelength range, such a device would be limited to a point sensor. we have developed a new sensor based on the 1.13 μm and 1.24 μm emission lines, coming from the same levels [xxxvi]. these lines present the same temperature behaviour as the green ones. as the lower level of these transitions is the 4i11/2-level and not the fundamental one (fig. 3), the signals are absorption free and their wavelengths correspond to a transparency region of the intermediate fibers. these arguments have permitted the development of an efficient quasi-distributed configuration without limitation on the sensing line length : the short lifetime of upper levels (1 μs) could allow realizing a sensors network. each sensitive head is separated from its neighbors by a 100-meter long transparent silica fiber in order to time-resolve the counter-propagative signals. conclusion energy transfer processes in rare-earth-doped materials have been studied since the middle of the 20th century. at the beginning, the applications of det were mainly conversion of infrared light to visible emission or detection of weak infrared signals with photomultipliers. a renewal interest appears with the development of optical fibers in which high power density can be achieved: single energy transfer allows improvement of high power fiber amplifiers and laser and det permits realizing point and quasi-distributed fiber sensors. local structure, valency states and spectroscopy of transition metal ions optical fiber materials with very broad-band gain are of great interest for many applications. tunability in re-doped fiber devices is already well established, but limited by shielding of the optically active electronic orbitals of re ions. optically active, unshielded orbitals are found in transition metal (tm) ions. some tm-doped bulk solid-state lasers materials, such as cr4+:yag, have demonstrated very good results as broad-band gain media [xxxvii]. tentatives with other tm ions, like ni2+ in vitroceramics fiber are also promising [xiv]. more recently, a 400-nm emission bandwidth was observed from a fiber whose cr-doped core was made of y2o3:a2o3:sio2 obtained by a rod-in-tube technique using a cr4+:yag rod as core material and a silica tube as cladding material [xxxviii]. little literature exists on chromium- and other tm-doped vitreous bulk silica, although this issue was addressed in the 70's [xxxix] to improve transmission of silica optical fibers. some reports on chromium-doped glasses have already shown evidence of absorption and near- infrared (nir) fluorescence due to cr4+ in these materials [xl,xli]. however their compositions and preparation techniques greatly differ from those of silica optical fibers. therefore, some basic studies on the optical properties of tm ions in silica-based optical fibers are needed. in particular, the final tm oxidation state(s) in the fiber core strongly depend(s) on the preparation process. also, the optical properties (absorption and luminescence) of one particular oxidation state of a tm ion varies from one host composition and structure to another, due to variations of the crystal-field (so-called ligand field in glass) [xlii]. hence the interpretation of absorption and emission spectroscopy is difficult. because no luminescence spectroscopy of the tm-doped silica fibers had been reported before, we have contributed to explore this field. we have studied the influence of the chemical composition of the doped region on the cr-oxidation states and on the spectroscopic properties of the samples. we have also studied the optical properties versus the experimental conditions (temperature and pump wavelength). we describe the experimental details specifically used for tm-doped fibers, then we summarize all results and interpretations. fabrication and characterization of chromium-doped samples the preforms and fibers were prepared as described in §0 chapitre:, using cr3+-salt alcoholic doping solution and oxygen or nitrogen (neutral) atmosphere for the drying-to- collapse stages. three different types of samples containing ge or/and al were prepared, referred to as cr(ge), cr(ge-al) and cr(al), respectively. the total chromium concentration ([cr]) was varied from below 50 mol-ppm to several thousands mol ppm. above several 100s mol ppm, preform samples had evidence of phase separation causing high background optical losses, whereas fibers (few 10s mol ppm) did not show phase separation and had low background losses (<1 db/m). the oxidation states of cr and their relative concentrations were analyzed by electron paramagnetic resonance, whereas the absolute content of all elements (including cr) was analyzed by plasma emission spectroscopy. absorption spectra were analyzed using the tanabe-sugano (t.-s.) formalism [xliii] to compare our assignments to optical transitions with reports on cr3+- and cr4+-doped materials. this formalism helps predicting the energy of electronic states of a tm in a known ligand field symmetry as a function of the field strength dq and the phenomenologic b and c so-called racah parameters (all in cm-1, fig. 5). the dq/b ratio allows the qualitative determination of some optical properties of tm ions, such as strength, energy and bandwidth of optical transitions. we have also estimated the absorption cross-sections using results from composition and valency measurements. absorption and emission spectroscopies including decay measurements were performed on both preforms and fibers, at room temperature (rt) and low temperature (lt, either 12 or 77 k), using various pump wavelengths: 673, 900 and 1064 nm. full details of the experimental procedures are given in [xliv,xlv,xlvi]. principal results by slightly modifying the concentration in germanium and/or aluminium in the core of the samples, their optical properties are greatly modified. in particular, we have shown that: i) only cr3+ and cr4+ oxidation states are stabilized. cr3+ is favoured by ge co-doping, and lies in octahedral site symmetry (o), as in other oxide glasses [xlvii]. cr4+ is present in all samples. this valency is promoted by al co-doping or when [cr] is high, and lies in a distorded tetrahedral site symmetry (cs) [xlviii,xlix]. the low-doped cr(al) samples contain only cr4+ and their absorption spectra are similar to those of aluminate [xl] and alumino-silicate glasses [xli] and even crystalline yag (y3al5o12) [l]. glass modifiers like al induce major spectroscopic changes, even at low concentrations (~1-2 mol%). this would help engineering the chromium optical properties in silica-based fibers, using possibly alternative modifiers. ii) the absorption spectra have been interpreted and optical transitions assigned for each present valency state (fig. 6). the absorption cross sections curves (abs) were estimated. for cr3+, abs(cr3+, 670 nm)= 43 x 10-24 m2 is consistent with reported values in other materials, such as ruby [li] and silica glass [xxxix], while abs(cr4+, 1000 nm)~3.5 x 10-24 m2 is lower than in reference crystals for lasers [lii] or saturable absorbers [liii], but consistent with estimated values in alumino-silicate glass [xli]. iii) using the t.-s. formalism, we found dq/b = 1.43 which is lower than the value were 3t2 and 1e levels cross (dq/b = 1.6, fig. 5). as a consequence, the expected emission is along the 3t23a2 transition as a broad featureless nir band. no narrow emission line from the 1e state is expected, in agreement with fluorescence measurements. dq/b is lower than those reported for cr4+ in laser materials like yag and forsterite [xlviii]. iv) the lt fluorescence from cr4+ spreads over a broad spectral domain, from 850 to 1700 nm, and strongly varies depending on core chemical composition, [cr] and p (pump wavelength). the observed bands were all attributed to cr4+ ions, in various sites. fig. 7 shows the fluorescence spectra of cr4+ in two different types of samples and in various experimental conditions. possible emission from other centers (cr3+, cr5+, cr6+) was discussed, but rejected [xlvi]. the fluorescence sensitivity to [cr] and p suggests that cr-ions are located in various host sites, and that several sites are simultaneously selected by an adequate choice of p (like in cr(ge-al)). it is also suggested that although al promotes cr4+ over cr3+ when [cr] is low, cr4+ is also promoted in ge-modified fibers at high [cr]. v) the strong decrease of fluorescence from lt to rt is attributed to temperature quenching caused by multiphonon relaxations, like in crystalline materials where the emission drops by typically an order of magnitude from 77 k to 293 k [xxxvii]. vi) the lt fluorescence decays are non-exponential (fig. 8) and depend on [cr] and s. the fast decay part is assigned to cr clusters or cr4+-rich phases within the glass. the 1/e- lifetimes ( at s = 1100 nm are all within the15-35 s range in al-containing samples, whereas  ~ 3-11 s in cr(ge) samples, depending on [cr]. the lifetime of isolated ions (iso), measured on the exponential tail decay curves (not shown) reach high values: iso~200 to 300 μs at s~1100 nm, iso~70 μs at s~1400 nm. in the heavily-doped cr(ge) samples, iso is an order of magnitude less. hence, cr4+ ions are hosted in various sites: the lowest energy ones suffer more non-radiative relaxations than the higher energy ones. also presence of al improves the lifetime, even at high [cr]. it is estimated that at rt, lifetime  would be of the order of 1 μs or less. this fast relaxation time, compared to re ions (~1 ms) has been implemented as a fiberized saturable absorber in a passively q-switched all-fiber laser [liv]. conclusion the observed lt fluorescence of cr4+ is extremely sensitive to glass composition, total cr concentration and excitation wavelength. using aluminum as a glass network modifier has advantages: longer excited state lifetime and broader fluorescence bandwidth than in germanium-modified silica. a combination of al and ge glass modification induces the broadest fluorescence emission in the nir range, to our knowledge, exhibiting a 550 nm- bandwidth. however, increasing the quantum efficiency is now necessary for practical fiber amplifiers and light sources. further investigations concluded to the necessity of local surrounding tm ions with a different material, i.e. having sensibly different chemical and physical properties compared to pure silica, in order to improve the local site symmetry and hence minimize nrd. preliminary implementation of this principle was reported recently, concerning cr3+ ions in post-heat-treated ga-modified silica fibers [lv]. when engineering of the local dopant environment will be possible, then practical tm-doped silica-based amplifying devices will be at hand. phonon interactions / non-radiative relaxations: improvement of tm3+ efficiency thulium-doped fibers have been widely studied in the past few years. because of tm3+ ion rich energy diagram, lasing action and amplification at multiple infrared and visible wavelengths are allowed. thanks to the possible stimulated emission peaking at 1.47 μm (3h4 => 3f4, see fig. 9), discovered by antipenko et al. [lvi], one of the most exciting possibilities of tm3+ ion is amplifying optical signal in the s-band (1.47–1.52 μm), in order to increase the available bandwidth for future optical communications. unfortunately, the upper 3h4 level of this transition is very close to the next lower 3h5 level so non-radiative de-excitations (nrd) are likely to happen in high phonon energy glass host, causing detrimental gain quenching. oxide modifiers influence on the 3h4-level lifetime to address this problem, we have studied the effect of some modifications of tm3+ ion local environment. keeping the overall fiber composition as close as possible to that of a standard silica fiber, we expect to control the rare-earth spectroscopic properties by co-doping with selected modifying oxides. we have studied the incorporation of modifying elements compatible with mcvd. geo2 and alo3/2 are standard refractive index raisers in silica. alo3/2 is also known to improve some spectroscopic properties of er3+ ion for c-band amplification [i] and to reduce quenching effect through clustering in highly rare-earth-doped silica [lvii]. both oxides have a lower maximum phonon energy than silica. we use high phonon energy po5/2 as opposite demonstration. geo2 and po5/2 concentrations are 20 and 8 mol%, respectively. alo3/2 concentration is varied from 5.6 to 17.4 mol%. tm3+ concentration is less than 200 mol ppm. to investigate the role of the modification of the local environment, decay curves of the 810 nm fluorescence from the 3h4 level were recorded. all decay curves measured are non- exponential. this can be attributed to several phenomena and will be discussed in this article. here, we study the variations of 1/e lifetimes () versus concentration of oxides of network modifiers (al or p) and formers (ge). the lifetime strongly changes with the composition of the glass host. the most striking results are observed within the tm(al) sample series:  linearly increases with increasing alo3/2 content, from 14 μs in pure silica to 50 μs in sample tm(al) containing 17.4 mol% of alo3/2. the lifetime was increased about 3.6 times. the lifetime of the 20 mol% geo2 doped fiber tm(ge) was increased up to 28 μs whereas that of the 8 mol% po5/2 doped fiber tm(p) was reduced down to 9 μs. we see that aluminum codoping seems the most interesting route among the three tested codopants. non-exponential shape of the 810-nm emission decay curves all fluorescence decay curves from the 3h4 level are non-exponential. we have investigated the reasons for this non-exponential shape in various silica glass compositions. we observed that the decay curve shape depends only on the al-concentration, even in the presence of ge or p in samples tm(ge) and tm(p), respectively [lviii]. it is thought that tm3+ ions are inserted in a glass which is characterized by a multitude of different sites available for the rare- earth ion, leading to a multitude of decay constants. this phenomenological model was first proposed by grinberg et al. and applied to cr3+ in glasses [lix]. here we apply this model, for the first time to our knowledge, to tm3+-doped glass fibers. in this method, a continuous distribution of lifetime rather than a number of discrete contributions is used. the advantage of this method is that no luminescence decay model or physical model of the material is required a priori. the luminescence decay is given by: i(t)  ai exp t / i   i  (2) where a() is the continuous distribution of decay constant. the procedure for calculating  and the fitting algorithm are described in detail in [lix]. for the fitting procedure, we considered 125 different values for i, logarithmically spaced from 1 to 1000 μs. by applying this procedure to all the decay curves, a good matching was generally obtained. for a given composition (fig. 10), we can notice two main distributions of the decay constant. with the aluminium concentration, they increase from 6 to 15 μs and from 20 to 50 μs, respectively. for the highest aluminium concentration (9 mol%, in tm(ge) and tm(p)), these two bands are still present (not shown in the figure). one is around 10 μs and the second one spreads from 30 to 100 μs, for both compositions (tm(ge) and tm(p)). according to the phenomenological model, the width of the decay constants distribution is related to the number of different sites. the large distribution around 80 μs is then due to a large number of sites available with different environments. it is however remarkable that this distribution at 80 μs is very similar in both sample types. from the tm3+ ion point of view (considering luminescence kinetics), tm(ge) and tm(p) glasses seem to offer the same sites. the meaning of the decay constant values is now discussed. lifetime constants obtained from the fitting can be correlated with the one expected for thulium located in a pure silica or pure al2o3 environment. the 3h4 lifetime is calculating by using this equation: 1 1 rad wnr (3) where rad corresponds to the radiative lifetime which is given to be 670 μs in silica [lx]. wnr is the non-radiative decay rate, expressed as [lxi]: wnr w0 exp (e 2ep)   (4) where w0 and  are constants depending on the material, e is the energy difference between the 3h4 and 3h5 levels and ep is the phonon energy of the glass. w0 and  were estimated for different oxide glasses [lxi, lxii]. the energy difference e was estimated by measuring the absorption spectrum of the fibers. when al concentration varies, this value is almost constant around 3700 cm-1 [lxiii]. with these considerations, the 3h4 expected lifetime can be calculated. in the case of silica glass, silica = 6 μs and for an al2o3 environment, alumina = 110 μs. these two values are in accordance with the ones we obtained from the fitting procedure. the distribution of decay constant around 10 μs corresponds to tm3+ ions located in almost pure silica environment while the second distribution is attributed to tm3+ located in al2o3 -rich sites. conclusion by adding oxide network modifiers or formers, we demonstrated that aluminium is the most efficient to improve the 3h4 level-lifetime. this was attributed to a lower local phonon energy. potential of the amplification in the s-band was then investigated. in the fiber with the highest aluminium concentration, gain curve was measured. although excitation wavelength (1060 nm), refractive index profile and thulium concentration were not optimized, a gain of 0.9 db was obtained at 1500 nm [lxiv]. with a numerical model of the tdfa that we developed [lxv], we estimated that a gain higher than 20 db is reachable in a silica-based tdfa. rare-earth-doped dielectric nanoparticles erbium-doped materials are of great interest in optical telecommunications due to the er3+ intra-4f emission at 1.54 μm. erbium-doped fiber amplifiers (edfa) were developed in silica glass because of the low losses at this wavelength and the reliability of this glass. developments of new rare-earth doped fiber amplifiers aim to control their spectroscopic properties: shape and width of the gain curve, optical quantum efficiency, .... standard silica glass modifiers, such as aluminum, give very good properties to available edfa. however, for more drastic spectroscopic changes, more important modifications of the rare-earth ions local environment are required. to this aim, we present a fiber fabrication route creating rare-earth doped calco- silicate or calco-phospho-silicate nanoparticles (np) embedded in silica glass. nanostructured fibers preparation in the chosen route, np are not prepared ex-situ and incorporated into the perform. to prepare them, we take advantage of the heat treatement occurring during the mcvd process. their formation is based on the basic principle of phase separation. on the basis of thermodynamical data such as activity coefficient, entropy of mixing, enthalpy of mixing and gibbs-free energy change, the phase diagram of the sio2-cao binary compound was derived using factstage software (fig. 11). a miscibility gap is found when the cao concentration is between 2 and 30 mol%. in this region, cao droplets are formed, like oil in water. such phenomenon is expected during perform fabrication as temperature reaches 2000°c during collapsing passes. for calcium doping, cacl2 salt was added to the er3+ containing soaking solution. four cacl2 concentrations were studied (0, 0.001, 0.1 and 1 mol/l). ge and p were also added by mcvd. when the ca concentration was increased in the doping solution, the aspect of the central core of the preform turned from transparent to milky. this variation is explained by the structural changes of the core. for preforms with calcium concentration higher than 0.01 mol/l, np were observed by transmission electron microscopy (tem) on preform samples (fig. 12). we can clearly observe polydisperse spherical np with an estimated mean diameter of 50 nm. smaller particles of 10 nm are visible. the size of the biggest particles was around 200 nm (not shown in fig. 12). when the ca concentration decreases, the size distribution of the particles is nearly identical but the density is lower. the composition of the core was investigated by energy dispersive x-ray analysis: the np contained equal amounts of ca, p and si cations, whereas only si cations was detected in the surrounding matrix. ge seemed to be homogeneously distributed over the entire glass. the most important finding is that er3+ ions and ca were detected only within the np. erbium emission characterizations spectroscopic characterizations on the emission line associated to the 4i13/2-4i15/2 transition at 1.54 μm were made at room temperature on er-doped samples with (sample a) and without (sample b) calcium. the results are shown in fig. 13 where we evidence the fact that the emission spectrum of sample a is broader than that of sample b. to explain these differences we have studied the er3+ local environment. exafs measurements at the er-liii edge (e=8358 ev) were carried out at the gilda-crg beamline at the european synchrotron radiation facility. in sample a the rare-earth is linked to o atoms in the first coordination shell and to si or p atoms (these two atoms can not be distinguished due to the similar backscattering amplitude and phase) in the second shell in a way similar to that already observed in silicate glasses [lxvi], phosphate glasses [lxvii]. also the structural parameters (about 7 o atoms at 2.26 å and si (or p) atoms at 3.6 å corresponding to a er-o-si (or p) bond angle of ≈140 deg) are in good agreement with the cited literature. si(or p) atoms are visible as they belong to the same sio4 (or po4) tetrahedron as the first shell o atoms but no further coordination shells are detected. this permits to state that an amorphous environment is realized around er3+ ions. on the other hand sample b presents a completely different exafs signal that is well comparable with the spectrum of erpo4. this means that er in this case is inserted in a locally well ordered phase of about a few coordination shells (around 4-5 å around the absorber). the fact that tem on this sample reveals a uniform sample is not in contradiction with this result; it just means that this phase is not spatially extended to form nm-sized np (in our tem analyses the spatial resolution is limited to few nm) but the ordering is extremely local, i.e. it is limited to only a few shells around the rare-earth ion. from these considerations, the broadening of the emission spectrum observed in sample a can be attributed to an inhomogeneous broadening due to er3+ ions located in a more disordered environment compare to sample b. here we see that the cumulated effects of ca and p within the er-doped np both amorphize the material structure around er3+ ions and increase the fluorescence inhomogeneous broadening. conclusion in this paragraph, we have demonstrated that through the phase separation mechanism, nanoparticles can be obtained in preforms by adding calcium. er3+ ions are found to be located only into these nanoparticles. an inhomogeneous broadening of the emission band is observed, associated to er3+ ions located in a more disordered environment compare to silica. this feature is particularly interesting in the production of materials for wavelength division multiplexing applications, such as erbium-doped fiber amplifier with a broader band gain. perspectives and conclusion the choice of a glass to develop new optical fiber component is most of the time a result of compromises. silica glass is the most widely used for its many advantages (reliability, low cost fabrication, ...). however, it suffers from different drawbacks, such as high phonon energy or low luminescent ions solubility, which affect quantum efficiency or emission bandwidth of luminescent ions, for example. we have shown in several cases that spectroscopic properties of dopants are not directly related to the average properties of the doped glass, but to their local environment. indeed, by slightly modifying the silica composition, we succeeded to control the chromium valence state and improve the thulium emission efficiency. moreover, we present interest of high doping level to take advantage of energy transfers. then, nanostructuration of doped fiber is proposed as a new route to 'engineer' the local dopant environment. all these results will benefit to optical fiber components such as lasers, amplifiers and sensors, which can now be realized with silica glass. references i desurvire, e. 1994. in erbium doped fiber amplifiers: principles and applications. wiley interscience. isbn 0-471-58977-2. ii desurvire, e., d. bayart, b. desthieux, s. bigo. 2002. in erbium doped fiber amplifiers: device and system developments. wiley interscience. isbn 0-471-41903-6. iii digonnet, m. 2001. in rare-earth-doped fiber lasers and amplifiers (2nd ed.) crc. isbn 0-824-70458-2. iv grattan, k.t.v., t. sun t. 2000. fiber optic sensor technology: an overview. sens. actuators a: phys. 82:40. v komukai, t., t. yamamoto, t. sugawa and y. miyajima, 1995. upconversion pumped thulium-doped fluoride fiber amplifier and laser operating at 1.47 μm. ieee j. quant. electron. 31:1880. vi minelly, j and a.ellison. 2002. applications of antimony-silicate glasses for fiber optic amplifiers. opt. fib. tech. 8:123. vii lin, h., s. tanabe, l. lin, y.y. hou, k. liu, d.l. yang, t.c. ma, j.y. yu and e.y.b. pun. 2007. near-infrared emissions with widely different widths in two kinds of er3+- doped oxide glasses with high refractive indices and low phonon energies. j. lum. 124:167. viii durteste, y., m. monerie, j.-y. allain, and h. poignant. 1991. amplification and lasing at 1.3 pm in praseodymium-doped fluoridezirconate fibers. electron. lett. 27:626 ix hewak, d.w., r.s. deol, j. wang, g. wylangowski, j.a. mederios neto, b.n. samson, r.i. laming, w.s. brocklesby, d.n. payne, a. jha, m. poulain, s. otero, s. surinach, and m.d. baro. 1993. low phonon-energy glasses for efficient 1.3 μm optical fiber amplifiers. electron. lett. 29:237. x murata, k., y. fujimoto, t. kanabe, h. fujita, and m. nakatsuka. 1999. bi-doped sio2 as a new laser material for an intense laser. fusion engineering and design 44:437. xi dianov, e.m., v.v. dvoyrin, v.m. mashinsky, a.a. umnikov, m.v. yashkov and a.n. gur'yanov. 2005. cw bismuth fiber laser. quantum electronics 35:1083. xii razdobreev, i., l. bigot, v. pureur, a. favre, g. bouwmans and m. douay. 2007. efficient all-fiber bismuth-doped laser. appl. phys. lett. 90:031103. xiii samson, b.n., p.a. tick, and n.f. borrelli. 2001. efficient neodymium-doped glass- ceramic fiber laser and amplifier. opt. lett. 26(3):145. xiv samson, b.n., l.r. pinckney, j. wang, g.h. beall, and n.f. borrelli. 2002. nickel-doped nanocrystalline glass-ceramic fiber. opt. lett. 27(15):1309. xv tick, p.a., n.f. borrelli, l.k. cornelius, and m.a. newhouse. 1995. transparent glass ceramics for 1300 nm amplifier applications. j. appl. phys. 78 (ll):6367. xvi see for example: nagel, s.r., j.b. macchesney and k.l. walker. 1985. modified chemical vapor deposition. in optical fiber communications: vol 1 'fiber fabrication', ed. t. li, orlando: academic press xvii townsend, j.e., s.b. poole and d.n. payne. 1987. solution doping technique for fabrication of rare earth doped optical fibers. electron. lett. 23:329. xviii vienne, g. 1996. fabrication and characterization of ytterbium:erbium codoped phosphosilicate fibers for optical amplifiers and lasers", phd thesis, southampton, uk. xix schultz, p.c. 1977. proc 11th int. cong. glass, 3:155. xx jackson, s.d. and t.a. king. 1999. theoretical modelling of tm-doped silica fiber lasers. j. light. technol. 17(5):948. xxi förster, t. 1948. zwischenmolekulare energiewanderung und fluoreszenz. ann. phys. (leipzig) b 2:55. xxii dexter, d.l. 1953. a theory of sensitized luminescence in solids. j. chem. phys. 21(5):836. xxiii grubb, s.g., w.f. humer, r.s. cannon, t.h. windhorn, s.w. vendetta, k.l. sweeney, p.a. leilabady, w.l. barnes, k.p. jedrzejewski and j.e. townsend. 1992. +21-dbm erbium power-amplifier pumped by a diode-pumped nd-yag laser. ieee photon. tech. lett. 4(6):553. xxiv barnes, w.l., s.b. poole, j.e. townsend, l. reekie, d.j. taylor and d.n. payne. 1989. er3+-yb3+ and er3+ doped fiber lasers. j. lightwave technol. 7(10):1461. xxv auzel, f. 1990. upconversion processes in coupled ion systems. j. lumin. 45(1-6):341. xxvi wright, j.c.. 1976. up conversion and excited state energy transfer in rare-earth doped materials. in topics in applied physics, vol. 15, ed. f.k. fong, springer verlag, n.y. xxvii reisfeld, r. and c.k. jorgensen. 1977. in lasers and excited states of rare earths. springer-verlag, berlin heidelberg. xxviii delevaque, e., t. georges, m. monerie, p. lamouler and j.-f. bayon. 1993. modeling of pair-induced quenching in erbium-doped silicate fibers. ieee photon. tech. lett. 5:73. xxix wagener, j.l., p.f. wysocki, m.j.f. digonnet and h.j. shaw. 1994. modeling of ion- pairs in erbium-doped fiber amplifiers. opt. lett. 19:347. xxx ainslie, b.j., s. p. craig and r. wyatt and k. moulding. 1989. optical and structural- analysis of a neodymium-doped silica-based optical fiber. mater. lett. 8:204. xxxi maurice, e., g. monnom, b. dussardier and d.b. ostrowsky. 1995. clustering induced non-saturable absorption phenomenon in heavily erbium-doped silica fibers. opt. lett. 20:2487. xxxii maurice, e., g. monnom, b. dussardier and d.b. ostrowsky. 1996. clustering effects on double energy transfer in heavily ytterbium-erbium-codoped silica fibers. j. opt. soc. am., 13:693. xxxiii berthou, h. and c.k. jörgensen. 1990. optical-fiber temperature sensor based on upconversion-excited fluorescence. opt. lett, 15:1100. xxxiv krug, p.a., m.g. sceats, g.r. atkins, s.c. guy and s.b. poole. 1991. intermediate excited-state absorption in erbium-doped fiber strongly pumped at 980 nm.. opt. lett, 16(24):1976. xxxv maurice, e., g. monnom, a. saissy, d.b. ostrowsky and g.w. baxter. 1994. thermalization effects between upper levels of green fluorescence in er-doped silica fibers. opt. lett, 19:990. xxxvi maurice, e., g. monnom , d.b. ostrowsky and g.w. baxter. 1995. 1.2 μm-transitions in erbium-doped fibers: the possibility of quasi-distributed temperature sensors. appl. optics, 34:4196. xxxvii sennaroglu, a., c.r. pollock and h. nathel. 1995. efficient continuous-wave chromium- doped yag laser. j. opt. soc. am. b. 12:930. xxxviii chen, j.-c., y.-s. lin, c.-n. tsai, k.-y. huang, c.-c. lai, and w.-z. su. 2007. 400- nm-bandwidth emission from a cr-doped glass fiber. ieee phot. technol. lett. 19(8):595. xxxix schultz, p.c.. 1974. optical absorption of the transition elements in vitreous silica, j. am. ceram. soc. 57:309. xl cerqua-richardson, k., b. peng and t. izumitani. 1992. spectroscopic investigation of cr4+-doped glasses. in osa proc. advanced solid-state lasers (1992), eds. l.l. chase and a.a. pinto (optical society of america), 13:52. xli hömmerich, u., h. eilers, w.m. yen, j.s. hayden and m.k. aston. 1994. near infrared emission at 1.35 m in cr doped glass. j. lum. 60&61:119. xlii henderson, b. and g.f. imbush, in optical spectroscopy of inorganic solids (clarendon, oxford, 1989). xliii sugano, s., y. tanabe and h. kamimura, in multiplets of transition-metal ions in crystals (academic press, new york, 1970). xliv felice, v., b. dussardier, j.k. jones, g. monnom and d.b. ostrowsky. 2001. chromium- doped silica optical fibers : influence of the core composition on the cr oxidation states and crystal field. opt. mat. 16:269. xlv felice, v., b. dussardier, j.k. jones, g. monnom and d.b. ostrowsky. 2000. cr4+-doped silica optical fibers : absorption and fluorescence properties. eur. phys. j. ap, 11:107. xlvi dussardier, b., y. guyot, v. felice, g. monnom and g. boulon. 2002. cr4+-doped silica- based optical fibers fluorescence from 0.8 μm to 1.7 μm. in proc. advanced solid state lasers, in trends in optics and photonics series (osa), isbn: 1-55752-697-4, 68:104. xlvii rasheed, f., k.p. o'donnell, b. henderson and d.b. hollis. 1991. disorder and the optical spectroscopy of cr3+-doped glasses: i. silicate glasses. j. phys.: condens. matter. 3:1915. xlviii anino, c., j. théry and d. vivien. 1997. new cr4+ activated compounds in tetrahedral sites for tunable laser applications. opt. mat. 8:121. xlix note: in the following the energy states are referred to by their irreductible representation in the td symmetry coordination (ground state is 3a2)). l moncorgé, r., h. manaa and g. boulon. 1994. cr4+ and mn5+ actives centers for new solid state laser materials. opt. mat. 4:139. li cronemeyer, d.c. 1966. optical absorption characteristics of pink ruby. j. opt. soc. am. 56:1703. lii sennaroglu, a., u. demirbas, s. ozharar and f. yaman. 2006. accurate determination of saturation parameters for cr4+-doped solid-state saturable absorbers. j. opt. soc. am. b 23(2):241. liii lipavsky, b., y. kalisky, z. burshtein, y. shimony and s. rotman. 1999. some optical properties of cr4+-doped crystals. opt. mat. 13:117. liv tordella, l., h. djellout, b. dussardier, a. saïssy and g. monnom. high repetition rate passively q-switched nd3+:cr4+ all-fiber laser. electron. lett., 39:1307. lv dvoyrin, v.v., v.m. mashinsky, v.b. neustruev, e.m. dianov, a.n. guryanov and a.a. umnikov. 2003. effective room-temperature luminescence in annealed chromium-doped silicate optical fibers. j. opt. soc. am. b. 20:280. lvi antipenko, b.m., a.a. mak, o.b. raba, k.b. seiranyan, and t.v. uvarova. 1983. "new lasing transition in the tm3+ ion", new lasing transition in the tm3+ ion. sov. j. quant. electron. 13(4):558. lvii arai, k., h. namikawa, k. kumata, t. honda, y. ishii, and t. handa. 1986. aluminium or phosphorus co-doping effects on the fluorescence and structural properties of neodymium-doped silica glass. j. appl. phys. 59(10):3430. lviii blanc, w., t.l. sebastian, b. dussardier, c. michel, b. faure, m. ude, and g. monnom. 2008. thulium environment in a silica doped optical fiber. j. non-cryst. solids 354(2- 9):435. lix grinberg, m., d.l. russell, k. holliday, k. wisniewski, and cz. koepke. 1998. continuous function decay analysis of a multisite impurity activated solid. opt. comm. 156(4-6):409. lx walsh, b.m. and n.p. barnes. 2004. comparison of tm:zblan and tm:silica fiber lasers: spectroscopy and tunable pulsed laser operation around 1.9 μm. appl. phys. b. 78 (3-4):325. lxi van dijk, j.m.f. and m.f.h. schuurmans. 1983. on the nonradiative and radiative decay rates and a modified exponential energy gap law for 4f–4f transitions in rare-earth ions. j. chem. phys. 78(9):5317. lxii layne, c.b., w.h. lowdermilk and m.j. weber. 1977. multiphonon relaxation of rare- earth ions in oxide glasses. phys. rev. b 16:10. lxiii faure, b., w. blanc, b. dussardier, and g. monnom. 2007. improvement of the tm3+:3h4 level lifetime in silica optical fibers by lowering the local phonon energy. j. non-cryst. solids 353(29):2767. lxiv blanc w., p. peterka, b. faure, b. dussardier, g. monnom, i. kasik, j. kanka, d. simpson, and g. baxter. 2006. characterization of a thulium-doped silica-based optical fiber for s-band amplification. spie proc. 6180:181. lxv peterka p., b. faure, w. blanc, m. karasek, and b. dussardier. 2004. theoretical modelling of s-band thulium-doped silica fiber amplifiers. opt. quant. electron. 36(1- 3):201. lxvi d'acapito, f., s. mobilio, l. santos, and r. almeida. 2001. local environment of rare- earth dopants in silica-titania-alumina glasses: an extended x-ray absorption fine structure study at the k edges of er and yb. appl. phys. lett. 78(18):2676. lxvii d'acapito, f., s. mobilio, p. bruno, d. barbier, and j. philipsen. 2001. luminescent properties of local atomic order of er3+ and yb3+ ions in aluminophosphate glasses. j. appl. phys. 90(1):265. figures captions fig. 1: schematic energy diagram of (a) single energy transfer between two ions, (b) double energy transfer. fig. 2 : absorption of the er-1 and er-2 fibers vs launched pump power. squares: experimental data; solid lines: cluster model for 0%, 10% and 52% of er3+-ions in clusters; vertical arrows: non-saturated absorption as a difference with simulation for 0% cluster. fig. 3 : energy scheme for the det process. level energies are in cm-1; their lifetimes between round brackets. fig. 4 : natural logarithm of measured intensity ratio (r) plotted against the inverse of temperature. fig. 5: normalized tanabe-sugano energy level diagram for cr4+ in tetrahedral ligand field (td symmetry) showing the energy states of interest, for c/b = 4.1. the free ion states are shown on the left of the ordinate axis. the dashed line (dq/b = 1.43) reveals the relative positions of the states found for cr4+ in the silica-based samples: the first excited state level is 3t2(3f). fig. 6:. background corrected absorption from (left) a cr(ge) preform ([cr] = 1400 ppm) and (right) a cr(al) fiber ([cr] = 40 ppm). circles: experimental data; solid lines: adjusted bands to cr3+ (left) and cr4+ (right) transitions, respectively; and resulting absorption spectra. assignments are indicated from the ground level cr3+:4a2 or cr4+:3a2 to the indicated excited level, respectively. the cr4+:3t2 level three-fold splitting is due to distorsion from perfect tetrahedral symmetry. the spin-forbidden cr3+:4a22e and cr4+:3a21e transitions are not visible and overlapping with the intense spin-allowed transitions. fig. 7: fluorescence spectra: (a) fiber cr(al):[cr] = 40 ppm, p = 900 nm, t=77 k, (b) preform cr(ge-al): [cr] = 300 ppm, p = 673 nm, t=12k. fig. 8: fluorescence decays from cr(al) samples, p = 673 nm, t=12 k: (a) s~1100 nm and [cr]=40 ppm, (b)s~1100 nm, [cr]=4000 ppm, and (c) s~1400 nm, [cr]=4000 ppm fig. 9: schematic energy diagram of tm3+ ion, showing the relevant multiplets. solid arrows: absorption and emission optical transitions; thick arrow: nrd (non-radiative de- excitation) across the energy gap between the 3h4 and 3h5 multiplets, e~ 3700 cm-1. fig. 10 : histograms of the recovered luminescence decay time distributions obtained for silica- based tm3+-doped fibers with phosphorus incorporated in the core and different al2o3 concentration. fig. 11: miscibility-gap in the derived phase-diagram of binary sio2-cao glass fig. 12: tem image from preform sample doped with ca and p. fig. 13 : room temperature emission spectra of er-doped preform with (sample a) and without (sample b) calcium. samples were excited at 980 nm. fig. 1: schematic energy diagram of (a) single energy transfer between two ions, (b) double energy transfer. 0 10 20 30 40 50 0 1 2 3 4 a (db) 0% 10% 52% er 2 fiber er 1 fiber nsa p (mw) in fig. 2 : absorption of the er-1 and er-2 fibers vs launched pump power. squares: experimental data; solid lines: cluster model for 0%, 10% and 52% of er3+-ions in clusters; vertical arrows: non-saturated absorption as a difference with simulation for 0% cluster. fig. 3 : energy scheme for the det process. level energies are in cm-1; their lifetimes between round brackets. 1 -0.5 0.5 1.5 ln ( r) 2 3 1000/t (k) fig. 4 : natural logarithm of measured intensity ratio (r) plotted against the inverse of temperature. 0 2 4 1 3 60 30 0 e/b dq/b 1.43 3f 1d 3p 3t1 (3p) 3t1 (3f) 3t2 (3f) 3a2 (3f) 1e (1d) fig. 5: normalized tanabe-sugano energy level diagram for cr4+ in tetrahedral ligand field (td symmetry) showing the energy states of interest, for c/b = 4.1. the free ion states are shown on the left of the ordinate axis. the dashed line (dq/b = 1.43) reveals the relative positions of the states found for cr4+ in the silica-based samples: the first excited state level is 3t2(3f). fig. 6:. background corrected absorption from (left) a cr(ge) preform ([cr] = 1400 ppm) and (right) a cr(al) fiber ([cr] = 40 ppm). circles: experimental data; solid lines: adjusted bands to cr3+ (left) and cr4+ (right) transitions, respectively; and resulting absorption spectra. assignments are indicated from the ground level cr3+:4a2 or cr4+:3a2 to the indicated excited level, respectively. the cr4+:3t2 level three-fold splitting is due to distorsion from perfect tetrahedral symmetry. the spin-forbidden cr3+:4a22e and cr4+:3a21e transitions are not visible and overlapping with the intense spin-allowed transitions. fig. 7: fluorescence spectra: (a) fiber cr(al):[cr] = 40 ppm, p = 900 nm, t=77 k, (b) preform cr(ge-al): [cr] = 300 ppm, p = 673 nm, t=12k. 0,001 0,01 0,1 1 0 100 200 300 400 time (μs) (a) (c) (b) fig. 8: fluorescence decays from cr(al) samples, p = 673 nm, t=12 k: (a) s~1100 nm and [cr]=40 ppm, (b)s~1100 nm, [cr]=4000 ppm, and (c) s~1400 nm, [cr]=4000 ppm 3h 6 3f 4 3h 4 3h 5 nrd 785 nm 810 nm 1470 nm (s-band) e fig. 9: schematic energy diagram of tm3+ ion, showing the relevant multiplets. solid arrows: absorption and emission optical transitions; thick arrow: nrd (non-radiative de- excitation) across the energy gap between the 3h4 and 3h5 multiplets, e~ 3700 cm-1. fig. 10 : histograms of the recovered luminescence decay time distributions obtained for silica- based tm3+-doped fibers with phosphorus incorporated in the core and different al2o3 concentration. fig. 11: miscibility-gap in the derived phase-diagram of binary sio2-cao glass fig. 12: tem image from preform sample doped with ca and p. fig. 13 : room temperature emission spectra of er-doped preform with (sample a) and without (sample b) calcium. samples were excited at 980 nm.

0911.1685

multi-objective optimisation method for posture prediction and analysis with consideration of fatigue effect and its application case

automation technique has been widely used in manufacturing industry, but there are still manual handling operations required in assembly and maintenance work in industry. inappropriate posture and physical fatigue might result in musculoskeletal disorders (msds) in such physical jobs. in ergonomics and occupational biomechanics, virtual human modelling techniques have been employed to design and optimize the manual operations in design stage so as to avoid or decrease potential msd risks. in these methods, physical fatigue is only considered as minimizing the muscle or joint stress, and the fatigue effect along time for the posture is not considered enough. in this study, based on the existing methods and multiple objective optimisation method (moo), a new posture prediction and analysis method is proposed for predicting the optimal posture and evaluating the physical fatigue in the manual handling operation. the posture prediction and analysis problem is mathematically described and a special application case is demonstrated for analyzing a drilling assembly operation in european aeronautic defence & space company (eads) in this paper.

introduction although the automation technique has been employed widely in industry, there are still lots of manual operations, especially in assembly and maintenance jobs due to the flexibility and the feasibility of human being (forsman et al., 2002). among these manual handling operations, there are occasionally several physical operations with high strength demands. among the workers in such operations, msd is one of the major health problems. the magnitude of the load, posture, personal factors, and sometimes vibration are potential exposures for msds (li and buckle, 1999). it is believed that one reason for msds is the physical fatigue resulted from the physical work. the aim of ergonomics is to generate working conditions that enhance safety, well-being and performance, and manual operation design and analysis is one of the key methods to improve manual work efficiency, safety, comfort, as well as job satisfaction. for manual handling operation design, the strength of the joint and muscle is of importance to guide the design of workspace or equipment to reduce work related injuries, and furthermore to help in personnel selection to increase work efficiency. human strength information can also be used in a human task simulation environment to define the load or exertion capabilities of each agent and, hence, decide whether a given task can be completed in a task simulation. it should be noticed that the physical strength does not remain immutable in a working process, and in fact it varies according to several conditions, such as environment, physical state and mental state. the diminution of the physical capacity along time is an obvious phenomenon in these manual operations. physical fatigue is defined as reduction of physical capacity, which is derived from the definition of muscle fatigue: "any reduction in the maximal capacity to generate force or power output" (vollestad, 1997). physical fatigue is mainly resulting from three reasons: magnitude of the external load, duration and frequency of the external load, and vibration. it was proved in (chen, 2000) that the movement strategy in industrial activities involving combined manual handling jobs, such as a lifting job, depends on the fatigue state of muscle, and it is obvious that the change of the movement strategy in the activities directly impacts the motion of the operation and then results in different loads in muscles and joints. if it goes worse, once the desired exertion is over the physical capacity, cumulative fatigue or injury might appear in the tissues as potential risks for msds. in order to make an appropriate design, the same problem has been encountered by countless organizations in a variety of industries: the human element is not being considered early or thoroughly enough in the life cycle of products, from design to recycling. more significantly, this does have a devastating impact on cost, time to market, quality and safety. using realistic virtual human is one method to take the early consideration of ergonomics issues in the design and it reduces the design cycle time and cost (badler, 1997; honglun et al., 2007). nowadays, there are several commercialized 3 human simulation tools available for job design and posture analysis, such as 3dsspp, jack, vsr and anybody. 3dsspp (three dimensional static strength prediction programme) is a tool developed in university michigan (chaffin et al., 1999). originally, this tool is developed to predict population static strengths and low back forces resulting from common manual exertions in industry. the biomechanical models used in 3dsspp are meant to evaluate very slow or static exertions (chaffin, 1997). it predicts static strength requirements for tasks such as lifts, presses, pushes, and pulls. the output includes the percentage of men and women who have the strength to perform the described job, spinal compression forces, and data comparisons to niosh guidelines. however, they do not allow dynamic exertions to be simulated, and there is no fatigue evaluation tool in this tool. jack (badler et al., 1993) is a human modelling and simulation software solution that helps organizations in various industries improve the ergonomics of product designs and refine workplace tasks. with jack, it is able to assign a virtual human in a task and analyze the posture and other performance of the task using existing posture analysis tools, like owsa (ovako working posture analyzing system) and so on. ptms (predetermined time measurement systems) are also integrated to estimate the standard working time of a specified task. in this virtual human tool, the fatigue term is considered in motion planning to avoid a path that has a high torque value maintained over a prolonged period of time. however, the reduction of the physical capacity is not modelled in the virtual human, although the work-rest schedule can be determined using its extension package. in vsr (virtual soldier research), another virtual human was developed for military application. in this research, the posture prediction is based on moo (multiple-objective optimisation) with three objective terms of human performance measures: potential energy, joint displacement and joint discomfort (yang et al., 2004). in santostm, fatigue is modelled based on the physiological principle mentioned in a series of publication (ding et al., 2000, 2002, 2003). because this muscle fatigue model is based on physiological mechanism of muscle, it requires dozens of variables to construct the mathematical model for a single muscle. meanwhile, the parameters for this muscle fatigue model are only available for quadriceps. in addition, in its posture prediction method, the fatigue effect is not integrated. anybody is a system capable of analyzing the musculoskeletal system of humans or other creatures as rigid-body systems. a modelling interface is designed for the muscle configuration, and optimisation method is used in the package to resolve the muscle recruitment problem in the inverse dynamics approach (damsgaard et al., 2006). in this system, the recruitment strategy is stated in terms of normalized muscle forces. "however, the scientific search for the muscle recruitment criterion is still 4 ongoing, and it may never be established." (damsgaard et al., 2006). furthermore, in the optimisation criterion, the capacities of the musculoskeletal system are assumed as constants, and no limitations from the fatigue are taken into account. in all the posture prediction methods mentioned above, especially in these optimisation methods, the physical capacity is treated as constant. for example, in anybody or other static optimisation methods, the muscle strength is proportional to the pcsa (physiological cross section area). in jack, the strength is the maximum achievable joint torque. in other words, the reduction of the physical capacity is not considered, and using these tools is not sufficient to predict or analyze the fatigue effect in a real manual operation. table 1. comparison of different available virtual human simulation tools 3dsspp(1,2) anybody(3) jack santotm(4) posture analysis √ √ √ √ joint effort analysis √ √ √ √ muscle force analysis √ posture prediction √ √ √ √ empirical data based √ optimization method based √ √ √ √ soo √ √ moo √ joint discomfort guided √ √ fatigue effect in optimization √ √ (1) 3dsspp is only suitable for static or quasi-static tasks. (2) the motion posture prediction is based on empirical data and optimization based differential inverse kinematics. (3) the objective function is programmable. (4) potential energy, joint displacement, joint discomfort and etc are used as objective functions. in manufacturing and assembly line work, repetitive movements constitute a major facet of several workplace tasks, and such movements lead to muscle fatigue. muscle fatigue generates influences on neuromuscular pathway, postural stability and global reorganization of posture (fuller et al., 2008). in the tools mentioned above, the fatigue effect can be inferred in posture analysis, but how the human reacts on physical fatigue by adjusting the posture in order to meet the physical requirements is not feasible in those tools. physical fatigue, which can be experienced by everyone in everyday, especially for those who are engaged in manual handling operations, should be taken into human simulation. a more realistic posture prediction can gain clearer understanding of human movement performance, and it is always a tempting goal for biomechanics and ergonomics researchers (zhang and chaffin, 2000). the predictive capacity, or the reality is provided by a model in computerized form, and these quantitative models should be able to predict realistically how people move and interact with systems. therefore, it should be necessary to integrate the feature of fatigue into posture prediction to predict the possible change of posture along with the reduction of the physical capacity. furthermore, the fatigue model should have a sufficient precision to reproduce the fatigue correctly. 5 in this paper, a posture analysis and posture prediction method is proposed to take account of the fatigue effect in the manual operations. at first, the general modelling procedure of virtual human is presented. the mathematical description of the posture prediction is formulated based on a muscle fatigue model in the following section. the overall framework involving the posture analysis method is shown to explain the workflow in a virtual working environment. at last, an application case in eads is demonstrated followed by results and discussions. 2. kinematic modelling and dynamic modelling of virtual human in this study, the human body is modelled cinematically as a series of revolute joints. modified denavit-hartenberg notation system (khalil and kleinfinger, 1986; khalil and dombre, 2002) is used to describe the movement flexibility of the joint. according to its function, one natural joint can be modelled by 1-3 revolute joints. each revolute joint has its own joint coordinate, labelled as i q , with joint limits: the upper limit u i q and the lower limit l i q . a set of generalized coordinates   1 q= t i n q q q   is defined as a vector to represent the kinematic chain. in fig. 1, the human body is geometrically modelled by 28 revolute joints to represent the main movement of the human body. the posture, velocity, and acceleration are expressed by the general coordinates q ,q , and q . it is feasible to achieve the kinematic analysis of the virtual human based on this kinematic model. by implementing existing inverse kinematic algorithms, it is able to predict the posture and trajectory of the human, particularly for the end effectors, i.e. both hands. in this operation, it is possible that all the joints are involved in the implementation of the inverse kinematics; therefore there are many possible solutions with such a high dof (28 in total for main joints). in industry, the sedentary operation occupies a large proportion for manual handling jobs, and even in some heavy operations, the upper extremity is mainly engaged to finish the task. therefore, in our application case, only both arms are kinematic and dynamic modelled to analyze the operation. 6 z2 z1 z3 z5 z4 z6 z7 z8 z9 z10 z11 z12 z13 z17 z14 z16 z18 z15 z21 z26 z19 z20 z25 z24 z22 z27 z28 z23 z0 x0 z10 right hand left hand figure 1. kinematic modelling of the human body no matter in static posture or in dynamic process, the movement and the external efforts can generate torques and forces at the joints. therefore, dynamic modelling of the human body is necessary for implementing inverse dynamic calculation. for each body segment, the most important dynamic parameters are the moment of inertia, gravity centre, and mass of the limb. such information can be achieved from some anthropometrical database and biomechanical database. 3. multi-objective optimisation for posture prediction the general description of the posture analysis problem based on multiple-objective optimisation (moo) is to find a set of q in order to minimize several objective functions in eq. (1) simultaneously: 7 1( ) min ( ) ( ) ( ) i n f f f f                   q ω q q q q   (1) subject to equality and inequality constraints in eq. (2). ( ) 0 1,2, , ( ) 0 1,2, , i i g i m h j e        q q   (2) with m is the number of inequality constraints and e is the number of equality constraints. two human performance measures are used to create the global objective function: fatigue and discomfort. of course, with the exception of these two performance measures, there are still several other objective functions, such energy expenditure (ren et al., 2007), joint displacement (yang et al., 2004), visibility and accessibility (chedmail et al., 2003) etc. in our current application, only fatigue and joint discomfort are taken into consideration for the posture prediction and evaluation, since the physical fatigue effect acting on the posture prediction is the main phenomena that should be verified. if several objective functions are involved in the posture prediction, it would be difficult to analyze the fatigue independently. fatigue 1 p dof i fatigue i i cem f            (3) in the literature, normalized muscle force is often used as a term to determine the muscle force. this term represents the minimization of muscle fatigue in the literature, and a similar measure has been used in (ayoub, 1998; ayoub and lin, 1995) for simulating the lifting activities. in our application, the summation of the normalized joint torques is used based on the same concept in eq. (3). dof is the total number of the revolute joints for modelling the human body. for each joint, the term normalized torque i i cem   represents the relative load of the joint. the summation of the relative load is one measure to minimize the fatigue of each joint. in traditional methods, i cem  is assumed constant in the operation. in order to integrate the fatigue effect, the fatigue process is mathematically modelled in a differential equation eq. (4). in this model, the temporal parameters and the physical parameters are taken into consideration, which represents the magnitude of physical load, duration, and frequency in the conventional ergonomics analysis methods. the descriptions of all the parameters in the equation are listed in table 2. 8 max cem cem i i i load i d k dt      (4) table 2: parameters in muscle fatigue and recovery model parameters unit description γmax nm maximum joint strength γcem nm joint strength at time instant t γ nm torque at the joint at time instant t k min-1 fatigue ratio, equals to 1 r min-1 recovery ratio, equals to 2.4 t min time the fatigue process is graphically shown in fig. 2. assume in a static posture, the load of the joint is constant load  . at the very beginning of an operation, the joint has the maximum strength max  . with time, the joint strength decreases from the maximum strength. the maximum endurance time (met) is the duration from the start to the time instant at which the strength decreases to the torque demand resulting from external load. load t joint capacity cem t time t safe potential risks max endurance time figure 2. fatigue effect on the joint strength this fatigue model is based on motor-units pattern of muscle (liu et al., 2002; vollestad, 1997). the joint torque capacity is the overall performance of muscles attached around the joint. in a muscle, there are mainly three types of muscle motor units: type i, type ii a, and type ii b. the fatigue resistance in ascending sequence is: type ii b < type ii a < type i. meanwhile, the muscle force generation capacity is: type i < type ii a < type ii b. muscle motor recruitment sequence starts from type i, and then goes to type ii a and at last type ii b. therefore, to fulfil the requirement of the larger 9 external force, more type ii b units are involved and then the faster muscle becomes fatigue. i load  can represent the influence from the external load. the fatigue resistance is determined by the composition of muscle units. when the capacity decreases, which means more and more type ii b units and type ii a units are getting fatigued, and at the same time type i units remain non fatigue and the overall fatigue resistance increases, and as a result the reduction process of the capacity decreases. this phenomenon is described by max cem i i   . this model has been mathematically validated by comparing the existing static met models in the literature (ma et al., 2009). high correlation has proved that this model is suitable for static posture or slow operation. the fatigue model for the dynamic operation has not yet been validated. recovery max ( ) t cem cem i i i d r d     (5) besides fatigue, the recovery of the physical capacity should also be modelled to predict the work-rest schedule in order to complete the design of manual handling operations. the recovery model in eq. (5) predicts the recuperation of the physical capacity and its original form is introduced in the literature (carnahan et al., 2001; wood, 1997). discomfort another objective function is joint discomfort. the discomfort measure is taken from vsr (yang et al., 2004). this measure evaluates the joint discomfort level from the rotational position of joint relative to its upper limit and its lower limit. the discomfort level is formulated in eq. (6) as follows, and it increases significantly as joint values approaches their limits. qu (eq. (7)) and ql (eq. (8)) are penalty terms correspondingly to the upper limit and lower limit of the joint. i  is the weighing value for each joint. the detailed notation of the variables in discomfort model is listed in table 3. table 3: parameters in joint discomfort model parameters unit description qi degree current position of joint i qu i degree upper limit of joint i ql i degree lower limit of joint i qn i degree neutral position of joint i g - constant, 106 qui - penalty term of upper limits qli - penalty term of lower limits γi - weighting value of joint i 10 2 1 1 ( ) dof norm discomfort i i i i i f q g qu g ql g            (6)   100 5.0 0.5sin 1 2 u i i i u l i i q q qu q q                       (7)   100 5.0 0.5sin 1 2 l i i i u l i i q q ql q q                       (8) an example calculated from the joint discomfort performance is graphically shown in fig.3. it is apparent that the joint discomfort reaches its minimum value at its neutral position and it increases when approaching its upper and lower limits. 0 20 40 60 80 100 120 140 160 180 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 joint movement range, qu=180, ql=0, qn=40 discomfort level of a joint neutral position figure 3. an example of the joint discomfort. objective function ( ) min ( ) ( ) fatigue discomfort f f f      q q q (9) the overall objective function uses fatigue measure and discomfort measure to determine the optimal geometrical configuration of the posture. the biomechanical aspect of the posture is evaluated by the fatigue objective function, and meanwhile, the geometrical constraints for the human body are measured by the discomfort measure. constraints 11 in this study, constraints from kinematic aspect and biomechanical aspect are used to determine the possible solution space. from kinematic aspect, the cartesian coordinates of the destination for the posture contributes to one constraint in eq. (10).   t x y z is the cartesian coordinates of the end-effector (right hand and left hand) of the aim of the reach. the function x can be described in direct kinematic approach. the transformation matrix between the end-effector and the reference coordinates can be modelled in the way of modified dh notation method. ( ) x y x z      q (10) joint limits (ranges of motion) are imposed in terms of inequality constraints in the form of eq. (11). l u i i i q q q   (11) from biomechanical aspect, theoretically there are mainly two constraints. one is the limitation of the joint strength (eq. (12)) and another one is equilibrium equation described in inverse dynamics in eq. (13). max 0 i i  (12) it should be noted that in eq. (12) the upper limit max i  is treated as unchangeable in conventional posture prediction methods. in our optimisation method, the upper limit is replaced by i cem  to update the physical capacity caused by fatigue. the joint strength depends on the posture of human body and personal factors; such as age and gender. in fig. 4, elbow joint flexion strength is shown for the 95% male adult population according to the literature (chaffin et al., 1999). the elbow flexion strength is related to the flexion angle of elbow s  and flexion angle of shoulder e  (shown in fig. 7). in the range of the joint, for a 50% population, the joint strength varies from 70 nm to 40 nm. for most of the population, the strength varies from 40 nm to almost 120 nm for the male. 12 figure 4. biomechanical joint flexion strength constraints of elbow in terms of equality constraints, another constraint is the inverse dynamics in eq. (13). with displacement, velocity and acceleration in general coordinates, the inverse dynamics formulates the equilibrium equation. in eq. (13), ( , , ) γ q q q  represents the term related to external loads, ( ) a q is the link inertia matrix, ( ) b q,q represents centrifugal and coriolis terms, and ( ) q q is the potential term. ( , , ) ( ) ( ) ( )    γ q q q a q q b q,q q q q    (13) in summary, the moo problem can be simplified as: for a static posture or in a relative slow motion, we can assume that q=0  , and q=0  , therefore, the joint torque depends only on the joint position and the external load. a set of solution satisfying all the constraints { ( ) ( ) } s q q 0, q 0 g h    can be found. in this case, we are trying to find a configuration q s  to achieve the minimization of both fatigue and discomfort objective functions. 4. framework and flowchart 13 owes objective work evaluation system virtual environment virtual human virtual interaction fatigue criteria posture criteria efficiency criteria comfort criteria environment human motion interaction motion capture haptic interfaces virtual reality simulated human motion human simulation fatigue analysis comfort analysis posutre analysis posture prediction algorithm virutal human status update figure 5. framework of the objective work evaluation system the posture analysis with consideration of fatigue is involved in an objective work evaluation system (owes) in fig. 5. the aim of the framework is to enhance the simulated human motion by using motion capture technique, and mainly two functions are designed: motion analysis and motion prediction. for motion analysis in such a system, manual handling operation is either captured by motion capture system or simulated by virtual human software in a virtual working environment. in this way, data- driven algorithm and computational approaches, two main methods for human modelling and simulation, can be integrated into the framework. the first method is developed based on experiment data and regression; therefore the most probable posture can be implemented for a specific data. however, a time-consuming data collection process is involved in such a method, such as motion tracking. the second one can be used for posture prediction, based on biomechanics and kinematics. with this tool, it is possible to predict the posture by formulating a set of equations. the interaction information is detected via haptic interfaces and recorded as external efforts on the joint, noted as   , ex ex j j f  . j is the index of the joint. both of the motion information and haptic interaction information are input into the work evaluation module. in such module, kinematic analysis can achieve the posture of the human body in each frame and the inverse dynamics is carried out to determine the corresponding effort at each joint   , j j f  . using predefined posture analysis criteria, 14 efficiency criteria, fatigue evaluation tool, etc, the different aspect of the manual handling operation can be assessed. in traditional system, there is no feedback from the analysis result to the prediction algorithm. in fact, the human does change its posture and trajectory according to different physical or mental status. in our framework, from the analysis result, the human status is updated, such as physical capacities. the updated status can be used further for posture prediction. therefore, the evaluation result, such as fatigue, needs to be taken back to the simulation to generate the much more realistic human simulation. 5. application case for drilling task task description in our research project, the application case is junction of two fuselage section with rivets from the assembly line of a virtual aircraft. one part of the job consists of drilling holes all around the section. the properties of this task can be described in natural language as: drilling holes around the fuselage circumference. the number of the holes could be up to 2000 under real work conditions. the drilling machine has a weight around 5 kg, and even up to 7 kg in the worst condition with consideration of the pipe weight. the drilling force applied to the drilling machine is around 49n. in general, it takes 30 seconds to finish a hole. the drilling operation is graphically shown in fig. 6. figure 6. drilling task in virtual aircraft factory in catia in this application case, there are several ergonomics issues and several physical exposures contribute to the difficulty and penalty of the job. it includes posture, heavy load from the drilling effort, the 15 weight of the drilling machine, and vibration. fatigue is mainly caused by the load on certain postures, and the vibration might result in damage to some other tissues of human body. to maintain the drilling work for a certain time, the load could cause fatigue in elbow, shoulder, and lower back. in this paper, the analysis is only carried out to evaluate the fatigue of right arm in order to verify the conception of the framework and the posture prediction method based on moo. the vibration is excluded from the analysis. further more, we assume that the worker carry the drilling machine symmetrically, the external loads are divided by two so as to simplify the calculation. the upper arm is modelled by five revolute joints in fig. 7. each revolute joint rotates around its z axis and the function of each joint is defined in table 4. 1 2 3 [ , , ] q q q is used to model the shoulder mobility. 4 5 [ , ] q q is used to describe the mobility around the elbow joint. s  is the flexion angle between shoulder and the body in the sagittal plane and the and e  is the angle between lower arm and upper arm in a flexion posture. x1 z2 z3 z4 z5 z0 z1 x0 x3, x4, x5 x2 rl3 shoulder e s elbow waist figure 7. kinematic modelling of human and two flexion angles of the arm table 4. five revolute joints in the arm kinematic model and their corresponding descriptions joints description 1 flexion and extension of shoulder joint 2 adduction and abduction of shoulder joint 3 supination and pronation of upper arm 4 flexion and extension of shoulder joint 5 supination and pronation of upper arm 16 the geometrical parameters of the limb are required in order to accomplish the kinematic modelling. such information can be obtained from anthropometrical database in the literature. take the arm as an example. the arm is segmented into two parts: upper arm and forearm (hand included). each part of the arm is simplified to a cylinder form and assumed a uniform distribution of density in order to calculate its moment of inertia. once the height of the virtual human is determined, according to anthropometry and biomechanics, both the length and the radius of the upper arm and lower arm can be estimated from eq. (14). the mass of each part can be achieved in occupational biomechanics by eq. (15). once the mass and cylinder radius and height are all known, its inertia moment around its long axis can be determined by a diagonal matrix in eq.(16). parameters unit description m kg mass of the virtual human h m height of the virtual human m kg mass of the segment f - subscript for forearm u - subscript for upper arm ig - moment of inertia of the segment h m length of the segment r m radius of the segment table 4: dynamic parameters and their descriptions in arm dynamic modelling 0.146 0.125 0.186 0.125 f f f u u u h h r h h h r h            (14) 0.451 0.051 0.549 0.051 f u m m m m          (15) 2 2 2 2 2 , , 4 12 4 12 2 g mr mh mr mh mr i diag          (16) results after kinematic and dynamic modeling of human arm, the posture analysis and posture prediction based on moo can be carried out. posture analysis: fatigue and recovery 17 in the left subfigure of fig. 8, the reduction of shoulder strength capacity is graphically presented using the fatigue model. in this case, the arm for the drilling work is configured by 30 s  and 90 e  . for maintaining the drilling posture, the torque generated by the external load at each joint remains constant. the joint load j  is represented by the horizontal solid line. the reduction of the strength capacity of the 95% male population is represented by the curves. for the male adult population, the strength of the joint locates in the range between 40 nm and 110 nm. the endurance time for such a drilling operation varies from 60 seconds to almost 450 seconds, and it proves that the strength variation is quite significant, and operation strategy and work-rest schedule should be designed with consideration of the individual variation (chaffin, 1997). furthermore, with the fatigue model, the reduction of the capacity is predictable for the manual operation. therefore, the posture prediction can be implemented based on the fatigue model. in the right subfigure of fig. 8, it is apparent that the same external load exerts different normalized load on the population. smaller joint capacity results in more rapid reduction of the capacity. 50 100 150 200 250 300 0 20 40 60 80 100 120 holding time of shoulder flexion for drilling hole tasks [s] reduction of shoulder flexion strength [nm] geometric configuration in s = 30o, e = 90o, mass of drilling machine 3.5 kg sj-2j sj-j sj sj+j sj+2j j load 50 100 150 200 250 300 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized reduction of shoulder flexion strength sj-2j sj-j sj sj+j sj+2j figure 8. reduction of the joint strength (shoulder) along time in the drilling task for a completed design of manual handling operation, work-rest schedule is also of great importance, especially for the manual handling work with relative high physical requirement. in fig. 9, a drilling process with 30 seconds for drilling a hole and 60 seconds for rest is shown. it can be observed that in 18 the capacity goes down in the work cycle and it recoveries in the following rest period. although there is a slight reduction of the capacity after one work-rest cycle, 95% of the population can maintain the drilling job for a long duration. 50 100 150 200 250 20 30 40 50 60 70 80 90 100 110 120 holding time of shoulder flexion for drilling hole tasks [s] reduction of shoulder flexion strength [nm] geometric configuration in s = 30o, e = 90o, mass of drilling machine 3.5 kg sj-2j sj-j sj sj+j sj+2j j figure 9. work-rest schedule predicted by fatigue and recovery model posture prediction: optimal posture for a drilling task in manual handling operation, the workspace parameters are important for determining the posture of the human body. in case of holding the drilling machine, the distance between the hole and shoulder is the most important geometrical constraint. in the scope between 0.4m and 0.7m, the geometrical configuration q can be determined, and then it is possible to calculate the fatigue measure and the discomfort measure. both measures are shown in fig. 10. it is obviously that the longer the distance is, the more the arm is extended, and as a result, the larger torque is applied to joints, which causes higher fatigue measure. simultaneously, the discomfort level changes with the distance. the larger the extension of the arm, the more the shoulder joint moves to its upper limit, however the elbow joint moves to its neutral position. the combination of both joints shows the declination along the distance. 19 0.45 0.5 0.55 0.6 0.65 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 fatigue performance measure distance from right shoulder to hole [m] discomfort performance measure fatigue performance measure at the very beginning fatigue performance measure after drilling 10 holes discomfort performance measure figure 10. fatigue and discomfort performance measures along the work distance the optimal posture can be determined using the moo method in fig. 11. weighted aggregation method is used in this case to covert the multi-objective problem into a single-objective method in order to achieve the pareto optimal in the pareto front represented by the solid curve. the single objective is mathematically formed in eq. (17). both measures are normalized. 1 2 1 min ( ) max( ) max( ) n discomfort fatigue j j j discomfort fatigue f f z w f w w f f      q (17) with 0 j w  and 1 1 n j j w    . each j w indicates the importance of each objective. this objective function can be further transformed to a straight line equation: z f w w f discomfort fatigue min 2 1    . if we assume that the fatigue and the discomfort have the same importance in the drilling case, the optimal position can be obtained at the intersection point between the solid straight line with slope k=- 1 and the pareto front in fig 11. however, the selection of the weighting value can have great influence on the optimal posture. the individual preference can be represented by the different weights of the two measures which results in straight lines with different slopes. in fig.11, two examples with slope k=-2 (dashed point line) and k=-0.5 (dashed line) are illustrated with different intersection points with the pareto front. those two points represent different posture strategies for posture control: the 20 former one with less discomfort, and the latter one with less joint stress. all the points in the pareto font are the feasible solutions for posture selection. the selection of posture depends on the physical status of individual, and the preference of the individual. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized discomfort of joints fd normalized fatigue of joints f f pareto front curve w1/w2=1 w1/w2=0.5 w1/w2=2 (fd, ff) solution: when w1/w2=1 solution: when w1/w2=2 solution: when w1/w2=0.5 solution zone figure 11. optimisation posture for the drilling task without fatigue optimal posture changed by fatigue meanwhile, the fatigue influences the posture. in order to evaluate the fatigue effect, we keep the same balance between fatigue and discomfort in our application. in fig. 12, the single objective function in eq. (17) along the distance from 0.4m to 0.7m is calculated and shown. the solid curve is the one without fatigue, and the dashed curve is the one with fatigue status after maintaining a drilling operation after 30 s. from the left subfigure, it is noticeable that the optimal distances for both situations are different, which maps to the different drilling posture. the optimal distance between the shoulder and the hole is smaller with fatigue then without fatigue. it proves that the manual handling strategy is making the arm close to the human body to maintain the same load when there is fatigue. in this posture, the user can handle the weight of the machine more easily. in the right subfigure, the pareto front in fatigue status is moved afterwards from the pareto front without fatigue as the fatigue measure increases resulting from reduction of the physical capacity. 21 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized discomfort of joints fd normalized fatigue of joints f f without fatigue with fatigue 0.45 0.5 0.55 0.6 0.65 0.7 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 distance from right shoulder to hole [m] summation of normalized discomfort and normalized fatigue without fatigue with fatigue figure 12. comparison of the optimized work distance in both non-fatigue and fatigue cases 6. discussion in this study, a fatigue model is integrated into a posture analysis and posture prediction method. with this model, it is possible to evaluate and design the posture for the manual handling operation by considering fatigue. the fatigue model can predict the reduction of the physical capacity in static posture or slow operation. the reduction of the physical capacity make the posture changed to maintain the external physical requirement. one limitation in our framework is that the posture analysis and prediction are only limited to the joint level until now, but not in muscle level. that is because it is believed that it is difficult to measure the force of each individual muscle, although the optimization method is employed to solve the underdetermined problem of the muscle skeleton system. the precision of the result is still questionable (freund and takala, 2001). from another point of view, the joint torque is generated and determined by a group of muscle attached around the joint. the coordination of the muscle group is very complex, and it is believed that calculating the joint torque can achieve a higher precision then calculating the individual muscle forces. meanwhile, in several ergonomics measurement, the met is also measured by the joint torque (mathiassen and ahsberg, 1999). 22 another limitation is that the result of the posture analysis is only applicable for static and slow operations, because the fatigue model is only validated by comparing with existing met models. for these static met models, all the measurement was carried out under static posture. dynamic motion and static posture are different in physiological principle, and fatigue and recovery phenomenon might occur alternatively and mix in a dynamic process. at last, the optimal posture is predicted in moo method. in such a method, the weighting values of each item are used to construct the overall objective function. however, it requires a priori knowledge about the relative importance of the objectives, and the trade-off between the fatigue and the discomfort can not be evaluated very well. "it is believed that the human body has certain strategy to lead the human motion, but it is dictated by just one performance measure; it may be necessary to combine various measures" (yang et al., 2004). two main problems rise for the motion prediction. one is how to model the performance measure. another one is how to combine all the performance measures together. human motion is very complex due to its large variability. each single performance measure is difficult to be validated in experiment. furthermore, for the combination, the correlation between different performance measures requires lots of effort to define and verify. moo method just provides a reference method in ergonomics simulation leading to a safer and better design of work. 7. conclusion and perspective in this paper, a new method based on moo method for posture prediction and analysis is presented. different from the other methods used in virtual human posture prediction methods, the effect from fatigue is taken into account. a fatigue model based on motor-units pattern is employed into the moo method to predict the reduction of the physical capacity. meanwhile, the work-rest schedule can be evaluated with the fatigue and recovery model. due to the validation of the fatigue model, this method is suitable for static or relative slow manual handling operation. at last, it is possible to predict the optimal posture of an operation to simulate the realistic motion. in the future, the fatigue for the dynamic working process will be validated and then integrated into the work evaluation system. 8. acknowledgments this research was supported by the eads and by the région des pays de la loire (france) in the context of collaboration between the ecole centrale de nantes (nantes, france) and tsinghua university (beijing, p.r.china). 23 references ayoub, m.m. (1998). a 2-d simulation model for lifting activities. computers and industrial engineering, 35(3-4), 619-622 ayoub, m.m., & lin, c.j. (1995). biomechanics of manual material handling through simulation: computational aspects. computers and industrial engineering, 29(1-4), 427-431 badler, n.i. (1997). virtual humans for animation, ergonomics, and simulation proceedings of the ieee workshop on non-rigid and articulated motion, 28-36. badler, n.i., phillips, c.b., & webber, b.l. (1993). simulating humans. new york: oxford university press, inc. carnahan, b.j., norman, b.a., & redfern, m.s. (2001). incorporating physical demand criteria into assembly line balancing. iie transactions, 33(10), 875-887 chaffin, d.b. (1997). development of computerized human static strength simulation model for job design. human factors and ergonomics in manufacturing, 7(4), 305-322 chaffin, d.b., andersson, g.b.j., & martin, b.j. (1999). occupational biomechanics (third ed.): wiley-interscience. chedmail, p., chablat, d., & le roy, c. (2003). a distributed approach for access and visibility task with a manikin and a robot in a virtual reality environment. ieee transactions on industrial electronics, 50, 693-698 chen, y.-l. (2000). changes in lifting dynamics after localized arm fatigue. international journal of industrial ergonomics, 25(6), 611-619 damsgaard, m., rasmussen, j., christensen, s.t., surma, e., & zee, m.d. (2006). analysis of musculoskeletal systems in the anybody modeling system. simulation modelling practice and theory, 14(8), 1100-1111 ding, j., wexler, a.s., & binder-macleod, s.a. (2000). a predictive model of fatigue in human skeletal muscles. journal of applied physiology, 89(4), 1322-1332 ding, j., wexler, a.s., & binder-macleod, s.a. (2002). a predictive fatigue model i: predicting the effect of simulation frequency and pattern of fatigue. ieee transactions on neural systems and rehabilitation engineering, 10(1), 48-58 ding, j., wexler, a.s., & binder-macleod, s.a. (2003). mathematical models for fatigue minimization during functional electrical stimulation. electromyography kinesiology, 13, 575-588 forsman, m., hasson, g.-a., medbo, l., asterland, p., & engstorm, t. (2002). a method for evaluation of manual work using synchronized video recordings and physiological measurements. applied ergonomics, 33(6), 533-540. freund, j., & takala, e.-p. (2001). a dynamic model of the forearm including fatigue. journal of biomechanics, 34(5), 597-605 24 fuller, j. lomond, k., fung, j. & côté, j. (2008). posture-movement changes following repetitive motion-induced shoulder muscle fatigue. journal of electromyography and kinesiology, 19(6), 1043- 1052 honglun, h., shouqian, s., & yunhe, p. (2007). research on virtual human in ergonomics simulation. computers & industrial engineering, 53(2), 350-356 khalil, w., & dombre, e. (2002). modeling, identification and control of robots: hermes science publications. khalil, w., & kleinfinger, j.f. (1986). a new geometric notation for open and closed-loop robots. proceedings of the ieee robotics and automation, 1174-1179. li, g., & buckle, p. (1999). current techniques for assessing physical exposure to work-related musculoskeletal risks, with emphasis on posture-based methods. ergonomics, 42(5), 674-695 liu, j.z., brown, r.w., & yue, g.h. (2002). a dynamical model of muscle activation, fatigue, and recovery. biophysical journal, 82(5), 2344-2359 ma, l., chablat, d., bennis, f., & zhang, w. (2009). a new simple dynamic muscle fatigue model and its validation. international journal of industrial ergonomics, 39(1), 211-220 mathiassen, s.e., & ahsberg, e. (1999). prediction of shoulder flexion endurance from personal factors. international journal of industrial ergonomics, 24(3), 315-329 ren, l., jones, r.k., & howard, d. (2007). predictive modelling of human walking over a complete gait cycle. journal of biomechanics, 40(7), 1567-1574 vollestad, n.k. (1997). measurement of human muscle fatigue. journal of neuroscience methods, 74(2), 219-227 wood, d.d. (1997). minimizing fatigue during repetitive jobs: optimal work-rest schedule. human factors 39(1), 83-101 yang, j., marler, r.t., kim, h., & arora, j.s. (2004). multi-objective optimization for upper body posture prediction. 10th aiaa/issmo multidisciplinary analysis and optimization conference zhang, x. & chaffin, d. (2000). a three-dimensional dynamic posture prediction model for simulating in-vehicle seated reaching movements: development and validation ergonomics, 43(9), 1314-1330

0911.1686

the globular cluster ngc 5286. ii. variable stars

we present the results of a search for variable stars in the globular cluster ngc 5286, which has recently been suggested to be associated with the canis major dwarf spheroidal galaxy. 57 variable stars were detected, only 19 of which had previously been known. among our detections one finds 52 rr lyrae (22 rrc and 30 rrab), 4 lpv's, and 1 type ii cepheid of the bl herculis type. periods are derived for all of the rr lyrae as well as the cepheid, and bv light curves are provided for all the variables. the mean period of the rrab variables is <pab> = 0.656 days, and the number fraction of rrc stars is n(c)/n(rr) = 0.42, both consistent with an oosterhoff ii (ooii) type -- thus making ngc 5286 one of the most metal-rich ([fe/h] = -1.67; harris 1996) ooii globulars known to date. the minimum period of the \rrab's, namely pab,min = 0.513 d, while still consistent with an ooii classification, falls towards the short end of the observed pab,min distribution for ooii globular clusters. as was recently found in the case of the prototypical ooii globular cluster m15 (ngc 7078), the distribution of stars in the bailey diagram does not strictly conform to the previously reported locus for ooii stars. we provide fourier decomposition parameters for all of the rr lyrae stars detected in our survey, and discuss the physical parameters derived therefrom. the values derived for the rrc's are not consistent with those typically found for ooii clusters, which may be due to the cluster's relatively high metallicity -- the latter being confirmed by our fourier analysis of the ab-type rr lyrae light curves. we derive for the cluster a revised distance modulus of (m-m)v = 16.04 mag. (abridged)

introduction ngc 5286 (c1343-511) is a fairly bright (mv = −8.26) and dense globular cluster (gc), with a central luminosity den- sity ρ0 ≈14,800l⊙/pc3 – which is more than a factor of six higher than in the case of ω centauri (ngc 5139), according to the entries in the harris (1996) catalog. in zorotovic et al. (2009, hereafter paper i) we presented a color-magnitude dia- gram (cmd) study of the cluster that reveals an unusual hor- izontal branch (hb) morphology in that it does not contain a prominent red hb component, contrary to what is normally found in gcs with comparable metallicity ([fe/h] = −1.67; harris 1996), such as m3 (ngc 5272) or m5 (ngc 5904). as a matter of fact, ngc 5286 contains blue hb stars reaching down all the way to at least the main sequence turnoff level in v. yet, unlike most blue hb gcs, ngc 5286 is known to contain a sizeable population of rr lyrae variable stars, ∗based on observations obtained in chile with the 1.3m warsaw telescope at the las campanas observatory, and the soar 4.1m telescope. 1 departamento de astronomía y astrofísica, pontificia universidad católica de chile, av. vicuña mackena 4860, 782-0436 macul, santiago, chile; e-mail: mzorotov, [email protected] 2 european southern observatory, alonso de cordova 3107, santiago, chile 3 john simon guggenheim memorial foundation fellow 4 on sabbatical leave at michigan state university, department of physics and astronomy, east lansing, mi 48824 5 department of physics and astronomy, michigan state university, east lansing, mi 48824 6 department of physics and astronomy, university of wisconsin, oshkosh, wi 54901 with at least 15 such variables being known in the field of the cluster (clement et al. 2001). in this sense, ngc 5286 resem- bles the case of m62 (ngc 6266; contreras et al. 2005), thus possibly being yet another member of a new group of gcs with hb types intermediate between m13 (ngc 6205)-like (a very blue hb with relatively few rr lyrae variables) and that of the oosterhoff i (oo i) cluster m3 (a redder hb, with a well-populated instability strip). ngc 5286 thus constitutes an example of the "missing link" between m3- and m13-like gcs (caloi, castellani, & piccolo 1987). previous surveys for variable stars in ngc 5286 (e.g., liller & lichten 1978; gerashchenko et al. 1997) have turned up relatively large numbers of rr lyrae stars. however, such studies were carried out either by photographic meth- ods, used comparatively few observations, or utilized reduc- tion methods that have subsequently been superseded by im- proved techniques, including robust multiple-frame photome- try (e.g., allframe; stetson 1994) and image subtraction (e.g., isis; alard 2000). this, together with the large central surface brightness of the cluster, strongly suggests that a large population of variable stars remains unknown in ngc 5286, especially towards its crowded inner regions. in addition, for the known or suspected variables, it should be possible to ob- tain light curves of much superior quality to those available, thus leading to better defined periods, amplitudes, and fourier decomposition parameters. indeed, to our knowledge, no modern variability study has ever been carried out for this cluster. a study of its variable star population appears especially interesting in view of its suggested association with the canis major dwarf spheroidal 2 m. zorotovic et al. galaxy (crane et al. 2003; forbes, strader, & brodie 2004), and the constraints that the ancient rr lyrae variable stars are able to pose on the early formation history of galax- ies (e.g., catelan 2004b, 2007, 2009; kinman, saha, & pier 2004; mateu et al. 2009). therefore, the time seems ripe for a reassessment of the variable star content of ngc 5286 – and this is precisely the main subject of the present paper. in §2, we describe the variable stars search techniques and the conversion from isis relative fluxes to standard magni- tudes. in §3, we show the results of our variability search, giv- ing the positions, periods, amplitudes, magnitudes, and colors for the detected variables. we show the positions of the vari- ables in the cluster cmd in §4. in §5, we provide the results of a fourier decomposition of the rr lyrae light curves, ob- taining several useful physical parameters. we analyze the cluster's oosterhoff type in §6, whereas §7 is dedicated to the type ii cepheid that we found in ngc 5286. §8 summarizes the main results of our investigation. all of the derived light curves are provided in an appendix. 2. observations and data reduction the images used in this paper are the same as described in paper i, constituting a set of 128 frames in v and 133 in b, acquired with the 1.3m warsaw university telescope at las campanas observatory, chile, in the course of a one-week run in april 2003. further details can be found in paper i. in addition, a few images were taken in feb. 2008 using the 4.1m southern astrophysical research (soar) telescope, located in cerro pachón, chile, to further check the positions of the variables in the crowded regions around the cluster center. the variable stars search was made using the image sub- traction package isis v2.2 (alard 2000). in order to convert the isis differential fluxes to standard magnitudes, we used daophot ii/allframe (stetson 1987, 1994) to obtain instrumental magnitudes for each of the variables in the b and v reference images of the isis reductions. first we obtained the flux of the variable star in the reference image, given by fref = 10 " c0−mref 2.5 " , (1) where mref is the instrumental magnitude of the star in the ref- erence image and c0 is a constant which depends on the pho- tometric reduction package (for daophot ii/allframe it is c0 = 25). then we derived instrumental magnitudes for each epoch from the differential fluxes ∆fi = fref −fi given by isis using the equation mi = c0 −2.5log(fref −∆fi). (2) finally, the equations to obtain the calibrated magnitudes (mi) from the instrumental magnitudes are of the following form: mi = mi +mstd −mref, (3) where mstd is the calibrated magnitude of the star in the ref- erence image (we used the standard magnitude data from pa- per i). 3. variable stars in our variability search, we found 57 variable stars: 52 rr lyrae (22 rrc, 30 rrab), 4 lpv's, and 1 type ii cepheid (more specifically, a bl herculis star). we identified 19 of the 24 previously catalogued variables, and discovered 38 new variables. a finding chart is provided in figure 1. of the 16 previously catalogued variables with known periods (v1-v16) we were able to find 15. v16 is the only one not present in our data, because it is not in the chip of the ccd that we have analyzed for variability. the other 8 pre- viously catalogued variables (v17-v24) were suggested by gerashchenko et al. (1997) based just on their position on the cmd. we found that only 4 of these stars (v17, v18, v20, and v21) are real variables in our survey. we do not detect any variable sources at the coordinates that they provide for the remaining 4 candidate variables in their study (v19, v22, v23 and v24). the recent images taken with a better spa- tial resolution at the 4.1m soar telescope reveal that v19 is very close to two other stars, and probably is not resolved in the images used by gerashchenko et al. (1997). v22 and v23 are close to the instability strip but they still fall in the blue part of the hb, so they are not variable stars. v24 is in the instability strip but very close to the blue part of the hb. it is possible that this star belongs to the blue hb and is contami- nated by a redder star. 3.1. periods and light curves periods were determined using the phase dispersion mini- mization (pdm; stellingwerf 1978) program in iraf. peri- ods, along with the coordinates and several important photo- metric parameters, are provided in table 1. in this table, col- umn 1 indicates the star's name. columns 2 and 3 provide the right ascension and declination (j2000 epoch), respectively, whereas column 4 shows our derived period. columns 5 and 6 list the derived amplitudes in the b and v bands, respectively, whereas columns 7 and 8 show the magnitude-weighted mean b and v magnitudes, corrected for differential reddening (see paper i for details). the corresponding intensity-mean av- erages are provided in columns 9 and 10 (also corrected for differential reddening). the average b−v color in magni- tude units and the intensity-mean color ⟨b⟩−⟨v⟩are given in columns 11 and 12, respectively, whereas column 13 lists the b−v color corresponding to the equivalent static star. finally, the last column indicates the star's variability type. to derive the color of the equivalent static star (i.e., the color the star would have if it were not pulsating), we first de- rived the magnitude-weighted mean color, and then applied an amplitude-dependent correction by interpolating on table 4 from bono, caputo, & stellingwerf (1995). we calculate the hb level, vhb = 16.63 ±0.04, as the aver- age ⟨v⟩magnitude of all the rr lyrae detected. magnitude data as a function of julian date and phase for the variable stars detected in our study are given in table 2. in this table, column 1 indicates the star's name, following the clement et al. (2001) designation (when available). column 2 indicates the filter used. column 3 provides the julian date of the observation, whereas column 4 shows the phase according to our derived period (from table 1). columns 5 and 6 list the observed magnitude in the corresponding filter and the asso- ciated error, respectively. light curves based on our derived periods (when available) are shown in the appendix. 4. color-magnitude diagram figure 2 shows the variable stars in the ngc 5286 cmd, decontaminated from field stars as described in paper i. we can see that all rr lyrae stars fall around the hb region, whereas the type ii cepheid is brighter than the hb. the four detected lpvs all fall close to the top of the rgb. these trends are precisely as expected if all the detected variables ngc 5286. ii. variable stars 3 table 1 photometric parameters for ngc 5286 variables id ra (j2000) dec (j2000) p ab av (b)mag (v)mag ⟨b⟩ ⟨v⟩ (b−v)mag ⟨b⟩−⟨v⟩ (b−v)st comments (h:m:s) (deg:m:s) (days) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) v01........... 13 : 46 : 21.5 −51 : 20 : 03.8 0.635 1.26 0.99 17.435 16.743 17.539 16.818 0.692 0.721 0.680 rrab v02........... 13 : 46 : 35.1 −51 : 23 : 14.8 0.611 0.82 0.71 17.517 17.002 17.578 17.047 0.515 0.530 0.507 rrab v03........... 13 : 46 : 54.1 −51 : 23 : 08.9 0.685 0.92 0.73 17.216 16.697 17.299 16.744 0.520 0.555 0.510 rrab v04........... 13 : 46 : 19.2 −51 : 23 : 47.2 0.352 0.60 0.46 17.067 16.639 17.114 16.668 0.428 0.446 0.423 rrc v05........... 13 : 46 : 33.4 −51 : 22 : 03.0 0.5873 ... 1.31 ... 17.383 ... 17.518 ... ... ... rrab v06........... 13 : 46 : 32.9 −51 : 23 : 04.3 0.646 1.38 1.08 16.850 16.363 16.990 16.485 0.487 0.504 0.476 rrab v07........... 13 : 46 : 29.4 −51 : 23 : 38.4 0.512 ... ... 17.08 : 16.58 : .... .... 0.50 : .... .... rrab v08...... 13 : 46 : 28.5 −51 : 23 : 10.2 2.33 1.24 1.15 15.569 15.099 15.730 15.259 0.471 0.471 ... bl her v09........... 13 : 46 : 39.0 −51 : 22 : 00.0 0.3003 0.63 0.48 17.266 16.873 17.314 16.904 0.393 0.411 0.387 rrc v10......... 13 : 46 : 24.0 −51 : 22 : 46.4 0.569 1.29 1.18 17.529 17.141 17.635 17.241 0.388 0.394 0.376 rrab v11......... 13 : 46 : 26.2 −51 : 23 : 35.7 0.652 0.94 0.69 17.120 16.530 17.194 16.519 0.590 0.676 0.580 rrab v12......... 13 : 46 : 09.1 −51 : 22 : 38.9 0.356 0.65 0.48 17.305 16.770 17.352 16.803 0.535 0.550 0.528 rrc v13......... 13 : 46 : 33.4 −51 : 23 : 33.8 0.294 0.65 0.49 16.871 16.518 16.919 16.548 0.353 0.371 0.346 rrc v14......... 13 : 46 : 23.2 −51 : 23 : 38.9 0.415 0.60 0.47 16.994 16.535 17.037 16.562 0.458 0.474 0.454 rrc v15......... 13 : 46 : 24.1 −51 : 22 : 56.8 0.585 1.78 1.19 17.539 16.825 17.768 16.919 0.713 0.849 0.689 rrab v17......... 13 : 46 : 34.6 −51 : 23 : 29.0 0.733 0.83 0.23 17.172 16.581 17.182 16.584 0.591 0.598 0.582 rrab v18......... 13 : 46 : 33.9 −51 : 23 : 16.0 0.781 0.20 ... 17.265 ... 17.280 ... ... ... ... rrab v20......... 13 : 46 : 25.2 −51 : 21 : 38.6 0.319 0.39 0.35 17.077 16.614 17.095 16.628 0.463 0.467 0.467 rrc v21......... 13 : 46 : 25.8 −51 : 24 : 02.7 0.646 0.89 0.73 17.176 16.673 17.246 16.718 0.503 0.528 0.493 rrab nv1........ 13 : 46 : 27.4 −51 : 25 : 46.1 0.366 0.54 0.43 17.156 16.695 17.199 16.722 0.461 0.476 0.459 rrc nv2........ 13 : 46 : 17.7 −51 : 23 : 56.8 0.354 0.60 0.50 17.284 16.758 17.336 16.789 0.526 0.547 0.521 rrc nv3........ 13 : 46 : 30.5 −51 : 23 : 32.5 0.755 0.55 0.30 17.292 16.501 17.321 16.503 0.791 0.819 0.788 rrab nv4........ 13 : 46 : 15.9 −51 : 23 : 31.7 0.786 0.28 0.21 17.099 16.450 17.108 16.455 0.649 0.653 0.660 rrab nv5........ 13 : 46 : 17.6 −51 : 20 : 33.0 0.357 0.60 0.45 17.166 16.659 17.207 16.685 0.507 0.522 0.502 rrc nv6........ 13 : 46 : 26.4 −51 : 23 : 27.6 0.566 1.45 1.12 16.888 16.511 17.028 16.604 0.377 0.425 0.366 rrab nv7........ 13 : 46 : 25.8 −51 : 21 : 34.6 0.339 0.48 0.41 16.986 16.560 17.017 16.585 0.426 0.431 0.427 rrc nv8........ 13 : 46 : 26.2 −51 : 23 : 12.4 0.80 0.30 0.30 17.103 16.463 17.113 16.473 0.640 0.640 0.649 rrab nv9........ 13 : 46 : 24.6 −51 : 23 : 11.6 0.745 ... ... 17.24 : 16.58 : ... ... 0.66 : ... ... rrab nv10...... 13 : 46 : 25.1 −51 : 23 : 02.2 0.339 0.68 0.54 16.922 16.436 16.984 16.454 0.485 0.529 0.477 rrc nv11...... 13 : 46 : 26.0 −51 : 23 : 02.7 0.536 0.85 0.71 16.813 16.347 16.889 16.402 0.466 0.487 0.457 rrab nv12...... 13 : 46 : 26.7 −51 : 23 : 02.5 0.905 0.52 0.41 16.876 16.251 16.906 16.269 0.625 0.637 0.622 rrab nv13...... 13 : 46 : 27.3 −51 : 22 : 58.2 0.583 0.95 1.35 16.514 16.083 16.585 16.215 0.431 0.370 0.421 rrab nv14...... 13 : 46 : 29.2 −51 : 22 : 07.8 0.284 0.64 0.44 17.319 16.803 17.366 16.826 0.516 0.540 0.509 rrc nv15...... 13 : 46 : 25.5 −51 : 22 : 53.1 0.742 0.34 0.52 17.128 16.477 17.138 16.505 0.651 0.633 0.657 rrab nv16...... 13 : 46 : 27.0 −51 : 22 : 50.7 0.366 0.54 0.32 17.125 16.618 17.158 16.629 0.507 0.529 0.505 rrc nv17...... 13 : 46 : 26.6 −51 : 22 : 49.3 0.322 0.49 0.50 16.923 16.490 16.946 16.431 0.433 0.415 0.433 rrc nv18...... 13 : 46 : 29.1 −51 : 22 : 46.9 0.362 0.57 0.44 17.257 16.410 17.295 16.431 0.846 0.864 0.843 rrc nv19...... 13 : 46 : 26.8 −51 : 22 : 44.2 0.658 0.94 1.70 16.902 16.624 16.924 16.706 0.278 0.218 0.268 rrab nv20...... 13 : 46 : 22.5 −51 : 22 : 43.2 0.3103 1.90 0.65 17.555 16.753 18.023 16.791 0.802 1.232 0.736 rrc nv21...... 13 : 46 : 29.4 −51 : 22 : 41.0 0.570 0.95 1.06 16.830 16.583 16.902 16.669 0.247 0.233 0.237 rrab nv22...... 13 : 46 : 27.5 −51 : 22 : 39.9 0.68 1.15 1.20 16.829 16.273 17.022 16.374 0.556 0.648 0.544 rrab nv23...... 13 : 46 : 26.7 −51 : 22 : 16.1 0.598 1.14 1.30 17.090 16.609 17.182 16.729 0.481 0.453 0.469 rrab nv24...... 13 : 46 : 29.1 −51 : 22 : 39.1 0.60 0.75 0.70 16.927 16.346 16.965 16.382 0.581 0.583 0.574 rrab nv25...... 13 : 46 : 24.2 −51 : 22 : 17.2 0.550 1.44 1.20 17.328 16.724 17.048 16.664 0.605 0.384 0.593 rrab nv26...... 13 : 46 : 26.0 −51 : 22 : 36.5 0.364 0.46 0.36 17.011 16.523 17.040 16.536 0.489 0.505 0.490 rrc nv27...... 13 : 46 : 30.1 −51 : 22 : 22.0 0.706 1.17 0.80 16.952 16.599 17.489 16.811 0.490 0.530 0.340 rrab nv28...... 13 : 46 : 29.8 −51 : 22 : 35.4 0.540 0.97 0.70 16.880 16.390 16.954 16.424 0.490 0.530 0.479 rrab nv29...... 13 : 46 : 28.8 −51 : 22 : 33.8 0.301 0.97 0.63 17.458 16.963 17.565 17.010 0.495 0.555 0.473 rrc nv30...... 13 : 46 : 29.1 −51 : 22 : 24.1 0.72 0.65 0.55 17.251 16.641 17.292 16.672 0.610 0.620 0.605 rrab nv31...... 13 : 46 : 25.5 −51 : 22 : 31.6 0.289 0.41 0.40 17.163 16.519 17.189 16.535 0.645 0.654 0.648 rrc nv32...... 13 : 46 : 25.2 −51 : 22 : 30.0 0.283 0.34 0.24 16.804 16.339 16.817 16.344 0.465 0.473 0.471 rrc nv33...... 13 : 46 : 27.9 −51 : 22 : 26.6 0.294 0.34 0.50 16.801 16.457 16.815 16.483 0.345 0.333 0.351 rrc nv34...... 13 : 46 : 26.7 −51 : 22 : 24.9 0.367 0.42 0.48 16.569 16.073 16.591 16.096 0.496 0.495 0.499 rrc nv35...... 13 : 46 : 24.9 −51 : 23 : 03.8 ... ... ... 14.70 : 13.38 : ... ... 1.32 : ... ... lpv nv36...... 13 : 46 : 29.1 −51 : 22 : 58.4 ... ... ... 15.30 : 13.40 : ... ... 1.90 : ... ... lpv nv37...... 13 : 46 : 28.8 −51 : 22 : 12.9 ... ... ... 15.70 : 14.60 : ... ... 1.10 : ... ... lpv nv38...... 13 : 46 : 28.8 −51 : 20 : 37.0 ... ... ... 15.30 : 13.74 : ... ... 1.56 : ... ... lpv are cluster members. note, however, that the cmd positions of the 4 lpvs are not precisely defined, since we lack ade- quate phase coverage for these stars. figure 3 is a magnified cmd showing only the hb region. to assess the effects of crowding, we use different symbol sizes for the variables in different radial annuli from the clus- ter center: small sizes for stars in the innermost cluster regions (r ≤0.29′) which are badly affected by crowding, medium sizes for stars with 0.29′ < r ≤0.58′, and large sizes for stars in the outermost cluster regions (r > 0.58′). as expected, the variables in the innermost cluster region present more scat- ter. apart from this, in general the detected rr lyrae fall inside a reasonably well-defined instability strip (is). how- ever, we can see that there is not a clear-cut separation in color between rrc's and rrab's. although the rrc's tend to be found preferentially towards the blue side of the is, as ex- pected, the rrab's are more homogeneously distributed. this can be again a scatter effect, because the rrab stars that are in the less crowded regions of the cluster (large and medium size circles) are more concentrated at the red part. 4.1. notes on individual stars v5: this star was only detected in our v images, so that we were unable to determine its color. it remains unclear to us 4 m. zorotovic et al. fig. 1.- finding chart for the ngc 5286 variables. top: the outer variables in the v image obtained with the 0.9m ctio telescope (note scale at the bottom right of the image). bottom left: the same image as before, but zoomed in slightly. bottom right: finding chart for the variables located closest to the cluster center. why isis was unable to detect this variable in the b data – note, from figure 1, that it is not located in an especially crowded region of the cluster. v7: our derived period, 0.512 d, is slightly longer than the previously reported period of 0.50667 d in liller & lichten (1978). as a matter of fact, v7 remains one of the shortest- period rrab stars among all known ooii gcs. because the period is so close to half a day, there is a consider- able gap in the phased light curve for v7 (as was also the case, though to a somewhat lesser degree, for the light curve of liller & lichten). we carefully checked the pdm peri- odogram of the star in search of acceptable longer periods, but could find none that fit our data, nor could we find a pe- riod that reduced significantly the spread seen in the phased light curve close to minimum light (which is particularly ob- vious in the v-band light curve). the liller & lichten light curve also shows substantial scatter close to minimum light, although in their case much of the scatter is clearly due to photometric error. v8: liller & lichten (1978) found a period of 0.7 d for this ngc 5286. ii. variable stars 5 table 2 photometry of the variable stars name filter jd phase mag e_mag (d) (mag) (mag) v01 v 2,452,736.54861 0.0000 16.2351 0.0031 v01 v 2,452,736.55367 0.0080 16.2779 0.0042 v01 v 2,452,736.55977 0.0176 16.2787 0.0051 v01 v 2,452,736.57490 0.0414 16.3361 0.0054 v01 v 2,452,736.58749 0.0612 16.3878 0.0059 v01 v 2,452,736.59664 0.0756 16.3996 0.0059 note. - this table is published in its entirety in the electronic edition of the astronomical journal. a portion is shown here for guidance regarding its form and content. fig. 2.- cmd for ngc 5286 (from paper i) including the variables ana- lyzed in this paper. crosses indicate the lpv's, circles the rr lyrae, and the open square the type ii cepheid. the positions of the variables are based on the intensity-mean magnitudes ⟨v⟩and on the colors of the equivalent static stars (b−v)st, as given in table 1. star and suggested that it is an rr lyrae-type variable. in our analysis we found an alias at 0.7 d for the period, but the best fit is obtained with a period of 2.33 d, which corresponds to a type ii cepheid of the bl her type. v18: as for v5, we only detected this star in one of the filters. in this case isis only detected it in our b images, so that we were unable to determine its color and also to perform fourier decomposition. v7 and nv9: we do not see the minimum and maximum, respectively, for these two variables. for that reason, we could not perform a fourier analysis, and the mean v and b values provided in table 1 are just approximate. 5. fourier decomposition light curves for rr lyrae variables were analyzed by fourier decomposition using the same equations as in corwin et al. (2003), namely mag = a0 + n x j=1 a j sin(2π jt/p+φj) (4) (for rrab variables) and fig. 3.- as in figure 2, but focusing on the hb region of the cmd. here we use different symbol sizes for the variables in different radial annuli from the cluster center: small symbols are related to stars inside the core radius (r ≤0.29′); medium-sized symbols refer to stars with 0.29′ < r ≤0.58′; and the larger symbols refer to stars in the outermost cluster regions (r > 0.58′). table 3 fourier coefficients: rrc stars id a21 a31 a41 φ21 φ31 φ41 v4 0.113 0.059 0.037 4.503 3.667 2.103 v9 0.226 0.056 0.053 4.735 2.905 1.283 v12 0.159 0.096 0.032 5.064 3.454 2.635 v13 0.194 0.063 0.058 4.748 2.451 1.248 v14 0.072 0.064 0.062 5.411 4.522 3.401 v20 0.059 0.016 0.026 4.723 3.832 3.128 nv1 0.117 0.091 0.053 4.768 3.635 3.295 nv2 0.123 0.051 0.036 4.621 3.406 2.433 nv5 0.083 0.059 0.043 4.766 4.134 2.693 nv7 0.151 :: 0.064 :: 0.012 :: 5.443 :: 4.886 :: 5.665 :: nv10 0.144 0.056 0.020 4.936 3.237 2.660 nv14 0.193 0.053 0.044 4.645 2.881 1.954 nv16 0.069 :: 0.094 :: 0.058 :: 4.448 :: 4.293 :: 2.468 :: nv17 0.115 :: 0.052 :: 0.085 :: 5.515 :: 3.764 :: 2.698 :: nv18 0.087 : 0.013 : 0.041 : 4.484 : 5.416 : 3.313 : nv20 0.164 0.098 0.072 4.844 3.104 1.823 nv26 0.068 : 0.072 : 0.035 : 5.041 : 4.154 : 3.058 : nv29 0.233 : 0.091 : 0.029 : 4.929 : 3.344 : 1.328 : nv31 0.152 : 0.002 : 0.046 : 4.557 : 1.086 : 0.590 : nv32 0.074 : 0.017 : 0.050 : 5.495 : 3.484 : 3.811 : nv33 0.168 : 0.075 : 0.073 : 4.704 : 2.230 : 0.747 : nv34 0.043 : 0.103 : 0.017 : 5.336 : 3.616 : 2.641 : mag = a0 + n x j=1 a j cos(2π jt/p+φj) (5) (for rrc variables), where again n = 10 was usually adopted. 5.1. rrc variables amplitude ratios a j1 ≡a j/a1 and phase differences φj1 ≡ φj −jφ1 for the lower-order terms are provided in table 3. in this table, a colon symbol (":") indicates an uncertain value, whereas a double colon ("::") indicates a very uncertain value. simon & clement (1993) used light curves of rrc vari- ables, as provided by their hydrodynamical pulsation mod- els, to derive equations to calculate mass, luminosity, effective temperature, and a "helium parameter" for rrc variables – all 6 m. zorotovic et al. table 4 fourier-based physical parameters: rrc stars id m/m⊙ log(l/l⊙) te(k) y [fe/h] ⟨mv ⟩ v4 0.563 1.725 7264 0.271 −1.690 0.747 v9 0.629 1.698 7358 0.275 −1.891 0.811 v12 0.598 1.744 7229 0.264 −1.681 0.737 v13 0.625 1.619 7568 0.298 −1.898 0.822 v14 0.495 1.751 7167 0.268 −1.069 0.711 v20 0.513 1.672 7382 0.289 −1.837 0.763 nv1 0.579 1.745 7218 0.265 −1.605 0.745 nv2 0.604 1.744 7231 0.264 −1.673 0.742 nv5 0.504 1.705 7290 0.280 −1.656 0.748 nv10 0.617 1.734 7259 0.266 −1.766 0.764 nv14 0.614 1.673 7418 0.282 −1.896 0.816 nv20 0.609 1.701 7342 0.275 −1.887 0.830 mean 0.601± 0.049 1.715± 0.040 7276± 110 0.273± 0.011 −1.713± 0.230 0.770± 0.039 as a function of the fourier phase difference φ31 ≡φ3 −3φ1 and the period. however, catelan (2004b, see his §4) pointed out that the simon & clement set of equations cannot, in their current form, provide physically correct values for luminosi- ties and masses, since they violate the constraints imposed by the ritter (period-mean density) relation. we still provide the derived values for the ngc 5286 rrc stars in this paper though, chiefly for comparison with similar work for other gcs. accordingly, we use simon & clement's (1993) equa- tions (2), (3), (6), and (7) to compute m/m⊙, log(l/l⊙), te, and y, respectively, for 12 of our rrc variables (i.e., those with the best defined fourier coefficients). we also use equa- tion (3) in morgan, wahl, & wieckhorst (2007) to compute [fe/h], and equation (10) in kovács (1998) to compute mv. the results are given in table 4. the unweighted mean values and corresponding standard errors of the derived mean mass, log luminosity (in so- lar units), effective temperature, "helium parameter", metal- licity (in the zinn & west 1984 scale), and mean absolute magnitude in v are (0.601 ± 0.049)m⊙, (1.715 ± 0.040), (7276±110) k, (0.273±0.011), (−1.71±0.23), and (0.770± 0.039) mag, respectively. 5.2. rrab variables amplitude ratios a j1 and phase differences φj1 for the lower-order terms are provided, in the case of the rrab's, in table 5. we also give the jurcsik-kovács dm value (jurcsik & kovács 1996, computed on the basis of their eq. [6] and table 6), which is intended to differentiate rrab stars with "regular" light curves from those with "anomalous" light curves (e.g., presenting the blazhko effect – but see also cacciari et al. 2005 for a critical discussion of dm as an in- dicator of the occurrence of the blazhko phenomenon). as before, a colon symbol (":") indicates an uncertain value, whereas a double colon ("::") indicates a very uncertain value. jurcsik & kovács (1996), kovács & jurcsik (1996, 1997), jurcsik (1998), kovács & kanbur (1998), and kovács & walker (1999, 2001) derived empirical expressions that relate metallicity, absolute magnitude, and temperature with the fourier parameters of rrab stars, in the case of sufficiently "regular" light curves (dm < 3). we accordingly use equations (1), (2), (5), and (11) in jurcsik (1998) to compute [fe/h], mv , v −k, and logte⟨v−k⟩, respectively, for the 12 rrab variables in our sample with dm < 3. the color indices b−vand v −i come from equations (6) and (9) of kovács & walker (2001); we then use equation (12) of kovács & walker (1999), assuming a mass of 0.7m⊙, to derive temperature values from equation (11) (for b−v) and equation (12) (for v −i) in kovács & walker (2001). these results are given in table 6. fourier parameters suggest a metallicity of [fe/h] = −1.52 ± 0.21 for ngc 5286 in the jurcsik (1995) scale; this corresponds to a value of [fe/h] = −1.68 in the zinn & west (1984) scale. this is consistent with the value derived using the rrc variables and is in excellent agreement with harris (1996, [fe/h] = −1.67), and also with the value obtained in paper i ([fe/h] = −1.70 ± 0.05, again in the zinn & west scale) on the basis of several different photometric parame- ters describing the shape and position of the cluster's red giant branch in the v, b−vdiagram. we find a mean absolute magnitude of ⟨mv ⟩= 0.717 ± 0.038 mag for the rrab stars. for the same set of 12 rrab stars used to derive this value, we also find ⟨v⟩= 16.64 ± 0.08 mag. this implies a distance modulus of (m −m)v = 15.92 ± 0.12 for ngc 5286, which is in excellent agree- ment with the value provided in the harris (1996) catalog, namely, (m −m)v = 15.95 mag. if one adopts instead for the hb an average absolute magnitude of mv = 0.60 mag at the ngc 5286 metallicity, as implied by equation (4a) in catelan & cortés (2008) – which is based on a calibration of the rr lyrae distance scale that uses the latest hippar- cos and hubble space telescope trigonometric parallaxes for rr lyr, and takes explicitly into account the evolutionary sta- tus of this star – one finds for the cluster a distance modu- lus of (m −m)v = 16.04 mag for ngc 5286. we caution the reader that the ngc 5286 rr lyrae stars could in principle be somewhat overluminous for the cluster's metallicity (in view of the cluster's predominantly blue hb), in which case the correct distance modulus could be even larger, by an amount that could be of the order of ∼0.1 mag (e.g., lee & carney 1999; demarque et al. 2000). finally, and as also pointed out by cacciari et al. (2005), we also warn the reader that intrin- sic colors and temperatures estimated from fourier decompo- sition are not particularly reliable, and should accordingly be used with due caution. the reader is also referred to kovács (1998) and catelan (2004b) for caveats regarding the validity of the results obtained based on the simon & clement (1993) relations for rrc stars. 6. the oosterhoff type of ngc 5286 figure 4 shows a histogram with our derived periods in ngc 5286. the bottom panel is similar to the upper panel, but with the rrc periods "fundamentalized" using the equa- tion logpf = logpc + 0.128 (e.g., catelan 2009, and refer- ences therein). note the lack of a sharply peaked distribu- ngc 5286. ii. variable stars 7 table 5 fourier coefficients: rrab stars id a21 a31 a41 φ21 φ31 φ41 dm v1 0.524 0.359 0.246 2.406 5.113 1.638 0.751 v2 0.379 0.257 0.108 2.293 4.754 1.181 2.534 v3 0.412 0.258 0.126 2.623 5.379 2.263 1.211 v5 0.472 0.303 0.231 2.329 5.040 1.459 5.957 v6 0.563 0.330 0.199 2.419 5.198 1.683 14.276 v7 ... ... ... ... ... ... ... v10 0.488 0.325 0.193 2.330 4.948 1.148 2.117 v11 0.491 0.226 0.176 2.484 5.228 1.844 16.137 v15 0.448 0.362 0.222 2.363 5.235 1.517 4.592 v17 0.267 0.078 0.019 2.653 5.840 3.345 1.502 v21 0.494 0.315 0.142 2.585 5.403 2.227 1.956 nv3 0.346 0.204 0.023 2.647 5.167 1.894 6.447 nv4 0.246 : 0.087 : 0.037 : 2.712 : 6.177 : 3.153 : 3.197 : nv6 0.458 0.352 0.237 2.260 4.839 1.164 2.117 nv8 0.330 0.135 0.021 2.775 5.838 3.085 0.379 nv9 ... ... ... ... ... ... ... nv11 0.334 0.210 0.107 2.176 4.616 1.198 3.550 nv12 0.295 : 0.080 : 0.106 : 3.511 : 0.525 : 4.805 : 3.143 : nv13 0.517 0.353 0.217 2.438 5.241 1.588 0.324 nv15 0.439 0.194 0.078 2.619 5.645 2.320 4.390 nv19 0.475 0.231 0.000 2.450 5.234 1.749 5.938 nv21 0.487 0.341 0.245 2.357 4.812 1.232 0.313 nv22 0.566 0.339 0.217 2.670 5.588 2.607 10.859 nv23 0.463 0.321 0.221 2.208 5.015 1.187 2.600 nv24 0.631 0.282 0.136 1.964 4.458 0.793 4.763 nv25 0.448 0.344 0.241 2.277 4.859 1.108 4.151 nv27 0.481 0.296 0.156 2.627 5.540 2.191 0.686 nv28 0.497 0.277 0.158 2.278 4.901 0.799 3.703 nv30 0.444 0.262 0.119 2.474 5.587 2.174 3.312 table 6 fourier-based physical parameters: rrab stars id [fe/h] ⟨mv ⟩ ⟨v −k⟩ logte⟨v−k⟩ ⟨b −v⟩ logte⟨b−v⟩ ⟨v −i⟩ logte⟨v−i⟩ v1 −1.587 0.707 1.200 3.801 0.355 3.804 0.450 3.837 v2 −1.938 0.728 1.239 3.798 0.355 3.802 0.443 3.843 v3 −1.501 0.692 1.256 3.794 0.366 3.801 0.475 3.829 v10 −1.452 0.747 1.105 3.811 0.327 3.815 0.398 3.848 v17 −1.139 0.752 1.321 3.786 0.406 3.790 0.534 3.810 v21 −1.529 0.687 1.267 3.793 0.376 3.798 0.489 3.826 nv6 −1.584 0.752 1.135 3.808 0.335 3.812 0.407 3.848 nv8 −1.509 0.645 1.385 3.780 0.409 3.785 0.551 3.810 nv13 −1.521 0.662 1.292 3.790 0.391 3.794 0.518 3.818 nv21 −1.140 0.732 1.069 3.814 0.322 3.819 0.396 3.845 nv23 −1.642 0.752 1.156 3.806 0.339 3.809 0.413 3.847 nv27 −1.397 0.677 1.239 3.796 0.370 3.800 0.484 3.825 mean −1.515± 0.213 0.717± 0.038 1.239± 0.093 3.797± 0.010 0.361± 0.029 3.802± 0.010 0.462± 0.054 3.833± 0.015 tion, contrary to what is seen in several gcs of both oost- erhoff types, but most notably in m3 (e.g., catelan 2004a; d'antona & caloi 2008, and references therein). in order to assign an oosterhoff type to ngc 5286, we must compare its rr lyrae properties with those found in other oosterhoff (1939, 1944) type i and ii gcs. in this sense, clement et al. (2001) found the mean rrab and rrc (plus rrd) periods for rr lyrae stars in galactic gcs to be 0.559 days and 0.326 days, respectively, for ooi clusters, and 0.659 days and 0.368 days, respectively, in the case of ooii clusters. in addition, catelan et al. (2009, in preparation) have recently shown that the minimum period of the ab-type pul- sators pab,min, when used in conjunction with ⟨pab⟩, provides a particularly reliable diagnostics of oosterhoff status. in this sense, the key quantities for the cluster can be sum- marized as follows: ⟨pab⟩= 0.656d, (6) ⟨pc⟩= 0.333d, (7) pab,min = 0.512d, (8) nc/nc+ab = 0.42. (9) one immediately finds that the value of ⟨pab⟩for the clus- ter points to an ooii status (see also fig. 5, which is based on the compilation presented in catelan 2009) – which is also favored by its relatively high c-type number fraction. on the other hand, the average period of the rrc's is lower than typ- ically found among ooii globulars, being more typical of ooi objects. however, as shown by catelan et al. (2009), there is a large overlap in ⟨pc⟩values between ooi and ooii globulars, thus making this quantity a poorer indicator of oosterhoff sta- tus than is often realized. the situation regarding pab,min is rather interesting, for the 8 m. zorotovic et al. fig. 4.- top: period histogram for the ngc 5286 rr lyrae stars. bottom: same as in the top, but fundamentalized the periods of the rrc's. fig. 5.- distribution of galactic gcs in the mean rrab period vs. metal- licity plane. the position of ngc 5286, as derived on the basis of our new measurements, is highlighted. value derived for ngc 5286 makes it one of the ooii clusters with the shortest pab,min values to date – though 0.512 d would still clearly be too long for an ooi cluster, which generally have pab,min < 0.5 d (catelan et al. 2009) – as opposed to ooii clusters, which typically have instead pab,min > 0.5 d. as can be seen from table 1 and figure 4 (top), after v7 (the star with p = 0.512 d), the next shortest-period star in ngc 5286 is nv11, with p = 0.536 d; this is indeed less atypical for an ooii object. the position of the cluster in a metallicity versus hb type diagram is displayed in figure 6. here hb type l ≡(b −r)/(b+v +r), where b, r, v are the numbers of blue, red, and variable (rr lyrae-type) hb stars, respec- tively; this quantity was derived for ngc 5286 in paper i. as discussed by catelan (2009), oosterhoff-intermediate gcs, such as found in several gcs associated with the dwarf satel- lite galaxies of the milky way, tend to cluster inside the (b-r)/(b+v+r) -1.0 -0.5 0.0 0.5 1.0 [fe/h] -2.5 -2.0 -1.5 -1.0 -0.5 0.0 "bulge, disk" gcs "old halo" gcs "young halo" gcs oosterhoff gap? +0.8 gyr 2.0 gyr n5286 fig. 6.- distribution of galactic gcs in the metallicity-hb type l plane. the region marked as a triangle represents the "oosterhoff gap," a seemingly forbidden region for bona-fide galactic gcs. ooi clusters tend to sit to the left of the oosterhoff gap, whereas ooii clusters are mostly found to its right. the overplotted lines are isochrones from catelan & de freitas pacheco (1993). adapted from catelan (2009). -0.6 -0.4 -0.2 0 0 0.5 1 fig. 7.- position of rr lyrae stars on the bailey (period-amplitude) dia- gram for v. filled circles show rrab's with dm < 3, open circles those with dm > 3, and crosses show the rrc's. solid lines are the typical lines for ooi clusters and dashed lines for ooii clusters, according to cacciari et al. 2005. as in figure 3, different symbol sizes are related to the radial distance from the cluster center. triangular-shaped region marked in this diagram – whereas galactic gcs somehow are not found in this same region, thus giving rise to the oosterhoff dichotomy in the galaxy. in this same plane, ooi clusters tend to fall to the left (i.e., redder hb types) of the triangular-shaped region, whereas ooii objects are more commonly found to its right. ngc 5286 falls rather close to the oosterhoff-intermediate region in this plane, but its position is indeed still consistent with ooii status (see also fig. 8 in catelan 2009). figures 7 and 8 show the positions of the rr lyrae stars on the bailey (period-amplitude) diagram for v and b mag- nitudes, respectively. circles indicate the rrab's, whereas crosses are used for the rrc's. as in figure 3, we use dif- ferent symbol sizes for the variables in different radial annuli ngc 5286. ii. variable stars 9 -0.6 -0.4 -0.2 0 0 0.5 1 1.5 fig. 8.- same as in figure 7, but for b. from the cluster center. also shown in this figure are typical lines for ooi and ooii clusters, which read as follows (see cacciari et al. 2005): aab b = −3.123 −26.331 logp−35.853 logp2, (10) aab v = −2.627 −22.046 logp−30.876 logp2, (11) for ab-type rr lyrae stars in ooi clusters. for rrab's in ooii clusters, in turn, the same lines can be used, but shifted in periods by ∆logp = +0.06.7 in the case of c-type stars, we derive reference lines on the basis of figures 2 and 4 of cacciari et al. (2005); these read as follows: ac b = −0.522 −2.290 logp, (12) ac v = −0.395 −1.764 logp, (13) for c-type rr lyrae stars in ooi clusters, and ac b = −0.039 −1.619 logp, (14) ac v = −0.244 −1.834 logp, (15) for c-type rr lyrae stars in ooii clusters (again based on presumably "evolved" rr lyrae stars in m3). this kind of diagram can be used as a diagnostic tool to investigate the oosterhoff classification of rrab stars. how- ever, the position of a star in this diagram can be strongly af- fected by the presence of the blazhko effect. in order to min- imize this problem, we make a distinction between variables with a jurcsik-kovács compatibility parameter value dm < 3 (filled circles) and dm > 3 (open circles). even considering only the rr lyrae stars with small dm and hence presum- ably "regular" light curves (according to the jurcsik-kovács criterion), we see that there is still a wide scatter among the rrab's in the bailey diagrams, with no clear-cut tendency for 7 cacciari et al. (2005) actually derived their ooii curves based on what appeared to be highly evolved stars in m3, and then verified that the same curves do provide a good description of rr lyrae stars in several bona-fide ooii gcs. stars to cluster tightly around either oosterhoff reference line, particularly in the b case. it is possible that at least some of the dispersion is caused by unidentified blends in the heavily crowded inner regions of the cluster, where most of the vari- ables studied in this paper can be found (see fig. 1). in fact, the variable stars in the innermost cluster regions (small cir- cles) show more scatter. if we only look at the variables that lie outside the core radius (the large and medium-sized cir- cles, respectively, with r > 0.29′), we can see that they cluster much more tightly around the oosterhoff ii line. it should be noted, in any case, that very recently corwin et al. (2008) have shown that the ab-type rr lyrae stars in the prototypical ooii cluster m15 (ngc 7078) similarly do not cluster around the ooii reference line derived on the basis of more metal- rich clusters, thus casting some doubt on the validity of these lines as indicators of oosterhoff status, at least at the more metal-poor end of the rr lyrae metallicity distribution. as far as the positions of the c-type rr lyrae stars in the bailey diagrams are concerned, we find that there is also a wide scatter, without any clearly defined tendency for the data to clump tightly around either of the oosterhoff reference lines – although the distribution does seem skewed towards shorter amplitudes at a given period, compared to the typical situation in ooii clusters. finally, we can also check how the average fourier-based physical parameters derived for the ngc 5286 variables rank the cluster in terms of oosterhoff status. this exercise is en- abled by a comparison with the data for several clusters of different oosterhoff types, as compiled in tables 6 and 7 of corwin et al. (2003). for the rrc's, we find that the mean masses, luminosities and temperatures are in fact more sim- ilar to those found for m3 (a prototypical ooi cluster) than they are for ooii globulars. part of the problem may be due to the fact that ngc 5286 is significantly more metal-rich than all ooii globulars used in the analysis; recall that φ31 is the only fourier parameter used in the simon & clement (1993) calibration of masses, luminosities, and temperatures, and that the impact of metallicity on the simon & clement re- lations has still not been comprehensively addressed (see §5 in clement, jankulak, & simon 1992, and also §4 in catelan 2009 for general caveats regarding the validity of those rela- tions). as a matter of fact, the recent study by morgan et al. (2007) clearly shows that, in the case of rrc variables, φ31 depends strongly on the metallicity. for the rrab's, in turn, both the derived temperatures and absolute magnitudes are fully consistent with an ooii classi- fication for the cluster. 7. the type ii cepheid we find one type ii cepheid (v8) with a period of 2.33 days and a visual amplitude of av = 1.15 mag, typical for a bl herculis star. we use equation (3) in pritzl et al. (2003) to obtain mv = −0.55 ± 0.07 mag for v8. using the intensity- weighted mean magnitude for v8 from table 1, we obtain for the cluster a distance modulus (m−m)v = 15.81 ±0.07 mag, slightly shorter than the values discussed in §5.2. however, as we can see from figure 9 in pritzl et al., a large dispersion in mv is indeed present for short-period type ii cepheids, thus possibly explaining the small discrepancy. 8. summary in this paper, we present the results of time-series photome- try for ngc 5286, a gc which has been tentatively associated with the canis major dwarf spheroidal galaxy. 38 new vari- 10 m. zorotovic et al. ables were discovered in the cluster, and 19 previously known ones were recovered in our study (including one bl her star that was previously catalogued as an rr lyrae). the popula- tion of variable stars consists of 52 rr lyrae (22 rrc and 30 rrab), 4 lpv's, and 1 type ii cepheid. from fourier decomposition of the rrab light curves, we obtained a value for the metalicity of the cluster of [fe/h] = −1.68 ± 0.21 dex in the zinn & west (1984) scale. we also derive a distance modulus of (m −m)v = 16.04 mag for ngc 5286, based on the recent rr lyrae mv −[fe/h] cali- bration of catelan & cortés (2008). using a variety of indicators, we discuss in detail the oost- erhoff type of the cluster, concluding in favor of an ooii clas- sification. the cluster's fairly high metallicity places it among the most metal-rich ooii clusters known, which may help ac- count for what appears to be a fairly unusual behavior for a cluster of this type, including relatively short values of pab,min and ⟨pc⟩, and unusual physical parameters, as derived for its c-type rr lyrae stars on the basis of fourier decomposition of their light curves. in regard to the cluster's suggested association to the ca- nis major dwarf spheroidal galaxy, we note that the metallic- ity and distance modulus derived in this work are very sim- ilar to the values previously accepted for the cluster (harris 1996), and thus the conclusions reached by previous authors (crane et al. 2003; forbes et al. 2004) regarding its possible association with this dwarf galaxy are not significantly af- fected by our new metallicity and distance estimates. in ad- dition, the position of the cluster in the hb morphology- metallicity plane is fairly similar to that found in several nearby extragalactic systems. as far as ngc 5286's rr lyrae pulsation properties are concerned, the present study shows them to be somewhat unusual compared with bona-fide galactic globular clusters, but still do not classify the clus- ter as an oosterhoff-intermediate system, as frequently found among the galaxy's dwarf satellites (e.g., catelan 2009, and references therein). it is interesting to note, in any case, that the canis major field, unlike what is found among other dwarf galaxies, appears to be basically devoid of rr lyrae stars (kinman et al. 2004; mateu et al. 2009), and also to be chiefly comprised of fairly high-metallicity ([m/h] ≥−0.7), young (t ≲10 gyr) stars (e.g., bellazzini et al. 2004). the present paper, along with paper i, show instead that ngc 5286 is an rr lyrae-rich, metal-poor globular cluster that is at least as old as the oldest globular clusters in the galactic halo. it is not immediately clear that such an object as ngc 5286 would be easily formed within a galaxy with the properties observed for the canis major main body – and this should be taken into account when investigating the physical origin and formation mechanism for the canis major overdensity and its associated tidal ring. we warmly thank i. c. leão for his help producing the finding chart, and an anonymous referee for several com- ments that helped improve the presentation of our results. mz and mc acknowledge financial support by proyecto fondecyt regular #1071002. support for mc is also pro- vided by proyecto basal pfb-06/2007, by fondap cen- tro de astrofísica 15010003, and by a john simon guggen- heim memorial foundation fellowship. has is supported by csce and nsf grant ast 0607249. references alard, c. 2000, a&as, 144, 363 bellazzini, m., ibata, r., monaco, l., martin, n., irwin, m. j., & lewis, g. f. 2004, mnras, 354, 1263 bono, g., caputo, f., & stellingwerf, r. f. 1995, apjs, 99, 263 cacciari, c., corwin, t. m., & carney, b. w. 2005, aj, 129, 267 caloi, v., castellani, v., & piccolo, f. 1987, a&as, 67, 181 catelan, m. 2004a, apj, 600, 409 catelan, m. 2004b, in variable stars in the local group, asp conf. ser., 310, ed. d. w. kurtz & k. r. pollard (san francisco: asp), 113 catelan, m. 2007, revmexaa conf. ser., 26, 93 catelan, m. 2009, ap&ss, 320, 261 catelan, m., & cortés, c. 2008, apj, 676, l135 catelan, m., & de freitas pacheco, j. a. 1993, aj, 106, 1858 clement, c. m., jankulak, m., & simon, n. r. 1992, apj, 395, 192 clement, c. m., muzzin, a., dufton, q., ponnampalam, t., wang, j., burford, j., richardson, a., rosebery, t., rowe, j., & hogg, h. s. 2001, aj, 122, 2587 contreras, r., catelan, m., smith, h. a., pritzl, b. j., & borissova, j. 2005, apj, 623, l117 corwin, t. m., catelan, m., smith, h. a., borissova, j., ferraro, f. r., & raburn, w. s 2003, aj, 125, 2543 corwin, t. m., borissova, j., stetson, p. b., catelan, m., smith, h. a., kurtev, r., & stephens, a. w. 2008, aj, 135, 1459 crane, j. d., majewski, s. r., rocha-pinto, h. j., frinchaboy, p. m., skrutskie, m. f., & law, d. r. 2003, apj, 594, 119 d'antona, f., & caloi, v. 2008, mnras, 390, 693 demarque, p., zinn, r., lee, y.-w., & yi, s. 2000, aj, 119, 1398 forbes, d. a., strader, j., & brodie, j. p. 2004, aj, 127, 3394 gerashchenko, a. n., kadla, z. i., & malakhova, yu. n. 1997, ibvs, 4418, 1 harris, w. e. 1996, aj, 112, 1487 jurcsik, j. 1995, aca, 45, 653 jurcsik, j. 1998, a&a, 333, 571 jurcsik, j., & kovács, g. 1996, a&a, 312, 111 kinman, t. d., saha, a., & pier, j. r. 2004, apj, 605, l25 kovács, g. 1998, msait, 69, 49 kovács, g., & jurcsik, j. 1996, apj, 466, l17 kovács, g., & jurcsik, j. 1997 a&a, 322, 218 kovács, g., & kanbur, s. m. 1998, mnras, 295, 834 kovács, g., & walker, a. r. 1999, apj, 512, 271 kovács, g., & walker, a. r. 2001, a&a, 371, 579 lee, j.-w., & carney, b. w. 1999, aj, 118, 1373 liller, m. h., & lichten, s. m. 1978, aj, 83, 41 mateu, c., vivas, a. k., zinn, r., miller, l. r., & abad, c. 2009, aj, 137, 4412 morgan, s. m., wahl, j. n., & wieckhorst, r. m. 2007, mnras, 374, 1421 oosterhoff, p. th. 1939, observatory, 62, 104 oosterhoff, p. th. 1944, bull. astron. inst. neth., 10, 55 pritzl, b. j., smith, h. a., stetson, p. b., catelan, m., sweigart, a. v., layden, a. c., & rich, r. m. 2003, aj, 126, 1381 simon, n, r., clement, c. m. 1993, apj, 410, 526 stellingwerf, r. f. 1978, apj, 224, 953 stetson, p. b. 1987, pasp, 99, 191 stetson, p. b. 1994, pasp, 106, 250 zinn, r., & west, m. j. 1984, apjs, 55, 45 zorotovic, m., catelan, m., zoccali, m., pritzl, b. j., smith, h. a., stephens, a. w., contreras, r., & escobar, m. e. 2009, aj, 137, 257 (paper i) ngc 5286. ii. variable stars 11 appendix light curves 12 m. zorotovic et al. ngc 5286. ii. variable stars 13 14 m. zorotovic et al. ngc 5286. ii. variable stars 15

0911.1687

supermagnetosonic jets behind a collisionless quasi-parallel shock

the downstream region of a collisionless quasi-parallel shock is structured containing bulk flows with high kinetic energy density from a previously unidentified source. we present cluster multi-spacecraft measurements of this type of supermagnetosonic jet as well as of a weak secondary shock front within the sheath, that allow us to propose the following generation mechanism for the jets: the local curvature variations inherent to quasi-parallel shocks can create fast, deflected jets accompanied by density variations in the downstream region. if the speed of the jet is super(magneto)sonic in the reference frame of the obstacle, a second shock front forms in the sheath closer to the obstacle. our results can be applied to collisionless quasi-parallel shocks in many plasma environments.

introduction.- when the angle between the nominal shock normal and the upstream magnetic field is small, the shock transition in a collisionless plasma is much more complex than in the quasi-perpendicular case [1]. the nonthermal nature of the upstream side of a quasi- parallel shock has been recognized for decades [2, 3, 4]. the downstream region, however, has only recently come under active research, both in astrophysical (supernovae [5]) and solar system (termination shock [6], earth's bow shock [7, 8]) contexts. the most detailed and extensive data of collisionless shock waves are from the earth's bow shock. in con- trast to remote observations and laboratory measure- ments, the near-earth space can be used to study in situ supersonic plasma flow past a magnetic obstacle- the flow of the solar wind around the magnetosphere of the earth. the magnetospheric boundary (the magne- topause) is usually located at a distance of 10 earth radii (1 re = 6371 km) in the solar direction. the bow shock is curved at magnetospheric scales while the structures in the solar wind and interplanetary magnetic field are large compared to the size of the magnetosphere. hence the locations of parallel and perpendicular regions of the bow shock vary depending on the direction of the interplane- tary magnetic field. consequently, we can access a wide range of plasma conditions via spacecraft observations. recent measurements have shown that the flow in the downstream region of a quasi-parallel shock is structured: nemecek et al. [9] have reported observations of transient ion flux enhancements in the earth's magnetosheath dur- ing radial interplanetary magnetic field. in subsequent studies, savin et al. [10] have found more than 140 events of anomalously high energy density. however, the source of these jets of high kinetic energy and ion flux has re- mained unclear. in this letter, we present a set of multi- spacecraft measurements from cluster [11] that allows us to suggest a formation mechanism for such jets. data.- we have analyzed cluster measurements from the evening of march 17, 2007, when the four spacecraft (c1-c4) were close to the nose of the magnetosphere. the spacecraft constellation was quite flat in the nom- inal plane of the magnetopause, since c3 and c4 were close to each other (950 km, 0.15 re apart), while the others were slightly more than 7000 km (>1 re) away. we have used data from the magnetic field experiment fgm from all four spacecraft, and from the ion experi- ment cis-hia from c1 and c3 [11]. information about the upstream conditions was provided by ace and wind satellites situated near the lagrangian point l1, as well as the geotail spacecraft, which at the time was in the foreshock region near the subsolar point. the free upstream solar wind flow was quite fast (v ∼530 km/s) and steady (see the upper panels of figure 1). the particle number density was around 2 cm−3 and hence the dynamic pressure (ρv 2, where ρ is the mass density) was low, close to 1 npa. the interplanetary magnetic field was approximately radial, i.e., the sunward magnetic field component bx [22] was dominant. moreover, the angle between the flow direc- tion and the magnetic field was less than 20◦. conse- quently, the bow shock was quasi-parallel at the subso- lar point. the upstream mach numbers [23] were all larger than 10: ma ∼12, ms ∼16, and mms ∼10. the location of the bow shock, as observed by geotail at (x, y, z)gse = ∼(14, −7, −3) re [22] when the shock moved over the spacecraft several times between 17:30 and 24:00 ut, matches well to the empirical model [12] for the measured upstream parameters. 2 b[nt] ace 2 3 4 5 θ[deg] 90 135 180 n[cm−3] 16:00 17:00 18:00 19:00 20:00 2 3 pdyn[npa] 1 2 b[nt] c1 cluster −40 −20 0 20 40 b[nt] c2 −40 −20 0 20 40 b[nt] c3 −40 −20 0 20 40 17−mar−2007 b[nt] c4 18:13 18:14 18:15 18:16 18:17 −40 −20 0 20 40 xgse ygse zgse abs m−sphere sheath shock jet fig. 1: upper panels: upstream solar wind data from the ace satellite (time-shifted by 44 minutes to account for the solar wind propagation to the magnetopause). first panel: magnitude of the interplanetary magnetic field and angle θ between the field direction vector and the xgse-axis [22]. second panel: plasma number density and dynamic pressure. ace was located at (x, y, z)gse = (237, 36.4, −18.6) re. the gray shading marks the period of interest. lower panels: magnetic field from all four cluster spacecraft in gse coor- dinates. the quartet was situated around (10.7, 1.5, 3) re. the color panels mark different plasma regions. white back- ground between color panels represents transition between two regions. the cluster quartet, moving on an outbound or- bit near the subsolar point, encountered the magne- topause the first time shortly after 17:00 ut and passed into the solar wind at 22:30 ut. between 17:00 and 20:00 ut cluster observed multiple magnetopause cross- ings. moreover, during this 3-hour period cluster ob- served several high speed jets (v ∼500 km/s) in the mag- netosheath behind the quasi-parallel bow shock. here we concentrate on the jet between 18:13 and 18:17 ut. as displayed in the lower panels of figure 1, all four spacecraft were inside the magnetosphere at the begin- ning of the interval. first the magnetopause moved in- wards passing over the cluster quartet at 250 km/s (ob- tained using 4-spacecraft timing). then the spacecraft observed a weak shock within the magnetosheath mov- ing in the same direction at 140 km/s. in the first panel of figure 2, the c1 measurements show that at this mo- ment the component of the plasma velocity parallel to the shock normal in the reference frame moving with the shock vn exceeds the magnetosonic speed vms (as well as the other characteristic speeds [23]). hence the mag- netosonic mach number mms > 1. (this is also the case with respect to the magnetopause.) the same was ob- served by c3 located about 8000 km away (not shown). after the shock cluster entered a cold, supermagne- tosonic jet with a plasma speed close to 500 km/s (see figure 2, first panel). at the location of c2, the shock and the magnetopause moved back across the spacecraft and it re-entered the magnetosphere for several seconds at 18:15:20 ut. the other spacecraft stayed in the su- personic jet for over a minute moving gradually back into normal sheath-type plasma. this transition can be seen in the ion velocity distributions (not shown): the narrow (∼1 mk) distribution of the jet was slowly replaced by a warmer, symmetric quasi-maxwellian after 18:16 ut. while in the jet, cluster observed a gradual increase in both plasma density and magnetic field: from low values of 7 cm−3 and 8 nt at the beginning of the jet to very high values of 22 cm−3 and 30 nt at the end (see fig- ure 2, second panel for c1 ion density). consequently, the dynamic pressure in the jet increased to over 6 npa, as compared to the nominal pressure of 1 npa. these en- hancements were accompanied by a substantial deflection of the bulk flow from its nominal direction, as illustrated by the third panel of figure 2. interpretation.- we propose the following mechanism to explain the formation of the jet: first, consider an oblique shock with radial upstream conditions (v1 ∥b1) as illustrated in the inset of figure 3. the rankine- hugoniot jump conditions for high ma give v1n = rv2n and v1t ∼v2t, where r is the shock compression ratio. we then consider the streamlines of plasma flow across a curved high ma shock as illustrated in figure 3. we infer, based on the analysis of the observations as will be discussed below, that the scale of the shock ripple under consideration is of the order of the spacecraft separation: 50 −100 ion inertial lengths, 7000−15000 km, 1 −3 re. as the shock primarily decelerates the component of the upstream velocity v1 parallel to the shock normal n, the shock crossing leads to efficient compression and deceler- ation in regions where the angle α between v1 and n is small. wherever α is large, however, the shock mainly deflects the flow while the plasma speed stays close to the upstream value. the plasma is still compressed so that the higher density together with the high speed leads to a jet of very high dynamic pressure. furthermore, if the speed v2 of this jet on the downstream side is still su- per(magneto)sonic in the reference frame of the obstacle, a second shock front forms closer to the obstacle. in addi- tion, depending on the ripple geometry, the flow behind the shock can converge causing local density enhance- ments, or diverge causing density depletions. let us compare figure 3 with the c1 measurements presented in figure 2 where, in the third panel, the bulk 3 [km/s] c1 −100 0 250 500 {−vz,vx}[km/s] c1 −500 −250 0 200 α[deg] c1 17−mar−2007 18:13 18:14 18:15 18:16 18:17 −90 −60 −30 0 n[cm−3] c1 0 10 20 30 vn (in shock frame) v va vs vms fig. 2: first panel: component of plasma velocity paral- lel to the local shock normal (−0.59, 0.52, −0.61)gse of the secondary shock (calculated with minimum variance analy- sis), in the reference frame moving with the secondary shock vn (black solid curve) and total plasma speed v (light blue solid curve). dashed curves show the characteristic speeds: alfv ́ en speed va (blue), sound speed vs (green), and magne- tosonic speed vms (red) [23]. second panel: plasma number density. third panel: bulk velocity projection to (−zgse, xgse) plane. fourth panel: the angle α calculated from the observed velocity deflection using the jump conditions for high ma and r = 4. the calculation is not expected to be valid at the edges of the jet where the shock is weak, and hence α is shown for the center only. all data are from c1. the color coding for different plasma regions is the same as in figure 1. v2 v1 α n z x obstacle downstream upstream 2nd shock c1 shock fig. 3: illustration of the effect of local shock curvature. the variation of the plasma number density in the downstream re- gion is illustrated by the shading: dark blue indicates density enhancement, light blue indicates density depletion. the tra- jectory of c1 in the reference frame moving with the ripple is sketched with the dashed line. the inset details the flow deflection when v1 is not parallel to n. flow direction is displayed in the (−zgse, xgse) plane. the observed pattern of the supermagnetosonic flow af- ter the secondary shock suggests that there is a ripple in the bow shock similar to the one of the illustration moving in the ∼zgse direction. this interpretation is supported by the observed density and flow speed pro- files. the fourth panel of figure 2 shows the upstream angle α for the supermagnetosonic jet calculated from the observations (considering both the downstream and up- stream data and taking r = 4). during the main velocity deflection, α ∼−65◦. the flow pattern in the ygse (not shown) reveals more of the three-dimensional structure of the ripple and will be considered elsewhere. the ob- servations of c3 are similar, though not identical to c1. given this and the fact that c2 was outside of the jet, we infer that the lower limit for the scale of the bow shock perturbation is of the order of the spacecraft separation (∼8000 km, 1.2 re). the ripples we propose to be the source of the high speed jets stem from the unstable nature of collisionless quasi-parallel shocks: reflected ions can stream against the upstream flow and interact with the incident plasma over long distances before returning (if at all) to the shock. this interaction triggers instabilities and creates waves that steepen into large structures convecting back to the shock front (see [1], and the references therein). the effect is most pronounced when b1 and v1 are aligned in the coordinate system of the obstacle. both observations and simulations have shown that ripples are inherent to quasi-parallel shocks: observa- tions of the ion reflection on the upstream side of the earth's bow shock [4] indicated that the direction of n varies when the shock is quasi-parallel at the subsolar point. such studies on the ion distributions have also shown that, at times, the solar wind does indeed pass through the shock layer without significant heating [13]. however, no connection between these two findings was made. furthermore, recent multi-spacecraft observations have characterised in detail the short, large amplitude magnetic structures (slams) [14, 15] convecting in the upstream of earth's quasi-parallel bow shock towards the shock front. slams have a scale size up to 1 re compa- rable to the ripples discussed here. in addition, measure- ments showed signatures that the shock transition itself is narrow consisting of only one to a few slams. the roughness of the parallel part of the shock front due to slams is clearly seen in the bow shock simulations of, e.g., blanco-cano et al. [16]. discussion and conclusions.- in previous studies savin et al. [10] have found several jets with high ki- netic energy density ( 1 2ρv 2), of which 33 jets had an energy density larger than 10 kev/cm3 (compared with 19 kev/cm3 in this event). nemecek et al. [9] have also reported what they call transient ion flux enhance- ments, with fluxes of 6 × 108/cm2s (∼8 × 108/cm2s in this event), during intervals of radial interplanetary mag- 4 netic field. both transient flux enhancements and high kinetic energy jets have properties similar to the jets re- ported here. neither nemecek et al. nor savin et al. could identify a clear source for the jets, but they could rule out, e.g., reconnection. here, we propose a gener- ation mechanism for high speed jets in the sheath that is in agreement with the measurements presented in this letter and those reported in previous studies. naturally, we cannot ascertain that all of the previously reported jets stem from the same shock geometry related origin. it has become evident that shocks are more structured than was previously recognized, so that a conventional plane wave description is not sufficient. in fact, the mech- anism proposed in this letter for spatial structuring of the downstream is valid for all rippled shocks regard- less of magnetic field obliquity, provided that the mach number is high. as voyager 1 and 2 crossed the helio- spheric termination shock [6], their observations revealed a rippled, supercritical (mms ∼10) quasi-perpendicular shock [17]. likewise, interplanetary shocks seem to be nonplanar [18] and also oblique ones may be rippling [19]. therefore we expect that the effects of the ripples, includ- ing supersonic jets, can be observed behind collisionless shocks in many plasma environments, and especially be- hind extended, varying shock fronts having quasi-parallel regions. in astrophysical context, the high speed jets and nonthermal structure can act as seeds for magnetic field amplification and particle acceleration [5], even for smooth upstream plasma. in magnetospheric con- text, the jets with their high dynamic pressure pro- vide a previously unidentified source for magnetopause waves during steady solar wind conditions. a locally perturbed magnetopause is consistent with the cluster measurements of c2 being within the magnetosphere while the other spacecraft were in the jet. in turn, the large magnetopause perturbation can affect the coupled magnetosphere-ionosphere dynamics [20]. note also that this letter presents observations of a weak shock within the magnetosheath during steady upstream conditions. previous studies of discontinuities within the sheath have been related to bow shock interaction with interplanetary shocks (see [21], and the references therein). in summary, we propose a generation mechanism for high speed jets in the downstream side of a quasi- parallel shock based on a set of multi-point measure- ments. quasi-parallel shocks are known to be rippled even during steady upstream conditions. the local cur- vature changes of the quasi-parallel shock can create fast bulk flows: in the regions where the upstream velocity is quasi-perpendicular to the local shock normal, the shock mainly deflects plasma flow while the speed stays close to the upstream value. together with the compression of the plasma, these localized streams can lead to jets with a kinetic energy density that is several times higher than the kinetic energy density in the upstream. we thank n. ness and d. j. maccomas for ace data, r. lepping and r. lin for wind data, and t. mukai and t. nagai for geotail data. we also thank cdaweb at nssdc and the cluster active archive for data access. h. h. thanks m. andr ́ e and e. k. j. kilpua for useful comments. the work of h. h. is supported by the vaisala foundation and the m. ehrnrooth foundation. the work of t. l., k. a., r. v., and m. p. is supported by the academy of finland. m. p. is also supported by eu-fp7 erc starting grant 200141-quespace. ∗[email protected] [1] d. burgess et al., space sc. rev. 118, 205 (2005). [2] j. r. asbridge, s. j. bame, and i. b. strong, j. geophys. res. 73, 5777 (1968). [3] m. m. hoppe and c. t. russell, nature 295, 41 (1982). [4] t. g. onsager et al., j. geophys. res. 95, 2261 (1990). [5] j. giacalone and j. r. jokipii, astrophys. j. 663, l41 (2007). [6] j. r. jokipii, nature 454, 38 (2008). [7] a. retin` o et al., nature physics 3, 236 (2007). [8] e. yordanova et al., phys. rev. lett. 100, 205003 (2008). [9] z. nemecek et al., geophys. res. lett. 25, 1273 (1998). [10] s. savin et al., jetp lett. 87, 593 (2008). [11] c. p. escoubet, r. schmidt, and m. l. goldstein, space sci. rev. 79, 11 (1997). [12] j. merka et al., j. geophys. res. 110, a04202 (2005). [13] j. t. gosling et al., j. geophys. res. 94, 10027 (1989). [14] e. a. lucek et al., ann. geophys. 20, 1699 (2002). [15] e. a. lucek et al., j. geophys. res. 113, a07s02 (2008). [16] x. blanco-cano, n. omidi, and c. t. russell, j. geo- phys. res. 114, a01216 (2009). [17] l. f. burlaga et al.,nature 454, 75 (2008). [18] m. neugebauer and j. giacalone, j. geophys. res. 110, a12106 (2005). [19] d. krauss-varban, y. li, and j. g. luhmann, in par- ticle acceleration and transport in the heliosphere and beyond, 7th annual astrophysics conference, edited by g. li, q. hu, o. verkhoglyadova, g. p. zank, r. p. lin, and j. g. luhmann (aip, 2008), pp. 307–313. [20] d. g. sibeck et al., j. geophys. res. 94, 2505 (1989). [21] l. prech, z. nemecek, and j. s. safrankova, geophys. res. lett. 35, l17s02 (2008). [22] geocentric solar ecliptic system (gse) coordinates: x- axis points from the earth towards the sun, y -axis op- posite to the planetary motion, and z-axis towards the ecliptic north pole. [23] characteristic speeds in plasma are alfv ́ en speed va = (b2/μ0ρm)1/2, sound speed vs = (γkbt/m)1/2, and magnetosonic speed vms = (v 2 a + v 2 s )1/2. thus the relevant mach numbers are alfv ́ en mach number ma = vn/va, sonic mach number ms = vn/vs, and magnetosonic mach number mms = vn/vms. here vn is the component of velocity parallel to the shock nor- mal in the reference frame moving with the shock front.

0911.1688

two-cooper-pair problem and the pauli exclusion principle

while the one-cooper pair problem is now a textbook exercise, the energy of two pairs of electrons with opposite spins and zero total momentum has not been derived yet, the exact handling of pauli blocking between bound pairs being not that easy for n=2 already. the two-cooper pair problem however is quite enlightening to understand the very peculiar role played by the pauli exclusion principle in superconductivity. pauli blocking is known to drive the change from 1 to $n$ pairs, but no precise description of this continuous change has been given so far. using richardson procedure, we here show that pauli blocking increases the free part of the two-pair ground state energy, but decreases the binding part when compared to two isolated pairs - the excitation gap to break a pair however increasing from one to two pairs. when extrapolated to the dense bcs regime, the decrease of the pair binding while the gap increases strongly indicates that, at odd with common belief, the average pair binding energy cannot be of the order of the gap.

introduction the first step towards understanding the microscopic grounds of superconductivity was made by fr ̈ ohlich1 who has realized that electrons in metals can form bound pairs due to their weak interaction with the ion lattice, which results in an effective electron-electron attraction. a few years later, cooper has considered2 a simplified quantum mechanical problem of two electrons with opposite spins and zero total momentum added to a "frozen" fermi sea, i.e., a sea of noninteracting electrons. within the cooper model, an attractive interaction between these two elec- trons is introduced, this interaction being localized in a finite-width layer above the "frozen" fermi sea. cooper has shown that such an attraction, no matter how weak, leads to the appearance of a bound state for the two ad- ditional electrons. this result was demonstrated for a single pair although it was fully clear that conventional superconductivity takes place in a macroscopic system of electrons paired by such an attraction. one year later, bardeen, cooper and schrieffer3 (bcs) have proposed an approximate solution of the quantum many-body problem for electrons with opposite spins at- tracting each other. a very important result of the bcs theory is the existence of a gap in the excitation spectrum above the ground state. in the bcs model, the potential layer, in which an attraction between electrons with op- posite spins acts, extends symmetrically on both sides of the fermi level. this implies that indeed a macroscopic number of electrons interact with each other. in order to avoid the difficult problem associated with the pauli exclusion principle between a given number of same spin electrons, the grand canonical ensemble was used. the original formulation of bcs theory is also based on a variational ansatz for the ground state wave function: the wave function is taken with all the electrons feeling the attraction, paired, i.e., "condensed" into the same quantum-mechanical state. it was, however, emphasized by schrieffer that electron pairs are not elementary bosons because they are con- structed from two elementary fermions4, so that their creation and destruction operators do not obey simple bosonic commutation relations. schrieffer also claimed that the large overlap which exists between pairs in the dense bcs configuration cuts any link with the two-body cooper model, the isolated pair picture thus having little meaning in the dense regime4. in spite of this claim, it is rather obvious that the many-electron bcs configuration can be reached from the one-cooper pair limit by sim- ply adding more and more electron pairs into the layer where the attraction acts, until the layer becomes half- filled. a canonical procedure of this kind would allow one to see the evolution of correlated electron pairs from the dilute to the dense regime and to understand deeper the role of the pauli exclusion principle in fermion pair- ing. notice that such an approach can also be considered as a useful and well-defined toy model for the crossover between local and extended pairs of attracting fermions, which in the present time attracts large attention within the field of ultracold gases5–7. the crossover problem is still open even for the simplest case of the "reduced" bcs potential for fermion-fermion interaction: a varia- tional solution has only been proposed long time ago by eagles8 and also by leggett9. it also uses a bcs-like ansatz for the ground state wave function. a possible way to tackle the problem in the canoni- cal ensemble, i.e., for a fixed number of electron pairs, is to use the procedure developed by richardson10,11. it al- lows us to write the form of the exact n-pair eigenstate of 2 the schr ̈ odinger equation in the case of the so-called "re- duced" bcs potential which is the simplest formulation of the electron-electron interaction mediated by the ion motion. the eigenstate, as well the energy of n pairs, read in terms of n parameters r1, ..., rn, which are solutions of n nonlinear algebraic equations. although richardson's approach greatly simplifies the problem by avoiding a resolution of a n-body schr ̈ odinger equation, the solution of these equations for r1, ..., rn in a com- pact form for general n remains an open problem. one of the difficulties is due to the fact that n is not a pa- rameter in these equations but only enters through the number of equations. this is rather unusual and makes the n-dependence of the system energy quite uneasy to extract. nowadays, richardson's equations are tackled numerically for small-size superconducting granules con- taining countable numbers of electron pairs12. we wish to add that the canonical approach has also been used in the form of a variational fixed-n projected bcs-like theories, see e.g. ref13. the goal of the present paper, is to extend the origi- nal cooper's work for one electron pair to two pairs: we analytically solve the two richardson's equations in the large sample limit. our work can be considered as an initial step towards the establishing of the precise link which exists between dilute and dense regimes of pairs, since it indicates a general trend for the evolution of the ground state energy with the increase of pair number, i.e., overlap between pairs. richardson's equations are here solved by three methods. they of course give iden- tical results but shine different light on these equations. the approaches to tackle richardson's equations, pro- posed in this paper, in fact constitute perspectives for the extension to a larger number of pairs and hopefully to the thermodynamical limit. the solution we obtain shows that the average pair binding energy is smaller in the two-pair configuration than for one pair. this result can be physically un- derstood by noting that electrons which are paired are fermions; therefore, by increasing the number of pairs, we decrease the number of states in the potential layer avail- able to form these paired states. the energy decrease we here find is actually quite general for composite bosons14. however, extrapolation of this understanding to the dense bcs configuration faces difficulty within the com- mon understanding of bcs results. indeed, it is gen- erally believed15,16 that the pair binding energy in the dense bcs limit is of the order of the superconducting gap ∆. at the same time, this gap is found as expo- nentially larger than the single pair binding energy ob- tained by cooper. according to the tendency we here revealed, the average pair binding energy in the dense regime should be smaller than that in the one-pair prob- lem. this discrepancy motivated us to focus on what is called pair binding energy and more generally "cooper pair" in the various understanding of the bcs theory. usually, pairs are said to have a binding energy of the order of ∆. however, such pairs are introduced not ab initio, but to provide a physical understanding of the bcs result for the ground state energy16,17. pairs with energy of the order of the gap are called "virtual pairs" by schrieffer4. they represent couples of electrons ex- cited above the normal fermi level for noninteracting electrons, as a result of the attraction between up and down spin electrons. since the fermi level is smeared out on a scale of ∆by the attraction, the number of such pairs is much smaller than the total number of elec- tron pairs feeling the attraction. the latter were named "superfluid pairs" by schrieffer4. by construction, the concept of "virtual pair" breaks a possible continuity be- tween the dilute and dense regimes of pairs in a somewhat artificial way. by contrast, staying within the framework of "superfluid pairs" greatly facilitates the physical un- derstanding of the role of pauli blocking in superconduc- tivity as well as in the bec-bcs crossover problem. our results in fact demonstrate the importance of a clear sep- aration between the various concepts of "cooper pair" found in the literature. we wish to mention that the results presented in this paper do not have straightforward experimental appli- cations. the main goal of this paper is to reveal the general trend for the evolution of energy spectrum when changing the number of pairs and to make a first step to- wards a fully controllable resolution of the n-pair prob- lem. however, even a two-pair configuration has a rela- tion to real materials having correlated pairs of fermions, because this configuration corresponds to a dilute regime of pairs, realized in some systems. conceptually, the overlap between pairs can be tuned either by changing fermion-fermion interaction or total number of pairs. we here show that by increasing the overlap between pairs, we block more and more states available for the construc- tion of paired states. for the first time, a dilute regime of pairs was addressed by eagles8 in the context of super- conducting semiconductors having a low carrier concen- tration. in particular, it was shown in this paper that the excitation spectrum in the dilute regime is controlled by the binding energy of an isolated pair (in agreement with our results) rather than by a more cooperative gap which appears, when pairs start to overlap. thus, this picture is quite similar to the isolated-pair model considered by cooper. there also is a variety of unconventional superconduc- tors which are characterized by rather short coherence length that implies pairs not overlapping so strongly as in conventional low-tc materials. for instance, it was argued in ref.18 that the bec-bcs crossover might be relevant for high-tc cuprates. some experiments seem to support this idea, for example ref19 where experi- mental data on the dependence of the superconducting transition temperature on fermi temperature are col- lected for various superconducting materials. this anal- ysis indicates that conventional low-tc superconductors stay apart from short-coherence length materials, includ- ing heavy fermion superconductors. thus, it was argued 3 that to understand these unconventional materials, it is appropriate to focus on the most basic aspect, i.e., on the short coherence length, rather than to introduce more ex- otic and less generic concepts20. it was also shown that the very recently discovered fe-based pnictides, which constitute a new class of high-tc superconductors, should be understood as low-carrier density metals resembling underdoped cuprates21, so that it is possible that the bec-bcs crossover phenomenon is relevant for these materials as well. quite recently, it was demonstrated in ref.22 that size quantization in nanowires made of conventional superconductors can result in a dramatic reduction of the coherence length bringing superconduct- ing state to the bec-bcs crossover regime. we finally would like to mention that the two-correlated pair prob- lem has received great attention within the ultracold gas field, see e.g. ref.5. all these examples demonstrate that, paradoxically, the cooper problem seems to be more rel- evant to modern physics than several decades ago. it is also worth mentioning that bcs hamiltonian, which only includes interaction between the up and down spin electrons with zero pair momentum, is oversimplified. nevertheless, fermionic pairs in the bec-bcs transition regime have not been described yet in a fully controlled manner even within this hamiltonian. one of the possi- ble strategies to tackle this crossover therefore is to find a precise solution of the problem for the simplest hamil- tonian and only after that, to turn to more elaborate hamiltonians. the paper is organized as follows. in section ii, we briefly recall the one-cooper pair problem to settle no- tations. in section iii, we present two solutions to the two-pair ground state, as well as a discussion of the pos- sible excited states. we conclude in section iv. in the appendix, we give another exact solution to the two-pairs richardson's equations which shines a different light to the problem. ii. the one-cooper pair problem let us briefly recall the one-cooper pair problem. we consider a fermi sea |f0⟩made of electrons with up and down spins. an attractive potential between electrons with opposite spins and opposite momenta acts above the fermi level εf0. this potential is taken as constant and separable to allow analytical calculations. in terms of free pair creation operators β† k = a† k↑a† −k↓it reads as v = −v x k′,k wk′wkβ† k′βk (1) v is a positive constant and wk = 1 for εf0 < εk < εf0 + ω. we add a pair of electrons with opposite spins to the "frozen" sea |f0⟩. when the pair has a nonzero momen- tum, it is trivial to see that a† p↑a† −p′↓|f0⟩with p ̸= p′ is eigenstate of h = h0 + v where h0 = p k,s εka† ksaks, its energy being (εp + εp′). if the pair has a zero total momentum, the h eigenstates are linear combinations of βk |f0⟩. we look for them as |ψ1⟩= x k g(k)β† k |f0⟩ (2) the schr ̈ odinger equation (h −e1) |ψ1⟩= 0 imposes g(k) to be such that [2εk −e1] g(k) −v wk x wk′g(k′) = 0 (3) for 2εk ̸= e1, the eigenfunction g(k) depends on k as wk/(2εk −e1), so that |ψ1⟩is only made of pairs within the potential layer, as physically expected. the eigen- values such that 2εk ̸= e1 for all k within the potential layer then follows from eq.(3) as 1 = v x k wk 2εk −e1 ≃v ρ0 2 z εf0 +ω εf0 2dε 2ε −e1 (4) ρ0 is the mean density of states in the potential layer. this leads for a weak potential, i.e., a dimensionless pa- rameter v = ρ0v small compared to 1, to e1 ≃2εf0 −εc (5) εc ≃2ωe−2/v (6) as seen below, it will be physically enlightening to rewrite this one-cooper pair binding energy as εc ≃(ρ0ω)(2e−2/v/ρ0) = nωεv (7) nω= ρ0ωis the number of empty pair states in the po- tential layer ωfrom which the cooper pair bound state is constructed, these states being all empty in the one- cooper pair problem. εv = 2e−2/v/ρ0 appears as a bind- ing energy unit induced by each of the empty pair states in the potential layer. εv only depends on the potential amplitude v and the density of states ρ0 in the potential layer. eq.(7) already shows that the wider the potential layer ω, the larger the number of empty states feeling the po- tential from which the cooper pair is made and, ulti- mately, the larger the binding energy εc. we can also note that the pair binding energy depends linearly on the number of states available to form a bound state. this remark is actually crucial to grasp the key role played by pauli blocking in superconductivity: indeed, this block- ing makes the number of empty states available to form a bound state decrease when the pair number increases - or when one pair is broken as in the case of excited states. iii. the two-cooper pair problem we now add two pairs having opposite spin electrons and zero total momentum to the fermi sea |f0⟩and we look for the eigenstates (h −e2) |ψ2⟩= 0 as |ψ2⟩= x g(k1, k2)β† k1β† k2 |f0⟩ (8) 4 the bosonic character of fermion pairs which leads to β† k1β† k2 = β† k2β† k1, allows us to enforce g(k1, k2) = g(k2, k1) without any lost of generality. the schr ̈ odinger equation fulfilled by g(k1, k2) is somewhat more compli- cated than for one pair. to get it, it is convenient to note that vβ† k1β† k2 |f0⟩= −v (1 −δk1k2)  wk1β† k2 + wk2β† k1  x wpβ† p |f0⟩ (9) the factor (1−δk1k2) being necessary for both sides of the above equation to cancel for k1 = k2. when used into (h −e2) |ψ2⟩= 0 projected upon ⟨f0| βk1βk2 we get 0 = (1 −δk1k2) [(2εk1 + 2εk2−e2) g(k1, k2) −v  wk1 x k̸=k2 wkg(k, k2) + (k1 ↔k2)    (10) the above equation makes g(k1, k1) undefined. this however is unimportant since the k1 = k2 contribution to |ψ2⟩anyway cancels due to the pauli exclusion prin- ciple. for k1 ̸= k2, the equation fulfilled by g(k1, k2) follows from the cancellation of the above bracket. with probably in mind a ( k1, k2) decoupling , richardson sug- gested to split e2 as e2 = r1 + r2 (11) with r1 ̸= r2, a requirement mathematically crucial as seen below. we can then note that 2εk1 + 2εk2 −e2 (2εk1 −r1) (2εk2 −r2) = 1 2εk1 −r1 + 1 2εk2 −r2 (12) with (r1, r2) possibly exchanged. this probably led richardson to see that the symmetrical function con- structed on the lhs of the above equation, namely g(k1, k2) = 1 (2εk1 −r1) (2εk2 −r2) +(r1 ↔r2) (13) is an exact solution of the schr ̈ odinger equation provided that r1 and r2 are such that 1 = v x wk 2εk −r1 + 2v r1 −r2 = (r1 ↔r2) (14) as obtained by inserting eq.(13) into (h −e2) |ψ2⟩= 0. note that the denominator in the above equation clearly shows why (r1, r2) are required to be different. the fundamental advantage of richardson's procedure is to replace the resolution of a 2-body schr ̈ odinger equation for g(k1, k2) by a problem far simpler, namely, the res- olution of two nonlinear algebraic equations. this procedure nicely extends to n pairs, the equa- tions for r1, ..., rn reading as eq.(14), with all possible r differences. however, to the best of our knowledge, the analytical resolution of these equations for arbitrary n has stayed an open problem, even when n = 2. we now show how we can tackle this resolution analytically, first through a perturbative approach, and then through two exact procedures. a. perturbative approach a simple way to tackle the richardson's equations an- alytically is to note that eq.(4) allows to replace 1 in the lhs of eq.(14) by the same sum with r1 replaced by e1. if we now add and substract the two richardson's equations, we get two equations in which the potential v has formally disappeared, namely x  wk 2εk −r1 + wk 2εk −r2  = 2 x wk 2εk −e1 (15) x  wk 2εk −r1 − wk 2εk −r2  = − 4 r1 −r2 (16) v is in fact hidden into e1. this is a wise way to put the singular v dependence of cooper pairs into the problem, at minimum cost. in view of eq.(15), we are led to expand the sums appearing in richardson's equations as x k wk 2εk −r1 = x k wk 2εk −e1 + e1 −r1 = ∞ x n=0 jn(r1 −e1)n (17) where j0 = 1/v while jn>0 is a positive constant given by jn = x k wk (2εk −e1)n+1 = ρ0 2 in nεn c (18) in ≃1 −e−2n/v (19) for v small. for this expansion to be valid, we must have |ri −e1| < 2εk −e1 for all k. this condition is going to be fulfilled for large samples, as possible to check in the end. it is convenient to look for ri through ci = (ri −e1) /εc with i = (1, 2). eqs.(15, 16) then give ∞ x n=1 in n (cn 1 + cn 2 ) = 0 (20) (c1 −c2) ∞ x n=1 in n (cn 1 −cn 2 ) = −2γc (21) the above formulation evidences that the richard- son's equations contain a small dimensionless parameter, namely γc = 4/nc (22) where nc = ρ0εc. indeed, nc is just the pair number from which pairs start to overlap. this makes nc large, 5 and consequently γc small compared to 1, in the large sample limit. for γc = 0 , the solution of the above equations reduces to c1 = c2 = 0, i.e., e2 = 2e1. the fact that the two-pair energy e2 differs from the energy of two single pairs 2e1 is physically due to pauli blocking, but mathematically comes from a small but nonzero value of γc. to solve eqs.(20, 21) in the small γc limit, it is conve- nient to set c1 = s + d and c2 = s −d. this allows us to rewrite eqs.(20, 21) as −d2 = (23) γc/2  i1+ i3 2 (d2+3s2)+* * *  +s [i2+i4(d2+s2)+* * * ] −s = i2 2 (d2 + s2) + i4 4 (d4 + 6d2s2 + s4) + * * * i1 + i3 3 (3d2 + s2) + * * * (24) their solution at lowest order in γc reads γc/2i1 ≃ −d2 ≃2si1/i2. when inserted into r1 + r2 = 2e1 + (c1 + c2)εc, this gives the two-pair energy as e2 ≃2e1 + γc i2 2i2 1 εc ≃2e1 + 2 ρ0  1 + 2e−2/v (25) using the expression of e1 given in eqs.(5,6), we can rewrite this energy as e2 ≃2  2εf0 + 1 ρ0  −εc  1 − 1 nω  ≃2  2εf0 + 1 ρ0  −εv (nω−1)  (26) compared to the energy of two single pairs 2e1 = 2(2εf0 −εc), we see that pauli blocking has two quite different effects. (i) it first increases the normal part of this energy as reasonable since the fermi level for free electrons increases. the first term in eq.(26) is nothing but 2εf0 +2(εf0 + 1 ρ0 ): one pair has a kinetic energy 2εf0 while the second pair has a slightly larger kinetic energy 2(εf0 + 1 ρ0 ), the fermi level increase when one electron is added, being 1/ρ0. (ii) another less obvious effect of the pauli exclusion principle is to decrease the average pair binding energy. indeed due to pauli blocking, (nω−1) pair states only are available to form a bound state in the two-pair configuration, while all the nωpair states are available in the case of a single cooper pair. b. exact approach the perturbative approach developed above, through the (ri −e1) expansion of the sum appearing in the richardson's equations, helped us to easily get the ef- fect of pauli blocking on the ground state of two cooper pairs. it is in fact possible to avoid this γc expansion as we now show. through the perturbative calculation, we have found that the difference r1 −r2 is imaginary at first order in γc. it is possible to prove that this difference is imaginary at any order in γc: the richardson's procedure amounts to add an imaginary part to the two-pair energy, in order to escape into the complex plane and avoid poles in sums like the one of eq.(4), the two "richardson's energies" then reading as r1 = r + ir′ and r2 = r −ir′, with r and r′ real. r is by construction real since r1 + r2 = 2r is the energy of the two cooper pairs. in order to show that r′ also is real, let us go back to eq.(16). in terms of (r, r′), this equation reads x wk x2 k + r′2 = 1 r′2 (27) where xk = 2εk −r is real. we then note that this equation also reads x wkx2 k |x2 k + r′2|2 = r′∗2 " 1 |r′|4 − x wk |x2 k + r′2|2 # (28) from it, we readily see that, since the lhs and the bracket are both real, r′∗2 must be real. r′∗2 can then either be positive or negative, e.i., r′ can be real or imag- inary, which produces (r1, r2) either both real or com- plex conjugate. to show that (r1, r2) cannot be both real, we go back to eq.(16). by noting that p wk is nothing but the num- ber nωof pairs in the potential layer, we can rewrite this equation as 0 = x wk  1 2εk −r1 − 1 2εk −r2 + 4 nω(r1 −r2)  (29) = 1 r1 −r2 x wk ak (2εk −r1)(2εk −r2) (30) where ak = (r1 −r2)2 + 4 nω (2εk −r1)(2εk −r2) (31) it is possible to rewrite the second term of ak using 4ab = (a + b)2 −(a −b)2. this leads to ak = (1 −1 nω )(r1 −r2)2 + 1 nω (4εk −r1 −r2)2 (32) since the number of pairs nωin the potential layer is far larger than 1, ak would be positive if (r1, r2) were both real. for (r1, r2) outside the potential layer over which the sum over k is taken, the sum in eq.(30) would be made of terms with a given sign, so that this sum cannot 6 cancels. consequently, solutions outside the potential layer must be complex conjugate whatever γc. for (r1, r2) complex conjugate, i.e., r′ real, the sum over k in eq.27, performed within a constant density of states, leads to 1 r′2 = ρ0 2 z εf0 +ω εf0 2dεk x2 k + r′2 = ρ0 2r′  arctan 2ω+ xf0 r′ −arctan xf0 r′  (33) where xf0 = 2εf0 −r. if we now take the tangent of the above equation, we find tan 2 ρ0r′ = 2ωr′ r′2 + xf0(2ω+ xf0) (34) turning to eq.(15), we find that it reads in terms of (r, r′) as x xk x2 k + r′2 = 1 v (35) if we again perform the integration over k with a constant density of states, this equation gives x2 f0 + r′2 (2ω+ xf0)2 + r′2 = e−4/v (36) r and r′ then appear as the solutions of two algebraic equations, namely eqs.(31) and (33). unfortunately, they do not have compact form solutions. it is however possible to solve these equations ana- lytically in the large sample limit. ρ0 then goes to in- finity so that nωand nc are both large. in this limit tan(2/ρ0r′) ≃2/ρ0r′ to lowest order in (1/ρ0). eq.(31) then gives r′2 ≃xf0(2ω+ xf0)/nω. for ρ0 infinite, i.e., nωinfinite, r′ reduces to zero so that, due to eq.(33), z = xf0/(2ω+ xf0) reduces to e−2/v. eq.(33) can then be rewritten as z2 ≃e−4/v 1 + z/nω 1 + 1/znω (37) since e−2/vnω= nc/2 is also large compared to 1, this gives the first order correction in 1/ρ0 to z ≃e−2/v as z ≃ e−2/v  1 + (e−2/v −e2/v)/2nω  . from xf0 = 2ωz/(1 − z) which for z small reduces to xf0 ≃2ω(z +z2), we end by dropping terms in e−4/v, with r at first order in 1/ρ0 given by r ≃2ǫf0 −2ωe−2/v  1 −1 nc − 1 nω  ≃2ǫf0 + 1 ρ0 −ǫc(1 − 1 nω ) (38) since e2 = 2r, this result is just the one obtained from the perturbative approach given in eq.(26). the major advantage of this exact procedure is to clearly show that the above result corresponds to the dominant term in both, the large sample limit by drop- ping terms in (1/ρ0)2 in front 1/ρ0, and the small poten- tial limit by dropping terms in e−4/v in front of e−2/v. as seen from the first expression of r in eq.(35), the pauli exclusion principle induces a double correction, in 1/nc and in 1/nωto the one-pair binding energy ǫc = 2ωe−2/v. however, the corrections in 1/nc ends by giving a poten- tial free correction to the 2-pair energy e2 because, in a non-obvious way, it in fact comes from a simple change in the free electron fermi sea filling, as seen from the second expression of r in eq.(35). c. excited state we now consider the 2-pair excited states with a bro- ken pair having a nonzero total momentum, as possibly obtained by photon absorption. such a pair does not feel the bcs potential, so that it stays uncorrelated. these excited states thus read  ψ1; k,k′ = x f(k1)β† k1a† k↑a† −k′↓|f0⟩ (39) to derive the equation fulfilled by f(k1), it is conve- nient to note that, for k ̸= k′, βpβ† k1a† k↑a† −k′↓|f0⟩ = δk1p (1 −δk1k −δk1k′) a† k↑a† −k′↓|f0⟩ (40) the bracket insuring cancellation for k1 = k or k′, as necessary due to the lhs. it is then easy to show, from the schr ̈ odinger equation h −e1, kk′
 ψ1; k,k′ = 0 pro- jected upon ⟨f0| a−k′↓ak↑βp that 0 = 1 −δpk −δpk′
2εp + εk + εk′ −e1, kk′
f(p) −v wp x q̸=k,k′ wqf(q)   (41) this makes f(p) undefined for p = k or k′. this is unim- portant since the corresponding contribution in  ψ1; k,k′ cancels due to the pauli exclusion principle. for p ̸= (k, k′) the equation fulfilled by f(p) is obtained by enforc- ing the bracket of the above equation to cancel. follow- ing the one-cooper pair procedure, we get the eigenvalue equation for one broken pair (k, −k′) plus one cooper pair as 1 v = x p̸=k,k′ wp 2εp + εk + εk′ −e1, kk′ (42) a first possibility is to have the two free electrons in the two lowest states of the potential layer, namely εk = εf0 and εk′ = εf0 + 1/ρ0. the p-state energy in the above 7 equation must then be larger than ε(2) f0 with ε(n) f0 = εf0 + n/ρ0; so that eq.(39) merely gives 1 v = ρ0 2 z εf0 +ω ε(2) f0 2dε 2ε + 2εf0 + 1/ρ0 −e1, kk′ (43) by writing εf0 + ωas ε(n) f0 + ω(n) with ω(n) = ω−n/ρ0, eqs.(4, 5) for the single pair energy readily give e1, kk′−  2εf0 + 1 ρ0  ≃2  εf0 + 2 ρ0  −2  ω−2 ρ0  e−2/v (44) another possibility is to put the two free electrons in the second and third lowest states of the potential layer, namely εk = εf0 + 1/ρ0 and εk′ = εf0 + 2/ρ0. the p- state energy in eq.(39) can then be equal to εf0 or larger than ε(3) f0 . in this case, eq.(39) gives 1 v = 1 2εf0 −e + ρ0 2 z εf0 +ω ε(3) f0 2dε 2ε −e (45) in which we have set e = e1, kk′ −2εf0 −3/ρ0. by e as 2ε(3) f0 −2ω(3)e−2/vx, the above equation gives x through 2 xn ′′ c −3 = log x 1 + xe−2/v (46) where n ′′ c = 2ω(3)e−2/vρ0 is close to nc, i.e., large compared to 1 in the large sample limit. this gives x ≃1 + 2/n ′′ c , so that the energy e1, kk′ would then be equal to e′ 1, kk′ ≃4εf0 + 7 ρ0 −2  ω−3 ρ0  e−2/v (47) this energy is larger than the one given in eq.(41) with the broken pair in the two lowest energy levels of the potential layer. such a conclusion stays valid for broken pair electrons in higher states: the minimum energy for a broken pair plus a correlated pair is given by e1, kk′ in eq.(41). the excitation gap to break one of the two cooper pairs into two free electrons ∆= e1, kk′ −e2, thus appears to be ∆= εc + 3 ρ0 = εc  1 + 3 nc  (48) we can then remember that the excitation gap for a single pair is equal to h εf0 + (εf0 + 1 ρ0 ) i −(2εf0−εc), i.e., εc + 1 ρ0 : the broken pair being again in the two lowest states of the potential layer, this brings an additional 1 ρ0 contribution to the average pair binding energy εc. eq.(45) thus shows that the gap increases when going from one to two pairs. this increase in fact comes from a mere kinetic energy increase induced by pauli blocking. it is worth noting that, while pauli blocking induces an increase of the gap, it produces a decrease from εc to εc (1 −1/nω) of the average pair binding energy when going from one to two correlated pairs. since nω= ρ0ω is far larger than nc = ρ0εc, the gap increase however is far larger than the binding energy decrease. the changes we obtain in the excitation gap and in the average pair binding energy when going from one to two pairs, are a strong indication that the gap in the dense bcs configuration cannot be simply linked to the pair binding energy, as commonly said. indeed, the pair binding energy is going to stay smaller than εc = 2ωe−2/v due to pauli blocking in the potential layer, while the experimental gap in the dense regime is known to be of the order of ωe−1/v which is far larger than εc. we wish to stress that, in addition to the excited states considered in this section, in which the broken pair ends by having a non-zero momentum, there also are excited states, not included into the present work. in these ex- cited states, the two pairs still have a zero momentum but correspond to r's located somewhere in the quasi- continuum spectrum of the one-electron states, i.e., in- between two one-electron levels. for such r's, it is not possible to straightforwardly replace summation by inte- gration in the richardson's equations as we did through- out the present paper. iv. conclusion we here extend the well-known one-pair problem, solved by cooper, and consider two correlated pairs of electrons added to a fermi sea of noninteracting elec- trons. the schr ̈ odinger equation for the two-pair ground state has been reduced by richardson to a set of two coupled algebraic equations. we here give three differ- ent methods to solve these two equations analytically in the large sample limit, providing a unique result. these methods are perspective for the extension to an arbi- trary number of pairs in order to hopefully cover the crossover between dilute and dense regimes of cooper pairs, as well as to apply them to nanoscopic supercon- ductors. although the two-pair problem we here solve, is only a first step toward the resolution of this quite funda- mental problem, it already allows us to understand more deeply the role of pauli blocking between electrons from which pairs are constructed. we show that this blocking leads to a decrease of the average pair binding energy in the two-pair system compared to the one-pair configura- tion. this decrease is due to the fact that by increasing the number of pairs, we decrease the number of available states to form bound pairs. this two-pair problem actually has some direct rela- tion to real physical systems, where correlated pairs are more local than in conventional bcs superconductors. we can mention underdoped cuprates, heavy fermion su- perconductors, pnictides, and ultracold atomic gases. it was shown long time ago8, in the context of supercon- ducting semiconductors with low carrier density, that the excitation spectrum of such dilute system of pairs is con- 8 trolled by the binding energy of an isolated pair rather than by a cooperative bcs gap. hence, this picture is very similar to the classical cooper model. within the two-pair configuration that we here solve using richard- son's procedure, we reach the same conclusion for the excitation spectrum. we also reveal how the compos- ite nature of correlated pairs affects their binding ener- gies through the pauli exclusion principles for elementary fermions from which the pairs are constructed. the extrapolation of the tendency we find to the dense bcs regime of pairs, indicates that the average pair bind- ing energy in this regime must be smaller than that of an isolated pair. at the same time, it is generally believed that the pair binding energy in the bcs configuration is of the order of the superconducting gap, which is much larger than the isolated pair binding energy. to under- stand this discrepancy, we must note that there are two rather different concepts of "pairs" in the many-particle bcs configuration. those with energy of the order of the gap are introduced not ab initio, but enforced to have a gap energy in order to provide a qualitative un- derstanding for the expression for the ground state en- ergy, found within the bcs theory. these entities, called "virtual pairs" by schrieffer4, correspond to pairs of elec- trons excited above the fermi sea of noninteracting elec- trons. these virtual pairs have to be contrasted with what schrieffer calls "superfluid pairs"4, made of all the electrons with opposite momenta feeling the attracting bcs potential, the number of these pairs being much larger than the number of "virtual pairs". staying within the framework of "superfluid pairs" greatly helps to un- derstand the dilute and dense regimes of pairs on the same footing. v. acknowledgements w. v. p. acknowledges supports from the french min- istry of education, rfbr (project no. 09-02-00248), and the dynasty foundation. vi. appendix in this appendix, we propose another exact approach to richardson's equations which may turn more conve- nient for problems dealing with a pair number larger than two. we start with eqs.(14) and calculate the sum by again assuming a constant density 2 v = z εf0 +ω εf0 2 dε 2ε −r1 + 4 ρ0(r1 −r2) (a1) with r1 possibly complex. instead of ri, we are going to look for ai = (2ω+2εf0 −ri)/(2εf0 −ri) with i = (1, 2). since ri = 2εf0 −2ω/(ai −1), the above equation yields 2 v = log a1 + 2 nω (a1 −1)(a2 −1) a1 −a2 (a2) where log denotes the principal value of the complex logarithmic function, i.e., the one that satisfies −π < im (log a1) ≤π by adding the same equation with (1, 2) exchanged, we readily get a1a2 = e4/v (a3) this equation is nothing but eq.(33), since for r1 = r+ ir′, we do have a1 = (2ω+xf0−r−ir′)/(xf0−r−ir′), with i changed into −i for (r1, a1) changed into (r2, a2). from eq.(a3), we conclude that a1 = e2/vt while a2 = e2/v/t. next, we note that, since the two-pair energy e2 = r1 + r2, which also reads e2 = 4εf0 −2ω a1 + a2 −2 e4/v + 1 −(a1 + a2) (a4) is real, (a1 +a2) must be real. this implies t+1/t = t∗+ 1/t∗or equivalently (t −t∗)(tt∗−1) = 0. consequently, t is either real or such that |t| = 1, i.e., t = eiφ. to choose between these two possibilities, we consider the difference of the two richardson's equations as written in eq.(a2). this difference which first appears as 0 = log a1 a2 + 4 nω (a1 −1)(a2 −1) a1 −a2 (a5) reads in terms of t as e2/v + e−2/v = t + 1 t −nω 2  t −1 t  log t (a6) for t real, (t −t−1)log t is always positive, except for t = 1 where it cancels. this shows that the rhs of eq.(a6), equal to 2 for t = 1, stays essentially smaller than 2 for nωfar larger than 1. since e2/v + e−2/v is far larger than 2 for v small, we conclude that eq.(a6) cannot be fulfilled for t real. the other possibility is t = eiφ with 0 < |φ| < π, so that log t = iφ we then have a1 = a∗ 2, i.e., r1 = r∗ 2. this shows that the two richardson's energies are complex conjugate, as found by the other exact approach. when used into eq.(a6), this t leads to φ sin φ + 2 nωcos φ = δc (a7) with δc = e2/v+e−2/v nω ≃ 2 nc . once eq.(a7) for φ is solved, the two-pair energy given in eq.(a4) follows from e2 = 4εf0 −4ω e2/v cos φ −1 e4/v + 1 −2e2/v cos φ (a8) the solutions of eq.(a7) cannot be expressed in a com- pact form in terms of classical functions. we however 9 see that the two dimensionless terms in eq.(a7), namely 2/nωand δc, are small. furthermore 2/nωis smaller than δc. the function in the lhs of eq.(a7) is increas- ing from 2/nωup to a maximum ≃1.82, then decreasing down to −2/nωas φ runs in [0, π], and still decreasing on [π, 2π]. this shows that eq.(a7) admits exactly one solution in the interval (0, π/2), another one in the inter- val (π/2, π) but no solution in [π, 2π]. changing φ in −φ would provide also two solutions on (−π, 0) but this just corresponds to exchange a1 and a2; so that they cannot be considered as distinct solutions. for these solutions, φ sin φ stays close to zero, so that φ is close to 0 or to π. for φ = 0, the rhs of the above equation reduces, for v small, to 4εf0 −4ωe−2/v which is just twice the energy of a single cooper pair as given in eq. (6). the effect of pauli blocking on this two-single pair energy results from a large but finite number of pair states nωin the poten- tial layer, as physically expected. we can note that, by contrast, φ = π would lead to e2 close to 4εf0 +4ωe−2/v. this solution has to be sorted out because it corresponds to r1 and r2 located in the complex plane very close to the real axis where the one-electron levels are positioned, so that the distance between them and this real axis is of the order of 1/ρ0. this prevents substitution of dis- crete summation by integration, in eq.(a1), as discussed above. for φ close to zero, eq.(a7) gives the leading term in 1/nωas φ2 ≃ e1/v −e−1/v)2 nω. the ratio in eq.(a8) then reads for φ small as 1 e2/v −1  1 −φ2 2 e2/v(e2/v + 1) (e2/v −1)2  ≃ 1 e2/v −1 −e2/v 2nω e4/v −e2/v −1 + e−2/v e2/v −1
3 (a9) when inserted into eq.(a8), we end with e2 ≃4εf0 −4ωe−2/v + 2ω nω  1 + 2e−2/v (a10) which is nothing but eq.(26). the main advantage of this second exact method is to have the two-pair energy reading in terms of φ which fol- lows from a single equation, namely eq.(a7). by con- trast, to get e2 through xf0 = 2εf0 −e2, as in the other exact method, we must solve two coupled equa- tions, namely eqs.(31) and (33). 1 h. frohlich, phys. rev. 79, 845 (1950). 2 l. n. cooper, phys. rev. 104, 1189 (1956). 3 j. bardeen, l.n. cooper, and j.r. schrieffer, phys. rev. 108, 1175 (1957). 4 j. r. schrieffer, theory of superconductivity, perseus books group, massachusetts (1999). 5 r. combescot, x. leyronas, and m. y. kagan, phys. rev. a 73, 023618 (2006). 6 i. bloch, j. dalibard, and w. zwerger, rev. mod. phys. 80, 885 (2008). 7 s. giorgini, l. p. pitaevskii, and s. stringari, rev. mod. phys. 80, 1215 (2008). 8 d. m. eagles, phys. rev. 186, 456 (1969). 9 a. j. leggett, j. de physique. colloques 41, c7 (1980). 10 r. w. richardson, phys. lett. 3, 277 (1963). 11 r. w. richardson and n. sherman, nucl. phys. 52, 221(1964). 12 j. dukelsky, s. pittel, and g. sierra, rev. mod. phys. 76, 643 (2004). 13 f. braun and j. von delft, phys. rev. lett. 81, 4712 (1998). 14 m. combescot, o. betbeder-matibet, and f. dubin, physics reports 463, 215 (2008). 15 e. m. lifshitz and l. p. pitaevskii, statistical physics, part 2, pergamon, oxford (1980). 16 a. l. fetter and j. d. walecka, quantum theory of many- particle systems, dover publications, new york (2003). 17 m. tinkham, introduction to superconductivity, dover publications, new york (2004). 18 m. randeria, in: a. griffin, d. snoke, and s. stringari (eds.), bose-einstein condensation. cambridge univer- sity press, campridge, pp. 355-392 (1995). 19 y. j. uemura, physica c 194, 282 (1997). 20 q. chen, j. stajic, s. tan, and k. levin, physics reports 412, 1 (2005). 21 d. j. singh and m.-h. du, phys. rev. lett 100, 237003 (2008); l. craco, m. s. laad, s. leoni, and h. rosner, phys. rev. b 78, 134511 (2008). 22 a. a. shanenko, m. d. croitoru, a. vagov, and f. m. peeters, arxiv:0910.2345 (2009).

0911.1689

the factor set of gr-categories of the type $(\pi,a)$

any $\gamma$-graded categorical group is determined by a factor set of a categorical group. this paper studies the factor set of the group $\gamma$ with coefficients in the categorical group of the type $(\pi,a).$ then, an interpretation of the notion of $\gamma-$operator $3-$cocycle is presented and the proof of cohomological classification theorem for the a $\gamma-$graded gr-category is also presented.

introduction the notion of a graded monoidal category was presented by fr ̈ ohlich and wall [4] by generalization some manifolds of categories with the action of a group γ. then, γ will be also regarded as a category with exactly one object, (say ∗), where the morphisms are the members of γ and the composition law is the group composition operation. a γ−grading on a category d is a functor gr : d →γ. the grading is called stable if for all c ∈obd, σ ∈γ, there is an equivalence u in d with domain c and gr(u) = σ. then σ is the grade of u. if (d, gr) is a γ−graded category, we define kerd to be the subcategory consisting of all morphisms of grade 1. a γ−monoidal category consists of: a stably γ−graded category (d, gr), γ−functors ⊗: d ×γ d − →d , i : γ − →d and natural equivalences (of grade 1) ax,y,z : (x⊗y )⊗z ∼ − →x⊗(y ⊗z), lx : i⊗x ∼ − →x, rx : x ⊗i ∼ − →x, where i = i(∗), satisfying coherence conditions of a monoidal category. for any γ−graded category (d, gr), authors wrote rep(d, gr) for the category of γ−fuctors f : γ →d, and natural transformations. an object of rep(d, gr) thus consists an object c of d with homomorphism γ →autd(c). homomorphism γ →autd(i) is the right inverse of the graded homomorphism autd(i) →γ. in other words, autd(i) is a split extension of the normal subgroup n of automorphisms of grade 1 by the subgroup which is isomorphic to γ. the extension defines an action of γ on n by σu = i(σ) ◦u ◦i(σ−1). in [1], authors considered the γ−graded extension problem of categories as a catego- rization of the group extension problem. the groups a, b in the short exact sequence: 0 →a →b →γ →1 are replaced with the categories c, d. a γ−monoidal extension of the monoidal category c is a γ−monoidal category d with a monoidal isomorphism j : c − →kerd. the construction and classification problems of γ−monoidal extension were solved by raising the main results of schreier-eilenberg-maclane on group extensions to categorical 1 level. with the notations of factor set and crossed product extension, authors proved that there exists a bijection ∆: h2(γ, c) ↔ext(γ, c) between the set of congruence classes of factor sets on γ with coefficients in the monoidal category c and the set of congruence classes of γ−extensions of c. the case c is a categorical group (also called a gr-category) was considered in [2]. then, the γ−equivariant structure appears on π−module a, where π = π0(c), a = autc(1) = π1(c). γ−extensions and γ−functors are classified by functors ch :γ cg →h3 γ ; z γ : z3 γ →γ cg, where γcg is the category of γ−extensions, z3 γ is the category in which any object is a triple (π, a, h), where (π, a) is a γ−pair and h ∈h3; h3 γ is the category obtained from z3 γ when h ∈z3 γ(π, a) is replaced with h ∈h3 γ(π, a). as we know, each categorical group is equivalent to a categorical group of the type (π, a) and the unit constraint is strict (in the sense lx = rx = idx). so we may solve the classification problem for this special case thanks to the desription of gr-functors of gr-categories of the type (π, a) [6]. we may better describe the factor set, and show that the γ−equivariant structure of a is a necessary condition of the factor set. thus, we may construct γ−operator 3−cocycles as a induced version of a factor set, instead of using complex construction as in [2]. by this way, we may obtain the classification theorem in a stronger form than the result in [2], that is the bijection: ω: sγ(π, a) →h3 γ(π, a) where sγ(π, a) is the set of congruence classes of γ−extensions of gr-categories of the type (π, a). the classification problem of γ−functors follows this method will be presented in another paper. 1 some notions let π be a group and a be a left π−module. a gr-category of the type (π, a) is a category s = s(π, a, ξ) in which objects are elements x ∈π, and morphisms are automorphisms aut(x) = {x} × a. the composition of two morphisms is defined by (x, u) ◦(x, v) = (x, u + v) the operation ⊗is defined by x ⊗y = xy (x, u) ⊗(y, v) = (xy, u + xv). the associative constraint ax,y,z is a normalized 3−cocycle (in the sense of group cohomol- ogy) ξ ∈z3(π, a), and the unit constraint is strict. then from now on, a gr-category of the type (π, a) refers to the one with above properties. 2 definition 1.1. let γ be a group and let (c, ⊗) be any monoidal category. we say that a factor set on γ with coefficients in (c, ⊗) is a pair (θ, f) consisting of: a family of monoidal autoequivalences f σ = (f σ, f f σ, c f σ) : c − →c, (σ ∈γ) and a family of isomorphisms of monoidal functors θσ,τ : f σf τ ∼ − →f στ, (σ, τ ∈γ) satisfying the conditions i) f 1 = idc ii) θ1,σ = idf σ = θσ,1 (σ ∈γ) iii) for all ∀σ, τ, γ ∈γ, the following diagrams are commutative f σf τf γ θσ,τf γ − − − − − →f στf γ f σθτ,γ   y   yθστ,γ f σf τγ θσ,τγ − − − − → f στγ in [6], authors described monoidal functors between monoidal categories of the type (π, a). thanks to this description, we will prove the necessary conditions of a factor set. definition 1.2. [6] let s = (π, a, ξ), s′ = (π′, a′, ξ′) be gr-categories. a functor f : s →s′ is called a functor of the type (φ, f) if f(x) = φ(x) , f(x, u) = (φ(x), f(u)) and φ : π →π′, f : a →a′ is a pair of group homomorphisms satisfying f(xa) = φ(x)f(a) for x ∈π, a ∈a. we have theorem 1.3. [6] let s = (π, a, ξ), s′ = (π′, a′, ξ′) be gr-categories and f = (f, e f , b f) be a gr-functor from s to s′. then, f is a functor of the type (φ, f). according to this theorem, any monoidal autoequivalence f σ is of the form f σ = (φσ, f σ). this remark is used frequently throughout this paper. definition 1.4. [2] let γ be a group, π be a γ−group. a γ−module a is a equivariant module on γ−group π if a is a π−module satisfying σ(xa) = (σx)(σa), for all σ ∈γ, x ∈π and a ∈a. 2 γ−graded extension of a gr−category of the type (π, a) for a given factor set (θ, f), we may construct a γ−graded crossed product extension of c, denoted by ∆(θ, f) as follows: c j − − − − →∆(θ, f) g − − − − →g where ∆(θ, f) is a category in which objects are objects of c and morphisms are pairs (u, σ) : a →b where σ ∈γ and u : f σ(a) →b is a morphism in c. the composition of two morphisms: a (u,σ) − − − − →b (v,τ) − − − − →c 3 is defined by: (u, σ) * (v, τ) = (u * f σ(v) * θσ,τ(a)−1, στ). this composition is associative and the unit exists thanks to cocycle and normalized condi- tions i), ii), iii) of (θ, f). a stably γ−grading on ∆(θ, f), g : ∆(θ, f) →γ is defined by g(u, σ) = σ, and the bijection j : c ∼ − − − − →ker(∆(α, f)) is defined by: j(a u − − − − →b) = (a (u,1) − − − − →b) proposition 2.1. if g : c →c′ and h : c′ →c are monoidal equivalence such that α : g ◦h ∼ = idc′, and β : h ◦g ∼ = idc, and d is a crossed product γ−extension of c by the factor set (θ, f), then the quadruple (g, h, α, β) induces: i) the factor set (θ′, f ′) of c′, ii) a γ−equivalence ∆(θ, f)) ↔∆(θ′, f ′)). proof. i) let f ′σ be the composition h ◦f σ ◦g and θ′σ,τ x = g(θσ,τ hx ◦f σ(βf τ hx)). one can verify that (θ′, f ′) is a factor set of c′. ii) we extend the functor g : c →c′ to a functor e g : ∆(θ, f)) ↔∆(θ′, f ′)) as follows: for the object x of c, let e gx = gx; for the morphism (u, σ) : x →y, where u : f σx →y, let e g(u, σ) = (g(u ◦f σ(βx)), σ). one can verify that e g is a γ−equivalentce from the above proposition, it is deduced that corollary 2.2. any γ−extension of a gr-category is equivariant to a γ−extension of a gr-category of the type (π, a). we now prove some necessary conditions for the existence of a factor set. theorem 2.3. let γ be a group and s = s(π, a, ξ) be a gr-category. if (θ, f) is a factor set of γ, with coefficients in s, then: i) there exists a group homomorphism φ : γ − →autπ ; f : γ − →auta, and a is equiped with a π−module γ−equivariant structure, induced by φ, f, ii) in definition 2.1, the condition i) of a factor set can be deduced from the remaining conditions. proof. i) according to theorem 1.1, any autoequivalence f σ, σ ∈γ, of a factor set is of the form (φσ, f σ) : s − →s. 4 since f f σx,y : f σ(xy) − →f σx.f σy, σ ∈γ is a morphism in (π, a), we have f σ(xy) = f σx.f σy, ∀x, y ∈π. this stated that φσ = f σ is a endomorphism in π. furthermore, f σ is an equivalence, so that φσ is an automorphism of group π, that is φσ ∈autπ. on the other hand, since θσ,τ x : f σf τx − →f στx is an arrow in (π, a), we have (f σf τ)(x) = f στ(x), ∀x ∈π. thus, φσφτ = φστ. this proved that φ : γ − → autπ σ 7− → φσ is a homomorphism of groups. then φ1 = φ(1) = idπ. let f f σx,y = (σ(xy), e f σ(x, y)), in which σ(xy) = φσ(xy); e f σ : π2 − →a and c f σ = (1, cσ) : f σ1 − →1 are maps. from the definition of the monoidal functor f σ, we have σx. e f σ(y, z) −e f σ(xy, z) + e f σ(x, yz) −e f σ(x, y) = a(σx, σy, σz) −f σ(a(x, y, z)) (1) (φσx)cσ + e f σ(x, 1) = 0 (2) cσ + e f σ(1, x) = 0 (3) we now observe isomorphisms of monoidal functors θσ,τ = (θσ,τ x ) where (θσ,τ x ) = (φσx, tσ,τ(x)) : f σf τ(x) − →f στ(x), in which tσ,τ : π − →a are maps. we have the following commutative diagrams f σf τ(x) (•,tσ,τ(x)) − − − − − − − →f στ(x) (•,f τf σ(a))   y   y(•,f στ(a)) f σf τ(x) (•,tσ,τ(x)) − − − − − − − →f στ(x) f σf τ(xy) (•, e f σ(f τx,f τy)+f σ e f τ(x,y)) − − − − − − − − − − − − − − − − − − →f σf τ(x).f σf τ(y) (•,tσ,τ(xy))   y   y(f στ x,tσ,τ(x))⊗(f σy,tσ,τ(y)) f στ(xy) (•, e f στ(x,y)) − − − − − − − − → f στ(x).f στ(y) f σf τ1 1 f στ1 ❄ (1,tσ,τ(1)) ❅ ❅ ❅ ❘ (1,f σ(cτ)+cσ) ✒ (1,cστ) from which we are led to f σf τ = f στ (4) 5 f στ(x)tσ,τ(y) −tσ,τ(xy) + tσ,τ(x) = e f στ(x, y) −e f σ(f τ x , f τ y ) −f σ( e f τ(x, y)) (5) f τ(cτ) + cτ −cστ = tσ,τ(1) (6) from the equality (4), we are led to a homomorphism f : γ − →auta given by f(σ) = f σ and so that f 1 = f(1) = ida. now, let σx = φσx, σc = f σc (7) for all σ ∈γ, x ∈π, c ∈a. thus, since f σ is a functor of the type (φσ, f σ), φσ(xb) = φσ(x)f σ(b), or σ(xb) = σ(x)σ(b), that is a is a π-module γ-equivariant. ii) from the condition ii) in the definition of a factor set, we are led to tσ,1 = t1,σ = 0 from the equality (5), for τ = 1, we obtain e f 1 x,y = 0, that is f f 1x,y = id. from the equality (3), for σ = 1, we have: c1 = 0, that is b f 1 = id. thus, f 1 is an identity monoidal functor. the theorem is proved. 3 enough strict factor set and induced 3-cocycle when the monoidal category c is replaced with the categorical group of the type (π, a) we obtain better descriptions than in the general case. for example, in theorem 3.2, authors proved that the condition i) of the definition of factor set of categorical groups of the type (π, a) is redundant. now, we continue "reducing" this concept in terms of other face. in this paper, we call a factor set (θ, f) enough strict if c f σ = idi for all σ ∈γ. definition 3.1. let γ be a group and c be a gr-category of the type (π, a). factor sets (θ, f) and (μ, g) on γ with coefficients in c are cohomologous if there exists a family of isomorphisms of monoidal functors uσ : (f σ, f f σ, c f σ) ∼ − →(gσ, f gσ, c gσ) (σ ∈γ) satisfying u1 = id(π,a) uστ.θστ = μσ,τ.uσgτ.f σuτ (σ, τ ∈γ) remark 3.2. if the two representatives (θ, f), (μ, g) are cohomologous, then f σ = gσ, σ ∈ γ. indeed, from the definition of cohomologous factor sets, there exists a family of iso- morphisms of monoidal functors uσ : (f σ, f f σ, b f σ) →(gσ, f gσ b gσ) (σ ∈γ). sinceuσ x : f σx →gσx is an arrow in (π, a), we have gσx = f σx. 6 furthermore, for any a ∈a, by the commutativity of the diagram f σx gσx f σx gσx ✲ uσ x ❄ f σ(x,a) ❄ gσ(x,a) ✲ uσ x by the commutativity of the diagram f σ(x, a) = gσ(x, a). extending lemma 1.1 [2] for a factor set, we have lemma 3.3. let s be a categorical group of the type (π, a). any factor set (θ, f) on γ with cofficients in s is cohomologous to an enough strict factor set (μ, g). proof. for each σ ∈γ, consider a family of isomorphisms in s: uσ x = ( idf σx if x ̸= 1 ( c f σ)−1 if x = 1 where 1 ̸= σ ∈γ, and u1 = id. then, we define gσ in a unique way such that uσ : gσ →f σ is a natural transfor- mation by setting gσ = f σ and: f gσx,y = (uσ x ⊗uσ y)−1 f f σx,y(uσ xy); c gσ = idi for such setting, clearly we have gσ = (gσ, f gσ, c gσ) : s →s is a monoidal equivalence. in particular, we have: c gσ = idi. this states the enough strictness of the family of functors gσ, as well as of the factor set (μ, g). now, we set μσ,τ : gσgτ →gστ the natural transformation which makes the following diagram gσgτ gστ f στ gσf τ f σf τ diagram 1 ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ✲ μσ,τ ◗◗ ◗ s gσuτ ✲ uστ ✲ uσf τ ✑✑ ✑ ✸ θσ,τ commute, for all σ, τ ∈γ. clearly, μσ,τ is a isomorphism of monoidal functors. we will prove that the family of μσ,τ satisfy the condition ii) of the definition of a factor set. we now prove that they satisfy the condition iii). consider the diagram: 7 gσgτgγ gσgτf γ gσf τf γ gσf τγ gσf τγ f σf τf γ f σf τγ gστgγ gστf γ f στf γ f στγ gστγ ✲ μσ,τgγ ✲ μσ,τf γ ✲ uσf τf γ ✲ θσ,τf γ ✲ uσf τγ ✲ θσ,τγ ✲ μσ,τγ ❄ gσgτuγ ❄ gσuτ f γ ❄ gσθτ,γ ✻ gσuτγ ❄ gστuγ ❄ uστf γ ❄ θστ,γ ✻ uστγ ❄ ✲ ✛ (i) (ii) (iii) (iv) (v) (vi) (vii) gσμτ,γ μστ,γ in this diagram, the region (i) commutes thanks to the naturality of μσ,τ; the regions (ii), (v), (vi), (vii) commute thanks to the diagram 1; the region (iii) commutes thanks to the naturality of uσ; the region (iv) commutes thanks to the definition of the factor set (θ, f). so the perimater commutes. this completes the proof. we now show that any factor set induces a γ−operator 3−cocycle based on the following definition. let a γ−pair (π, a), that is, a π−module γ−equavariant a, cohomology groups hn γ(π, a) studied in [3]. we recall that cohomology group hn γ(π, a), with n ≤3, can be computed as the cohomology group of the struncated cochain complex: ∼ cγ (π, a) : 0 − − − − →c1 γ(π, a) ∂ − − − − →c2 γ(π, a) ∂ − − − − →z3 γ(π, a) − − − − →0, in which c1 γ(π, a) consists of normalized maps f : π →a, c2 γ(π, a) consists of nor- malized maps g : π2 ∪(π × γ) →a and z3 γ(π, a) consists of normalized maps h : π3 ∪(π2 × γ) ∪(π × γ2) →a satisfying the following 3−cocycle conditions: h(x, y, zt) + h(xy, z, t) = x(h(y, z, t)) + h(x, yz, t) + h(x, y, z) (8) σ(h(x, y, z)) + h(xy, z, σ) + h(x, y, σ) = h(σ(x), σ(y), σ(z)) + (σ(x))(h(y, z, σ)) + h(x, yz, σ) (9) σ(h(x, y, τ)) + h(τ(x), τ(y), σ) + h(x, σ, τ) + (στ(x))(h(y, σ, τ)) = h(x, y, στ) + h(xy, σ, τ) (10) σ(h(x, τ, γ)) + h(x, σ, τγ) = h(x, στ, γ) + h(γ(x), σ, τ) (11) for all x, y, z, t ∈π; σ, τ, γ ∈γ. for each f ∈c1 γ(π, a), the coboundary ∂f is given by (∂f)(x, y) = x(f(y)) −f(xy) + f(y), (12) (∂f)(x, σ) = σ(f(x)) −f(σ(x)), (13) and for each g ∈c2 γ(π, a), ∂g is given by: (∂g)(x, y, z) = x(g(y, z)) −g(xy, z) + g(x, yz) −g(x, y), (14) (∂g)(x, y, σ) = σ(g(x, y)) −g(σ(x), σ(y)) −σ(x)(g(y, σ)) + g(xy, σ) −g(x, σ), (15) (∂g)(x, σ, τ) = σ(g(x, τ)) −g(x, στ) + g(τ(x), σ). (16) 8 proposition 3.4. any enough strict factor set (θ, f) on γ with coefficients in s = (π, a, ξ) induces an element h ∈z3 γ(π, a). proof. suppose f σ = (f σ, f f σ, id). then, we can write f f σx,y = (φσ(xy), e f σ(x, y)) = (σ(xy), g(x, y, σ)) where e f : π2 × γ →a is a function. for the family of isomorphisms of monoidal functors θσ,τ = (θσ,τ x ), we are able to write θσ,τ x = (φστx, tσ,τ(x)) = (φστx, t(x, σ, τ)) where t : π × γ2 →a is a function. from functions ξ, e f, t , in which ξ is associated with the associative constraint a of (π, a), we determine the function h as follows: h : π3 ∪(π2 × γ) ∪(π × γ2) →a where h = ξ ∪e f ∪t, in the sense h |π3= ξ; h |π2×γ= e f; and h |π×γ2= t the above determined h is a γ−operator 3−cocycle. indeed, the equalities (1), (5) turn into: −σx.h(y, z, σ) + h(xy, z, σ) + h(x, y, σ) −h(x, yz, σ) = h(σx, σy, σz) −σ(h(x, y, z)) (17) (στ)x.h(y, σ, τ) −h(xy, σ, τ) −h(x, σ, τ) = h(x, y, στ) −h(τx, τx, σ) −σh(x, y, τ) (18) moreover, from the relations of 3−cocycle ξ, we obtain: xh(y, z, t) −h(xy, z, t) + h(x, yz, t) −h(x, y, zt) + h(x, y, z) = 0 (19) the cocycle condition θστ,γ.θσ,τf γ = θσ,τγ.f σθτ,γ yeilds h(x, στ, γ) + h(γx, σ, τ) = h(x, σ, τγ) + σh(x, τ, γ) (20) for all x, y, z ∈π; σ, τ ∈γ. it follows that h satisfies the relations of a 3−cocycle in z3 γ(π, a). however, we have to prove the normalized property of h. first, since the unit constraints of (π, a) are strict and the factor set (θ, f) is enough strict, the equalities (2), (3), (6) turn into: h(x, 1, σ) = e f σ(x, 1) = 0 = e f σ(1, x) = h(1, x, σ) h(1, σ, τ) = tσ,τ(1) = 0 since c f 1 = id, we have h(x, y, 1γ) = e f 1(x, y) = 0 since the normalized property of associative constraint a, h(1, y, z) = h(x, 1, z) = h(x, y, 1) = 0 thanks to ii) in the definition of a factor set, we have h(x, 1γ, τ) = h(x, σ, 1γ) = 0 let h(θ,f ) = h, we have h(θ,f ) ∈z3 γ(π, a). this completes the proof of theorem. 9 4 classification theorem let (μ, g) be another enough strict factor set on γ with coefficients in ((π, a, ξ), which is cohomologous to (θ, f). thus, π−module γ−equivariant structure a and the element h(μ,g) ∈h3 γ(π, a) defined by (μ, g) has the following property: proposition 4.1. let two enough strict factor sets (μ, g), (θ, f) on γ with coefficients in categorical group s of the type (π, a) be cohomologous. then, they determine the same structure of π−module γ−equivariant on a and 3−cocyles inducing h(θ,f ), h(μ,g) are coho- mologous. proof. according to remark 3.2, f σ = gσ(σ ∈γ. then, they induce the same π-module γ-equivariant structure, according to the relation (7) and theorem 2.3. now, we prove that elements h(θ,f ) and h(μ,g) are cohomologous. we denote h(μ,g) = h′. hence, by determining of h(μ,g) referred in proposition 3.4, we have (σ(xy), h′(x, y, σ)) = g hσx,y; (στx, h′(x, σ, τ) = μσ,τx, for all x, y ∈π, σ, τ ∈γ let u : π × γ →a be the function defined by u(x, σ) = uσ x. it determines an extending 2−cochain of u, denoted by g, with g|π2 : π2 →a is the null map. since (uστ.θσ,τ)x = (μσ,τ.uσgτ.f σuτ)x, we have g(x, στ) + h(x, σ, τ) = h′(x, σ, τ) + g(τx, σ) + σg(x, τ) (21) since f gσx,y * uσ x⊗y = (uσ x ⊗uσ y) * f f σx,y, we have h′(x, y, σ) −h(x, y, σ) = g(x, σ) + σxg(y, σ) −g(xy, σ) (22) since c f σ = d hσ = id, we have uσ 1 = d hσ( c f σ)−1 = id1. hence g(1, σ) = 0 for all σ ∈γ (23) since u1 = id((π,a),⊗), we have g(x, 1γ) = 0 for all x ∈g (24) by the determining of g and relations (21)-(24), we have g ∈c2 γ(π, a) and h(θ,f ) − h(μ,g) = ∂g. this completes the proof of proposition. thus, any factor set (θ, f) on γ with coefficients in categorical groups of the type (π, a) determines a structure of g−module γ−equivariant a and an element h(θ,f ) ∈ h3 γ(π, a) uniquely. now, we consider the problem: giving an element h ∈z3 γ(π, a), does there exist a factor set (θ, f) on γ with coefficients in (π, a), inducing h. according to the definition of γ−operator cocycle, we have functions h |π3= ξ; h |π2×γ= e f; and h |π×γ2= t hence, we may determine the factor set (θ, f): f σx = σx; f σ(x, c) = (σx, σc) that is f σ(c) = σc c f σ = id1, f f σx,y = (σ(xy), e f(x, y, σ)) θστx = (στx, t(x, σ, τ)), for all σ, τ ∈γ, x ∈π, c ∈a. clearly, the above determined factor set induces h. we now state the main result of the paper: 10 theorem 4.2. there exists a bijection ω: sγ(π, a) →h3 γ(π, a), where sγ(π, a) is the set of congruence classes of γ−extensions of categorical groups of the type (π, a). proof. any element of sγ(π, a) may have a crossed product γ−extension ∆((θ, f), c) be a presentative, where c is a categorical group (π, a, ξ). according to proposition 3.3, it is possible to assume that (θ, f) is enough strict. then, (θ, f) induces 3−cocycle h = h(θ,f ) (proposition 3.4). according to proposition 4.1, the correspondence cl(θ, f) →cl(h(θ,f )) is a map. thanks to the above remark, this correspondence is a surjection. we now prove that it is an injection. let ∆and ∆′ be crossed product γ−extension of gr-categories s = s(π, a, ξ), s′ = s′(π, a, ξ′) by factor sets (θ, f), (θ′, f ′). moreover, 3−cocycles inducing h, h′ are cohomologous. we will prove that ∆and ∆′ are equivariant γ−extensions. according to the determining of h, h′, h |π3= ξ, h′ |π3= ξ′, ξ′ = ξ + δg, where g : π2 →a is a map. then, gr-categories s and s′ are gr-equivalent, so there exists a gr-equivalence (k, e k) : s →s′, where e kx,y = (•, g(x, y)). we may extend the gr-functor (k, e k) to a γ−functor (k∆, g k∆) : ∆→∆′ as follows: k∆(x) = k(x) ; k∆(a, σ) = (k(a), σ) ; g k∆= e k. it is easy to see that (k∆, g k∆) is a γ−equivalence. hence, ωis an injection. references [1] a.m. ceggara, a.r.garzn and j.a.ortega graded extensions of monoidal categories. journal of algebra. 241 (2001),620-657. [2] a. m. cegarra, j. m. garca - calcines and j. a. ortega, on grade categorical groups and equivariant group extensions. canad. j. math. vol. 54(5), 2002 pp. 970 - 997. [3] a. m. cegarra, j. m. garca - calcines and j. a. ortega,cohomology of groups with oprators. homology homotopy appl. (1) 4(2002), 1 - 23. [4] a. frolich and c. t. c wall, graded monoidal categories. compositio math. 28 (1974). 229-285. [5] s. mac lane, homology, springer- verlag, berlin and new york, 1963. [6] n. t. quang, on gr-functors between gr-categories: obstruction theory for gr-functors of the type (φ, f), arxiv: 0708.1348 v2 [math.ct] 18 apr 2009. 11 address: department of mathematics hanoi national university of education 136 xuan thuy street, cau giay district, hanoi, vietnam. email: [email protected] 12

0911.1690

canonical quantization of a dissipative system interacting with an anisotropic non-linear absorbing environment

a canonical quantization scheme is represented for a quantum system interacting with a nonlinear absorbing environment. the environment is taken anisotropic and the main system is coupled to its environment through some coupling tensors of various ranks. the nonlinear response equation of the environment against the motion of the main system is obtained. the nonlinear langevin-schr\"{o}dinger equation is concluded as the macroscopic equation of motion of the dissipative system. the effect of nonlinearity of the environment is investigated on the spontaneous emission of an initially excited two level-atom imbedded in such an environmrnt.

introduction the simplest way of describing a damped system in classical dynamics is by adding a resisting force, generally velocity-dependent, to the equation of motion of the system. frequently the magnitude of the resisting force may be closely presented, over a limited range of velocity, by the law fd = avn, where v is the velocity of the damped system and a and n are constants. for example for the friction force n = 0, viscous force n = 1 and for high speed motion n = 2 [1]. such an approach is no longer possible in quantum mechanics, because one can not find a unitary time evolution operator for both the states and the observables, consistently. in order to take into account the dissipation in a quantum system, there are usually two approaches. the first approach is a phenomenological way, by which the effect of dissipation is taken into account by constructing a suitable lagrangian or hamiltonian for the system [2, 3]. following this method the first hamiltonian was proposed by caldirola [4] and kanai [5] and afterward by others [6, 7]. there are difficulties about the quantum mechanical solu- tions of the caldirola-kanai hamiltonian. for example quantization using this way violates the uncertainty relations or canonical commutation rules. the uncertainty relations vanishes as time tends to infinity [8]-[11]. the second approach is based on the assumption that the damping forces is caused by an irreversible transfer of energy from the system to a reservoir [12, 13]. in this method , modeling the absorptive environment by a col- lection of harmonic oscillators and choosing a suitable interaction between the system and the oscillators, a consistent quantization is achieved for both the main system and the environment[14]-[26]. in the heisenberg picture, one can obtain the linear langevin-schr ̈ odinger equation, as the macroscopic equation of motion of the main system.[14, 15]. in the present work, following the second approach, a fully canonical quantization is introduced for a system moving in an anisotropic non-linear absorbing environment. the dissipative system is the prototype of some important problems which the present approach can be applied to cover such problems straightforwardly. the paper is organized as follows: in section 2, a lagrangian for the total system (the main system and the environment) is proposed and a classical treatment of the dissipative system is achieved. in section 3, the lagrangian introduced in the section 2 is used for a canonical quantization of both the 2 main system and the non-linear environment. in section 4, the present quan- tization is used to investigate the effect of the nonlinearity of the environment on the spontaneous decay rate of an initially excited two-level atom embedded in the absorbing environment. finally, the paper is closed with a summary and some concluding remarks in section 5. 2 three-dimensional quantum dissipative sys- tems when an absorbing environment responds non-linearly against the motion of a system, the non-linear langevin-schr ̈ odinger equation is usually appeared as the macroscopic equation of motion of the system. as an example, when the electromagnetic field is propagated in an absorbing non-linear polarizable medium, the vector potential satisfies the non-linear langevin-schr ̈ odinger equation. in this section, the motion of a three-dimensional system in the presence of an anisotropic non-linear absorbing environment is classically treated. for this purpose , the environment is modeled by a continium of three dimensional harmonic oscillators labeled by a continuous parameter ω. the total lagrangian is proposed as l(t) = le + ls + lint. (1) which is the sum of three pars. the part le is the lagrangian of the envi- ronment le(t) = z ∞ 0 dω 1 2 ̇ x(ω, t) * ̇ x(ω, t) −1 2ω2 x(ω, t) * x(ω, t)  . (2) where x(ω, t) is the dynamical variable of the oscillator labeled by ω. the second part ls in (1) is the lagrangian of the main system. taking the system as a particle with mass, m, moving under an external potential v (q), one can write ls = 1 2m ̇ q(t) * ̇ q(t) −v (q). (3) the last part lint in the total lagrangian (1) is the interaction term between the system and its absorbing environment and includes both the linear and 3 nonlinear contributions as follows lint = z ∞ 0 dω f (1) ij (ω) ̇ qi(t) xj(ω, t) + z ∞ 0 dω z ∞ 0 dω′f (2) ijk(ω, ω′) ̇ qi(t) xj(ω, t)xk(ω′, t) + z ∞ 0 dω z ∞ 0 dω′ z ∞ 0 dω′′ f (3) ijkl(ω, ω′, ω′′) ̇ qi(t) xj(ω, t)xk(ω′, t)xl(ω′′, t) + . . . * * * (4) where f (1), f (2), f (3), * * * are the coupling tensors of the main system and its environment. as it is seen from (4) the coupling tensor f (1) describes the linear contribution of the interaction part and the sequence f (2), f (3), .... describe, respectively, the first order of the non-linear interaction part, the second order of the non-linear interaction part and so on. the interaction la- grangian (4) is the generalization of the lagrangian that previously has been applied to quantize the electromagnetic field in the presence of anisotropic linear magnetodielectric media [27]. the coupling tensors f (1), f (2), f (3), ... in (4) are the key parameters of this quantization scheme. as it will be seen, in the next section, the susceptibil- ity tensors of the environment (of the various ranks) are expressed in terms of the coupling tensors. also the noise forces are obtained in terms of the coupling tensors and the dynamical variables of the environment at t = −∞. 2.1 the classical lagrangian equations the classical equations of motion of the total system can be obtained using the principle of the hamilton's least action, δ z dt l(t) = 0. these equations are the euler-lagrange equations. for the dynamical variables, x(ω, t), the 4 euler-lagrange equations are as d dt δl δ( ̇ xi(ω, t)) ! − δl δ(xi(ω, t)) = 0 i = 1, 2, 3 ⇒ ̈ xi(ω, t) + ω2xi(ω, t) = ̇ qj(t)f (1) ji (ω) + z ∞ 0 dω′ ̇ qj(t) h f (2) jik(ω, ω′) + f (2) jki(ω′, ω) i xk(ω′, t) + z ∞ 0 dω′ z ∞ 0 dω′′ ̇ qj(t) h f (3) jikl(ω, ω′, ω′′) + f (3) jkil(ω′, ω, ω′′) + f (3) jkli(ω′, ω′′, ω) i xk(ω′, t)xl(ω′′, t) + * * ** * * (5) also the lagrange equations for the freedom degrees of the main system are obtained as follows d dt δl δ( ̇ qi(t)) ! − δl δ(qi(t)) = 0 i = 1, 2, 3 ⇒m ̈ q(t) + ▽v (q) = − ̇ r(t) (6) where ri(t) = z ∞ 0 dω f (1) ij (ω) xj(ω, t) + z ∞ 0 dω z ∞ 0 dω′f (2) ijk (ω, ω′)xj(ω, t) xk(ω′, t) + z ∞ 0 dω z ∞ 0 dω′f (3) ijkl(ω, ω′, ω′′) xj(ω, t) xk(ω, t) xl(ω′′, t) + * * * (7) in eq. (6) − ̇ r(t) is the force exerted on the main system due to its motion inside the absorbing environment. it will be seen that the force − ̇ r(t) can be separated into two parts. one part is the damping force which is dependent on the various powers of the velocity of the main system. the second part is the noise forces which has sinusodial time dependence. both the damping and the noise forces are necessary for a consistent quantization of a dissipative system. without the noise forces the quantization of a dissipative system encounter inconsistency. according to the fluctuation- dissipation theorem the absence of any of these two parts leads to the vanishing of the other part. 5 3 canonical quantization in order to represent a canonical quantization, the canonical conjugate mo- menta corresponding to the dynamical variables x(ω, t) and q should be computed using the lagrangian (1). these momenta are as follows qi(ω, t) = δl δ( ̇ xi(ω, t)) = ̇ xi(ω, t) i = 1, 2, 3 (8) pi(t) = δl δ( ̇ qi) = m ̇ qi + ri(t) i = 1, 2, 3. (9) having the canonical momenta, both the dissipative system and the envi- ronment can be quantized in a standard fashion by imposing the following equal-time commutation rules [qi(t) , pj(t)] = ıħδij (10) [xi(ω, t) , qj(ω′, t)] = ıħδijδ(ω −ω′) (11) using the lagrangian (1) and the expressions for the canonical momenta given by (8) and (9), the hamiltonian of the total system clearly can be written as h(t) = [p(t) −r(t)]2 2m + v (q) + 1 2 z ∞ 0 dω h q2(ω, t) + ω2x2(ω, t) i (12) where the cartesian components of r(t) is defined by (7). the hamiltonian (12) is the counterpart of the hamiltonian of the quantized electromagnetic field in the presence of magnetodielectric media[27]-[29]. using the commu- tation relations (10), (11) and applying the total hamiltonian (12), it can be shown that the combination of the heisenberg equations of motion of the canonical variables x(ω, t) and q(ω, t) leads to the eq.(5). similarly, one can obtain eq.(6) as the equation of motion of q(t) in the heisenberg picture. let us introduce the annihilation and creation operators of the environment as follows bi(ω, t) = s 1 2ħω [ωxi(ω, t) + ıqi(ω, t)] . (13) from the commutation relations (11) it is clear that the ladder operators bi(ω, t) and b† i(ω, t) obey the commutation relations h bi(ω, t), b† j(ω′, t) i = δijδ(ω −ω′) (14) 6 the hamiltonian (12) can be rewritten in terms of the creation and annihi- lation operators bi(ω, t) and b† i(ω, t) as follows h = (p −r(t))2 2m + v (q) + hm (15) where hm = 3 x i=1 z dω ħω b† i(ω, t)bi(ω, t) (16) is the hamiltonian of the absorbing environment in the normal ordering form and ri(t) = z ∞ 0 dω s ħ 2ωf (1) ij (ω) h bj(ω, t) + b† j(ω, t) i + z ∞ 0 dω z ∞ 0 dω′ ħ 2 √ ωω′ f (2) ijk(ω, ω′) h bj(ω, t)bk(ω′, t)+ + b† j(ω, t)b† k(ω′, t) +bj(ω, t)b† k(ω′, t) + b† j(ω, t)bk(ω′, t) i + * * * (17) are the cartesian components of the operator r, where the summation should be done over the repeated indices. 3.1 the response equation of the environment the response equation of the absorbing environment is the base of separating the force − ̇ r(t) , in the right hand of (6), into two parts, that is, the damping force and the noise force. if eq.(5) is solved for x(ω, t) and then, the obtained solution is substituted into the definition of r(t) given by (7), one can obtain the response equation of the environment. the differential equations (5) are a continuous collection of coupled non-linear differential equations for the dynamical variables x(ω, t). the exact solution of this equation is impossible unless an iteration method to be used. for simplicity here we apply the first order of approximation and neglect the terms containing the coupling tensors f (2), f (3), ... in the right hand of (5) and write the solution of eq.(5), approximately, as x(ω, t) = xn(ω, t) + z t −∞dt′sin ω(t −t′) ω (f (1))†(ω) * ̇ q(t′), (18) 7 where (f (1))† ij(ω) = f (1) ji (ω) and xn(ω, t) is the solution of homogeneous equation ̈ xn(ω, t) + ω2xn(ω, t) = 0. in fact xn(ω, t) is asymptotic form of x(ω, t) for very large negative times and can be written as xni(ω, t) = s ħ 2ω h bin i (ω)e−ıωt + b†in i (ω)eıωti (19) where bin(ω) and b†in(ω) are some time independent annihilation and creation operators which obviously satisfy the same commutation relations (14). the approximated solution (18) yields the response equation of the environment, such that, the susceptibility tensors appearing in it, satisfy the various sym- metry properties reported by the literature [30]. now substituting x(ω, t) from (18) in (7), the response equation of the non- linear absorbing environment is found as follows r(t) = r(1) + r(2) + .... r(1) i (t) = z +∞ −∞dt χ(1) ij (t −t′) ̇ qj(t′) + r(1) n i(t) r(2) i (t) = z +∞ −∞dt′ z +∞ −∞dt′′χ(2) ijk(t −t′, t −t′′) ̇ qj(t′) ̇ qk(t′′) + r(2) n i(t) (20) where χ(1) is the susceptibility tensor of the environment in the linear regime and is defined by χ(1) ij (t) =          z ∞ 0 dωsin ωt ω1 f (1) in (ω)f (1) jn (ω) t > 0 0 t ≤0 (21) and χ(2) ijk causes the first order of the nonlinearity of the response equation, where for for t1, t2 ≥0 are given by χ(2) ijk(t1, t2) = z ∞ 0 dω1 z ∞ 0 dω2 sin ω1t1 ω1 sin ω2t2 ω2 f (2) inm(ω1, ω2) f (1) jn (ω1) f (1) km(ω2) (22) and χ(2) ijk is zero for t1, t2 < 0. in (21) and (22) the summation should be done over the repeated indices m, n. from the definition (21) it is clear 8 that χ(1) is a symmetric tensor, χ(1) ij = χ(1) ji . there are also some symmetry features for the non-linear susceptibility tensors of the various orders. these symmetry properties can be satisfied by imposing some conditions on the coupling tensors f (2), f (3), ... . for example the susceptibility tensor χ(2), should satisfy the symmetry property [30] χ(2) ijk(t1, t2) = χ(2) ikj(t2, t1) (23) where is fulfilled provided that the coupling tensor f (2) obey the symmetry condition f (2) ijk(ω, ω′) = f (2) ikj(ω′, ω), (24) similarly inserting the approximated solution x(ω, t) from (18) into (7), one can obtain the (n −1)'th susceptibility tensor of the environment in the non-linear regime for t1, t2, ...tn ≥0 as the following χ(n) i i1...in(t1, t2, ..., tn) = z ∞ 0 dω1 z ∞ 0 dω2.... z ∞ 0 dωn sin ω1t1 ω1 sin ω2t2 ω2 ....sin ωntn ωn × f (n) ij1j2....jn(ω1, ω2, ... ωn) f (1) i1j1(ω1) f (1) i2j2(ω2).... f (1) injn(ωn) (25) and χ(n) ii1...in(t1, t2, ..., tn) is identically zero for t1, t2, ...tn < 0. the susceptibil- ity tensor χ(n) ii1...in(t1, t2, ..., tn) should satisfy the symmetry relations [30] χ(n) i i1...ik,...il,...,in(t1, t2, ..., tk, ...tl, ..., tn) = χ(n) i i1...il,...ik,...,in(t1, t2, ..., tl, ...tk, ..., tn) (26) where this symmetry relation is clearly fulfilled by imposing the symmetry conditions f (n) ij1j2.......,jk,...,jl,...,jn(ω1, ω2, ..., ωk, ..., ωl, ..., ωn) = f (n) ij1j2.......,jl,...,jk,...,jn(ω1, ω2, ..., ωl, ..., ωk, ..., ωn) (27) on the n'th coupling tensor in the interaction lagrangian (4). in eq. (20) r(1) n (t) and r(2) n (t) are the noise forces in the linear regime and the first order of non-linearity , respectively, and using the symmetry 9 relation (24) are obtained as r(1) n i(t) = z ∞ 0 dω f (1) ij (ω)xj n(ω, t) r(2) n i(t) = z ∞ 0 dω z ∞ 0 dω′ f (2) ijk(ω, ω′) xj n(ω, t) xk n(ω′, t) + z ∞ 0 dω z ∞ 0 dω′ f (2) inm(ω, ω′) f (1) jm(ω′) z t −∞dt′ sin ω′(t −t′) ω′ × h xn n(ω, t) ̇ qj(t′) + ̇ qj(t′) xm n (ω′, t) i (28) where the summation should be done over the repeated indices and xn(ω, t) is the asymptotic solution (19). it is remarkable that for some known susceptibility tensors χ(1), χ(2), ..., χ(n), the coupling tensors f (1), f (2), ...f (n) satisfying the definitions (21),(22) and (25) are not unique. in fact if the coupling tensors f (1), f (2), ..., f (n) sat- isfy (21),(22) and (25) for the given susceptibility tensors, also the coupling tensors f ′(1), f ′(2), ..., f ′(n) defined by f ′(1) ij = f (1) im ajm f ′(n) i i1i2....in = f (n) i j1j2...jn ai1j1ai2j2....ainjn (29) satisfy (21),(22) and (25), where a is an orthogonal matrix aimamj = δij. the various choices of the coupling tensors f (1), f (2), ..., f (n) which is related to each other by the orthogonal transformation (29) do not change the physi- cal observables. the commutation relations between the dynamical variables of the total system remain unchanged under the orthogonal transformation (29). for example in the next section it is shown that the decay rate of an initially excited two-level atom, embedded in a non-linear absorbing environ- ment, are independent of the various choices of the coupling tensors which is related to each other by the transformation (29). now combination of the response equation (20) and equation (6) yields the non-linear lagevin-schr ̈ odinger equation m ̈ qi(t) + z +∞ −∞dt′ ̇ χ(1) ij (t −t′) ̇ qj(t′) + + z +∞ −∞dt′ z +∞ −∞dt′′ ̈ χ(2) ijk(t −t′, t −t′′) ̇ qj(t′) ̇ qk(t′′) + .... +∂v (⃗ q) ∂qi = − ̇ r(1) n i(t) − ̇ r(2) n i(t) + ... (30) 10 as the macroscopic equation of motion of the main system in the anisotropic non-linear absorbing environment. the velocity dependent terms in the left hand of this equation are the damping forces exerted on the main system. the noise forces − ̇ r(1) n i, − ̇ r(2) n i, ... in the right hand of (30) are necessary for a consistent quantization of the dissipative system. as a realization, if this quantization method would be applied for the electromagnetic field in the presence of an absorbing non-linear dielectric medium, the vector potential would satisfy the equation (30). in that case, the tensors χ(1), χ(2), .... would play the role of the electric susceptibility tensors and − ̇ r(1) n , − ̇ r(2) n , ... would be the noise polarization densities of various orders. 4 the effect of nonlinearity of the environ- ment on the spontaneous emission of a two- level atom imbedded in an absorbing envi- ronment in this section the effect of non-linearity of the absorbing environment is investigated on the spontaneous emission of a two-level atom embedded in such an environment. to calculate the spontaneous decay rate of an initially excited two-level atom, the quantization scheme in the preceding section is used and the theory of damping based on the density operator method is applied [31]. neglecting the second power of the operator r in (15) the hamiltonian of the total system can be written as h = h0 + h′ h0 = hs + hm = p2 2m + v (q) + 3 x i=1 z dω ħω b† i(ω)bi(ω) h′ = −p * r (31) let us suppose the main system is a one electron atom with two eigenstates |1⟩and |2⟩correspond to two eigenvalues e1 and e2, respectively (e2 > e1). 11 the hamiltonian (31) can now be rewritten as [31], [32]. h = h0 + h′ h0 = ħω0σ†σ + 3 x i=1 z dω ħω b† i(ω)bi(ω) ω0 = e2 −e1 ħ h′ = ı mω0r * h dσ −d∗σ†i (32) where m is the electron mass of the atom, σ = |1⟩⟨2| , σ† = |2⟩⟨1| , are the pauli operators and d = ⟨1|r|2⟩, where r is the position vector of the electron with respect to the center of mass of the atom. dropping the energy noncoserving terms correspond to rotating wave approximation and regarding the relation (17), the interaction term h′ up to the first order of nonlinearity in the interaction picture is expressed as h′ i(t) = e ıh0t ħ h′(0) e −ıh0t ħ = ımω0 z ∞ 0 dω s ħ 2ω f (1) ij (ω) h diσ b† j(ω) eı(ω−ω0)t −h.c i +ımω0 z ∞ 0 dω z ∞ 0 dω′ ħ 2 √ ωω′ f (2) ijk(ω, ω′) h diσb† j(ω)b† k(ω′)e−ı(ω0−ω−ω′)t +diσbj(ω)b† k(ω′)e−ı(ω0+ω−ω′)t + diσb† j(ω)bk(ω′)e−ı(ω0−ω+ω′)t −d∗ i σ†bj(ω)bk(ω′)eı(ω0−ω−ω′)t −d∗ i σ†bj(ω)b† k(ω′)eı(ω0−ω+ω′)t −d∗ i σ†b† j(ω)bk(ω′)eı(ω0+ω−ω′)ti (33) where the symmetry relation (23) has been used. let the combined density operator of the atom together with the environment is denoted by ρsr in the interaction picture. then, the reduced density operator of the atom alone, denoted by ρs, is obtained by taking the trace of ρsr with respect to the coordinates of the environment, that is ρs = trr[ρsr]. since it is assumed that h′ i(t) is sufficiently small, according to the density operator approach for the damping theory [31], the time evolution of the reduced density operator ρs is the solution of equation ̇ ρs(t) = −ı ħtrr[h′ i(t) , ρs(0) ⊗ρr(0)] −1 ħ2trr z t 0 dt′ [h′ i(t), , [h′ i(t′) , ρs(t) ⊗ρr(0)]] (34) 12 up to order of h′2 i , where ρr(0) is the density operator of the environment at t = 0. in this formalism the environment is taken in equilibrium. also the markovian approximation has been applied replacing ρs(t′) by ρs(t) in the integrand in eq.(34). to calculate the spontaneous emission of the atom, the initial states of the atom and the environment are taken as ρr(0) = |0⟩⟨0| ρs(0) = |2⟩⟨2| (35) where |0⟩is the vacuum state of the environment. now substituting h′ i(t) from (33) into (34) and regarding (35) the time evolution of the reduced density operator ρs is obtained as ̇ ρs = mω0 2 z ∞ 0 dω ω f (2) ijj (ω, ω)[diσe−ıω0t + h.c] −m2ω2 0 2ħ z ∞ 0 dω ω d∗ i f (1) ij (ω) f (1) lj (ω) dl z t 0 dt′ e−ı(ω−ω0)(t−t′) σ†σρs(t) + h.c  +m2ω2 0 ħ z ∞ 0 dω ω d∗ i f (1) ij (ω) f (1) lj (ω) dl σρs(t)σ† z t 0 dt′ cos(ω −ω0)(t −t′) −m2ω2 0 2 z ∞ 0 dω1 z ∞ 0 dω2 1 ω1ω2 di1 f (2) i1j1j1(ω1, ω1) f (2) i2j2j2(ω2, ω2) di2 × z t 0 dt′ σρs(t) σe−ıω0(t+t′) + h.c  +m2ω2 0 2 z ∞ 0 dω1 z ∞ 0 dω2 1 ω1ω2 d∗ i1 f (2) i1j1j1(ω1, ω1) f (2) i2j2j2(ω2, ω2) di2 × [σ†ρs(t) σ + σρs(t) σ†] z t 0 dt′ cos ω0(t −t′) +m2ω2 0 z ∞ 0 dω1 z ∞ 0 dω2 1 ω1ω2 d∗ i1 f (2) i1j1j2(ω1, ω2) f (2) i2j1j2(ω1, ω2) di2 ×σρs(t) σ† z t 0 dt′ cos(ω0 −ω1 −ω2)(t −t′) −m2ω2 0 4 z ∞ 0 dω1 z ∞ 0 dω2 1 ω1ω2 d∗ i1 f (2) i1j1j1(ω1, ω1) f (2) i2j2j2(ω2, ω2) di2 × z t 0 dt′ e−ıω0(t−t′) σσ†ρs(t) + h.c  13 −m2ω2 0 4 z ∞ 0 dω1 z ∞ 0 dω2 1 ω1ω2 d∗ i1 f (2) i1j1j1(ω1, ω1) f (2) i2j2j2(ω2, ω2) di2 × z t 0 dt′ e−ıω0(t−t′) ρs(t)σ†σ + h.c  −m2ω2 0 2 z ∞ 0 dω1 z ∞ 0 dω2 1 ω1ω2 d∗ i1 f (2) i1j1j2(ω1, ω2) f (2) i2j1j2(ω1, ω2) di2 × z t 0 dt′ e−ı(ω0−ω1−ω2)(t−t′) σ†σρs(t) + h.c  where the the repeated indices implies that the summation should be done over them. then, the equation of motion of the matrix elements ρs11 = ⟨1|ρs|1⟩, ρs22 = ⟨2|ρs|2⟩and ρs12 = ρ∗ s21 = ⟨1|ρs|2⟩now is obtained as ̇ ρs11 = m2ω2 0 ħ z ∞ 0 dω ω d∗ i f (1) ij (ω) f (1) lj (ω) dl ρs22(t) z t 0 dt′ cos(ω −ω0)(t −t′) +m2ω2 0 z ∞ 0 dω1 z ∞ 0 dω2 1 ω1ω2 d∗ i1 f (2) i1j1j2(ω1, ω2) f (2) i2j1j2(ω1, ω2) di2 ×ρs22(t) z t 0 dt′ cos(ω0 −ω1 −ω2)(t −t′) +m2ω2 0 2 z ∞ 0 dω1 z ∞ 0 dω2 1 ω1ω2 d∗ i1 f (2) i1j1j1(ω1, ω1) f (2) i2j2j2(ω2, ω2) di2 ρs22(t) ×[ρs22(t) −ρs11(t)] z t 0 dt′ cos ω0(t −t′) (36) ̇ ρs22 = −m2ω2 0 ħ z ∞ 0 dω ω d∗ i f (1) ij (ω) f (1) lj (ω) dl ρs22(t) z t 0 dt′ cos(ω −ω0)(t −t′) −m2ω2 0 z ∞ 0 dω1 z ∞ 0 dω2 1 ω1ω2 d∗ i1 f (2) i1j1j2(ω1, ω2) f (2) i2j1j2(ω1, ω2) di2 ×ρs22(t) z t 0 dt′ cos(ω0 −ω1 −ω2)(t −t′) −m2ω2 0 2 z ∞ 0 dω1 z ∞ 0 dω2 1 ω1ω2 d∗ i1 f (2) i1j1j1(ω1, ω1) f (2) i2j2j2(ω2, ω2) di2 ρs22(t) ×[ρs22(t) −ρs11(t)] z t 0 dt′ cos ω0(t −t′) (37) 14 ̇ ρs12 = ̇ ρ∗ s21 = mω0 2 z ∞ 0 dω ω f (2) ijj (ω, ω) die−ıω0t −m2ω2 0 2 z ∞ 0 dω1 z ∞ 0 dω2 1 ω1ω2 di1 f (2) i1j1j1(ω1, ω1) f (2) i2j2j2(ω2, ω2) di2 ×ρs22(t) z t 0 dt′e−ıω0(t+t′) (38) for sufficiently large times the integrals appeared in the equations (36) and (37) can be approximated by 1 π z t 0 dt′ cos(ω −ω0)(t −t′) ∼δ(ω −ω0) 1 π z t 0 dt′ cos(ω0 −ω1 −ω2)(t −t′) ∼δ(ω0 −ω1 −ω2) 1 π z t 0 dt′ cos ω0(t −t′) ∼δ(ω0) = 0 (39) hence the time evolution of the matrix elements ρs11 and ρs22 for sufficiently large times is reduced to ̇ ρs11 = γ ρs22 ̇ ρs22 = −γ ρs22 (40) where γ = πmω2 0 ħ d∗ i f (1) ij (ω)f (1) lj (ω)dl +πm2ω2 0 z ∞ 0 dω 1 ω(ω −ω0) d∗ i1 f (2) i1j1j2(ω, ω −ω0) f (2) i2j1j2(ω, ω −ω0) di2 (41) is the decay rate of the spontaneous emission of the initially excited two level atom up to the first order of nonlinearity. the first term in (41) is the decay rate in the absence of nonlinearity effects and the second term is the first contribution related to nonlinear effects of the environment. it may be noted from (40) that ̇ ρs11 + ̇ ρs22 = 0 which implies the conservation of the probability. an important point is that the decay rate γ is invariant under the various coupling tensors which is related to each other by the transformation (29). this should be so, because the decay rate γ is a physical observable. 15 5 summary a fully canonical quantization of a quantum system moving in an anisotropic non-linear absorbing environment was introduced. the main dissipative sys- tem was coupled with the environment through some coupling tensors of various ranks. the coupling tensors have an important role in this theory. based on a response equation, the forces against the motion of the main system were resolved into two parts, the damping forces and the noise forces. the response equation of the environment was obtained using the heisenberg equations describing the time evolution of the coordinates of the system and the environment. some susceptibility tensors of various ranks were attributed to the environment. the susceptibility tensors in the linear and non-linear regimes were defined in terms of the coupling tensors of the system and its environment. it was shown that, by imposing some symmetry conditions on the coupling tensors, the susceptibility tensors obey the symmetry proper- ties reported in the literature. a realization of this quantization method is the quantized electromagnetic field in the presence of a non-linear absorbing dielectric. finally the effect of the nonlinearity of the environment was in- vestigated on the spontaneous decay rate of a two-level atom imbedded in the non-linear environment. references [1] m. razavy, classical and quantum dissipative systems, imperial col- lege press(2005). [2] j. messer, acta phys. austriaca 50, 75 (1979). [3] h. dekker, phys. rep. 80, 1 (1981). [4] p. caldirola, nuovo cimento 18, 393 (1941). [5] e. kanai, prog. theoret. phys. 3, 440 (1948). [6] p. havas, nuovo cim. suppl. 5, 363 (1957). [7] h. h. denman, am. j. phys. 34, 1147 (1966). [8] w. e. brittin, phys. rev. 77,396 (1950). 16 [9] p. havas, bull. am. phys. soc. 1, 337 (1956). [10] g. valentini, rend. 1st. lomb. so: a 95, 255 (1961). [11] m. razavy, can. j. phys. 50, 2037 (1972). [12] h. haken, rev. mod. phys. 47, 67 (1975). [13] g. nicolis, i. prigogine, self-organization in non-equilibirium system, wiely, new york, (1977). [14] g. w. ford, j. t. lewis, r. f. o,connell, phys. rev. a 37, 4419(1988). [15] g. w. ford, j. t. lewis, r. f. o,connell, phys. rev. lett. 55, 2273(1985). [16] a. o. caldeira, a. j. leggett, phys. rev. lett. 46, 211(1981). [17] a. o. caldeira, a. j. leggett, ann. phys. (n.y.) 149, 374(1983). [18] a. h. castro neto, a. o. caldeira, phys. rev. lett. 67, 1960(1991). [19] b. l. hu, j. p. paz, y. zhang, phys. rev. d 45, 2843(1992). [20] jie-lou liao, e. pollak, chem. phys. 268, 295 (2001). [21] a. o. caldeira, a. j. leggett, physica 121a, 585 (1983): 130a, 374(1985). [22] a. o. caldeira, a. j. leggett, ann. phys. (n.y.) 153, 445(1984). [23] a. j. leggett, s. chakravarty, a. t. dorsey, m. p. a. fisher, a. grag, w. zwerger, rev. mod. phys. 59, 1 (1987). [24] h. g. shuster, v. r. veria, phys. rev. b 34, 189(1986). [25] g. w. ford, m. kac, p. mazur, j. math. phys. 6, 504(1964). [26] f. kheirandish, m.amooshahi, mod. phys. lett. a, vol.20, no. 39, 3025(2005). [27] f. kheirandish, m.amooshahi, phys. rev. a 74, 042102(2006). 17 [28] m.amooshahi, f. kheirandish, j. phys. a: math. theor. 41, 275402(2008). [29] m.amooshahi, j. math. phys. 50, 062301(2009). [30] guang s. he, song h. liu, physics of nonlinear optics, world scientific (1999). [31] m. o. scully, m. s. zubairy, quantum optics, cambridge, university press (1997). [32] p. w. milonni, the quantum vacuum, academic press(1994). 18

0911.1691

vertical partitioning of relational oltp databases using integer programming

a way to optimize performance of relational row store databases is to reduce the row widths by vertically partitioning tables into table fractions in order to minimize the number of irrelevant columns/attributes read by each transaction. this paper considers vertical partitioning algorithms for relational row-store oltp databases with an h-store-like architecture, meaning that we would like to maximize the number of single-sited transactions. we present a model for the vertical partitioning problem that, given a schema together with a vertical partitioning and a workload, estimates the costs (bytes read/written by storage layer access methods and bytes transferred between sites) of evaluating the workload on the given partitioning. the cost model allows for arbitrarily prioritizing load balancing of sites vs. total cost minimization. we show that finding a minimum-cost vertical partitioning in this model is np-hard and present two algorithms returning solutions in which single-sitedness of read queries is preserved while allowing column replication (which may allow a drastically reduced cost compared to disjoint partitioning). the first algorithm is a quadratic integer program that finds optimal minimum-cost solutions with respect to the model, and the second algorithm is a more scalable heuristic based on simulated annealing. experiments show that the algorithms can reduce the cost of the model objective by 37% when applied to the tpc-c benchmark and the heuristic is shown to obtain solutions with cost close to the ones found using the quadratic program.

introduction in this paper we consider oltp databases with an h-store [? ] like architecture in which we would aim for maximizing the number of single-sited transactions (i.e. transactions that can be run to completion on a single site). given a database schema and a workload we would like to reduce the cost of evaluating the workload. in row-stores, where each row is stored as a contiguous segment and access is done in quantums of whole rows, a significant amount of super- fluous columns/attributes (we will use the term attribute in the following) are likely to be accessed during evaluation of a workload. it is easy to see that this superfluous data access may have a negative impact on performance so in an ∗email: [email protected] 1 optimal world the amount of data accessed by each query should be minimized. one approach to this is to perform a vertical partitioning of the tables in the schema. a vertical partitioning is a, possibly non-disjoint, distribution of at- tributes and transactions onto multiple physical or logical sites. (notice, that vertical and horizontal partitioning are not mutually exclusive and can perfectly be used together). the optimality of a vertical partitioning depends on the con- text: olap applications with lots of many-row aggregates will likely benefit from parallelizing the transactions on multiple sites and exchanging small sub- results between the sites after the aggregations. oltp applications on the other hand, with many short-lived transactions, no many-row aggregates and with few or no few-row aggregates would likely benefit from gathering all attributes read by a query locally on the same site: inter-site transfers and the synchronization mechanisms needed for non-single-sited or parallel queries (e.g. undo and redo logs) are assumed to be bottlenecks in situations with short transaction dura- tions. ? ] and ? ] discuss the benefits of single-sitedness in high-throughput oltp databases in more details. this paper presents a cost model together with two algorithms that find either optimal or close-to-optimal vertical partitionings with respect to the cost model. the two algorithms are based on quadratic programming and simulated annealing, respectively. for a given partitioning and a workload, the cost model estimates the number of bytes read/written by access methods in the storage layer and the amount of data transfer between sites. our model is made with a specific setting in mind, captured by five headlines: oltp the database is a transaction processing system with many short lived transactions. aggregates no many-row aggregates and few (or no) aggregates on small row- subsets. preserve single-sitedness we should try to avoid breaking single-sitedness as a large number of single-sited transactions will reduce the need for inter-site transfers and completely eliminate the need for undo and redo logs for these queries if the partitioning is performed on an h-store like dmbs [? ]. workload known transactions used in the workload together with some run- time statistics are assumed to be known when applying the algorithms. furthermore, following the consensus in the related work (see section 1.3) we simplify the model by not considering time spent on network latency (if all vertical partitions are placed locally on a single site, then time spend on network latency is trivially zero anyway). a description of how to include latency in the model at the expense of increased complexity can be found in appendix a. 1.1 outline of approach the basic idea is as follows. we are given an input in form of a schema together with a workload in which queries are grouped into transactions, and each query is described by a set of statistical properties. for each query q in the workload and for each table r accessed by q the input provides the average number nr of rows from table r that is retrieved 2 from or written to storage by query q. together with the (average) width wa of each attribute a from table r this generally gives a good estimate for how much attribute a costs in retrievals/writes by access methods for each evaluation of query q, namely w ′ a,q = wa * nr. given a set of sites, the challenge is now to find a non-disjoint distribution of all attributes, and a disjoint distribution of transactions to these sites so that the costs of retrievals, writes and inter-site transfers, each defined in terms of w ′ a,q as explained in details below, is minimized. this means, that the primary executing site of any given query is assumed to be the site that hosts the transaction holding that query. as mentioned above, our algorithms will not break single-sitedness for read queries and therefore no additional costs are added to the execution of read queries by applying this algorithm. in contrast, since the storage costs (the sum of retrieval, write and inter-site transfer costs) for a query is minimized and each tuple become as narrow as possible, the total costs of evaluating the queries (e.g. processing joins, handling intermediate subresults, etc.) are assumed to be, if not minimized, then reduced too. 1.2 contributions this paper contributes with the following: • an algorithm optimized for h-store like architectures, preserving single- sitedness for read queries and in which load balance among sites versus minimization of total costs can be prioritized arbitrarily, • a more scalable heuristic, and • a micro benchmark of a) both algorithms based on tpc-c and a set of random instances, b) a comparison between the benefits of local versus remote partition location, and c) a comparison between disjoint and non- disjoint partitioning. 1.3 related work a lot of work has been done on data allocation and vertical partitioning but to the best of our knowledge, no work solves the exact same problem as the present paper: distributing both transactions and attributes to a set of sites, allowing attribute replication, preserving single-sitedness for read queries and prioritizing load balancing vs. total cost minimization. we therefore order the references below by increasing estimated problem similarity and do not mention work dedicated on vertical partitioning of olap databases. in ? ? ] reduced the cost of information retrieval by vertically partitioning records into a primary and a secondary record segment. this was done by constructing a bi-partite graph with two node sets: one set with a node for each attribute and one set with a node for each transaction. by connecting attribute and transaction nodes with a weighted edge according to their affinity, a min-cut algorithm could be applied to construct the partitioning. ? ] assumed a set of horizontal and vertical fragments of a database was known in advance and produced a disjoint distribution of these fragments onto a set of network-connected processors using a greedy first-fit bin packing heuristic. 3 similarly, ? ] distributed a set of predefined fragments to a set of sites, but used a linearized quadratic program to compute the solution. ? ] took as input a geographically distributed database together with statis- tics for a query pattern on this database and produced as output a non-disjoint distribution of whole database tables to the physical sites so that the total amount of transfer was minimized. they modelled the problem as a linearized quadratic program which was solved in practice using heuristics. the costs of joins were minimized by first transferring join keys and then transferring the relevant attributes for the relevant rows to a single collector site. ? ] constructed a disjoint partitioning with non-remote partition placement. they used an attribute affinity matrix to represent a complete weighted graph and generated partitions by finding a linearly connected spanning tree and con- sidering a cycle as a fragment. ? ] generated a non-remote, disjoint partitioning minimizing the amount of disk access by recursively applying a binary partitioning. the partitioning decisions were based on an integer program and with strong assumptions on a system-r like architecture when estimating the amount of disk access. ? ] also constructed a disjoint partitioning with non-remote partition place- ment. they used a two-phase strategy where the first phase generated all rel- evant attribute groups using association rules [? ] considering only one query at a time, and the second phase merged the attribute groups that were useful across queries. ? ] presented an algorithm for generating disjoint partitioning by either minimizing costs or by ensuring that exactly k vertical fragments were produced. inter-site transfer costs were not considered. the partitioning was produced using a bottom-up strategy, iteratively merging two selected partitions with the best "merge profit" until only one large super-partition existed. the k- way partitioning was found at the iteration having exactly k partitions and the lowest-cost partitioning was found at the iteration with the lowest cost. ? ] minimized the amount of disk access by constructing a non-remote and non-disjoint vertical partitioning. two binary partitioning algorithms based on the branch-and-bound method were presented with varying complexity and accuracy. the partitionings were formed by recursively applying the binary partitioning algorithms on the set of "reasonable cuts". ? ] did not present an algorithm but gave an interesting objective function for evaluating vertical partitionings. the function was based on the square-error criterion as given in [? ] for data clustering, but did not cover placement of transactions which, in our case, has a large influence on the expected costs. ? ] considered the vertical partitioning problem for three different environ- ments: a) single site with one memory level, b) single site with several memory levels, and c) multiple sites. the partitions could be both disjoint and non- disjoint. a clustering algorithm grouped attributes with high affinity by using an attribute affinity matrix together with a bond energy algorithm [? ]. three basic algorithms for generating partitions were presented which, depending on the desired environment, used different prioritization of four access and transfer cost classes. 4 1.4 outline of paper in section 2 we derive a cost model together with a quadratic program defining the first algorithm. section 3 describes a heuristic based on the cost model found in section 2, and section 4 discusses a couple of ideas for improvements. computational results are shown in section 5. 2 a linearized qp approach in this section we develop our base model, a quadratic program (qp), which will later be extended to handle load balancing and then linearized in order to solve it using a conventional mixed integer program (mip) solver. 2.1 the base model in a vertical partitioning for a schema and a workload we would like to minimize the sum a + pb (1) where a is the amount of data accessed locally in the storage layer, b is the amount of data needed to be transferred over the network during query updates and p is a penalty factor. we assume that each transaction has a primary executing site. for each transaction t ∈t , each table attribute a ∈a, and each site s ∈s consider two decision variables xt,s ∈{0, 1} and ya,s ∈{0, 1} indicating if transaction t is executed on site s and if attribute a is located on site s, respectively. all transactions must be located at exactly one site (their primary executing site), that is x t∈t xt,s = 1 , ∀s ∈s (2) and all attributes must be located at at least one site, that is x a∈a ya,s ≥1 , ∀s ∈s. to determine the size of a and b from equation (1) introduce five new static binary constants describing the database schema: • αa,q indicates if attribute a itself is accessed by query q • βa,q indicates if attribute a is part of a table that q accesses • γq,t indicates if query q is used in transaction t • δq indicates if query q is a write query • φa,t indicates if any query in transaction t reads attribute a single-sitedness should be maintained for reads. that is, if a read query in transaction t accesses attribute a then a and t must be co-located: xt,sφa,t = 1 ⇒ya,s = 1 , ∀t ∈t , a ∈a 5 or equivalently ya,s −xt,sφa,t ≥0 , ∀t ∈t , a ∈a. in order to estimate the cost of reading, writing and transferring data, in- troduce the following weights: • wa denotes the average width of attribute a • fq denotes the frequency of query q • na,q denotes for query q the average number of rows retrieved from or written to the table holding attribute a then the cost of reading or writing a in query q is estimated to wa,q = wa*fq*na,q and the cost of transferring attribute a over the network is estimated to pwa,q. notice, that wa,q is only an estimate due to fq and na,q. consider the amount of local data access, a, and let a = ar + aw where ar and aw is the amount of read and write access, respectively. for a given site r and query q, ar is the sum of all attribute weights wa,q for which 1) q is a read query, 2) attribute a is stored on r, 3) the transaction that executes query q is executed on r and 4) q accesses any attribute in the table fraction that holds a. as we maintain single-sitedness for reads, βa,q can be used to handle 4), resulting in ar = x a,t,s,q wa,qβa,qγq,t(1 −δq)xt,sya,s. accounting for local access of write queries, aw, is less trivial. consider the following three approaches: access relevant attributes an attribute a at site s should be accounted for if and only if there exists an attribute a′ on s that q updates so that a and a′ are attributes of the same table. while this accounting is the most accurate of the three it is also the most expensive as it implies an element of the form ya,sya′,s in the objective function which adds an undesirable amount of |a|2|s| variables and 3|a|2|s| constraints to the problem when linearized (see section 2.3). access all attributes we can get around the increased complexity by assum- ing that write queries q always writes to all sites containing table fractions of tables accessed by q, regardless of whether q actually accesses any of the attributes of the fractions. while this is correct for insert statements (assuming that inserts always write complete rows) it is likely an overes- timation for updates: imagine a lot of single-attribute updates on a wide table where the above method would have split the attribute in question to a separate partition. this overestimation will imply that the model will partition tables that are updated often or replicate attributes less often than the accounting model described above. access no attributes another approach to simplify the cost function is to completely avoid accounting for local access for writes and solely let the network transfer define the write costs. with this underestimation of write costs, attributes will then tend to be replicated more often than in the first accounting model. 6 in this paper we choose the second approach, which gives a conservative overes- timate of the write costs as we then obtain more accurate costs for inserts and avoid extending the model with undesirably many variables and constraints. intuitively speaking, this choice implies that read queries will tend to partition the tables for best possible read-performance, and the write queries will tend to minimize the amount of attribute replication. we now have aw = x q,a,s wa,qβa,qδqya,s and thus a = x a,t,s,q wa,qβa,qγq,t(1 −δq)xt,sya,s + x q,a,s wa,qβa,qδqya,s. (3) b accounts for the amount of network transfer and since we enforce single- sitedness for all reads b is solely the sum of transfer costs for write queries. we assume that write queries only transfer the attributes they update and does not transfer to the site that holds their own transaction: b = x a,t,s,q wa,qαa,qγq,tδq(1 −xt,s)ya,s. by noticing that p a,t,s,q αa,qγq,tya,s = p a,s,q αa,qya,s we can construct the minimization problem as min p t,a,s c1(a, t)xt,sya,s + p a,s c2(a)ya,s s.t. p s xt,s = 1 ∀t p s ya,s ≥1 ∀a ya,s −xt,sφa,t ≥0 ∀a, t xt,s, ya,s ∈{0, 1} ∀t, a, s (4) where c1(a, t) = x q wq,aγq,t(βa,q(1 −δq) −pαa,qδq) and c2(a) = x q wa,qδq(βa,q + pαa,q). both c1 and c2 are completely induced by the schema, query workload and statistics and can therefore be considered static when the partitioning process starts. 2.2 adding load balancing we are interested in extending the model in (4) to also handle load balancing of the sites instead of just minimizing the sum of all data access/transfer. from equation (3) define the work of a single site s ∈s as x a,t c3(a, t)xt,sya,s + x a c4(a)ya,s (5) where c3(a, t) = p q wa,qγq,tβa,q(1 −δq) and c4(a) = p q wa,qβa,qδq. introduce the variable m and for each site s let the value of (5) be a lower bound for m. 7 adding m to the objective function is then equivalent to also minimizing the work of the maximally loaded site. in order to decide how to prioritize cost minimization versus load balancing in the model, introduce a scalar 0 ≤λ ≤1 and weight the original cost from (4) and m by λ and (1 −λ), respectively. the new objective is then λ x a,t,s c1(a, t)xt,sya,s + λ x a,s c2(a)ya,s + (1 −λ)m (6) where m is constrained as follows: x a,t c3(a, t)xt,sya,s + x a,q c4(a)ya,s ≤m , ∀s ∈s. notice that while we are now minimizing (6), the objective of (4) should still be considered as the actual cost of a solution. 2.3 linerarization we use the technique discussed in [? , chapter iv, theorem 4] to linearize the model. this is done by replacing the quadratic terms in the model with a variable ut,a,s and adding the following new constraints: ut,a,s ≤xt,s ∀t, a, s ut,a,s ≤ya,s ∀t, a, s ut,a,s ≥xt,s + ya,s −1 ∀t, a, s for ut,a,s ≥0, notice that ut,a,s = 1 if and only if xt,s = ya,s = 1 and that ut,a,s is guaranteed to be binary if both xt,s and ya,s are binary (thus, there is no need for requiring it explicitly in the model). now, the model in (4) extended with load balancing looks as follows when linearized: min λ p t,a,s c1(a, t)ut,a,s + λ p a,s c2(a)ya,s + (1 −λ)m s.t. p s xt,s = 1 ∀t p s ya,s ≥1 ∀a ya,s −xt,sφa,t ≥0 ∀a, t p a,t c3(a, t)ua,t,s + p a,q c4(a)ya,s ≤m ∀s ut,a,s −xt,s ≤0 ∀t, a, s ut,a,s −ya,s ≤0 ∀t, a, s ut,a,s −xt,s −ya,s + 1 ≥0 ∀t, a, s xt,s, ya,s ∈{0, 1} ∀t, a, s ut,a,s ≥0 ∀t, a, s (7) 2.4 complexity the objective function in quadratic programs can be written on the form 1 2ztqz + cz + d where in our case z = (x1,1, . . . , x|t |,|s|, y1,1, . . . , x|a|,|s|) is a vector containing the decision variables, q is a cost matrix, c is a cost vector and d a constant. 8 q can be easily defined from (6) by dividing q into four quadrants, letting the sub-matrices in the upper-left and lower-right quadrant equal zero and letting the upper-right and lower-left submatrices be defined by c1(a, t). q is indefinite and the cost function (6) therefore not convex. as shown by ? ] finding optimum when q is indefinite is np-hard. 3 the sa solver – a heuristic approach we develop a heuristic based on simulated annealing (see [? ]) and will refer to it as the sa-solver from now on. the base idea is to alternately fix x and y and only optimize the not-fixed vector, thereby simplifying the problem. in each iteration we search in the neighborhood of the found solution and accept a worse solution as base for a further search with decreasing probability. let xt,s hold an assignment of transactions to sites and define the neighbor- hood x′ of x as a change of location for a subset of the transactions so that for each t ∈t we still have p s x′ t,s = 1. similarly, let ya,s hold an assignment of attributes to sites but define the neighborhood y′ of y as an extended replication of a subset of the attributes. that is, for each a ∈a in that subset we have ya,s = 1 ⇒y′ a,s and p s y′ a,s > p s ya,s. we found that altering the location for a constant number of 10% of both transactions/attributes yielded the best results. the heuristic now looks as pictured in algorithm 1. notice, that the algorithm 1 the heuristic based on simulated annealing (sa). it iteratively fixes x and y and accepts a worse solution from the neighborhood with decreasing probability. 1: initialize temperature τ > 0 and reduction factor ρ ∈]0; 1[ 2: set the number l of inner loops 3: initialize x randomly so that (2) is satisfied 4: fix ←"x" 5: s ←findsolution(fix) 6: while not frozen do 7: for i ∈{1, . . ., l} do 8: x ←neighborhood of x 9: y ←neighborhood of y 10: s′ ←findsolution(fix) 11: ∆←cost(s′) −cost(s) 12: p ←a randomly chosen number in [0; 1] 13: if ∆≤0 or p < e−∆/τ then 14: s ←s′ 15: end if 16: fix ←the element in {"x","y"} \ {fix} 17: end for 18: τ ←ρ * τ 19: end while linearization constraints is not needed since either x or y will be constant in each iteration. this reduces the size of the problem considerably. 9 4 further improvements consider a table with n attributes together with two queries: one accessing at- tribute 1 through k and one accessing attribute k through n. then it is sufficient to find an optimal distribution for the three attribute groupings {1, . . ., k −1}, {k} and {k + 1, . . . , n}, considering each group as an atomic unit and thereby reducing the problem size. in general, it is only necessary to distribute groups of attributes induced by query access overlaps. ? ] refer to these attribute overlaps as reasonable cuts. even though this will not improve the worst-case complexity, this reduction may still have a large performance impact on some instances. also, assuming that transactions follow the 20/80 rule (20% of the transac- tions generate 80% of the load), the problem can be solved iteratively over t starting with a small set of the most heavy transactions. 5 computational results we assume that the context is a database with a very high transaction count like the memory-only database h-store [? ] (now voltdb1) and thus need to compare ram access versus network transfer time when deciding an appro- priate network penalty factor p. a pci express 2.0 bus transfers between 32 gbit/s and 128 gbit/s while the bandwidth of pc3 ddr3-sdram is at least 136 gbit/s so the bus is the bottleneck in ram accesses. we assume that the network is well configured and latency is minimal. therefore the network penalty factor could be estimated to p ∈[3; 128] if either a gigabit or 10-gigabit network is used to connect the physical sites. we assume the use of a 10-gigabit network and therefore set p = 8 in our tests unless otherwise stated. we furthermore mainly focus on minimizing the total costs of execution and therefore set λ low. if λ is kept positive the model will, however, choose the more load balanced layout if there is a cost draw between multiple layouts. we set λ = 0.1 in our tests unless otherwise stated. all tests were run on a macbook pro with a 2.4 ghz intel core 2 duo and 4gb 1067 mhz ddr3 ram, running mac os x 10.5. the gnu linear programming kit2 (glpk) 4.39 was used as mip solver, using only a single thread. the test implementation is available upon request. 5.1 initial temperature the temperature τ used in the heuristic described in section 3 determines how willing the algorithm is to accept a worse solution than the currently best found. let c∗and c denote the objective for the best solution so far and the currently generated solution, respectively. in the computational results provided here we accept a worse solution with 50% probability in the first set of iterations if c−c∗ c < 5%. referring to the notation used in algorithm 1, we have 50% = e0.05c∗/τ and thus an initial temperature of τ = −0.05c∗/ ln 0.5. 1http://voltdb.com 2http://gnu.org/software/glpk 10 5.2 the tpc-c v5 instance we perform tests on the tpc-c version 5.10.1 benchmark3. the tpc-c specifi- cation describes transactions, queries and database schema but does not provide the statistics needed to create a problem instance. we therefore made some sim- plified assumptions: all queries are assumed to run with equal frequency and all queries (not transactions) are assumed to access a single row except in the obvious cases where aggregates are used or there are being iterated over the result. in these cases we assume that the query accesses 10 rows. thereby, the new-order transaction for example, are assumed to access 11 rows in average. we model update queries as two sub-queries: a read-query accessing all the attributes used in the original query and a write-query only accessing the attributes actually being written (and thus whose update needs to be distributed to all replicas). 5.3 random instances to the best of our knowledge there is no standard library of typical oltp instances with schemas, workloads and statistics so in order to explore the char- acteristics of the algorithms we perform some experiments on a set of randomly generated instances instead as it showed up to be a considerable administrative and bureaucratic challenge (if possible at all) to collect appropriate instances from "real life" databases. the randomly generated instances vary in several parameters in order to clarify which characteristics that influence the potential cost reduction by applying our vertical partitioning algorithms. the parame- ters include: number of transactions in workload, number of tables in schema, maximum number of attributes per table, maximum number of queries per transaction, percentage of queries being updates, maximum number of different tables being referred to from a single query, maximum number of individual at- tributes being referred to by a single query, the set of allowed attribute widths. we define classes of problem instances by upper bounds on all parameters. in- dividual instances are then generated by choosing the value of each parameter evenly distributed between 1 and its upper bound. that is, if e.g. the maximum allowed number of attributes in tables is k, the number of table attributes for each table in the generated instance will be evenly distributed between 1 and k with a mean of k/2. 5.4 results in the following we perform a series of tests and display the results in tables where each entry holds the found objective of (4) for the given instance. table 1 explores the influence of a set of parameters in the randomly gener- ated instances by varying one parameter at a time while fixing the rest. we test two classes of instances using the sa solver: a smaller with #tables = |t | = 20 and a larger with #tables = |t | = 100. the results suggest that the largest workload reduction is obtained for instances having relatively few queries per transaction, few updates, many attributes per table and/or a moderate number of attribute references per query. the number of table references per query and 3http://www.tpc.org/tpcc 11 the allowed attribute widths, however, only seem to have moderate influence on the result. #tables = |t | = 20 #tables = |t | = 100 |s| = 1 |s| = 2 |s| = 3 |s| = 1 |s| = 2 |s| = 3 a max queries per transaction 1 0.585 0.309 0.278 3.194 1.784 1.471 3 1.567 1.478 1.386 5.743 4.550 4.189 5 1.305 1.054 0.972 8.840 7.569 6.983 b percent updates queries 0 1.747 1.369 1.110 5.959 4.235 3.510 10 1.567 1.478 1.386 5.743 4.550 4.189 30 1.349 1.244 1.263* 5.106 4.555 4.462 c max attributes per table 5 0.520 0.520* 0.520* 2.583 2.772* 2.712* 15 1.567 1.478 1.386 5.743 4.550 4.189 35 1.643 0.968 0.850 14.970 7.341 5.355 d max table references per query 2 0.602 0.430 0.356 3.447 3.022 2.865 5 1.567 1.478 1.386 5.743 4.550 4.189 10 2.246 1.607 1.516 8.147 6.063 5.623 e max attribute references per query 5 0.678 0.288 0.199 5.176 2.526 1.969 15 1.567 1.478 1.386 5.743 4.550 4.189 25 1.115 0.988 1.008* 5.641 5.909* 5.684* f allowed attribute widths {2, 4, 8} 1.194 1.080 1.030 4.456 3.488 3.500* {4, 8} 1.567 1.478 1.386 5.743 4.550 4.189 {4, 8, 16} 2.387 2.160 2.060 8.912 6.977 7.000 table 1: comparing the effect of parameter changes. results were found using the sa solver. we test three possible values for each parameter, varying one parameter at the time and fixing all other parameters at their default value (marked with bold). the costs are shown in units of 106. tests are divided into two classes having both the number of transactions and schema tables equal to 20 (left) and 100 (right), respectively. the results suggest that the largest workload reduction, unsurprisingly, is obtained for instances having relatively few queries per transaction, few updates, many attributes per table and/or a moderate number of attribute references per query. the number of table references per query and the allowed attribute widths, however, only seem to have moderate influence on the result. table 3 compares the qp and sa solvers on the tpc-c benchmark and a set of randomly generated larger instances, divided into two classes with either large or low potential for cost reduction. the random instances are described in table 2 where the columns here refer to the single-letter labels for the pa- rameters shown in table 1. as seen in table 3 the sa solver is generally faster than the qp solver but the qp solver obtains lower costs when the in- stances are small. expectedly, the instances in class "rndb. . . " with many attribute references per query but few queries per table gains little or no cost reduction by applying the algorithms. tpc-c, on the other hand, gets a cost reduction of 37% and the random instances in class "rnda. . . ", with many at- tributes per table and relatively few attribute references per query, get a cost reduction between 25% and 85%. none of the algorithms found a cost reduction for the instances rndat4x100 and rndat8x100 because of the "overweight" of transactions compared to the number of attributes in the schemas. table 4 depicts an actual partitioning of tpc-c constructed by the qp solver for three sites. table 5 illustrates the effect of disjoint versus nondisjoint partitioning, that is, partitioning without and with attribute replication. as seen, greater cost re- duction can be obtained when allowing replication but in exchange to increased 12 name a b c d e f |t | #tables rndat4x15 3 10 30 3 8 {2, 4, 8, 16} 15 4 rndat8x15 3 10 30 3 8 {2, 4, 8, 16} 15 8 rndat8x15u50 3 50 30 3 8 {2, 4, 8, 16} 15 8 rndat16x15 3 10 30 3 8 {2, 4, 8, 16} 15 16 rndat32x15 3 10 30 3 8 {2, 4, 8, 16} 15 32 rndat4x100 3 10 30 3 8 {2, 4, 8, 16} 100 4 rndat8x100 3 10 30 3 8 {2, 4, 8, 16} 100 8 rndat16x100 3 10 30 3 8 {2, 4, 8, 16} 100 16 rndat32x100 3 10 30 3 8 {2, 4, 8, 16} 100 32 rndbt4x15 3 10 5 6 28 {2, 4, 8, 16} 15 4 rndbt8x15 3 10 5 6 28 {2, 4, 8, 16} 15 8 rndbt16x15 3 10 5 6 28 {2, 4, 8, 16} 15 16 rndbt16x15u50 3 50 5 6 28 {2, 4, 8, 16} 15 16 rndbt32x15 3 10 5 6 28 {2, 4, 8, 16} 15 32 rndbt4x100 3 10 5 6 28 {2, 4, 8, 16} 100 4 rndbt8x100 3 10 5 6 28 {2, 4, 8, 16} 100 8 rndbt16x100 3 10 5 6 28 {2, 4, 8, 16} 100 16 rndbt32x100 3 10 5 6 28 {2, 4, 8, 16} 100 32 table 2: random instances used when comparing the qp and sa solvers in table 3. the instances in the upper part (rnda. . . ) are expected to get a large cost reduction while instances in the lower part (rndb. . . ) are expected to get a small cost reduction. the columns refer to the single- letter labels for the parameters shown in table 1. qp sa instance |a| |t | |s| cost time (s) cost time (s) |s| = 1 tpc-c v5 92 5 2 0.133 1 0.138 5 0.208 tpc-c v5 92 5 3 0.132 6 0.132 5 0.208 tpc-c v5 92 5 4 0.132 33 0.132 5 0.208 rndat4x15 54 15 4 (0.332) 1800 0.396 10 0.933 rndat8x15 105 15 4 (0.324) 1800 0.327 18 0.808 rndat16x15 225 15 4 (0.267) 1800 0.309 41 1.180 rndat32x15 492 15 4 (0.315) 1800 0.217 89 1.491 rndat64x15 1023 15 4 (0.269) 1800 0.268 190 1.452 rndat4x100 54 100 4 (8.001) 1800 8.246 79 7.946 rndat8x100 105 100 4 (7.681) 1800 8.018 150 7.454 rndat16x100 225 100 4 - t/o 6.525 321 8.741 rndat32x100 492 100 4 - t/o 4.501 728 8.916 rndat64x100 1023 100 4 - t/o 4.119 1531 9.591 rndbt4x15 12 15 4 0.303 65 0.303 3 0.303 rndbt8x15 27 15 4 (0.448) 1800 0.424 6 0.440 rndbt16x15 49 15 4 (0.333) 1800 0.334 9 0.385 rndbt32x15 98 15 4 (0.319) 1800 0.319 16 0.361 rndbt64x15 210 15 4 (0.221) 1800 0.221 31 0.229 rndbt4x100 54 100 4 (4.484) 1800 2.251 18 2.251 rndbt8x100 105 100 4 (4.323) 1800 2.419 37 2.419 rndbt16x100 225 100 4 (2.001) 1800 1.774 62 1.774 rndbt32x100 492 100 4 (2.419) 1800 1.999 124 1.999 rndbt64x100 1023 100 4 - 1800 2.473 270 2.473 table 3: comparing the qp algorithm with the simulated annealing based heuristic (sa), allowing attribute replication and with remote partition placement. costs are shown in units of 106. the sa algorithm had a 30 second time limit for each iteration and if the limit was reached it pro- ceeded with another neighborhood. the qp algorithm had a time bound of 30 minutes and an mip tolerance gap of 0.1%. where the time limit was reached, the best found cost (if any) is written in parentheses. "t/o" indicates that no integer solution was found within the time limit. 13 site 1 transaction payment customer.c balance customer.c city customer.c credit customer.c credit lim customer.c data customer.c discount customer.c d id customer.c first customer.c id customer.c last customer.c middle customer.c phone customer.c since customer.c state customer.c street 1 customer.c street 2 customer.c w id customer.c zip district.d city district.d id district.d name district.d state district.d street 1 district.d street 2 district.d w id district.d ytd district.d zip history.h amount history.h c d id history.h c id history.h c w id history.h data history.h date history.h d id history.h w id orderline.ol dist info orderline.ol number stock.s order cnt stock.s remote cnt stock.s ytd warehouse.w city warehouse.w id warehouse.w name warehouse.w street 1 warehouse.w street 2 warehouse.w ytd warehouse.w zip site 2 transaction stocklevel customer.c city customer.c delivery cnt customer.c payment cnt customer.c since customer.c ytd payment district.d id district.d next o id district.d w id item.i im id orderline.ol d id orderline.ol i id orderline.ol o id orderline.ol w id stock.s i id stock.s quantity stock.s w id site 3 transaction delivery transaction neworder transaction orderstatus customer.c balance customer.c credit customer.c discount customer.c d id customer.c first customer.c id customer.c last customer.c middle customer.c w id district.d id district.d next o id district.d tax district.d w id item.i data item.i id item.i name item.i price neworder.no d id neworder.no o id neworder.no w id order.o all local order.o carrier id order.o c id order.o d id order.o entry d order.o id order.o ol cnt order.o w id orderline.ol amount orderline.ol delivery d orderline.ol d id orderline.ol i id orderline.ol o id orderline.ol quantity orderline.ol supply w id orderline.ol w id stock.s data stock.s dist 01 stock.s dist 02 stock.s dist 03 stock.s dist 04 stock.s dist 05 stock.s dist 06 stock.s dist 07 stock.s dist 08 stock.s dist 09 stock.s dist 10 stock.s i id stock.s quantity stock.s w id warehouse.w id warehouse.w tax table 4: the result of a vertical partitioning of the tpc-c benchmark using the qp solver for three sites. each column represents the contents of a site and is divided into three sub-sections: a header, a section holding the transaction names and a longer section holding the attributes assigned to the respective site. 14 computation time. w. replication w/o replication instance |a| |t | |s| cost time (s) cost time (s) ratio tpc-c v5 92 5 1 0.208 0 0.208 0 - tpc-c v5 92 5 2 0.133 1 0.207 1 64% tpc-c v5 92 5 3 0.132 6 0.207 2 64% tpc-c v5 92 5 4 0.132 33 0.207 3 64% rndat4x15 54 15 2 4.855 28 6.799 1 71% rndat8x15 105 15 2 4.710 517 5.809 6 81% rndat8x15 27 15 2 4.244 4 4.402 0 96% rndat16x15 49 15 2 3.410 34 3.852 0 89% table 5: computational results from solving the tpc-c benchmark and a few random instances with the qp solver. costs are shown in units of 105. the table shows that costs can be reduced by allowing attribute replication and that tpc-c does not benefit noticeably from being partitioned and distributed to more than two sites. the ratio column displays the ratio between the replicated and non-replicated cost. table 6 compares two different kinds of partition placements: 1) all partitions being located at one single site (thereby avoiding inter-site transfers) and 2) partitions being located at remote sites. these two situations can be simulated by setting p = 0 and p > 0, respectively. the benefits of local placements are given by the amount of updates in the workload as only updates cause inter- site transfers. more updates implies larger costs for remote placements. for a somewhat extreme case, instance "rndat8x15u50", with 50% of the queries being updates, the costs are about 33% lower when placing the partitions locally. local remote instance |a| |t | |s| cost (qp) cost (sa) cost (qp) cost (sa) tpc-c v5 92 5 1 1.916 1.916 1.916 1.916 tpc-c v5 92 5 2 1.210 1.208 1.221 1.273 tpc-c v5 92 5 3 1.208 1.208 1.220 1.220 rndat4x15 54 15 2 4.709 4.742 4.855 4.888 rndat8x15 105 15 2 4.424 4.808 4.710 5.187 rndat8x15u50 105 20 2 3.189 3.313 4.778 4.873 rndbt8x15 27 15 2 4.365 4.332 4.244 4.730 rndbt16x15 49 15 2 3.335 3.387 3.410 3.404 rndbt16x15u50 49 20 2 5.066 5.220 5.438 5.438 table 6: comparing the costs of local (p = 0) versus remote (p > 0) location of partitions and with attribute replication allowed. costs are in units of 105. write-rarely instances or instances in class "rndb. . . " do not benefit noticeably by placing all partitions locally, even the instances with 50% update queries, however instances in class "rnda. . . " with a large update ratio do. the reason is that only updates cause inter-site transfer. that the costs of the local placement for rndbt8x15 is larger than when placed remotely is since λ > 0. 15 6 conclusion we have constructed a cost model for vertical partitioning of relational oltp databases together with a quadratic integer program that distributes both at- tributes and transactions to a set of sites while allowing attribute replication, preserving single-sitedness for read queries and in which load balancing vs. total cost minimization can be prioritized arbitrarily. we also presented a more scalable heuristic which seems to deliver good results. for both algorithms we obtained a cost reduction of 37% in our model of tpc-c and promising results for the random instances. even though the latter theoretically can be constructed with arbitrary high/low benefits from vertical partitioning, the test runs on our selected subset of random instances seem to indicate that 1) our heuristic scales far better than the qp-solver, and 2) it can obtain valuable cost reductions on many real-world oltp databases, as we tried to select the parameters realistically. one thing we miss, however, is an official oltp testbed – a library con- taining realistic oltp workloads, schemas and statistics. such a collection of realistic instances could serve as base for several insteresting and important studies for understanding the nature and characteristics of oltp databases. acknowledgements the author would like to acknowledge daniel abadi for competent and valuable discussions and feedback. also, rasmus pagh, philippe bonnet and laurent flindt muller have been very helpful with insightful comments on preliminary versions of the paper. 16 a latency this section describes how to extend the algorithms to also estimate costs of network latency for queries accessing attributes on remote sites. we assume, that all remote access (if any) for queries are done in parallel and with a constant number of requests per query per remote site. let pl denote a latency penalty factor and introduce a new binary variable ψq for each query q indicating with ψq = 1 if q accesses any remotely placed attributes. letting n denote the number of remotely accessed attributes by q we have n > 0 ⇒ψq = 1 and n = 0 ⇒ψq = 0, or equivalently (ψq −1)n = 0 and ψq −n ≤0. this results in the following two classes of new constraints: (ψq −1) x a,s δqαa,qγq,t(1 −xt,s)ya,s = 0 , ∀q, t and ψq − x a,s δqαa,qγq,t(1 −xt,s)ya,s ≤0 , ∀q, t the total latency in a given partitioning can now be estimated by the sum pl p q fqψq which can be added to the cost objective function (4). 17

0911.1692

energy spectrum of cosmic ray muons in ~ 100 tev energy region reconstructed from the bust data

differential and integral energy spectra of cosmic ray muons in the energy range from several tev to ~ 1 pev obtained by means of the analysis of multiple interactions of muons (pair meter technique) in the baksan underground scintillation telescope (bust) are presented. the results are compared with preceding bust data on muon energy spectrum based on electromagnetic cascade shower measurements and depth-intensity curve analysis, with calculations for different muon spectrum models, and also with data of other experiments.

introduction energy spectrum of muons plays an important role in the physics of high energy cosmic rays. its characteristics depend on the primary cosmic ray spectrum and composition, and also on the processes of primary particle interactions with nuclei of air atoms. therefore information on muon energy spectrum may be used, on the one hand, for extraction of independent estimates of primary spectrum and composition if to suppose that interaction model is known, and, on the other hand, under certain assumptions about primary cosmic ray spectrum and composition, for the search of possible changes in characteristics of hadron interactions above the energy limit reached in accelerator experiments. the region of muon energies above 100 tev is of a special interest. in this region, the contribution of "prompt" muons from decays of charmed and other short-lived particles at reasonable suppositions about cross sections of their production can appear. this has to give some excess of such muons. but on the other hand the influence of the knee in the primary energy spectrum on muon spectrum shape is expected if the knee has really astrophysical origin. this effect leads to the decrease of muon flux at such energies. in the alternative case, if the spectrum of primary particles does not change its slope and appearance of the knee is connected with interaction model changes, the inclusion of new physical processes or states of matter is required. these processes can change the whole picture of eas development, and standard estimations of eas energies can appear wrong. as shown in [1], in this case the excess of muons will be increasing with energy rather sharply. usually, possible contribution of any fast (in comparison with decays of pions and kaons) processes to generation of muons is taken into account by means of introduction of the parameter r in the formula describing the inclusive spectrum of high-energy muons in the atmosphere [2]: dnμ deμ = 0.14e−γμ μ ×               1 1 + 1.1eμ cos θ 115 + 0.054 1 + 1.1eμ cos θ 850 + r               [cm−2 s−1 sr−1 gev−1]. (1) ∗corresponding author. address: nevod, mephi, kashirskoe sh. 31, moscow 115409, russia. email address: [email protected] (a.g. bogdanov) august 5, 2011 here r is the ratio of the number of these prompt muons to the number of charged pions with the same energy at production; muon energy is measured in gev; γμ = 2.7. the slope of the energy spectrum of prompt muons is about a unit less than that from decays of pions and kaons. in contrast to high energy muons from π-, k-decays, the flux of prompt muons does not exhibit sec θ enhancement with the increase of zenith angle. unfortunately, cross sections of charmed particle production in a necessary range of kinematic variables are poorly known, and existing theoretical estimates of the r value have a large spread. nevertheless, the expected range of energies where the fluxes of prompt muons and muons from π-, k-decays become comparable is near 100 tev [3]; differential spectra of prompt muons and "usual" muons (from pion and kaon decays) are equal to each other at this energy for r ≈10−3. if the observed knee in extensive air showers (eas) energy spectrum at pev energies is related with the inclusion of new physical processes (or formation of a new state of matter) with production in a final state of very high energy (vhe) muons [4], then their contribution to muon energy spectrum may be estimated by the formula: divheμ deμ = aγ1(e0/106 gev)−γ1 e0 × n2 μ fμ h 1 −(γ1/γ2) (ek/e0)(γ2−γ1)/γ2i, (2) where eμ and primary particle energy e0 are related as eμ nμ/fμ = ∆(e0) = e0 −ek (e0/ek)γ1/γ2 , e0 > ek. (3) here γ1 and γ2 are integral eas energy spectrum slopes below and above the knee energy ek; nμ is a typical multiplicity of produced vhe muons; fμ is a fraction of the difference between primary particle energy and measured eas energy which is carried away by vhe muons; a = 1.5 × 10−10 cm−2 s−1 sr−1 is the integral intensity of primary particles with energy above 1 pev. appearance of vhe muons is also expected at energies about 100 tev; however, their relative contribution should increase with energy more rapidly compared to muons from charmed particle decays, and this feature is the only one which could allow separate two hypotheses on possible reasons of changes in the muon energy spectrum behavior. there are few experimental data in the energy region close to 100 tev (including the data obtained with bust), and they have a very wide spread (see, e.g., review [5]). this spread is most probably caused by various uncertainties of the methods used for investigations of the muon energy spectrum in the range ≥10 tev. unfortunately, the most direct method of muon energy spectrum study – magnetic spectrometer technique – did not allow reach energies above 10 tev because of both technical (the necessity to ensure high magnetic field induction simultaneously with manifold increase of magnetized volume) and physical (increase of probability of secondary electron contamination in the events with the increase of muon energy) reasons. therefore, two other methods of muon spectrum investigations – calorimeter measurements of muon-induced cascade shower spectrum and depth- intensity curve analysis – were mainly used. method based on the depth-intensity measurements has serious uncertainties in estimation of the surface muon energy related with ambiguities in rock density and its composition, their non-uniformity in depth, and, in case of the mountain overburden, with errors in slant depth evaluation. besides, this method has a principal upper limitation for accessible muon energies, since at depths more than about 12 km w.e. (in standard rock) the intensity of atmospheric muons becomes lower than the background flux of muons locally produced by neutrinos in the surrounding material. taking into account energy loss fluctuations, such depth corresponds to effective muon threshold energy about 100 tev. the method based on calorimetric measurements of the spectrum of electromagnetic cascades induced via muon bremsstrahlung does not have upper physical limit. however, possibilities of investigations of muon spectrum at high energies are limited by low probability of the production of bremsstrahlung photons with energies comparable to muon energy (εγ ∼eμ), rapidly decreasing muon intensity, and consequently, by the necessity of corresponding increase of the detector mass. a special case of this technique represents the burst-size technique, when the cascade is detected in one point (in one layer of the detector). such approach is used in the analysis of horizontal air showers (has) which may be produced deep in the atmosphere only by muons (or neutrinos). however, many questions appear in interpretation of measurements of this kind: is the shower produced by single muon or by several particles? how to reject the background contribution from usual (hadron-induced) eas? what is the effective target thickness for such observations? as a rule, there are no simple answers for these questions. 2 it is important to mark that, in contrast to magnetic spectrometer technique where the energies (the momenta) of individual particles are measured and differential energy spectrum may be directly constructed, two other methods provide essentially integral estimates: intensity of muons penetrating to the observation depth in depth-intensity mea- surements, and the amount of muons with energies exceeding the energy of bremsstrahlung photon in measurements of cascade shower spectrum. at that, effective muon energies do not strongly exceed the energy threshold (typically, about 2 times). since the methods discussed above encounter serious difficulties of principal or technical character, other methods are needed to ensure a breakthrough in the energy region ∼100 tev and higher. from this point of view, the most promising seems to be pair meter technique [6, 7]. this method of muon energy evaluation is based on measurements of the number and energies of secondary cascades (with ε << eμ) originated as a result of multiple successive interactions of muon in a thick layer of matter, mainly due to direct electron-positron pair production. at sufficiently high muon energies, in a wide range of relative energy transfers ε/eμ ∼10−1 −10−3 pair production becomes the dominating muon interaction process, and its cross section rapidly increases with eμ. a typical ratio of muon energy and the energy of these secondary cascades is determined by muon and electron mass ratio and is of the order of 100. an important advantage of this technique is the absence of principal upper limitation for measured muon energies (at least up to 1016 −1017 ev, where the influence of landau-pomeranchuk-migdal effect on direct electron pair production cross section may become important). in case of a sufficient setup thickness (≥500 radiation length) and large number of detecting layers (of the order of hundred) pair meter technique allows estimating individual muon energies; possibilities of the method for relatively thin targets depend on the shape of the investigated muon energy spectrum. in the present paper, bust data are analyzed on the basis of a modification of multiple interaction method elab- orated for realization of the pair meter technique in thin setups. the results are compared with earlier bust data on muon spectrum obtained by means of electromagnetic cascade shower measurements and depth-intensity curve analysis. 1. measurements of depth-intensity curve and spectrum of electromagnetic cascades at bust bust [8] is located in an excavation under the slope of mt. andyrchy (north caucasus) at effective rock depth 850 hg/cm2 which corresponds to about 220 gev threshold energy of detected muons. the telescope (fig. 1) rep- resents a four-floor building with the height 11 m and the base 17 × 17 m2. the floors and the walls are entirely covered with scintillation detectors (the total number 3152) which form 8 planes (4 vertical and 4 horizontal, two of the latter being internal ones). the upper horizontal plane contains 576 = 24 × 24 scintillation detectors, the other three 400 = 20 × 20 detectors each. the distance between neighboring planes in a vertical is 3.6 m. total thickness of one layer (construction materials and scintillator) is about 7.2 radiation length. each of the detectors represents an aluminum tank with sizes 0.7 × 0.7 × 0.3 m3 filled with liquid scintillator on the basis of white spirit viewed by a 15 cm diameter pmt (feu-49) through pmma illuminator. most probable energy deposition in the detector at passage of a near-vertical muon is 50 mev. the anode output of pmt serves for measurements of the energy deposition in the plane in the range from 12.5 mev to 2.5 gev and for the formation of master pulses for various physical programs. pulse channel with operating threshold of 12.5 mev (since 1991, 10 mev threshold) is connected to 12th dynode and provides coordinate information ("yes-no" type). the signal from 5th dynode of pmt is used to measure the energy deposition in the detector in the range from 0.5 to 600 gev by means of the logarithmic converter of the pulse amplitude to duration. bust was created for investigations of cosmic ray muons and neutrinos as a telescope, but in principle it can detect as single muons (and muon bundles) so muon-induced cascade showers. therefore, for the analysis of data concerning muon energy spectrum, three methods can be used: depth-intensity relation, measurements of the spectrum of electromagnetic cascades, and pair meter technique. results of the analysis of the bust data on the depth-intensity dependence are given in [9]. the underground muon intensity was measured in two zenith angle intervals (50◦−70◦and 70◦−85◦) for slant depth between 1000 and 12000 hg/cm2. up to 6000 hg/cm2 in both zenith angle intervals the measured intensities agreed with the expectation for a usual muon spectrum (from pion and kaon decays). however at greater depths some excess of muons at moderate zenith angles (50◦−70◦) was observed which was interpreted by the authors as an indication for the appearance of 3 5 8 7 6 horizontal planes of scintillation detectors 24 24 20 20 20 20 20 20 figure 1: high-energy muon passing through baksan underground scintillation telescope (geant4 simulation). numbers of the planes (5 to 8) correspond to sequence of their construction. prompt muons from charmed particle decays. the estimated contribution of prompt muons corresponded to the value of the parameter r = (1.5 ± 0.5) × 10−3. results of investigations of muon spectrum by means of measurements of the spectrum of electromagnetic cas- cades in bust are described in [10]. in this work, the muon energy spectrum ih(> eh) at the depth of setup location was derived from the spectrum of energy depositions in the telescope which was used as a 4-layer sampling calorime- ter. for re-calculation to the surface muon energy spectrum i0(> eμ) the authors used the solution of the kinetic equation for muon flux passing through a thick layer of matter. some excess of the number of cascades in the tale of the spectrum found in this experiment could be caused as by methodical so by physical (inclusion of prompt muons) reasons. authors noted that a similar flattening of the spectrum was also observed in a number of other experiments, but at different energies, which evidences in favor of methodical reasons of its appearance. as a whole, results of the analysis of the bust data on the depth-intensity curve and electromagnetic cascade spectrum poorly agree with each other. below, results of independent analysis of the available bust data based on ideas of the pair meter technique are described. 2. application of method of multiple interactions in bust in order to evaluate individual muon energies (assuming that they have a usual power type integral spectrum ∼e−2.7 μ ) by means of the pair meter technique with a reasonable accuracy, it is necessary to detect several (≥5) muon interactions in the setup with total target thickness of several hundred radiation length and ∼100 detecting layers. if the number of layers and the setup thickness are low, the pair meter technique turns into the method of "plural" (in a limiting case, twofold) interactions. in this situation, evaluation of energies of individual muons is practically impossible; however, energy characteristics of the muon flux may be investigated on a statistical basis. the sensitivity of such method depends on the shape of muon energy spectrum and, as estimates show, for a more flat spectrum than the usual one, for example nμ ∼e−1.7 μ (prompt muons, vhe muons, eas muons in the range eμ << e0), it is sufficient to detect only two interactions even in the setup with the thickness of the order of several tens radiation length. a significant volume of experimental data accumulated at bust (more than 10 years of observations in combi- nation with about 200 m2 sr geometric acceptance of the telescope) allows infer conclusions on the behavior of the muon spectrum in the region of very high energies on the basis of the method of multiple interactions, in spite of a small number of layers in the telescope (four) and low setup thickness (∼30 radiation length). 4 in fact, the structure of bust allows distinctively select not more than two successive interactions of muon in the telescope (fig. 2). in the longitudinal profile of energy depositions (in horizontal planes) in such events, a minimum ("deep", emin) in one of the inner planes and two maximums ("humps") above and below it must be observed. it is convenient to denote as e1 the energy deposition measured in the higher maximum, e2 the deposition in the second one; then the depth of the deep may be characterized by the ratio k2 = e2/emin. k
= e
/ e   e  top plane of bust (5) inner plane (7) bottom plane (6) inner plane (8) e
e   run 516828 frame 10093 event 9736-418339 12 gev 1.2 gev 46 gev 3.4 gev 12 gev 0.5 gev 64 gev 4.1 gev               e, gev figure 2: twofold interactions of muons in bust. left: examples of detected and simulated events; horizontal telescope planes are plotted. hit detectors ("yes-no") are shown in grey; colors correspond to different energy depositions (the scale below). right: longitudinal profile of energy depositions in the telescope and definition of phenomenological parameters of the event. simulation of the bust response for the passage of single muons was performed by means of geant4 toolkit [11, 12]. before production of large-scale simulations, comprehensive tests of the correctness of muon electromagnetic interaction processes implementation in geant4 in a wide range of energies and for various materials were done. the number of simulated events for muon energies above 350 gev (at ground surface) was comparable to the expected number of such muons for the observation period (at "usual" energy spectrum), and for energies more than 1 tev, 10 tev, and 100 tev exceeded the expected muon statistics in about 5, 40, and 500 times, respectively. in every simulated event, information on energy depositions in scintillation detectors and on muon interactions with energy transfers more than 1 gev was recorded. analysis of simulation results has shown that qualitatively the selection parameters e1, e2, k2 influence the event samples in a following way: • the shift in e2 is nearly proportional to the shift in muon energy; • increase of the minimal value of the relative depth of the deep k2 suppresses contribution of nuclear showers (from inelastic muon interaction with nuclei) which may imitate multiple interactions; • increase of the threshold in e1 decreases the number of muons with moderate energies (∼tev), while most of high-energy events (hundreds tev) are retained. 5 among possible versions of muon energy estimation in the pair meter technique, sufficiently effective and conve- nient is the use of rank statistics of energies transferred in muon interactions: transferred energies ε j in an individual event are arranged in a decreasing order, and n-th value εn is then used to estimate muon energy [6]. energy depo- sitions measured in scintillation planes of the telescope, which determine the longitudinal profile of the event, are not simply related with the transferred energies. this is caused by random location of interaction points relative to detector planes, superposition of cascades from different interactions, fluctuations of cascade development, etc. how- ever, analysis of simulated events allows conclude that the energy deposition e2 in the second in value maximum is determined mainly by the second in energy cascade, related with production of e+e−pair by muon (relative energy transfers ε/eμ ∼10−2 −10−3), while the largest cascade (associated with the largest energy deposition e1) with a high probability is caused by muon bremsstrahlung or inelastic muon interaction (with ε/eμ ≥0.1). since the spectra of rank statistics are nearly similar to the spectrum of muons, it is expedient to use for the following analysis the distributions of events in the value of e2, and to vary other parameters of event selection: e1 (≥5 gev, ≥20 gev, ≥ 40 gev, etc.) and k2 (≥1, 2, 5, ...). 3. analysis of experimental data on multiple interactions of muons experimental data accumulated at bust during 12.5 years in 1983-1995 and 2 years (2003-2004)after restoration of amplitude measurement system [13] have been analyzed. periods of reliable operation of all systems responsible for energy deposition measurements were selected on the basis of a careful statistical analysis of the data. as a result, the total "live" time of registration amounted to 3.3 × 108 s (more than 10 years), and the total number of events after preliminary selection (with total energy deposition ≥10 gev in horizontal planes of the telescope) was about 10 millions. in more details, event selection criteria are described in [14]. only information of horizontal telescope planes was used. the total number of experimental events with twofold muon interactions selected with conditions e1, e2 ≥5 gev and muon tracks crossing all four horizontal planes equals to 1831; the corresponding statistics of simulated events amounts to 26951 events. experimental distributions of the events n(e2) were compared with geant4 simulation results for different selec- tion criteria (e1 ≥5 gev and k2 ≥1; e1 ≥20 gev and k2 ≥2, etc.) and four different muon energy spectrum models (fig. 3): 1. usual muon spectrum from π-, k-decays in the atmosphere (equation (1) with r = 0 and γμ = 2.7); 2. usual spectrum with addition of prompt muons at the level of r = 1 × 10−3; 3. the same, but with three times higher prompt muon contribution, r = 3 × 10−3; 4. usual spectrum with inclusion of vhe muons according to equation (2) with following parameters: nμ = 1, fμ = 0.025, ek = 5 pev, γ1 = 1.7, and γ2 = 2.0. 10 3 10 4 10 5 10 6 10 -3 10 -2 10 -1 model 4 model 3 model 2 e 3 dn /de , cm -2 s -1 sr -1 gev 2 e , gev model 1 figure 3: differential muon energy spectra for vertical direction (4 models). 6 experimental and calculated integral distributions of the events in e2 are presented in fig. 4. as a whole, within statistical uncertainties the data and calculations for a usual muon spectrum are in a good agreement in the range 5 gev ≤e2 ≤30 gev. however, at large values of e2 (more than 80 gev) the expected number of events is several times less (and at the tale of the distribution, almost ten times) than the observed in the experiment. let us note that namely in the region e2 ∼100 gev and higher the multiple interaction method in bust becomes the most sensitive to the changes of muon spectrum shape. 10 1 10 2 10 -1 10 0 10 1 10 2 10 3 n (> e 2 ), events e 2 , gev e x p e rim e n t: ca lcu la tio n s: m o d e l 1 m o d e l 2 m o d e l 3 m o d e l 4 a) e 1 > 5 gev, k 2 > 1 10 1 10 2 10 -1 10 0 10 1 10 2 10 3 n (> e 2 ), events e 2 , gev e x p e rim e n t: ca lcu la tio n s: m o d e l 1 m o d e l 2 m o d e l 3 m o d e l 4 b) e 1 > 20 gev, k 2 > 2 figure 4: integral distributions of experimental events (the points) and expected spectra (the curves) in e2 for 4 different muon spectrum models (see the text) and two sets of selection criteria (a,b). at comparison of the corresponding to fig. 4a differential distribution in e2 with the expected one under assump- tion of a usual muon spectrum (from π-, and k-decays) the value of χ2 appears equal to 32.9 (at 8 degrees of freedom) which implies the rejection of such hypothesis on the spectrum shape with about 99.9% confidence. situation re- mains nearly the same after inclusion of prompt muons with r = 10−3 (spectrum model 2, χ2 = 24.7). much better agreement is reached at comparison of the data with calculation results for sufficiently large fraction of prompt muons (model 3, r = 3 × 10−3) or addition of vhe muons (model 4); corresponding values of χ2 in these cases are equal to 17.4 and 15.6, respectively. it is important to note that the observed excess of events with large values of e2 is retained at different approaches to data analysis and different selection criteria (compare fig. 4a and fig. 4b). four events with highest values of e2 (more than 80 gev) are presented in fig. 5. all these events are detected inside the telescope in all horizontal planes and have a clear topology. therefore, in spite of low statistics, the deviation of experimental distributions from calculations performed in frame of generation of muons only in π-, k-decays seems to be significant, and evidences for a possible existence of the fluxes of vhe or prompt muons with the considered parameters. 4. muon energy spectrum in order to pass from the experimental distributions of event characteristics to the muon energy spectrum, it is necessary to determine which intervals of muon energies give the main contribution to generation of registered events, to choose effective estimates for them (mean, logarithmic mean, or median muon energies) and to define the conversion procedure. distributions of muon energies giving contribution to the events with several threshold values e2 at fixed parameter k2 calculated for 4 different assumptions on muon spectrum shape (spectrum models 1-4 described in the preceding section) are plotted in fig. 6 (a,b,c,d). these distributions are rather wide even for a usual spectrum of muons (fig. 6a), and, in presence of the additional muon flux with a more hard spectrum, at high e2 values become bimodal (figs. 6b-6d). the appearance of the second hump in the region of muon energies of hundreds tev and higher is caused by a good sensitivity of the multiple interaction method namely to this, more hard, part of the muon spectrum. in order to illustrate the decisive role of direct electron pair production process in the multiple interaction method, calculations for muon spectrum with the addition of vhe muons (model 4) were repeated with the exclusion of pair 7                               !"                          #   !"                               # #              $ % &  ' !"                                 &  '   &  '    &  '  #  &  '   &  '    &  '   &  '    &  '   &  '  # &  ' # &  '    &  '   &  '   &  '    &  '  # &  ' e ( e ( e ( e ( e ) e ) e ) e ) figure 5: experimental events with highest values of e2. energy depositions measured in scintillation planes are indicated. production. the obtained distributions (fig. 7) appeared insensitive to the additional vhe muon flux (compare with fig. 6d), and effective muon energies in this case would not exceed several tens tev even for high values of e2. energy spectra of muons from the bust data on multiple interactions were obtained in a following way. at first, for certain sets of selection criteria (e1 and k2 parameters) the differential and integral distributions of the observed events nobs with an equal step in common logarithm of e2 were constructed, namely, the number of events in every bin ∆lg(e2, gev) = 0.7-0.9, 0.9-1.1, ..., 2.3-2.5 for the differential distribution, and the total number of events with lg(e2, gev) ≥0.7, ≥0.9, etc. for the integral one were counted. expected model distributions nmod of the events in lge2, and also energy distributions of muons giving the con- tribution to events in a certain interval ∆lg e2 or ≥lge2, corresponding mean, logarithmic mean, median energies e∗ μ for differential distributions and effective threshold muon energies e∗ μ0 for integral ones were computed on the basis of the results of geant4 telescope response simulations for the respective combination of selection criteria (e1 and k2) and four models of surface muon energy spectrum discussed above. dependences of logarithmic mean, mean, and median muon energies on e2 (for differential in e2 event distri- butions) are presented in fig. 8 (a,b, and c respectively) for different spectrum models. these dependences clearly demonstrate the main advantage of the multiple interaction method, namely, the possibility to advance in muon energy region of hundreds tev and even few pev in case of the presence of substantial flux of muons with a hard spectrum at these energies. since the energy deposition in scintillator layers of bust constitutes about 10% of cascade energy [10], the ratio between effective muon energy e∗ μ and e2 reaches in this case the order of 103. in fig. 8d, the depen- dences of effective threshold energy e∗ μ0 (estimated via logarithmic mean values) for the integral distributions in e2 are shown. qualitatively, it is seen that the dependences in figs. 8a and 8d only weakly differ from each other. finally, the estimates of differential and integral muon spectra are found in a following way: d ̃ nμ(e∗ μ)/deμ = dnμ(e∗ μ)/deμ × ndif obs(e2)/ndif mod(e2), (4) ̃ nμ(≥e∗ μ0) = nμ(≥e∗ μ0) × nint obs(e2)/nint mod(e2), (5) where dnμ(e∗ μ)/deμ and nμ(≥e∗ μ0) are differential and integral muon energy spectra for the respective spectrum model calculated at corresponding effective muon energy (e∗ μ and ≥e∗ μ0). 8 10 2 10 3 10 4 10 5 10 6 10 7 10 8 0.00 0.05 0.10 0.15 0.20 0.25 dw/dlge e , gev e 2 > 10 gev e 2 > 40 gev e 2 > 100 gev a) model 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 0.00 0.05 0.10 0.15 0.20 0.25 dw/dlge e , gev e 2 > 10 gev e 2 > 40 gev e 2 > 100 gev b) model 2 10 2 10 3 10 4 10 5 10 6 10 7 10 8 0.00 0.05 0.10 0.15 0.20 0.25 dw/dlge e , gev e 2 > 10 gev e 2 > 40 gev e 2 > 100 gev c) model 3 10 2 10 3 10 4 10 5 10 6 10 7 10 8 0.00 0.05 0.10 0.15 0.20 0.25 dw/dlge e , gev e 2 > 10 gev e 2 > 40 gev e 2 > 100 gev d) model 4 figure 6: energy distributions of muons giving contribution to events with different threshold values e2 for 4 models of muon energy spectrum. 10 2 10 3 10 4 10 5 10 6 10 7 10 8 0.00 0.05 0.10 0.15 0.20 0.25 dw/dlge e , gev e 2 > 10 gev e 2 > 40 gev e 2 > 100 gev model 4, e + e - = 0 figure 7: energy distributions of muons giving contribution to events with different threshold values e2 for the model 4 of the muon spectrum with the "switched-off" pair production process (compare fig. 6d). 9 10 1 10 2 10 0 10 1 10 2 10 3 10 4 logarithmic mean e *, tev e 2 , gev model 1 model 2 model 3 model 4 a) 10 1 10 2 10 0 10 1 10 2 10 3 10 4 mean e *, tev e 2 , gev model 1 model 2 model 3 model 4 b) 10 1 10 2 10 0 10 1 10 2 10 3 10 4 median e *, tev e 2 , gev model 1 model 2 model 3 model 4 c) 10 1 10 2 10 0 10 1 10 2 10 3 10 4 logarithmic mean e 0 *, tev e 2 m in , gev model 1 model 2 model 3 model 4 d) figure 8: dependences of logarithmic mean (a), mean (b), and median (c) muon energies on e2 for differential in e2 distributions. frame (d): effective threshold muon energies for integral in e2 distribution. the curves correspond to four different spectrum models. differential muon energy spectra for vertical direction reconstructed from the experimental data according to the described procedure at four different assumptions on muon spectrum model are presented in fig. 9. results are shown for one of the combinations of the selection criteria with highest statistics (e1 ≥5 gev, k2 ≥1). since there is no generally accepted definition of the effective energy of muons responsible for the observed events, the points corresponding to all three versions (mean, logarithmic mean, and median energies) are given in the figure. the curves in each frame represent the assumed spectrum models. 5. discussion the following conclusions can be made from the analysis of the results presented in fig. 9. if one assumes that the muon spectrum is formed only due to decays of pions and kaons in the atmosphere (i.e. "usual" muon spectrum, fig. 9a), then a strong dependence of spectrum reconstruction results on the choice of the effective muon energy (mean, logarithmic mean, median energy) appears as a large spread of reconstructed points. furthermore, muon intensity estimated in frame of this assumption in the range of several tens tev (considering median or logarithmic mean energy) or around 100 tev (according to mean energy) is practically ten times higher than the expected one, and seriously contradicts results of other experiments compilation of which is given in [5]. the spread of experimental points relative to the model spectrum curves decreases as the contribution of additional muon flux with a more hard energy spectrum increases (fig. 9b and 9c). at the same time, the agreement is improving 10 10 3 10 4 10 5 10 6 10 7 10 -4 10 -3 10 -2 10 -1 10 0 a) model 1 experiment: e mean e logarithmic mean e median calculation: e 3 dn /de , cm -2 s -1 sr -1 gev 2 e , gev 10 3 10 4 10 5 10 6 10 7 10 -4 10 -3 10 -2 10 -1 10 0 b) model 2 experiment: e mean e logarithmic mean e median calculation: e 3 dn /de , cm -2 s -1 sr -1 gev 2 e , gev 10 3 10 4 10 5 10 6 10 7 10 -4 10 -3 10 -2 10 -1 10 0 c) model 3 experiment: e mean e logarithmic mean e median calculation: e 3 dn /de , cm -2 s -1 sr -1 gev 2 e , gev 10 3 10 4 10 5 10 6 10 7 10 -4 10 -3 10 -2 10 -1 10 0 d) model 4 experiment: e mean e logarithmic mean e median calculation: e 3 dn /de , cm -2 s -1 sr -1 gev 2 e , gev figure 9: differential muon energy spectra reconstructed from bust data on multiple interactions at different assumptions on muon spectrum model (a,b,c,d) with different choice of effective muon energy: mean (circles), logarithmic mean (diamonds), and median (triangles). also in the range of moderate muon energies (tens tev); in other words, the dependence of results on the choice of effective muon energy (mean, logarithmic mean, median energy) at muon spectrum reconstruction disappears. the best agreement of the data with the expectation in a wide range of energies (from few tev to few pev) is observed for the spectrum with addition of the flux of vhe muons with parameters indicated above (fig. 9d, spectrum model 4); the r.m.s. deviation of the points from the curve in this case is minimal. in fig. 10, the integral muon energy spectra measured at bust by means of different methods are compared. one of the possible reasons of the difference between the results obtained from the depth-intensity curve and electromag- netic cascade spectrum measurements may be related with different procedures used for muon spectrum reconstruction from the experimental data. thus, in the paper [9], in order to pass from the depth-intensity dependence (after eval- uation of the r parameter) to the integral energy spectrum of muons at the surface (taking into account muon energy loss fluctuations) the mean energy of that part of the spectrum which is responsible for muon flux intensity at a given depth was used. authors note that the mean energy of prompt muons (from charmed particle decays), due to a more flat spectrum, is about twice more than the mean energy of usual muons; therefore weighted average of mean energies for two components of the flux was used for the conversion. in the paper [10], the transition from the measured spectrum of energy depositions of electromagnetic cascades in the telescope to the muon energy spectrum was performed for median energies of muons responsible for events in a given energy deposition bin. a deep in the reconstructed muon energy spectrum around 10 tev (fig. 10) 11 10 3 10 4 10 5 10 6 10 -3 10 -2 10 -1 m o d e l 4 m o d e l 3 m o d e l 2 bust - depth-intensity bust - calorimeter bust - pairmeter (model 3) bust - pairmeter (model 4) e 2 n (>e ), cm -2 s -1 sr -1 gev 2 e , gev m o d e l 1 figure 10: integral energy spectrum of muons for vertical direction reconstructed from the bust data on depth-intensity curve (circles, [9]), spectrum of electromagnetic cascades (triangles, [10]), and by means of multiple interaction method for two models (3 and 4) used in muon spectrum reconstruction from experimental data. the curves represent calculations for different spectrum models. most probably is related with some methodical reasons, since it is difficult to suggest any physical explanation of its appearance. as to the absolute value of muon flux measured by this method, it is necessary to note that the systematic uncertainty in muon intensity could reach about 25%, since, as it was indicated in [10], the accuracy of the absolute energy calibration of energy deposition measurements was about 10%. for the reconstruction of the integral muon energy spectrum from the data on multiple interactions of muons in bust, the spectrum models 3 and 4 were used (open and solid diamonds in fig. 10, respectively). the estimates of effective threshold muon energies were obtained on the basis of logarithmic mean values as optimal ones for quasi- power spectra of particles [6]. as it is seen from the figure, no deviations from the usual spectrum is observed up to energies of at least ∼10 tev for model 3 and ∼30 tev for model 4, while around 100 tev and higher a considerable excess in comparison with the spectrum of muons from π-, k-decays appears. at that, the muon energy reconstruction by means of model 4 gives better agreement of experimental points with theoretical curve than by means of model 3. a natural question may arise at discussions of the obtained results: how was it possible to register pev muons with a relatively small-size setup (∼200 m2) bearing in mind that their flux is extremely low? and these doubts are correct for usual decay muons; in this case, during the period of the experiment (more than 10 years), at best, one or two muons with energy above 1 pev could cross the telescope. however, for more flat spectra (muons from charmed particles or vhe muons from new generation processes) the expected number of such muons may reach several tens. and, even taking into account relatively low probability of generation of events with twofold muon interactions in such a thin setup as bust (of the order of ∼10−1), the possibility of registration of pev muons becomes quite real. in fig. 11, the differential muon energy spectrum obtained from the bust data by means of multiple interaction method is compared with results of other experiments taken from the compilation [5]. as it is seen from the figure, the present data are the first ones at energies above 100 tev and, in spite of low statistics, evidence for a change of muon spectrum behavior namely in this region. one may expect that this energy region will be accessible soon for investigations by means of cascade shower spectrum measurements at icecube [15], and the use of pair meter technique at such scale setups would also allow to explore the range of pev muon energies. 12 p * d (p) ( cm + , s + - sr + - (gev/c) , ) muon momentum (gev/c) baksan (present work) 4 3 2 1 10 3 10 4 10 5 0.01 0.1 10 5 10 6 , figure 11: differential muon energy spectra for vertical direction measured in various experiments (compilation from [5]). the curves correspond to different models of prompt muon contribution considered in [5]. present bust results obtained by means of multiple interaction method are added (open and solid diamonds for models 3 and 4 correspondingly). conclusion method of multiple interactions of muons based on the ideas of the pair meter technique gives possibility to use bust data for estimation of the energy spectrum of cosmic ray muons in a wide energy region from several tev to hundreds tev. the analysis shows that no serious deviations from the usual spectrum formed as a result of pion and kaon decays are observed up to muon energies ∼20 tev for model 3 and ∼50 tev for model 4, if the existence of an additional flux of muons with a more hard spectrum is taken into account. at energies ≳100 tev this additional flux exceeds the expected contribution of muons from charmed particles corresponding to the parameter r ∼10−3, and may be explained with r ∼3 × 10−3, which suggests a more fast increase of charmed particle yield compared to recent theoretical predictions. however, the best description of the experimental data can be achieved by assuming an additional contribution of vhe muons from new physical processes related with the appearance of the observed knee in cosmic ray energy spectrum. references [1] a.a. petrukhin, in proc. xith rencontres de blois frontiers of matter, blois, france, 1999. ed. by j. tran thanh van (the gioi publishers, vietnam, 2001), p. 401. [2] k. nakamura et al. (particle data group), j. phys. g 37, 075021 (2010). [3] l.v. volkova, o. saavedra, astropart. phys., 32, 136 (2009). [4] a.a. petrukhin, isvhecri 2010, batavia, usa, 2010, arxiv:1101.1900v1 [astro-ph.he]. [5] e.v. bugaev et al., phys. rev. d, 58 (1998) 05401; arxiv:hep-ph/9803488 v3, jan 2000. [6] r.p. kokoulin, a.a. petrukhin, nucl. instr. meth. a, 263 (1988) 468. [7] r.p. kokoulin, a.a. petrukhin, sov. j. part. nucl., 21 (1990) 332. [8] a.e. chudakov et al., proc. 16th icrc, kyoto, 1979, v. 10, p. 276. [9] yu.m. andreev et al., proc. 21st icrc, adelaide, 1990, v. 9, p. 301. [10] v.n. bakatanov et al. sov. j. nucl. phys., 55 (1992) 1169. [11] s. agostinelli et al., nucl. instr. meth. a, 506 (2003) 250. [12] j. allison et al., ieee trans. nucl. science, 53 (2006) 270. [13] a.f. yanin et al., instr. experim. techn., 47, no. 3 (2004) 330. [14] a.g. bogdanov et al., physics of atomic nuclei, 72 (2009) 2049. [15] f. halzen, s. klein, rev. sci. instrum 81 (2010) 081101. 13

0911.1694

regularizing portfolio optimization

the optimization of large portfolios displays an inherent instability to estimation error. this poses a fundamental problem, because solutions that are not stable under sample fluctuations may look optimal for a given sample, but are, in effect, very far from optimal with respect to the average risk. in this paper, we approach the problem from the point of view of statistical learning theory. the occurrence of the instability is intimately related to over-fitting which can be avoided using known regularization methods. we show how regularized portfolio optimization with the expected shortfall as a risk measure is related to support vector regression. the budget constraint dictates a modification. we present the resulting optimization problem and discuss the solution. the l2 norm of the weight vector is used as a regularizer, which corresponds to a diversification "pressure". this means that diversification, besides counteracting downward fluctuations in some assets by upward fluctuations in others, is also crucial because it improves the stability of the solution. the approach we provide here allows for the simultaneous treatment of optimization and diversification in one framework that enables the investor to trade-off between the two, depending on the size of the available data set.

introduction markowitz' portfolio selection theory [1, 2] is one of the pillars of theoretical finance. it has greatly influenced the thinking and practice in investment, capital allocation, index tracking, and a number of other fields. its two major ingredients are (i) seeking a trade- offbetween risk and reward, and (ii) exploiting the cancellation between fluctuations of (anti-)correlated assets. in the original formulation of the theory, the underlying process was assumed to be multivariate normal. accordingly, reward was measured in terms of the expected return, risk in terms of the variance of the portfolio. the fundamental problem of this scheme (shared by all the other variants that have been introduced since) is that the characteristics of the underlying process generating the distribution of asset prices are not known in practice, and therefore averages are replaced by sums over the available sample. this procedure is well justified as long as the sample size, t (i.e. the length of the available time series for each item), is sufficiently large compared to the size of the portfolio, n (i.e. the number of items). in that limit, sample averages asymptotically converge to the true average due to the central limit theorem. unfortunately, the nature of portfolio selection is not compatible with this limit. institutional portfolios are large, with n's in the range of hundreds or thousands, while considerations of transaction costs and non-stationarity limit the number of available data points to a couple of hundreds at most. therefore, portfolio selection works in a region, where n and t are, at best, of the same order of magnitude. this, however, is not the realm of classical statistical methods. portfolio optimization is rather closer to a situation which, by borrowing a term from statistical physics, might be termed the "thermodynamic limit", where n and t tend to infinity such that their ratio remains fixed. it is evident that portfolio theory struggles with the same fundamental difficulty that is underlying basically every complex modeling and optimization task: the high number of dimensions and the insufficient amount of information available about the system. this difficulty has been around in portfolio selection from the early days and a plethora of methods have been proposed to cope with it, e.g. single and multi-factor models [3], bayesian estimators [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], or, more recently, tools borrowed from random matrix theory [18, 19, 20, 21, 22, 23]. in the thermodynamic regime, estimation errors are large, sample to sample fluctuations are huge, results obtained from one sample do not generalize well and can be quite misleading concerning the true process. the same problem has received considerable attention in the area of machine learning. we discuss how the observed instabilities in portfolio optimization (elaborated in section 2) can be understood and remedied by looking at portfolio theory from the point of view of machine learning. portfolio optimization is a special case of regression, and therefore can be understood as a machine learning problem (see section 3). in machine learning, as regularizing portfolio optimization. 3 well as in portfolio optimization, one wishes to minimize the actual risk, which is the risk (or error) evaluated by taking the ensemble average. this quantity, however, can not be computed from the data, only the empirical risk can. the difference between the two is not necessarily small in the thermodynamic limit, so that a small empirical risk does not automatically guarantee small actual risk [24]. statistical learning theory [24, 25, 26] finds upper bounds on the generalization error that hold with a certain accuracy. these error bounds quantify the expected generalization performance of a model, and they decrease with decreasing capacity of the function class that is being fitted to the data. lowering the capacity therefore lowers the error bound and thereby improves generalization. the resulting procedure is often referred to as regularization and essentially prevents over-fitting (see section 4). in the thermodynamic limit, portfolio optimization needs to be regularized. we show in section 5 how the above mentioned concepts, which find their practical application in support vector machines [27, 28], can be used for portfolio optimization. support vector machines constitute an extremely powerful class of learning algorithms which have met with considerable success. we show that regularized portfolio optimization, using the expected shortfall as a risk measure, is almost identical to support vector regression, apart from the budget constraint. we provide the modified optimization problem which can be solved by linear programming. in section 6, we discuss the financial meaning of the regularizer: minimizing the l2 norm of the weight vector corresponds to a diversification pressure. we also discuss alternative constraints that could serve as regularizers in the context of portfolio optimization. taking this machine learning angle allows one to organize a variety of ideas in the existing literature on portfolio optimization filtering methods into one systematic and well developed framework. there are basically two choices to be made: (i) which risk measure to use, and (ii) which regularizer. these choices result in different methods, because different optimization problems are being solved. while we focus here on the popular expected shortfall risk measure (in section 5), the variance has a long history as an important risk measure in finance. several existing filtering methods that use the variance risk measure essentially implement regularization, without necessarily stating so explicitly. the only work we found in this context [7] that mentiones regularization in the context of portfolio optimization has not been noticed by the ensuing, closely related, literature. it is easy to show that when the l2 norm is used as a regularizer, then the resulting method is closely related to bayesian ridge regression, which uses a gaussian prior on the weights (with the difference of the additional budget constraint). the work on covariance shrinkage, such as [8, 9, 10, 11], falls into the same category. other priors can be used [17], which can be expected to lead to different results (for an insightful comparison see e.g. [29]). using the l1 norm has been popularized in statistics as the "lasso" (least absolute shrinkage and selection operator) [29], and methods that use any lp norm are also known as the "bridge" [30]. regularizing portfolio optimization. 4 2. preliminaries – instability of classical portfolio optimization. portfolio optimization in large institutions operates in what we called the thermodynamic limit, where both the number of assets and the number of data points are large, with their ratio a certain, typically not very small, number. the estimation problem for the mean is so serious [31, 32] as to make the trade-offbetween risk and return largely illusory. therefore, following a number of authors [8, 9, 33, 34, 35], we focus on the minimum variance portfolio and drop the usual constraint on the expected return. this is also in line with previous work (see [36] and references therein), and makes the treatment simpler without compromising the main conclusions. an extension of the results to the more general case is straightforward. nevertheless, even if we forget about the expected return constraint, the problem still remains that covariances have to be estimated from finite samples. it is an elementary fact from linear algebra that the rank of the empirical n × n covariance matrix is the smaller of n and t. therefore, if t < n, the covariance matrix is singular and the portfolio selection task becomes meaningless. the point t = n thus separates two regions: for t > n the portfolio problem has a solution, whereas for t < n, it does not. even if t is larger than n, but not much larger, the solution to the minimum variance problem is unstable under sample fluctuations, which means that it is not possible to find the optimal portfolio in this way. this instability of the estimated covariances, and hence of the optimal solutions, has been generally known in the community, however, the full depth of the problem has only been recognized recently, when it was pointed out that the average estimation error diverges at the critical point n = t [37, 38, 39]. in order to characterize the estimation error, kondor and co-workers used the ratio q2 0 between (i) the risk, evaluated at the optimal solution obtained by portfolio optimization using finite data and (ii) the true minimal risk. this quantity is a measure of generalization performance, with perfect performance when q2 0 = 1, and increasingly bad performance as q2 0 increases. as found numerically in [38] and demonstrated analytically by random matrix theory techniques in [40], the quantity q0 is proportional to (1 −n/t)−1/2 and diverges when t goes to n from above. the identification of the point n = t as a phase transition [36, 41] allowed for the establishment of a link between portfolio optimization and the theory of phase transitions, which helped to organize a number of seemingly disparate phenomena into a single coherent picture with a rich conceptual content. for example, it has been shown that the divergence is not a special feature of the variance, but persists under all the other alternative risk measures that have been investigated so far: historical expected shortfall, maximal loss, mean absolute deviation, parametric var, expected shortfall, and semivariance [36, 41, 42, 43]. the critical value of the n/t ratio, at which the divergence occurs, depends on the particular risk measure and on any parameter that the risk measure may depend on (such as the confidence level in expected shortfall). regularizing portfolio optimization. 5 however, as a manifestation of universality, the power law governing the divergence of the estimation error is independent of the risk measure [36, 41, 42], the covariance structure of the market [39], and the statistical nature of the underlying process [44]. ultimately, this line of thought led to the discovery of the instability of coherent risk measures [45]. 3. statistical reasons for the observed instability in portfolio optimization as mentioned above, for simplicity and clarity of the treatment we do not impose a constraint on the expected return, and only look for the global minimum risk portfolio. this task can be formalized as follows: given a fixed budget, customarily taken to be unity, given t past measurements of the returns of n assets: xk i , i = 1, . . . , n, k = 1, . . . , t, and given the risk functional f(w*x), find a weighted sum (the portfolio), w * x,‡ such that it minimizes the actual risk r(w) = ⟨f(w * x)⟩p(x), (1) under the constraint that p i wi = 1. the central problem is that one does not know the distribution p(x), which is assumed to underly the generation of the data. in practice, one then minimizes the empirical risk, replacing ensemble averages by sample averages: remp(w) = 1 t t x k=1 f(w * x(k)) (2) now, let us interpret the weight vector as a linear model. the model class given by the linear functions has a capacity h, which is a concept that has been introduced by vapnik and chervonenkis in order to measure how powerful a learning machine is [24, 25, 26]. (in the statistical learning literature, a learning machine is thought of as having a function class at its disposal, together with an induction principle and an algorithmic procedure for the implementation thereof [46]). the capacity measures how powerful a function class is, and thereby also how easy it is to learn a model of that class. the rough idea is this: a learning machine has larger capacity if it can potentially fit more different types of data sets. higher capacity comes, however, at the cost of potentially over-fitting the data. capacity can be measured, for example, by the vapnik-chervonenkis (vc-) dimension [24], which is a combinatoric measure that counts how many data points can be separated in all possible ways by any function of a given class. to make the idea tangible for linear models, focus on two dimensions (n = 2). for each number of points, n, one can choose the geometrical arrangement of the points in the plane freely. once it is chosen, points are labeled by one of two labels, say "red" and "blue". can a line separate the red points from the blue points for any of the 2n different ways in which the points could be colored? the vc-dimension is the largest number of points for which this can be done. two points can trivially be separated by a line. three points that are not arranged collinear can still be separate for any of ‡ notation: bold face symbols are understood to denote vectors. regularizing portfolio optimization. 6 the 8 possible labelings. however, for four points this is no longer the case, since there is no geometrical arrangement for which one could not find a labeling that can not be separated by a line. the vc-dimension is 3, and in general, for linear models in n dimensions, it is n + 1 [46, 47]. in the regime in which the number of data points are much larger than the capacity of the learning machine, h/t << 1, a small empirical risk guarantees small actual risk [24]. for linear functions through the origin that are otherwise unconstrained, the vc- dimension grows with n. in the thermodynamic regime, where n/t is not very small, minimizing the empirical risk does not necessarily guarantee a small actual risk [24]. therefore it is not guaranteed to produce a solution that generalizes well to other data drawn from the same underlying distribution. in solving the optimizing problem that minimizes the empirical risk, eq. (2) in the regime in which n/t is not very small, portfolio optimization over-fits the observed data. it thereby finds a solution that essentially pays attention to the seeming correlations in the data which come from estimation noise due to finite sample effects, rather than from real structure. the solution is thus different for different realizations of the data, and does not necessarily come close to the actual optimal portfolio. 4. overcoming the instability the generalization error can be bounded from above (with a certain probability) by the empirical error plus a confidence term that is monotonically increasing with some measure of the capacity, and depends on the probability with which the bound holds [48]. several different bounds have been established, connected with different measures of capacity, see e.g. [47]. poor generalization and over-fitting can be improved upon by decreasing the capacity of the model [25, 26], which helps to lower the generalization error. support vector machines are a powerful class of algorithms that implement this idea. we suggest that if one wants to find a solution to the portfolio optimization problem in the thermodynamic regime, then one should not minimize the empirical risk alone, but also constrain the capacity of the portfolio optimizer (the linear model). how can portfolio optimization be regularized? portfolio optimization is essentially a regression problem, and therefore we can apply statistical learning theory, in particular the work on support vector regression. note first that the capacity of a linear model class for which the length of the weight vector is restricted to ∥w∥2 ≤a has an upper bound which is smaller than the capacity of unconstrained linear models [25, 26]. the capacity is minimized when the length of the weight vector is minimized [25, 26]. vapnik's concept of structural risk minimization [48] results in the support vector algorithm [27, 28] which finds the model with the smallest capacity that is consistent with the data, that is the model with smallest ∥w∥2. this leads to a convex constrained optimization problem [27, 28] which can be solved using linear programming. regularizing portfolio optimization. 7 5. regularized portfolio optimization with the expected shortfall risk measure. while the original markowitz' formulation [1] measures risk by the variance, many other risk measures have been proposed since. today, the most widely used risk measure, both in practice and in regulation, is value at risk (var) [49, 50]. var has, however, been criticized for its lack of convexity, see e.g. [51, 52, 53], and an axiomatic approach, leading to the introduction of the class of coherent risk measures, was put forward [51]. expected shortfall, essentially a conditional average measuring the average loss above a high threshold, has been demonstrated to belong to this class [54, 55, 56]. expected shortfall has been steadily gaining popularity in recent years. the regularization we propose here is intended to cure its weak point, the sensitivity to sample fluctuations, at least for reasonable values of the ratio n/t. choose the risk functional f(z) = zθ(z −αβ), where αβ is a threshold, such that a given fraction β of the (empirical) loss-distribution over z lies above αβ. one now wishes to minimize the average over the remaining tail distribution, containing the fraction ν := 1 −β, and defines the expected shortfall as es = min ǫ " ǫ + 1 νt t x k=1 1 2 −ǫ −w * x(k) + | −ǫ −w * x(k)|
# . (3) the term in the sum implements the θ-function, while ν in the denominator ensures normalization of the tail distribution. it has been pointed out [57] that this optimization problem maps onto solving the linear program: min w,ξ,ǫ " 1 t t x k=1 ξk + νǫ # (4) s.t. w * x(k) + ǫ + ξk ≥0; ξk; ≥0 (5) x i wi = 1. (6) we propose to implement regularization by including the minimization of ∥w∥2. this can be done using a lagrange multiplier, c, to control the trade-off– as we relax the constraint on the length of the weight vector, we can, of course, make the empirical error go to zero and retrieve the solution to the minimal expected shortfall problem. the new optimization problem reads: min w,ξ,ǫ " 1 2∥w∥2 + c 1 t t x k=1 ξk + νǫ !# (7) s.t. −w * x(k) ≤ǫ + ξk; (8) ξk ≥0; ǫ ≥0; (9) x i wi = 1. (10) regularizing portfolio optimization. 8 the problem is mathematically almost identical to a support vector regression (svr) algorithm called ν-svr. there are two differences: (i) the budget constraint is added, and (ii) the loss function is asymmetric. expected shortfall is an asymmetric version of the ǫ-intensive loss, used in support vector regression, defined as the maximum of {0; |f(x) −y| −ǫ}, where f(x) is the interpolant, and y the measured value (response). in that sense ǫ measures an allowable error below which deviations are discarded.§ the use of asymmetric risk measures in finance is motivated by the consideration that investors are not afraid of upside fluctuations. however, to make the relationship to support vector regression as clear as possible, we will first solve the more general symmetrized problem, before restricting our treatment to the completely asymmetric case, corresponding to expected shortfall. in addition, one may argue that focusing exclusively on large negative fluctuations might not be advisable even from a financial point of view, especially when one does not have sufficiently large samples. in a relatively small sample it may happen that a particular item, or a certain combination of items, dominates the rest, i.e. produces a larger return than any other item in the portfolio at each time point, even though no such dominance exists on longer time scales. the probability of such an apparent arbitrage increases with the ratio n/t, and when it occurs it may encourage an investor acting on a lopsided risk measure to take up very large long positions in the dominating item(s), which may turn out to be detrimental on the long run. this is the essence of the argument that has led to the discovery of the instability of coherent and downside risk measures [43, 45]. according to the above, let us consider the general case where positive deviations are also penalized. the objective function, eq. (7), then becomes min w,ξ,ǫ " 1 2∥w∥2 + c 1 t t x k=1 (ξk + ξ∗ k) + νǫ !# , (11) and additional constraints have to be added to eqs. (8) to (10): w * x(k) ≤ǫ + ξ∗ k; ξ∗ k ≥0. (12) this problem corresponds to ν-svr, a well understood regression method [60], with the only difference that the budget constraint, eq. (10) is added here. in the finance context the associated loss might be called symmetric tail average (sta). solving the regularized expected shortfall minimization problem, eqs. (7)–(10) is a special case of solving the regularized sta minimization problem, eq. (11) with the constraints eqs. (8)–(10) and (12). therefore, we solve the more general problem first (section 5.1), before providing, in section 5.2, the solution to the regularized expected shortfall, eqs. (7)–(10). § the mathematical similarity between minimum expected shortfall without regularization and the eν- svm algorithm [58] was pointed out, but incorrectly, in [59]. there is an important difference between the two optimization problems. in eν-svm, the length of the weight vector, ∥w∥, is constrained, which implements capacity control. in the pure expected shortfall minimization, eq. (4), this is not done. instead, the total budget p i wi is fixed. this difference is not correctly identified in the proof of the central theorem (theorem 1) in [59]. regularizing portfolio optimization. 9 5.1. regularized symmetric tail average minimization the solution to the regularized symmetric tail average problem, eq. (11) with the constraints eqs. (8)–(10) and (12), is found in analogy to support vector regression, following [60], by writing down the lagrangean, using lagrange multipliers, {α, α∗, γ, λ, η, η∗}, for the constraints. the solution is then a saddle point, i.e. minimum over primal and maximum over dual variables. the lagrangean is different from the one that arises in ν-svr in that it is modified by the budget constraint: l[w, ξ, ξ∗, ǫ, α, α∗, γ, λ, η, η∗] = 1 2∥w∥2 + c t t x k=1 (ξk + ξ∗ k) + cνǫ −λǫ + γ x i wi −1 ! + t x k=1 α∗ k(w * x(k) −ǫ −ξ∗ k) − t x k=1 αk(w * x(k) + ǫ + ξk) − t x k=1 (ηkξk + η∗ kξ∗ k) (13) = f[w] + ǫ cν −λ − t x k=1 (αk + α∗ k) ! −γ (14) + t x k=1  ξk c t −αk −ηk  + ξ∗ k c t −α∗ k −η∗ k  with f[w] = w * 1 2w − t x k=1 (αk −α∗ k)x(k) −γ1 !! , (15) where 1 denotes the unit vector of length n. setting the derivative of the lagrangian w.r.t. w to zero gives: wopt = t x k=1 (αk −α∗ k)x(k) −γ1 (16) this solution for the optimal portfolio is sparse in the sense that, due to the karush- kuhn-tucker conditions (see e.g. [61]), only those points contribute to the optimal portfolio weights, for which the inequality constraints in (8), and the corresponding constraints in eq. (12), are met exactly. the solution of wopt contains only those points, and effectively ignores the rest. this sparsity contributes to the stability of the solution. regularized portfolio optimization (rpo) operates, in contrast to general regression, with a fixed budget. as a consequence, the lagrange multiplier γ now appears in the optimal solution, eq. (16). compared to the optimal solution in support vector (sv) regression, wsv, the solution vector under the budget constraint, wrpo, is shifted by γ: wrpo = wsv −γ1. (17) let us now consider the dual problem. the dual is, in general, a function of the dual variables, which are here {α, α∗, γ, λ, η, η∗}, although we will see in regularizing portfolio optimization. 10 the following that some of these variables drop out. the dual is defined as d := minw,ξ,ξ∗,ǫ l[w, ξ, ξ∗, ǫ, α, α∗, γ, λ, η, η∗], and the dual problem is then to maximize d over the dual variables. we can replace the minimization over w by evaluating the lagrangian at wopt. for that we have to evaluate f[wopt] = −1 2∥wopt∥2 (18) =  −1 2 t x k=1 (αk −α∗ k)x(k) −γ1 !2 . (19) for the other terms in the lagrangian, we have to consider different cases: (i) if  cν −λ −pt k=1(αk + α∗ k)  < 0, then l can be minimized by letting ǫ →∞, which means that d = −∞. (ii) if  cν −λ −pt k=1(αk + α∗ k)  ≥ 0: the term ǫ  cν −λ −pt k=1(αk + α∗ k)  vanishes. reason: if equality holds, this is trivially true, and if the inequality holds strictly then l can be minimized by setting ǫ = 0. similarly, for the other constraints (the notation (∗) means that this is true for variables with and without the asterisk): (i) if  c t −α(∗) k −η(∗) k  < 0, then l can be minimized by letting ξ(∗) k →∞, which means that d = −∞. (ii) if  c t −α(∗) k −η(∗) k  ≥0, then ξk  c t −α(∗) k −η(∗) k  = 0. reason: if the inequality holds strictly then l can be minimized by ξ(∗) k = 0. if equality holds then it is trivially true. by a similar argument, the term γ in eq. (14) disappears in the dual. altogether we have that either d = −∞, or d(α, α∗, γ) = min ξ,ξ∗,ǫ f[wopt(α, α∗, γ)] = −1 2∥wopt∥2 (20) and t x k=1 (α∗ k + αk) ≤cν −λ (21) and α(∗) k + η(∗) k ≤c t . (22) note that the variables ξ(∗) k , η(∗) k , ǫ, λ do not appear in f[wopt(α, α∗, γ)]. the dual problem is therefore given by max α,α∗,γ  −1 2 t x k=1 (αk −α∗ k)x(k) −γ1 !2 . (23) s.t. {αk, α∗ k} ∈  0, c t  (24) t x k=1 (α∗ k + αk) ≤cν. (25) regularizing portfolio optimization. 11 we can analytically maximize over γ and obtain for the optimal value γ = 1 n t x k=1 (αk −α∗ k) n x i=1 x(k) i −1 ! (26) the optimal projection (= optimal portfolio) is given by wopt * x = t x k=1 (αk −α∗ k)x(k) * x −1 n t x k=1 (αk −α∗ k) n x i=1 x(k) i −1 ! 1 * x. (27) for n →∞the second term vanishes and the solution is the same as the the solution in support vector regression. note that the kernel-trick (see e.g. [47]), which is used in support vector machines to find nonlinear models hinges on the fact that only dot products of input vectors appear in the support vector expansion of the solution. as a consequence of the budget constraint, one can no longer use the kernel-trick (compare eq. (27)). as long as we disregard derivatives, this is not a problem for portfolio optimization. keep in mind, however, that the budget constraint introduces this otherwise undesirable property. support vector algorithms typically solve the dual form of the problem (for a recent survey see [62]), which is in our case given by max α,α∗,γ −1 2 " t x k=1 t x l=1 (αk −α∗ k)(αl −α∗ l ) x(k)x(l) −1 n n x i=1 x(k) i n x i=1 x(l) i !# (28) s.t. {αk, α∗ k} ∈  0, c t  ; t x k=1 (α∗ k + αk) ≤cν. for n →∞the problem becomes identical to ν-svr, which can be solved by linear programming, for which software packages are available [63]. for finite n, it can still be solved with existing methods, because it is quadratic in the αk's. solvers such as the ones discussed in [64] and [62] can be used, but have to be adapted to this specific problem. the regularized symmetric tail average minimization problem (eq. (11) with the constraints eqs. (8)–(10) and (12)) is, as we have shown here, directly related to support vector regression which uses the ǫ-insensitive loss function. the ǫ-insensitive loss is stable to local changes for data points that fall outside the range specified by ǫ. this point is elaborated in section 3 in [60], and relates this method to robust estimation of the mean. it can also be extended to robust estimation of quantiles [60] by scaling of the slack variables ξk by μ and ξ∗ k by 1 −μ, respectively. this scaling translates directly to the portfolio optimization problem, which is an extreme case: downside risk measures penalize only loss, not gain. the asymmetry in the loss function corresponds to μ = 1. regularizing portfolio optimization. 12 5.2. regularized expected shortfall. by this final change we arrive at the regularized portfolio optimization problem, eqs. (7)–(10), which we originally set out to solve. this is now easily solved in analogy to the previous paragraphs: the slack variables ξ∗ k disappear, together with the respective lagrange multipliers which enforce constraints, including α∗ k. the optimal solution is now wopt = t x k=1 αkx(k) −γ1, (29) with γ = 1 n t x k=1 αk n x i=1 x(k) i −1 ! . (30) the dual problem is given by max αk −1 2 " t x k=1 t x l=1 αkαl x(k)x(l) −1 n n x i=1 x(k) i n x i=1 x(l) i !# s.t. αk ∈  0, c t  ; t x k=1 αk ≤cν. (31) which, like its symmetric counterpart, eq. (28), can be solved by adjusting existing algorithms. the formalism provides a free parameter, c, to set the balance between the original risk function and the regularizer. its choice may depend on a number of factors, such as the investors time horizon, the nature of the underlying data, and, crucially, on the ratio n/t. intuitively, there must be a maximum allowable value cmax(n/t) for c, such that when one puts more emphasis on the data, c > cmax(n/t), then over fitting will occur with high probability. it would be desirable to know an analytic expression for (a bound on) cmax(n/t). in practice, cross-validation methods are often employed in machine learning to set the value of c. those methods are not free of problems (see, for example, the treatment in [65]), and the optimal choice of this parameter remains an open problem. 6. regularization corresponds to portfolio diversification. above, we have controlled the capacity of the linear model by minimizing the l2 norm of the portfolio weight vector. in the finance context, minimizing ∥w∥2 = x i w2 i ≃ 1 neff (32) corresponds roughly to maximizing the effective number of assets, neff, i.e. to exerting a pressure towards portfolio diversification [66]. we conclude that diversification of the portfolio is crucial, because it serves to counteract the observed instability by acting as a regularizer. regularizing portfolio optimization. 13 other constraints that penalizes the length of the weight vector could alternatively be considered as a regularizer, in particular any lp norm. the budget constraint alone, however, does not suffice as a regularizer, since it does not constrain the length of the weight vector. adding a ban on short selling, wi ≥0, to the budget constraint, p i wi = 1, limits the allowable solutions to a finite volume in the space of weights and is equivalent to requiring that p i |wi| ≤1.∥it thereby imposes a limit on the l1 norm, that is on the sum of the absolute amplitudes of long and short positions. one may argue that it may be a good idea to use the l1 norm instead of the l2 norm, because that may make the solution sparser. however, the l1 norm has a tendency to make some of the weights vanish. indeed, it has been shown that in the orthonormal design case (using the variance as the risk measure) an l1 regularizer will set some of the weights to zero, while an l2 regularizer will scale all the weights [29]. the spontaneous reduction of portfolio size has also been demonstrated in numerical simulations [67]: as one goes deeper and deeper into the regime where t is significantly smaller than n, under a ban on short selling, more and more of the weights will become zero. the same "freezing out" of the weights has been observed in portfolio optimization [68] as an empirical fact. it is important to stress that the vanishing of some of the weights does not reflect any structural property of the objective function, it is just a random effect: as clearly demonstrated by simulations [67], for a different sample a different set of weights vanishes. the angle of the weight vector fluctuates wildly from sample to sample. (the behavior of the solutions is similar for other limit systems as well.) this means that the solutions will be determined by the limit system and the random sample, rather than by the structure of the market. so the underlying instability is merely "masked", in that the solutions do not run away to infinity, but they are still unstable under sample fluctuations when t is too small. as it is certainly not in the interest of the investor to obtain a portfolio solution which sets weights to zero on the basis of unreliable information from small samples, the above observations speak strongly in favor of using the l2 norm over the l1 norm. 7. conclusion we have made the observation that the optimization of large portfolios minimizes the empirical risk in a regime where the data set size is similar to the size of the portfolio. in that regime, a small empirical risk does not necessarily guarantee a small actual risk [24]. in this sense naive portfolio optimization over-fits the data. regularization can overcome this problem by reducing the capacity of the considered model class. regularized portfolio optimization has choices to make, not only about the risk function, but also about the regularizer. here, we have focussed on the increasingly popular expected shortfall risk measure. using the l2 norm as a regularizer leads to a convex optimization problem which can be solved with linear programming. we ∥this point has been made independently by [17]. regularizing portfolio optimization. 14 have shown that regularized portfolio optimization is then a variant of support vector regression. the differences are an asymmetry, due to the tolerance to large positive deviations, and the budget constraint, which is not present in regression. our treatment provides a novel insight into why diversification is so important. the l2 regularizer implements a pressure towards portfolio diversification. therefore, from a statistical point of view, diversification is important as it is one way to control the capacity of the portfolio optimizer and thereby to find a solution which is more stable, and hence meaningful. in summary, the method we have outlined in this paper allows for the unified treatment of optimization and diversification in one principled formalism. it shows how known methods from modern statistics can be used to improve the practice of portfolio optimization. 8. acknowledgements we thank leon bottou for helpful discussions and comments on the manuscript. this work has been supported by the "cooperative center for communication networks data analysis", a nap project sponsored by the national office of research and technology under grant no. kckha005. ss thanks the collegium budapest for hosting her during this collaboration, and the community at the collegium for providing a creative and inspiring atmosphere. [1] h. markowitz. portfolio selection. journal of finance, 7:77–91, 1952. [2] h. markowitz. portfolio selection: efficient diversification of investments. j. wiley and sons, new york, 1959. [3] e. j. elton and m. j. gruber. modern portfolio theory and investment analysis. wiley, new york, 1995. [4] j.d. jobson and b. korkie. improved estimation for markowitz portfolios using james-stein type estimators. proceedings of the american statistical association (business and economic statistics), 1:279–284, 1979. [5] p. jorion. bayes-stein estimation for portfolio analysis. journal of financial and quantitative analysis, 21:279–292, 1986. [6] p.a. frost and j.e. savarino. an empirical bayes approach to efficient portfolio selection. journal of financial and quantitative analysis, 21:293–305, 1986. [7] r. macrae and c. watkins. safe portfolio optimization. in h. bacelar-nicolau, f. c. nicolau, and j. janssen, editors, proceedings of the ix international symposium of appliedstochastic models and data analysis: quantitative methods in business and industry society, asmda-99, 14-17 june 1999, lisbon, portugal, page 435. ine, statistics national institute, portugal, 1999. [8] r. jagannathan and t. ma. risk reduction in large portfolios: why imposing the wrong constraints helps. journal of finance, 58:1651–1684, 2003. [9] o. ledoit and m. wolf. improved estimation of the covariance matrix of stock returns with an application to portfolio selection. journal of empirical finance, 10(5):603–621, 2003. [10] o. ledoit and m. wolf. a well-conditioned estimator for large-dimensional covariance matrices. j. multivar. anal., 88:365–411, 2004. [11] o. ledoit and m. wolf. honey, i shrunk the sample covariance matrix. j. portfolio management, 31:110, 2004. [12] v. demiguel, l. garlappi, and r. uppal. optimal versus naive diversification: how efficient is the 1/n portfolio strategy? review of financial studies, 2007. regularizing portfolio optimization. 15 [13] l. garlappi, r. uppal, and t. wang. portfolio selection with parameter and model uncertainty: a multi-prior approach. review of financial studies, 20:41–81, 2007. [14] v. golosnoy and y. okhrin. multivariate shrinkage for optimal portfolio weights. the european journal of finance, 13:441–458, 2007. [15] r. kan and g. zhou. optimal portfolio choice with parameter uncertainty. journal of financial and quantitative analysis, 42:621–656, 2007. [16] g. frahm and ch. memmel. dominating estimators for the global minimum variance portfolio, 2009. deutsche bundesbank, discussion paper, series 2: banking and financial studies. [17] v. demiguel, l. garlappi, f. j. nogales, and r. uppal. a generalized approach to portfolio optimization: improving performance by constraining portfolio norms. management science, 55:798–812, 2009. [18] l. laloux, p. cizeau, j.-ph. bouchaud, and m. potters. noise dressing of financial correlation matrices. phys. rev. lett., 83:1467–1470, 1999. [19] v. plerou, p. gopikrishnan, b. rosenow, l.a.n. amaral, and h.e. stanley. universal and non- universal properties of cross-correlations in financial time series. phys. rev. lett., 83:1471, 1999. [20] l. laloux, p. cizeau, j.-p. bouchaud, and m. potters. random matrix theory and financial correlations. international journal of theoretical and applied finance, 3:391, 2000. [21] v. plerou, p. gopikrishnan, b. rosenow, l. a. n. amaral, t. guhr, and h. e. stanley. a random matrix approach to cross-correlations in financial time-series. phys. rev. e, 65:066136, 2000. [22] z. burda, a. goerlich, and a. jarosz. signal and noise in correlation matrix. physica, a343:295, 2004. [23] m. potters and j.-ph. bouchaud. financial applications of random matrix theory: old laces and new pieces. acta phys. pol., b36:2767, 2005. [24] v. vapnik and a. chervonenkis. on the uniform convergence of relative frequencies of events to their probabilities. theory of probability and its applications, 16(2):264–280, 1971. [25] v. vapnik. the nature of statistical learning theory. springer verlag, new york, 1995. [26] v. vapnik. statistical learning theory. john wiley and sons, new york, 1998. [27] b. e. boser, i. m. guyon, and v. n. vapnik. a training algorithm for optimal margin classifiers. in d. haussler, editor, proc. 5th annual acm workshop on computational learning theory, pages 144–152. acm press, 1992. [28] c. cortes and v. vapnik. support vector networks. machine learning, 20:273–297, 1995. [29] r. tibshirani. regression shrinkage and selection via the lasso. j. royal. statist. soc b., 58(1):267–288, 1996. [30] i. frank and j. friedman. a statistical view of some chemometrics regression tools. technometrics, 35:109–148, 1993. [31] v.k. chopra and w.t. ziemba. the effect of errors in means, variances, and covariances on optimal portfolio choice. journal of portfolio management, 19:611, 1993. [32] r.c. merton. on estimating the expected return on the market: an exploratory investigation. journal of financial economics, 8:323361, 1980. [33] y. okhrin and w. schmied. distribution properties of portfolio weights. journal of econometrics, 134:235–256, 2006. [34] a. kempf and c. memmel. estimating the global minimum variance portfolio. schmalenbach business review, 58:332348, 2006. [35] g. frahm. linear statistical inference for global and local minimum variance portfolios. statistical papers, 2008. doi: 10.1007/s00362-008-0170-z. [36] i. kondor, s. pafka, and g. nagy. noise sensitivity of portfolio selection under various risk measures. journal of banking and finance, 31:1545–1573, 2007. [37] s. pafka and i. kondor. noisy covariance matrices and portfolio optimization. eur. phys. j., b 27:277–280, 2002. [38] s. pafka and i. kondor. noisy covariance matrices and portfolio optimization ii. physica, a regularizing portfolio optimization. 16 319:487–494, 2003. [39] s. pafka and i. kondor. estimated correlation matrices and portfolio optimization. physica, a 343:623–634, 2004. [40] z. burda, j. jurkiewicz, and m. a. nowak. is econophysics a solid science? acta physica polonica, b 34:87–132, 2003. [41] m. mezard s. ciliberti, i. kondor. on the feasibility of portfolio optimization under expected shortfall. quantitative finance, 7:389–396, 2007. [42] m. mezard s. ciliberti. risk minimization through portfolio replication. eur. phys. j., b 57:175– 180, 2007. [43] i. varga-haszonits and i. kondor. the instability of downside risk measures. j. stat. mech., p12007, 2008. [44] i. varga-haszonits and i. kondor. noise sensitivity of portfolio selection in constant conditional correlation garch models. physica, a385:307–318, 2007. [45] i. kondor and i. varga-haszonits. feasibility of portfolio optimization under coherent risk measures, 2008. submitted to quantitative finance. [46] b. sch ̈ olkopf. support vector learning. gmd-bericht ; 287. oldenbourg, m ̈ unchen, germany, 1997. dissertation: berlin, techn. univ., diss., 1997. [47] b. sch ̈ olkopf, c. j.c. burges, and a. j. smola. advances in kernel methods - support vector learning. mit press, cambridge, ma, 1999. [48] v. vapnik and a. chervonenkis. theory of pattern recognition. nauka, 1974. [in russian. german translation available from akademie-verlag, berlin, 1979.]. [49] p. jorion. var: the new benchmark for managing financial risk. mcgraw-hill, new york, 2000. [50] j.p. morgan and reuters. riskmetrics. technical document available at http://www.riskmetrics.com. [51] p. artzner, f. delbaen, j.-m. eber, and d. heath. coherent measures of risk. mathematical finance, 9:203–228, 1999. [52] p. embrechts. extreme value theory: potential and limitations as an integrated risk measurement tool. derivatives use, trading and regulation, 6:449–456, 2000. [53] c. acerbi, c. nordio, and c. sirtori. expected shortfall as a tool for financial risk management., 2001. unpublished. [54] c. acerbi. spectral measures of risk: a coherent representation of subjective risk aversion. journal of banking and finance, 26(7):1505–1518, 2002. [55] c. acerbi and d. tasche. on the coherence of expected shortfall. journal of banking and finance, 26(7):1487–1503, 2002. [56] c. acerbi. coherent representations of subjective risk-aversion. in g. szeg ̈ o, editor, risk measures for the 21st century. john wiley and sons., 2004. [57] r. t. rockafellar and s. uryasev. optimization of conditional value-at-risk. journal of risk, 2(3):21–41, 2000. [58] f. perez-cruz, j. weston, d.j.l. herrmann, and b. sch ̈ olkopf. extension of the nu-svm range for classification. in advances in learning theory: methods, models and applications, volume 190 of nato science series iii: computer and systems sciences, pages 179–196. ios press, amsterdam, 2003. [59] a. takeda and m. sugiyama. ν-support vector machine as conditional value-at-risk minimization, 2008. [60] b. sch ̈ olkopf, a. j. smola, r. c. williamson, and p. l. bartlett. new support vector algorithms. neural computation, 12(5):1207–1245, 05 2000. [61] d. p. bertsekas. nonlinear programming. athena scientific, belmont, ma, 1995. [62] l ́ eon bottou and chih-jen lin. support vector machine solvers. in l ́ eon bottou, olivier chapelle, dennis decoste, and jason weston, editors, large scale kernel machines, pages 301–320. mit press, cambridge, ma., 2007. regularizing portfolio optimization. 17 [63] robert j. vanderbei. loqo users manual.x. software available at http://www.princeton.edu/ rvdb/loqo/loqo.html. [64] antoine bordes, seyda ertekin, jason weston, and l ́ eon bottou. fast kernel classifiers with online and active learning. journal of machine learning research, 6:1579–1619, september 2005. [65] y. bengio and y. grandvalet. no unbiased estimator of the variance of k-fold cross-validation. in s. becker, l. saul, and b. sch ̈ olkopf, editors, advances in neural information processing systems 16 (nips'03). mit press, cambridge, ma., 2004. [66] j.-ph. bouchaud and m. potters. theory of financial risk - from statistical physics to risk management. cambridge university press, cambridge, uk, 2000. [67] n. gulyas and i. kondor. portfolio instability and linear constraints, 2007. submitted to physica a. [68] b. scherer and r. d. martin. introduction to modern portflio optimization with nuopt and s-plus. springer, 2005.

0911.1695

h\"ansch--couillaud locking of mach--zehnder interferometer for carrier removal from a phase-modulated optical spectrum

we describe and analyse the operation and stabilization of a mach--zehnder interferometer, which separates the carrier and the first-order sidebands of a phase-modulated laser field, and which is locked using the h\"ansch--couillaud method. in addition to the necessary attenuation, our interferometer introduces, via total internal reflection, a significant polarization-dependent phase delay. we employ a general treatment to describe an interferometer with an object which affects the field along one path, and we examine how this phase delay affects the error signal. we discuss the requirements necessary to ensure the lock point remains unchanged when phase modulation is introduced, and we demonstrate and characterize this locking experimentally. finally, we suggest an extension to this locking strategy using heterodyne detection.

introduction for many experiments and laser-locking schemes it is necessary to create light which is phase-coherent with and frequency shifted relative to a master laser oscilla- tor. common approaches include acousto-optical mod- ulation [1], electro-optical modulation [2], and current modulation of the laser [3–5]. the latter two of these approaches give, typically, a phase-modulated spectrum. it is often useful to separate the frequency components of such a spectrum, and there are several devices which can perform this task [6,7]. we describe one of these- a mach–zehnder interferometer-and a locking-scheme based on the method of h ̈ ansch and couillaud [8]; we include a description of a relative phase delay between the linear polarization components, which is expected for many real devices, and we take care to ensure the lock point does not change when phase modulation is introduced. this property is essential in our application, where we use a sideband from an electro-optically mod- ulated laser field to drive raman transitions between hyperfine states in cold alkali-earth atoms (similar to ref. [9]), and in which we change the modulation fre- quency and depth during the experiment. for related uses, see references [10] and [11]. we will provide a sim- ple and general analysis, and a demonstration of a robust embodiment in a realistic experimental setting. 2. theoretical framework consider a light field ⃗ e0 incident on an interferometer, as depicted in fig. 1. one path passes unperturbed to the output beam-splitter, while the second path is subject to a phase delay φ, and passes through an object, described by ˆ o, before recombining with the first path. the output field is hence ⃗ et = 1 2  ⃗ e0 −eiφ ˆ o ⃗ e0  ; (1) the minus sign is a consequence of the phase change un- der reflection. fig. 1. prototype interferometer showing incident field ⃗ e0, beam-splitters, output fields ⃗ et and ⃗ es (the latter after a quarter-wave plate λ/4) and an internal object which affects the field which follows the longer path. we pass this field through a quarter-wave plate (with fast-axis at 45◦to the vertical) and analyze the resulting electric field: ⃗ es = ˆ q 1 2  ⃗ e0 + eiφ ˆ o ⃗ e0  , where ˆ q = 1 √ 2  1 i i 1  (2) in the standard jones matrix representation [12]. we de- fine the error signal s as the difference between the linear polarization components of the field ⃗ es, and the trans- mission t as the horizontal component of the field ⃗ et : s =   ⃗ es * ˆ x  2 −   ⃗ es * ˆ y  2 and t =   ⃗ et * ˆ x  2 ; (3) we have discarded the vertical polarization from the out- put, which reduces the power but means it is possible to achieve complete extinction. by choosing the incident field and the internal object this framework can be used to describe a variety of lock- ing schemes. the original h ̈ ansch–couillaud scheme, al- though here phrased in terms of interferometers rather than a cavity, corresponds to horizontally polarized light and a rotated linear polarizer [8]. we now apply this framework to describe an internal object which attenuates and also delays one polarization. the linearly polarized incident field ⃗ e0, and the object ˆ o are described by the following: ⃗ e0 = e0  cosθ sin θ  and ˆ o =  1 0 0 (1 −α)eiβ  (4) 1 where α is the attenuation, β is the phase delay, and θ is the angle by which the linearly polarized light is inclined from vertical. in the absence of attenuation, the two polarizations will be subjected only to different delays, and the dif- ference between the two displaced interference patterns could be used as an error signal; here the field before the quarter-wave plate would be used to derive s. if the phase delay β were zero, we would recover a rotated version of the h ̈ ansch–couillaud scheme. also, if the attenuation was total (α = 1), any phase delay would be inconsequential and we would again recover this original scheme. in the intermediate case of partial attenuation (α < 1) and non-zero phase delay (β ̸= 0), we find that the er- ror signal crosses zero at the maximum of transmission, but has the non-zero value 1 2(1 −α) sin β sin 2θ at the transmission minimum. its gradient about the extrema is ± 1 4  1 −(1 −α) cos β  sin 2θ. in this intermediate regime, the introduction of phase modulation, as discussed in the next section, affects the positions at which the error signal crosses zero. as described later, we found that a phase delay was unavoidable in our device; therefore, in order to recover the h ̈ ansch–couillaud scheme, we in- troduced total attenuation of the vertical polarization component. 3. phase modulated light we may analyze a modulated field ⃗ e(t) = ⃗ e0ei r t 0 ω(t′)dt′ in terms of its fourier components. for the case of sim- ple phase modulation, where r t 0 ω(t′)dt′ = ω0t+m cosωt (ω0: unmodulated frequency; ω: modulation frequency; m: modulation depth), we expand using the jacobi– anger identity, treat the sidebands as independent fields, and sum their contributions to obtain the error signal spm ω0τ
= +∞ x n=−∞ |jn(m)|2 s ω0τ + nωτ
, (5) where the subscript 'pm' indicates that phase modula- tion is present; τ = φ/ω0 is the optical path delay in our device, jn(m) is the nth order bessel function, and we have assumed that all frequency components interact in the same way with the optical elements. we operate the device near the condition ωτ = π which separates the carrier from the first-order sidebands or, more generally, ensures odd and even numbered side- bands exit from opposite ports of the interferometer. un- less the signal crosses zero at the transmission minimum, the position where it does cross will shift when phase modulation is introduced. hence we must ensure the sig- nal is zero at the transmission extrema while maintaining a non-zero gradient; this is satisfied for β = 0 or α = 1. to achieve this, we can compensate for any differential phase shift β in our device using a waveplate, or we can introduce complete attenuation of the vertical polariza- -0.2 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 25 photodiode signal (arb. units) piezo voltage offset (volts) transmission signal fig. 2. transmission t (black) and error signal s (red) with (solid) and without (dashed) phase modulation of the input light, as the path difference is scanned using a piezo-electric stack. the dashed vertical lines mark the positions where the error signals coincide, and these are very close to s = 0. they are, however, displaced from the transmission extrema, but there is adequate scope for optimization for a given input field. tion using a linear polarizer. the overall signal becomes spm(ω0τ) = s ω0τ
+∞ x n=−∞ (−1)n |jn(m)|2 . (6) under these conditions, any change in phase modula- tion depth does not affect the positions where the error signal crosses zero; the gradient changes, but for modu- lation depths m ≲0.38π it maintains the same sign. 4. experimental implementation a mach–zehnder interferometer was constructed from readily available components, including two non- polarizing bk7 beam-splitter cubes, and a bk7 right- angled prism [13]. the cubes were glued using low- expansion uv curing glue [14], with care taken to ensure their faces were parallel, and the pair was mounted on a kinematic mount. the prism was glued to a translation stage; a screw was available for coarse path-difference ad- justment, and a piezo-electric stack was used for small adjustments and locking feedback. incident light was spatially filtered and collimated (to ensure the incident wavefronts were flat) and aligned into the device. beam overlap was found visually, and then maximized by scan- ning the path difference using the piezo and observing the contrast ratio of power exiting each of the output ports. in the absence of a relative phase delay between the polarizations, one could generate an error signal using the slight polarization selectivity of reflections by the nominally polarization insensitive beam-splitter cubes. however, we found that a significant relative phase de- lay of β ≈78◦was introduced by the two total internal 2 0 0.2 0.4 0.6 0.8 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 photodiode signal (arb. units) frequency relative to carrier (ghz) maximized carrier minimized carrier 0 0.1 -0.8 -0.7 -0.6 0 0.1 0.6 0.7 0.8 fig. 3. phase-modulated spectra after filtering by the mach–zehnder interferometer, showing maximum and minimum carrier transmission. the modulation fre- quency (2.7 ghz) is larger than the free-spectral range of the scanning fabry–p ́ erot cavity (2 ghz) used to obtain this trace, and so the sidebands appear at ±700 mhz for the lower- and upper-sidebands, respectively; these are magnified in the insets. by comparing the amplitudes, we estimate the modulation depth to be m ≈0.2π, and by comparing with an unfiltered reference trace (obtained by blocking one path of the interferometer), we see that, for the case of minimized carrier transmission, the carrier is attenuated by more than 20 db while approximately 1 db of sideband power is lost. this trace was smoothed using a 5 mhz bandwidth moving-average filter. reflections inside our right-angled prism [12] and, to re- cover the h ̈ ansch–couillaud method, it was necessary to introduce a linear polarizer to ensure complete attenua- tion of the vertical polarization (i.e. α = 1). fig. 2 shows the interferometer transmission meas- ured with and without the phase-modulated optical side- bands that we wish to separate. the modulation fre- quency is ω= 2π × 2.725 ghz and the wavelength is λ = 780 nm; the device is operated near to a path differ- ence cτ = cπ/ω≈55 mm, which corresponds to the con- dition ωτ = π. the photodiodes sample different parts of the beam cross-section, and also have different respon- sivities. it may be necessary to adjust the photodiode balance and the offset, but as demonstrated by this un- optimized trace, a real device operates approximately as predicted. from the decrease in visibility and error sig- nal, we estimate a modulation depth of m ≈0.2π. this agrees with the more direct measurement with a scan- ning fabry–p ́ erot cavity, as shown in fig. 3. the imper- fect behaviour of the device is accounted for partly by the unequal reflectivities of the beam-splitter cubes, but a more significant problem is the spatial overlap of the fields in this free-space device; we would expect improved performance and more ideal behaviour if the device was implemented using single-mode optical fibers and fiber- based beam-splitters. 0 10 20 30 40 50 0.01 0.1 1 10 100 power spectral density (db, arb. units) fourier frequency (hz) unlocked locked fig. 4. fast fourier transform of the recorded output when the system is locked (black) and unlocked (red). the plot has been smoothed using a 10 mhz bandwidth moving-average filter. we constructed a feedback circuit with an integrated high-voltage output, similar to that in ref. [15], and recorded the transmission with and without feedback; the fourier transforms of these are shown in fig. 4. the bandwidth of the circuit is ∼100 hz, and for low fre- quencies (< 10 hz) the circuit reduces drift by several orders of magnitude. the very slightly increased noise at high frequency is expected for a feedback circuit. 5. alternative: heterodyne detection another method by which we could ensure the error signal would remain unchanged when phase modulation was introduced would be to mix the photodiode signals with a frequency-shifted reference, derived from the un- modulated field, and extract electronically a signal cor- responding to the beat note of this reference mixed with the carrier. furthermore, it would be possible to tune the reference field frequency close to, and hence make the error signal depend upon, any chosen frequency com- ponent in a complex, more powerful spectrum. alterna- tively, with a sufficiently fast photodiode, one may tune the detection electronics to select a desired optical fre- quency component. this reference field could be created by using an acousto-optical modulator (aom) to pick offa small fraction of the laser field before it passes through, for ex- ample, an electro-optical phase modulator; see fig. 5. as an example, consider an interferometer which separates positive and negative first-order sidebands from a 3ghz phase-modulated field. using a 100mhz aom-shifted reference field and a fast photodiode, one could detect beat-frequencies at 2900mhz, 100mhz, and 3100mhz; using standard electronics, a specific frequency compo- nent could be extracted and passed to the locking cir- cuit. here, feedback could be made to depend on one sideband, leaving the other sideband and a significant fraction of the carrier. the resulting spectrum would be 3 fig. 5. a proposed h ̈ ansch–couillaud scheme using heterodyne detection. an acousto-optical modulator (aom) extracts a reference before sideband modula- tion. light passes through the interferometer, composed of non-polarizing beam-splitter cubes and a right-angled prism, and is detected by photodiodes labeled (±). the reference light is incident on the detectors and the lin- ear polarizers (solid black lines) are at 45◦relative to the beam-splitter cube. the two half-wave plates labeled λ/2 serve to introduce the small rotation necessary for the h ̈ ansch–couillaud scheme before the mach–zehnder, and to rotate the reference light by 45◦so that equal power from this reference falls on the photodiodes af- ter the polarizing beamsplitter cube in the heterodyne detection setup. well suited to driving stimulated raman transitions. 6. conclusions we have described and demonstrated a mach–zehnder interferometer used to separate carrier and first-order sidebands from a phase-modulated laser field, which we have locked using the h ̈ ansch–couillaud method. the aim of this article was twofold. firstly, we provided a simple model that allows the interferometer to be under- stood in terms of its constituent birefringent and reflec- tive elements. h ̈ ansch and couillaud did not consider a differential phase-change between the polarisations, but this arises naturally in our real device and so we have ex- tended their model. we then applied this model to the analysis of the interferometer used in our experiments; our results show that the technique is, despite its sim- plicity, appropriate for this commonly-encountered situ- ation. the polarization-dependent phase delay, originating from total internal reflection in the corner-reflector of the mach–zehnder interferometer, affects the error sig- nal, causing an offset at the transmission minimum and leaving the locking scheme sensitive to intensity fluctua- tions. these effects can be eliminated by extinguising one polarization using an internal linear polarizer, leaving the lock point approximately fixed as phase modulation is introduced. the slight residual offset of the lock points from the transmission extrema observed experimentally was readily corrected by adjusting the photodiode bal- ance, and the interferometer was easily optimized for a given input spectrum. an alternative approach accom- plishes this indifference to phase modulation using het- erodyne detection and a frequency-shifted reference light field. references 1. p. bouyer, t. l. gustavson, k. g. haritos, and m. a. kasevich, "microwave signal generation with optical in- jection locking," optics letters 21, 1502 (1996). 2. k. szymaniec, "injection locking of diode lasers to frequency modulated source," optics communications 144, 50–54 (1997). 3. k. y. lau, c. harder, and a. yariv, "direct modula- tion of semiconductor lasers at f > 10 ghz by low- temperature operation," applied physics letters 44, 273 (1984). 4. j. ringot, y. lecoq, j. garreau, and p. szriftgiser, "generation of phase-coherent laser beams for raman spectroscopy and cooling by direct current modulation of a diode laser," the european physical journal d 7, 285 (1999). 5. c. affolderbach, a. nagel, s. knappe, c. jung, d. wiedenmann, and r. wynands, "nonlinear spec- troscopy with a vertical-cavity surface-emitting laser (vcsel)," applied physics b: lasers and optics 70, 407–413 (2000). 6. d. haubrich, m. dornseifer, and r. wynands, "lossless beam combiners for nearly equal laser frequencies," rev. sci. inst. 71, 338–340 (2000). 7. r. p. abel, u. krohn, p. siddons, i. g. hughes, and c. s. adams, "faraday dichroic beam splitter for raman light using an isotopically pure alkali-metal-vapor cell," optics letters 34, 3071 (2009). 8. t. w. h ̈ ansch and b. couillaud, "laser frequency sta- bilization by polarization spectroscopy of a reflecting reference cavity," optics communications 35, 441–444 (1980). 9. m. kasevich and s. chu, "measurement of the gravita- tional acceleration of an atom with a light-pulse atom interferometer," applied physics b photophysics and laser chemistry 54, 321–332 (1992). 10. i. dotsenko, w. alt, s. kuhr, d. schrader, m. muller, y. miroshnychenko, v. gomer, a. rauschenbeutel, and d. meschede, "application of electro-optically generated light fields for raman spectroscopy of trapped cesium atoms," applied physics b 78, 711–717 (2004). 11. j. schneider, o. gl ̈ ockl, g. leuchs, and u. l. andersen, "quadrature measurements of a bright squeezed state via sideband swapping," optics letters 34, 1186 (2009). 12. e. hecht, optics (addison wesley, 2001), 4th ed. 13. suitable components are available from many suppliers, including thorlabs (bs011 and ps911). 14. a suitable low-expansion glue is manufactured by dy- max; part number op-67-ls. 15. v. v. yashchuk, d. budker, and j. r. davis, "laser fre- quency stabilization using linear magneto-optics," rev. sci. inst. 71, 341–346 (2000). 4

0911.1696

a quantum lovasz local lemma

the lovasz local lemma (lll) is a powerful tool in probability theory to show the existence of combinatorial objects meeting a prescribed collection of "weakly dependent" criteria. we show that the lll extends to a much more general geometric setting, where events are replaced with subspaces and probability is replaced with relative dimension, which allows to lower bound the dimension of the intersection of vector spaces under certain independence conditions. our result immediately applies to the k-qsat problem: for instance we show that any collection of rank 1 projectors with the property that each qubit appears in at most $2^k/(e \cdot k)$ of them, has a joint satisfiable state. we then apply our results to the recently studied model of random k-qsat. recent works have shown that the satisfiable region extends up to a density of 1 in the large k limit, where the density is the ratio of projectors to qubits. using a hybrid approach building on work by laumann et al. we greatly extend the known satisfiable region for random k-qsat to a density of $\omega(2^k/k^2)$. since our tool allows us to show the existence of joint satisfying states without the need to construct them, we are able to penetrate into regions where the satisfying states are conjectured to be entangled, avoiding the need to construct them, which has limited previous approaches to product states.

introduction and results in probability theory, if a number of events are all independent of one another, then there is a positive (possibly small) probability that none of the events will occur. the lov ́ asz local lemma (proved in 1975 by erd ̈ os and lov ́ asz) allows one to relax the independence condition slightly: as long as the events are "mostly" independent of one another and are not individually too likely, then there is still a positive probability that none of them occurs. in its simplest form it states theorem 1 ([el75]). let b1, b2, . . . , bn be events with pr(bi) ≤p and such that each event is mutually independent of all but d of the others. if p * e * (d + 1) ≤1 then pr(vn i=1 bc i ) > 0. the lov ́ asz local lemma (lll) is an extremely powerful tool in probability theory as it sup- plies a way of dealing with rare events and of showing that a certain event holds with positive probability. it has found an enormous range of applications (see, e.g., [as04]), for instance to graph colorability [el75], lower bounds on ramsey numbers [spe77], geometry [mp87], and al- gorithms [mt09]. for many of these results there is no known proof which does not use the local lemma. one notable application of the lll is to determine conditions under which a k-cnf formula is satisfiable. if each clause of such a formula φ involves a disjoint set of variables, then it is obvious that φ is satisfiable. one way to see this is to observe that a random assignment violates a clause with probability p = 2−k and hence the probability that all m clauses are satisfied by a random assignment is (1 −p)m > 0. but what if some of the clauses share variables, i.e., if they are "weakly dependent"? this question is readily answered by using the lll: corollary 2. let φ be a k-sat formula in cnf-form. if every variable appears in at most 2k/(e * k) clauses then φ is satisfiable. this corollary follows from thm. 1 by letting bi be the event that the i-th clause is not satisfied for a random assignment, which happens with probability p = 2−k, and noting that each clause depends only on the d ≤(2k/e) −k other clauses that share a variable with it. in particular this corollary gives a better understanding of sat, the prototype np-complete problem in classical complexity theory. in the last decade enormous advances have been made in the area of quantum complexity, the theory of easy and hard problems for a quantum computer. in particular, a natural quantum analog of k-sat, called k-qsat, was introduced by bravyi [bra06]: instead of clauses we have projectors π1, . . . , πm, each acting non-trivially on k qubits, and we have to decide if all of them can be satisfied jointly. more precisely, we ask if there is a state |ψ⟩on all qubits such that πi|ψ⟩= 0 for all 1 ≤i ≤m (in physics language: we ask if the system is frustration-free). this problem1 was shown to be qma1-complete for k ≥4 [bra06] and as such has received considerable attention [liu06, liu07, bs07, bt08, lmss09, llm+09, bmr09]. note that the question is easy for a set of "disjoint" projectors: if no two projectors share any qubits, then clearly |ψ⟩= |ψ1⟩⊗* * * ⊗|ψm⟩is a satisfying state, where |ψi⟩is such that πi|ψi⟩= 0, just like in the case of disjoint k-sat. it is thus very natural to ask if there still is a joint satisfy- ing state when the projectors are "weakly" dependent, i.e., share qubits only with a few other projectors. one might speculate that a quantum local lemma should provide the answer. 1when defined with an appropriate promise gap for no-instances 2 motivated by this question we ask: is there a quantum local lemma? what will take the role of notions like probability space, probability, events, conditional probability and mutual indepen- dence? what properties should they have? and can we prove an analogous statement to cor. 2 for k-qsat? our results: we answer all these questions in the positive by first showing how to generalize the notions of probability and independence in a meaningful way applicable to the quantum setting and then by proving a quantum local lemma. we then show that it implies a statement analogous to cor. 2 for k-qsat with exactly the same parameters as in the classical case. as we describe later in this section, we then combine our results with recent advances in the study of random qsat to substantially widen the satisfiable range and to provide greatly improved lower bounds on the conjectured threshold between the satisfiable and the unsatisfiable region. let us first focus on the conceptual step of finding the right notions of probability and indepen- dence. in the quantum setting we deal with vector spaces and the probability of a certain event to happen is determined by its dimension. it is thus very natural to have the following corre- spondence of classical and "quantum" notions, using the apparent similarity between events and linear spaces: definition 3. we define the following, in correspondence with the classical notions: probability space ω → vector space v event a ∈ω → subspace a ⊆v complement ac = ω\ a → orthogonal subspace a⊥ probability pr(a) → relative dimension r(a) := dim a dim v union and disjunction a ∨b, a ∧b → a + b = {a + b|a ∈a, b ∈b}, a ∩b conditioning pr(a|b) = pr(a∧b) pr(b) → r(a|b) := r(a∩b) r(b) = dim(a∩b) dim(b) a, b independent pr(a ∧b) = pr(a) * pr(b) → a, b r-independent r(a ∩b) = r(a) * r(b) this definition by analogy brings us surprisingly far. it can be verified (see sec. 2) that many useful properties hold for r, like (i) 0 ≤r ≤1, (ii) monotonicity: a ⊆b ⇒r(a) ≤r(b), (iii) the chain rule (iv) an "inclusion/exclusion" formula and (v) r(a) + r(a⊥) = 1. there are, however, two important differences between probability and relative dimension. one concerns the complement of events. for probabilities, the conditional version of property (v) holds: pr(a|b) + pr(ac|b) = 1. for r we can easily find counterexamples to the state- ment r(a|b) + r(a⊥|b) = 1 (for instance two non-equal non-orthogonal lines a and b in a two-dimensional space, where r(a|b) + r(a⊥|b) = 0). it is this property that is used in most proofs of the local lemma, and one of the difficulties in our proof of a quantum lll (qlll) is to circumvent its use. the second difference concerns our notion of r-independence. in probability theory, if a and b are independent, then so are ac and b. again, this is not true any more for r and easy counterex- amples can be found (see sec. 2). it is thus important to find the right formulation of a quantum local lemma concerning mutual independence of events. keeping these caveats in mind and us- ing our notion of relative dimension, we prove a general quantum lll (see sec. 3), which in its simplest form gives: theorem 4. let x1, x2, . . . , xn be subspaces, where r(xi) ≥1 −p and such that each subspace is mutu- ally r-independent of all but d of the others. if p * e * (d + 1) ≤1 then r(tn i=1 xi) > 0. 3 note that in contrast to the classical lll in thm. 1 which is stated in terms of the "bad" events bi, here we are working with the "good" events. while in the classical case these two formulations are equivalent, this is no longer the case for our notion of r-independence. an immediate application of our qlll is to k-qsat, where we are able to show the exact analogue of cor. 2. corollary 5. let {π1, . . . , πm} be a k-qsat instance where all projectors have rank 1. if every qubit appears in at most 2k/(e * k) projectors, then the instance is satisfiable. it follows by defining (with a slight abuse of notation) subspaces xi = π⊥ i of satisfying states for πi. noticing that r(xi) = 1 −2−k and that projectors are mutually r-independent whenever they do not share qubits, and observing that an equivalent formulation of the k-qsat-problem is to decide whether dim(tm i=1 π⊥ i ) > 0, thm. 4 gives the desired result (see secs. 2 and 3 for details and more applications to k-qsat). random qsat: over the past few decades a considerable amount of effort was dedicated to un- derstanding the behavior of random k-sat formulas [ks94, mpz02, mmz05]. research in this area has witnessed a fruitful collaboration among computer scientists, physicists and mathematicians, and is motivated in part by an attempt to better understand the class np, as well as some recent surprising applications to hardness of approximation (see, e.g., [fei02]). the main focus in this area is an attempt to understand the phase transition phenomenon of random k-sat, namely, the sharp transition from being satisfiable with high probability at low clause density to being unsatisfiable with high probability at high clause density. the existence of this phase transition at a critical density αc was proven by friedgut in 1999 [fri99];2 however only in the k = 2 case its value is known exactly (αc = 1 [cr92, goe92, bbc+01]). a long line of works for k = 3 have narrowed it down to 3.52 ≤αc ≤4.49 [kkl03, hs03, dkmp08] (with evidence that αc ≈4.267 [mpz02]), and in the large k limit it has been shown that 2k ln 2 −o(k) ≤αc ≤2k ln 2 [ap04]. the quantum analogue of this question, namely understanding the behavior of random k-qsat instances, has recently started attracting attention. as in the classical case, the motivation here comes from an attempt to understand qma1, the quantum analogue of np (of which k-qsat is a complete problem), as well as the possibility of applications to hardness of approximation, but also from the hope to obtain insight into phase transition effects in other quantum physical systems. the definition of a random k-qsat instance is similar to the one in the classical case. fix some α > 0. then a random k-qsat instance on n qubits of density α is obtained by repeating the following m = αn times: choose a random subset of k qubits and pick a random rank-1 projector on them. an equivalent way to describe this is to say that we choose a random k-uniform hypergraph from the ensemble gk(n, m), in which m = αn k-hyperedges are picked uniformly at random from the set of all possible k-hyperedges on n vertices (with repetitions) and then a random rank-1 projector is chosen for each hyperedge. in a first work on the random k-qsat model, laumann et al. [lmss09] fully characterize the k = 2 case and show a threshold at density αq c = 1/2 using a transfer matrix approach introduced by bravyi [bra06]. curiously, the satisfying states in the satisfiable region are product states. they also establish the first lower and upper bounds on a possible (conjectured [bmr09]) threshold. in 2actually, it is still not known whether the critical density converges for large n; see [fri99] for details on this technical (but nontrivial) issue. 4 a recent breakthrough bravyi, moore and russell [bmr09] have dramatically improved the upper bound to 0.574 * 2k, below the large k limit of ln 2 * 2k ≈0.69 * 2k for the classical threshold! recently, laumann et al. [llm+09] have given substantially improved lower bounds, essen- tially showing the following. theorem 6. [llm+09] if there is a matching of projectors to qubits such that (i) each projector is matched to a qubit on which it acts nontrivially and (ii) no qubit is matched to more than one projector, then the k-qsat instance is satisfiable. such a matching exists with high probability for random instances of qsat if the density is below some critical value c(k) (hence c(k) ≤αq c), with c(3) ≈0.92 and c(k) →1 for large k. there remained a distressingly large gap between the best rigorous lower (< 1) and upper (≈0.574 * 2k) bounds for a satisfiable/non-satisfiable threshold of random k-qsat. using our quantum lll we are able to dramatically improve the lower bound on such a threshold. to get a better intuition on the kind of bounds the quantum lll can give in this setting, let us first look at a simple toy example: random k-qsat instances picked according to the uniform distribution on d-regular k-hypergraphs gk(n, d) (so m = dn/k and their density is α = d/k). it is easy to see that a matching as assumed in thm. 6 only exists iff k ≥d, so this technique shows satisfiability only below density 1. our cor. 5, on the other hand, immediately implies that the instance is satisfiable as long the density α ≤2k/(e * k2). it is this order of magnitude that we manage to achieve also in the random k-qsat model described above. we show theorem 7. a random k-qsat instance of density α ≤2k/(12 * e * k2) is satisfiable with high probability for any k ≥1. hence αq c ≥2k/(12 * e * k2). all previous lower bound proofs [lmss09, llm+09] were based on constructing tensor product states which satisfy all constraints. in fact it is conjectured [llm+09] that c(k) is the critical density above which entangled states would necessarily appear as satisfying states. to our knowledge no technique has allowed to deal with entangled satisfying states in this setting. using the quantum lll allows us to show the existence of a satisfying state without the need to generate it, and in particular the satisfying state need not be a product state (and probably is not). we conjecture that the improvement in our bound, which is roughly exponential in k, is due to this difference. the main difficulty we encounter in the proof of thm. 7 (see sec. 4) is that even though the average degree in gk(n, m = αn) is of the right order of magnitude (≈2k/k) to apply the quantum lll (cor. 5), the maximum degree can deviate vastly from it (its expected size is roughly logarith- mic in n), and hence prevent a direct application of the quantum lll. the key insight is that we can split the graph into two parts, one essentially consisting of high degree vertices that deviate by too much from the average degree and the other part containing the remaining vertices. we then show that the first part obeys the matching conditions of thm. 6 [llm+09] and hence has a sat- isfying state, and the second part obeys the maximum degree requirements of the quantum lll and is hence also satisfiable. the challenge is to "glue" these two satisfying solutions together. for this we need to make sure that each edge in the second part intersects the first part in at most one qubit (by adding all other edges to the first part, while carefully treating the resulting dependen- cies). we can then create a new (k −1)-local projector of rank 2 for each intersecting edge, which reflects the fact that one qubit of this edge is already "taken". this allows to effectively decouple the two parts. 5 discussion and open problems: we have shown a general quantum lll. an obvious open question is whether it has more applications for quantum information. we call our generalization of the lov ́ asz local lemma "quantum" in view of the applications we have given. however, stricto sensu there is nothing quantum in our version of the lll; it is a statement about subspaces and the dimensions of their intersections. as such it seems to be very versatile and we hope that it will find a multitude of other applications, not only in quantum information, but also in geometry or linear algebra. more generally, our lll holds for any set of objects with a valuation r and operations t and + that obey properties (i)-(iv) (see lemma 8) and might be applicable even more generally. since the lll has so many applications, we hope that our "geometric" lll becomes equally useful. the standard proof of the classical lll is non-constructive in the sense that it asserts the exis- tence of an object that obeys a system of constraints with limited dependence, but does not yield an efficient procedure for finding an object with the desired property. in particular, it does not provide an efficient way to find the actual satisfying assignment in cor. 2. a long line of research [bec91, alo91, mr98, cs00, sri08, mos08] has culminated in a very recent breakthrough result by moser [mos09] (see also [mt09]), who gave an algorithmic proof of the lll that allows to effi- ciently construct the desired satisfying assignment (and more generally the object whose existence is asserted by the lll [mt09]). moser's algorithm itself is a rather simple random walk on assign- ments; an innovative information theoretic argument proves its correctness (see also [for09]). this opens the exciting possibility to draw an analogy for a (possibly quantum) algorithm to construct the satisfying state in instances of qsat which are known to be satisfiable via our qlll, and we hope to explore this connection in future work. structure of the paper: in sec. 2 we study properties of relative dimension r and of r-independence, allowing us to prove a general qlll in sec. 3. sec. 4 extends our results to the random k-qsat model and presents our improved bound on the size of the satisfiable region. 2 properties of relative dimension here we summarize and prove some of the properties of the relative dimension r and of r-independence as defined in def. 3, which will be useful in the proof of the quantum lll in the next section. lemma 8. for any subspaces x, y, z, xi ⊆v the following hold (i) 0 ≤r(x) ≤1. (ii) monotonicity: x ⊆y →r(x) ≤r(y). (iii) chain rule: r(tn i=1 xi|y) = r(x1|y) * r(x2|x1 ∩y) * r(x3|x1 ∩x2 ∩y) * . . . * r(xn| tn−1 i=1 xi ∩y). (iv) inclusion/exclusion: r(x) + r(y) = r(x + y) + r(x ∩y). (v) r(x) + r(x⊥) = 1 and r(x|y) + r(x⊥|y) ≤1. (vi) r(x|z) + r(y|z) −r(x ∩y|z) ≤1. 6 proof. properties (i), (ii), (iii) and (v) follow trivially from the definition. property (iv) follows from dim(x) + dim(y) = dim(x ∩y) + dim(x + y), which is an easy to prove statement about vector spaces (see e.g. [kos97], thm. 5.3). property (vi) follows from (ii) and (iv): inclusion/exclusion (iv) gives r(x ∩z) + r(y ∩z) = r(x ∩z + y ∩z) + r(x ∩y ∩z) ≤r(z) + r(x ∩y ∩z), where the last inequality follows from the monotonicity property (ii) using x ∩z + y ∩z ⊆z. dividing by r(z) gives the desired result. we also need to extend our definition of r-independence (def. 3) to the case of several sub- spaces, in analogy to the case of events. definition 9 (mutual independence). an event a (resp. subspace x) is mutually independent (resp. mutually r-independent) of a set of events (resp. subspaces) {y1, . . . , yl} if for all s ⊆[l], pr(a| vl i=1 yi) = pr(a) (resp. r(x| tl i=1 yi) = r(x)). note that unlike in the case of probabilities, it is possible that two subspaces a and b are mutu- ally r-independent but ac and b are not mutually r-independent. one example for this are the fol- lowing subspaces of r4: a = span({(1, 0, 0, 0), (0, 1, 0, 0)} and b = span({(1, 0, 0, 0), (0, 1, 1, 0)}. we have r(a|b) = r(a) = 1/2 but r(a⊥|b) = 0 while r(a⊥) = 1/2. let us now relate the notion of mutual r-independence to the situation in k-qsat instances. we first associate a subspace with a projector, in the natural way. definition 10 (projectors and associated subspace). a k-local projector on n-qubits is a projector of the form π ⊗in−k, where π is a projector on k qubits q1, . . . , qk and in−k is the identity on the remaining qubits. we say that π acts on q1, . . . , qk. for a projector π, let its satisfying space be xπ⊥:= ker π = {|ψ⟩| π|ψ⟩= 0}. when there is no risk of confusion we denote xπ⊥by π⊥and its complement by π. recall that in statements like cor. 5 we would like to say that two projectors are mutually r-independent if they do not share any qubits. this is indeed the case, as the following lemma shows. lemma 11. assume a projector π does not share any qubits with projectors π1, . . . , πl. then xπ⊥is mutually r-independent of {xπ⊥ 1 , . . . , xπ⊥ l}. proof. let us split the hilbert space h of the entire system into h = h1 ⊗h2, where h1 is the space which consists of the qubits π acts on non-trivially (and π1, . . . , πlact as identity) and the remaining space h2. by assumption there are projectors π and π1, . . . , πlsuch that π = π ⊗in−k and πi = ik ⊗πi. for every s ⊆[l], r(π| \ i∈s πi) = dim(π t i∈s πi) dim(t i∈s πi = dim(π ⊗t i∈s πi) dim(i ⊗t i∈s πi) = dim(π) dim(t i∈s πi) dim(h1) dim(t i∈s πi) = r(π). remark: in exactly the same way one can show that π is mutually r-independent of {π⊥ 1 , . . . , π⊥ l} and that both π and π⊥are mutually r-independent of {π1, . . . , πl}. hence the property of not sharing qubits (or, for subspaces, having a certain tensor structure), which in particular implies mutual r-independence, is in some sense a stronger notion of independence than r-independence. 7 to prove our quantum lll we only require the weaker notion of r-independence, which poten- tially makes the quantum lll more versatile and applicable in settings where there is no tensor structure. 3 the quantum local lemma we begin by stating the classical general lov ́ asz local lemma. to this end we need to be more precise about what we mean by "weak" dependence, introducing the notion of the dependency graph for both events and subspaces (see e.g. [as04] for the case of events), where we use relative dimension r as in def. 3. definition 12 (dependency graph for events/subspaces). the directed graph g = ([n], e) is a de- pendency graph for (i) the events a1, . . . , an if for every i ∈[n], ai is mutually independent of {aj|(i, j) / ∈e}, (ii) the subspaces x1, . . . , xn if for every i ∈[n], xi is mutually r-independent of {xj|(i, j) / ∈e}. with these notions in place we can state the general lov ́ asz local lemma (sometimes also called the asymmetric lll). theorem 13 ([el75]). let a1, a2, . . . , an be events with dependency graph g = ([n], e). if there exists 0 ≤y1, . . . , yn < 1, such that pr(ai) ≤yi * ∏(i,j)∈e(1 −yj), then pr( n ^ i=1 ac i) ≥ n ∏ i=1 (1 −yi). in particular, with positive probability no event ai holds. we prove a quantum generalization of this lemma with exactly the same parameters. as men- tioned before, we have to modify the formulation of the lll to account for the unusual way r-independence behaves under complement. we are now ready to state and prove our main re- sult. theorem 14 (quantum lov ́ asz local lemma). let x1, x2, . . . , xn be subspaces with dependency graph g = ([n], e). if there exist 0 ≤y1, . . . , yn < 1, such that r(xi) ≥1 −yi ∏ (i,j)∈e (1 −yj), (1) then r(tn i=1 xi) ≥∏n i=1(1 −yi). note that when r is replaced by pr and t by v we recover the lll thm. 13. our proof uses properties that hold both for pr and r, in particular we also prove thm. 13. one can say that we generalize the lll to any notion of probability for which the properties (i)-(iv) of lemma 8 hold (these are the only properties of r we need in the proof). proof of theorem 14: we modify the proof in [as04] in order to avoid using the property pr(a|b) + pr(ac|b) = 1 which does not hold for r. to show thm. 14, it is sufficient to prove the following lemma. 8 lemma 15. for any s ⊂[n], and every i ∈[n], r(xi| t j∈s xj) ≥1 −yi. thm. 14 now follows from the chain rule (lemma 8.iii): r( n \ i=1 xi) = r(x1)r(x2|x1)r(x3|x1 ∩x2) . . . r(xn| n−1 \ j=1 xj) ≥ n ∏ i=1 (1 −yi) . we prove the lemma by complete induction on the size of the set s. for the base case, if s is empty, we have r(xi) ≥1 −  yi ∏ (i,j)∈e (1 −yj)  ≥1 −yi. inductive step: to prove the statement for s we assume it is true for all sets of size < |s|. fix i and define d = s ∩{j|(i, j) ∈e} and i = s\d (i and d are the independent and dependent part of s with respect to the i'th element). let xi = t j∈i xj and xd = t j∈d xj. then 1 −r(xi| \ j∈s xj) = 1 −r(xi|xi ∩xd) = 1 −r(xi ∩xd|xi) r(xd|xi) = r(xd|xi) −r(xi ∩xd|xi) r(xd|xi) . (2) to show the lemma we need to upper bound this expression by yi. we first upper bound the numerator: r(xd|xi) −r(xi ∩xd|xi) ≤1 −r(xi|xi) = 1 −r(xi) ≤yi ∏ (i,j)∈e (1 −yj), where for the first inequality we use lemma 8.vi, then the fact that xi and xi are r-independent, and the assumption on r(xi), eq. (1) in thm. 14. now, we lower bound the denominator of eq. (2). suppose d = {j1, . . . , j|d|}, then r  \ j∈d xj|xi  = r xj1|xi
* . . . * r  xj|d||xj1 ∩. . . ∩xj|d|−1 ∩xi  ≥∏ j∈d 1 −yj ≥∏ j:(i,j)∈e 1 −yj. the equality follows from the chain rule (lemma 8.iii), the first inequality follows from the induc- tive assumption, and the second inequality follows from the fact that d = {j|(i, j) ∈e} ∩s ⊆ {j|(i, j) ∈e}, and that yj < 1. for many applications we only need a simpler version of the quantum lll, often called the symmetric version, which we have already stated in thm. 4. proof of theorem 4: thm. 4 follows from thm. 14 in the same way the symmetric lll of thm. 1 follows from the more general lll of thm. 13 [as04]; we include it here for completeness: if d = 0 then r(tn i=1 xi) = πn i=1r(xi) > 0 by the chain rule (lemma 8.iii) and mutual r-independence of all subspaces. for d ≥1, by the assumption there is a dependency graph g = ([n], e) for the subspaces x1, . . . , xn in which for each i; |{j|(i, j) ∈e}| ≤d. taking yi = 1/(d + 1) (< 1) and using that for d ≥1, (1 − 1 d+1)d > 1 e we get r(xi) ≥1 −p ≥1 − 1 e(d + 1) ≥1 − 1 d + 1(1 − 1 d + 1)d ≥1 −yi(1 −yi)|{j|(i,j)∈e}|, 9 which is the necessary condition eq. (1) in thm. 14. hence r( n \ i=1 xi) ≥(1 − 1 d + 1)n > 0. (3) note that eq. (3) also allows us to give a lower bound on the dimension of the intersecting subspace, which might be useful for some applications. we can now move to the implications of the qlll for "sparse" instances of qsat and prove cor. 5. it is a special case of this slightly more general corollary. corollary 16. let {π1, . . . , πm} be a k-qsat instance where all projectors have rank at most r. if every qubit appears in at most d = 2k/(e * r * k) projectors, then the instance is satisfiable. proof. by assumption, each projector shares qubits with at most k(d −1) other projectors. as we have already shown in lemma 11, each π⊥ i is mutually r-independent from all but d = k(d −1) of the other π⊥ j . with p = r * 2−k we have r(π⊥ i ) ≥1 −p. the corollary follows from thm. 4 because p * e * (d + 1) ≤r * 2−k * e(k(2k/(e * r * k) −1) + 1) ≤1. 4 an improved lower bound for random qsat this section is devoted to the proof of thm. 7. as mentioned in the introduction, in random k-qsat we study a distribution over instances of k-qsat with fixed density, defined as follows. definition 17 (random k-qsat). random k-qsat of density α is a distribution over instances {π1, . . . , πm} on n qubits, where m = αn, obtained as follows: 1. construct a k-uniform hypergraph g with n vertices and m edges (the constraint hypergraph) by choosing m times, uniformly and with replacement, from the (n k) possible k-tuples of vertices. 2. for each edge i (1 ≤i ≤m) pick a k-qubit state |vi⟩acting on the corresponding qubits uniformly from all such states (according to the haar measure) and set πi = |vi⟩⟨vi| ⊗in−k. remark: (gk(n, m) vs. gk(n, p)) the distribution on hypergraphs obtained in the first step is denoted by gk(n, m) and has been studied extensively (see, e.g., [bol01, as04]). a closely related model is the so called erd ̈ os-renyi gk(n, p) model, where each of the (n k) k-tuples is independently chosen to be an edge with probability p. for p = m/(n k) the expected number of edges in gk(n, p) is m and these two distributions are very close to each other. in most cases proving that a certain property holds in one implies that it holds in the other (see [bol01]). there seems to be no con- sensus whether to define the random k-sat and k-qsat models with respect to the distribution gk(n, m) or gk(n, p); for instance the upper bounds on the random k-qsat threshold of [bmr09] are shown in the gk(n, m) model, whereas the lower bounds [lmss09, llm+09] are given in the gk(n, p) model. this, however, does not matter, as properties such as being satisfiable with high probability will always hold for both models. as mentioned, for α = c * 2k/k2, even though a graph from gk(n, m) has average degree davg = kα = c * 2k/k, and hence on average each qubit appears in c * 2k/k projectors, we cannot apply 10 the qlll and its cor. 5 directly: the degrees in gk(n, m) are distributed according to a poisson distribution with mean davg and hence we expect to see some high degree vertices (in fact the expected maximum degree at constant density is expected to be roughly logarithmic in n [bol01]). the idea behind the proof of thm. 7 is to single out the "high-degree" part vh of the graph and to treat it separately. the key is to show (i) that the matching conditions of laumann et al.'s thm. 6 is fulfilled by vh on one hand and (ii) to demonstrate how to "glue" the solution on vh with the one provided by qlll on the remaining graph. we first show how to glue two solutions, which also clarifies the requirements for h. lemma 18 (gluing lemma). let p = {π1, . . . , πm} be an instance of k-qsat with rank-1 projectors. assume that there is a subset of the qubits vh and a partition of the projectors into two sets h and l, where h (possibly empty) consists of all projectors that act only on qubits in vh, such that 1. the reduced instance given by h (restricted to qubits in vh) is satisfiable. 2. each qubit / ∈vh appears in at most 2k/(4 * e * k) projectors from l. 3. each projector in l has at most one qubit in vh. then p is satisfiable. proof. let |φh⟩be a satisfying state for h on the qubits vh (if h = ∅this can be any state). to extend it to the whole instance, we need to deal with the projectors in l acting on a qubit from vh. let l = {π1, . . . , πl}. from l we construct a new "decoupled" instance l′ = {q1, . . . , ql} of k-qsat with projectors of rank at most 2 that have no qubits in vh. if πi ∈l does not act on any qubit in vh, we set qi := πi. otherwise, order the k qubits on which πi acts such that the first one is in vh. πi can be written as πi = |vi⟩⟨vi| ⊗in−k, where |vi⟩is a k-qubit state. we can decompose |vi⟩= a0|0⟩⊗|v0 i ⟩+ a1|1⟩⊗|v1 i ⟩, where the first part of the tensor product is the qubit in vh and |v1 i ⟩and |v2 i ⟩are (k −1)-qubit states on the remaining qubits. define qi = |v1 i ⟩⟨v1 i | + |v2 i ⟩⟨v2 i | ⊗in−k+1. call vl′ the set of qubits on which the projectors in l′ act on. note that by construction vl′ is disjoint from vh, and that vh ∪vl′ is the set of all qubits in p; hence h and l′ are "decoupled". claim 19. assume there is a satisfying state |φl′⟩for l′ on vl′. then |φ⟩= |φh⟩⊗|φl′⟩is a satisfying state for p. proof. by construction, |φ⟩satisfies all the projectors from h and all projectors in l that do not have qubits in vh. to see that it also satisfies any projector πi from l with a qubit in vh, observe that |φl′⟩is orthogonal to both |v1 i ⟩and |v2 i ⟩. hence no matter how |φl′⟩is extended on the qubit of vh in πi, the resulting state is orthogonal to |vi⟩. it remains to show that l′ is satisfiable. this follows immediately from cor. 16: we observe that each projector in l′ can be viewed as a k-local projector of rank at most 4; and by the assumption each qubit in vl′ appears in at most 2k/(4 * e * k) projectors of l′. the gluing lemma 18 only depends on the underlying constraint hypergraph. we can hence give the construction of the "high degree" part of the instance purely in terms of hypergraphs, and will from now on associate subsets of edges with the corresponding subsets of projectors. motivated by the gluing lemma, our goal is to separate a set of "high degree" vertices vh (above 11 a certain cut-off degree d) with induced edges h such that each edge outside h has at most one vertex in vh. we achieve this by starting with the high degree vertices and iteratively adding all those edges that intersect in more than one vertex. definition 20 (construction of vh). let g = g([n], e) be a k-uniform hypergraph and d > 0. con- struct sets of vertices v0, v1, . . . ⊆[n] and edges e1, e2 . . . ⊆e iteratively in the following steps, starting with all sets empty: 0) let v0 = {v ∈v| deg(v) > d}. 1) for all e ∈e \ e0, if e has 2 or more vertices in v0, then add e to e1, and add to v1 all vertices in e not already in v0. . . . i) for all e ∈e \ (e0 ∪. . . ∪ei−1), if e has 2 or more vertices in si−1 j=0 vj, add e to ei, and add to vi all the vertices in ei which are not already in si−1 j=0 vj. stop at the first step s such that es = ∅. let vh := ss i=0 vi, h := ss i=1 ei and l := e \ h. by construction all the vi are disjoint and similarly for the ei. the process of adding edges stops at some step s (es = ∅), because e \ (e0 ∪. . . ∪es−1) keeps shrinking until this happens. note that h consists precisely of all those edges in e that have only vertices in vh (i.e. g(vh, h) is the hypergraph induced by g on vh). to show that a random k-qsat instance of density α is satisfiable with high probability, we only need to show that the construction of vh, h and l of def. 20 fulfills the conditions of the gluing lemma 18 with high probability. we set d = 2k/(4 * e * k) in def. 20, so that conditions 2. and 3. are fulfilled by construction. to finish the proof of thm. 7 it thus suffices to show that the instance given by h on qubits in vh is satisfiable. to show this we build on laumann et al.'s thm. 6. lemma 21. for a random k-qsat instance with density α ≤2k/(12 * e * k2), the reduced instance h obtained in the construction of def. 20 with d = 2k/(4 * e * k) fulfills the matching conditions of thm. 6 with high probability. proof. the proof of this key lemma proceeds in two parts. the first one (lemma 22) shows that any hypergraph induced by a small enough subset of vertices in a hypergraph from gk(n, αn) fulfills the matching conditions. the second part (lemma 23) then shows that vh is indeed small enough with high probability. lemma 22 (small subgraphs have a matching). let g be a random hypergraph distributed according to gk(n, αn) and let γ = (e(e2 * α)1/(k−2))−1. with high probability, for all w ⊂v with |w| < γn, the induced hypergraph on w obeys the matching conditions of thm. 6. proof. there is simple intuition why small sets obey the matching conditions - the density inside a small induced graph is much smaller than the density of g: for simplicity set α = 2k−1 and γ = 1/(2 + 2δ) for some δ > 0. imagine fixing w ⊂v of size γn and then picking the graph g according to gk(n, p) with p = αn/(n k) ≈ α nk−1 = ( 2 n)k−1. the induced graph on w is distributed according to gk(γn, p) and hence its density is α′ = p * (γn k )/γn ≈p * (γn)k−1 = (1 + δ)−(k−1) ≪1. 12 at such low densities the matching conditions are fulfilled with high probability (see the remark below thm. 6). we proceed to prove the somewhat stronger statement that the matching condi- tions hold for all small subsets. let us first examine the matching conditions. we can construct a bipartite graph b(g), where on the left we put the edges of g and on the right the vertices of g. we connect each edge on the left with those vertices on the right that are contained in that edge. then the matching conditions of thm. 6 are equivalent to saying that there is a matching in b(g) that covers all left vertices. by hall's theorem [hal35, die97], such a matching exists iff for all t, every subset of t edges on the left is connected to at least t vertices on the right. hence, there is a "bad" subset w ⊂v with |w| < γn not obeying the matching conditions iff for some t < γn there is a subset of vertices of size t −1 that contains t edges. let us compute the probability of such a bad event to happen. first, fix a subset s ⊆v of size t −1 and let us compute the probability that it contains t edges. the probability that a random edge lands in s is at most ((t −1)/n)k. since in gk(n, m) all m edges are picked independently, we get pr[s contains t edges] ≤ m t  t −1 n kt . by the union bound over all subsets s of size t −1 (there are ( n t−1) of them) and all t we get the following bound pr[∃"bad" w] ≤ γn ∑ t=1  n t −1 m t  t −1 n kt ≤ γn ∑ t=1 n t αn t   t n kt ≤ ne t t αne t t  t n kt = γn ∑ t=1 e2α  t n k−2!t =: γn ∑ t=1 at. note that the sum is clearly dominated by the first term (t = 1). more precisely we have ∀1 ≤t < γn −1 at+1 at = e2α t + 1 t (k−2)t t + 1 n k−2 ≤e2αek−2γk−2 =: r < 1, where for the last inequality we have used the bound on γ. hence ∑γn t=1 at ≤∑γn t=1 a1rt−1 = 1 1−ra1, and we get pr[∃"bad" w] ≤ 1 1−r e2α nk−2 →0. lemma 23 (vh is small). let g be a hypergraph picked from gk(n, αn) and let vh be the set of vertices generated by the procedure in definition 20 with d = 2k/(4 * e * k). then for k ≥12 and αk ≤d/3, with high probability |vh| ≤(ǫ0 + o(1))n for some ǫ0 satisfying ǫ0 < γ where γ is the constant from lemma 22. remark: as is standard in the model of random k-sat and random k-qsat, if we look at the large k limit we will always first take the limit n →∞for fixed k and then k →∞. hence we will always treat k (and d and α) as a constant in o(*) and o(*) terms. proof. throughout the proof we will set α to its maximum allowed value of d/(3k). the statement of lemma 23 for smaller α then follows by monotonicity. 13 for the proof of this lemma, we first replace gk(n, αn) by a slightly different model of random hypergraphs g′ k(n, α′n). in g′ k(n, α′n), we first generate a random sequence of vertices of length kα′n with each vertex picked i.i.d. at random. we then divide the sequence into blocks of length k and, for each block that contains k different vertices, we create a hyperedge. (for blocks that contain the same vertex twice, we do nothing.) the expected number of blocks containing the same vertex twice is o((k 2)α′) = o(1). there- fore, we can choose α′ = α + o(1) and, with high probability, we will get at least αn edges (and each of those edges will be uniformly random). this means that it suffices to prove the lemma for g′ k(n, α′n). for this model, we will show that |vi| satisfies the following bounds: claim 24. there is an ǫ0 < γ 2 and ǫi := 2−iǫ0 such that for all i : 0 ≤i ≤l with l := ⌈3 2 log n⌉, with probability at least 1 −2i n2, |vi| ≤ǫin. (4) this implies that vl is empty with probability at least 1 −o( 1 √n). in this case, |vh| = ∑l−1 i=0 |vi|. with probability at least 1 −2l+1 n2 = 1 −o( 1 √n), (4) is true for all i. then, |vh| = l−1 ∑ i=0 |vi| ≤2ǫ0n < γn, which completes the proof of the lemma. in what follows we will repeatedly use azuma's inequality [azu67, hoe63, as04]: let y0, . . . , yn be a martingale, where |yi+1 −yi| ≤1 for all 0 ≤i < n. for any t > 0, pr(|xn −x0| ≥t) ≤exp(−t2 2n). (5) we now prove claim 24, by induction on i. we start with the base case i = 0. here, we will also bound r0, the number of edges incident to v0, and show pr [r0 ≥ǫ0dn] ≤ 1 2n2 . (6) the i = 0 case. recall that v0 = {v|deg(v) ≥d}. by linearity of expectation, e[|v0|] = npr(deg(v) ≥d). the degree of a vertex is a sum of independent 0-1 valued random variables with expectation slightly less than α′k. in the large n limit, this becomes a poisson distribution with mean ≤α′k = d/3 + o(1). using the tail bound for poisson distributions (see, e.g., [as04] thm. a.1.15), we obtain pr(deg(v) ≥d) ≤(e2/27)d/3. note that for k ≥12 we have ( e2 27)d/3 ≤5 8ǫ0, where we set ǫ0 = α′ 12d2k = d/(3k) + o(1) 12d2k ≤ 1 12dk2 < γ 2 . then, e[|v0|] ≤5 8ǫ0n. to bound e[r0], observe that r0 ≤∑v∈v0 deg(v) and hence e[r0] ≤n pr(deg(v) ≥d) * e[deg(v)|deg(v) ≥d] ≤5 8ǫ0n * e[deg(v)|deg(v) ≥d] ≤5 6ǫ0nd, where for the last inequality we have used e[deg(v)|deg(v) ≥d] ≤4 3d, which follows from the following simple fact: 14 fact 25. let x be a random variable distributed according to a poisson distribution with mean λ. then for k > 1, e[x|x ≥kλ] ≤(k + 1)λ. proof. e[x|x ≥kλ] = ∑∞ j=kλ j * pr(x = j) pr(x ≥kλ) = 1 pr(x ≥kλ) ∞ ∑ j=kλ je−λ λj j! = 1 pr(x ≥kλ) λ ∞ ∑ j=kλ e−λ λj−1 (j −1)! = λ  1 + pr(x = kλ −1) pr(x ≥kλ)  ≤λ  1 + pr(x = kλ −1) pr(x = kλ)  = λ  1 + kλ λ  = (1 + k)λ. to prove (4) and (6), we use azuma's inequality eq. (5). let x0, x1, . . . , xkα′n be the martingale defined in the following way. we pick the vertices of the sequence defining g at random one by one and let xi be the expectation of |v0| (resp. r0) when the first i vertices of the sequence are already chosen and the rest is still uniformly random. picking one vertex in any particular way changes the size of |v0| by at most 1 and of r0 by at most d (when the degree of a vertex crosses the threshold d to be in v0). therefore, for v0, |xi −xi−1| ≤1 (|xi −xi−1| ≤d for the bound on r0). for v0, by azuma's inequality pr[||v0| −e[|v0|]| ≥t] = pr[|xkα′n −x0| ≥t] ≤e− t2 2kα′n . to make this probability less than 1/n2, we chose t = 2 √ kα′√ n ln n. then, with probability at least 1 −1 n2 , |v0| ≤e[|v0|] + o( p n log n) ≤5 8ǫ0n + o( p n log n) ≤ǫ0n, which gives bound (4). similarly, to show bound (6) for r0, we choose t = 2d √ kα′p n(ln n + 1). then, we get that with probability at least 1 − 1 2n2, r0 ≤e[r0] + o( p n log n) ≤5 6ǫ0nd + o( p n log n) ≤ǫ0nd. the i > 0 case. we will first condition on the event f that bound (6) holds and bounds (4) hold for all previous i. moreover, we fix the following objects: • the sets v0, . . . , vi−1; • the edges in e1, . . . , ei−1; • the degrees of all vertices v ∈v0 ∪. . . ∪vi−1; conditioning on v0, . . . , vi−1 and their degrees is equivalent to fixing the number of times that each v ∈v0 ∪. . . ∪vi−1 appears in the sequence defining the graph g according to g′ k(n, α′n). furthermore, conditioning on e1, . . . , ei−1 means that we fix some blocks of the sequence to be equal to edges in e1, . . . , ei−1. we can then remove those blocks from the sequence and adjust the degrees of the vertices that belong to those edges. conditioning on e1, . . . , ei−1 also means that we condition on the fact that there is no other block containing two vertices from v0 ∪. . . ∪vi−2. we now consider a random sequence of vertices satisfying those constraints. let b be the total number of blocks (after removing e0, . . . , ei−1) and call mj the number of blocks that contain one element of vj for 0 ≤j ≤i −1. (the mj are fixed since the vj are fixed.) the sequence of vertices on the b blocks is uniformly random among all sequences with a fixed number of occurrences of elements in vj (a total of mj) and such that no two of them occur in the same block. note that 15 an edge from ei must have at least one of its vertices in vi−1. we have m0 + . . . + mi−2 blocks containing one vertex from v0 ∪. . . ∪vi−2 each. for each of those blocks, the probability that one of the mi−1 occurrences of v ∈vi−1 ends up in it is at most (k −1) mi−1 kb −m0 −. . . −mi−2 . (7) for any other block, the probability that two or more occurrences of v ∈vi−1 are in it is at most k 2  mi−1(mi−1 −1) (kb −m0 −. . . −mi−2)(kb −m0 −. . . −mi−2 −1) ≤ k 2   mi−1 kb −m0 −. . . −mi−2 2 . (8) observe that ej+1 + mj ≤dvj for j ≥1 since each vertex in vj is incident to less than d edges. moreover, e1 + m0 ≤r0. note that this implies that kb −(m0 + m1 + . . . + mi−2) ≥ kα′n −k [r0 + d(v1 + . . . + vi−2)]. recall that we are conditioning on the event f that the bounds in (4) and (6) hold, and hence we can further bound kb −(m0 + m1 + . . . + mi−2) ≥kα′n − k [ǫ0nd + d(ǫ1n + . . . + ǫi−2n)] ≥kα′n −2kdǫ0n. for our choice of ǫ0 ≤ α′ 12kd2 we hence obtain kb −(m0 + . . . + mi−2) ≥α′ k 2n. by combining (7) and (8), using the union bound for all relevant blocks, we get e[|ei|] ≤ (k −1) m0 + . . . + mi−2 α′ k 2n + α′n k 2  mi−1 (α′ k 2n)2 ! mi−1 ≤2mi−1  m0 + . . . + mi−1 α′n  ≤2d2vi−1  r0/d + v1 + . . . + vi−1 α′n  . since we are conditioning on the event f that (4) and (6) hold, we can bound r0 and vj and obtain e[|ei|] ≤2d2vi−1 ǫ0n + ǫ1n + . . . + ǫi−1n α′n  ≤2d2vi−1 2ǫ0 α′ ≤vi−1 3k , where we have substituted ǫ0 = α′ 12d2k. together with the observation that |vi| ≤k|ei| we have hence shown in our setting that e[|vi|] ≤vi−1/3. (9) the large deviation bound (4) again follows from azuma's inequality (5). we pick the sequence of kb vertices (after removing e0, . . . , ei−1) vertex by vertex and let xi to be the expectation of |vi| after picking the i first vertices of the sequence. then, x0, x1, . . . , xkb form a martingale and choosing one vertex of the sequence affects |vi| by at most k. therefore, |xi −xi−1| ≤k when bounding |vi|. we now apply azuma's inequality (5) with t = 2k p kα′n(ln n + 1) and obtain, in our setting of fixed sets vj, fixed degrees of their elements and fixed sets ej for 0 ≤j ≤i −1, and conditioning on the event f, pr(||vi| −e[|vi||] ≥o( p n log n)) ≤ 1 2n2 . 16 using eq. (9), the induction hypothesis and the fact that we are conditioning on bound (4) to hold, we get that with probability at least 1 − 1 2n2 , |vi| ≤e[|vi|] + o( p n log n) ≤vi−1 3 + o( p n log n) ≤ǫi−1n 3 + o( p n log n) ≤ǫin. since this holds for all fixed sets vj, fixed degrees of their elements and fixed sets ej for 0 ≤j ≤ i −1, it also holds when we remove this conditioning (while still conditioning on the event f). by the union bound, f does not hold with probability at most 2i−1 2 n2 . hence, with probability at least 1 −2i n2, vi ≤ǫin and we have shown the bound in (4). this terminates the proof of lemma 21 for all k ≥12. for smaller values of k our bound of α ≤2k/(12 * e * k2) is smaller than the bound obtained by laumann et al. [llm+09], and hence the lemma also holds. hence we have shown thm. 7. remark: note that in the limit of large k our results can be tightened to give a bound of α ≤ (d −o( p d log d))/k = 2k/(4 * e * k2) −o( √ 2k/2√ k) for the satisfiable region. the analysis essentially changes only for the bound on e[|v0|] in the beginning of the i = 0 base case, where we have to use the tail bound for the poisson distribution for smaller deviations. acknowledgments the authors would like to thank the erwin schr ̈ odinger international institute in vienna, where part of this work was done, for its hospitality. we thank noga alon, chris laumann and oded regev for valuable discussions. references [alo91] n. alon. a parallel algorithmic version of the local lemma. random structures and algorithms, 2(4):367–378, 1991. [ap04] d. achlioptas and y. peres. the threshold for random k-sat is 2ˆ k log 2-o (k). amer- ican mathematical society, 17(4):947–974, 2004. [as04] n. alon and j.h. spencer. the probabilistic method. wiley-interscience, 2004. [azu67] k. azuma. weighted sums of certain dependent random variables. tohoku mathemat- ical journal, 19(3):357–367, 1967. [bbc+01] b. bollob ́ as, c. borgs, j.t. chayes, j.h. kim, and d.b. wilson. the scaling window of the 2-sat transition. random structures and algorithms, 18(3):201–256, 2001. [bec91] j. beck. an algorithmic approach to the lov ́ asz local lemma. random structures and algorithms, 2(4):343–365, 1991. [bmr09] s. bravyi, c. moore, and a. russell. bounds on the quantum satisfibility threshold. arxiv preprint arxiv:0907.1297, 2009. 17 [bol01] b. bollob ́ as. random graphs. cambridge university press, 2 edition, 2001. [bra06] s. bravyi. efficient algorithm for a quantum analogue of 2-sat. arxiv preprint quant- ph/0602108, 2006. [bs07] s. beigi and p.w. shor. on the complexity of computing zero-error and holevo capac- ity of quantum channels. arxiv preprint arxiv:0709.2090, september 2007. [bt08] s. bravyi and b. terhal. complexity of stoquastic frustration-free hamiltonians. arxiv preprint arxiv:0806.1746, june 2008. [cr92] v. chv ́ atal and b. reed. mick gets some (the odds are on his side). in proceedings of the 33rd annual symposium on foundations of computer science, pages 620–627. ieee computer society, 1992. [cs00] a. czumaj and c. scheideler. coloring non-uniform hypergraphs: a new algorithmic approach to the general lov ́ asz local lemma. in proceedings of the 11th symposium on discrete algorithms, 17:30-39, 2000. [die97] r. diestel. graph theory (graduate texts in mathematics). springer heidelberg, 1997. [dkmp08] j. diaz, l. kirousis, d. mitsche, and x. perez-gimenez. a new upper bound for 3-sat. arxiv preprint arxiv:0807.3600, july 2008. [el75] p. erd ̈ os and l. lov ́ asz. problems and results on 3-chromatic hypergraphs and some related questions. infinite and finite sets, 2:609–627, 1975. [fei02] uriel feige. relations between average case complexity and approximation complex- ity. in proc. 34th annual acm symp. on theory of computing (stoc), pages 534–543, 2002. [for09] l. fortnow. a kolmogorov complexity proof of the lov ́ asz local lemma. http://blog.computationalcomplexity.org/2009/06/kolmogorov-complexity-proof-of-lov.html, 2009. [fri99] e. friedgut. sharp thresholds of graph properties, and the k-sat problem. journal of the american mathematical society, 12(4):1017–1054, october 1999. [goe92] a. goerdt. a threshold for unsatisfiability. in mathematical foundations of computer science 1992, pages 264–274. 1992. [hal35] p. hall. on representatives of subsets. j. london math. soc, 10(1):26–30, 1935. [hoe63] w. hoeffding. probability inequalities for sums of bounded random variables. journal of the american statistical association, pages 13–30, 1963. [hs03] m.t. hajiaghayi and g.b. sorkin. the satisfiability threshold of random 3-sat is at least 3.52. arxiv preprint math/0310193, 2003. [kkl03] a.c. kaporis, l.m. kirousis, and e. lalas. selecting complementary pairs of literals. electronic notes in discrete mathematics, 16:47–70, october 2003. 18 [kos97] a.i. kostrikin. linear algebra and geometry. taylor & francis, 1997. [ks94] s. kirkpatrick and b. selman. critical behavior in the satisfiability of random boolean expressions. science, 264(5163):1297–1301, may 1994. [liu06] y.k. liu. consistency of local density matrices is qma-complete. in approximation, randomization, and combinatorial optimization. algorithms and techniques, pages 438– 449. 2006. [liu07] y.k. liu. the complexity of the consistency and n-representability problems for quan- tum states. arxiv preprint arxiv:0712.3041, december 2007. [llm+09] c.r. laumann, a.m. l ̈ auchli, r. moessner, a. scardicchio, and s. l sondhi. on prod- uct, generic and random generic quantum satisfiability. arxiv preprint arxiv:0910.2058, october 2009. [lmss09] c.r. laumann, r. moessner, a. scardicchio, and s.l. sondhi. phase transitions and random quantum satisfiability. arxiv preprint arxiv:0903.1904, march 2009. [mmz05] m. m ́ ezard, t. mora, and r. zecchina. clustering of solutions in the random satisfia- bility problem. physical review letters, 94(19):197205, may 2005. [mos08] r.a. moser. derandomizing the lov ́ asz local lemma more effectively. arxiv preprint arxiv:0807.2120, july 2008. [mos09] r.a. moser. a constructive proof of the lov ́ asz local lemma. in proceedings of the 41st annual acm symposium on theory of computing, pages 343–350. acm new york, ny, usa, 2009. [mp87] p. mani-levitska and j. pach. decomposition problems for multiple coverings with unit balls. manuscript, 1987. [mpz02] m. mezard, g. parisi, and r. zecchina. analytic and algorithmic solution of random satisfiability problems. science, 297(5582):812–815, 2002. [mr98] m. molloy and b. reed. further algorithmic aspects of the local lemma. in proceedings of the 30th annual acm symposium on theory of computing, pages 524-529, 1998. [mt09] r.a. moser and g. tardos. a constructive proof of the general lov ́ asz local lemma. arxiv preprint arxiv:0903.0544, 2009. [spe77] j. spencer. asymptotic lower bounds for ramsey functions. discrete math, 20(1):69–76, 1977. [sri08] a. srinivasan. improved algorithmic versions of the lov ́ asz local lemma. in pro- ceedings of the nineteenth annual acm-siam symposium on discrete algorithms, pages 611–620, 2008. 19

0911.1697

time-varying autoregressions in speech: detection theory and applications

this article develops a general detection theory for speech analysis based on time-varying autoregressive models, which themselves generalize the classical linear predictive speech analysis framework. this theory leads to a computationally efficient decision-theoretic procedure that may be applied to detect the presence of vocal tract variation in speech waveform data. a corresponding generalized likelihood ratio test is derived and studied both empirically for short data records, using formant-like synthetic examples, and asymptotically, leading to constant false alarm rate hypothesis tests for changes in vocal tract configuration. two in-depth case studies then serve to illustrate the practical efficacy of this procedure across different time scales of speech dynamics: first, the detection of formant changes on the scale of tens of milliseconds of data, and second, the identification of glottal opening and closing instants on time scales below ten milliseconds.

introduction t his article presents a statistical detection framework for identifying vocal tract dynamics in speech data across different time scales. since the source-filter view of speech production motivates modeling a stationary vocal tract using the standard linear-predictive or autoregressive (ar) model [2], it is natural to represent temporal variation in the vocal tract using a time-varying autoregressive (tvar) process. consequently, we propose here to detect vocal tract changes via a generalized likelihood ratio test (glrt) to determine whether an ar or tvar model is most appropriate for a given speech data segment. our main methodological contribution is to derive this test and describe its asymp- totic behavior. our contribution to speech analysis is then to consider two specific, in-depth case studies of this testing framework: detecting change in speech spectra, and detecting glottal opening and closing instants from waveform data. earlier work in this direction began with the fitting of piecewise-constant ar models to test for nonstationarity [3], [4]. however, in reality, the vocal tract often varies slowly, rather than as a sequence of abrupt jumps; to this end, [5]–[8] studied time-varying linear prediction using tvar models. in a more general setting, kay [9] recently proposed a version of based upon work supported in part by darpa grant hr0011-07-1- 0007, dod air force contract fa8721-10-c-0002, and an nsf graduate research fellowship. a preliminary version of this material appeared in the 10th annual conference of the international speech communication association (interspeech 2009) [1]. the opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the united states government. d. rudoy and p. j. wolfe are with the statistics and information sciences laboratory, harvard university, oxford street, cambridge, ma 02138 (e- mail: {rudoy, patrick}@seas.harvard.edu) t. f. quatieri is with the lincoln laboratory, massachusetts institute of technology, lexington, ma 02173 usa. (e-mail: [email protected]). the rao test for ar vs. tvar determination; however, when available, likelihood ratio tests often outperform their rao test counterparts for finite sample sizes [10]. nonparametric approaches to detecting spectral change in acoustic signals were proposed by the current authors in [11], [12]. detecting spectral variation across multiple scales is an important first step toward appropriately exploiting vocal tract dynamics. this can lead to improved speech analysis algo- rithms on time scales on the order of tens of milliseconds for speech enhancement [11], [13]–[15], classification of time- varying phonemes such as unvoiced stop consonants [7], and forensic voice comparison [16]. at the sub-segmental time scale (i.e., less than one pitch period), sliding-window ar analysis has been used to capture vocal tract variation and to study the excitation waveform as a key first step in applications including inverse filtering [17], speaker identification [18], synthesis [19], and clinical voice assessment [20]. in the first part of this article, we develop a general detection theory for speech analysis based on tvar models. in section ii, we formally introduce these models, derive their corresponding maximum-likelihood estimators, and develop a glrt appropriate for speech waveforms. after providing ex- amples using real and synthetic data, including an analysis of vowels and diphthongs from the timit database [21], we then formulate in section iii a constant false alarm rate (cfar) test and characterize its asymptotic behavior. in section iv, we discuss the relationship of our framework to classical methods, including the piecewise-constant ar approach of [3]. next, we consider two prototype speech analysis applica- tions: in section v, we apply our glrt framework to detect formant changes in both whispered and voiced speech. we then show how to detect glottal opening and closing instants via the glrt in section vi. we evaluate our results on the more difficult problem of detecting glottal openings [22] using ground-truth data obtained by electroglottograph (egg) analysis, and also show performance comparable to methods based on linear prediction and group delay for the task of identifying glottal closures. we conclude and briefly discuss future directions in section vii. ii. time-varying autoregressions and testing a. model specification recall the classical pth-order linear predictive model for speech, also known as an ar(p) autoregression [2]: ar(p): x[n] = p x i=1 aix[n −i] + σw[n], (1) where the sequence w[n] is a zero-mean white gaussian process with unit variance, scaled by a gain parameter σ > 0. arxiv:0911.1697v2 [stat.ap] 18 apr 2010 revised manuscript 2 a more flexible pth-order time-varying autoregressive model is given by the following discrete-time difference equation [5]: tvar(p): x[n] = p x i=1 ai[n]x[n −i] + σw[n]. (2) in contrast to (1), the linear prediction coefficients ai[n] of (2) are time-dependent, implying a nonstationary random process. the model of (2) requires specification of precisely how the linear prediction coefficients evolve in time. here we choose to expand them in a set of q +1 basis functions fj[n] weighed by coefficients αij as follows: ai[n] = q x j=0 αijfj[n], for all 1 ≤i ≤p. (3) we assume throughout that the "constant" function f0[n] = 1 is included in the chosen basis set, so that the classical ar(p) model of (1) is recovered as ai ≡αi0 * 1 whenever αij = 0 for all j > 0. many choices are possible for the functions fj[n]-legendre [23] and fourier [5] polynomials, discrete prolate spheroidal functions [6], and even wavelets [24] have been used in speech applications. the functional expansion of (3) was first studied in [23], [25], and subsequently applied to speech analysis by [5]–[7], among others. coefficient trajectories ai[n] have also been modeled as sample paths of a suitably chosen stochastic pro- cess (see, e.g., [26]). in this case, however, estimation typically requires stochastic filtering [13] or iterative methods [27] in contrast to the least-squares estimators available for the model of (3), which are described in section ii-c below. b. ar vs. tvar generalized likelihood ratio test (glrt) we now describe how to test the hypothesis h0 that a given signal segment x = (x[0] x[1] * * * x[n −1])t has been generated by an ar(p) process according to (1), against the alternative hypothesis h1 of a tvar(p) process as specified by (2) and (3) above. we introduce a glrt to examine evidence of change in linear prediction coefficients over time, and consequently in the vocal tract resonances that they represent in the classical source-filter model of speech. according to the functional expansion of (3), the tvar(p) model of (2) is fully described by p(q + 1) expansion coeffi- cients αij and the gain term σ. for convenience we group the coefficients αij into q + 1 vectors αj, 0 ≤j ≤q, as αj ≜ α1j α2j * * * αpj
t . we may then partition a vector α ∈rp(q+1)×1 into blocks associated to the ar(p) portion of the model αar, and the remainder αtv, which captures time variation: α ≜ αt ar | αt tv
t = αt 0 | αt 1 αt 2 * * * αt q
t . (4) recalling that the tvar(p) model (hypothesis h1) reduces to an ar(p) model (hypothesis h0) precisely when αj = 0 for all j > 0, we may formulate the following hypothesis test: model : tvar(p) with parameters α, σ2; hypotheses : ( h0 : αj = 0 for all j > 0, h1 : αj ̸= 0 for at least one j > 0. (5) estimate αtv, αar under h1 estimate, αar under h0 x estimate σ2 under h1 estimate σ2 under h0 t(x) - + ( ) 1 2 ˆ ( )ln h n p σ − ( ) 0 2 ˆ ( )ln h n p σ − fig. 1. computation of the glrt statistic t(x) according to section ii-c. each of these two hypotheses in turn induces a data likelihood in the observed signal x ∈rn×1, which we denote by phi(*) for i = 1, 2. the corresponding generalized likelihood ratio test comprises evaluation of a test statistic t(x), and rejection of h0 in favor of h1 if t(x) exceeds a given threshold γ: t(x) ≜2 ln supα,σ2 ph1(x; α, σ2) supα0,σ2 ph0(x; α0, σ2) h1 ≷ h0 γ. (6) c. evaluation of the glrt statistic the numerator and denominator of (6) respectively im- ply maximum-likelihood (ml) parameter estimates of α = (αt ar | αt tv)t and α0 in (4) under the specified tvar(p) and ar(p) models, along with their respective gain terms σ2. intuitively, when h0 is in force, estimates of αtv will be small; we formalize this notion in section iii-a by showing how to set the test threshold γ to achieve a constant false alarm rate. as we now show, conditional ml estimates are easily obtained in closed form, and terms in (6) reduce to estimates of σ2 under hypotheses h0 and h1, respectively. given n observations, partitioned according to x = (xp xn−p)t ≜ x[0] * * * x[p −1] | x[p] * * * x[n −1]
t , the joint probability density function of α, σ2 is given by: p(x ; α, σ2) = p(xn−p | xp ; α, σ2)p(xp ; α, σ2). (7) here the notation | reflects conditioning on random variables, whereas ; indicates dependence of the density on non-random parameters. as is standard practice, we approximate the un- conditional data likelihood of (7) by the conditional likelihood p(xn−p | xp ; α, σ2), whose maximization yields an estimator that converges to the exact (unconditional) ml estimator as n →∞(see, e.g., [28] for this argument under h0). gaussianity of w[n] implies the conditional likelihood p(xn−p | xp; α, σ2) = 1 (2πσ2)(n−p)/2 exp − n−1 x n=p e2[n] 2σ2 ! , where e[n] ≜x[n] −pp i=1 pq j=0 αijfj[n]x[n −i] is the associated prediction error. the log-likelihood is therefore ln p(xn−p | xp; α, σ2) = −n −p 2 ln(2πσ2) −∥xn−p −hxα∥2 2σ2 (8) where the (n−p+1)th row of the matrix hx ∈r(n−p)×p(q+1) is given by the kronecker product (x[n −1] * * * x[n −p]) ⊗ (f0[n] f1[n] * * * fq[n]) for any p ≤n ≤n −1. revised manuscript 3 100 200 300 400 500 −4 −2 0 2 4 synthetic signal x[n] sample number 100 200 300 400 500 −1.5 −1 −0.5 0 0.5 1 true ar trajectories a[n] sample number 100 200 300 400 500 −1.5 −1 −0.5 0 0.5 1 sample number coefficient magnitude fitted ar trajectories true model fitted ar model fitted tvar model −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 imaginary axis real axis pole trajectories 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 probability of detection frequency jump: π/80 rad 5ms 10ms 15ms 20ms 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 frequency jump: 3π/80 rad 5ms 10ms 15ms 20ms 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 frequency jump: 5π/80 rad 5ms 10ms 15ms 20ms 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 probability of false alarm frequency jump: 7π/80 rad 5ms 10ms 15ms 20ms fig. 2. example of glrt detection performance for a "formant-like" synthetic tvar(2) signal. left: a test signal and its tvar coefficients are shown at top, with pole trajectories and ar vs. tvar estimates below. right: operating characteristics of the corresponding glrt (p = 2 tvar coefficients, q = 4 legendre polynomials, fs = 16 khz) shown for various frequency jumps and data lengths. maximizing (8) with respect to α therefore yields the least- squares solution of the following linear regression problem: xn−p = hxα + σw, (9) where w ≜(w[p] * * * w[n −1])t . consequently, the condi- tional ml estimate of α follows from (8) and (9) as b α = ht x hx
−1 ht x xn−p. (10) the estimator of (10) corresponds to a generalization of the covariance method of linear prediction-to which it exactly reduces when the number q of non-constant basis functions employed is set to zero [5]; we discuss the corresponding generalization of the autocorrelation method in section iv-b. the conditional ml estimate of σ2 is obtained by substitut- ing (10) into (8) and maximizing with respect to σ2, yielding c σ2 = 1 n −p n−1 x n=p   x[n]x[n]− p x i=1 q x j=0 c αijfj[n]x[n]x[n−i]   . (11) under h0 (the time-invariant case), the estimator of (11) reduces to the familiar c σ2 = b rxx[0] −pp i=1 αi0b rxx[i], where rxx[τ] is the autocorrelation function of x[n] at lag τ. in summary, the conditional ml estimates of αar, αtv and σ2 under h1 are obtained using (10) and (11), respectively. estimates of αar and σ2 under h0 are obtained by setting q = 0 in (10) and (11). substituting these estimates into the glrt statistic of (6), we recover the following intuitive form for t(x), whose computation is illustrated in fig. 1: t(x) = (n −p) ln  c σ2h0/c σ2h1  . (12) d. evaluation of glrt detection performance to demonstrate typical glrt behavior, we first consider an example detection scenario involving a "formant-like" signal synthesized by filtering white gaussian noise through a second-order digital resonator. the resonator's center fre- quency is increased by δ radians halfway through the duration of the signal, while its bandwidth is kept constant; an example 560-sample signal with δ = 7π/80 radians is shown in fig. 2. detection performance in this setting is summarized in the right-hand panel of fig. 2, which shows receiver operating characteristic (roc) curves for different signal lengths n and frequency jump sizes δ. these were varied in the ranges n ∈{80, 240, 400, 560} samples (10 ms increments) and δ ∈{π/80, 3π/80, 5π/80, 7π/80} radians (200 hz incre- ments), and 1000 trial simulations were performed for each combination. to generate data under h0, δ was set to zero. in agreement with our intuition, detection performance improves when δ is increased while n is fixed, and vice versa-simply put, larger changes and those occurring over longer intervals are easier to detect. moreover, even though the span of the chosen legendre polynomials does not include the actual piecewise-constant coefficient trajectories, the norm of their projection onto this basis set is sufficiently large to trigger a detection with high probability. we next consider a large-scale experiment designed to test the sensitivity of the test statistic t(x) to vocal tract variation in real speech data. to this end, we fitted ar(10) and tvar(10) models (with q = 4 legendre polynomials) to all instances of the vowels /eh/, /ih/, /ae/, /ah/, /uh/, /ax/ (as in "bet,' "bit," "bat," "but," "book," and "about") and the diphthongs /ow/, /oy/, /ay/, /ey/, /aw/, /er/ (as in "boat," "boy," "bite," "bait," "bout," and "bird") in the training portion of the timit database [21]. data were downsampled to 8 khz, and values of t(x) were averaged across all dialects, speakers, and sentences (50, 000 vowel and 25, 000 diphthong instances in total). per-phoneme averages are reported in table i, and indicate considerably stronger detections of vocal tract variation in diphthongs than in vowels-and indeed a two-sample t test revised manuscript 4 table i vocal tract variation in timit vowels & diphthongs. vowel eh ih ae ah uh ax t(x) 67.5 60.5 94.6 63.8 58.9 32.1 diphthong ow oy ay ey aw er t(x) 134.1 302.4 187.4 130.6 161.6 133.0 easily rejects (p-value ≊0) the hypothesis that the average values of t(x) for the two groups are equal. this finding is consistent with the physiology of speech production, and demonstrates the sensitivity of the glrt in practice. iii. analysis of detection performance to apply the hypothesis test of (5), it is necessary to select a threshold γ as per (6), such that the null hypothesis of a best-fit ar(p) model is rejected in favor of the fitted tvar(p) model whenever t(x) > γ. below we describe how to choose γ to guarantee a constant false alarm rate (cfar) for large sample sizes, and give the asymptotic (in n) distribution of the glrt statistic under h0 and h1, showing how these results yield practical consequences for speech analysis. a. derivation of glrt asymptotics and cfar test under suitable technical conditions [29], likelihood ratio statistics take on a chi-squared distribution χ2 d(0) as the sample size n grows large whenever h0 is in force, with the degrees of freedom d equal to the number of parameters restricted under the null hypothesis. in our setting, d = pq since the pq coefficients αtv are restricted to be zero under h0, and we may write that t(x) ∼χ2 pq(0) under h0 as n →∞. thus, we may specify an allowable asymptotic constant false alarm rate for the glrt of (5), defined as follows: lim n→∞pr {t(x) > γ; h0} = pr  χ2 pq(0) > γ . (13) since the asymptotic distribution of t(x) under h0 depends only on p and q, which are set in advance, we can determine a cfar threshold γ by fixing a desired value (say, 5%) for the right-hand side of (13), and evaluating the inverse cumulative distribution function of χ2 pq(0) to obtain the value of γ that guarantees the specified (asymptotic) constant false alarm rate. when x is a tvar process so that the alternate hypothesis h1 is in force, t(x) instead takes on (as n →∞) a noncentral chi-squared distribution χ2 d(λ). its noncentrality parameter λ > 0 depends on the true but unknown parameters of the model under h1; thus in general t(x) n→∞ ∼ χ2 pq(λ), ( λ = 0 under h0, λ > 0 under h1. (14) it is easily shown by the method of [9] that the expression for λ in the case at hand is given by λ = αt tv(f t f ⊗σ−2r)αtv, (15) where * denotes the schur complement with respect to the first p × p matrix block of its argument, the (j + 1)th column of the matrix f ∈ r(n−p)×(q+1) is given by fj[p] fj[p + 1] * * * fj[n −1]
t , and r is given by: r ≜      rxx[0] rxx[1] * * * rxx[p −1] rxx[1] rxx[0] * * * rxx[p −2] . . . . . . ... . . . rxx[p −1] rxx[p −2] * * * rxx[0]     . here {rxx[0], rxx[1], . . . , rxx[p −1]} is the autocorrelation sequence corresponding to αar (given, e.g., by the "step-down algorithm" [28]). the expression of (15) follows from the fact that f t f ⊗σ−2r is the fisher information matrix for our tvar(p) model; its schur complement arises from the composite form of our hypothesis test, since the parameters αar, σ2 are unrestricted under h0. more generally, we may relate this result to the underlying tvar coefficient trajectories ai[n], arranged as columns of a matrix a, with each column-wise mean trajectory value a corresponding entry in a matrix ̄ a. letting ̃ a ≜a− ̄ a denote the centered columns of a, and noting both that f t f ⊗r = f t f ⊗r and that f (f t f )−1f t a = a when h1 is in force, properties of kronecker products [30] can be used to show that (15) may be written as λ = σ−2 tr( ̃ ar ̃ at ). (16) thus λ depends on the centered columns of a, which contain the true but unknown coefficient trajectories ai[n] minus their respective mean values. b. model order selection the above results yield not only a practical cfar threshold-setting procedure, but also a full asymptotic descrip- tion of the glrt statistic of (6) under both h0 and h1. in light of this analysis, it is natural to ask how the tvar model order p should be chosen in practice, along with the number q of non-constant basis functions. in deference to the large literature on the former subject [2], we adopt here the standard "2 coefficients per 1 khz of speech bandwidth" rule of thumb. intuitively, the choice of basis functions should be well matched to the expected characteristics of the coefficient tra- jectories ai[n]. to make this notion quantitatively precise, we appeal to the results of (14)–(16) as follows. first, the statis- tical power of our test to successfully detect small departures from stationarity is measured by the quantity pr  χ2 d(λ) > γ . a result of [31] then shows that for fixed γ, the power function pr  χ2 d(λ) > γ is: 1) strictly monotonically increasing in λ, for fixed d; 2) strictly monotonically decreasing in d for fixed λ. each of these properties in turn yields a direct and important consequence for speech analysis: • test power is maximized when λ attains its largest value: for fixed p and q, the noncentrality parameter λ of (16) determines the power of the test as a function of σ2 and the true but unknown coefficient trajectories a. • overfitting the data reduces test power: choosing p or q to be larger than the true data-generating model will revised manuscript 5 0 0.2 0.4 0.6 0.8 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 probability of false alarm probability of detection effect of model order on detection performance p = 2 p = 4 p = 6 p = 10 0 0.2 0.4 0.6 0.8 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 q = 1 q = 4 q = 7 q = 10 fig. 3. the effect of overfitting on the detection performance of the glrt statistic for the synthetic signal of fig. 2. an increase in the model order-p (left) and q (right)-decreases the probability of detection at any cfar level. result in a quantifiable loss in power, as λ will remain fixed while the degrees of freedom increase. the first of these consequences follows from property 1 above, and reveals how test power depends on the energy of the centered tvar trajectories ̃ a = a − ̄ a for fixed ̄ a and p, q, σ2. to verify the second consequence, observe that the product ̃ ar ̃ at remains unaffected by an increase in either p or q beyond that of the true tvar(p) model. then by property 2, the corresponding increase in the degrees of freedom pq will lead to a loss of test power. this analysis implies that care should be taken to ade- quately capture the energy of tvar coefficient trajectories while guarding against overfitting; this formalizes our ear- lier intuition and reinforces the importance of choosing a relatively low-dimensional subspace formed by the span of low-frequency basis functions whose degree of smoothness is matched to the expected tvar(p) signal characteristics under h1. this conclusion is further illustrated in fig. 3, which considers the effects of overfitting on the "formant-like" synthetic example of section ii-d, with p = 2, n = 100 sam- ples, δ = 7π/80 radians, and piecewise-constant coefficient trajectories. not only is the effect of overfitting p apparent in the left-hand panel, but the detection performance also suffers as the degree q of the legendre polynomial basis is increased, as shown in the right-hand panel. iv. relationship to classical approaches we now relate our hypothesis testing framework to two classical approaches in the literature. first, we compare its performance to that of brandt's test [3], which has seen wide use both in earlier [4], [32] and more recent studies [15], [33], [34], for purposes of transient detection and automatic segmentation for speech recognition and synthesis. second, we demonstrate its advantages relative to the autocorrelation method of time-varying linear prediction [5], showing that data windowing can adversely affect detection performance in this nonstationary setting. a. classical piecewise-constant ar approach a related previous approach is to model x as an ar process with piecewise-constant parameters that can undergo 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 probability of detection piecewise−constant signal rocs t'r(x) t(x) t'(x) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 probability of false alarm piecewise−linear signal rocs t(x) t'(x) 0 50 100 150 200 250 300 2000 2100 2200 2300 2400 2500 ω[n] time−varying center frequency 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 argmaxr t' r(x) sample number ml changepoint location fig. 4. comparing the detection performance of the statistic of (6) and that of (17): (top-left) comparison using the piecewise-constant signal (n = 100, δ = 5π/80) of section ii-d with p = 2 and q = 2 legendre polynomials used for computing (6); (top-right) piecewise-linear center frequency of the digital resonator used to generate the 2nd synthetic example; (bottom-left) comparison using the piecewise-linear signal (n = 300) with p = 2 and q = 3 legendre polynomials used for computing (6); (bottom-right) histogram of the changepoint r that maximizes the test statistic t ′ r(x) for each instantiation of the signal with piecewise-linear tvar coefficient trajectories. at most a single change [35]. the essence of this ap- proach, first employed in the speech setting by [3], is to split x into two parts according to x = (xr | xn−r) = (x[0] * * * x[r −1] | x[r] * * * x[n −1])t for some fixed r, and to assume that under h0, x is modeled by an ar(p) process with parameters α0, whereas under h1, xr and xn−r are described by distinct ar(p) processes with parameters αr and αn−r, respectively. in this context, testing for change in ar parameters at some known r can be realized as a likelihood ratio test; the associated test statistic t ′ r(x) is obtained by applying the covariance method to x, xr, and xn−r in order to estimate α0, αr, and αn−r, respectively. however, since the value of r is unknown in practice, t ′ r(x) must also be maximized over r, yielding a test statistic t ′(x) as follows: t ′(x) ≜max r t ′ r(x) 2p ≤r < n −2p, with (17) t ′ r(x) ≜ sup αr, αn−r ph1(xr; αr)ph1(xn−r; αn−r) sup α0 ph0(x; α0) . (18) we compared the detection performance of the glrt statistic of (6) with that of (17) on both the piecewise-constant signal of fig. 2 and a piecewise-linear tvar(2) signal to illustrate their respective behaviors-the resulting roc curves are shown in fig. 4. in both cases, it is evident that the tvar- based statistic of (6) has more power than that of (17), in part due to the extra variability introduced by maximizing over all values of r in (17)-especially those near the boundaries of its range. even in the case of the piecewise-constant signal, correctly matched to the assumptions underlying (17), the revised manuscript 6 tvar-based test is outperformed only when r is known a priori, and (18) is used. this effect is particularly acute in the small sample size setting-an important consideration for the single-pitch-period case study of section vi. this example demonstrates that any estimates of r can be misleading under model mismatch. as shown in the bottom- right panel of fig. 4, the detected changepoint is often esti- mated to be near the start or end of the data segment, but no "true" changepoint exists since the time-varying center frequency is continuously changing. thus piecewise-constant models are only simple approximations to potentially complex tvar coefficient dynamics; in contrast, flexibility in the choice of basis functions implies applicability to a broader class of time-varying signals. note also that computing (17) requires brute-force evalua- tion of (18) for all values of r, whereas (6) need be calculated once. moreover, t ′(x) fails to yield chi-squared (or any closed-form) asymptotics [35], thus precluding the design of a cfar test and any quantitative evaluation of test power. b. classical linear prediction and windowing recall that our glrt formulation of section ii, stemming from the tvar model of (2), generalized the covariance method of linear prediction to the time-varying setting. the classical autocorrelation method also yields least-squares esti- mators, but under a different error minimization criterion than that corresponding to conditional maximum likelihood. to see this, consider the tvar model x[n] = p x i=1 ai[n −i]x[n −i] + σw[n], (19) in lieu of (2). grouping the coefficients αij into p vectors e αi ≜ (αi0 αi1 * * * αiq)t , 1 ≤i ≤p, induces a partition of the expansion coefficients given by e α ≜(e αt 1 e αt 2 * * * e αt p )t - a permutation of elements of α in (4). the autocorrelation estimator of e α is then obtained by minimizing the prediction error over all n ∈z, while assuming that x[n] = 0 for all n / ∈ [0, . . . , n −1], and is equivalent to the least-squares solution of the following linear regression problem: x = f hx e α + σ e w, (20) where e w = w[0] * * * w[n −1])t
and the nth row of f hx ∈rn×p(q+1) is given by (f0[n −1]x[n −1] * * * f0[n − p]x[n −p] * * * fq[n −1]x[n −1] * * * fq[n −p]x[n −p]). the autocorrelation estimate of e α then follows from (20) as:1 b e α = (f ht x f hx)−1f ht x x. (21) moreover, when the autocorrelation method is used for spectral estimation in the stationary setting, x is often pre- multiplied by a smooth window. to empirically examine the role of data windowing in the time-varying setting, we generated a short 196-sample synthetic tvar(2) signal x 1as noted by [5], f ht x f hx is a block-toeplitz matrix comprised of p2 symmetric blocks of size (q + 1) × (q + 1)-this special structure arises as a direct consequence of the synchronous form of the tvar trajectories in (19). thus, the multichannel levinson-durbin recursion [36] may be used to invert f ht x f hx directly. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 probability of false alarm probability of detection covariance−based test autocorrelation−based test autcorrelation (hamming window) 0 5 10 15 20 25 30 0 0.02 0.04 0.06 0.08 histogram of t(x) t(x) autocorrelation−based test autocorrelation (hamming window) fig. 5. comparison of covariance- and autocorrelation-based test statistics, based on 5000 trials with a short (196-sample) data record. top: roc curves showing the effects of data windowing on detection performance. bottom: detail of how windowing changes the distribution of t(x) under h0. using q = 0 (h0) and q = 2 (h1) non-constant legendre polynomials, and fitted x using ar(3) and tvar(3) models- with the extra autoregressive order expected to capture the effects of data windowing. we then generated an roc curve associated with the glrt statistic of (12), shown in the top panel of fig. 5, along with roc curves corresponding to an evaluation of (12) following the autocorrelation-rather than the covariance-method, both with and without windowing. the bottom panel of fig. 5 shows the empirical distributions of both autocorrelation-based test statistics under h0, and indicates how windowing has the inadvertent effect of hinder- ing detection performance in this setting. we have observed the effects of fig. 5 to be magnified for even shorter data records, implying greater precision of the covariance-based glrt approach, which also has the advantage of known test statistic asymptotics under correct model specification. v. case study i: detecting formant motion we now introduce a glrt-based sequential detection al- gorithm to identify vocal tract variation on the scale of tens of milliseconds of speech data, and undertake a more refined analysis than that of section ii-d to demonstrate its efficacy on both whispered and voiced speech. our results yield strong empirical evidence that appropriately specified tvar models can capture vocal tract dynamics, just as ar models are known to provide a time-invariant vocal tract representation that is robust to glottal excitation type. a. sequential change detection scheme our basic approach is to divide the waveform into a sequence of k short-time segments {x1, x2, . . . , xk} using shifts of a single n0-sample rectangular window, and then to merge these segments, from left to right, until spectral change is detected via the glrt statistic of (6). the procedure, detailed in algorithm 1, begins by merging the first pair of adjacent short-time segments x1 and x2 into a longer segment revised manuscript 7 algorithm 1 sequential formant change detector 1) initialization: set γ via (13), input waveform data x • compute k short-time segments {x1, . . . , xk} of x using shifts of a rectangular window • set k = 1, xl = x1, xr = x2 • set a marker array c[k] = 0 for all 1 ≤k < k 2) while k < k • set xm = xl + xr and compute t(xm) via (6) • if t(xm) < γ (no formant motion within xm) – set xl = xm, c[k] = 0 else (formant motion detected within xm) – set xl = xk, c[k] = 1 • set xr = xk+1, k = k + 1 3) return the set of markers {k : c[k] = 1} xm and computing t(xm); failure to reject h0 implies that xm is stationary. thus, the short-time segments remain merged and the next pair considered is (xm, x3). this procedure continues until h0 is rejected, indicating the presence of change within the merged segment under consideration. in this case, the scheme is re-initialized, and adjacent short-time segments are once again merged until a subsequent change in the spectrum is detected. in algorithm 1, the cfar threshold γ of (13) is set prior to observing any data, by appealing to the asymptotic distribution of t(x) under h0 developed in section iii-a. in principle, the time resolution to within which change can be detected is limited only by n0. using arbitrarily short windows, however, increases the variance of the test statistic and results in an increase in false alarms-a manifestation of the fourier uncertainty principle. decreasing γ also serves to increase the (constant) false alarm rate, and leads to spurious labeling of local fluctuations in the estimated coefficient trajectories (e.g., due to the position of the sliding window relative to glottal closures) as vocal tract variation. b. evaluation with whispered speech in order to evaluate the glrt in a gradually more realistic setting, we first consider the case of whispered speech to avoid the effects of voicing, and apply the formant change detection scheme of algorithm 1 to whispered utterances containing slowly-varying and rapidly-varying spectra, respectively. the waveform used in the first experiment comprises a whispered vowel /a/ (as in "father") followed by a diphthong /ai/ (as in "liar"). it was downsampled to 4 khz in order to focus on changes in the first two formants, and algorithm 1 was applied to this waveform as well as to its 0–1 khz and 1–2 khz subbands (containing the first and second formants, respectively). results are summarized in fig. 6, and clearly demonstrate that the glrt is sensitive to formant motion. all three spectrograms indicate that spectral change is first detected near the boundary of the vowel and diphthong-precisely when the vocal tract configuration starts to change. subsequent consec- utive changes are found when sufficient formant change has fig. 6. result of applying algorithm 1 (16 ms rectangular windows, p = 4, q = 2 legendre polynomials, 1% cfar) to detect formant movement in the whispered waveform /a ai/. spectrograms corresponding to subbands containing the first formant only (a) second formant only (b) and both formants (c) were computed using 16 ms hamming windows with 50% overlap, and are overlaid with formant tracks computed by wavesurfer [37]. black lines demarcate times at which formant motion was detected; the time-domain waveform overlaid with these boundaries is also shown (d). been observed relative to data duration-a finding consistent with our earlier observation in section ii-d that more data are required to detect small changes in the ar coefficient trajectories, and by proxy the vocal tract, at the same level of statistical significance (i.e., same false alarm rate). next observe that whereas three "changepoints" are found when the waveform contains two moving resonances, a total of three "changepoints" are marked in the single-resonance waveforms shown in figs. 6(a) and 6(b). intuitively, each of these signals can be thought of as having "less" spectral change than the waveform shown in fig. 6(c), which contains both formants. thus, since the corresponding amounts of spectral change are smaller, longer short-time segments are required to detect formant movement-as indicated by the delays in detecting the vowel-diphthong transition seen in figs. 6(a) and (b) relative to (c). we next conducted a second experiment to demonstrate that the glrt can also detect a more rapid onset of spectral change as compared to, e.g., the relatively slow change in the spectrum of the diphthong. to this end we applied algorithm 1 to a sustained whispered vowel (/i/ as in "beet"), followed by the plosive /t/ at 10 khz. the results, shown in fig. 7, indicate that no change is detected during the sustained vowel, whereas the plosive is clearly identified. finally, we have observed change detection results such as these to be robust to not only reasonable choices of p (roughly 2 coefficients per 1 khz of speech bandwidth) and q (1–10), but also to the size of the initial window length (10–40 ms), and the constant false alarm rate (1–20%). revised manuscript 8 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 spectrogram time frequency 0 500 1000 1500 2000 2500 3000 3500 4000 4500 time−domain waveform sample number amplitude fig. 7. algorithm 1 (16 ms windows, p=10, q =4 legendre polynomials, 1% cfar), applied to detect formant movement in the whispered waveform /i t/. its spectrogram (top) is overlaid with formant tracks computed by wavesurfer [37] and black lines demarcating the time instants at which formant motion was detected; the time-domain signal is also shown (bottom). c. extension to voiced speech we next conducted an experiment to show that the tvar- based glrt is robust to the presence of voicing. we repeated the first experiment of section v-b above using a voiced vowel-diphthong pair /a ai/ over the range 0–4 khz. the same parameter settings were employed, except for the addition of two poles to take into account the shape of the glottal pulse during voicing [2]. algorithm 1 yields the results shown in fig. 8(a), which parallel those shown in fig. 6 for the whis- pered case. indeed, the first change occurs at approximately the vowel-diphthong boundary, with subsequent "changepoints" marked when sufficient formant movement has been observed. the similarities in these results are due in part to the fact that the analysis windows employed in both cases span at least one pitch period. to wit, consider the synthesized voiced phoneme /a/ and the associated glrt statistic of (6) shown in the top and bottom panels of fig. 8(b), respectively. even though the formants of the synthesized phoneme are constant, the value of t(x) undergoes a stepwise decrease from over the 1% cfar threshold when < 1 pitch period is observed, to just above the 50% cfar threshold when < 1.5 periods are observed-and finally stabilizes to a level below the 50% cfar threshold after more than two periods are seen. in contrast, the glrt statistic computed for the associated whispered phoneme, generated by filtering white noise by a vocal tract parameterized by the same formant values and shown in the bottom panel of fig. 8(b), remains time-invariant. these results indicate that the periodic excitation during voicing has negligible impact on the glrt statistic when longer (i.e., > 2 pitch periods) speech segments are used, and explain the robustness of the glrt statistic t(x) to the presence of voicing in the experiments of this section. on the other hand, the glrt is sensitive to the glottal flow when shorter speech segments are employed, suggesting that it can be also used effectively on sub-segmental time scales, as we show in section vi. vi. case study 2: sub-segmental speech analysis we now demonstrate that our glrt framework can be used not only to detect formant motion across multiple pitch (a) spectrogram of the voiced waveform /a ai/ is overlaid with formant tracks computed by wavesurfer [37] and black lines demarcating the time instants at which formant motion was detected; the time domain signal is shown for reference (bottom). 0 50 100 150 200 250 300 350 400 450 500 amplitude synthetic voiced phoneme segment 0 50 100 150 200 250 300 350 400 450 500 0 10 20 30 40 50 60 test statistics samples t(x) t(x): whispered t(x): voiced 50% cfar 1% cfar (b) formant-synthesized voiced phoneme /a/ (top) and associated glrt statistic (bottom, green) are shown along with 1% (solid black) and 50% cfar (dashed-black) thresholds. window lengths of 5 −35 ms at 1 ms (16-sample) increments with p = 6, q = 3 legendre polynomials were used to calculate t(x). values of t(x) for a whispered /a/ (bottom, blue) generated using the same formant values are shown for comparison. fig. 8. detecting vocal tract dynamics in voiced speech (a) and the impact of the quasi-periodic glottal flow on the glrt statistic t(x) (b). periods, as discussed above in section v, but also to detect vocal tract variations within individual pitch periods. since the vocal tract configuration is relatively constant during the glottal airflow closed phase, and undergoes change at its boundaries [18], a hypothesis test for vocal tract variation provides a natural way to identify both glottal opening and closing instants within the same framework. we show below that this framework is especially well suited to detecting the gradual change associated with glottal openings, and can also be used to successfully detect glottal closures. glottal closure identification is a classical problem (see, e.g., [19] for a recent review), with mature engineering solutions typically based on features of the linear prediction residual or the group delay function (see, e.g., [17], [19], [22] and references therein). in contrast, the slow onset of the open phase results in a difficult detection problem, and glottal revised manuscript 9 airflow velocity at the glottis waveform 50 100 150 200 250 300 350 400 450 typical egg derivative closed phase closed phase open phase fig. 9. glottal openings and closures demarcated over two pitch periods of a typical vowel, shown with idealized glottal flow (top), speech (middle), and egg derivative (bottom) waveforms as a function of time. opening detection has received relatively little attention in the literature [22], with preliminary results reported only in recent conference proceedings [38], [39]. a. physiology of sub-segmental variations figure 9 illustrates the idealized open and closed glottal phases associated with a typical vowel, along with the corre- sponding waveform and derivative electroglottograph (degg) data indicating approximate opening and closing instants [20], [40]. in each pitch period, the glottal closure instant (gci) is defined as the moment at which the vocal folds close, and marks the start of the closed phase-an interval during which no airflow volume velocity is measured at the glottis (top panel), and the acoustic output at the lips takes the form of exponentially-damped oscillations (middle panel). nominally, the glottal opening instant (goi) indicates the start of the open phase: the vocal folds gradually begin to open until airflow velocity reaches its maximum amplitude, after which they begin to close, leading to the next gci. time-invariance of the vocal tract suggests the use of linear prediction to estimate formant values during the closed phase [2], and then to use changes in these values across sliding windows to determine goi and gci locations [18]. indeed, as the vocal folds begin to open at the goi, the vocal tract gradually lengthens, resulting in a change in the frequency and bandwidth of the first formant [41]-an effect that can be explained by a source-filter model with a time- varying vocal tract. furthermore, the assumption that short- term statistics of the speech signal undergo maximal change in the vicinity of a gci implies that such regions will exhibit large linear-prediction errors. b. detection of glottal opening instants we first give a sequential algorithm to detect gois via the glrt statistic t(x). to study the efficacy of the proposed method, we assume that the timings of the glottal closures are available, and use these to process each pitch period independently. in addition to evaluating the absolute error rates of our proposed scheme using recordings of sustained vowels, we also compare it with the method of [17]-a standard prediction-error-based approach that remains in wide use, and effectively underlies more recent approaches such as [38]. 1) sequential goi detection procedure: in contrast to the "merging" procedure of algorithm 1, our basic approach here is to scan a sequence of short-time segments xw, induced by shifts of an n0-sample rectangular window initially left- aligned with a glottal closure instant, until spectral change is detected via the glrt statistic of (6). at each iteration, the window slides one sample to the right, and t(xw) is evaluated; this procedure continues until t(xw) exceeds a specified cfar threshold γ, indicating that spectral change was detected, and signifying the beginning of the open phase. in this case, the goi location is declared to be at the right edge of xw. on the other hand, a missed detection results if a goi has not been identified by the time the right edge of the sliding window coincides with the next glottal closure instant. the exact procedure is summarized in algorithm 2. algorithm 2 sequential glottal opening instant detector 1) initialization: input one pitch period of data x between two consecutive glottal closure locations g1 and g2 • set wl = g1, wr = wl + n0, and set γ via (13) • set xw = (x[wl] * * * x[wr]) 2) while t(xw) < γ and wr < g2 • increment wl and wr (slide window to right) • recompute xw and evaluate t(xw) 3) if wr < g2, then return wr as the estimated glottal opening location, otherwise report a missed detection. since each instantiation of algorithm 2 is confined to a single pitch period, the parameters n0, p, and q must be chosen carefully. to ensure robust estimates of the tvar coefficients, the window length n0 cannot be too small; on the other hand, if it exceeds the length of the entire closed-phase region, then the goi cannot be resolved. likewise, choosing a small number of tvar coefficients results in smeared spectral estimates, whereas using large values of p leads to high test statistic variance and a subsequent increase in false alarms; this same line of reasoning also leads us to keep q small. thus, in all the experiments reported in section vi-b, we employ n0 = 50-sample windows, p = 4 tvar coefficients and the first 2 legendre polynomials as basis functions (q = 1). we also evaluated the robustness of our results with respect to these settings, and observed that using window lengths of 40– 60 samples, 3–6 tvar coefficients, and 2–4 basis functions also leads to reasonable results in practice. 2) evaluation: we next evaluated the ability of algorithm 2 to identify the glottal opening instants in five sustained vowels uttered by a male speaker (109 hz average f0), synchronously recorded with an egg signal (center for laryngeal surgery and voice rehabilitation, massachusetts general hospital), and subsequently downsampled to 16 khz. the speech filing system [42] was used to extract degg peaks and dips, which revised manuscript 10 100 200 300 400 −1 −0.5 0 0.5 1 speech waveform x[n] 221 241 261 281 301 −4 −2 0 2 4 ar coefficient trajectories a[n] 100 200 300 400 −0.05 0 0.05 0.1 0.15 egg derivative degg[n] 221 241 261 281 301 0 5 10 15 20 glrt test statistic sample number t(x) 15% cfar fig. 10. algorithm 2 applied to detect the goi in a pitch period of the vowel /a/ (top left), shown together with its egg derivative (bottom left). the sliding window is left-aligned with the gci (solid black line); estimated ar coefficients (top right) and the glrt statistic t(x) of (6) (bottom right) are then computed for each subsequent window position. the detected goi (dashed black line) corresponds to the location of the first determined change (at the 15% cfar level) in vocal tract parameters. in turn provided a means of experimentally measured ground truth for our evaluations. a typical example of goi detection is illustrated in fig. 10, which shows the results of applying algorithm 2 to an excised segment of the vowel /a/. the detected goi in this example was declared to be at the right edge of the first short- time segment xw for which t(xw) exceed the 15% cfar threshold γ, and is marked by a dashed black line in all four panels of fig. 10. as can be seen in the bottom-right panel, the estimated goi coincides precisely with a dip in the degg waveform. moreover, as the top-right panel shows, this location corresponds to a significant change in the estimated coefficient trajectories, likely due both to a change in the frequency and bandwidth of the first formant (resulting from nonlinear source-filter interaction [18], [41]), as well as an increase in airflow volume velocity (from zero) at the start of the open phase. detection rates were then computed over 75 periods of each vowel, and detected gois were compared to degg dips in every pitch period that yielded a goi detection. the resultant detection rates and root mean-square errors (rmse, conditioned on successful detection) are reported in table ii, along with a comparison to the prediction-error-based approach of [17], which we now describe. table ii goi detection accuracy (ms, no. missed detections). /a/ /e/ /i/ /o/ /u/ glrt rmse (ms) 0.69 1.03 1.00 1.15 0.69 wmg [17] rmse (ms) 1.04 1.78 1.13 1.97 1.10 glrt missed det. 0 5 0 0 0 wmg [17] missed det. 8 18 4 6 4 3) comparison with approach of wong, markel, and gray (wmg) [17]: the approach of [17] involves first computing a 1400 1600 1800 2000 2200 2400 degg waveform degg[n] 1400 1600 1800 2000 2200 2400 ar coefficient trajectories a[n] 1400 1600 1800 2000 2200 2400 algorithm 2 t(x) 1400 1600 1800 2000 2200 2400 approach of wong, markel and gray [19] sample number η(x) fig. 11. comparison of algorithm 2 and the approach of [17] for goi detection. the egg derivative for 8 periods of the vowel /a/, and estimated ar coefficients, are shown for all sliding window positions (top two panels) along with the associated values of t(x) and η(x) (bottom two panels). true and estimated goi locations are indicated by solid and dashed black lines, respectively. note the variability in the dynamic range of η(x) from one pitch period to the next, and the missed detection (2nd pitch period, bottom panel). normalized error measure η(xw) for each short-time segment xw (induced by a sliding window as in algorithm 2), and then identifying the goi instant with the right edge of xw when a large increase in η(xw) is observed. the measure η(xw) is obtained by fitting a time-invariant ar(p) model to xw (using (10) with q = 0), calculating the norm of the resultant prediction error, and normalizing by the energy of short-time segment xw. figure 11 provides a comparison of this approach to that of algorithm 2, over 8 periods of the vowel /a/. here algorithm 2 is implemented with a 15% cfar level, but the threshold for η(x) must be set manually, since no theoretical guidelines are available [17]. indeed, as illustrated in the bottom panel of fig. 11, variability in the dynamic range of η(x) across pitch periods implies that any fixed threshold will necessarily introduce a tradeoff between detection rates and rmse. in this example, lowering the threshold to intersect with η(x) in the second pitch period-and thereby removing the missed detection-results in a 25% increase in rmse. the denominator of the glrt statistic t(xw) depends on the same prediction error residual used to calculate η(xw); however, as indicated by fig. 11, it remains much more stable across pitch periods. thus, while the approach of [17] relies on large absolute changes in ar residual energy to detect glottal openings, that of algorithm 2 explicitly takes into account the ratio of ar to tvar residual energies-resulting in improved overall performance. indeed, though thresholds were set individually for each vowel of table ii, and manually revised manuscript 11 adjusted to obtain the best rmse performance while keeping the number of missed detections reasonably small, algorithm 2 with a 15% cfar threshold exhibits both superior detection rates and rmse. c. detection of glottal closure instants although our main focus here is on goi detection, the glrt statistic of (6) may also be employed to detect glottal closures. indeed, under the assumption stated earlier that the speech signal undergoes locally maximal change in the vicinity of a gci, a simple gci detection algorithm immediately sug- gests itself: compute (6) for every location of an appropriate sliding analysis window, and declare the glottal closure to occur at the midpoint of the window with the largest associated value of t(x). in this formulation, t(x) is being treated simply as a signal with features that may be helpful in finding the gci locations; no test threshold need be set. a typical result is shown in the third panel of fig. 12, obtained using the same parameter settings (50-sample window, p = 4, q = 2) as in the goi detection scheme of section vi-b. we compared this method to two others based on linear prediction and group delay, as described above. first, we implemented the alternative likelihood-ratio epoch detection (lred) approach of [32], which tests for a single change in ar parameters. second we used the "front end" of the popular dypsa algorithm for goi detection [22], comprising the generation of gci candidates and their weighting by the ideal phase-slope function deviation cost as implemented in the voicebox online toolbox [43]. table iii summarizes the gci estimation results under the same conditions as reported in table ii. all three methods are comparable in terms of accuracy, though the glrt approach proposed here can be used-with the same parameter settings-for both gci and goi detection. table iii gci detection accuracy (ms). vowel/ gci rmse /a/ /e/ /i/ /o/ /u/ glrt (n0 = 50, p = 4, q = 2) 0.47 1.05 0.73 0.97 1.03 lred [32] (n0 = 72, p = 6) 1.02 0.69 1.00 0.65 1.12 dypsa front end [22](n0 = 50) 0.61 0.68 0.70 0.79 1.10 results from both our approach and the dypsa front end can in turn be propagated across pitch periods (using, e.g., dynamic programming [22]) to inform a broader class of group-delay methods [19], though we leave such a system- level comparison as the subject of future work. vii. discussion the goal of this article has been to develop a statis- tical framework based on time-varying autoregressions for the detection of nonstationarity in speech waveforms. this generalization of linear prediction was shown to yield efficient fitting procedures, as well as a corresponding generalized likelihood ratio test. our study of glrt detection performance yielded several practical consequences for speech analysis. incorporating these conclusions, we presented two algorithms 100 200 300 400 500 600 700 x[n] 100 200 300 400 500 600 700 degg[n] 100 200 300 400 500 600 700 t(x) fig. 12. using the glrt statistic of (6) to find gci locations in a segment of the vowel /a/. the speech waveform (top), the egg derivative (middle) and the glrt statistic (bottom) are overlaid with the true (solid, black line) and estimated (dashed, black line) glottal closure locations in each pitch period. to identify changes in the vocal tract configuration in speech data at different time scales. at the segmental level we demon- strated the sensitivity of the glrt to vocal tract variations corresponding to formant changes, and at the sub-segmental scale, we used it to identify both glottal openings and closures. methodological extensions include augmenting the tvar model presented here to explicitly account for the quasi- periodic nature of the glottal flow (or its time derivative), and deriving a glrt statistic corresponding to (6) in the case where only noisy waveform measurements are available. important next steps in applying the hypothesis testing frame- work to practical speech analysis include further development of the glottal closure and opening detection schemes of the previous section, which were here applied independently in each pitch period. incorporating the dynamic programming approach of [22] will likely serve to improve performance, as will incorporating the glrt statistic as part of global frame- to-frame cost function in such a framework. acknowledgements the authors wish to thank daryush mehta at the center for laryngeal surgery and voice rehabilitation at massachusetts general hospital for providing recordings of audio and egg data, nicolas malyska for helpful discussions, and the anony- mous reviewers for suggestions that have improved the paper. references [1] d. rudoy, t. f. quatieri, and p. j. wolfe, "time-varying autoregressive tests for multiscale speech analysis," in proc. 10th ann. conf. intl. speech commun. ass., 2009. [online]. available: http://sisl.seas. harvard.edu [2] t. f. quatieri, discrete-time speech signal processing: principles and practice. upper saddle river, nj: prentice-hall, 2002. [3] a. v. brandt, "detecting and estimating parameter jumps using ladder algorithms and likelihood ratio tests," proc. ieee intl. conf. acoust. speech signal process., vol. 8, pp. 1017–1020, 1983. revised manuscript 12 [4] r. andre-obrecht, "a new statistical approach for the automatic seg- mentation of continuous speech signals," ieee trans. acoust. speech signal process., vol. 36, pp. 29–40, 1988. [5] m. g. hall, a. v. oppenheim, and a. s. willsky, "time-varying parametric modeling of speech," signal process., vol. 5, pp. 267–285, 1983. [6] y. grenier, "time-dependent arma modeling of nonstationary signals," ieee trans. acoust. speech signal process., vol. 31, pp. 899–911, 1983. [7] k. s. nathan and h. f. silverman, "time-varying feature selection and classification of unvoiced stop consonants," ieee trans. speech audio process., vol. 2, pp. 395–405, 1994. [8] k. schnell and a. lacroix, "time-varying linear prediction for speech analysis and synthesis," in proc. ieee intl. conf. acoust. speech signal process., 2008, pp. 3941–3944. [9] s. m. kay, "a new nonstationarity detector," ieee trans. signal process., vol. 56, pp. 1440–1451, 2008. [10] s. kay, fundamentals of statistical signal processing: detection the- ory. upper saddle river, nj: prentice-hall, 1998. [11] d. rudoy, p. basu, t. f. quatieri, b. dunn, and p. j. wolfe, "adaptive short-time analysis-synthesis for speech enhancement," in proc. ieee intl. conf. acoust. speech signal process., 2008, pp. 4905–4908. [online]. available: http://sisl.seas.harvard.edu [12] d. rudoy, p. basu, and p. j. wolfe, "superposition frames for adaptive time-frequency representations and fast reconstruction," ieee trans. signal process., vol. 58, pp. 2581–2596, 2010. [13] j. vermaak, c. andrieu, a. doucet, and s. j. godsill, "particle methods for bayesian modeling and enhancement of speech signals," ieee trans. speech audio process., vol. 10, pp. 173–185, 2002. [14] r. c. hendriks, r. heusdens, and j. jensen, "adaptive time segmentation for improved speech enhancement," ieee trans. audio speech lang. process., vol. 14, pp. 2064–2074, 2006. [15] v. tyagi, h. bourlard, and c. wellekens, "on variable-scale piecewise stationary spectral analysis of signals for asr," speech commun., vol. 48, pp. 1182–1191, 2006. [16] g. s. morrison, "likelihood-ratio forensic voice comparison using parametric representations of the formant trajectories of diphthongs," j. acoust. soc. am., vol. 125, pp. 2387–2397, 2009. [17] d. y. wong, j. d. markel, and a. h. gray, "least squares glottal inverse filtering from the acoustic speech waveform," ieee trans. acoust. speech signal process., vol. 27, pp. 350–355, 1979. [18] m. d. plumpe, t. f. quatieri, and d. a. reynolds, "modeling of the glottal flow derivative waveform with application to speaker identifica- tion," ieee trans. speech audio process., vol. 7, pp. 569–586, 1999. [19] m. brookes, p. a. naylor, and j. gudnasson, "a quantitative assessment of group delay methods of identifying glottal closures in voiced speech," ieee trans. audio speech lang. process., vol. 8, pp. 1017–1020, 2006. [20] d. g. childers and j. n. larar, "electroglottography for laryngeal function assessment and speech analysis," ieee trans. on biomed. eng., vol. 31, pp. 807–817, 1984. [21] j. s. garofolo, l. lamel, w. fisher, j. fiscus, d. pallett, n. dahlgren, and v. zue, timit acoustic-phonetic continuous speech corpus. philadelphia, pa: linguistic data consortium, 1993. [22] p. a. naylor, a. kounoudes, j. gudnasson, and m. brookes, "estimation of glottal closure instants in voiced speech using the dypsa algorithm," ieee trans. audio speech lang. process., vol. 15, pp. 34–43, 2007. [23] l. a. liporace, "linear estimation of non-stationary signals," j. acoust. soc. am., vol. 58, pp. 1268–1295, 1975. [24] m. k. tsatsanis and g. b. giannakis, "time-varying system identifica- tion and model validation using wavelets," ieee trans. signal process., vol. 41, pp. 3512–3523, 1993. [25] t. s. rao, "the fitting of non-stationary time series models with time dependent parameters," j. roy. stat. soc. b, vol. 32, pp. 312–322, 1970. [26] g. kitagawa and w. gersch, "a smoothness priors time-varying ar coefficient modeling of nonstationary covariance time series," ieee trans. automat. control, vol. 30, pp. 48–56, 1985. [27] t. hsiao, "identification of time-varying autoregressive systems using maximum a posteriori estimation," ieee trans. signal process., vol. 56, pp. 3497–3509, 2008. [28] s. kay, modern spectral estimation: theory and application. upper saddle river, nj: prentice-hall, 1988. [29] m. kendall, a. stuart, j. k. ord, and s. arnold, kendall's advanced theory of statistics. hodder arnold, 1999, vol. 2a. [30] j. w. brewer, "kronecker products and matrix calculus in system theory," ieee trans. circuits syst., vol. 25, pp. 772–781, 1978. [31] s. d. gupta and m. d. perlman, "power of the noncentral f-test: effect of additional variates on hotelling's t2 test," j. am. statist. ass., vol. 69, pp. 174–180, 1974. [32] e. moulines and r. d. francesco, "detection of the glottal closure by jumps in the statistical properties of the speech signal," speech commun., vol. 9, pp. 401–418, 1990. [33] j. p. bello, l. daudet, s. abdallah, c. duxbury, and m. davies, "a tutorial on onset detection in music signals," ieee trans. speech audio process., vol. 13, pp. 1035–1048, 2005. [34] s. jarifia, d. pastora, and o. rosec, "a fusion approach for automatic speech segmentation of large corpora with application ot speech synthe- sis," speech communication, vol. 50, pp. 67–80, 2008. [35] r. e. quandt, "tests of the hypothesis that a linear regression system obeys two separate regimes," j amer. stat. assoc., vol. 55, pp. 324–330, 1960. [36] s. l. marple, digital spectral analysis. englewood cliffs, nj: prentice- hall, 1987. [37] k. sj ̈ olander and j. beskow. (2005) wavesurfer 1.8.5 for windows. kth royal institute of technology. [online]. available: http: //www.speech.kth.se/wavesurfer/wavesurfer-185-win.zip [38] t. drugman and t. dutoit, "glottal closure and opening instant detection from speech signals," in proc. 10th ann. conf. intl. speech commun. ass., 2009. [39] m. p. thomas, j. gudnason, and p. a. naylor, "detection of glottal closing and opening instants using an improved dypsa framework," in proc. 17th eur. signal process. conf, glasgow, scotland, 2009. [40] n. henrich, c. d'alessandro, b. doval, and m. castellengo, "on the use of the derivative of electroglottographic signals for characterization of nonpathological phonation," j. acoust. soc. am., vol. 115, pp. 1321– 1332, 2004. [41] t. v. ananthapadmanabha and g. fant, "calculation of true glottal flow and its components," speech commun., vol. 1, pp. 167–184, 1982. [42] m. huckvale. (2000) speech filing system: tools for speech research. university college london. [online]. available: http: //www.phon.ucl.ac.uk/resource/sfs [43] m. brookes. (2006) voicebox: a speech processing toolbox for matlab. imperial college london. [online]. available: http: //www.ee.imperial.ac.uk/hp/staff/dmb/voicebox/voicebox.html

0911.1698

finite temperature properties of a supersolid: a rpa approach

we study in random-phase approximation the newly discovered supersolid phase of ${}^4$he and present in detail its finite temperature properties. ${}^4$he is described within a hard-core quantum lattice gas model, with nearest and next-nearest neighbour interactions taken into account. we rigorously calculate all pair correlation functions in a cumulant decoupling scheme. our results support the importance of the vacancies in the supersolid phase. we show that in a supersolid the net vacancy density remains constant as function of temperature, contrary to the thermal activation theory. we also analyzed in detail the thermodynamic properties of a supersolid, calculated the jump in the specific heat which compares well to the recent experiments.

introduction one of the biggest accomplishments of theoretical con- densed matter physics is the ability to classify various phases and phase transitions by their mathematical or- der. these mathematical orders are usually expressed by order parameters reflecting certain limiting behaviours of two particle correlation functions. this concept makes it possible to describe the properties of physically very dif- ferent systems in a common language and establish a link between them. without this concept, the counter-intuitive idea of a su- persolid, i.e. a solid that also exhibits a superflow, would not have been conceivable. in this language supersolidity, firstly proposed by andreev and lifshitz[1] in 1969 and picked up by leggett and chester in 1970[3,2] is the clas- sification of systems that simultaneously exhibit diagonal and off-diagonal long range order. yet the idea of a super- solid was reluctantly received by the scientific community, because early experiments failed to detect any such effect in helium-4. apart from john goodkind[4] who had pre- viously seen suspicious signals in the ultrasound signals of solid helium, physicists were surprised when in 2004 kim and chan [5,6] announced the discovery of supersolid he- lium. although equipped with a head start of more than 30 years the theoretical understanding of the supersolid state lags vastly behind, not least because supersolidity as we observe it today is nothing like what the pioneers in early 1970's had anticipated. it is now evident that defectons and impurities play a cru- cial role in the formation of the supersolid state. however, the data and the results of various experiments draw a rather complex picture of the supersolid phase. the nor- mal solid to supersolid transition is not of the usual first or second order type transition but bears remarkable resem- blance to the kosterlitz thouless transition well-known in two dimensional systems. furthermore annealing experi- ments show that 3he impurities play a significant role but the data remain all but conclusive, as the measured su- perfluid density varies by order of magnitudes among the different groups. recently a change in the shear modulus of solid helium was found and the measured signal almost identically mimics the superfluid density measured by kim and chan [5,6]. some popular theories give plausible ex- planations of some aspects of the matter. that, such as the vortex liquid theory suggested by phil anderson [9], is in good agreement with properties of the phase tran- sition; theories based on defection networks are capable of describing the change in the shear modulus. however, we feel that a satisfying and comprehensive theory is still missing. part of the problem stems from the complexity of the models. many models are only at sufficient accuracy solvable with numerical monte carlo methods. the re- sults, doubtlessly useful, are seldom intuitive and the lack of analytical results do not meet our notion of the under- standing of a phenomenon. other approaches on the other hand are limited in their analytical significance and rather represent a phenomenological ansatz. in this work we attempt to fill a part of this gap in the the- oretical understanding of the supersolid phase. we present a theory of supersolidity in a quantum-lattice gas model (qlg) beyond simple mean-field approaches. following the approach of k.s. liu and m.e. fisher[12] we map the qlg model to the anisotropic heisenberg model (ahm). 2 a.j. stoffel, m. gul ́ acsi: finite temperature properties of a supersolid: a rpa approach the method of green's functions proves to be very suc- cessful in the description of ferromagnetic and antiferro- magnetic phases and we use this method to investigate the supersolid phase which corresponds to a canted antiferro- magnetic phase. applying the random-phase approxima- tion (rpa), we broke down using cumulant decoupling, the higher order green's functions to obtain a closed set of solvable equations. this method gives a fully quantum mechanical and analytical solution of a supersolid phase. we will see that quantum fluctuations have a significant effect on the net vacancy density of the supersolid and we will also see that the superfluid state is unstable against zero-energy quasi-particle excitations. we also derive im- portant thermodynamic properties of this model and de- rive formulas for interesting properties such as the power law exponents and the jump of the specific heat across the normal solid to supersolid line. the paper is organized as follows: in section 2 we intro- duce the generic hamiltonian of a bosonic many body sys- tem and discretize it to a quantum lattice gas model. this model is equivalent to the anisotropic heisenberg model in an external field and we will identify the corresponding phases in section 3. in section 4 we re-derive the classical mean-field solution and discuss briefly their significance before we in section 5 recapitulate the green's functions for the anisotropic heisenberg model in the random-phase approximation at zero temperature. the ground state of the system is obtained by solving the corresponding self- consistency equation. in the following two sections we de- rive basic thermodynamic properties as well as ordinary differential equations to calculate the first and second or- der phase transition lines. in section 8 we analyse the quasi-particle energy spectrum and in the last section we calculate phase diagrams for various parameter sets and analyse the properties of the supersolid phase and the cor- responding transitions. 2 generic hamiltonian and anisotropic heisenberg model if we neglect possible 3he impurities supersolid helium-4 is a purely bosonic system whose dynamics and structure are governed by a generic bosonic hamiltonian: h = z d3xψ†(x)(−1 2m∇2 + μ)ψ(x) +1 2 z d3xd3x′ψ†(x)ψ†(x′)v (x −x′)ψ(x)ψ(x′) (1) here ψ†(x) are the particle creation operators and ψ(x) destruction operators and obey the usual bosonic commu- tation relations. hamiltonians such as in eq. (1) are not, due to the vast size of the many body hilbert-space, di- agonalisable even for simple potentials v (x). the accom- plishment of any successful theory is to find an approxi- mation that sustains the crucial mechanism while reduc- ing the mathematical complexity to a traceable level. here we follow the method of matsubara and matsuda [14] who successfully introduced the quantum lattice gas (qlg) model to describe superfluid helium. we believe that the quantum lattice gas model is particularly useful for super- solids as the spatial discretization of this model serves as a natural frame for the crystal lattice of (super)-solid helium and a bipartite lattice elegantly simplifies the problem of breaking translational invariance symmetry for states that exhibit diagonal long range order. according to matsubara and tsuneto [13] the generic hamil- tonian eq. (1) in the discrete lattice model reads: h = μ x i ni + x ij uij(a† i −a† j)(ai −aj) + x ij vijninj (2) here uij are solely finite for nearest neighbor and next nearest neighbor hopping and otherwise zero. the val- ues of unn and unnn are such that the kinetic energy is isotropic up to the 4th order. as atoms do not penetrate each other there can exist only one atom at a time on a lat- tice site and consequently a† and a are the creation and annihilation operators of a hard core boson commuting on different lattice sites, but obey the anti-commutator relations on identical sites. equation (2) is the hubbard model in 3 dimensions for hard core bosons. neither fully bosonic nor fermionic, the lack of wicks theorem inhibits the application of pertubative field theory though hard core bosonic systems are algebraic identical to spin sys- tems. a simple transformation, where the bosonic opera- tors are substituted by spin-1/2 operators generating the su(2) lie algebra, maps the present qlg model onto the anisotropic heisenberg model: h = hz x i sz i + x ij j∥ ijsz i sz j + x ij j⊤ ij (sx i sx j + sy i sy j )(3) here the correspondence between the qlg parameters and the spin coupling constants is given by: j∥ ij = vij j⊤ ij = −2uij hz = −μ + x j j⊤ ij − x j j∥ ij (4) 3 phases the self-consistency equations in the random-phase ap- proximation as we will derive in a later chapter are very lengthy and therefore it is our primary goal to keep the present model as simple as possible while still being able to describe the crucial physics. for this reason we will define the anisotropic heisenberg model on a bipartite bcc lat- tice which consists of two interpenetrating sc sub-lattices as figure 1 shows. in this lattice geometry the perfectly solid phase is composed of a fully occupied (on-site) sub- lattice a while sublattice b refers to the empty interstitial and is consequently vacant. as there is no spatial density a.j. stoffel, m. gul ́ acsi: finite temperature properties of a supersolid: a rpa approach 3 a b fig. 1. the bcc lattice consists of two interpenetrating sc sub-lattices. in the perfect solid phase one sub-lattice (i.e. sub- lattice a) serves as on-site centers and is occupied while sub- lattice b represents the empty interstitial and is vacant (for simplicity, we only present the two dimensional case). table 1. possible magnetic phases and the corresponding phases of the hubbard model. all phases are defined by their long range order. the columns, from left to right, are the spin configurations, magnetic phases, odlro, dlro and corre- sponding 4he phases, respectively. ↑↑ fe no no normal liquid րր cfe yes no superfluid րւ caf yes yes supersolid ↑↓ af no yes normal solid variation in the liquid phases both sublattices are equally occupied, and the mean occupation number simply cor- responds to the particle density. we are aware that the above choice of sc sublattices does not reflect the true 4he crystal structure which is hexagonal closed-packed (hcp). nevertheless we believe that crucial physical properties such as phase transition and critical constants do not de- pend on the specific geometry as long as no other effects such as frustration are evoked. table 3 charts the various magnetic phases and identifies the corresponding phases of the 4he system. according to their spin configurations we identify the four magnetic phases to be of ferromagnetic (fe), canted ferromagnetic (cfe), canted anti-ferromagnetic (caf) and antiferro- magntic (af) orders. the off-diagonal long range order parameter m1 and the diagonal long range order m2 are: m1 = ⟨sx a⟩+ ⟨sx b⟩ m2 = ⟨sz a⟩−⟨sz b⟩ (5) which readily identify the corresponding phases of the he- lium system. 0.5 1 1.5 2 t [k] 2 4 6 8 hz caf cfe fe af fig. 2. the phase diagram for j⊤ 1 = 1.498, j⊤ 2 = 0.562, j∥ 1 = −3.899 and j∥ 2 = −1.782 as calculated by mf. 4 mean-field solution in our previous work [10] we have already re-derived the classical mean-field solution of the anisotropic heisenberg model at zero temperature. as this model provides an easy and intuitive access to the model we extend the approx- imation to finite temperatures as was done by k.s liu and m.e. fisher [12] and briefly discuss its solution and phase diagram. the mean-field hamiltonian is obtained by substituting the spin- 1 2 operators with their expectation values: hmf = −hz(⟨sz a⟩+ ⟨sz b⟩) −2j∥ 1 ⟨sz a⟩⟨sz b⟩−j∥ 2 (⟨sz a⟩⟨sz a⟩+ ⟨sz b⟩⟨sz b⟩) −2j⊤ 1 ⟨sx a⟩⟨sx b⟩−j⊤ 2 (⟨sx a⟩⟨sx a⟩+ ⟨sx b⟩⟨sx b⟩) (6) here j∥ 1 = −q1j∥ i∈a,j∈b, j∥ 2 = −q2j∥ i∈a,j∈a , j⊤ 1 = −q1j⊤ i∈a,j∈b and j⊤ 2 = −q1j⊤ i∈a,j∈a where q1 = 6 and q2 = 8 are the number of nearest and next nearest neigh- bors. the mean value of sy drops out as the randomly broken symmetry sx ↔sy (odlro) allows for ⟨sy⟩= 0. the standard method of deriving the corresponding self- consistency equations is to minimize the helmholtz's free energy f = h −t s. the entropy s is given by the pseudo spin entropy of the system: s = −1 2[(1 2 + sa) ln(1 2 + sa) + (1 2 −sa) ln(1 2 −sa) +(1 2 + sb) ln(1 2 + sb) + (1 2 −sb) ln(1 2 −sb)](7) where sa = p ⟨sza⟩2 + ⟨sxa⟩2 and sb = p ⟨szb⟩2 + ⟨sxb⟩2. in the canted anti-ferromagnetic and the canted ferro- magnetic states there are 4 self-consistency equations in the ferromagnetic and anti-ferromagnetic phases where ⟨sx⟩= ⟨sy⟩= 0 they are reduced by two. these equations are readily obtained by differentiating the free energy with 4 a.j. stoffel, m. gul ́ acsi: finite temperature properties of a supersolid: a rpa approach respect to ⟨sz⟩and ⟨sx⟩respectively. the resulting equa- tions can be rearranged to yield: p ⟨sza⟩2 + ⟨sxa⟩2 = tanh(βωa) 2 p ⟨szb⟩2 + ⟨sxb⟩2 = tanh(βωb) 2 (8) hz + 2⟨sza⟩(j∥ 2 −j⊤ 2 ) + 2⟨szb⟩j∥ 1 = 2j⊤ 1 ⟨sxb⟩ ⟨sxa⟩⟨sza⟩ hz + 2⟨szb⟩(j∥ 2 −j⊤ 2 ) + 2⟨sza⟩j∥ 1 = 2j⊤ 1 ⟨sxa⟩ ⟨sxb⟩⟨szb⟩(9) where ωa = [(2j⊤ 1 ⟨sx b⟩+ 2j⊤ 2 ⟨sx a⟩)2 + (2j∥ 1 ⟨sz b⟩+ 2j∥ 2⟨sz a⟩+ hz)2] 1 2 ωb = [(2j⊤ 1 ⟨sx a⟩+ 2j⊤ 2 ⟨sx b⟩)2 + (2j∥⟨sz b⟩+ 2j∥⟨sz a⟩+ hz)2] 1 2 (10) the two equations (eq. (9)) are dismissed in the ferromag- netic and the anti-ferromagnetic phases as the transversal fields ⟨sx a⟩and ⟨sx b⟩become zero. at zero temperature t = 0 the free energy and ⟨h⟩coincide. this allows us to deduce the phases at absolute zero in the limiting cases (limits of hz) from equation (6). in the limit of hz →∞ the hamiltonian reduces to an effective one particle model: h = hz(⟨sza⟩+ ⟨szb⟩) (11) consequently the system will be in the energetically favor- able ferromagnetic phase. in the opposite limit, hz →0 and with antiferromagnetic coupling j∥ 1 < 0 the term of nearest neighbor interaction h = j∥ 1 ⟨sza⟩⟨szb⟩ (12) is the only term that significantly lowers the energy. there- fore the ground-state is the anti-ferromagnetic state. at t=0 and a suitable choice of parameters the canted fer- romagnetic and the canted anti-ferromagnetic phases are realized in-between these limits as seen in figure 2. how- ever with increasing temperature the regions of canted ferromagnetic and canted anti-ferromagnetic phases de- plete and at sufficiently high temperatures only the anti- ferromagnetic and ferromagnetic phases survive. as we can see from the phase diagram in figure 2 most tran- sitions are of second order. only for parameter regimes where the caf does not appear the resulting cfe-af is of first order. the boundary lines are defined by ordi- nary differential equations and generally need to be calcu- lated numerically. nonetheless there exist analytical ex- pressions for the locations of the transitions at absolute zero as derived by matsuda and tsuneto [13]. the ferro- magnetic to canted ferromagnetic transition point is de- termined by equations (9) if we set ⟨sza⟩= ⟨szb⟩= 1 2 and ⟨sxa⟩= ⟨sxb⟩= 0: hz f e−cf e = j⊤ 1 + j⊤ 2 −j∥ 1 −j∥ 2 (13) equally the canted anti-ferromagnetic to anti-ferromagnetic transition is defined by ⟨sza⟩= −⟨szb⟩= 1 2 and ⟨sxa⟩= ⟨sxb⟩= 0: hz caf −af = q (−j∥ 1 + j∥ 2 −j⊤ 2 )2 −(j⊤ 1 )2 (14) the canted ferromagnetic and the canted anti-ferromagnetic phases coexist where the order parameter of the dlro ,m1 = ⟨sza⟩−⟨szb⟩approaches zero. we replace ⟨sza⟩ and ⟨szb⟩in equation (9) with m1 and m2 = ⟨sza⟩+⟨szb⟩ and retain only linear terms of m1. subtracting and adding both equations respectively yields: hz + 2m2(j∥ 2 −j⊤ 2 + j∥ 1 ) = 2j⊤ 1 m2 2m1(j∥ 2 −j⊤ 2 −j∥ 1 ) = −2j⊤ 1 m14m2 2 + 1 4m2 2 −1 (15) we used that p ⟨sza⟩2 + ⟨sxa⟩2 = 1 4 at t=0. the solu- tion of these two equations determine the corresponding transition point which is given by: hz cf e−caf = j∥ 1 + j∥ 2 −j⊤ 1 −j⊤ 2 −j∥ 1 + j∥ 2 + j⊤ 1 −j⊤ 2 × q (−j∥ 1 + j∥ 2 −j⊤ 2 )2 −(j⊤ 1 )2 (16) 5 green's functions although the classical mean-field theory is quite insight- ful and gives an accurate description of the variously or- dered phases its fails to take quantum fluctuations and quasi-particle excitations into account. hence, in order to overcome these shortcomings and to derive a fully quan- tum mechanical approximation we solve the anisotropic heisenberg model in the random-phase approximation which is based on the green's function technique. at finite tem- perature the retarded and advanced tyablikov [15,16] com- mutator green's function defined in real time are: gμν ijret(t) = −iθ(t)⟨|[sμ i (t), sν j ]|⟩ gμν ijadv(t) = iθ(−t)⟨|[sμ i (t), sν j ]|⟩ (17) the average ⟨⟩involves the usual quantum mechanical as well as thermal averages. the most successful technique of solving many body green's function involves the method of equation of motion which is given by: i∂tgxy ijret(t) = δ(t)⟨[sx i , sy j ]⟩−iθ(t)⟨[[sx i , h], sy j ]⟩ i∂tgxy ijadv(t) = δ(t)⟨[sx i , sy j ]⟩+ iθ(−t)⟨[[sx i , h], sy j ]⟩ (18) the commutator [sx i , h] can be eliminated by using the heisenberg equation of motion giving rise to higher, third order green's functions on the rhs. in order to obtain a closed set of equations we apply the cumulant decoupling procedure which splits up the third order differential equa- tion into products of single operator expectation values a.j. stoffel, m. gul ́ acsi: finite temperature properties of a supersolid: a rpa approach 5 and two spin green's functions. the cumulant decoupling [17] is based on the assumption that the last term of the following equality is negligible: ⟨ˆ a ˆ b ˆ c⟩= ⟨ˆ a⟩⟨ˆ b ˆ c⟩+ ⟨ˆ b⟩⟨ˆ a ˆ c⟩ +⟨ˆ c⟩⟨ˆ a ˆ b⟩−2⟨ˆ a⟩⟨ˆ b⟩⟨ˆ c⟩ +⟨( ˆ a −⟨ˆ a⟩)( ˆ b −⟨ˆ b⟩)( ˆ c −⟨ˆ c⟩)⟩ (19) the set of differential equations is not explicitly dependent of the temperature, i.e. temperature dependence solely comes with thermal averaging of the single operator expec- tation values. therefore the solution is formally identical to the zero temperature solution and the detailed deriva- tion of the green's function in their full form can be found in ref. [11]. the averages of the spin operator, appearing in the cu- mulant decoupling scheme, have to be determined self- consistently. two self-consistent equations can be derived from correlation functions corresponding to the green's functions. the self-consistency equations of the canted ferromagnetic (superfluid) and canted antiferromagnetic (supersolid) phases are structurally different from the fer- romagnetic (normal solid) and antiferromagnetic (normal fluid) phases, as the off-diagonal long range order increases the number of degrees of freedom by two and hence two additional conditions, resulting from the analytical prop- erties of the commutator green's functions, apply. thus, the self-consistency equations of the canted phases, can be written as a function of the temperature, the external magnetic field and the spins in x-direction: f c a(⟨sx a⟩, ⟨sx b⟩, hz, t ) = 0 f c b(⟨sx a⟩, ⟨sx b⟩, hz, t ) = 0 (20) similar the self-consistency equations for the ferromag- netic and anti-ferromagnetic phases: f nc a (⟨sz a⟩, ⟨sz b⟩, hz, t ) = 0 f nc b (⟨sz a⟩, ⟨sz b⟩, hz, t ) = 0 (21) these equations involve a 3 dimensional integral over the first brillouin zone. as explained in the work [10] on the zero temperature formalism this integral can be reduced to a two dimensional integral by introducing a general- ized density of state (dos) and gives the model a wider applicability. 6 thermodynamic properties the relation between then qlg and the anisotropic heisen- berg model is such that the chemical potential μ corre- sponds to the external field hz, i.e. the grand canonical partition function in the qlg corresponds to the canoni- cal partition function in the anisotropic heisenberg model where the number of spins is fixed. consequently, thermo- dynamic potentials of the two models are related as: θqlg −μn = θheisenberg (22) here θ refers to an arbitrary thermodynamic potential. in the same way as the ground state minimizes the in- ternal energy u = ⟨h⟩at absolute zero, the free energy f = u −t s is minimized at finite temperatures. we wish to stress that the internal energy in the present approxi- mation cannot be derived accurately from the expectation value of the hamiltonian in the following way: u = hz x i ⟨sz i ⟩+ x ij j∥ ij⟨sz i sz j ⟩ + x ij j⊤ ij (⟨sx i sx j ⟩+ ⟨sy i sy j ⟩) (23) the cumulant decoupling, though a good approximation to the anisotropic heisenberg model, is also an exact solu- tion of an unknown effective hamiltonian heff. therefore thermodynamically consistent results are only obtained if the effective hamiltonian is substituted in the equation above, i.e. u = ⟨heff⟩. here, as we do not know the ex- plicit form of this effective model, we have to integrate the free energy from thermodynamic relations: df = ⟨sz a⟩+ ⟨sz b⟩ 2 dhz + sdt (24) this equation allows us to select the ground state in re- gions where two or more phases exist self-consistently ac- cording to the random-phase approximation. the entropy of the spin system is given by s = z d3k  1 1 + exp(−βω(k)) log( 1 1 + exp(−βω(k))) + 1 1 + exp(βω(k)) log( 1 1 + exp(βω(k)))  (25) this formula is of purely combinatorial origin and reflects the fact that the hard-core boson system is equivalent to a fermionic system given by the jordan-wigner transforma- tion. the ω(k) terms refer to the energies of the spin-wave excitations. in the solid phases both branches have to be taken into account. the entropy of the anisotropic heisen- berg model given for a fixed number of spins corresponds to the entropy of the qlg model at a constant volume. therefore, in order to obtain the usual configurational en- tropy of the qlg, we have to divide by the number of particles per unit cell: (sconf = 2s na+nb ). here na = 1/2 −sz a and nb = 1/2 −sz b are the particle occupation numbers on lattice sites a and b. as the na- ture of the normal solid to supersolid phase transition is not yet satisfactorily understood recent experiments [19] have focused on the behavior of the specific heat across the transition line in the hope of shedding light on the matter. the specific heat at constant temperature and constant pressure respectively are given by: cv = t ∂sconf ∂t  t,v,n , cp = t ∂sconf ∂t  t,p,n (26) although the external magnetic field hz in the spin model is an observable the corresponding quantity in the qlg 6 a.j. stoffel, m. gul ́ acsi: finite temperature properties of a supersolid: a rpa approach model, namely the chemical potential is not. therefore we are also interested in attaining a formula for the pressure associated with a certain chemical potential. the relation- ship is most easily derived from the following maxwell re- lation: ∂p ∂μ  t,v = ∂n ∂v  t,μ = #lattice sites v (1 −ǫ) (27) where ǫ := ⟨sz a⟩+ ⟨sz b⟩. note that, using this equation in order to obtain the pressure at any specific temperature we need a reference point, i.e. a chemical potential where the corresponding pressure is known. this point is given by μ →∞which corresponds to n →0 and, hence, p →0. consequently, in order to obtain the pressure for a specific chemical potential μ′ we have to integrate over the interval [∞, μ′]. 7 boundary lines the state of the system at any point in t and hz is given by the self-consistency equations and the free energy as can be derived from eq. (24). nevertheless the resulting computations come at high computational cost and there- fore it seems most feasible to derive odes which deter- mine the first and second order transition lines. first we will derive the ordinary differential equations which define the more abundant second order transitions. the normal fluid (fe) and the normal solid (cfe) phases are deter- mined by equations (21) and the supersolid (caf) and superfluid (cfe) are defined by equations (20) and con- dition equation (9). consequently on the ss-ns and sf- nf transition line, where both the normal (fe and cfe) and the super (cfe and caf) phases coexists following equations hold: f n a (⟨sza⟩, ⟨szb⟩, hz, t ) = 0 f n b (⟨sza⟩, ⟨szb⟩, hz, t ) = 0 hz + 2⟨sza⟩(j∥ 2 −j⊤ 2 ) + 2⟨szb⟩j∥ 1 = 2j⊤ 1 ⟨sxb⟩ ⟨sxa⟩⟨sza⟩ hz + 2⟨szb⟩(j∥ 2 −j⊤ 2 ) + 2⟨sza⟩j∥ 1 = 2j⊤ 1 ⟨sxa⟩ ⟨sxb⟩⟨szb⟩ (28) on the ss-ns boundary line the quotient ⟨sxa⟩/⟨sza⟩is not known a priori and therefore we eliminate it in the equation above yielding: f n a (⟨sza⟩, ⟨szb⟩, hz, t ) = 0 f n b (⟨sza⟩, ⟨szb⟩, hz, t ) = 0 f(⟨sza⟩, ⟨szb⟩, hz) := (hz + 2⟨sza⟩(j∥ 2 (0) −j⊤ 2 ) + 2⟨szb⟩j∥ 1 ) × (hz + 2⟨szb⟩(j∥ 2 −j⊤ 2 ) + 2⟨sza⟩j∥ 1 ) −(2j⊤ 1 ⟨szb⟩)2 = 0 (29) we introduce a variable s which parametrizes the bound- ary curve. if we for instance choose ds=dt we get the following set of ordinary differential equations, defining the nf-sf and the ns-ss transition lines:       ∂f n a ∂⟨sza⟩ ∂f n a ∂⟨szb ⟩ ∂f n a ∂hz ∂f n a ∂t ∂f n b ∂⟨sza⟩ ∂f n b ∂⟨szb ⟩ ∂f n b ∂hz ∂f n b ∂t ∂f ∂⟨sza⟩ ∂f ∂⟨szb ⟩ ∂f ∂hz 0 0 0 0 1       *      ∂⟨sza⟩ ∂s ∂⟨szb ⟩ ∂s ∂hz ∂s ∂t ∂s     =    0 0 0 1   (30) upon crossing the sf-ss transition line, coming from the superfluid phase the set of possible solutions branches off into two phases, the supersolid and a non-physical (com- plex valued) superfluid phase. therefore any matrix of or- dinary differential equations will render a singularity and consequently we have to approach the transition line in a limiting process: f s a(⟨sxa⟩, ⟨sxb⟩, hz, t ) = 0 f s b(⟨sxa⟩, ⟨sxb⟩, hz, t ) = 0 lim ǫ→0(⟨sxa⟩−⟨sxb⟩−ǫ) = 0 (31) the resulting ode is      ∂f s a ∂⟨sxa⟩ ∂f s a ∂⟨sxb ⟩ ∂f s a ∂hz ∂f s a ∂t ∂f s b ∂⟨sxa⟩ ∂f s b ∂⟨sxb ⟩ ∂f s b ∂hz ∂f s b ∂t 1 −1 0 0 0 0 0 1     *      ∂⟨sxa⟩ ∂s ∂⟨sxb ⟩ ∂s ∂hz ∂s ∂t ∂s     =    0 0 ǫ 1   (32) which defines the superfluid to supersolid transition. as mentioned previously, there are certain parameter regimes where not all four possible phases are appearing and con- sequently a first order transition (mostly superfluid to su- persolid) will occur. in the previous chapter we have seen that such a transition line is difficult to locate. however, a tricritical point frequently appears in the phase diagram and at this point the first order transition evolves into a second order transition. this tricritical point can be taken as a initial value for a differential equation defining the corresponding first order transition line. the relevant ode may be derived from a clausius clapey- ron type equation. on the transition line both phases have equal free energy. hence: ∆s ∆(⟨sza⟩+ ⟨szb⟩) = ∂t ∂hz (33) where s refers to the spin entropy as derived in the pre- vious section (eq. (25)) and ∆refers to the difference of either entropy or spin mean-field of the superfluid and the normal solid phases. this equation together with ds = dt and the total derivative of the two self-consistency equa- tions for the normal solid and one equation for the super- fluid phase form a set of 5 odes determining ⟨sx⟩in the superfluid phase and ⟨sza⟩and ⟨szb⟩in the normal solid phase along the boundary line in the t −hz plane:         ∂f s ∂⟨sx⟩0 0 ∂f s ∂hz ∂f s ∂t 0 ∂f n a ∂⟨sza⟩ ∂f n a ∂⟨szb ⟩ ∂f n a ∂hz ∂f n a ∂t 0 ∂f n b ∂⟨sza⟩ ∂f n b ∂⟨szb ⟩ ∂f n b ∂hz ∂f n b ∂t 0 0 0 ∆s ∆⟨sza+b⟩ 0 0 0 0 1         (34) a.j. stoffel, m. gul ́ acsi: finite temperature properties of a supersolid: a rpa approach 7 8 excitation spectrum the superfluid phase features, due to spontaneously bro- ken u(1) symmetry, the well know gapless goldstone bosons, i.e. linear phonons. the supersolid phase additionally ex- hibits a second, gapped branch which is due to the break down of discrete translational symmetry. figure 4 reveals a zero frequency mode in the supersolid phase at [100] of the first brillouin zone and consequently the superfluid to supersolid phase transition is character- ized by a collapsing roton minimum at [100]. the disper- sion relation in the superfluid (cfe) phase is given by: ω(k) = 2{(j⊤ 1 (γ1(k) −1) + j⊤ 2 (γ2(k) −1)) ×  ⟨sz⟩2(j⊤ 1 (γ1(k) −1) + j⊤ 2 (γ2(k) −1))− ⟨sx⟩2(j⊤ 1 + j⊤ 2 −j∥ 1 γ1(k) −j∥ 2 γ2(k)) i }1/2 (35) from this equation we can see that the energy possibly goes to zero at [100] (corresponds to γ1(k) = −1 and γ2(k) = 1) when following condition is fulfilled: j⊤ 1 + j⊤ 2 + j∥ 1 −j∥ 2 < 0 (36) hence we obtain a further condition (supplementary to eq. (16)) for the existence of the superfluid to supersolid transition. equation (35) allows for the existence of a second region of the reciprocal lattice space where the dispersion rela- tion might go soft. for γ1(k) = 0 and γ2(k) = −1 which corresponds to [111] we obtain following condition: −2j⊤ 2 j⊤ 1 + j⊤ 2 + j∥ 2 > 0 (37) it is also interesting to study the behavior of the excitation spectrum with increasing temperature. in a conventional superfluid the long wave-length behavior is given by: ω(k) = no(t )v (0) m k (38) here v(0) is the interaction potential at zero momentum and m is the particles' mass. the density of the conden- sate no(t ) typically decreases with increasing temperature and for that reason we expect lower energies with increas- ing temperature. in figure 3 we can see that the quasi- particle energies indeed decrease with increasing temper- ature. apart from the region in the vicinity of [100] the energies at higher temperatures lie significantly lower than those ones closer to absolute zero. this is important as it will contribute to the variation of thermodynamic quan- tities such as the entropy or the specific heat. figure 4 depicts the variation of the excitation spectrum with in- creasing temperature in the supersolid phase. in this phase the low lying phonon branch mostly depletes with increas- ing temperature, although there exist a region between [110] and [111] where the zero temperature spectrum is significantly higher. contrary to the phonon branch the gapped mode lifts the energy around the long wave length εk [100] [110] [111] [000] [000] fig. 3. excitation spectrum in the superfluid phase for for j⊤ 1 = 1.498k, j⊤ 2 = 0.562k, j∥ 1 = −3.899k, j∥ 2 = −1.782k and hz = 3. solid line refers to t = 0, dotted to t = 0.5, dashed line to t = 1 and the long dashed line to t = 1.3. εk [000] [100] [110] [111] [000] fig. 4. excitation spectrum in the supersolid phase for for j⊤ 1 = 1.498k, j⊤ 2 = 0.562k, j∥ 1 = −3.899k, j∥ 2 = −1.782k and hz = 0.8. the solid lines refer to t = 0 and the dashed lines to t = 1. limit [000] and around [100]. in comparison with the super- fluid dispersion relation the excitation spectrum changes its form/shape rather than scaling down with increasing temperature as in the superfluid phase. we observe that the supersolid phase exhibits a more complex and diverse structure than the superfluid or normal solid phases alone. 9 discussion 9.1 on finite temperature properties in this section we will discuss finite temperature properties of the anisotropic heisenberg model and its solution in the random-phase approximation. in order to be able to compare the temperature dependence of the model in the random-phase approximation with the classical mean-field 8 a.j. stoffel, m. gul ́ acsi: finite temperature properties of a supersolid: a rpa approach 0 0.5 1 1.5 2 t [k] 0 2 4 6 8 hz sf (cfe) nf (fe) ss (caf) ns (af) fig. 5. the phase diagram for j⊤ 1 = 1.498k, j⊤ 2 = 0.562k, j∥ 1 = −3.899k and j∥ 2 = −1.782k in the rpa (solid lines) and in classical mf (dashed lines). mf overestimates the tem- perature by about 30%. approximation we chose the set of parameters that was extensively scrutinized by liu and fisher [12]: j⊤ 1 = 1.498k j⊤ 2 = 0.562k j∥ 1 = −3.899k j∥ 2 = −1.782k (39) as mentioned in the previous section, liu and fisher [12] have chosen this set of parameters because it provides ar- guably the best fit to the phase diagram of helium-4. since we believe that the validity of the quantum lattice gas model is too limited to appropriately reproduce the be- havior of helium-4 over the whole range of temperature and pressure we do not discuss most properties in the pressure-temperature space but rather present the major part of the results in the more comprehensible magnetic field -temperature coordinates. only where the theory can be compared to relevant experimental data, such as the heat capacity at constant pressure we work in the corre- sponding representation. the phase diagram of the anisotropic heisenberg model in t and hz coordinates is given in figure 5. the dashed lines correspond to the phase diagram of the mean-field ap- proximation. we see that the diagrams are quantitatively quite similar but in the mean-field solution the tempera- ture is somewhat overestimated giving an approximately 30% higher temperature for the tetra-critical point. as mentioned before the critical magnetic fields hz c, due to quantum fluctuations, are lower in the random-phase ap- proximation. this effect is most distinctive on the super- solid to superfluid transition line as there the deletion of the spin magnitude is strongly pronounced. the net vacancy density ǫ, density of vacancies minus den- sity of interstitials sparked interest as the question arose [18] as to whether the number of vacancies follow predic- tions of thermal activation theory or are due to the effects 0 0.5 1 1.5 2 2.5 3 t/tc 0 0.02 0.04 0.06 0.08 net vacancy density fig. 6. net vacancy density for four different pressures. the dashed line shows the curve expected if the normal solid would exist down to zero temperature 0.5 1 1.5 hz 0 0.05 0.1 0.15 0.2 ε = s z a + s z b fig. 7. net vacancy density as a function of the external mag- netic field (equates to the chemical potential) for four different temperatures. the solid line refers to t=0k, the dashed line to t=0.4k, the long dashed line to t=0.7k and the dotted line refers to t=1k. of an incommensurate crystal. figure 6 shows the net va- cancy density in the supersolid and the normal solid phase at constant pressures. in isobar curves, curves of constant pressure, the magnetic field hz is controlled through equa- tion (27). we see that the net vacancy density is nearly constant in the supersolid phase. only in the normal solid phase the net vacancy density increases exponentially with increasing temperature in total agreement with classical thermal activation theory. the almost temperature inde- pendent behavior of the net vacancy density in the su- persolid phase is an important finding of the quantum lattice gas model and should be observable in high reso- lution experiments if this effect is real. in figure (7) we plotted the net vacancy density as a function of hz (or equivalently μ the chemical potential) across the normal solid, the supersolid and the superfluid phases for various temperatures, namely t = 0k, t = 0.4k, t = 0.7k and t = 1.0k. the term net vacancy density does not a.j. stoffel, m. gul ́ acsi: finite temperature properties of a supersolid: a rpa approach 9 0 0.2 0.4 0.6 0.8 t/tc 0 0.01 0.02 0.03 0.04 -∆f/k 0.1 1e-07 1e-06 1e-05 0.0001 0.001 0.01 fig. 8. free energy in the supersolid phase. the leading con- tribution comes from a t 4.2-term. the inset shows the free energy in double logarithmic scale. the dashed line is the fit to the t 4.2-term. the leading correction comes from a t 5-term, indicated by the long dashed line. fig. 9. [color online] 3d plot of the spin entropy. the four phases are clearly distinguishable as the entropy is non-smooth across the transition lines. 0.5 1 1.5 2 t/tc 0.2 0.4 0.6 0.8 1 1.2 1.4 cp fig. 10. specific heat at constant pressure. 0 0.025 0.05 0.075 0.1 0.125 ε = s z a+s z b 0 0.5 1 1.5 t [k] fig. 11. curves show the net vacancy density as a function of temperature on the supersolid to normal solid transition lines. the solid line refers to set 1: j⊤ 1 = 1.498k, j⊤ 2 = 0.562k, j∥ 1 = −3.899k and j∥ 2 = −1.782k and the dashed line to set 2: j⊤ 1 = 0.5k, j⊤ 2 = 0.5k, j∥ 1 = −2.0k and j∥ 2 = −0.5k. have a physical meaning in the superfluid phase and here the quantity ǫ rather refers to the particle density of the fluid. the particle density in the superfluid phase is lin- ear in hz and independent of the temperature t , which follows immediately from condition eq. (9) as ⟨sx a⟩/⟨sx b⟩ is equal to one in the fluid phase. in the supersolid phase the dependence of the net vacancy density on the chemi- cal potential is stronger than in the superfluid phase. the chemical potential (magnetic field hz) is roughly inversely proportional to the pressure. meaning that the superfluid phase exhibits a higher compressibility than the supersolid phase, which is a quite remarkable result. interestingly the net vacancy density in the supersolid phase also increases linearly with the chemical potential. this is due to the ratio of the superfluid order parameter ⟨sx a⟩/⟨sx b⟩varies as the square root of the magnetic field in the vicinity of the transition: ⟨sx a⟩/⟨sx b⟩∝1 − q hz −hz caf −af (40) the exponent 1 2 is typical for mean-field type approxima- tions and appears close to all transition lines. figure 11 shows the net vacancy density of the model on the su- persolid to normal solid transition line as a function of temperature, and also reveals the mean-field type square root law dependence. at zero temperature in the solid phase the net vacancy density is equal to zero, hence the crystal is incompress- ible. in real systems compressibility usually occurs as a result of change in the lattice constant a, characterized by the grueneisen parameter. the quantum lattice gas model does not take this effect into account since the lattice con- stant is treated as a constant. however measurements have shown that the lattice constant in the (supersolid) helium is almost a constant indicating that the net vacancy den- sity is the crucial parameter. also of interest are the free energy and the entropy. figure (8) shows the free energy of 10 a.j. stoffel, m. gul ́ acsi: finite temperature properties of a supersolid: a rpa approach the anisotropic heisenberg model in the supersolid (caf) phase at constant pressure. at low t the free energy usu- ally follows a power law: f ∝t α (41) the coefficient α is a universal property of the model which is constant over the entire regime of a phase. the exponent is most easily acquired in the log-log plot, shown in the inset of the plot. in this logarithmic scale α is given by the slope of the curve and the leading contribution is given by, approximately α = 4.2 (42) in the supersolid region. this is close in value to the usual t 4-term attributed to the linear phonon modes, and fol- lows from equation (25) with ω(k) linear in k. here the t 4-term is solely due to the superfluid mode and in real solids an additional t 4-term contribution, accounting for the lattice phonon-modes, will appear. the logarithmic plot also reveals the leading correction to the free energy given by α = 5. the non-configurational entropy of the system over the en- tire range of temperature and magnetic field hz is given in figure 9. all four phases are visible and as expected from a thermodynamically equilibrated system the entropy is monotonically increasing with respect to temperature. figure 10 depicts the configurational specific heat at con- stant pressure in the supersolid and the normal solid phase. the jump in the specific heat at the critical temperature tc, in agreement with the second order phase transition, appears to be smeared out due to numerical inaccuracies as the specific heat is the second derivation of the free energy which had to be integrated of the interval [hz, ∞]. the jump in the specific heat may also be calculated from following formula which is an analogy to the clausius- clapeyron equation: ∆∂2f ∂hz2 dhz2 dt 2  t l + ∆∂2f ∂hz∂t dhz dt  t l + ∆∂2f ∂t 2 = 0 (43) ∆ch = −t ∆∂2f ∂t 2 = t ∆ ∂(⟨sza⟩+ ⟨szb⟩) ∂hz  ∂hz ∂t 2 t l +t ∆ ∂(⟨sza⟩+ ⟨szb⟩) ∂t  ∂hz ∂t  t l (44) as we have cp = ch −t ∂2f ∂t ∂hz ( ∂hz ∂t )p , we have for the specific heat at constant pressure: ∆cp = t ∆ ∂(⟨sza⟩+ ⟨szb⟩) ∂hz  ∂hz ∂t 2 t l +t ∆ ∂(⟨sza⟩+ ⟨szb⟩) ∂t  ∂hz ∂t  t l − ∂hz ∂t  p  (45) 0 0.1 0.2 0.3 0.4 0.5 0.6 t [k] 0.8 1 1.2 1.4 1.6 1.8 2 hz 0 0.1 0.2 0.3 0.84 0.85 0.86 0.87 nf sf ns ss fig. 12. the phase diagram for j⊤ 1 = 0.5k, j⊤ 2 = 0.5k, j∥ 1 = −1.0k and j∥ 2 = 0.5k. the supersolid phase vanishes below t=0.323k. the resulting sf-ns transition is first order. for the values corresponding to figure 10 we obtain an estimated jump of 0.02 which is in good agreement with the curve. 9.2 first order boundary lines in parameter regimes where the supersolid phase does not appear in certain temperature regions there consequently appears a first order phase transition between the super- fluid and the normal solid phase. liu and fisher [12] com- pared the free energies of the competing phases to estab- lish the transition line. this procedure is not applicable in the random-phase approximation, as was outlined in the section on thermodynamic properties. other than the mean-field approximation where the hamiltonian is given by equation (6) the effective hamiltonian of the random- phase approximation is not known. therefore we have to integrate the first order transition line from a clau- sius clapeyron like equation as derived in the section on boundary lines. a set of parameters which exhibits such a first order transition at low temperatures is given by: j⊤ 1 = 0.5 j⊤ 2 = 0.5 j∥ 1 = −1.0 j∥ 2 = 0.5 (46) the corresponding phase diagram is shown in figure 12. according to mean-field eq. (14) and eq. (16) the sec- ond order superfluid to supersolid transition as well as the supersolid to normal solid transition is at absolute zero at hz = 0.86603, implying that the supersolid phase does not exist at zero temperature. at higher tempera- tures (above t > 0.323k) the supersolid phase does exist. at t = 0.323k where the sf-ss and the ss-ns transi- tion lines intersect there occurs a tricritical point. on this tricritical point the superfluid and normal solid phases co- exist (as well as the supersolid phase) and the first order a.j. stoffel, m. gul ́ acsi: finite temperature properties of a supersolid: a rpa approach 11 0 0.5 1 1.5 2 t [k] 0.8 1 1.2 1.4 1.6 hz nf sf ss ns tricritical points fig. 13. the phase diagram for j⊤ 1 = 0.38k, j⊤ 2 = 2.5k, j∥ 1 = −1.2k and j∥ 2 = 2.4k. the supersolid phase does not appear above t=1.785k. the two tricritical points are con- nected by a first order sf-ns phase transition. phase transition line can be calculated from the ordinary differential equation as has been derived in the section on boundary lines (eq. (34)). the tricritical point is as such the starting point for the integration of the ode. there also exist a regime of parameters where the super- solid phase is suppressed at higher temperatures as can be seen in figure 13 which corresponds to: j⊤ 1 = 0.38 j⊤ 2 = 2.5 j∥ 1 = −1.2 j∥ 2 = 2.4 (47) the sf-ss and the ss-ns transition lines converge before the nf-sf line is reached, hence no tetracritical point such as in figure 5 is present. the two tricritical points are connected by a first order superfluid to normal solid transition line. 9.3 beyond the model in section 8 we have seen that the supersolid phase ap- pears if and only if the roton dip at γ1 = −1 and γ2 = 1, i.e. [001] of the first brillouin zone collapses. additionally we have seen that the model also allows for a collapsing minimum at [111] if condition eq. (37) is met. a set of parameters that fulfills this condition is given by: j⊤ 1 = 0.5k j⊤ 2 = 0.5k j∥ 1 = −2k j∥ 2 = −1.5k (48) note that the nearest neighbor and the next nearest neigh- bor constants in this configuration j⊤ 1 and j⊤ 2 , corre- sponding to the kinetic energy, are relatively weak and 0 0.2 0.4 0.6 0.8 t [k] 0 1 2 3 4 5 hz nf sf unknown phase fig. 14. phase diagram for j⊤ 1 = 0.5k,j⊤ 2 = 0.5k, j∥ 1 = −2k and j∥ 2 = −1.5k. the superfluid phase becomes unstable due to a imaginary quasi-particle spectrum (dashed line). rpa does not yield any stable phase beneath that line. are both of equal strength, leading to a highly anisotropic kinetic energy. figure 14 shows the corresponding phase diagram according to the random-phase approximation. the normal fluid to superfluid transition line starts at hz = 4.5 (49) at absolute zero and decreases with increasing tempera- ture. according to eq. (14) and eq. (16) the critical exter- nal fields hz defining the superfluid to supersolid and the supersolid to normal solid transitions are, due to negative values under the square root, imaginary and hence phys- ically not relevant. consequently in the classical mean- field approximation the superfluid phase extends down to hz = 0 and a first order superfluid to normal solid tran- sition does not occur as the relatively large negative j∥ 2 increases the free energy of a possible solid phase. the random-phase approximation however draws a slightly different picture. analogous to the classical mean-field solution the random-phase approximation also yields a phase transition near hz = 4.5. but unlike the classical mean-fields solution, the superfluid phase here does not survive all the way down to hz = 0. due to the partic- ular choice of parameters the superfluid phase becomes unstable at around hz = 2; i.e the quasi-particle spectrum turns imaginary at γ1(k) = 0 and γ2(k) = −1 ([111]). in- terestingly beyond this line no other stable phase exists in the present approach; there is no set of spin fields ⟨sxa⟩, ⟨sxb⟩, ⟨sza⟩and ⟨szb⟩that solves the self-consistency equations (20) or (21). to understand why the present approach breaks down and how the phase below hz = 2 might look like we first inves- tigate the physical meaning of the collapsing roton mini- mum at [100] (equivalently [010] and [001]) leading to the supersolid phase consisting of two sc sub-lattices as al- ready analyzed in the previous sections: the roton dip at [100] corresponds to a density wave given by: cos(2πx/a) + cos(2πy/a) + cos(2πz/a) 3 (50) 12 a.j. stoffel, m. gul ́ acsi: finite temperature properties of a supersolid: a rpa approach fig. 15. two dimensional projection of the lattice structure of a (super)-solid phase on three sub-lattices triggered by a collapse of the roton minimum at [111] of the first brillouin zone. this density wave takes on value one on sub-lattice a and minus one on sub-lattice b, hence it reproduces the peri- odicity of the supersolid crystal. in the same way a collapsing roton dip on the main diag- onals [111] as given here refers to following density wave: cos(πx/a) cos(πy/a) cos(πz/a) (51) this density wave yields zero on sub-lattice b and alter- natingly one and minus one on sub-lattice a (neighbors have opposite signs). consequently this phase refers to a (super)-solid phase exhibiting three sub-lattices a, b and c where the mean-fields take on three different values (see fig. 15). it would be quite interesting to study the pos- sibilities and properties of such a phase and we leave as future work the extension of the present approach to ac- count for a three sub-lattice phase and the investigation its properties. 10 conclusion in this paper we extended the mean-field theory of the qlg model by liu and fisher [12] at finite temperatures and employed a random-phase approximation, where we derived green's functions using the method of equation of motion. we applied the cumulant decoupling procedure to split up emergent third order green's functions in the eom. in comparison to the mf theory employed by liu and fisher [12] the rpa is a fully quantum mechanically solution of the qlg model and therefore takes quantum fluctuations as well as quasi-particle excitations into ac- count. the computational results show that these quasi- particle excitations are capable to render the superfluid phase unstable and thus evoke a phase transition. quan- tum fluctuations account for additional vacancies and in- terstitials even at zero temperature and most interestingly the net vacancy density is altered in the supersolid phase. a potential shortcoming of the rpa is the lack of knowl- edge of the effective hamiltonian and therefore the inter- nal and free energy. usually the free energy is needed to compute first order transition lines as the ground state is given by the lowest energy state. we bypassed this obsta- cle by deriving a clausius-clapeyron like equation which defines the first order superfluid to normal solid transi- tion line. the entropy which is an input parameter of this equation is calculated from the spin wave excitation spectrum. the jump in the specific heat across the sec- ond order superfluid to supersolid transition line reveals information about the nature of the transition. however the specific heat is the second derivative of the free en- ergy and thus the jump is smeared out by the numerical calculations. consequently we derived an equation which gives an optional estimation of the jump and is in good agreement with the numerical estimate. most important our theory predicts a net vacancy density which, in the supersolid phase is significantly different from thermal ac- tivation theory. in the normal solid phase the net vacancy density roughly follows the predictions of thermal acti- vation theory, although quantum mechanical effects give a measurable contribution. across the phase transition, however, in the supersolid phase the net vacancy density stays rather constant as t increases. references 1. a. f. andreev and i. m. lifshitz, zh. eksp. teor. fiz. 56,(1969) 2057 [jetp 29, (1969) 1107]. 2. g. v. chester, phys. rev. a 2, (1970) 256. 3. a. j. leggett, phys. rev. lett. 25, (1970) 1543. 4. p. c. ho, i. p. bindloss and j. m. goodkind, j. low temp. physics 109, (1997) 409. 5. e. kim, m.h.w. chan, nature 427, (2004) 225. 6. e. kim, m.h,w. chan, science 305, (2004) 1941. 7. y. aoki, j. c. graves, h. kojima, phys. rev. lett. 99, (2007) 015301. 8. day, j.r. beamish, j. nature 450, (2007) 853856. 9. p.w. anderson, nature phys. 3,(2007) 160. 10. a. stoffel and m. gulacsi, manuscript submitted to phys. rev. b. 11. a. stoffel and m. gulacsi, manuscript in preparation. 12. k.-s. liu, m.e. fisher, j. low. temp. phys. 10, (1973) 655. 13. h. matsuda, t. tsuneto, prog. theoret. phys. suppl. 46, (1970) 411. 14. t. matsubara and h. matsuda,progr. theoret. phys. 16, (1956) 569;17, (1957). 15. n.n. bogolyubov, s.v. tyablikov, doklady akad. nauk. s.s.s.r. 126, 53 (1959) [translation: soviet phys. -doklady 4,(1959) 604]. 16. s.v. tyablikov, ukrain. mat. yhur. 11, (1959) 287. 17. p.e. bloomfield and e.b. brown, phys. rev. b 22, (1980) 1353. 18. p. w. anderson, w. f. brinkman and david a. huse, science 310,(2005) 1164. 19. x. lin, a. clark and m. chan, nature 449,(2007) 1025.

0911.1699

review of quasi-elastic charge-exchange data in the nucleon-deuteron breakup reaction

the available data on the forward charge exchange of nucleons on the deuteron up to 2gev per nucleon are reviewed. the value of the inclusive nd->pnn/np->pn cross section ratio is sensitive to the fraction of spin-independent neutron-proton backward scattering. the measurements of the polarisation transfer in d(\vec{n},\vec{p})nn or the deuteron analysing power in p(\vec{d},pp)n in high resolution experiments, where the final nn or pp pair emerge at low excitation energy, depend upon the longitudinal and transverse spin-spin np amplitudes. the relation between these types of experiments is discussed and the results compared with predictions of the impulse approximation model in order to see what new constraints they can bring to the neutron-proton database.

introduction the charge exchange of neutrons or protons on the deuteron has a very long his- tory. the first theoretical papers that dealt with the subject seem to date from the beginning of 1950s with papers by chew [1, 2], gluckstein and bethe [3], and pomeranchuk [4]. the first two groups were strongly influenced by the measure- ments of the differential cross section of the d(n, p) reaction that were then being undertaken at ucrl by powell [5]. apart from coulomb effects, by charge sym- metry the cross section for this reaction should be the same as that for d(p, n). the spectrum of the emerging neutron in the forward direction here shows a very strong peaking for an energy that is only a little below that of the incident pro- ton beam. there was therefore much interest in using the reaction as a means of producing a good quality neutron beam up to what was then "high" energies, i.e., a few hundred mev. the theory of this proposal was further developed by watson [6], shmushkevich [7], migdal [8], and lapidus [9]. since we have recently reviewed the phenomenology of the d(n, p) and d(p, n) charge exchange [10], the theory will not be treated here in any detail. the aim of the present paper is rather to discuss the database of the existing inclusive and exclusive measurements and make comparisons with the information that is available from neutron-proton elastic scattering data. the proton and neutron bound in the deuterons are in a superposition of 3s1 and 3d1 states and their spins are parallel. on the other hand, if the four- momentum transfer t = −q2 between the incident neutron and final proton in the nd →p{nn} reaction is very small, the pauli principle demands that the two emerging neutrons be in the spin-singlet states 1s0 and 1d2. in impulse (single- scattering) approximation, we would then expect the transition amplitude to be proportional to a spin-flip isospin-flip nucleon-nucleon scattering amplitude times a form factor that represents the overlap of the initial spin-triplet deuteron wave function with that of the unbound (scattering-state) nn wave function. the peaking observed in the energy spectrum of the outgoing proton is due to the huge neutron-neutron scattering length, which leads to a very strong final state interaction (fsi) between the two neutrons. a detailed evaluation of the proton spectrum from the d(n, p)nn reaction would clearly depend upon the deuteron and nn wave functions, i.e., upon low energy nuclear physics. however, a major advance was made by dean [11, 12]. he showed that, if one integrated over all the proton energies, there was a closure sum rule where all the dependence on the nn wave function vanished. dσ dt  nd→p{nn} = (1 −f(q)) dσ dt  si np→pn + [1 −1 3f(q)] dσ dt  sf np→pn , (1.1) where f(q) is the deuteron form factor. here the neutron-proton differential cross section is split into two parts that represent the contribution that is independent 2 of any spin transfer (si) between the initial neutron and final proton and one where there is a spin flip (sf). if the beam energy is high, then in the forward direction q ≈0, f(0) = 1, and eq. (1.1) reduces to dσ dt  nd→p{nn} = 2 3 dσ dt  sf np→pn . (1.2) there are modifications to eq. (1.1) through the deuteron d-state though these do not affect the forward limit of eq. (1.2) [11, 12, 13]. as a consequence, the ratio rnp(0) = dσ dt  nd→p{nn} , dσ dt  np→pn = 2 3 dσ dt  sf np→pn , dσ dt  np→pn (1.3) is equal to two thirds of the fraction of spin flip in np →pn between the incident neutron and proton outgoing in the beam direction. it is because the ratio of two unpolarised cross sections can give information about the spin dependence of neutron-proton scattering that so many groups have made experimental studies in the field and these are discussed in section 3. of course, for this to be a useful interpretation of the cross section ratio the energy has to be sufficiently high for the dean sum rule to converge before any phase space limitations become impor- tant. the longitudinal momentum transfer must be negligible and terms other than the np →pn impulse approximation should not contribute significantly to the evaluation of the sum rule. although the strong nn fsi helps with these concerns, all the caveats indicate that eq. (1.3) would provide at best only a qualitative description of data at the lower energies. the alternative approach is not to use a sum rule but rather to measure the excitation energy in the outgoing dineutron or diproton with good resolution and then evaluate the impulse approximation directly by using deuteron and nn scattering wave functions, i.e., input information from low energy nuclear physics. this avoids the questions of the convergence of the sum rule and so might yield useful results down to lower energies. a second important feature of the d(p, n)pp reaction in these conditions is that the polarisation transfer between the initial proton and the final neutron is expected to be very large, provided that the excitation energy epp in the final two-proton system is constrained to be only a few mev [14, 15]. in fact the reaction has been used by several groups to furnish a polarised neutron beam [16, 17, 18] but also as a means to study neutron-proton charge exchange observables, as described in section 4. bugg and wilkin [13, 19] realised that in the small epp limit the deuteron tensor analysing powers in the p(⃗ d, {pp})n reaction should also be large and with a significant angular structure that was sensitive to the differences between the neutron-proton spin-flip amplitudes. this realisation provided an impetus for the study of high resolution p(⃗ d, {pp})n experiments that are detailed in section 5. 3 the inclusive (p, n) or (n, p) measurements of section 3 and the high resolution ones of sections 4 and 5 are in fact sensitive to exactly the same physics input. to make this explicit, we outline in section 2 the necessary np formalism through which one can relate the forward values of rnp or rpn, the polarisation transfer in d(⃗ n, ⃗ p )nn and the deuteron tensor analysing power in p(⃗ d, {pp})n in impulse approximation to the longitudinal and transverse polarisation transfer coefficients in neutron-proton elastic scattering. predictions for the observables are made there using an up-to-date phase shift analysis. data are available on the rnp and rpn parameters in, respectively, inclusive d(n, p)nn and d(p, n)pp reactions at energies that range from tens of mev up to 2 gev and the features of the individual experiments are examined in section 3, where the results are compared to the predictions of the phase shift analysis. polarisation transfer data have become steadily more reliable with time, with firmer control over the nn excitation energies and better calibrated polarisation measurements so that the data described in section 3 now extend from 10 mev up to 800 mev. four experimental programmes were devoted to the study of the cross sec- tion and tensor analysing powers of the p(⃗ d, {pp})n reaction using very different experimental techniques. their procedures are described in section 5 and the re- sults compared with the predictions of the plane wave impulse approximation. in general this gives a reasonable description of the data out to a three-momentum transfer of q ≈mπ by which point multiple scatterings might become important. these data are however only available in an energy domain where the neutron- proton database is extensive and reliable and the possible extensions are also outlined there. the comparison between the sum-rule and high resolution approaches is one of the subjects that is addressed in our conclusions of section 6. the consistency between the information obtained from the d(⃗ n, ⃗ p )nn and p(⃗ d, {pp})n reactions in the forward direction is striking and the belief is expressed that this must contribute positively to our knowledge of the neutron-proton charge exchange phenomenology. 2 neutron-proton and nucleon-deuteron observables we have shown that the input necessary for the evaluation of the forward charge exchange observables can be expressed as combinations of pure linearly indepen- dent np →np observables evaluated in the backward direction [10]. although the expressions are independent of the scattering amplitude representation, for our purposes it is simplest to use the results of polarisation transfer experiments. the nn formalism gives two series of polarisation transfer parameters that are mutu- ally dependent [20]. using the notation xsrbt for experiments with measured spin orientations for the scattered (s), recoil (r), beam (b), and target (t) particles, 4 we have either the polarisation transfer from the beam to recoil particles, dσ dt k0rb0 = 1 4tr  σ2rmσ1bm† , (2.1) or the polarisation transfer from the target to the scattered particle dσ dt ks00t = 1 4tr  σ1smσ2tm† . (2.2) here σ1s, σ1b, σ2t, and σ2r are the corresponding pauli matrices and m is the scattering matrix. the unpolarised invariant elastic scattering cross section dσ dt = π k2 dσ dω= 1 4tr  mm† , (2.3) where k is the momentum in the cm frame and t is the four-momentum transfer. a first series of parameters describes the scattering of a polarised neutron beam on an unpolarised proton target, where the polarisation of the final outgo- ing protons is measured by an analyser through a second scattering. the spins of the incident neutrons can be oriented either perpendicularly or longitudinally with respect to the beam direction, with the final proton polarisations being mea- sured in the same directions. at θcm = π there are two independent parameters, k0nn0(π) and k0ll0(π), referring respectively to the transverse (n) and longitudi- nal (l) directions. it was shown in ref. [10] that the forward d(n, p)n/p(n, p)n cross section ratio can be written in terms of these as rnp(0) = 1 6 {3 −2k0nn0(π) −k0ll0(π)} . (2.4) a second series of parameters describes the scattering of an unpolarised neu- tron beam on a polarised proton target, where it is the polarisation of the final outgoing neutron that is determined. this leads to the alternative expression for rpn(0): rnp(0) = 1 6 {3 −2kn00n(π) + kl00l(π)} , (2.5) where kn00n(π) = k0nn0(π) but kl00l(π) = −k0ll0(π). other equivalent relations are to be found in ref. [20] it cannot be stressed enough that the small angle (n, p) charge exchange on the deuteron is sensitive to the spin transfer from the incident neutron to the outgoing proton and not that to the outgoing neutron. the latter observables are called the depolarisation parameters d which, for example, are given in the case of a polarised target by dσ dt d0r0t = 1 4tr  σ2rmσ1tm† . (2.6) 5 if one were to evaluate instead of eq. (2.5) the combination rnp(0) = 1 6 {3 −2d0n0n(π) −d0l0l(π)} , (2.7) then one would get a completely independent (and wrong) answer. using the said sp07 phase shift solution at 100 mev one finds that rnp(0) = 0.60 while rnp(0) = 0.13. hence one has to be very careful with the statement that the np →np spin dependence in the backward direction is weak or strong. it depends entirely on which particles one is discussing. in plane wave impulse approximation, the one non-vanishing deuteron ten- sor analysing power in the p(d, {pp})n reaction in the forward direction can be expressed in terms of the same spin-transfer parameters, provided that the exci- tation energy in the pp system is very small such that it is in the 1s0 state [13, 10]: ann(0) = 2(k0ll0(π) −k0nn0(π)) 3 −k0ll0(π) −2k0nn0(π) * (2.8) in an attempt to minimise confusion, observables in the nucleon-deuteron sector will be labelled with capital letters and only carry two subscripts. in the same approximation, the longitudinal and transverse spin-transfer pa- rameters in the d(⃗ p,⃗ n)pp between the initial proton and the final neutron emerg- ing in the beam direction are similarly given by kll(0) = − 1 −3k0ll0(π) + 2k0nn0(π) 3 −k0ll0(π) −2k0nn0(π)  , knn(0) = − 1 + k0ll0(π) −2k0nn0(π) 3 −k0ll0(π) −2k0nn0(π)  . (2.9) independent of any theoretical model, these parameters are related by [14, 21] kll(0) + 2knn(0) = −1. (2.10) equally generally, in the 1s0 limit the forward longitudinal and transverse deuteron tensor analysing powers are trivially related; all(0) = −2ann(0) , (2.11) and these are in turn connected to the spin-transfer coefficients through [21] all(0) = −(1 + 3kll(0))/2 or ann(0) = −(1 + 3knn(0))/2. (2.12) we stress once again that, although eqs. (2.8,2.9) are model dependent, eqs. (2.10), (2.11), and (2.12) are exact if the final pp system is in the 1s0 state. the variation of the np backward elastic cross section with energy and the values of rnp(0), ann(0), and kll(0) have been calculated using the energy 6 table 1: values of the np backward differential cross section in the cm sys- tem dσ/dω, and in invariant normalisation dσ/dt. also shown are the forward d(n, p)n/p(n, p)n ratio rnp(0), the longitudinal polarisation transfer parameter kll(0) in the d(⃗ p,⃗ n)pp reaction, and the deuteron analysing power ann(0) in the p(⃗ d, {pp})n reaction at the same energy per nucleon. these have all been eval- uated from the plane wave impulse approximation using the energy dependent psa of arndt et al., solution sp07 [22]. tn dσ/dω dσ/dt rnp(0) kll(0) ann(0) (gev) mb/sr mb/(gev/c)2 0.010 78.74 52728 0.404 -0.370 -0.027 0.020 42.92 14371 0.433 -0.273 0.045 0.030 29.84 6661 0.466 -0.167 0.125 0.040 23.56 3944 0.498 -0.085 0.186 0.050 20.11 2693 0.525 -0.030 0.227 0.060 18.04 2013 0.547 0.000 0.250 0.070 16.71 1599 0.565 0.014 0.260 0.080 15.81 1323 0.579 0.014 0.261 0.090 15.17 1129 0.591 0.006 0.255 0.100 14.68 983 0.600 -0.008 0.244 0.120 13.98 780 0.613 -0.048 0.214 0.150 13.27 592 0.627 -0.118 0.162 0.200 12.46 417 0.639 -0.231 0.077 0.250 11.88 318 0.645 -0.327 0.005 0.300 11.45 255 0.645 -0.405 -0.054 0.350 11.19 214 0.644 -0.472 -0.104 0.400 11.02 184 0.639 -0.530 -0.148 0.450 10.88 162 0.631 -0.582 -0.186 0.500 10.62 142 0.621 -0.630 -0.223 0.550 10.10 123 0.608 -0.678 -0.259 0.600 9.45 105 0.596 -0.726 -0.295 0.650 9.07 93.4 0.588 -0.762 -0.321 0.700 8.96 85.8 0.586 -0.773 -0.330 0.750 8.95 79.9 0.588 -0.769 -0.327 0.800 8.93 74.7 0.592 -0.761 -0.321 0.850 8.98 69.9 0.596 -0.754 -0.315 0.900 8.81 65.5 0.601 -0.748 -0.311 0.950 8.73 61.5 0.605 -0.744 -0.308 1.000 8.65 57.9 0.609 -0.740 -0.305 1.050 8.57 54.7 0.613 -0.737 -0.303 1.100 8.50 51.7 0.616 -0.735 -0.302 1.150 8.44 49.1 0.620 -0.735 -0.301 1.200 8.40 46.8 0.623 -0.736 -0.302 1.250 8.38 44.9 0.626 -0.739 -0.304 1.300 8.39 43.2 0.629 -0.740 -0.308 7 dependent gw/vpi psa solution sp07 [22] and are listed in table 1. the relations between the observables used in refs. [22] and [20] are to be found in the said program. the gw/vpi psa for proton-proton scattering can be used up to 3.0 gev but, according to the authors, the predictions are at best qualitative above 2.5 gev [22]. because this is an energy dependent analysis, one cannot use the said program to estimate the errors of any observable. although the equiv- alent psa for neutron-proton scattering was carried out up to 1.3 gev, very few spin-dependent observables have been measured above 1.1 gev. let us summarise the present status of the np database at intermediate en- ergies. about 2000 spin-dependent np elastic scattering data points, involving 11 to 13 independent observables, were determined at saturne 2 over large angular intervals mainly between 0.8 and 1.1 gev [23, 24]. a comparable amount of np data in the region from 0.5 to 0.8 gev was measured at lampf [25] and in the energy interval from 0.2 to 0.56 gev at psi [26]. the triumf group also contributed significantly up to 0.515 gev [27]. the saturne 2 and the psi data were together sufficient, not only to imple- ment the psa procedure, but also to perform a direct amplitude reconstruction at several energies and angles. it appears that the spin-dependent data are more or less sufficient for this procedure at the lower energies, whereas above 0.8 gev there is a lack of np differential cross section data, mainly at intermediate angles. 3 measurements of unpolarised quasi-elastic charge-exchange observables 3.1 the (n, p) experiments the first measurement of the d(n, p) differential cross section was undertaken at ucrl by powell in 1951 [5]. these data at 90 mev were reported by chew [2], though only in graphical form, and from this one deduces that rnp(0) = 0.40 ± 0.04. a year later cladis, hadley, and hess, working also at the ucrl synchrocyclotron, published data obtained with the 270 mev neutron beam [28]. their value of 0.71 ± 0.02 for the ratio of their own deuteron/hydrogen data is clearly above the permitted limit of 2/3 by more than the claimed error bar. this may be connected with the very broad energy spectrum of the incident neutron beam, which had a fwhm ≈100 mev. at the dubna synchrocyclotron the first measurements were carried out by dzhelepov et al. [29, 30] in 1952 - 1954 with a 380 mev neutron beam. somewhat surprisingly, the authors considered that their result, rnp(0) = 0.20 ± 0.04, to be compatible with the ucrl measurements [5, 28]. in fact, later more refined experiments [31] showed that the dzhelepov et al. value was far too low and it should be discarded from the database. 8 at the end of that decade larsen measured the same quantity at lrl berkeley at the relatively high energy of 710 mev and obtained rnp(0) = 0.48 ± 0.08 [32]. however, no previous results were mentioned in his publication. in his contribution to the 1962 cern conference [33], dzhelepov presented the angular dependence of rnp(θ) at 200 mev. although he noted that the authors of the experiment were yu. kazarinov, v. kiselev and yu. simonov, no reference was given and we have found no publication. reading the value from a graph, one obtains rnp(0) = 0.55 ± 0.03. one advantage of working at very low energies, as was done in moscow [34], is that one can obtain a neutron beam from the 3h(d, n)4he reaction that is almost monochromatic. at 13.9 mev there is clearly no hope at all of fulfilling the conditions of the dean sum rule so that the value given in table 2 was obtained with a very severe cut. instead, the group concentrated on the final state interaction region of the two neutrons which, in some ways, is similar to the approach of the high resolution experiments to be discussed in section 5. by comparing the data with the d(p, n)pp results of ref. [35], it was possible to see the effects of the coulomb repulsion when the two protons were detected in the fsi peak. though the value obtained by measday [36] at 152 mev has quite a large error bar, rnp(0) = 0.65 ± 0.10, this seems to be mainly an overall systematic effect because the variation of the result with angle is very smooth. these results show how rnp(θ) approaches two thirds as the momentum transfer gets large and the pauli blocking becomes less important. the 794 mev measurement from lampf [37] is especially detailed, with very fine steps in momentum transfer. extrapolated to t = 0 it yields rnp(0) = 0.56± 0.04. however, the authors suggest that the true value might be a little higher than this due to the cut that they imposed upon the lowest proton momentum considered. by far the most extensive d(n, p)nn data set at medium energies was obtained by the freiburg group working at psi, the results of which are only available in the form of a diploma thesis [31]. however, the setup used by the group for neutron- proton backward elastic scattering is described in ref. [38]. the psi neutron beam was produced through the interaction of an intense 589 mev proton beam with a thick nuclear target. this delivered pulses with widths of less than 1 ns and bunch spacings of 20 or 60 ns. combining this with a time-of-flight path of 61 m allowed for a good selection of the neutron momentum, with an average resolution of about 3% fwhm. data were reported at fourteen neutron energies from 300 to 560 mev, i.e., above the threshold for pion production so that the results could be normalised using the cross section for np →dπ0, which was measured in parallel [38]. over this range rnp(0) showed very little energy dependence, with an average value of 0.62 ± 0.01, which is quite close to the upper limit of 2/3. at the jinr vblhe dubna a high quality quasi-monoenergetic polarised neutron beam was extracted in 1994 from the synchrophasotron for the purposes 9 of the ∆σl(np) measurements [39, 40], though this accelerator was stopped in 2005. polarised deuterons are not yet available from the jinr nuclotron but, on the other hand, intense unpolarised beams with very long spills could be obtained from this machine. since the final ∆σl set-up included a spectrometer, the study of the energy dependence of rnp(0) could be extended up to 2.0 gev through the measurement of seven points [41]. that at 550 mev agrees very well with the neighbouring psi point [31] while the one at 800 mev is consistent with the lampf measurement [37]. since the values of rnp(0) above 1 gev could not have been reliably predicted from previous data, the nuclotron measurements in the interval 1.0 < tn < 2 gev can be considered to be an important achievement in this field. it would be worthwhile to complete these experiments by measurements in smaller energy steps in order to recognise possible anomalies or structures. it is also desirable to extend the investigated interval up to the highest neutron energy at the nuclotron (≈3.7 gev) since such measurements are currently only possible at this accelerator. the data on rnp(0) from the d(n, p)nn experiments discussed above are sum- marised in table 2, where the kinetic energy, facility, year of publication, and reference are also listed. several original papers show the values of the angu- lar distribution of the charge exchange cross section on the deuteron. in such cases, the rnp(0) listed here were obtained using the predictions for the free for- ward np charge-exchange cross sections taken from the said program (solution sp07) [22]. these values are shown in table 1. 3.2 the (p, n) experiments although high quality proton beams have been available at many facilities, the evaluation of a rpn(0) ratio from d(p, n)pp experiments requires the division of this cross section by that for the charge exchange on a nucleon target. where necessary, we have done this using the predictions of the sp07 said solution [22] given in table 1. given also the difficulties in obtaining absolute normalisations when detecting neutrons, we consider that in general the results obtained using neutron beams are likely to be more reliable. the low energy data of wong et al. [35] at 13.5 mev do show evidence of a peak for the highest momentum neutrons but this is sitting on a background coming from other breakup mechanisms that are probably not associated with charge exchange. the value given in table 3 without an error bar is therefore purely indicative. in 1953 hofmann and strauch [42], working at the harvard university ac- celerator, published results on the interaction of 95 mev protons with several nuclei and measured the d(p, n) reaction for the first time. an estimation of the charge-exchange ratio from the plotted data gives rpn(0) = 0.48 ± 0.03. the measurements at 30 and 50 mev were made using the time-of-flight fa- cility of the rutherford laboratory (rhel) proton linear accelerator [43]. the 10 table 2: the rnp(0) data measured using the d(n, p)nn reaction. the total estimated uncertainties quoted do not take into account the influence of the different possible choices on the cut on the final proton momentum. tn rnp(0) facility year ref. (mev) 13.9 0.19 moscow 1965 [34] 90.0 0.40 ± 0.04 ucrl 1951 [5] 152.0 0.65 ± 0.10 harvard 1966 [36] 200.0 0.55 ± 0.03 jinr dlnp 1962 [33] 270.0 0.71 ± 0.02 ucrl 1952 [28] 299.7 0.65 ± 0.03 psi 1988 [31] 319.8 0.64 ± 0.03 psi 1988 [31] 339.7 0.64 ± 0.03 psi 1988 [31] 359.6 0.63 ± 0.03 psi 1988 [31] 379.6 0.64 ± 0.03 psi 1988 [31] 380.0 0.20 ± 0.04 inp dubna 1955 [29] 399.7 0.61 ± 0.03 psi 1988 [31] 419.8 0.62 ± 0.03 psi 1988 [31] 440.0 0.63 ± 0.03 psi 1988 [31] 460.1 0.61 ± 0.03 psi 1988 [31] 480.4 0.61 ± 0.03 psi 1988 [31] 500.9 0.59 ± 0.03 psi 1988 [31] 521.1 0.60 ± 0.03 psi 1988 [31] 539.4 0.62 ± 0.03 psi 1988 [31] 550.0 0.59 ± 0.05 jinr vblhe 2009 [41] 557.4 0.63 ± 0.03 psi 1988 [31] 710.0 0.48 ± 0.08 lrl 1960 [32] 794.0 0.56 ± 0.04 lampf 1978 [37] 800.0 0.55 ± 0.02 jinr vblhe 2009 [41] 1000 0.55 ± 0.03 jinr vblhe 2009 [41] 1200 0.55 ± 0.02 jinr vblhe 2009 [41] 1400 0.58 ± 0.04 jinr vblhe 2009 [41] 1800 0.57 ± 0.03 jinr vblhe 2009 [41] 2000 0.56 ± 0.05 jinr vblhe 2009 [41] 11 neutron spectrum, especially at 30 mev, does not show a clear separation of the charge-exchange impulse contribution from other mechanisms and the dean sum rule is far from being saturated. the same facility was used at the higher energies of 95 and 144 mev, where the target was once again deuterated polythene [44]. this allowed the spectrum to be studied up to a proton-proton excitation en- ergy epp ≈14 mev when neutrons from reactions on the carbon in the target contributed. it was claimed that the cross sections obtained had an overall nor- malisation uncertainty of about ±10% and that the impulse approximation could describe the data within this error bar. the highest energy (p, n) data were produced at lampf [45], where the charge-exchange peak was clearly separated from other mechanisms, including pion production, and the conditions for the use of the dean sum rule were well satisfied. their high value of rpn(0) = 0.66 ± 0.08 at 800 mev would be reduced to 0.61 if the np data of table 1 were used for normalisation instead of those available in 1976. the approach by the ucl group working at tp = 135 mev at harwell was utterly different to the others. they used a high-pressure wilson cloud chamber triggered by counters, which resulted in a large fraction of the 1740 photographs containing events [46]. this led to the 1048 events of proton-deuteron collisions that were included in the final data analysis. instead of detecting the neutron from the d(p, n)pp reaction, the group measured both protons. in a sense therefore the experiment is similar to that of the dubna bubble chamber group [47], but in inverted kinematics. due to the geometry of the counter selection system, the apparatus was blind to protons that were emitted in a cone of laboratory angles θlab < 10◦with energies above 6 mev. although the corrections for the associate losses are model dependent, these should not affect the neutrons emerging at small angles and the results were integrated down to a neutron kinetic energy that was 8 mev below the maximum allowed. the differential cross sections were compared to the plane wave impulse approximation calculations of castillejo and singh [48]. the results from the various d(p, n)pp experiments are summarised in table 3. 3.3 the unpolarised dp →ppn reaction in principle, far more information is available if the two final protons are measured in the deuteron charge exchange reaction and not merely the outgoing neutron. this has been achieved by using a beam of deuterons with momentum 3.35 gev/c incident on the dubna hydrogen bubble chamber. because of the richness of the data contained, the experiment has had a very long history with several reanalyses [49, 50, 51, 52, 47]. of the seventeen different final channels studied, the largest number of events (over 105) was associated with deuteron breakup. these could be converted very reliably into cross sections by comparing the sum over all channels with 12 table 3: the rpn(0) data measured using the d(p, n)pp reaction. the total estimated uncertainties quoted do not take into account the influence of the different possible choices on the cut on the final neutron momentum. tkin rpn(0) facility year ref. (mev) 13.5 0.18 livermore 1959 [35] 30.1 0.14 ± 0.04 rhel 1967 [43] 50.0 0.24 ± 0.06 rhel 1967 [43] 95.0 0.48 ± 0.03 harvard 1953 [42] 94.7 0.59 ± 0.03 harwell 1967 [44] 135.0 0.65 ± 0.15 harwell 1965 [46] 143.9 0.60 ± 0.06 harwell 1967 [44] 647.0 0.60 ± 0.08 lampf 1976 [45] 800.0 0.66 ± 0.08 lampf 1976 [45] the known total cross section. corrections were made for the loss of elastic dp scattering events at very small angles. the dp →ppn events were divided into two categories, depending upon whether it was the neutron or one of the two protons that had the lowest momentum in the deuteron rest frame. this identification of the charge-retention or charge-exchange channels is expected to be subject to little ambiguity for small momentum transfers. with this definition, the total cross section for deuteron charge exchange was found to be 5.85 ± 0.05 mb. the big advantage of the bubble chamber approach is that one can check many of the assumptions that are made in the analysis. the crucial one is, of course, the separation into the charge-exchange and charge-retention events. in the latter case the distribution of "spectator" momenta psp falls smoothly with psp but in the charge-exchange sample there is a surplus of events for psp ≳200 mev/c that may be associated with the virtual production of a ∆(1232) that de-excites through ∆n →pp. perhaps a fifth of the charge-exchange cross section could be due to this mechanism [51] but, fortunately, such events necessarily involve significant momentum transfers and would not influence the extrapolation to q = 0. after making corrections for events that have larger opening angles [47], the data analysis gives a value of dσ dt (dp →{pp}n)     t=0 = 2 3 dσsf dt (dp →{pp}n)     t=0 = 30 ± 4 mb/(gev/c)2, (3.1) where σsf is the cross section corresponding to the spin flip from the initial proton to the final neutron and the 2/3 factor comes from the dean sum rule. some of 13 the above error arises from the estimation of the effects of the wide angle proton pairs and in the earlier publication of the group [52], where the same data set was treated somewhat differently, a lower value of 25±3 mb/(gev/c)2 was obtained. the dubna bubble chamber measurement can lead to a relatively precise value of the average of the spin-spin amplitudes-squared. using eq. (3.1) one obtains very similar information to that achieved with the high resolution dp → {pp}n measurements to be discussed in section 5 and with very competitive error bars. on the other hand, if the primary aim is to derive estimates for the spin- independent contribution to the forward np charge-exchange cross section, then it loses some of the simplicity and directness of the d(n, p)nn/p(n, p)n comparison. this is because one has to evaluate the ratio of two independently measured numbers, each of which has its own normalisation uncertainty. the problem is compounded by the fact that, as we have seen from the direct (n, p) measurements of rnp(0), the contribution of the spin-independent amplitude represents only a small fraction of the total. in the earlier publications by the dubna group, the necessary normalisation denominator was taken from the elastic neutron-proton scattering measurements of shepard et al. at the pennsylvania proton accelerator [53]. these were made at sixteen energies and over wide angular ranges. however they disagreed strongly with all other existing np data, not only in the absolute values, but also in the shapes of angular distributions. this problem was already apparent at low energies, starting 182 mev. as a result, these data have long been discarded by physicists working in the field and they have been removed from phase shift analysis databases, e.g. from the saclay-geneva psa in 1978 [54]. a much more reliable np →pn data set was provided by the er54 group of bizard et al. [55], numerical values of which are to be found in refs. [56, 57]. fit- ting these data with two exponentials, gives a forward cross section of dσ/dt|t=0 = 54.7 ± 0.2 mb/(gev/c)2, which the dubna group used in their final publica- tion [47]. it is very different from the shepard et al. result [53] of 36.5 ± 1.4 mb/(gev/c)2, which the group quoted in their earlier work [52]. this dif- ference, together with the changed analysis corrections, accounts for the diverse values of rnp(0) from the same experiment that are given in table 4. 3.4 data summary the values of rnp(0) and rpn(0) from tables 2 and 3 are shown in graphical form in fig. 1, with only the early dubna point [29] being omitted. the p(d, 2p) values in table 4 represent the results of increased statistics and a different analysis and only the point from the last publication is shown [47]. the first comparison of such data with np phase shift predictions was made in 1991 in a thesis from the freiburg group [58], where both the gw/vpi [59] and saclay-geneva [54] were studied. the strong disagreement with the results of the psi measurements [31] was due to the author misinterpreting the relevant 14 table 4: summary of the available experimental data on the rnp(0) ratio mea- sured with the dubna bubble chamber using the dp →{pp} n reaction. the kinetic energy quoted here is the energy per nucleon. the error bars reflect both the statistical and systematic uncertainties. although the data sets are basically identical, the 2008 analysis [47] is believed to be the most reliable. tkin rnp(0) facility year ref. (mev) 977 0.43 ± 0.22 jinr vblhe 1975 [49] 977 0.63 ± 0.12 jinr vblhe 2002 [52] 977 0.55 ± 0.08 jinr vblhe 2008 [47] quantity as being rnp(0) of eq. (2.7) instead of rnp(0) of eq. (2.4). the correct predictions from the current gw/vpi phase shift analysis ob- tained on the basis of eq. (2.4) are shown in fig. 1 up to the limit of their validity at 1.3 gev. the small values of rnp(0) at low energies is in part due to the much greater importance of the spin-independent contribution there, as indicated by the phase shift predictions. there are effects arising also from the limited phase space but, when they are included (dashed curve), they change the results only marginally. a much greater influence is the cut that authors have to put onto the emerging neutron or proton to try to isolate the charge-exchange contribution from that of other mechanisms. this procedure becomes far more ambiguous at low energies when relatively severe cuts have to be imposed. the data in fig. 1 seem to be largest at around the lowest psi point [31], where they get close to the allowed limit of 0.67. in fact, if the glauber shadow- ing effect is taken into account [60], this limit might be reduced to perhaps 0.63. as already shown by the phase shift analysis, the contribution from the spin- independent term is very small in this region. on the other hand, in the region from 1.0 to 1.3 gev the phase shift curve lies systematically above the experi- mental data. since the conditions for the dean sum rule seem to be best satisfied at high energies, this suggests that the said solution underestimates the spin- independent contribution above 1 gev. it has to be noted that the experimental np database is far less rich in this region. 15 figure 1: experimental data on the rnp(0) ratio taken in the forward direction. the closed circles are from the (n, p) data of table 2, the open circles from the (p, n) data of table 3, and the cross from the (d, 2p) datum of table 4. these results are compared to the predictions of eq. (2.4) using the current said solution [22], which is available up to a laboratory kinetic energy of 1.3 gev. the dashed curve takes into account the limited phase space available at the lower energies. 4 polarisation transfer measurements in d(⃗ p, ⃗ n)pp it was first suggested by phillips [14] that the polarisation transfer in the charge exchange reaction d(⃗ p,⃗ n)pp should be large provided that the excitation energy epp in the final pp system is small. under such conditions the diproton is in the 1s0 state so that there is a spin-flip transition from a jp = 1+ to a 0+ configuration of the two nucleons. this spin-selection argument is only valid for the highest neutron momentum since, as epp increases, p- and higher waves enter and the polarisation signal reduces [15]. nevertheless, the reaction has been used successfully by several groups to produce polarised neutron beams [16, 17, 18]. in the 1s0 limit, there are only two invariant amplitudes in the forward di- rection and, as pointed out in eq. (2.10), the transverse and longitudinal spin- transfer coefficients knn and kll are then related by kll(0) + 2knn(0) = −1. 16 one obvious experimental challenge is to get sufficient energy resolution through the measurement of the produced neutron to guarantee that the residual pp sys- tem is in the 1s0 state. the other general problem is knowing sufficiently well the analysing power of the reaction chosen to measure the final neutron polarisation. some of the earlier experiments failed on one or both of these counts. the first measurement of knn(0) for d(⃗ p,⃗ n)pp seems to have been performed at the rochester synchrocyclotron at 200 mev in the mid 1960s [61]. a neutron polarimeter based upon pn elastic scattering was used, with the analysing power being taken from the existing nucleon-nucleon phase shifts. however, the resolu- tion on the final proton energies was inadequate for our purposes, with an energy spread of 12 mev fwhm coming from the primary beam and the finite target thickness. a similar experiment was undertaken at 30 and 50 mev soon afterwards at the rhel proton linear accelerator [62]. the results represent averages over the higher momentum part of the neutron spectra. a liquid 4he scintillator was used to measure the analysing power in neutron elastic scattering from 4he, though the calibration standard was uncertain by about 8%. although falling largely outside the purpose of this review, it should be noted that there were forward angle measurements of knn(0) at the triangle uni- versities nuclear laboratory at five very low energies, ranging from 10.6 to 15.1 mev [63]. this experiment also used a 4he polarimeter that in addition served to measure the neutron energy with a resolution of the order of 200 kev. although all the data at the lowest epp were consistent with knn(0) ≈−0.2, a very strong dependence on the pp excitation energy was found, with knn(0) passing through zero in all cases for epp < 2 mev. hence, after unfolding the resolution it is likely that the true value at epp = 0 is probably slightly more negative than −0.2. the strong variation with epp is reproduced in a simple implementation of the faddeev equations that was carried out, though without the inclusion of the coulomb interaction [64]. the rcnp experiment at 50, 65, and 80 mev used a deuterated polyethy- lene target [65]. the calibration of the neutron polarimetry was on the basis of the charge exchange from 6li to the 0+ ground state of 6be, viz 6li(⃗ p,⃗ n)6begs. although at the time the polarisation transfer parameters for this reaction had not been measured, they were assumed to be the same as for the transition to the first excited (isobaric analogue) state of 6li. this was subsequently shown to be a valid assumption by a direct measurement of neutron production with a 6li target [66]. on the other hand, the resolution in epp was of the order of 6 mev, which arose mainly from the measurement of the time of flight over 7 m. as a consequence, the authors could not identify clearly the strong dependence of knn(0) on epp that was seen in experiments where the neutron energy was better measured [63, 67, 68]. such a dependence would have been more evident in the data if there had not been a contribution at higher epp from the 12c in the target. 17 the most precise measurements of the polarisation transfer parameters at low energies were accomplished in experiments at psi at 56 and 70 mev [67, 68]. one of the advantages of their setup was the time structure of the psi injector cyclotron, where bursts of width 0.7 ns, separated by 20 ns, were obtained at 72 mev, increasing to about 1.2 ns, separated by 70 ns, at 55 mev. this allowed the production of a near-monoenergetic neutron beam for use in other low energy experiments [69]. beams with a good time structure were also obtained after acceleration of the protons to higher energies and these were necessary for the measurements of rnp(0) [31]. the target size was small compared to the time-of-flight path of ≈4.3 m in the initial experiment [67] so that the total timing resolution of typically 1.4 ns led to one in epp of a few mev. the polarisation of the proton beam was very well known and that of the recoil neutron was measured by elastic scattering of the neutrons from 4he. apart from small coulomb corrections, the analysing power of 4he(⃗ n, n)4he should be identical to that of the proton in 4he(⃗ p, p)4he, for which reliable data existed. the results at both 54 and 71 mev showed that the polarisation transfer parameters change very strongly with the measured neutron energy and hence with epp. this must go a long way to explain the anomalous results found by the rcnp group [65]. at 54 mev both knn and kll were measured and, when extrapolated to the 1s0 limit of maximum neutron energy, the values gave kll +2knn = (−0.1164±0.013)+2(−0.4485±0.011) = −1.013±0.026 , (4.1) in very satisfactory agreement with the 1s0 identity of eq. (2.10). the subsequent psi measurement at 70.4 mev made significant refinements in two separate areas [68]. the extension of the flight path to 11.6 m improved the resolution in the neutron energy by about a factor of three, which allowed a much more detailed study to the epp dependence of knn to be undertaken. the neutron polarimeter used the p(⃗ n, p)n reaction and an independent calibration was carried out by studying the 14c(⃗ p,⃗ n)14n2.31 reaction in the forward direction. the 2.31 mev level in question is the first excited state of 14n, which is the isospin analogue of the jp = 0+ ground state of 14c. in such a case there can be no spin flip and the polarisation of the recoil neutron must be identical to that of the proton beam. in order to isolate this level cleanly, the neutron flight path was increased further to 16.4 m for this target. the results confirmed those of the earlier experiment [67] and, in particular, showed that even in the forward direction knn(0) varied significantly with the energy of the detected neutron. the dependence of the parameterisation of the results on epp is shown in fig 2. near the allowed limit, epp is equal to the deviation of the neutron energy from its kinematically allowed maximum. a strong variation of the polarisation transfer parameter with epp is predicted when using the faddeev equations [68, 70], though these do not give a perfect de- scription of the data. these calculations represent full multiple scattering schemes 18 figure 2: fit to the measured values of knn of the d(⃗ p,⃗ n)pp reaction in the forward direction at a beam energy of 70.4 mev as a function of the excitation energy in the pp final state [68]. with all binding corrections and off-shell dependence of the nucleon-nucleon am- plitudes. nevertheless it is important to note that the kll(0) prediction for very low epp is quite close to that of the plane wave impulse approximation. on the other hand, the fact that both the data and a sophisticated theoretical model show the strong dependence on epp brings into question the hope that the 1s0 proton-proton final state remains dominant in the forward direction for low beam energies. this is one more reason to doubt the utility of the dean sum rule to estimate rnp(0) at low energies. the validity of the plane wave impulse approximation for the unpolarised d(p, n)pp reaction at 135 mev has also been tested at iucf [71]. the conclusions drawn here are broadly similar to those from an earlier study at 160 mev [72]. in the forward direction the plane wave approach reproduces the shape of the dependence on epp out to at least 5 mev, though the normalisation was about 20% too low. on the other hand, the group evaluated the model using an s-state hulth ́ en wave function for the deuteron and so it is not surprising that some renormalisation was required. the epp dependence follows almost exclusively from the pp wave function, which was evaluated realistically. the comparison with more sophisticated faddeev calculations was, of course, hampered by the difficulty of including the coulomb interaction, which is particularly important 19 for low epp [14]. the values of knn(0) obtained at iucf at 160 mev [72] show a weaker dependence on epp than that found in the experiments below 100 mev [67, 68]. nevertheless these data do indicate that the influence of p-waves in the final pp system is not negligible for epp ≈10 mev. the early measurements of knn(0) and kll(0) at lampf [17, 18] were hampered by the poor knowledge of the neutron analysing power in ⃗ np elas- tic scattering that was used in the polarimeter. this was noted by bugg and wilkin [13], who pointed out that, although the data were taken in the forward direction and with good resolution, they failed badly to satisfy the identity of eq. (2.10). they suggested that both polarisation transfer parameters should be renormalised by overall factors so as to impose the condition. in view of this argument and the results of the subsequent lampf experiment [73], the values reported from these experiments in table 5 have been scaled such that kll(0) + 2knn(0) = −0.98 (to allow for some dilution from the p-waves in the pp system) and the error bars increased a little to account for the uncertainty in this procedure. the above controversy regarding the values of the forward polarisation trans- fer parameters in the 500 – 800 mev range was conclusively settled by a subse- quent lampf experiment by mcnaughton et al. in 1992 [73]. following an idea suggested by bugg [74], the principle was to produce a polarised neutron beam through the d(⃗ p,⃗ n)pp reaction, sweep away the charged particles with a bend- ing magnet, and then let the polarised neutron beam undergo a second charge exchange through the d(⃗ n, ⃗ p )nn reaction. by charge symmetry, the values of kll(0) for the two reactions are the same and, if the energy loss in both cases is minimised, the beam polarisation pb and final proton polarisation pp are related by pp = [kll(0)]2 pb . (4.2) the beauty of this techniques is that only proton polarisations had to be measured with different but similarly calibrated instruments. also, because the square occurs in eq. (4.2), the errors in the evaluation of kll are reduced by a factor of two. the energy losses were controlled by time-of-flight measurements and very small corrections were made for the fact that the two reactions happened at slightly different beam energies. the overall precision achieved in this experiment was typically 3% and the results clearly demonstrated that there had been a significant miscalibration in much of the earlier lampf neutron polarisation standards. the group also sug- gested clear renormalisations of the measured polarisation transfer parameters. since several of the authors of the earlier papers also signed the mcnaughton work, this lends a seal of approval to the procedure. the longitudinal polarisation transfer in the forward direction was measured later at lampf at 318 and 494 mev [75] with neutron flight paths of, re- 20 spectively, 200 and 400 m so that the energy resolution was typically 750 kev (fwhm). this allowed the authors to use the 14c(⃗ p,⃗ n)14n2.31 reaction to cal- ibrate the neutron polarimeter, a technique that was taken up afterwards at psi [68]. including these results, we now have reliable values of either kll(0) or knn(0) from low energies up to 800 mev. 21 4.1 data summary the values of knn(0) and kll(0) measured in the experiments discussed above are presented in table 5 and shown graphically in fig. 3. the results are com- pared in the figure with the predictions tabulated in table 1 of the pure 1s0 plane wave impulse approximation of eq. (2.9) that used the said phase shifts [22] as input. wherever possible the data are extrapolated to epp = 0. this is especially important at low energies and, if this causes uncertainties or there are doubts in the calibration standards, we have tried to indicate such data with open symbols, leaving closed symbols for cases where we believe the data to be more trustworthy. figure 3: forward values of the longitudinal and transverse polarisation transfer parameters kll(0) and knn(0) in the d(⃗ p,⃗ n)pp reaction as functions of the proton kinetic energy tn. in general we believe that greater confidence can be placed in the data represented by closed symbols, which are from refs. [73] (stars), [75] (circles), [72] (triangle), [67, 68] (squares), and the average of the five tunl low energy points [63] (inverted triangle). the open symbols come from refs. [62] (diamonds), [65] (triangle), [61] (circle), [18] (crosses), and [17] (star), with the latter two being renormalised as explained in table 5. the curve is the plane wave 1s0 prediction of eq. (2.8), as tabulated in table 1. the impulse approximation curve gives a semi-quantitative description of all the data, especially the more "reliable" results. at low energies we expect that this approach would be at best indicative but it is probably significant that the curve falls below the mcnaughton et al. results [73] in the 500 to 800 mev range, where the approximation should be much better. it is doubtful whether the glauber correction [60, 13] can make up this difference and this suggests that the current values of the said neutron-proton charge-exchange amplitudes [22] might require some slight modifications in this energy region. similar evidence is 22 found from the measurements of the deuteron analysing power, to which we now turn. 23 table 5: measured values of the longitudinal and transverse polarisation transfer parameters for the d(⃗ p,⃗ n)pp reaction in the forward direction. the total esti- mated uncertainties quoted do not take into account the influence of the different possible choices on the cut on the final neutron momentum. data marked ∗ have been renormalised to impose kll(0) + 2knn(0) = −0.98 and the error bar increased slightly. tn kll(0) knn(0) facility year ref. (mev) 10.6 - −0.17 ± 0.06 tunl 1980 [63] 12.1 - −0.20 ± 0.07 tunl 1980 [63] 13.1 - −0.14 ± 0.05 tunl 1980 [63] 14.1 - −0.12 ± 0.06 tunl 1980 [63] 15.1 - −0.22 ± 0.09 tunl 1980 [63] 30 - −0.13 ± 0.03 rhel 1969 [62] 50 - −0.23 ± 0.07 rhel 1969 [62] 50 - −0.27 ± 0.05 rcnp 1986 [65] 54 −0.116 ± 0.013 −0.449 ± 0.011 psi 1990 [67] 65 - −0.31 ± 0.03 rcnp 1986 [65] 70.4 - −0.457 ± 0.011 psi 1999 [68] 71 - −0.480 ± 0.013 psi 1990 [67] 80 - −0.37 ± 0.04 rcnp 1986 [65] 160 - −0.43 ± 0.04 iucf 1987 [72] 203 - −0.27 ± 0.11 rochester 1987 [61] 305 −0.411 ± 0.010 - lampf 1992 [73] 318 −0.41 ± 0.01 - lampf 1993 [75] 485 −0.579 ± 0.011 - lampf 1992 [73] 494 −0.59 ± 0.01 - lampf 1993 [75] 500 −0.60 ± 0.03∗ −0.19 ± 0.04∗ lampf 1985 [18] 635 −0.686 ± 0.012 - lampf 1992 [73] 650 −0.79 ± 0.03∗ −0.10 ± 0.03∗ lampf 1985 [18] 722 −0.717 ± 0.013 - lampf 1992 [73] 788 −0.720 ± 0.017 - lampf 1992 [73] 800 −0.68 ± 0.05∗ −0.15 ± 0.04∗ lampf 1981 [17] 800 −0.78 ± 0.04∗ −0.10 ± 0.04∗ lampf 1985 [18] 24 5 deuteron polarisation studies in high resolu- tion (⃗ d, 2p) experiments we have pointed out through eq. (2.12) that in the 1s0 limit the deuteron (⃗ d, 2p) tensor analysing power in the forward direction can be directly evaluated in terms of the (⃗ p,⃗ n) polarisation transfer coefficient. therefore, instead of measuring beam and recoil polarisations, much of the same physics can be investigated by measuring the analysing power with a polarised deuteron beam without any need to detect the polarisation of the final particles. this is the approach advocated by bugg and wilkin [19, 13]. unlike the sum-rule methodology applied by a dubna group [47], only the small part of the p(⃗ d, 2p)n final phase space where epp is at most a few mev needs to be recorded. for this purpose one does not need the large acceptance offered by a bubble chamber and four separate groups have undertaken major programmes using different electronic equipment. we now discuss their results. 5.1 the spes iv experiments the franco-scandinavian collaboration working at saclay studied the p(⃗ d, 2p)n reaction at 0.65, 1.6, and 2.0 gev by detecting both protons in the high resolution spes iv magnetic spectrometer [76, 77, 78, 79]. the small angular acceptance (1.7◦× 3.4◦) combined with a momentum bite of ∆p/p ≈7% gave access only to very low pp excitation energies and monte carlo simulations showed that the peak of the epp distribution was around 650 kev. under these circumstances any contamination from p-waves in the pp system can be safely neglected. on the other hand, the small angular acceptance meant that away from the forward direction the data were primarily sensitive to ann. on account of the small acceptance, the deflection angle in the spectrometer was adjusted to measure the differential cross section and ann at discrete values of the momentum transfer q. the results for the laboratory differential cross section and ann obtained at 1.6 gev for both the p(⃗ d, 2p)n and quasi-free d(⃗ d, 2p)nn reactions are shown in fig. 4. also shown in the figure are the authors' theoretical predictions of the plane wave impulse approximation and also ones that included the glauber double-scattering term [60, 13]. these give quite similar results for momentum transfers below about 150 mev/c but produce important changes for larger q, especially in the deuteron analysing power. the neutron-proton charge exchange amplitudes used were the updated versions of the analysis given in ref. [80] that were employed in other theoretical estimates [13, 81, 82, 83]. the predictions were averaged over the spes iv angular acceptance and, in view of the rapid change in the transition form factor with q, this effect can be significant. the validity of this procedure was tested by reducing the horizontal acceptance by a 25 figure 4: the measurements of the p(⃗ d, 2p)n laboratory differential cross section and deuteron tensor analysing power at 1.6 gev by the franco-scandinavian group [79] are compared to their theoretical impulse approximation estimates without the double scattering correction (dashed curve) and with (solid line). the experimental cross section data (stars) have been normalised to the solid line at q = 0.7 fm−1. it should be noted that the ratio of the data on deuterium (open circles) to those on hydrogen is not affected by this uncertainty. factor of two [78]. the acceptance of the spes iv spectrometer for two particles was very hard to evaluate with any precision and the hydrogen data were normalised to the theoretical prediction at q = 0.7 fm−1 that included the glauber correction. on the other hand, the ratio of the cross section with a deuterium and hydrogen target could be determined absolutely and, away from the forward direction, was found to be 0.68±0.04. this is reduced even more for small q, precisely because of the pauli blocking in the unobserved nn system, similar to that we discussed for the evaluation of rnp(0). since for small epp the np spin-independent amplitude cannot contribute and the spin-orbit term vanishes at q = 0, the extra reduction factor should be precisely 2/3, which is consistent with the value observed. a high precision (unpolarised) d(d, 2p)nn experiment was undertaken at kvi (groningen) to investigate the neutron-neutron scattering length [84]. in this case the pp and nn systems were both in the 1s0 region of very small excitation energies. the shape of the nn excitation energy spectrum was consistent with that predicted by plane wave impulse approximation with reasonable values of the nn scattering length. the primary aim of the franco-scandinavian group was the investigation of spin-longitudinal and transverse responses in medium and heavy nuclei and 26 also to extend these studies to the region of ∆(1232) excitation in the ⃗ dp → {pp}∆0. nevertheless, it is interesting to ask how useful these data could be for the establishment or checking of neutron-proton observables. the (d, 2p) transition form factor decreases very rapidly with momentum transfer because of the large deuteron size. as a consequence, the glauber double scattering term, which shares the momentum transfer between two collisions, becomes relatively more important. estimates of this effect are more model dependent [13, 79] and, as is seen from fig. 4, it may be dangerous to rely on them beyond about q ≈150 mev/c. absolute cross sections were not measured in these experiments and there were only two points in the safe region of momentum transfer and these represented averages over significant ranges in q. the central values of q marked on fig. 4 were evaluated from a monte carlo simulation of the spectrometer that used the theoretical model as input. as a consequence, the results give relatively little information on the magnitudes of the spin-flip compared to the non-spin-flip amplitudes. it is perhaps salutary to note that at larger q the estimate of the cross section without the double scattering correction describes the data better than that which included it. however, the reverse is true for the analysing power. the major contribution to the np database comes from the measurement of ann at small q. since the beam polarisation was known with high precision, this provides a robust relation between the magnitudes of the three spin-flip ampli- tudes but only at two average values of q. neutron-proton scattering has been extensively studied in the 800 mev region [22], and so it is not surprising that this p(⃗ d, 2p)n experiment gave results that are completely consistent with its pre- dictions. the dip in ann in both the theoretical estimates and the experimental data is due primarily to the expected vanishing of the distorted one-pion-exchange contribution to one of the spin-spin amplitudes for q ≈mπ. 5.2 the rcnp experiments almost simultaneously with the start of the spes iv experiments [76], an rcnp group studied the deuteron tensor analysing power ann in the p(⃗ d, 2p)n reaction at the much lower energies of td = 70 mev [85]. the primary motivation was to compare the forward angle data with the results of the polarisation transfer parameter knn that had been measured previously by the same group [65]. for small angles a magnetic spectrograph was used, which restricted the excitation energy of the final protons to be less than 200 kev. at larger angles, where the cross section is much smaller, a si telescope array with a larger acceptance was employed and the selection epp < 1 mev was imposed in the off-line analysis. in all cases, the only significant background arose from the random coincidence of two protons from the breakup of separate deuterons. this is particularly important for small angles due to the spectator momentum distribution in the 27 deuteron. additional data were taken at 56 mev, but solely in the forward direction. at such low energies, the plane wave impulse approximation based upon the neutron-proton charge exchange amplitudes may provide only a semi-quantitative description of the experimental data; there are likely to be significant contribu- tions from direct diagrams. nevertheless, as can be seen in fig. 5, the estimates given in the paper [85] that were made using the then existing (sp86) said phase shift solution [22] were reasonable near the forward direction and would be even closer if modern np solutions were used. at larger angles there is significant disagreement between the data and model and the authors show that part of this could be rectified if the np input amplitudes were evaluated at the mean of the incident and outgoing energies. this feature has been implemented in the more refined impulse approximation calculations of ref. [81], where the theory was evaluated in the brick-wall frame. figure 5: measurements of the deuteron tensor analysing power ann for the p(⃗ d, 2p)n reaction at td = 70 mev by the rcnp collaboration as a function of momentum transfer q [85]. in all cases epp < 1 mev. the results are compared to the authors' own theoretical plane wave impulse approximation estimates that were based upon the said sp86 phase shift solution [22] . 28 the group was disappointed to find that in the forward direction the relation of eq. (2.12) between their own (⃗ p,⃗ n) spin transfer data [65] and their deuteron tensor analysing power was far from being satisfied. this could not be explained by the difference in beam energy or the smearing over small angles. because the (d, 2p) results were obtained under the clean 1s0 conditions of epp < 200 kev, the problem must be laid at the door of the much poorer energy resolution associated with the detection of neutrons. it was only the later psi experiment [68] which showed that the spin-transfer parameter varied very strongly with epp and, as argued in section 4, this is probably the resolution of the discrepancy. 5.3 the emric experiments the aims and the equipment of the emric collaboration [81, 82, 83], also working at saclay, were very different and much closer to the original ideas of bugg and wilkin [19, 13]. the driving force was the desire to use the (⃗ d, 2p) reaction as the basis for the construction of a deuteron tensor polarimeter that could be used to measure the polarisation of the recoil deuteron in electron-deuteron elastic scattering. for this purpose the device had to have a much larger acceptance than that available at spes iv and be compact, so that it could be transported to and implemented in experiments at an electron machine. the emric apparatus was composed of an array of 5 × 5 csi scintillator crystals (4 × 4 × 10 cm3), optically coupled to phototubes, which provided in- formation on both energy and particle identification. placed at 70 cm from the liquid hydrogen target, it subtended an angular range of ±7◦so that several overlapping settings were used in order to increase the angular coverage. since the orientation of the deuteron polarisation could be rotated through the use of a solenoid, away from the forward direction this gave access to both transverse deuteron tensor analysing powers, the sideways ass as well as the normal ann, under identical experimental conditions. in the initial experiment at a deuteron beam energy of td = 200 mev [82], the angular resolution achieved with the csi crystals was only ±1.6◦but in the second measurement at td = 350 mev the system was further equipped with two multiwire proportional chambers that improved it to 0.1◦. having identified fast protons using a pulse-shape analysis technique based on the time-decay properties of the csi crystals, their energies could be measured with a resolution of the order of 2%. the missing mass of a proton pair yielded a clean neutron signal with a fwhm = 14 mev/c2, the only contamination coming from events where not all the energy was deposited in the csi array. the compact system allowed measurements over the wide angular and epp ranges that are necessary for the construction of a polarimeter with a high figure of merit. however, for the present discussion we concentrate our attention purely on the data where epp < 1 mev, for which the dilution of the analysing power signal by the proton-proton p waves is small. the emric results for the differential 29 cross section and two tensor analysing powers at 350 mev are shown in fig. 6. due to a slip in the preparation of the publication [83], both the experimental data and the impulse approximation model were downscaled by a factor of two [86], which has been corrected in the figure shown here. one should take into account that there are systematic errors (not shown) arising from the efficiency corrections that are estimated to be typically of the order of 20%, though they are larger at the edges of emric [87]. this might account for the slight oscillations of the data around the theoretical prediction in fig. 6. figure 6: measurements of the p(⃗ d, 2p)n differential cross section and two deuteron tensor analysing powers for epp < 1 mev at a beam energy of td = 350 mev by the emric collaboration [83] are compared to the theoretical plane wave impulse approximation estimates of ref. [81]. the values of both the experimental cross section data and theoretical model have been scaled up by a factor of two to correct a presentational oversight in the publication [83]. the plane wave impulse approximation calculation of ref. [81] describes the data quite well, though one has to note that the presentation is on a logarithmic scale and that there are at least 20% normalisation uncertainties. the data represented three settings of the emric facility and their fluctuations around the predictions could be partially due to minor imperfections in the acceptance corrections. the model is also satisfactory for the analysing powers out to at least q ≈150 mev/c, from which point the ann data remain too negative. however, as we argued with the spes iv results of fig. 4, it is at about this value of q that the glauber double scattering correction becomes significant. we can therefore conclude that the good agreement of the ass and ann data in the "safe" region of q ≲150 mev/c is confirmation that the ratios of the different spin-spin contributions given by the bugg amplitudes of ref. [80] are 30 quite accurate. nevertheless, their overall strength is checked far less seriously by these data because of the normalisation uncertainty and the logarithmic scale of fig. 4. the emric experiment [83] was the only one of those discussed that was capable of investigating the variation of the deuteron analysing power ann with excitation energy and, in view of the strong effects found for the d(⃗ p,⃗ n)pp polari- sation transfer parameters at 56 and 70 mev [67, 68], it would be interesting to see if anything similar happened for ann. extrapolating the td = 200 mev results to the forward direction, it is seen that ann ≈0.23, 0.17 and 0.10 for the three bins of excitation energy epp < 1 mev, 1 < epp < 4 mev, and 4 < epp < 8 mev, respectively. this variation is smaller than that found for knn [67, 68]. on the other hand, since the (longitudinal) momentum transfer remains very small in the forward direction, the plane wave impulse approximation predicts very little change with epp. the aim of the group was to show that the (⃗ d, 2p) reaction had a large and well understood polarisation signal and this was successfully achieved. the expe- rience gained with the emric device laid the foundations for the development of the polder polarimeter [88, 86], which was subsequently used to separate the contributions from the deuteron monopole and quadrupole form factors at jlab [89]. 5.4 the anke experiments a fourth experimental approach is currently being undertaken using the anke magnetic spectrometer that is located at an internal target position forming a chi- cane in the cosy cooler synchrotron. this machine is capable of accelerating and storing protons and deuterons with momenta up to 3.7 gev/c, i.e., kinetic energies of tp = 2.9 gev and td = 2.3 gev. the (⃗ d, 2p) measurements form part of a much larger spin programme that will use combinations of polarised beams and targets [90]. only results from a test experiment at td = 1170 mev are presently available [91, 92], and these are described below. there are several problems to be overcome before the p(⃗ d, 2p)n reaction could be measured successfully at anke. the horizontal acceptance for the reaction is limited to laboratory angles in the range of approximately −2◦< θhor < 4◦and much less in the vertical direction. this constrains severely the range of momen- tum transfers that can be studied. furthermore, the axis of the spin alignment of the circulating beam is vertical and, unlike the emric case [83], there is insuffi- cient place for a solenoid to rotate the polarisation. as a consequence, the values 31 of ann and ass cannot be extracted under identical condition. furthermore, the polarisations of the beam have to be checked independently at the anke energy. finally, unlike the external beam experiments of spes iv or emric, the luminosity inside the storage ring has also to be established at the anke position. most of the above difficulties can be addressed by using the fact that one can observe and measure simultaneously in anke the following reactions: ⃗ dp → {pp}n, ⃗ dp →dp, ⃗ dp → 3heπ0, and ⃗ dp →pspdπ0, where psp is a fast spectator proton. what cannot, of course, be avoided is the cut in the momentum trans- fer which at td = 1170 mev means that the deuteron charge exchange reaction has good acceptance only for q ≲150 mev/c. however, we already saw in the spes iv case that for larger momentum transfers the double scattering correc- tions become important and, as a result, the extraction of information on np amplitudes becomes far more model dependent. the luminosity, and hence the cross section, was obtained from the mea- surement of the dp →pspdπ0 reaction, for which the final spectator proton and produced deuteron fall in very similar places in the anke forward detector to the two protons from the charge exchange reaction. using only events with small spectator momenta, and interpreting the reaction as being due to that induced by the neutron in the beam deuteron, np →dπ0, reliable values could be obtained for the luminosity. this approach had the subsidiary advantage that to some ex- tent the glauber shadowing correction [60] cancels out between the dp →pspdπ0 and dp →{pp}n reactions. the cosy polarised ion source that feeds the circulating beam was pro- grammed to provide a sequence of one unpolarised state, followed by seven com- binations of deuteron vector and tensor polarisations. although these were mea- sured at low energies, it had to be confirmed that there was no loss of polarisation through the acceleration up to td = 1170 mev. this was done by measuring the analysing powers of ⃗ dp →dp, ⃗ dp → 3heπ0, and ⃗ dp →pspdπ0 and comparing with results given in the literature [93]. as expected, there was no discernable depolarisation. due to the geometric limitations, the acceptance of the anke forward de- tector varies drastically with the azimuthal production angle φ. the separation between ann and ass depends upon studying the variation of the cross section with φ. an accurate knowledge of the acceptance is not required for this purpose because one can work with the ratio of the polarised to unpolarised cross section where, to first order, the acceptance effects drop out. the monte carlo simula- tion of the acceptance was sufficiently good to give only a minor contribution to the error in the unpolarised cross section itself. the claimed overall cross section uncertainty of 6% is dominated by that in the luminosity evaluation. the limited anke acceptance also cuts into the epp spectrum and the collab- oration only quote data integrated up to a maximum of 3 mev. the results shown in fig. 7 were obtained with a cut of epp < 1 mev, as were the updated theoret- 32 ical predictions from ref. [81], where the current said np elastic amplitudes at 585 mev were used as input [22]. q [mev/c] 0 50 100 150 b / (mev/c)] μ /dq [ σ d 0 1 2 3 4 q [mev/c] 0 50 100 150 tensor analysing powers -1.5 -1.0 -0.5 0.0 0.5 1.0 nn a ss a figure 7: measurements of the p(⃗ d, 2p)n differential cross section and two deuteron tensor analysing powers for epp < 1 mev at a beam energy of td = 1170 mev by the anke collaboration [91, 92] are compared to the theo- retical plane wave impulse approximation estimates of ref. [81]. the agreement between the plane wave impulse approximation and the ex- perimental data is very good for all three observables over the full momentum transfer range that is accessible at anke. since there have been many neutron- proton experiments in this region, it is to be believed that the np elastic scattering amplitudes are very reliable at 585 mev. extrapolating the results to q = 0 and using the impulse approximation model, one finds that ann = −0.26±0.02. this is to be compared to the said value of −0.28, though no error can be deduced directly on their prediction [22]. all this suggests that the methodology applied by the anke collaboration is sufficient to deliver useful np amplitudes at higher energies, where less is known experimentally. compared to the spes iv and emric experiments, there are finer divisions in momentum transfer and hence more points in the safe q region. apart from taking data up to the maximum cosy energy of td ≈2.3 gev, there are plans to measure the deuteron charge exchange reaction with a po- larised beam and target [90]. the resulting values of the two transverse spin correlation parameters will allow the relative phases of the spin-flip amplitudes to be determined. to go higher in energy, it will be necessary to use a proton beam on a deu- terium target, detecting both slow recoil protons from the p⃗ d →{pp}n in the silicon tracking telescopes with which anke is equipped [94]. the drawback here is that the telescopes require a minimum momentum transfer so that the energies of the protons can be measured and this is of the order of 150 mev/c at low epp. this technique has already been used at celsius to generate a tagged 33 neutron beam on the basis of the pd →npp reaction at 200 mev by measuring both slow recoil protons in silicon microstrip detectors [95]. 5.5 data summary in table 6 we present the experimental values of the deuteron tensor analysing power in the ⃗ dp →{pp}n reaction extrapolated to the forward direction. the error bars include some attempt to take into account the uncertainty in the angular extrapolation. the resulting data are also shown in fig. 8. table 6: measured values of the forward deuteron tensor analysing power ann in the ⃗ dp →{pp}n reaction in terms of the kinetic energy per nucleon tn. the errors include some estimate for the extrapolation to θ = 0◦. tn ann(0) facility year ref. (mev) 28 0.015 ± 0.021 rcnp 1987 [85] 35 0.134 ± 0.018 rcnp 1987 [85] 100 0.23 ± 0.03 emric 1993 [83] 175 0.15 ± 0.03 emric 1993 [83] 325 −0.05 ± 0.03 spes iv 1995 [79] 585 −0.26 ± 0.03 anke 2009 [92] 800 −0.27 ± 0.04 spes iv 1995 [79] 1000 −0.32 ± 0.04 spes iv 1995 [79] in the forward direction the plane wave impulse approximation predictions of eq. (2.8) for the forward analysing power should be quite accurate provided that the excitation energy in the final diproton is small so that it is in the 1s0 state. this condition is well met by the data described here, where epp is always below 1 mev [79, 85, 83, 92]. this prediction, which is also tabulated in table 1, describes the trends of the data very well in regions where the neutron-proton phase shifts are well determined. we also show in the figure the values of ann deduced using eq. (2.12) from the d(⃗ p,⃗ n)pp measurements summarised in table 5. only those data are retained where the neutron polarisation was well measured and the pp excitation energy was small, though generally not as well determined as when the two final protons were detected. the consistency between the (⃗ d, pp) and (⃗ p,⃗ n) data is striking and it is interesting to note that they both suggest values of ann that are slightly lower in magnitude at high energies than those predicted by the np phase shifts of the said group [22]. the challenge now is to continue measuring these data into the more unchartered waters of even higher energies. 34 although we have concentrated here on the results for the forward analysing power, it is clear that this represents only a small part of the total data set as demonstrated by the results of figs. 4, 6, and 7. 35 figure 8: values of the forward deuteron tensor analysing power in the ⃗ dp → {pp}n reaction as a function of the kinetic energy per nucleon tn. the directly measured experimental data (closed symbols) from spes iv (squares) [79], em- ric (closed circles) [83], anke (star) [92], and rcnp (triangles) [85] were all obtained with a pp excitation energy of 1 mev or less. the error bars include some estimate of the uncertainty in the extrapolation to θ = 0. the open sym- bols were obtained from measurements of the polarisation transfer parameter in d(⃗ p,⃗ n)pp by using eq. (2.12). the data are from refs. [73] (circles), [75] (squares), [72] (cross), and [67, 68] (triangles). the curve is the plane wave 1s0 prediction of eq. (2.8), as tabulated in table 1. 36 6 conclusions originally the deuteron was thought of merely as a useful substitute for a free neutron target. as an example of this, it has been shown that at large momentum transfers the spin-dependent parameters measured in free np scattering and quasi- free in pd collisions give very similar results [24]. the situation is very different at low momentum transfers where it is not clear which of the nucleons is the spectator or, indeed, whether the concept of calling one of the nucleons a spectator makes any sense at all. however, a more interesting effect comes about in the medium energy neutron charge exchange on the deuteron, nd →p{nn}, when the excitation energy enn in the two neutron system is very low. under such conditions the pauli principle demands that the two neutrons should be in a 1s0 state and there then has to be spin-flip isospin-flip transition from the spin-triplet np in the deuteron to the singlet nn system. the rate for the charge-exchange deuteron breakup nd →p{nn} would then depend primarily on the spin-spin np →pn amplitudes. the above remarks only assume a practical importance because of an "acci- dent" in the low energy nucleon-nucleon interaction. in the nn system there is an antibound (or virtual) state pole only a fraction of an mev below threshold. although the pole position is displaced slightly in the pp case by the coulomb repulsion, it results in huge pp and nn scattering lengths. in the nd →pnn re- action, it leads to the very characteristic peak at the hard end of the momentum spectrum of the produced proton. since we know that these events are the result of the spin-flip interaction, we clearly want to use them to investigate in greater depth this interaction. there are two distinct ways to try to achieve our aims and we have tried to review them both in this article. these are the inclusive (sum-rule) approach of section 3 and the high resolution polarisation experiments of sections 4 and 5. in impulse approximation, at zero momentum transfer, the d(n, p)nn inter- action only excites spin-singlet final states and dean [11, 12] has shown that the inclusive measurement of the proton momentum spectrum can then be in- terpreted in terms of the spin-flip np amplitudes through the use of a sum rule. though the shape of the proton momentum spectrum must depend upon the details of the low energy nn interaction and also on the deuteron d-state, the integral over all momenta would not, provided that the sum rule has converged before any of the limitations imposed by the three-body phase space have kicked in. the inclusive approach has many positive advantages, in addition to being independent of the low energy nucleon-nucleon dynamics. in a direct comparison of the production rates of protons in the d(n, p)nn and p(n, p)n reactions using the same apparatus, many of the sources of systematic errors drop out in the evaluation of the cross section ratio rnp(0). these are primarily effects associated with the neutron flux and uncertainties in the proton detection system. 37 there are, however, no similar benefits when working with a proton beam, where one measures instead d(p, n)pp. here one can only construct the rpn(0) ratio by dividing by a p(n, p)n cross section that has been measured in an in- dependent experiment. this is probably the reason why there are fewer entries in table 3 compared to table 2. we must therefore stress that, in general, the d(np)nn determinations of rnp(0) are much to be preferred over those of d(p, n)pp. on the face of it, the determination of rpn(0) through the measurement of the two fast protons from the p(d, pp)n reaction in a bubble chamber looks like a very hard way to obtain a result [47]. in addition to having to use independent data to provide the normalisation cross section in the denominator, the reaction is first measured exclusively in order afterwards to construct an inclusive distribution. on the other hand, a full kinematic determination allows one to check many of the assumptions made in the analysis and, in particular, those related to the isolation of the charge-exchange impulse approximation contribution from those of other possible feynman diagrams. a major difficulty in any of the inclusive measurements is ensuring that the phase space is sufficiently large that the sum rule has been saturated without being contaminated by other driving mechanisms. this means that the low energy determinations of rnp(0) are all likely to underestimate the "true" value and there could be some effects from this even through the energy range of the psi experiments [31]. even more worrying is the fact that at low energies the rapid variation of knn(0) with epp, as measured in the d(⃗ p,⃗ n)pp reaction [68], shows that there are significant deviations from plane wave impulse approximation with increasing epp. these deviations are probably too large to be ascribed to effects arising from the variation of the longitudinal momentum transfer with epp. this brings into question the whole sum rule approach at low energies. the alternative high resolution approach of measuring the 1s0 peak of the final state interaction requires precisely that, i.e., high resolution. this can be achieved in practice by measuring the (n, p) reaction with a very long time-of- flight path [75] or by measuring the protons in the dp →{pp}n reaction with either a deuteron beam [79, 83, 92] or a very low density deuterium target [46]. the resulting data are then sensitive to the low energy np interaction in the deuteron and the pp interaction in the 1s0 final state. however, such interactions are well understood and lead to few ambiguities in the charge exchange predic- tions. establishing a good overall normalisation can present more of a challenge. in addition to obvious acceptance and efficiency uncertainties, if one evaluates a cross section integrated up to say epp = 3 mev then one has to measure the 3 mev with good absolute precision, which is non-trivial for a deuteron beam in the gev range. hence it might be that at high energies the inclusive measure- ments could yield more precise determinations of absolute values of rnp(0) [41] than could be achieved by using high resolution experiments. on the other hand, measuring just the fsi peak with good resolution allows one more easily to follow the variation with momentum transfer and there are also 38 fewer kinematic ambiguities. more crucially, the spin information from the (⃗ n, ⃗ p ) or (⃗ d, {pp}) reactions enables one to separate the different spin contributions to the small angle charge exchange cross section. it could of course be argued that this is not just a benefit for an exclusive reaction since, if the dubna bubble chamber experiments [47] had been carried out with a polarised deuteron beam, then these would also have been able to separate the contributions from the two independent forward spin-spin contributions through the use of the generalised dean sum rule [13, 10]. it is, however, much more feasible to carry out (d, {pp}) measurements with modern electronic equipment and the hope is that, through the use of polarised beams and targets, they will lead to evaluations of the relative phases between the three independent np →pn spin-spin amplitudes out to at least q ≈mπ [90]. we have been very selective in this review, concentrating our attention on the forward values of the nd →pnn/np →pn cross section ratio, the (⃗ n, ⃗ p ) po- larisation transfer, and the deuteron tensor analysing power in nucleon-deuteron charge-exchange break-up collisions. in the latter cases, we have specialised to the kinematic situations where two of the final nucleons emerge in the 1s0 state. under these conditions there are strong connections between the three types of experiment described and this we have tried to stress. however, there is clearly much additional information in the data at larger angles, which we have here gen- erally neglected. we have also avoided discussing the extensive data that have been taken on nuclear targets, where the selectivity of the (⃗ n, ⃗ p ) or (⃗ d, {pp}) reactions can be used to identify particular classes of final nuclear states. at the higher energies, these states could even include the excitation of the ∆(1232) isobar. despite the successful measurements, none of the rnp(0) data nor those from the exclusive polarised measurements have so far been included in any of the existing phase shift analyses. they have merely been used as a posteriori checks on their predictions. we have argued that they could also provide valuable input into the direct neutron-proton amplitude reconstruction in the backward direc- tion [10]. for any of these purposes it would be highly desirable to control further the range of validity of the models used to interpret the data and, in particu- lar, to examine further the effects of multiple scattering. there remain therefore theoretical as well as experimental challenges to be overcome. acknowledgements we are grateful to j. ludwig for furnishing us with a partial copy of ref. [31]. sev- eral people gave us further details regarding their own experiments. these include d. chiladze, m.j. esten, v. glagolev, a. kacharava, s. kox, m.w. mcnaughton, t. motobayashi, i. sick, and c.a. whitten. there were also helpful discussions with d.v. bugg, z. janout, n. ladygina, i.i. strakovsky, e. strokovsky, and 39 yu. uzikov. one of the authors (cw) wishes to thank the institute of ex- perimental and applied physics of the czech technical university prague, and its director stanislav posp ́ ıˇ sil, for hospitality and support during the prepara- tion of this paper. this work has been supported by the research programme msm 684 077 0029 of the ministry of education, youth, and sport of the czech republic. 40 references [1] g.f. chew, phys. rev. 80, 196 (1950). [2] g.f. chew, phys. rev. 84, 710 (1951). [3] r.l. gluckstein and h. bethe, phys. rev. 81, 761 (1950). [4] i. pomeranchuk, doklady akad. nauk 77, 249 (1951). [5] w.m. powell, report ucrl 1191, berkeley (1951). [6] k.m. watson, phys. rev. 88, 1163 (1952). [7] i.m. shmushkevich, thesis, leningrad physico-technical inst. (1953). [8] a.b. migdal, j. exp. theor. phys. (ussr) 28 (1955) 3; trans. sov. phys. jetp 1, 2 (1955). [9] l.i. lapidus, j. exp. theor. phys. (ussr) 32 (1957) 1437; trans. sov. phys. jetp 5, 1170 (1957). [10] f. lehar and c. wilkin, eur. phys. j. a 37, 143 (2008). [11] n.w. dean, phys. rev. d 5, 1661 (1972). [12] n.w. dean, phys. rev. d 5, 2832 (1972). [13] d.v. bugg and c. wilkin, nucl. phys. a 167, 575 (1987). [14] r.j.n. phillips, rep. prog. phys. 22, 562 (1959); idem proc. phys. soc. a 74, 652 (1959); idem nucl. phys. 53, 650 (1964). [15] g.v. dass and n.m. queen, j. phys. a 1, 259 (1968). [16] a.s. clough et al., phys. rev. c 21, 988 (1980). [17] p.j. riley et al., phys. lett. 103b, 313 (1981). [18] j.s. chalmers et al., phys. lett. 153b, 235 (1985). [19] d.v. bugg and c. wilkin, phys. lett. 154b, 243 (1985). [20] j. bystrick ́ y, f. lehar, and p. winternitz, j. phys. (paris) 39, 1 (1978). [21] g.g. ohlsen, rep. prog. phys. 35, 717 (1972). [22] r.a. arndt, i.i. strakovsky, and r.l. workman, phys. rev. c 62, 034005 (2000); r.a. arndt, w.j. briscoe, i.i. strakovsky, and r.l. work- man, phys. rev. c 76, 025209 (2007); said solutions available from http://gwdac.phys.gwu.edu. 41 [23] d. adams et al., acta polytechnica (prague) 36, 11 (1996). [24] a. de lesquen et al., eur. phys. j. c 11, 69 (1999). [25] m.w. mcnaughton et al.. phys. rev. c 53, 1092 (1996) and references therein. [26] a. ahmidouch et al., eur. phys. j. c 2, 627 (1998) and references therein. [27] d.v. bugg et al., phys. rev. c 21, 1004 (1980) and references therein. [28] j.r. cladis, j. hadley, and w.n. hess, phys. rev. 86, 110 (1952). [29] v.p. dzhelepov et al., izvestia akad. nauk 19, 573 (1955). [30] v.p. dzhelepov et al., nuovo cim. suppl. 3, 61 (1956). [31] b. pagels, untersuchung der quasielastischen ladungsaustauschreaktion nd →pnn im neutronenergiebereich von 290 bis 570 mev, diploma the- sis, universit ̈ at freiburg im breisgau (1988). [32] r.r. larsen, nuovo cim. a 18, 1039 (1960). [33] v.p. dzhelepov, recent investigations on nucleon-nucleon scattering at the dubna synchrocyclotron, proc. int. conf. high energy phys., ed. j. prentki (cern, geneva, 1962), p. 19. [34] v.k. voitovetskii, i.l. korsunskii, and yu.f. pazhin, nucl. phys. 69, 531 (1965). [35] c. wong et al., phys. rev. 116, 164 (1959). [36] d.f. measday, phys. lett. 21, 66 (1966). [37] b.e. bonner et al., phys. rev. c 17, 664 (1978). [38] j. franz, e. r ̈ ossle, h. schmitt, and l. schmitt, physica scripta t87, 14 (1999). [39] v.i. sharov et al., eur. phys. j. c 37, 79 (2004). [40] f. lehar, phys. part. nuclei 36, 501 (2005). [41] v.i. sharov et al., eur. phys. j. a 39, 267 (2009). [42] j.a. hofmann and k. strauch, phys. rev. 90, 559 (1953). [43] c.j. batty, r.s. gilmore, and g.h. stafford, phys. lett. 16, 137 (1965). [44] a. langsford et al., nucl. phys. a 99, 246 (1967). 42 [45] c.w. bjork et al., phys. lett. 63b, 31 (1976). [46] m.j. esten, t.c. griffith, g.j. lush, and a.j. metheringham, rev. mod. phys. 37, 533 (1965); idem nucl. phys. 86, 289 (1966). [47] v.v. glagolev et al., cent. eur. j. phys. 6, 781 (2008). [48] l. castillejo and l.s. singh, proc. conf. on nuclear forces and the few body problem (pergamon press, london, 1960) p. 193. [49] b.s. aladashvili et al., nucl. phys. b 86, 461 (1975). [50] b.s. aladashvili et al., j. phys. g 3, 7 (1977). [51] b.s. aladashvili et al., j. phys. g 3, 1225 (1977). [52] v.v. glagolev et al., eur. phys. j. a 15, 471 (2002). [53] p.f. shepard, t.j. devlin, r.e. mishke, and j. solomon, phys. rev. d 10, 2735 (1974). [54] j. bystrick ́ y, c. lechanoine-leluc, and f. lehar, j. phys. (paris) 48, 199 (1987). [55] g. bizard et al., nucl. phys. b 85, 14 (1975). [56] j. bystrick ́ y and f. lehar, nucleon-nucleon scattering data, ed. h. behrens und g. ebel, (fachinformationszentrum karlsruhe), 1978 edition, nr.11-1 (1978), 1981 edition, nr.11-2 and nr.11-3 (1981). [57] j. bystrick ́ y et al., in landolt-b ̈ ornstein, vol. 9 (springer, berlin, 1980). [58] r. binz, untersuchung der spinabh ̈ angigen neutron-proton wechselwirkung im energiebereich von 150 bis 1100 mev, phd thesis, universit ̈ at freiburg im breisgau (1991). [59] r.a. arndt, j.s. hyslop iii, and l.d. roper, phys. rev. d 35, 199 (1987). [60] r.j. glauber and v. franco, phys. rev. 156, 1685 (1967). [61] n.w. reay, e.h. thorndike, d. spalding, and a.r. thomas, phys. rev. 150, 806 (1966). [62] l.p. robertson et al., nuc. phys. a 134, 545 (1969) [63] p.w. lisowski, r.c. byrd, r.l. walter, and t.b. clegg, nucl. phys. a 334, 45 (1980). [64] m. jain and g. doolen, phys. rev. c 8, 124 (1973). 43 [65] h. sakai et al., phys. lett. b 177, 155 (1986). [66] r. henneck et al., phys. rev. 37, 2224 (1988). [67] m.a. pickar et al., phys. rev. c 42, 20 (1990). [68] m. zeier et al., nucl. phys. a 654, 541 (1999); m. zeier, phd thesis, uni- versit ̈ at basel, 1997. [69] r. henneck et al., nucl. instr. meth. phys. res. a 259, 329 (1987). [70] w. gl ̈ ockle et al., phys. rep. 274, 107 (1996). [71] b.d. anderson et al., phys. rev. c 54, 1531 (1996). [72] h. sakai et al., phys. rev. c 35, 344 (1987). [73] m.w. mcnaughton et al., phys. rev. c 45, 2564 (1992). [74] d.v. bugg, private communication noted in ref. [73]. [75] d.j. mercer et al., phys. rev. lett. 71, 684 (1993). [76] c. ellegaard et al., phys. rev. lett. 59, 974 (1987). [77] c. ellegaard et al., phys. lett. b 231, 365 (1989). [78] t. sams, ph.d. thesis, niels bohr institute, copenhagen (1990), available at www.nbi.dk/ sams. [79] t. sams et al., phys. rev. c 51, 1945 (1995). [80] r. dubois et al., nucl. phys. a 377, 554 (1982). [81] j. carbonell, m.b. barbaro, and c. wilkin, nucl. phys. a 529, 653 (1991). [82] s. kox et al., phys. lett. b 266, 265 (1991). [83] s. kox et al., nucl. phys. a 556, 621 (1993). [84] c. b ̈ aumer et al., phys. rev. c 71, 044003 (2005). [85] t. motobayashi et al., nucl. phys. a 481, 207 (1988). [86] j.-s. r ́ eal, ph.d. thesis, university of grenoble (1994). [87] s. kox, private communcation (june 2009). [88] s. kox et al., nucl. instr. meth. phys. res. a 346, 527 (1994). [89] d. abbott et al., phys. rev. lett. 84, 5053 (2000). 44 [90] a. kacharava, f. rathmann, c. wilkin, spin physics from cosy to fair, cosy proposal 152 (2005), arxiv:nucl-ex/0511028. [91] d. chiladze et al., phys. lett. b 637, 170 (2006). [92] d. chiladze et al., eur. phys. j. a 40, 23 (2009). [93] d. chiladze et al., phys. rev. st accel. beams 9, 050101 (2006). [94] r. schleichert et al., ieee trans. nucl. sci. 50, 301 (2003). [95] t. peterson et al., nucl. instr. meth. phys. res. a 527, 432 (2004). 45

0911.1700

four-dimensional spin foam perturbation theory

we define a four-dimensional spin-foam perturbation theory for the ${\rm bf}$-theory with a $b\wedge b$ potential term defined for a compact semi-simple lie group $g$ on a compact orientable 4-manifold $m$. this is done by using the formal spin foam perturbative series coming from the spin-foam generating functional. we then regularize the terms in the perturbative series by passing to the category of representations of the quantum group $u_q(\mathfrak{g})$ where $\mathfrak{g}$ is the lie algebra of $g$ and $q$ is a root of unity. the chain-mail formalism can be used to calculate the perturbative terms when the vector space of intertwiners $\lambda\otimes \lambda \to a$, where $a$ is the adjoint representation of $\mathfrak{g}$, is 1-dimensional for each irrep $\lambda$. we calculate the partition function $z$ in the dilute-gas limit for a special class of triangulations of restricted local complexity, which we conjecture to exist on any 4-manifold $m$. we prove that the first-order perturbative contribution vanishes for finite triangulations, so that we define a dilute-gas limit by using the second-order contribution. we show that $z$ is an analytic continuation of the crane-yetter partition function. furthermore, we relate $z$ to the partition function for the $f\wedge f$ theory.

introduction spin foam models are state-sum representations of the path integrals for bf theories on sim- plicial complexes. spin foam models are used to define topological quantum field theories and quantum gravity theories, see [1]. however, there are also perturbed bf theories in various di- mensions, whose potential terms are powers of the b field, see [10]. the corresponding spin-foam perturbation theory generating functional was formulated in [10], but further progress was hin- dered by the lack of the regularization procedure for the corresponding perturbative expansion and the problem of implementation of the triangulation independence. the problem of implementation of the triangulation independence for general spin foam per- turbation theory was studied in [2], and a solution was proposed, in the form of calculating the perturbation series in a special limit. this limit was called the dilute-gas limit, and it was given by λ →0, n →∞, such that g = λn is a fixed constant, where λ is the perturbation theory parameter, also called the coupling constant, n is the number of d-simplices in a simpli- cal decomposition of a d-dimensional compact manifold m and g is the effective perturbation 2 j. faria martins and a. mikovi ́ c parameter, also called the renormalized coupling constant. however, the dilute-gas limit could be used in a concrete example only if one knew how to regularize the perturbative contributions. the regularization problem has been solved recently in the case of three-dimensional (3d) euclidean quantum gravity with a cosmological constant [9], following the approach of [3, 8]. the 3d euclidean classical gravity theory is equivalent to the su(2) bf-theory with a b3 perturbation, and the corresponding spin foam perturbation expansion can be written by using the ponzano–regge model. the terms in this series can be regularized by replacing all the spin-network evaluations with the corresponding quantum spin-network evaluations at a root of unity. by using the chain–mail formalism [23] one can calculate the quantum group perturbative corrections, and show that the first-order correction vanishes [9]. consequently, the dilute-gas limit has to be modified so that g = λ2n is the effective perturbation parameter [9]. another result of [9] was to show that the dilute gas limit cannot be defined for an arbitrary class of triangulations of the manifold. one needs a restricted class of triangulations such that the number of possible isotopy classes of a graph defined from the perturbative insertions is bounded. in 3d this can be achieved by using the triangulations coming from the barycentric subdivisions of a regular cubulation of the manifold [9]. in this paper we are going to define the four-dimensional (4d) spin-foam perturbation theory by using the same approach and the techniques as in the 3d case. we start from a bf-theory with a b ∧b potential term defined for a compact semi-simple lie group g on a compact 4-manifold m. in section 2 we define the formal spin foam perturbative series by using the spin-foam generating functional method. we then regularize the terms in the series by passing to the category of representations for the quantum group uq(g) where g is the lie algebra of g and q is a root of unity. in sections 4 and 5 we then use the chain–mail formalism to calculate the perturbative contributions. the first-order perturbative contribution vanishes, so that we define the dilute-gas limit in section 6 by using the second-order contribution. we calculate the partition function z in the dilute-gas limit for a class of triangulations of a 4-dimensional manifold which are arbitrarily fine and have a controllable local complexity. we conjecture that such a class of triangulations always exists for any 4-dimensional manifold, and can be given by the triangulations corresponding to the barycentric subdivisions of a fixed cubulation of the manifold. we then show that z is given as an analytic continuation of the crane–yetter partition function. in section 7 we relate the path-integral for the f ∧f theory with the spin foam partition function and in section 8 we present our conclusions. 2 spin foam perturbative expansion let g be the lie algebra of a semisimple compact lie group g. the action for a perturbed bf-theory in 4d can be written as s = z m bi ∧fi + λ gij bi ∧bj
, (2.1) where b = bili is a g-valued two-form, li is a basis of g, f = da + 1 2[a, a] is the curvature 2-form for the g-connection a on a principal g-bundle over m, xi = xi and gij is a symmetric g-invariant tensor. here if x and y are vector fields in the manifold m then [a, a](x, y ) = [a(x), a(y )]. we will consider the case when gij ∝δij, where δij is the kronecker delta symbol. in the case of a simple lie group, this is the only possibility, while in the case of a semisimple lie group one can also have gij which are not proportional to δij. for example, in the case of the so(4) group one can put gab,cd = ǫabcd, where ǫ is the totally antisymmetric tensor and 1 ≤a, . . . , d ≤4. we will also use the notation tr (xy ) = xiyi and ⟨xy ⟩= gijxiy j. four-dimensional spin foam perturbation theory 3 consider the path integral z(λ, m) = z dadbei r m(bi∧fi+λ⟨b∧b⟩). (2.2) it can be evaluated perturbatively in λ by using the generating functional z0(j , m) = z dadbei r m(bi∧fi+bi∧ji), (2.3) (where jili is an arbitrary 2-form valued in g) and the formula z(λ, m) = exp  −iλ z m gij δ δji ∧ δ δjj  z0(j , m)    j =0. (2.4) the path integrals (2.3) and (2.4) can be represented as spin foam state sums by discretizing the 4-manifold m, see [10]. this is done by using a simplicial decomposition (triangulation) of m, t(m). it is useful to introduce the dual cell complex t ∗(m) [24] (a cell decomposition of m), and we will denote the vertices, edges and faces of t ∗(m) as v, l and f, respectively. a vertex v of t ∗(m) is dual to a 4-simplex σ of t(m), an edge l of t ∗(m) is dual to a tetra- hedron τ of t(m) and a face f of t ∗(m) is dual to a triangle ∆of t(m). the action (2.1) then takes the following form on t(m) s = x ∆ tr (b∆ff) + λ 5 x σ x ∆′,∆′′∈σ ⟨b∆′b∆′′⟩, where ∆′ and ∆′′ are pairs of triangles in a four-simplex σ whose intersection is a single vertex of σ and b∆= r ∆b. the variable ff is defined as eff = y l∈∂f gl, where f is the face dual to a triangle ∆, l's are the edges of the polygon boundary of f and gl are the dual edge holonomies. one can then show that z0(j , m) = x λf,ιl y f dim λf y v a5(λf(v), ιl(v), jf(v)), (2.5) where the amplitude a5(λf(v), ιl(v), jf(v)), also called the weight for the 4-simplex σ, is given by the evaluation of the four-simplex spin network whose edges are colored by ten λf(v) irreps and five ιl(v) intertwiners, while each edge has a d(λ)(ej ) insertion. here d(λ)(ej ) is the repre- sentation matrix for a group element ej in the irreducible representation (irrep) λ, see [10, 18]. note that a vertex v is dual to a 4-simplex σ, so that the set of faces f(v) intersecting at v is dual to the set of ten triangles of σ. similarly, the set of five dual edges l(v) intersecting at v is dual to the set of five tetrahedrons of σ. the sum in (2.5) is over all colorings λf of the set of faces f of t ∗(m) by the irreps λf of g, as well as over the corresponding intertwiners ιl for the dual complex edges l. equivalently, λf label the triangles of t(m), while ιl label the tetrahedrons of t(m). in the case of the su(2) group and j = 0, the amplitude a5 gives the 15j symbol, see [1, 13]. for the general definition of 15j-symbols a5(λf(v), ιl(v)) see [16, 17]. then z0(j = 0, m) can be written as z0(m) = x λf ,ιl y f dim λf y v a5(λf(v), ιl(v)), (2.6) 4 j. faria martins and a. mikovi ́ c which after quantum group regularization (by passing to a root of unity) becomes a manifold invariant known as the crane–yetter invariant [6]. the formula (2.4) is now given by the discretized expression z(λ, m, t) = exp  −iλ 5 x σ x ∆,∆′∈σ gij ∂2 ∂j i f ∂j j f′  z0(j , m)    j =0, (2.7) where t denotes the triangulation of m. the equation (2.7) can be rewritten as z(λ, m, t) = x λf,ιl y f dim λf exp −iλ x σ ˆ vσ ! y v a5(λf(v), ιl(v)), (2.8) where the operator ˆ vσ is given by ˆ vσ = 1 5 x ∆,∆′∈σ gijl(λ) i ⊗l(λ′) j ≡1 5 x f,f′;v∈f∩f′ gijl(λf ) i ⊗l (λf′) j . (2.9) this operator acts on the σ-spin network evaluation a5 by inserting the lie algebra basis ele- ment l(λ) for an irrep λ into the spin network edge carrying the irrep λ. the expression (2.9) follows from (2.5), (2.7) and the relation ∂d(λ)(ej ) ∂j i    j =0 = l(λ) i . following (2.9), let us define a g-edge in a 4-simplex spin network as a line connecting the middle points of two edges of the spin network, such that this line is labelled by the tensor gij. we associate to a g-edge the linear map x ij gijl(λf ) i ⊗l (λf′) j , where λf and λf′ are the labels of the spin network edges connected by the g-edge and l(λf ) i denotes the action of the basis element li of g in the representation λf. the action of the operator gijl(λf ) i ⊗l (λf′) j in a single 4-simplex of the j = 0 spin foam state sum (2.5) can be represented as the evaluation of a spin network γσ,g obtained from the 4-simplex spin network γσ by adding a g-edge between the two edges of γσ labeled by λf and λf′. when gij ∝δij and the intertwiners cλλ′a, from λ ⊗λ′ to a, where a is the adjoint repre- sentation, form a one-dimensional vector space, γσ,g becomes the 4-simplex spin network with an insertion of an edge labeled by the adjoint irrep, see fig. 1. this simplification happens because the matrix elements of l(λ) i can be identified with the components of the intertwiner cλλ′a, since these intertwiners are one-dimensional vector spaces, i.e. l(λ) i
αβ = cλλa α β i , (2.10) so that x i l(λ) i
αβ l(λ′) i
α′β′ = x i cλλa α β icaλ′λ′ i α′ β′ . (2.11) then the right-hand side of the equation (2.11) represents the evaluation of the spin network in fig. 2. the condition (2.10) is not too restrictive since it includes the su(2) and so(4) groups. we need to consider this particular case in order to be able to use the chain–mail techniques. four-dimensional spin foam perturbation theory 5 figure 1. a 15j symbol (4-simplex spin network) with a g-edge insertion (dashed line). here a is the adjoint representation. figure 2. spin network form of equation (2.11). the action of ( ˆ vσ)n in a5 is given by the evaluation of a γσ,n spin network which is obtained from the γσ spin network by inserting n g-edges labeled by the adjoint irrep. these additional edges connect the edges of γσ which correspond to the triangles of the 4-simplex σ where the operators l(λ) i ⊗l(λ′) i from ˆ vσ act. let z(m, t) = ∞ x n=0 inλnzn(m, λ, t), then z0(m) = z0(j , m)    j =0. the state sum z0 is infinite, unless it is regularized. the usual way of regularization is by using the representations of the quantum group uq(g) at a root of unity, which, by passing to a finite-dimensional quotient, yields a modular hopf algebra [26]. there are only finitely many irreps with non-zero quantum dimension in this case, and the corresponding state sum z0 has the same form as in the lie group case, except that the usual spin network evaluation used for the spin-foam amplitudes has to be replaced by the quantum spin network evaluation. in this way one obtains a finite and triangulation independent z0, usually known as the crane–yetter invariant [6]. this 4-manifold invariant is determined by the signature of the manifold [23, 26]. the same procedure of passing to the quantum group at a root of unity can be applied to the perturbative corrections zn, but in order to obtain triangulation independent results, the dilute gas limit has to be implemented [2, 9]. 6 j. faria martins and a. mikovi ́ c 2.1 the chain–mail formalism and observables of the crane–yetter invariant the chain–mail formalism for defining the turaev–viro invariant and the crane–yetter inva- riant was introduced by roberts in [23]. in the four-dimensional case, the construction of the related manifold invariant z0(m) had already been implemented by broda in [4]. however, the equality with the crane–yetter invariant, as well as the relation of z0(m) with the signature of m appears only in the work of roberts [23]. we will follow the conventions of [3]. let m be a four-dimensional manifold. suppose we have a handle decomposition [12, 14, 24] of m, with a unique 0-handle, and with hi handles of order i (where i = 1, 2, 3, 4). this gives rise to the link chlm in the three-sphere s3, with h2 + h1 components (the "chain–mail link"), which is the kirby diagram of the handle decomposition [12, 14]. namely, we have a dotted unknotted (and 0-framed) circle for each 1-handle of m, determining the attaching of the 1-handle along the disjoint union of two balls, and we also have a framed knot for each 2-handle, along which we attach it. this is the four- dimensional counterpart of the three-dimensional chain–mail link of roberts, see [23, 3]. the crane–yetter invariant z0(m), which coincides with the invariant z0(j , m)j =0, defined in the introduction, see equation (2.6), can be represented as a multiple of the spin-network eva- luation of the chain mail link chlm, colored with the following linear combination of quantum group irreps (the ω-element): ω= x λ (dimq λ)λ, see [23]. explicitly, by using the normalizations of [3]: z0(m) = η−1 2 (h2+h1+h3−h4+1)⟨chlm, ωh2+h1⟩, (2.12) where η = x λ (dimq λ)2. roberts also proved in [23] that z0(m) = κs(m), where κ is the evaluation of a 1-framed unknot colored with the ω-element and s(m) denotes the signature of m. given a triangulated manifold (m, t), consider the natural handle decomposition of m ob- tained from thickening the dual cell decomposition of m; see [23, 24]. then a handle de- composition of m (with a single 0-handle), such that (2.12) explicitly gives the formula for z0(m) = z0(j , m)j =0, appearing in equation (2.5), is obtained from this handle decomposi- tion by canceling pairs of 0- and 1-handles [12, 14, 24], until a single 0-handle is left; in this case, in the vicinity of each 4-simplex, the chain–mail link has the form given in fig. 3. this explicit calculation appears in [23, 3] and essentially follows from the lickorish encircling lemma [13, 15]: the spin-network evaluation of a graph containing a unique strand (colored with the represen- tation λ) passing through a zero framed unknot colored with ωvanishes unless λ is the trivial representation. a variant of the crane–yetter model z0 in (2.12) is achieved by inspecting its observables, addressed in [3]. consider a triangulated 4-manifold m. consider the handle decomposition of m obtained from thickening the dual complex of the triangulation, and eliminating pairs of 0- and 1-handles until a single 0-handle is left. any triangle of the triangulation of m therefore yields a 2-handle of m. now choose a set s with ns triangles of m, which will span a (possibly singular) surface σ2 of m. color each t ∈s with a representation λt. the associated observable of the crane–yetter invariant is: z0(m, s, λs) = η−1 2(h2+h1+h3−h4+1)⟨chlm; ωh2+h1−ns, λs⟩ y t∈s dimq(λt), (2.13) four-dimensional spin foam perturbation theory 7 figure 3. portion of the chain-mail link corresponding to a 4-simplex; this may have additional meridian circles (corresponding to 1-handles) since we also eliminate pairs of 0- and 1-handles, until a single 0- handle is left. where ⟨chlm; ωh2+h1−ns, λs⟩ denotes the spin-network evaluation of the chain–mail link chlm, where the components associated with the triangles t ∈s are colored by λs and the remaining components with ω. we can see chlm as a pair (ls, ks), where ks denotes the components of the chain–mail link given by the triangles of s and ls the remaining components of the chain–mail link. we thus have chlm = ls ∪ks. let zwrt(n, γ) denote the witten–reshetikhin–turaev invariant of the colored graph γ embedded in the 3-manifold n, in the normalization of [3]. then theorem 2 of [3] says that: z0(m, s, λs) = zwrt ∂(m \ ˆ σ2), k′ s
κs(m\ˆ σ2) η χ(σ2) 2 −ns y t∈s dimq(λt). (2.14) here ˆ σ2 is an open regular neighborhood of σ2 in m, s denotes the signature of the manifold and χ denotes the euler characteristic. the link k′ s is the link in ∂(m \ ˆ σ2) along which the 2- handles associated to the triangles of σ2 would be attached, in order to obtain m. this theorem of [3] essentially follows from the fact that the pair (ls, ks) is a surgery presentation of the pair ∂(m \ ˆ σ2), k′ s
, a link embedded in a manifold, apart from connected sums with s1 × s2. 3 the first-order correction recall that there are two possible ways of representing the crane–yetter invariant z0: as a state sum invariant (2.6) and as the evaluation of a chain–mail link (2.12). it follows from (2.8) that z1 can be written as nz′ 0 where n is the number of 4-simplices of t(m) and z′ 0 is the state sum given by a modification of the state sum z0 where a single 4-simplex is perturbed by the operator ˆ vσ. in order to calculate z′ 0 consider a 4-manifold m with a triangulation t whose dual complex is t ∗. given a 4-simplex σ ∈t we define an insertion i as being a choice of a pair of triangles of σ which do not belong to the same tetrahedron of σ and have therefore a single vertex in common (following (2.7) we will distinguish the order in which the triangles appear). given the colorings λf of the triangles of σ (or of the dual faces f) and the colorings ιl of the tetrahedrons of σ (or of the dual edges l), then a5(λf, ιl, i) is the evaluation of the spin network of fig. 1. 8 j. faria martins and a. mikovi ́ c figure 4. portion of the graph cmli m corresponding to a 4-simplex with an insertion. all strands are to be colored with ω, unless they intersect the insertion, as indicated. we then have: z1(m, t) = 1 5 x σ x iσ x λf ,ιl a5(λf(σ), ιl(σ), iσ) y f dim λf y v̸=v(σ) a5(λf(v), ιl(v)), (3.1) where v(σ) is the vertex of t ∗corresponding to σ. this sum is over the set of all 4-simplices σ of t(m), as well as over the set iσ of all insertions of σ and over the set of all colorings (λf, ιl) of the faces and the edges of t ∗(m) (or equivalently, a sum over the colorings of the triangles and the tetrahedrons of t(m).) the infinite sum in (3.1) is regularized by passing to the category of representations of the quantum group uq(g), where q is a root of unity. in order to calculate z1(m, t) in this case, let us represent it as an evaluation of the chain–mail link chl(m, t) [23] in the three-sphere s3. as explained in subsection 2.1, the invariant z0(m) can be represented as a multiple of the evaluation of the chain mail link chlm colored with the linear combination of the quan- tum group irreps ω= p λ(dimq λ)λ, see equation (2.12). analogously, by extending the 3- dimensional approach of [9], a chain-mail formulation for the equation (3.1) can be given. con- sider the handle decomposition of m obtained by thickening the dual cell decomposition of m associated to the triangulation t of m. one can cancel pairs of 0- and 1-handles, until a single 0-handle is left. let chlm be the associated chain-mail link (the kirby diagram of the handle decomposition). we then have z1(m, t) = 1 5 x i x λ,λ′ η−1 2 (h2+h1+h3−h4+1) dimq λ dimq λ′⟨chli m, ωh2+h1−2, λ, λ′⟩, (3.2) where, as before, an insertion i is the choice of a pair of triangles t1 and t2 in some 4-simplex of m, such that t1 and t2 have only one vertex in common. given an insertion i, the graph chli m is obtained from the chain-mail link chlm by inserting a single edge (colored with the adjoint representation of g) connecting the components of chlm (colored with λ and λ′) corresponding to t1 and t2; see fig. 4. chli m can be considered as a pair (li, γi) where li denotes the components of chli m not incident to the insertion i (which are exactly h2 +h2 −2) and γi denotes the component of chli m containing the insertion i. hence we use the notation ⟨chli m, ωh2+h1−2, λ, λ′⟩to mean the evaluation of the pair (li, γi) where all components four-dimensional spin foam perturbation theory 9 figure 5. spin network γ1(λ, λ′). here a is the adjoint representation. of li are colored with ωand the two circles of γi are colored with λ and λ′, with an extra edge connecting them, colored with the adjoint representation a. consider an insertion i connecting the triangles t1 and t2, which intersect at a single vertex. equation (3.2) coincides, apart from the inclusion of the single insertion, with the observables of the crane–yetter invariant [3] (subsection 2.1) for the pair of triangles colored with λ and λ′; see equation (2.13). by using the discussion in section 6 and theorem 2 of [3] (see equation (2.14)) one therefore obtains, for each insertion i and each pair of irreps λ, λ′: η−1 2(h2+h1+h3−h4+1)⟨chli m, ωh2+h1−2, λ, λ′⟩= zwrt(s3, γ(λ, λ′))η−3 2z0(m), (3.3) where γ(λ, λ′) is the colored link of fig. 5. in addition zwrt denotes the witten–reshetikhin– turaev invariant [27, 22] of graphs in manifolds, in the normalization of [3, 8]. note that in the notation of equation (2.14), σ2 = t1 ∪t2 is two triangles which intersect at a vertex, thus χ(σ2) = 1 and also its regular neighborhood ˆ σ2 is homeomorphic to the 4-disk, thus s(m) = s(m \ ˆ σ2). equation (3.3) follows essentially from the fact that the pair chli m = (li, γi) is a surgery presentation [14, 12] of the pair (s3, γ(λ, λ′)), apart from connected sums with s1 × s2; c.f. theorem 3, below. to see this, note that (after turning the circles associated with the 1-handles of m into dotted circles) the link li is a kirby diagram for the manifold m minus an open regular neighborhood σ′ of the 2-complex σ made from the vertices and edges of the triangulation of m, together with the triangles t1 and t2. since t1 and t2 intersect at a single vertex, any regular neighborhood ˆ σ2 of the (singular) surface σ2 spanned by t1 and t2 is homeomorphic to the 4-disk d4. therefore σ′ is certainly homeomorphic to the boundary connected sum ♮k i=1(d3×s1)
♮d4, whose boundary is #k i=1(s2×s1)
#s3, for some positive integer k. here # denotes the connected sum of manifolds and ♮denotes the boundary connected sum of manifolds. the circles c1, c2 ⊂γi associated with the triangles t1 and t2 define a link which lives in ∂ˆ σ2 ∼ = s3 ⊂ #k i=1(s2 × s1)
#s3. the two circles c1 and c2 define a 0-framed unlink in s3, with each individual component being unknotted. let us see why this is the case. we will turn the underlying handle decomposition of m upside down, by passing to the dual handle decomposition of m, where each i-simplex of the triangulation of m yields an i-handle of m; see [12, p. 107]. consider the bit p ⊂m of the handle-body yielded by the 2-complex σ, thus p is (like σ′) a regular neighborhood of σ. maintaining the 0-handle generated by the vertex t1∩t2, eliminate some pairs of 0- and 1-handles, in the usual way, until a single 0-handle of p is left. clearly ∂p ∗= ∂(m \ σ′), where ∗denotes the orientation reversal. the circles c1 and c2, in ∂p ∗, correspond now (since we considered the dual handle decomposition) to the belt-spheres of the 2-handles of p (attached along ct1 and ct2) and associated with the triangles t1 and t2. since c1 and c2 are 0-framed meridians going around ct1 and ct2 (see [12, example 1.6.3]) it therefore follows that these circles are unlinked and are also, individually, unknotted; see fig. 6. given this and the fact that the insertion i colored with a also lives in s3, it follows that chli m = (li, γi) is a surgery presentation of the pair (s3, γ(λ, λ′)), apart from the connected sums, distant from γ(λ, λ′), with s1 × s2. since the evaluation of the tadpole spin network is zero, it follows that zwrt(s3, γ(λ, λ′))= 0, and consequently 10 j. faria martins and a. mikovi ́ c figure 6. the kirby diagram for p in the vicinity of the triangles t1 and t2. we show the belt-spheres c1 and c2 of the 2-handles of p (attaching along ct1 and ct2) associated with the triangles t1 and t2. theorem 1. for any triangulation t of m we have z1(m, t) = 0. 4 the second-order correction since z1 = 0, we have to calculate z2 in an appropriate limit such that the partition function z is different from z0 and such that z is independent of the triangulation [9]. let n be the number of 4-simplices. from (2.8) we obtain z2(m) = 1 2 x λf ,ιl y f dim λf  x σ ˆ v 2 σ + x σ̸=σ′ ˆ vσ ˆ vσ′  y v a5(λf(v), ιl(v)), (4.1) where ˆ vσa5(λf(v), ιl(v)) = 1 5 x insertions i of σ a5(λf(v), ιl(v), i), (4.2) if σ is dual to v, see fig. 1. on the other hand, ˆ vσa5(λf(v), ιl(v)) = a5(λf(v), ιl(v)) if v is not dual to σ. in order to to solve the possible framing and crossing ambiguities arising from the equation (4.1), a method analogous to the one used in [9] can be employed. note that there are exactly 30 insertions in a 4-simplex σ, corresponding to pairs of triangles of σ with a single vertex in common. this is because there are exactly three triangles of σ having only one vertex in common with a given triangle of σ. analogously to the first-order correction, z2 can be written as z2(m, t) = 1 2η−1 2(h2+h1+h3−h4+1) n x k=1 ⟨chlm, ωh2+h1, ˆ v 2 k ⟩ + 1 2η−1 2 (h2+h1+h3−h4+1) x 1≤k̸=l≤n ⟨chlm, ωh2+h2, ˆ vk, ˆ vl⟩, (4.3) where the first sum denotes the contributions from two insertions ˆ v in the same 4-simplex σk and the second sum represents the contributions when the two insertions ˆ v act in different 4-simplices σk and σl. as in the previous section, we will use the handle decomposition of m with an unique 0-handle naturally obtained from the thickening of t ∗(m). note that each ⟨chlm, ωh2+h1, ˆ v 2 k ⟩corresponds to a sum over all the possible choices of pairs of insertions in the 4-simplex σk. the value of ⟨chlm, ωh2+h1, ˆ v 2 k ⟩is obtained from the evaluation of the chain-mail link chlm colored with ω, which contains g-edges carrying the adjoint representation, as in the calculation of the first-order correction. four-dimensional spin foam perturbation theory 11 figure 7. colored graph γ′ 2(λ, λ′, λ′′), a wedge graph. here a is the adjoint representation. a configuration c is, by definition, a choice of insertions distributed along a set of 4-simplices of m. given a positive integer n and a set r of 4-simplices of m, we denote by cn r the set of configurations with n insertions distributed along r. by expanding each v into a sum of insertions, the equation (4.3) can be written as: z2(m, t) = 1 50η−1 2(h2+h1+h3−h4+1) n x k=1 x c∈c2 {σk} ⟨chlc m, ωh2+h1⟩ + 1 50η−1 2(h2+h1+h3−h4+1) x 1≤k̸=l≤n x c∈c2 {σk,σl} ⟨chlc m, ωh2+h2⟩. (4.4) note that each graph chlc m splits naturally as (lc, γc), where the first component contains the circles non incident to any insertion of c. the second sum in equation (4.4) vanishes, because it is the sum of terms proportional to:  x λ,λ′ ⟨γ1(λ, λ′)⟩   2 z0(m) (4.5) and to x λ,λ′,λ′′ ⟨γ′ 2(λ, λ′, λ′′)⟩z0(m), (4.6) where γ1 is the dumbbell spin network of fig. 5, and γ′ 2 is a three-loop spin network, see fig. 7. these spin networks arise from the cases when the pair (lc, γc) is a surgery presentation of the disjoint union s3, γ1
⊔ s3, γ1
and of s3, γ′ 2
, respectively, apart from connected sums with s1 × s2; see theorem 3 below. the former case corresponds to a situation where the two insertions act in pairs of triangles without a common triangle, and the latter corresponds to a situation where the two pairs of triangles have a triangle in common, which necessarily is a triangle in the intersection σk ∩σl of the 4-simplices σk and σl. the evaluations in (4.5) and (4.6) vanish since the corresponding spin networks have tadpole subdiagrams. the first sum in (4.4) also gives the terms proportional to the ones in equations (4.5) and (4.6). these terms correspond to two insertions connecting two pairs made from four distinct triangles of σk and to two insertions connecting two pairs of triangles made from three distinct triangles of σk, respectively. all these terms vanish. the non-vanishing terms in equation (4.4) arise from a pair of insertions connecting the same two triangles in a 4-simplex. there are exactly 30 of these. therefore, by using theorem 3 of section 5, we obtain: z2(m, t) = 3n 5 η−2 x λ,λ′ dimq λ dimq λ′⟨γ2(λ, λ′)⟩z0(m), where γ2 is a two-handle dumbbell spin network, see fig. 8. we thus have: 12 j. faria martins and a. mikovi ́ c figure 8. colored graph γ2(λ, λ′), a double dumbbell graph. as usual a is the adjoint representation. theorem 2. the second-order perturbative correction z2(m, t) divided by the number of n of 4-simplices of the manifold is triangulation independent. in fact: z2(m, t) n = 3 5η−2 x λ,λ′ dimq(λ) dimq(λ′)⟨γ2(λ, λ′)⟩z0(m). here ⟨γ2(λ, λ′)⟩denotes the spin-network evaluation of the colored graph γ2(λ, λ′). note that ⟨γ2(λ, λ′)⟩= θ(a, λ, λ)θ(a, λ′, λ′) dimq a , which is is obviously non-zero, and therefore z2(m, t) ̸= 0. 5 higher-order corrections for n > 2, the contributions to the partition function will be of the form η−1 2(h2+h1+h3−h4+1)⟨chlm, ωh1+h2, ( ˆ v1)k1 * * * ( ˆ vn)kn ⟩, where k1 + * * * + kn = n. by using the equation (4.2), each of these terms splits as a sum of terms of the form: 1 5n η−1 2(h2+h1+h3−h4+1)⟨chlc m, ωh1+h2⟩= ⟨c⟩, (5.1) where c is a set of n insertions (a "configuration") distributed among the n 4-simplices of the chosen triangulation of m, such that the 4-simplex σi has ki insertions. insertions are added to the chain-mail link chlm as in fig. 4, forming a graph chlc m (for framing and crossing ambiguities we refer to [9]). note that each graph chlc m splits as (lc, γc) where lc contains the components of chlm not incident to any insertion. as in the n = 1 and n = 2 cases, equation (5.1) coincides, apart from the extra insertions, with the observables of the crane–yetter invariant defined in [3]; see subsection 2.1. therefore, by using the same argument that proves theorem 2 of [3] we have: theorem 3. given a configuration c consider the 2-complex σc spanned by the k triangles of m incident to the insertions of c. let σ′ c be a regular neighborhood of σc in m. then m can be obtained from m \ σ′ c by adding the 2-handles corresponding to the faces of σ (and some further 3- and 4-handles, corresponding to the edges and vertices of σ). these 2-handles attach along a framed link k in ∂(m \ σ′), a manifold diffeomorphic to ∂(σ′ c) with the reverse orientation. the insertions of c can be transported to this link k defining a graph kc in ∂(σ′∗ c) = ∂(m \ σ′ c). we have: ⟨c⟩= x λ1,...,λk zwrt ∂(σ′∗ c), kc; λ1, . . . , λk
κs(m\σ′)η χ(σ) 2 −k dimq λ1 * * * dimq λk, where s(m \ σ′) denotes the signature of the manifold m \ σ′ and χ denotes the euler charac- teristic. four-dimensional spin foam perturbation theory 13 note that, up to connected sums with s1 × s2, the pair chlc m = (lc, γc) is a surgery presentation of (∂(m \ σ′ c), kc) = (∂(σ′∗ c), kc). unlike the n = 1 and n = 2 cases, it is not possible to determine the pair (∂(σ′∗ c), kc) for n ≥3 without having an additional information about the configuration. in fact, considering the set of all triangulations of m, an infinite number of diffeomorphism classes for (m \σ′, kc) is in general possible for a fixed n; see [9] for the three dimensional case. this makes it complicated to analyze the triangulation independence of the formula for zn(m, t) for n ≥3. since z(m, t) = x n λnzn(m, t), where zn(m, t) = x k1+***+kn=n 1 k1! * * * kn!η−1 2 (h2+h1+h3−h4+1)⟨chlm, ωh1+h2, ( ˆ v1)k1 * * * ( ˆ vn)kn ⟩, in order to resolve the triangulation dependence of zn, let us introduce the quantities zn = lim n→∞ zn(m, t) n n 2 z0(m) ; (5.2) see [9, 2]. the limit is to be extended to the set of all triangulations t of m, with n being the number of 4-simplices of m, in a sense to be made precise; see [9]. from sections 3, 4 and theorem 2 it follows that z1 = 0, z2 = 3η−2 5 dimq a x λ dimq λθ(a, λ, λ) !2 . note that the values of z1 and z2 are universal for all compact 4-manifolds. the expression for z2 is finite because there are only finitely many irreps for the quantum group uq(g), of non-zero quantum dimension, when q is a root of unity. 6 dilute-gas limit we will now show how to define and calculate the limit in the equation (5.2). let m be a 4- manifold and let us consider a set s of triangulations of m, such that for any given ǫ > 0 there exists a triangulation t ∈s such that the diameter of the biggest 4-simplex is smaller than ǫ, i.e. the triangulations in s can be chosen to be arbitrarily fine. we want to calculate the limit in equation (5.2) only for triangulations belonging to the set s. furthermore, we suppose that s is such that (c.f. [9]): restriction 1 (control of local complexity-i). together with the fact that the triangulations in s are arbitrarily fine we suppose that: there exists a positive integer l such that any 4-simplex of any triangulation t ∈s intersects at most l 4-simplices of t. let us fix n and consider z2n(m, t) when n →∞. the value of z2n will be given as a sum of contributions of configurations c such that n1 insertions of ˆ v act in a 4-simplex σ1, n2 of insertions of ˆ v act in the 4-simplex σ2 ̸= σ1 and so on, such that n1 + n2 + * * * + nn = 2n and nk ≥0. 14 j. faria martins and a. mikovi ́ c a configuration for which any 4-simplex has either zero or two insertions, with all 4-simplices which have insertions being disjoint will be called a dilute-gas configuration. there will be 15n n! n!(n −n)! −δ(n, t), dilute-gas configurations, where δ is the number of pairs of 4-simplices in t with non-empty intersection. from restriction 1 it follows that δ(n, t) = o(n) as n →∞. each dilute-gas configuration contributes zn 2 z0(m) to z2n(m, t) and we can write z2n z0 =  n! n!(n −n)! −o(n)  zn 2 + x non-dilute c ⟨c⟩ z0 , (6.1) where ⟨c⟩= 1 5n η−1 2(h2+h1+h3−h4+1)⟨chlc m, ωh1+h2⟩, denotes the contribution of the configura- tion c. let us describe the contribution of the non-dilute configurations c more precisely. recall that a configuration c is given by a choice of n insertions connecting n pairs of triangles of m, where each pair belongs to the same 4-simplex of m and the triangles have only one common vertex. given a configuration c with n insertions, consider a (combinatorial) graph γc with a vertex for each triangle appearing in c and for each insertion and edge connecting the corresponding vertices. the graph γc is obtained from γc by collapsing the circles of γc of it into vertices. however note that γc is merely a combinatorial graph, whereas γc is a graph in s3, which can have a complicated embedding. if γc has a connected component homeomorphic to the graph made from two vertices and an edge connecting them, then ⟨c⟩vanishes, since in this case the embedded graph whose surgery presentation is given by (chlc m, γc) will have a tadpole. in fact, looking at theorem 3, one of the connected components of (∂(σ′∗), kc) will be (s3, γ1), where γ1 is the graph in fig. 5. consider a manifold with a triangulation with n 4-simplices, satisfying restriction 1. the number of possible configurations c with l insertions with make a connected graph γc is bounded by n(10l)l−1(l−1)!. in particular the number of non-dilute configurations v with 2n insertions and yielding a non-zero contribution is bounded by max l1+***+l2n=2n li̸=1 ∃i : li≥3 b 2n y i=1 n(10l)lk−1(lk −1)! = o n n−1
. (6.2) this is simply the statement that if a graph γc has k connected components, then it has o(n k) possible configurations. since k ≤n −1 for a non-dilute configuration, the bound (6.2) follows. we now need to estimate the value of ⟨c⟩for a non-dilute configuration c. we will need to make the following restriction on the set s. we refer to the notation introduced in theorem 3. restriction 2 (control of local complexity-ii). the set of s of triangulations of m is such that given a positive integer n then the number of possible diffeomorphism classes for the pair (∂(σ′ c), kc) is finite as we vary the triangulation t ∈s and the configuration c with n inser- tions. in the three-dimensional case a class of triangulations s satisfying restrictions 1 and 2 was constructed by using a particular class of cubulations of 3-manifolds (which always exist, see [5]) and their barycentric subdivisions, see [9]. these cubulations have a simple local structure, with only three possible types of local configurations, which permits a case-by-case analysis as the cubulations are refined through barycentric subdivisions. in the case of four-dimensional four-dimensional spin foam perturbation theory 15 cubulations, no such list is known, although it has been proven that a finite (and probably huge) list exists. therefore the approach used in the three-dimensional case cannot be directly applied to the four-dimensional case. however, it is reasonable to assume that triangulations coming from the barycentric subdivisons of a cubulation of m satisfy restriction 2. more precisely, given a cubulation □of m, let ∆□be the triangulation obtained from □by taking the cone of each i-face of each cube of □, starting with the 2-dimensional faces. consider the class s = {∆□(n)}∞ n=0, where □(n) is the barycentric subdivision of order n of □. then we can see that restriction 1 is satisfied by this example, and we conjecture that s also satisfies restriction 2. the restriction 2 combined with theorem 3 implies that the value of ⟨c⟩in equation (6.1) is bounded for a fixed n, considering the set of all triangulations in the set s and all possible configurations with 2n insertions. since the number of non-dilute configurations c which have a non-zero contribution is of o(n n−1), it follows that x non-dilute c ⟨c⟩ z0 = o n n−1
. therefore lim n→∞ z2n(m, t) n nz0(m) = zn 2 n! , or z2n(m, t) z0(m) ≈zn 2 n! n n, (6.3) for large n. in the case of z2n+3, the dominant configurations for triangulations with a large number n of 4-simplices consist of configurations c (as before called dilute) whose associated combinatorial graph γc has as connected components a connected closed graph with three edges and (n −1) connected closed graphs with two edges. we can write: z2n+3 z0 = x dilute c ⟨c⟩+ x non-dilute c ⟨c⟩. since the number of dilute configurations is of o(n n), while the second sum is of o(n n−1), due to the restrictions 1 and 2, we then obtain for large n z2n+3 z0 = o(n n). more precisely z2n+3 z0 ≈z3(z2)nn (n −1)! n!(n −1 −n)!, or z2n+3 z0 ≈z3(z2)n n n n! , (6.4) for large n, where z3 is the sum of two terms. the first term is 30 6 × 53 x λ,λ′ dimq λ dimq λ′⟨γ3(λ, λ′)⟩, 16 j. faria martins and a. mikovi ́ c figure 9. colored graph γ3(λ, λ′), a triple dumbbell graph. as usual a is the adjoint representation. figure 10. colored graph γ′ 3(λ, λ′, λ′′). as usual a is the adjoint representation. where γ3 is the triple dumbbell graph of fig. 9, corresponding to three insertions connecting the same pair of triangles of the underlying 4-simplex (there are exactly 30 of these). the second term is 30 6 × 53 x λ,λ′,λ′′ dimq λ dimq λ′ dimq λ′′⟨γ′ 3(λ, λ′, λ′′)⟩, where γ′ 3 appears in fig. 10. this corresponds to three insertions making a chain of triangles, pairwise having only a vertex in common (there are exactly 30 insertions like these for each 4-simplex). 6.1 large-n asymptotics let us now study the asymptotics of z(m, t) for n →∞. we will denote z(m, t) as z(λ, n) and z0(m) as z0(λ0), in order to highlight the fact that the crane–yetter state sum z0(m) can be understood as a path integral for the bf-theory with a cosmological constant term λ0 r m⟨b ∧b⟩, such that λ0 is a certain function of an integer k0, which specifies the quantum group at a root of unity whose representations are used to construct the crane–yetter state sum. in the case of a quantum su(2) group it has been conjectured that λ0 = 4π/k0, see [25]. consequently z(λ) = z da db ei r m⟨b∧f +λb∧b⟩= z dadbei r m⟨b∧f +λ0b∧b⟩ei(λ−λ0) r m ⟨b∧b⟩, which means that our perturbation parameter is λ −λ0 instead of λ. let us consider the partial sums zp (λ, n) = p x n=0 inzn(n)(λ −λ0)n, where p = √ n. in this way we ensure that each perturbative order n in zp is much smaller than n when n is large. we can then use the estimates from the previous section, and from (6.3) four-dimensional spin foam perturbation theory 17 and (6.4) we obtain zp (λ, n) z0(λ0) ≈ p/2 x m=0 i2m(λ −λ0)2m n m m! zm 2 + (p −3)/2 x m=0 i2m+3(λ −λ0)2m+3z3(z2)m n m m! ≈  1 + iz3(λ −λ0)3 ∞ x m=0 (−1)m gm m! zm 2 =  1 + iz3(λ −λ0)3 e−gz2, where g = (λ −λ0)2n. (6.5) given that z(λ, n) = lim p →∞zp (λ, n) for |λ −λ0| < r, where r is the radius of convergence of z(λ, n) = ∞ x n=0 in(λ −λ0)nzn(n), (6.6) then z(λ, n) ≈z√ n(λ, n) for large n. therefore z(λ, n) z0(λ0) ≈  1 + iz3(λ −λ0)3 exp (−gz2) , (6.7) for λ ≈λ0, where g = (λ −λ0)2n. in the limit n →∞, λ →λ0 and g constant we obtain z(λ, n) z0(λ0) →exp(−gz2). we can rewrite this result as z(m, g) = e−gz2z0(m), (6.8) where z(m, g) is the perturbed partition function in the dilute-gas limit. this value is triangu- lation independent and it depends on the renormalized coupling constant g. note that (6.7) can be rewritten as z∗(λ, g) z0(λ0) ≈  1 + iz3(λ −λ0)3 exp (−gz2) , (6.9) where we have changed the variable n to variable g = n(λ −λ0)2 and z∗(λ, g) = z  λ, g (λ −λ0)2  . (6.10) the approximation (6.9) is valid for λ →λ0 and g (λ −λ0)2 →∞. 18 j. faria martins and a. mikovi ́ c the result (6.8) can be understood as the lowest order term in the asymptotic expansion (6.9) where g is a constant. however, one can have a more general situation where g = f(λ −λ0) such that f(λ −λ0) (λ −λ0)2 →∞, (6.11) for λ →λ0. in this case z∗(λ, g) ≈e−z2f(λ−λ0)  1 + iz3(λ −λ0)3 z0(λ0), (6.12) which opens the possibility that z∗(λ, g) ≈[z0(k)]k=φ(λ) , (6.13) where z0(k) is the real number extension of the crane–yetter state sum and k = φ(λ) is the relation between k and λ. in the case of a quantum su(2) group at a root of unity z0(m, k) = e−iπs(m)r(k), (6.14) where s(m) is the signature of m and r(k) = k(2k + 1)/4(k + 2), see [23]. the value of λ which corresponds to k is conjectured to be k = 4π/λ, see [25], and this is an example of the function φ. the relation λ ∝1/k could be checked by calculating the large-spin asymptotics of the quantum 15j-symbol for the case of the root of unity, in analogy with the three-dimensional case, where by computing the asymptotics of the quantum 6j-symbol one can find that λ = 8π2/k2, see [21]. the quantum 15j-symbol asymptotics is not known, but for our purposes it is sufficient to know that λ →0 as k →∞. equations (6.12), (6.13) and (6.14) imply f(λ −λ0) ≈iπs(m) z2 [h(λ) −h(λ0)] + iz3 z2 (λ −λ0)3, (6.15) where h(λ) = r(φ(λ)). the solution (6.15) is consistent with the condition (6.11), since h′(λ0) ̸= 0. however, f has to be a complex function, although the original definition (6.5) suggests a real function. this means that λ has to take complex values in order for (6.13) to hold, i.e. we need to perform an analytic continuation of the function in (6.9). also note that the definition (6.6) and the fact that z1(n) = 0 (theorem 1) imply that z′(λ0, n) = 0, but this does not imply that lim λ→λ0 dz∗(λ, f(λ −λ0)) dλ = 0 since z(λ, n) and z∗(λ, g) are different functions of λ due to (6.10) and g = f(λ −λ0). there- fore the approximation (6.13) is consistent. this also implies that we can define a triangulation independent z(λ, m) as the real number extension of the function z0(k, m), see (6.14). there- fore z(λ, m) = z0(k, m)|k=φ(λ) = ̃ z0(λ, m). (6.16) four-dimensional spin foam perturbation theory 19 7 relation to ⟨f ∧f ⟩theory it is not difficult to see that the equations of motion for the action (2.1) are equivalent to the equations of motion for the action ̃ s = z m ⟨f ∧f⟩, because the b field can be expressed algebraically as bi = −gijf j/(2λ). at the path-integral level, this property is reflected by the following consideration. one can formally perform a gaussian integration over the b field in the path integral (2.2), which gives the following path integral z = d(λ, m) z da exp  i 4λ z m ⟨f ∧f⟩  , (7.1) where d(λ, m) is the factor coming from the determinant factor in the gaussian integration formula. more precisely, if we discretize m by using a triangulation t with n triangles, then the path integral (2.2) becomes a finite-dimensional integral z y l,i dai l z y f,i dbi f exp  i x ∆ tr (b∆ff) + iλ 5 x σ x ∆′,∆′′∈σ ⟨b∆′b∆′′⟩  . (7.2) the integral over the b variables in (7.2) can be written as z +∞ −∞ * * * z +∞ −∞ y k,l dbkleiλ(b,qb)+i(b,f), (7.3) where m = dim a, b = (b11, . . . , bmn) and f = (f11, . . . , fmn) are vectors in rmn, (x, y ) is the usual scalar product in rmn and q is an mn × mn matrix. the integral (7.3) can be defined as the analytic continuation λ →iλ, f →if of the formula z +∞ −∞ * * * z +∞ −∞ y k,l dbkle−λ(b,qb)+(f,b) = r πmn λmn det qe(f,q−1f)/4λ, (7.4) so that when n →∞such that the triangulations become arbitrarily fine, we can represent the limit as the path integral (7.1). since r m⟨f ∧f⟩is a topological invariant of m, which is the second characteristic class number c2(m), see [7], we can write z(m, λ) = e(m, λ)eic2(m)/λ, where e(m, λ) = d(m, λ) z da and d(m, λ) denotes the (λmn det q)−1/2 factor from (7.4). as we have shown in the previous section, z(λ, m) = ̃ z0(m, λ), so that in the case of su(2) e(m, λ) = e−ic2(m)/λ−iπs(m)h(λ). (7.5) therefore one can calculate the volume of the moduli space of connections on a principal bundle provided that the relation k = φ(λ) is known for the corresponding quantum group. 20 j. faria martins and a. mikovi ́ c 8 conclusions the techniques developed for the 3d spin foam perturbation theory in [9] can be extended to the 4d case, and hence the 4d partition function has the same form as the corresponding 3d partition function in the dilute gas limit, see (6.8). the constant z2 depends only on the group g and an integer, and z2 is related to the second-order perturbative contribution, see section 5. the constant z2 appears because the constant z1 vanishes for the same reason as in the 3d case, which is the vanishing of the tadpole spin network evaluation. the result (6.8) implies that z(m, g) is not a new manifold invariant, but it is propor- tional to the crane–yetter invariant. given that the renormalized coupling constant g is an arbitrary number, a more usefull way of representing our result is the asymptotic for- mula (6.12). this formula allowed us to conclude that z(m, λ) can be identified as the crane– yetter partition function evaluated at k = φ(λ), see (6.13) and (6.16). the formula (6.12) also applies to the spin foam perturbation expansion in 3d, where z2 and z3 are given as the state sums of the corresponding 3d graphs, see [9]. therefore the formula (6.12) is the jus- tification for the conjecture made in [9], where the triangulation independent z(λ, m) was identified with the turaev–viro partition function ztv (m, k) for k = 4π2/λ2 in the su(2) case. the relation (6.16) was useful for determining the volume of the moduli space of connections on the g-principal bundle for arbitrary values of λ, given that z(m, λ) is related to the path integral of the ⟨f ∧f⟩theory, see (7.5). however, it still remains to be proved the conjec- ture that k ∝1/λ for g = su(2), while for the other groups the function k = φ(λ) is not known. note that the result (6.8) depends on the existence of a class of triangulations of m which are arbitrarily fine, but having a finite degree of local complexity. as explained in section 5 it is reasonable to assume that such a class exists, and can be constructed by considering the triangulations coming from the barycentric subdivisions of a fixed cubulation of m. our approach applies to lie groups whose vector space of intertwiners λ ⊗λ →a is one- dimensional for each irreducible representation λ. this is true for the su(2) and so(4) groups, but it is not true for the su(3) group. this can probably be fixed by adding extra information to the chain-mail link with insertions at the 3-valent vertices. also note that we only considered the gij ∝δij case. this is sufficient for simple lie groups, but in the case of semi-simple groups one can have non-trivial gij. especially interesting is the so(4) case, where gij ∝ǫabcd. in the general case one will have to work with spin networks which will have l(λ) and gij insertions, so that it would be very interesting to find out how to generalize the chain–mail formalism to this case. one of the original motivations for developing a four-dimensional spin-foam perturbation theory was a possibility to obtain a nonperturbative definition of the four-dimensional euclidean quantum gravity theory, see [10] and also [19, 20]. the reason for this is that general relativity with a cosmological constant is equivalent to a perturbed bf-theory given by the action (2.1), where g = so(4, 1) for a positive cosmological constant, while g = so(3, 2) for a negative cosmological constant and gij = ǫabcd in both cases, see [19, 20]. however, the gij in the gravity case is not a g-invariant tensor, since it is only invariant under a subgroup of g, which is the lorentz group. consequently this perturbed bf theory is not topological. in the euclidean gravity case one has g = so(5), and the invariant subgroup is so(4) since gij = ǫabcd. one can then formulate a spin foam perturbation theory along the lines of section 3. however, the chain–mail techniques cannot be used, because gij is not a g-invariant tensor and therefore one lacks an efficient way of calculating the perturbative contributions. in order to make further progress, a generalization of the chain–mail calculus has to be found in order to accommodate the case when gij is invariant only under a subgroup of g. four-dimensional spin foam perturbation theory 21 acknowledgments this work was partially supported fct (portugal) under the projects ptdc/mat / 099880/2008, ptdc/mat/098770/2008, ptdc/mat/101503/2008. this work was also partially supported by cma/fct/unl, through the project pest-oe/mat/ui0297/2011. references [1] baez j., an introduction to spin foam models of quantum gravity and bf theory, in geometry and quantum physics (schladming, 1999), lecture notes in phys., vol. 543, springer, berlin, 2000, 25–93, gr-qc/9905087. [2] baez j., spin foam perturbation theory, in diagrammatic morphisms and applications (san francisco, ca, 2000), contemp. math., vol. 318, amer. math. soc., providence, ri, 2003, 9–21, gr-qc/9910050. [3] barrett j.w., faria martins j., garc ́ ıa-islas j.m., observables in the turaev–viro and crane–yetter models, j. math. phys. 48 (2007), 093508, 18 pages, math.qa/0411281. [4] broda b., surgical invariants of four-manifolds, hep-th/9302092. [5] cooper d., thurston w., triangulating 3-manifolds using 5 vertex link types, topology 27 (1988), 23–25. [6] crane l., yetter d.a., a categorical construction of 4d topological quantum field theories, in quantum topology, ser. knots everything, vol. 3, world sci. publ., river edge, nj, 1993, 120–130, hep-th/9301062. [7] eguchi t., gilkey p.b., hanson a.j., gravitation, gauge theories and differential geometry, phys. rep. 66 (1980), 213–393. [8] faria martins j., mikovi ́ c a., invariants of spin networks embedded in three-manifolds, comm. math. phys. 279 (2008), 381–399, gr-qc/0612137. [9] faria martins j., mikovi ́ c a., spin foam perturbation theory for three-dimensional quantum gravity, comm. math. phys. 288 (2009), 745–772, arxiv:0804.2811. [10] freidel l., krasnov k., spin foam models and the classical action principle, adv. theor. math. phys. 2 (1999), 1183–1247, hep-th/9807092. [11] freidel l., starodubtsev a., quantum gravity in terms of topological observables, hep-th/0501191. [12] gompf r.e., stipsicz a.i., 4-manifolds and kirby calculus, graduate studies in mathematics, vol. 20, american mathematical society, providence, ri, 1999. [13] kauffman l.h., lins s.l., temperley–lieb recoupling theory and invariants of 3-manifolds, annals of math- ematics studies, vol. 134, princeton university press, princeton, nj, 1994. [14] kirby r.c., the topology of 4-manifolds, lecture notes in mathematics, vol. 1374, springer-verlag, berlin, 1989. [15] lickorish w.b.r., the skein method for three-manifold invariants, j. knot theory ramifications 2 (1993), 171–194. [16] mackaay m., spherical 2-categories and 4-manifold invariants, adv. math. 143 (1999), 288–348, math.qa/9805030. [17] mackaay m., finite groups, spherical 2-categories, and 4-manifold invariants, adv. math. 153 (2000), 353– 390, math.qa/9903003. [18] mikovi ́ c a., spin foam models of yang–mills theory coupled to gravity, classical quantum gravity 20 (2003), 239–246, gr-qc/0210051. [19] mikovi ́ c a., quantum gravity as a deformed topological quantum field theory, j. phys. conf. ser. 33 (2006), 266–270, gr-qc/0511077. [20] mikovi ́ c a., quantum gravity as a broken symmetry phase of a bf theory, sigma 2 (2006), 086, 5 pages, hep-th/0610194. [21] mizoguchi s., tada t., three-dimensional gravity from the turaev–viro invariant, phys. rev. lett. 68 (1992), 1795–1798, hep-th/9110057. [22] reshetikhin n., turaev v.g., invariants of 3-manifolds via link polynomials and quantum groups, invent. math. 103 (1991), 547–597. 22 j. faria martins and a. mikovi ́ c [23] roberts j., skein theory and turaev–viro invariants, topology 34 (1995), 771–787. [24] rourke c.p., sanderson b.j., introduction to piecewise-linear topology, reprint, springer study edition, springer-verlag, berlin – new york, 1982. [25] smolin l., linking topological quantum field theory and nonperturbative quantum gravity, j. math. phys. 36 (1995), 6417–6455, gr-qc/9505028. [26] turaev v.g., quantum invariants of knots and 3-manifolds, de gruyter studies in mathematics, vol. 18, walter de gruyter & co., berlin, 1994. [27] witten e., quantum field theory and the jones polynomial, comm. math. phys. 121 (1989), 351–399.

0911.1701

search for chaos in neutron star systems: is cyg x-3 a black hole?

the accretion disk around a compact object is a nonlinear general relativistic system involving magnetohydrodynamics. naturally the question arises whether such a system is chaotic (deterministic) or stochastic (random) which might be related to the associated transport properties whose origin is still not confirmed. earlier, the black hole system grs 1915+105 was shown to be low dimensional chaos in certain temporal classes. however, so far such nonlinear phenomena have not been studied fairly well for neutron stars which are unique for their magnetosphere and khz quasi-periodic oscillation (qpo). on the other hand, it was argued that the qpo is a result of nonlinear magnetohydrodynamic effects in accretion disks. if a neutron star exhibits chaotic signature, then what is the chaotic/correlation dimension? we analyze rxte/pca data of neutron stars sco x-1 and cyg x-2, along with the black hole cyg x-1 and the unknown source cyg x-3, and show that while sco x-1 and cyg x-2 are low dimensional chaotic systems, cyg x-1 and cyg x-3 are stochastic sources. based on our analysis, we argue that cyg x-3 may be a black hole.

introduction x-ray binary systems vary on timescales rang- ing from months to milli-seconds (see, e.g., (chen et al. 1997; paul et al. 1997; nowak et al. 1999; cui 1999; gleissner et al. 2004; axelsson 2008)). detailed analysis of their temporal variability and fluctuation provides important insights into the geom- etry and physics of emitting regions and the accretion process. however, the origin of variability is still not clear. it could be due to varying external parameters, like the infalling mass accretion rate. it could also be due to possible instabilities in the inner regions of the accretion disk where the flow is expected to be nonlinear and turbulent. uttley et al. (2005) (see also timmer et al. 2000 and thiel et al. 2001) argued that the non-linear behavior of a system can be understood from the log-normal distribution of the fluxes and the rms-flux relation. this implies that the temporal behavior of the system may be driven by underlying stochastic variations. by studying the underlying non- linear behavior, important constraints can be obtained on these various possibilities. an elegant way of obtaining the constraint is to per- form the nonlinear time series analysis of observed data and to compute the correlation dimension d2 in a non- subjective manner. this technique has already been used to diverse situations ((grassberger & procaccia 1983a; grassberger & procaccia 1983b; schreiber 1999; aberbandel 1996; serre et al. 1996; misra et al. 2004; misra et al. 2006; harikrishnan et al. 2006), and refer- ences therein). by obtaining d2 as a function of the embedding dimension m, one can infer the origin of the variability. for example, d2 ≈m for all m corresponds to the system having stochastic fluctuation which favors 1 bidya [email protected] 2 [email protected] 3 [email protected] the idea that x-ray variations are driven by variations of some external parameters. on the other hand, a saturated d2 to a finite (low) value, beyond a certain m, implies a deterministic chaos which argues in favor of inner disk instability. however, to implement the algorithm successfully, the system in question should provide enough data. the technique was used earlier to understand the nonlinear nature of a black hole system cyg x- 1 (unno et al. 1990) and an active galactic nucleus (agn) ark 564 (gliozzi et al. 2002), but due to in- sufficient data points the analyses were hampered and no concrete conclusions were made about d2. later on, another black hole system grs 1915+105 was analyzed (misra et al. 2004; misra et al. 2006; harikrishnan et al. 2006) which was shown to display low dimensional chaos in certain temporal classes, while stochastic in other classes. however, so far none of the neutron star systems have been analyzed in detail in order to understand the origin of nonlinearity. decades back, voges et al. (1987) attempted to understand the chaotic nature of her x-1, but the analysis was hampered by low signal to noise ratio (norris & matilsky 1989). since then the investigation of chaotic signature in neutron stars remains unattended. can a neutron star system not be deterministic? indeed several features of x-ray binary systems consisting of a neutron star, such as their magne- tosphere and khz quasi-periodic oscillation (qpo) and its possible relation to the spin frequency of the neutron star, favor the idea that they exhibit nonlinear reso- nance (e.g. (blaes et al. 2007; mukhopadhyay 2009)). while the qpo itself is a mysterious feature whose origin is still unclear, its possible link to the spin frequency of the neutron star4 indicates the origin 4 however, some authors (mendez & belloni 2007) suggested that the khz qpos may not be related to the spin. 2 karak, dutta, mukhopadhyay of qpo to be from nonlinear phenomena. several lmxbs having a neutron star exhibit twin khz qpos (mendez et al. 1998; van der klis 2006). for some of them, e.g. 4u 1636-53 (jonker et al. 2002), ks 1731-260 (smith et al. 1997), 4u 1702- 429 (markwardt et al. 1999), 4u 1728-34 (van straaten et al. 2002), the spin frequency of the neutron star has been predicted from observed data. however, for the source sco x-1, which exhibits no- ticeable time variability (mendez & van der klis 2000), while we observe twin khz qpos, we do not know the spin frequency yet (but see (mukhopadhyay 2009)). for another neutron star cyg x-2, we do observe khz qpos (wijnands et al. 1998) as well. several black holes also exhibit qpos, e.g. grs 1915+105 (belloni et al. 2001; mcclintock & remillard 2006), cyg x-1 (angelini et al. 1994). in the present paper, we first aim at analyzing the time series of two neutron star sources sco x-1 and cyg x-2 to understand if a neutron star is a deterministic nonlinear (chaotic) system. then we try to manifest the knowl- edge of nonlinear (chaotic/random) property of compact sources to distinguish a black hole from a neutron star. subsequently, knowing their difference based on the said property, we try to identify the nature of a unknown source (whether it is a black hole or a neutron star). while the nature of some sources, as mentioned above, has already been predicted based on alternate method, for some others, e.g. cyg x-3, ss433, it has not yet been confirmed. for the present purpose, we therefore concentrate on three additional sources cyg x-1, cyg x-2 and cyg x-3. while cyg x-1 has been predicted to be a black hole and cyg x-2 be a neutron star, the nature of cyg x-3 is not confirmed yet. some authors (ergma & yungelson 1998; schmutz et al. 1996; szostek & zdziarski 2008) argued for a black hole nature of cyg x-3, on the basis of its jet, the time variations in the infrared emission lines, the bepposax x-ray spectra and so on. however, earlier it was argued for a neutron star (chadwick et al. 1985) by measuring its 1000 gev γ-rays which suggests a pulsar period of 12.5908 ± 0.0003 ms. by analyzing the time series and computing the correlation dimension d2, here we aim at pinpointing the nature of cyg x-3: whether a black hole or a neutron star. in the next section, we briefly outline the procedure to be followed in understanding the nonlinear nature of a compact object from observed data and to implement it to analyze the neutron star source sco x-1. in §3, we then describe nonlinear behaviors of cyg x-1, cyg x-2 and cyg x-3. subsequently, in §4, we compare all the results and argue for a black hole nature of cyg x-3. finally, we summarize in §5. 2. procedure and nonlinear nature of sco x-1 the method to obtain d2 is already established (see, e.g., (grassberger & procaccia 1983b; misra et al. 2004; harikrishnan et al. 2006)). therefore, here we discuss it briefly. we consider pca data of the rxte satellite (see table 1 for the observations ids) from the archive for our analysis. we process the data using the ftools software. we extract a few continuous data streams of 2500 −3500 sec long. the time resolution used to gen- erate lightcurves is ∼0.1 −1 sec. this is the range of optimum resolution, at least for the sources we consider, to minimize noise without losing physical information of the sources. a finer time resolution would be poisson noise dominated and a larger binning might give too few data points to derive physical parameters from it (see misra et al. 2004, 2006, for details). then we calculate the correlation dimension accord- ing to the grassberger & procaccia (1983a,b) algorithm. from the time series s(ti) (i = 1,2,...,n), we construct an m dimensional space (called embedding space ), in which any vector has the following form: x(ti) = [s(ti), s(ti + τ), ........., s(ti + (m −1)τ)], (1) where τ is the time delay chosen in such a way that each component of the vector x(ti) is independent of each other. for a particular choice of embedding dimension m, we compute the correlation function: cm(r) = 1 n(nc −1) n x i=1 nc x j=1,j̸=i θ(r −|xi −xj|), (2) which is basically the average number of points within a hypersphere of diameter r, where θ is a heaviside step function, n the total number of points and nc the num- ber of centers. if the system has a strange attractor, then one can show that for a small value of r d2(m) = d log cm(r) d log r . (3) in this numerical calculation, we divide the whole phase space into m cubes of length r around a point and we count the average number of data points in these cubes to calculate cm(r). the edge effects, which come due to the finite number of data points, have been avoided by calculating cm(r) in the range rmin < r < rmax, where rmin is the value of r for cm(r) just greater than one and rmax can be found by restricting the m cubes to be within the embedding space. in fig. 1, we show the variation of log(cm(r)) with log(r) for different values of m for sco x-1 data. d2(m) can be calculated from the linear part of the log(cm(r)) vs. log(r) curve and its value depends on the value of m. for a stochastic system, d2 ≈m for all m. on the other hand, for a chaotic or deterministic system, initially d2(m) increases linearly with the increase of m, then it reaches a certain value and saturates. this satu- rated value of d2 is taken to be the correlation dimension of the system which is a non-integer. the standard de- viation gives the error in d2. we first concentrate upon the neutron star source sco x-1. in figs. 2a,b we show respectively the lightcurve and the variation of d2 as a function of m and find that d2 saturates to a value 2.6 (± 0.8). as this is a non-integer, the system might be chaotic. on the other hand, we know that the lorenz attrac- tor is an example of an ideal chaos with d2 = 2.05. therefore, sco x-1 may be like a lorenz system. but due to noise its d2 seems appearing slightly higher (misra et al. 2004; misra et al. 2006) than the actual value. however, one should be cautious about the fact that sco x-1 is a bright source (much brighter than other sources considered later). hence, the dead time effect on search for chaos in neutron stars: is cyg x-3 a black hole? 3 the detector might affect the actual value of saturated d2 and the computed value might be slightly different than the actual one. however, this can not rule out the signature of chaos in sco x-1, particularly because the corresponding count rates are confined in the same order of magnitude and hence the dead time effect, if any, is expected to affect all the count rates in a similar way. however, a saturated d2 is necessary but not a suf- ficient evidence for chaos. existence of color noise (for which the power spectrum p(ν) ∝ν−α, where the power spectral indices α = 0, 1 and 2 correspond to "white", "pink" and "red" noise respectively) into a stochastic system might lead to a saturated d2 of low value as well (e.g. osborne & provenzale 1989; theiler et al 1992; misra et al. 2006; harikrish- nan et al. 2006). therefore, it is customary to an- alyze data by alternate approach(s) to distinguish it from a pure noisy time series (kugiumtzis 1999). one of the techniques is the surrogate data analysis (e.g. (schreiber & schmitz 1996)), which has been described earlier in detail and implemented for a black hole (misra et al. 2006; harikrishnan et al. 2006). in brief, surrogate data is random data generated by taking the original signal and reprocessing it so that data has the same/similar fourier power spectrum and autocorrela- tion along with the same distribution, mean and vari- ance as of the original data, but has lost all deterministic characters. then the same analysis is carried out to the original data and the surrogate data to identify any dis- tinguishable feature(s) between them. the scheme pro- posed by schreiber & schmitz (1996), known as iter- ative amplitude-adjusted fourier transform (iaaft), is more consistent to generate surrogate data. figures 2c-f compare results for the original data with the surrogate data. it is clear that while distributions and power spectra are same/similar for both the data sets, d2 is much higher for the surrogate data which suggests existence of low dimensional chaos in sco x-1 with d2 ∼2.6. this confirms, for the first time to best of our knowledge, a neutron star source to display chaotic behavior. as the existence of chaos is a plausible signa- ture of instability in the inner region of accretion flows which is nonlinear and turbulent, as mentioned in §1, the corresponding qpo, which is presumably an inner disk phenomenon as well, is expected to be governed by non- linear resonance mechanisms (e.g. mukhopadhyay 2009). in table 1, we enlist the average counts < s >, its root mean square (rms) variation √ < s2 > −< s >2/ < s >, the expected poisson noise < pn > ≡ √ < s >, and the ratio of the expected poisson noise to the rms value for all sources. it clearly shows a strong correla- tion between the inferred behavior of the systems and the ratio of the expected poisson noise to the rms value. 3. nonlinearity of cyg x-1,2,3 we now look into three additional compact sources: cyg x-1 (black hole), cyg x-2 (neutron star) and cyg x- 3 (nature is not confirmed yet), and apply the same analysis as in the case of sco x-1. figure 3b shows that d2 for cyg x-1 seems not to saturate and ap- pears very high. however, there is no surprise in it because its variability is similar to the temporal class χ of the black hole grs 1915+105 which was shown to be poisson noise dominated and stochastic in nature (misra et al. 2004). indeed, earlier analysis of cyg x-1 data, while it could not conclusively quantify the under- lying chaotic behavior due to insufficient data, revealed very high dimensional chaos. moreover a large < pn > (as well < pn > /rms) for cyg x-1, compared to that for sco x-1 given in table 1, reveals the system to be noise dominated. this ensures cyg x-1 to be different from sco x-1. however, the variation of d2 as a func- tion of m for the original data does not deviate notice- ably from that of corresponding surrogate data, as shown in fig. 3c, which argues that cyg x-1 is not a chaotic system. figures 4b,c show that d2 for cyg x-2 saturates to a low value ∼4, which is significantly different than that of corresponding surrogate data. the power spectra and distributions, on the other hand, for original and surro- gate data are same/similar (as shown in sco x-1, not repeated further). the saturated d2 for cyg x-2 is al- most double than that of lorenz system, possibly due to high poisson noise to rms ratio (see table-1). this sug- gests the corresponding system to be a low dimensional chaos. from figs. 5b,c we see that for cyg x-3 the varia- tions of d2 as a function of m for original and surrogates data are similar to that of cyg x-1. this confirms that the behavior of the unknown source cyg x-3 is similar to that of the black hole source cyg x-1 (see, however, (axelsson et al. 2008)). note from table 1 that cyg x-1, x-2, x-3 are significantly noise dominated compared to sco x-1. although noise could not suppress the chaotic signature in the neutron star cyg x-2, its saturated d2 is higher than that of sco x-1. on the other hand, even though the poisson noise to rms ratio in cyg x-1 is lower than that in cyg x-2 (but poisson noise itself is higher in cyg x-1), its d2 never saturates, which confirms the source to be non-chaotic; the apparent stochastic signa- ture is not due to poisson noise present into the system. 4. comparison between cyg x-1, cyg x-2 and cyg x-3 finally, we compare the variations of d2 for all three cases of cygnus in fig. 6. remarkably we find that d2 values for cyg x-1 and cyg x-3 practically overlap, appearing much larger compared to that for cyg x-2 which is shown to be a low dimensional chaotic source. on the other hand, cyg x-2 is a confirmed neutron star and cyg x-1 a black hole. therefore, cyg x-3 may be a black hole. 5. summary the source cyg x-3, whose nature is not confirmed yet, seems to be a black hole based on the analysis of its non- linear behavior. on the other hand, we have shown, for the first time to best of our knowledge, that neutron star systems could be chaotic in nature. the signature of de- terministic chaos, which argues in favor of inner disk in- stability, into an accreting system has implications in un- derstanding its transport properties particularly in ke- plerian accretion disk (winters et al. 2003). note that in keplerian accretion disks transport is necessarily due to turbulence in absence of significant molecular viscos- ity. the signature of chaos confirms instability and then plausible turbulence. on the other hand, for a rotating neutron star having a magnetosphere, signature of chaos 4 karak, dutta, mukhopadhyay suggests their qpos to be nonlinear resonance phenom- ena (mukhopadhyay 2009). the absence of chaos and related/plausible signature of instability in cyg x-1 and cyg x-3 suggests the underlying accretion disk to be sub-keplerian (narayan & yi 1995; chakrabarti 1996) in nature which is dominated significantly by gravita- tional force. this work is partly supported by a project (grant no. sr/s2/hep12/2007) funded by department of science and technology (dst), india. also the financial sup- port to one of the authors (jd) has been acknowledged. the authors would like to thank arnab rai choudhuri of iisc and the anonymous referee for carefully reading the manuscript, constructive comments and suggestions. references aberbandel, h. d. l. 1996, analysis of observed chaotic data (springer: new york) angelini, l., white, n. e., stella, l. 1994, in new horizon of x-ray astronomy, ed. f. makino, & t. ohashi (tokyo: universial academy press), 429 axelsson, m. 2008, aipc, 1054, 135 axelsson, m., larsson, s. & hjalmarsdotter, l. 2008, mnras, 394, 1544 belloni, t., m ́ endez, m., s ́ anchez-fern ́ andez, c. 2001, apj, 372, 551 blaes, o. m., srmkov ́ a, e., abramowicz, m. a., kluniak, w., torkelsson, u. 2007, apj, 665, 642 chadwick, p. m., dipper, n. a., dowthwaite, j. c., gibson, a. i. & harrison, a. b. 1985, nature, 318, 642 chakrabarti, s. k. 1996, apj, 464, 664 chen, x., swank, j. h. & taam, r. e. 1997, apj, 477, l41. cui, w. 1999, apj, 524, l59 ergma, e. & yungelson, l. r. 1998, a&a, 333, 151 gleissner, t., wilms, j., pottschmidt, k., uttley, p., nowak, m. a., & staubert, r. 2004, a&a, 414, 1091 gliozzi, m., brinkmann, w., r ̈ ath, c., papadakis, i. e., negoro, h. & scheingraber, h. 2002, a&a, 391, 875 grassberger, p. & procaccia, i. 1983, physica d, 9, 189 grassberger, p. & procaccia, i 1983, phys. rev. lett., 50, 346 harikrishnan, k. p., misra, r., ambika, g. & kembhavi, a. k. 2006, physica d, 215, 137 jonker, p. g., mendez, m., & van der klis, m. 2002, mnras, 336, l1 kugiumtzis, d. 1999, phys. rev. e, 60, 2808 markwardt, craig b., strohmayer, tod e. & swank, jean h, 1999, apj 512, l125 mcclintock, j. e., & remillard, r. a. 2006, in compact stellar x-ray sources, ed. w. h. g. lewin & m. van der klis, (cambridge: cambridge univ. press) mendez, m., & belloni, t. 2007, mnras, 381, 790 mendez, m., van der klis, m., wijnands, r., ford, e. c., van paradijis, j., & vaughan, b. a. 1998, apj, 505, l23 mendez, m. & van der klis 2000, mnras 318, 938 misra, r., harikrishnan, k. p., ambika, g. & kembhavi, a. k. 2006, apj, 643, 1114 misra, r., harikrishnan, k. p., mukhopadhyay, b., ambika, g. & kembhavi, a. k. 2004, apj, 609, 313 mukhopadhyay, b. 2009, apj, 694, 387 narayan, r. & yi, i. 1995, apj, 452, 710 norris, j. p. & matilsky, t. a. 1989, apj, 346, 912 nowak, m. a., vaughan, b. a., wilms, j., dove, j. b. & begelman, m. c. 1999, apj, 510, 874 osborne, a. r. & provenzale, a. 1989, phy. d, 35, 357 paul, b., agrawal, p. c., rao, a. r., vahia, m. n., yadav, j. s., marar, t. m. k., seetha, s., kasturirangan, k. 1997, a&a 320 l37 schmutz, w., geballe, t. r. & schild, h. 1996, a&a, 311, 25 schreiber, t. 1999, phys. rep., 308, 1 schreiber, t. & schmitz, a. 1996, phys. rev. lett., 77, 635 serre, t., kollath, z. & buchler, j. r. 1996, a&a, 311, 833 smith, d. a., morgan, e. h., & bradt, h. 1997, apj, 479, 137 szostek, a. & zdziarski, a. a. 2008, mnras, 386, 593 theiler, j., eubank, s., longtin, a., galdrikian, b., doyne, f. j. 1992, physica d, 58, 77 unno, w., et al. 1990, pasj, 42, 269 timmer, j, schwarz, u & voss, h. u et al. 2000, phys. rev. e, 61, 1342 thiel, m., romano, m & schwarz, u et al. 2001, a&a suppl. 276, 187 uttley, p., mchardy, i. m., vaughan, s. 2005, mnras, 359, 345 van der klis, m. 2006, adspr, 38, 2675 van straaten, s., van der klis, m., di salvo, t., & belloni, t. 2002, apj, 568, 912 voges, w., atmanspacher, h., & scheingraber, h. 1987, apj, 320, 794 winters, w. f., balbus, s. a., & hawley, j. f. 2003, mnras, 340, 519 wijnands, r., homan, j., van der klis, m., kuulkers, e., van paradijs, j., lewin, w. h. g., lamb, f. k., psaltis, d. & vaughan, b. 1998, apj, 493, l87 search for chaos in neutron stars: is cyg x-3 a black hole? 5 4 5 6 7 8 9 10 11 −15 −10 −5 0 log (cm) log (r) m = 2 m = 4 m = 6 m = 8 m = 12 m = 14 m = 10 fig. 1.- variation of log (cm ) as a function of log( r) for different embedding dimensions. the linear scaling range is used to calculate the correlation dimension. table 1 observed data source obs. i. d. < s > rms < p n > < p n > /rms behavior sco x-1 91012-01-02-00 58226 0.074 0.004 0.054 c cyg x-1 10512-01-09-01 10176 0.261 0.031 0.119 nc/s cyg x-2 10063-10-01-00 4779 0.075 0.014 0.191 c cyg x-3 40061-01-07-00 3075 0.125 0.057 0.455 nc/s columns:- 1: name of the source, 2: rxte observational identification number from which the data has been extracted. 3: the average count in the lightcurve < s > 4: the root mean square variation in the lightcurve, rms. 5: the expected poisson noise variation, < p n >≡ √ < s >. 6: the ratio of the expected poisson noise to the actual root mean square variation 7: the behavior of the system (c: chaotic behavior; s: stochastic behavior; nc: nonchaotic behavior) 6 karak, dutta, mukhopadhyay 38.65745 38.65755 38.65765 38.65775 45 50 55 60 65 70 t/107 rate/103 (a) 5.0 5.5 6.0 6.5 0 50 100 150 200 n rate/104 (d) 0 2 4 6 8 10 12 0 2 4 6 8 10 12 m d2 (b) 10 −3 10 −2 10 −1 10 0 10 −8 10 −6 10 −4 10 −2 ν power (e) 10 −3 10 −2 10 −1 10 0 10 −8 10 −6 10 −4 10 −2 ν power (f) 0 2 4 6 8 10 0 2 4 6 m d2 (c) fig. 2.- sco x-1: (a) variation of count rate as a function of time in units of 107 sec (lightcurve), without subtracting the initial observation time. (b) variation of correlation dimension, along with error bars, as a function of embedding dimension for original data. the solid line along the diagonal of the figure indicates an ideal stochastic curve. (c) variation of correlation dimension as a function of embedding dimension for original (points) and corresponding surrogate (dashed lines) data. (d) variation of number of count rate as a function of count rate itself in units of 104 sec−1 (distribution) for original (solid line) and surrogate (points) data. power-spectra for (e) original and (f) surrogate data. search for chaos in neutron stars: is cyg x-3 a black hole? 7 7.7709 7.7710 7.7711 0 10 20 30 t/107 rate/103 (a) 0 0.5 1.0 1.5 2.0 2.5 0 500 1000 1500 2000 n rate/104 (d) 0 2 4 6 8 10 12 0 2 4 6 8 10 12 m d2 (b) 10 0 10 −5 10 −4 10 −3 10 −2 ν power (f) 10 −2 10 −1 10 0 10 1 10 −5 10 −4 10 −3 10 −2 ν power (e) 0 2 4 6 8 10 0 2 4 6 8 m d2 (c) fig. 3.- cyg x-1: same as fig. 2. 8 karak, dutta, mukhopadhyay 32.3262 32.3263 32.3264 32.3265 3 4 5 6 t/107 rate/103 (a) 0 2 4 6 8 10 12 0 2 4 6 8 10 12 m d2 (b) 0 2 4 6 8 10 0 2 4 6 8 m d2 (c) fig. 4.- cyg x-2: (a) variation of count rate as a function of time in units of 107 sec (lightcurve). (b) variation of correlation dimension, along with error bars, as a function of embedding dimension for original data. the solid line along the diagonal of the figure indicates an ideal stochastic curve. (c) variation of correlation dimension as a function of embedding dimension for original (points) and corresponding surrogate (dashed lines) data. 39.0451 39.0452 39.0453 39.0454 39.0455 0 1 2 3 4 5 t/107 rate/103 (a) 0 2 4 6 8 10 12 0 2 4 6 8 10 12 m d2 (b) 0 2 4 6 8 10 0 2 4 6 8 m d2 (c) fig. 5.- cyg x-3: same as fig. 4. search for chaos in neutron stars: is cyg x-3 a black hole? 9 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 m d2 cyg x−1 cyg x−2 cyg x−3 fig. 6.- comparison of the variation of correlation dimension as a function of embedding dimension between cyg x-1 (open circle), cyg x-2 (star), cyg x-3 (cross).

0911.1703

lorenz cycle for the lorenz attractor

in this note we study energetics of lorenz-63 system through its lie-poisson structure.

introduction in 1955 e. lorenz [1] introduced the concept of energy cycle as a powerful instrument to understand the nature of atmospheric circulation. in that context conversions between potential, kinetic and internal energy of a fluid were studied using atmospheric equations of motion under the action of an external radiative forcing and internal dissipative processes. following these ideas, in this paper we will illustrate that chaotic dynamics governing lorenz-63 model can be described introducing an appropriate energy cycle whose components are kinetic, potential energy and casimir function derived from lie-poisson structure hidden in the system; casimir functions, like enstrophy or potential vorticity in fluid dynamical context, are very useful in analysing stability conditions and global description of a dynamical system. a typical equation describing dissipative-forced dynamical systems can be written in einstein notation as: { } , i i ij j i x x h x f = −λ + & n i ... 2 , 1 = (1) equations (1) have been written by kolmogorov, as reported in [2], in a fluid dynamical context, but they are very common in simulating natural processes as useful in chaos synchronization [3]. here, antisymmetric brackets represent the algebraic structure of hamiltonian part of a system described by function h , and a cosymplectic matrix j [4], { } g f j g f k i ik ∂ ∂ = , . (2) positive definite diagonal matrix λ represents dissipation and the last term f represents external forcing. such a formalism, as mentioned before, is particularly useful in fluid dynamics [5], where navier-stokes equations show interesting properties in their hamiltonian part (euler equations). moreover, finite dimensional systems as (1) represent the proper reduction of fluid dynamical equations [6], in terms of conservation of the symplectic structures in the infinite domain [7]. method of reduction, contrary to the classical truncation one, leads to the study of dynamics on lie algebras, i.e to the study of lie-poisson equations on them, which are extremely interesting from the physical viewpoint and with a mathematical aesthetical appeal [8,9]. given a group g and a real-valued function (possibly time dependent), r g → * : e t h , which plays the role of the hamiltonian, in the local co-ordinates i x the lie-poisson equations read as h x c x k j j ik i ∂ = & , (3) where tensor j ik c represents the constants of structure of the lie algebra g and the cosymplectic matrix assumes the form ( ) j ik ik j j c x = x . it is straightforward to show that, in this formalism g is endowed with a poisson bracket characterized by expression (2) for functions ( ) * g ∞ ∈c g f, . casimir functions c are given by the kernel of bracket (2), i.e. { } ( ) * g ∞ ∈ ∀ = c g g c , 0 , , therefore they represent the constants of motion of the hamiltonian system, { } , 0 c c h = = & ; moreover they define a foliation of the phase space [10] . ii. lorenz system and its geometry here we will be interested in ( ) 3 so g = with i i jk ij x j ε = , where ijk ε stands for the levi-civita symbol; in case of a quadratic hamiltonian function, 1 2 ik i k k x x = ω (4) they represent the euler equations for the rigid body, with casimir 1 2 ij i j c x x δ = and relative foliation geometry 2 s . in a previous paper [11] it has been shown that also the famous lorenz-63 system [12] ⎪ ⎩ ⎪ ⎨ ⎧ − = − + − = + − = 3 2 1 3 2 1 3 1 2 2 1 1 x x x x x x x x x x x x β ρ σ σ & & & (5) where 8 10, , 28 3 σ β ρ = = = , can be written in the kolmogorov formalism as in (1), ( ) ⎪ ⎩ ⎪ ⎨ ⎧ + − − = − − − = + − = σ ρ β β σ σ σ 3 2 1 3 2 1 3 1 2 2 1 1 x x x x x x x x x x x x & & & . (6) assuming the following axially symmetric gyrostat [13] hamiltonian with rotational kinetic energy k and a linear potential ( ) k k k u x x ω = will be written as h k u = + (7) with: 1 2 3 1 0 0 0 2 0 0 0 2 ω = ⎡ ⎤ ⎢ ⎥ ω = ω = ⎢ ⎥ ⎢ ⎥ ω = ⎣ ⎦ inertia tensor, 1 2 3 0 0 0 1 0 0 0 σ β λ = ⎡ ⎤ ⎢ ⎥ λ = λ = ⎢ ⎥ ⎢ ⎥ λ = ⎣ ⎦ dissipation, internal forcing given by an axisymmetric rotor [ ] 0,0,σ = ω and external forcing ( ) 0,0, β ρ σ = − + ⎡ ⎤ ⎣ ⎦ f . in order to distinguish effects of different terms in the energy cycle we leave the notation [ ] 0,0,ω = ω assuming that for numerical values in lorenz attractor ω σ = . casimir function ( ) c t , that is constant in the conservative case, will give a useful geometrical vision to the understanding of dynamical behaviour of (6). this is given by studying fixed points of ( ) i i c t x c = ∂ & & , which defines an invariant triaxial ellipsoid 0 ξ with center 0,0, 2 ρ σ + ⎧ ⎫ − ⎨ ⎬ ⎩ ⎭ and axes , , 2 2 2 f f f a b c β βσ β = = = , having equation 0 ij i j i i x x f x −λ + = . (8) fixed points of system (6) ( ) ( ) { } 1 , 1 , 1 β ρ β ρ σ = ± − ± − − − ± x and ( ) { } 0,0, ρ σ = − + 0 x belong to 0 ξ . computing the flux density of vector field = u x & through this manifold ( ) i i u φ = ∂ξ u , because of reflection symmetry i i x x →− , 1,2 i = of equations (6), two symmetric regions are identified by respectively, 0 φ < and 0 φ > . lorenz attractor ψintersects the manifold in these regions at maxima and minima of casimir function, entering the ellipsoid trough ( ) min c twice where 0 φ < , right r ψ and left lobe l ψ , and symmetrically exiting trough ( ) max c twice where 0 φ > , ( ) ( ) { } min ,max c c ∩ = 0 ψ ξ (fig.1). fig.1 invariant ellipsoid 0 ξ intersecting the attractor in the set ( ) ( ) { } min ,max c c ∩ = 0 ψ ξ . casimir maxima (red) and minima (blue) are shown. black stars represent 2 of the 3 fixed points. note that 0 x lies on the southern pole. in order to show points on 0 ξ , only part of trajectory is shown. the particular choice of parameters 8 10, 28, 3 σ ρ β = = = has moreover the peculiarity for ( ) max c to be an ordered set [9]. this property gives the opportunity to find a range of variability for the maximum radius of the casimir sphere ( ) r t c = , ( ) min 0 max r r r t r r ± = ≤ ≤ = , (9) where ( ) ( ) 2 2 1 1 r β ρ σ ± = − + + and 0 r ρ σ = + ; moreover ellipsoid 0 ξ defines a natural poincarè section for lorenz equations with interesting properties in the associated return map for casimir maxima [14]. iii. energy cycle in order to introduce an energy cycle into system (1), we first consider pure conservative case. here dynamics lies on 2 s and we note that introduction of potential u produces a deformation on the geodesics trajectory [15] of the riemannian metric given by the quadratic form (4). adding the spin rotor 3 x ω , the centre of the ellipsoid of revolution is shifted from the origin, that remains the centre of the casimir sphere of radius 2c . as regards fixed points, given at 0 ω = by two isolated centers, namely the poles ( ) 2 ,0,0 c ± of 2 s , and all points belonging to 'equatorial circle' ( ) 0, 2 sin , 2 cos c c θ θ , introduction of potential ( ) 3 u x reduces this set to four equilibrium points located at ( ) 1,2 0,0, 2 f c = ± and ( ) 2 3,4 2 ,0, f c ω ω = ± − − . this last pair starts to migrate into the south pole of 2 s as ω grows, and disappears for 2c ω > giving rise to an oyster bifurcation [16], leaving two stable centers 1,2 f . lie-poisson structure of the system permits to analyze nonlinear stability of these two points introducing pseudoenergy functions [17] 1,2 1,2 i h c λ = + , where 1,2 λ are the solutions of equation 1,2 1,2 i i f f h c λ ∂ = −∂ (10) at fixed points ( ) 1,2 0,0, 2 f c = ± . computation of quadratic form 1,2 , i j f i ∂ shows 1 f as a maximum and 1 f as a minimum for 2c ω > . we point out that introduction of potential reduces number of fixed points on the sphere to the minimal number 2; therefore it stabilizes the system's dynamics. total energy e , identified with hamiltonian, does not change and in terms of k and u , a simple energy cycle, similar to the classical oscillator, can be described using bracket formalism (4) with relative rules { } { } ( ) , , , u h u k u k = =c and { } { } ( ) , , , k h k u k u = =c ( ) ( ) , , 0 k k u u u k c ⎧ = ⎪ ⎪ = ⎨ ⎪ = ⎪ ⎩ & & & c c , (11) where ( ) , u k c is positive if energy is flowing from u to k ; the net rate of conversion of potential into kinetic energy factor is of the form ( ) ( ) 1 2 1 2 , u k x x ω = ω −ω c (12) due to the linear dependence of u on 3 x . as a result, a symmetry between quadrants i and iii holds since ( ) 1 2 0 , 0 x x u k > ⇒ < c corresponding to a net conversion of kinetic energy into potential one k u → ; the opposite happens in quadrants ii and iv where ( ) 1 2 0 , 0 c x x u k < ⇒ > and u k → . following the ideas of extending the algebraic formalism of hamiltonian dynamics to include dissipation [18] , we introduce a lyapunov function ( ) ( ) ( ) * 3 l x c so ∞ ∈ , together with a symmetric bracket , ik i k f l f g l f = = ∂ ∂ & (13) where ( ) ik g x generally is a symmetric negative matrix. taken alone, formalism (13) gives rise to a gradient system dynamics , i i ik k x l x g l = = ∂ & (14). including lie-poisson structure (2), it is possible to study equations (11) adding various kinds of dissipation models depending on the choice of 'metric tensor' ( ) ik g x ( ) ( ) , , , , , k k u l k u u k l u c l c ⎧ = + ⎪ ⎪ = + ⎨ ⎪ = ⎪ ⎩ & & & c c (15). because of ( ) , 0 c h = c last equation in (15) describes the contraction of the manifold where motion takes place. in order to find a dissipation process that naturally takes into account the compact and semisimple structure of ( ) 3 so , we use the so called cartan-killing dissipation derived form cartan-killing metric [18[ 1 2 n m ik im kn g ε ε = (16) physically with this choice, for l c α = and α + ∈r , dynamics reduces to an isotropic linear damping ik ik g δ = − of both energy and casimir functions, miming a rayleigh dissipation; in this way trajectories approache a stable fixed point at the origin ( ) 0 0,0,0 = x . because of λ term of lorenz equations, we can introduce a lyapunov function with anisotropic dissipation of the form 1 2 ik i k l x x = λ (17); moreover in the spirit of formalism (13), an external torque representing forcing can be easily included in the symmetric bracket by a translation l l g → + where 3 g f x = − ⋅ (18). it is interesting to note that on the ellipsoid 0 ξ , both ( ) l g ∇ + and lorenz field u = x & are orthogonal to c ∇ , but since determinant ( ) , , 0 c l g ∇ ∇ + ≠ u , the 3- vectors do not belong to the same plane. energy cycle for lorenz attractor can be finally written as ( ) ( ) ( ) 3 , , 2 ij jk i k k u k x x g u u k u f c l g β ω ⎧ = − −λ ω −ω ⎪ ⎪ = − + ⎨ ⎪ = − + ⎪ ⎩ & & & c c (19), i.e. defining forcing terms as 3 3 3 , , , k u c f g k f x f g u f f g c fx ω ⎧ = = ω ⎪ = = ⎨ ⎪ = = ⎩ (20). in this formalism the first two equations of (19) describe energy variations of a particle dynamics constrained to move on a spherical surface of variable radius ( ) 2 r t c = . it is easy to verify that for isotropic dissipation l c α = , even in presence of forcing, equations (19) describe a purely dissipative dynamics. in spherical coordinates after simplification, it becomes 2 2 sin r r f α θ = − − & (21). for lorenz parameters, combined effects of conservative part, anisotropic dissipation and forcing components of (1), makes dynamics of the spherical radius deterministic, bounded, recurrent and sensitive to initial conditions, as shown in fig.2, in other words chaotic. motion on a variable but topologically stable manifold justifies the presence of last equation in (19) that takes into account the background field ( ) c t . fig.2 sensitivity to initial conditions for two numerically different casimir functions top: time evolution for two casimir radii ( ) ( ) 2 r t c t = . bottom: ( ) r t δ ; with ( ) 0 0.008 r t δ ≈ if we consider steady state conditions, where all three time derivatives in (19) are set to zero, we note that last equation represents ellipsoid 0 ξ of equation (8) and the only point solution lying on it is given by fixed point ( ) { } 0,0, ρ σ = − + 0 x corresponding to the asymptotic max r in (9). here ( ) , 0 u k = c , ( ) 2 2 k c ρ σ = = + and potential reaches its minimum value ( ) u σ ρ σ = − + . in order to study behaviour of casimir function and its conversion terms, it will be useful to rewrite last equation of (19) in terms of lyapunov function l and forcing g , both contained in a function ( ) ( ) w 2l g = − + x . as a matter of facts, substituting ( ) 2 w c l g = − + = && & & & in (19), we have: . ( ) ( ) , , , c c w w k w u l g w = + + + & (22) from which applying antisymmetric properties of lie- poisson bracket, conversion terms for u and k are written as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 1 2 1 2 1 2 , 2 , , 2 , 2 , , 2 1 0 c c c c c c ijk j k i w k k l k g x x x f x x w u u l u g x x ε ω σ ⎛ ⎞ = + = ω λ + ω −ω ⎜ ⎟ ⎝ ⎠ = + = − + ∑ (23) note that all of terms in (23) are linear functions of ( ) , u k c . for lorenz parameters it results 0 ijk j k i ε ω λ > ∑ , ( ) 1 0 ω σ − > and ( ) 1 2 0 f ω −ω > from which energy cycle reads as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 2 2 1 2 3 1 2 2 1 2 , , 1 1 1 2 2 1 , 2 3 ij jk i k k u k x x g u u k u f f w u u k l g lg β ω β β σ σ σ ω ω ⎧ ⎪ = − −λ ω −ω ⎪ ⎪ ⎪ = − + ⎨ ⎪ ⎛ ⎞ ⎡ ⎤ − ω + − ω + − ω ⎪ ⎣ ⎦ = + − + ω −ω ⋅ + + + ⎜ ⎟ ⎪ ⎜ ⎟ ω −ω ⎪ ⎝ ⎠ ⎩ & & & c c c (24) this reformulation of energy cycle takes into account dissipation and forcing in conversion terms between , , k u w where w c = & at least formally plays the role of internal energy of the system. iv. conversion factors apart from dissipation and forcing terms from (24) it is clear that energy cycle for lorenz attractor can be studied by analysing the behaviour of conversion term ( ) , u k c shown in fig.3. fig.3. kinetic energy, potential energy and their relative conversion term ( ) , u k c are plotted on lorenz attractor. colour ranges from black to bright copper, going from low to high numerical values. for ( ) , u k c , regions of negative values 1 2 0 x x < , or transition regions where ( ) , 0 u k > c , are shown in orange. they cover about 12% of the attractor. given that dynamics of lorenz system is well described as a sequence of traps and jumps between the two lobes of the strange attractor; we observe what follows. when trapped in a lobe, system state experiences spiral-like trajectory, whose centre is in a fixed point ( ) ( ) { } 1 , 1 , 1 β ρ β ρ σ ± = ± − ± − − − x and whose radius increases in time as energy and casimir maxima, till touching the boundaries planes 1 0 x = or 2 0 x = where a transition to the opposite lobe occurs. after lobe transition, trajectory starts from regions close to the opposite unstable fixed point ( ) ( ) { } 1 , 1 , 1 β ρ β ρ σ = − − − − xm m m , moving towards boundary planes and so on. looking at the attractor as a fractal object, 12% of its whole structure lies in transition regions where 1 2 0 x x < . the above described dynamical behaviour above described is ruled by laws (23) and (24) as shown in fig.4. fig.4 time evolution for energy cycle variables. top sign of ( ) 1 x t represents jumps and traps. jumps are associated to spikes in conversion terms. second row: ( ) c t expansions and contractions of the casimir sphere, in its maxima, are modulated by the sign of conversion terms values. middle and fourth lines, conversion terms ( ) , u k c and ( ) , w k c show an opposite phase dynamics. bottom ( ) , w u c . note inequality ( ) ( ) ( ) , , , u k w u w k < < c c c reminding that ( ) ( ) , 2 , w u u l = c c , a conversion from kinetic to potential energy k u → occurs in trapping regions 1 2 0 x x > where a number of loops around ± x of increasing radius occurs; in the same regions a conversion w u → occurs, (fig.5). fig.5 potential energy, "internal energy" and their relative conversion term ( ) , w u c are plotted on lorenz attractor. colour ranges from black to bright copper, going from low to high numerical values. regions where ( ) , 0 w u > c , are shown in orange. as regards to term ( ) , w k c , coordinate 3 x drives the behaviour of ( ) , k l c , giving rise to both conversions k l ↔ inside a lobe of the attractor, while k g → is the only possible conversion, (fig.6). fig.6. kinetic energy, "internal energy" and their relative conversion term ( ) , c w k are plotted on lorenz attractor. colour ranges from black to bright copper, going from low to high numerical values. regions where ( ) , 0 w k > c are shown in orange. in trapping regions function w c = & act as a source for total energy h u k = + , ( ) , 0 w k > c and ( ) , 0 w u > c . in this phase, casimir sphere, centred in { } 0,0,0 = 0 x , expands. in regions 1 2 0 x x < a drastic change in energy cycle occurs, depending on the bounded nature of the system. here potential energy is transformed into kinetic energy u k → and lyapunov function into potential energy, l u → ; therefore an implosion of casimir sphere occurs. also, because of (23) l k ↔ and g k → . in lobe-jumping regions function w c = & acts as a sink for both potential and kinetic energy, since ( ) , 0 w k < c and ( ) , 0 w u < c . concerning numerical values of conversion terms, the following inequalities hold: ( ) ( ) 3 , 2 1 , ijk j k i k l x k g f ε ⎛ ⎞ = ω λ < ⎜ ⎟ ⎝ ⎠ ∑ c c , (25) ( ) ( ) ( ) , 1 1 2 1 , k u w u σ − = < − c c , (26) ( ) ( ) ( )( ) ( ) , 3 2 1 2 1 , w k w u β ρ σ ω σ − + ≤ < − c c , (27) (since max z ρ σ = + ): ( ) ( ) ( ) , , , k u w k w u < < c c c . (28) conversion rules are reassumed in the following diagram where dashed lines represent cycle in jumping regions 1 2 0 x x < and continuous lines refer to trapping lobes 1 2 0 x x > of the attractor. v. mechanical ape\upe and predictability in the spirit of 1955 lorenz work on general circulation of the atmosphere [1], we introduce for system (6) quantities respectively known as available potential energy max min ape u u = − and unavailable potential energy min upe u = ; they represent, respectively, the portion of potential energy that can be converted into kinetic energy, and the portion that cannot. in atmospheric science ape is a very important subject since its variability determines transitions in the atmospheric circulation. in analogy with the theory of margules [19], in which a fluid inside a vessel is put into motion and free surface oscillates up and down, while potential energy is constrained from below by the potential energy of the fluid at rest we start by considering the conservative system : 1 2 2 1 3 1 3 1 2 x x x x x x x x x σ σ = ⎧ ⎪ = − − ⎨ ⎪ = ⎩ & & & (29) fixing casimir and energy values ( ) 0 0 , c e , equation ( ) , 0 c u k = gives at 1 0 x = ⇒ max 0 k c = ω and min 0 u e c = −ω ; at 2 0 x = ⇒ ( )( ) ( ) 2 2 max 0 0 2 1 1 u c e ω ω ω ⎡ ⎤ = − + − ω − − ω − ⎣ ⎦ and min 0 max k e u = − , where 2 3 ω = ω = ω for lorenz system, therefore we get: ( )( ) 2 2 0 0 0 0 0 0 2 1 1 c e ape e c upe e c ω ω ω ⎧ − + − ω − − ⎪ = − + ω ⎨ ω − ⎪ = −ω ⎩ (30) in case of full lorenz system, energy and casimir are not conserved even though their associated surfaces intersect instantaneously; therefore introducing dissipation and forcing one can still consider the evolution of ape and upe as state functions. fig .7 shows a graphical representation of the two quantities over lorenz attractor; ape increases as 3 x decreases while upe follows the opposite way. fig.7 mechanical ape and upe plotted on lorenz attractor. this behaviour coincides with that of predictability regions over the attractor computed using breeding vectors technique [20] and shown in fig.8. giving an initial perturbation δ 0 x , red vector growth g over n=8 steps is computed as 1 log g n δ δ ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ 0 x x , regions of the lorenz attractor within which all infinitesimal uncertainties decrease with time [21] are located in regions of ( ) max upe and ( ) min ape . fig.8 breeding vectors map for lorenz attractor, solution of equations (6). colour ranges from black to bright copper meaning low to high predictability. finally we note that the set ( ) ( ) { } min ,max ape ape ∩ = ape ψ ξ , where ape ξ represents the surface ( ) 0 d ape dt = has as ordered subset ( ) max ape . in this view, lorenz map for ape maxima ape shown in fig.9 assumes the meaning of 'handbook' for regime transitions; for low values of ape system is trapped withine one lobe until it reaches the minimum necessary value to jump into the opposite one. fig.9 left: ordered set for ape maximum values lying on ellipsoid ape ξ (not shown); colour ranges from black to bright copper, going from low to high numerical values. right: lorenz map for ape maxima. vi. energetics and dynamics physically, introduction of function w has the following meaning: ellipsoid 0 ξ contains sets ( ) min c l,r ψ and ( ) max c l,r ψ for system (6) and acts as a boundary for regions of maximum forcing and dissipation. inside solid ellipsoid ξ , 2 g l > and forcing drives motion, also because of 0 w c = > & which implies that casimir sphere continually expands. and for ( ) , 0 c u k < trajectory of (6) links a point ( ) min c ∈ l,r ψ x to a point ( ) max c ∈ l,r ψ y into the same lobe. otherwise, for ( ) , 0 c u k > lorenz equations link a minimum ( ) min c ∈ l,r ψ x of , l r ψ into a maximum ( ) max c ∈ r,l ψ y of the opposite lobe. outside ξ , (more precisely into the region ( ) ∪ ∩ l r ψ ψ ξ), 2l g > , 0 w < and casimir sphere continually implodes; here dissipation constrains motion to be globally bounded inside a sphere of radius max r ρ σ = + , being ( ) 0 tr −λ < . for ( ) , 0 c u k < , dynamics links a point ( ) max c ∈ l,r ψ y to a point ( ) min c ∈ l,r ψ x in the same lobe; for ( ) , 0 c u k > , instead, a point ( ) max c ∈ l,r ψ y will be linked to a point ( ) min c ∈ r,l ψ x of the opposite lobe. fig.10 shows these links. fig. 10 traps and jumps. trajectories start from right lobe r ψ casimir maxima (shown in red). solid line track ends up on the same set while dash-dot one reaches maxima on the other lobe l ψ . black spots indicate initial conditions for both the trajectories, minima for casimir are shown in blue, while black stars represent fixed points ± 0 x ,x it is remarkable to note that using formalism above described, it is possible to proof the outward spiral motion around fixed points ± x . for a trajectory ( ) , l ⊂ 1 2 l,r p p ψ linking two points ( ) ( ) ( ) 1 1 2 2 , max t t c ∈ l,r ψ p p ( ) 2 1 , 0 c t t u k dτ < ⇒ ∫ ( ) 2 1 0 t t u u f d β ω τ + − < ∫ & (31) after integration and indicating by 3 x ω the time average of potential energy along ( ) l 1 2 p ,p we get ( ) ( ) ( ) ( ) 3 3 3 2 1 0 x x x t t β ρ σ − < − + − < − 2 1 p p ( 32 ) and then ( ) ( ) ( ) ( ) 3 3 3 3 x x x x ± > > > 1 2 0 x p p x . in order to better understand the statistics of persistence in the lobes, let's consider the effect of conversion factor (12). system (6) can be written as a particular case of the following set of equations ( ) 1 1 2 2 1 3 1 2 3 1 2 3 x x x x x x x x x x x x σ ω ω β β ρ ω ⎧ = − + ⎪ = − − − ⎨ ⎪ = − − + ⎩ & & & (33). fig.11 shows that under variation of parameter ω the resulting attractor, while conserving its global topology and fractal properties, will explore a greater volume domain due to the increased external forcing. fig.11 statistical behaviour as a function of ω term. left: from top to bottom (as ω increases), ( ) 1 , 1,2,3 k x t k = shows less and less persistence of trajectory inside a lobe; right: signs of conversion term over the attractor: regions where ( ) , 0 u k > c (in orange) approach unstable points ± x as ω increases and region where ( ) , 0 u k < c (in blue) decreases in size. the most important effect, however, is given by a significant change in persistence statistics; more precisely regions ( ) , 0 c u k > will expand to inner regions of attractor close to fixed points ± x , increasing the probability of jumping to the opposite lobe. vii. conclusions up to now, lorenz system has been studied under many viewpoints in literature; in this paper energy cycle approach has been fully exploited. the nature of lie-poisson structure in lorenz equation has been shown to be fruitful, for example in finding a geometrical invariant, ellipsoid 0 ξ , whose physical meaning is the boundary of action between forcing and dissipation. in this manner, kinetic-potential energy transfer term ( ) , c u k keeps track of dynamical behaviour of trapping and jumping, also giving information about global predictability of the system as illustrated by direct comparison of conversion factors with classical results on predictability. acknowledgements. authors warmly thank prof. e.kalnay for having provided codes in ref.[20] in order to compute breeding vector analysis. refences [1] e.lorenz, tellus 7, 157-167 (1955) [2] v.i. arnold, proc.roy. soc., a 434 19-22 (1991) [3] a.d'anjous, c.sarasola and f.j. torrealdea, journ. of phys.:conference series 23 238-251. (2005) [4] j.e. marsden and t. ratiu, introduction to mechanics and symmetry, springer, berlin, 1994 [5] p.j. morrison, rev. mod. phys. 70 467-521 (1998) [6] a.pasini, v.pelino and s.potestà, phys.lett. a 241 (1998) 77-83 [7] v.zeitlin, phys.rev.lett 93 no 26 264501-1-264501-3 (2004) [8] v.pelino and a.pasini, , phys.lett. a 291 389-396 (2001) [9] v.pelino and f.maimone, phys rev.e 76, (2007) [10] v.i. arnold and b.a. khesin, topological methods in hydrodynamics, springer, berlin 1988 [11] a.pasini and v.pelino, phys.lett. a 275 435-446 (2000) [12] e.n. lorenz, j.atmos. sci. 20 130 (1963) [13] a.elipe,v. lanchares, cel. mech dyn astr 101 49-68 (2008) [14] m.gianfelice,f.maimone, v.pelino, s.vaienti, invariant densities for expanding lorenz- like maps, in preparation. [15] k.suzuki,y.watanabe,t.kambe, j.phys.a math.gen 31 6073-6080 (1998) [16] a.elipe, m.arribas and a.riaguas, j.phys.a:math.gen. 30 587-601 (1997) [17] r.salmon, lectures on geophysical fluid dynamics oxford (1998) [18] p.j. morrison, journal of physics 169 1-12 (2009) [19] j.marshall, r.a. plumb, atmosphere, ocean and cliamte dynamics, academic press (2007) [20] e.evans, n.batti,j.kinney,l.pann,m.pena,s.chih,e.kalny,j.hansen, bams 519-524 (2004) [21] l.a.smith, c.ziehmann, k.fraedrich q.j.r meteorol.soc. 125, 2855-2886 (1999)

0911.1704

universal correlations and coherence in quasi-two-dimensional trapped bose gases

we study the quasi-two-dimensional bose gas in harmonic traps at temperatures above the kosterlitz-thouless transition, where the gas is in the normal phase. we show that mean-field theory takes into account the dominant interaction effects for experimentally relevant trap geometries. comparing with quantum monte carlo calculations, we quantify the onset of the fluctuation regime, where correlations beyond mean-field become important. although the density profile depends on the microscopic parameters of the system, we show that the correlation density (the difference between the exact and the mean-field density) is accurately described by a universal expression, obtained from classical-field calculations of the homogeneous strictly two-dimensional gas. deviations from universality, due to the finite value of the interaction or to the trap geometry, are shown to be small for current experiments. we further study coherence and pair correlations on a microscopic scale. finite-size effects in the off-diagonal density matrix allows us to characterize the cross-over from kosterlitz-thouless to bose-einstein behavior for small particle numbers. bose-einstein condensation occurs below a characteristic number of particles which rapidly diverges with vanishing interactions.

introduction in recent years, several experiments [1, 2] studied two- dimensional ultra-cold atomic gases from the normal phase down in temperature to the kosterlitz–thouless transition [3] and into the low-temperature superfluid phase. the interference of two simultaneously prepared two-dimensional gases evidenced the presence of vortices [1]. related experiments investigated interaction and correlation effects [2, 4] in the density profile and in co- herence patterns. for a quantitative description of the kosterlitz–thouless transition and of the interaction ef- fects, it proved necessary to account for the quasi-two- dimensional nature of the gas, that is to include thermal excitations in the strongly confined z-axis in addition to the weak trapping potential in the xy-plane [5, 6]. in weakly interacting two-dimensional bose gases, the kosterlitz–thouless phase transition occurs at rel- atively high phase-space density (number of atoms per phase-space cell λ2 t = 2πħ2/mt ). this density is ncλ2 t ≃log(ξn/ ̃ g) [7–9], where ̃ g characterizes the two- dimensional interaction strength, t is the temperature, m the mass of the atoms, and n the density. the coef- ficient ξn = 380 ± 3 was determined numerically using classical-field simulations [9]. for the ens experiment of hadzibabic et al. [1], the critical phase-space den- sity is ncλ2 t ∼8 in the center of the trap, whereas in the nist experiment of clad ́ e et al. [2], ncλ2 t is close to 10. for phase-space densities between one and the ∗electronic address: [email protected] †electronic address: [email protected] critical number, the gas is quantum degenerate yet nor- mal. the atoms within one phase-space cell are indistin- guishable. they lose their particle properties and acquire the characteristics of a field. the mean-field description of particles interacting with a local atomic density n(r) may further be modified through correlations and fluctu- ations. quantum correlations can be several times larger than the scale λt. this gives rise to "quasi-condensate" behavior inside the normal phase. in this paper, we study the quantum-degenerate regime at high phase-space density in the normal phase. we first discuss the peculiar quasi-two-dimensional thermo- dynamic limit where, as the number of particles in the gas is increased, the interactions and the lattice geome- try are scaled such that a finite fraction of all particles are in the excited states of the system. in this thermo- dynamic limit, the kosterlitz–thouless transition takes place at a temperature comparable to the bose–einstein transition temperature in the non-interacting case, and the local-density approximation becomes exact. we first clarify the relation between different recent versions of quasi-two-dimensional mean-field theory [6, 10, 11] in the local-density approximation (lda), and also determine the finite-size corrections to the lda. we compare mean- field theory to a numerically exact solution obtained by path-integral quantum monte carlo (qmc) calculations with up to n ≳105 interacting particles in a harmonic trap with parameters chosen to fit the experiments. we concentrate on the correlation density, the difference be- tween the exact density and the mean-field density at equal chemical potential, and show that it is essentially a universal function, independent of microscopic details. within classical-field theory, the correlation density is obtained from a reparametrization of known results for 2 the strictly two-dimensional homogeneous system [12]. the classical-field results hold for small interaction pa- rameters ̃ g →0, but our full qmc solution accounts for corrections. we compute the correlation density by qmc and show that it is largely independent of the trap geometry, the temperature, and the interaction strength. we also study off-diagonal coherence properties, and the density–density correlation function of the quasi-two- dimensional gas. it is well known that even at high temperature, bosonic bunching effects enhance the pair- correlation function on length scales below λt which for the ideal bose gas approaches the characteristic value 2n2 at vanishing separation. in our case, interference in the z-direction reduces the in-plane density fluctuations even for an ideal gas and within mean-field theory, and the reduction of the pair correlations from 2n2 no longer proves the presence of beyond-mean-field effects. we finally discuss finite-size effects in the quasi-two- dimensional bose gas. for the density profile, they are not very large, but we point out their great role for off-diagonal correlations. the latter are responsible for a cross-over between the physics of bose–einstein con- densation at small particle number and the kosterlitz– thouless physics for larger systems; both regimes are of relevance for current experiments. this cross-over takes place at a particle number n ∼ ̃ g−2 which grows very rapidly as the interaction in the gas diminishes. ii. system parameters and mean-field description a. quasi-two-dimensional thermodynamics we consider n bosons in a three-dimensional pancake- shaped harmonic potential with parameters ωx = ωy = ω and ωz ≫ω at inverse temperature β = 1/t . the z variable is separate from x and y, and we denote the three-dimensional vectors as ⃗ r = (r, z), and write two- dimensional vectors as r = (x, y), and r = |r|. the quasi-two-dimensional regime of the bose gas [6] is defined through a particular thermodynamic limit n → ∞, where the temperature is a fixed fraction t ≡t/t 2d bec of the bose–einstein transition temperature of the ideal two-dimensional bose gas, t 2d bec = √ 6nħω/π. although the two-dimensional trapped bose gas undergoes a bose– einstein transition only for zero interactions, t 2d bec still sets the scale for the kosterlitz–thouless transition in the interacting gas[6, 8]. in the quasi-two-dimensional regime, a finite fraction of atoms remains in excited states in z. the excitation energy is scaled as ħωz ∝t 2d bec, which implies that ωz increases as n 1/2 in the thermo- dynamic limit. interatomic collisions are intrinsically three- dimensional. here, we consider the experimentally relevant case where the range of the scattering potential r0 is much smaller than the typical lateral extension lz = (mωz/ħ)−1/2, and also where r0 is much smaller than the inter-particle distances. the interactions are then described by the three-dimensional s-wave scattering length as, and one may characterize the quasi-two-dimensional gas through a bare effective two-dimensional interaction strength, ̃ g, ̃ g = mg ħ2 z dz [ψ0(z)]4 (1) g = 4πħ2as m , (2) where ψ0(z) is the unperturbed ground state of the con- fining potential and g is the usual three-dimensional coupling constant. for a harmonic confinement, ̃ g = √ 8πas/lz, and as/lz must be kept constant in the ther- modynamic limit to obtain a fixed two-dimensional in- teraction strength. quasi-two-dimensional scattering amplitudes depend logarithmically on energy, ǫ, in terms of a universal func- tion of as/lz and of ǫ/ħωz ∼t/ωz, which are both kept constant in the quasi-two-dimensional thermodynamic limit. the logarithmic energy dependence yields small corrections of order (as/lz)2 [13–15] to the bare interac- tion ̃ g. they can be neglected in the following. the scaling behavior in the quasi-two-dimensional limit corresponds to the following reduced variables: ̃ r = r/lt ̃ z = z/lz t = t/t 2d bec ̃ ωz = ħωz/t 2d bec ̃ n = nλ2 t ̃ g = mg/( √ 2πlzħ2), (3) where lt = (t/mω2)1/2 is the thermal extension in the plane. the quasi-two-dimensional limit consists in tak- ing n →∞, with t, ̃ ωz and ̃ g all constant. in this limit, lt /λt = t p 3n/π3 ≫1, so that macroscopic and mi- croscopic length scales separate, and the scaling of the three-dimensional density, n3d, is at constant ̃ n3d = n3dλ2 tlz. (4) in reduced variables, the normalization condition n = r dr n(r) = (lt /λt)2 r d ̃ r ̃ n( ̃ r) is expressed as z ∞ 0 d ̃ r ̃ r ̃ n( ̃ r) = π2 6t2 . (5) the local-density approximation becomes exact in the quasi-two-dimensional limit. in the ens experiment, 87rb atoms are trapped at temperatures t ≈50 −100 nk. the in-plane trap- ping frequencies are ω/(2π) ≈50hz whereas the con- finement is of order ωz/(2π) ≈3khz. with n ∼2 * 104 atoms trapped inside one plane, typical parameters are t 2d bec ≈300nk (using ħ/kb ≃7.64 * 10−3nks), so that ̃ ωz ≈0.44 −0.55. the scattering length as = 5.2nm leads to an effective coupling constant ̃ g = 0.13, using ħ/m ≃6.3 * 10−8m2s−1a−1, where a is the atomic mass number. in the nist experiment, sodium atoms at t ≈ t 2d bec ≈100nk are confined by harmonic trapping poten- tials with ω/(2π) ≈20hz, and ωz/(2π) ≈1khz. this is 3 described by reduced parameters ̃ g = 0.02 and ̃ ωz = 0.50. the critical densities are ̃ nc ≈log(380/ ̃ g) ≈8.2 for the ens parameters, somewhat lower than the nist value ̃ nc ≈9.9. using the quasi-two-dimensional mean-field es- timates of ref. [6], the kosterlitz–thouless temperatures are located at tkt ≡tkt/t 2d bec ≈0.69 and tkt ≈0.74, respectively. the quasi-two-dimensional limit describes a kinemat- ically two-dimensional gas, whose extension in the z di- rection is of the order of the thermal wavelength λt. as ̃ ωz ≡λ2 t/(2πtl2 z) is decreased, a system at finite n turns three-dimensional. this is already the case for the ideal quasi-two-dimensional gas (with ̃ g = 0) where the bose– einstein transition temperature crosses over from two- dimensional to three-dimensional behavior as a function of ̃ ωz, with asymptotic behavior given by tbec ∼    h ζ(2) ζ(3) i1/3 ̃ ω1/3 z −1 6 ζ(2) ζ(3) ̃ ωz for ̃ ωz ≪1 1 − 1 2ζ(2)3/2 exp (− ̃ ωz) for ̃ ωz ≫1 (6) (see [6]). in eq. (6), the first term for ̃ ωz ≪1 de- scribes three-dimensional bose–einstein condensation in an anisotropic trapping potential. for the interacting bose gas, the nature of the kosterlitz–thouless transition in two dimensions dif- fers from the bose–einstein transition of the three- dimensional gas. for large ̃ ωz, universal features of the kosterlitz–thouless transition are preserved, but the density profiles and the value of the kosterlitz–thouless transition temperature depend on ̃ ωz and ̃ g [5, 6]. for small confinement strength ̃ ωz, a dimensional cross-over between the two-dimensional kosterlitz–thouless transi- tion and the three-dimensional bose–einstein condensa- tion takes place at particle numbers such that the level spacing in the confined direction is comparable to the (two-dimensional) correlation energies, t ̃ ωz ≲ ̃ g ̃ n/π. the quasi-two-dimensional limit differs from the "ex- perimentalist's" thermodynamic limit where the atom number is increased in a fixed trap geometry, and at constant temperature. in this situation, the ratio be- tween the microscopic and the macroscopic length scales, lt /λt = t/(ħω √ 2π), remains constant and finite. the number of particles in any region of nearly constant den- sity remains also finite so that, in contrast to the quasi- two-dimensional thermodynamic limit, corrections to the lda persist. b. n-body and mean-field hamiltonians the gas specified in section ii a is described by the hamiltonian h =h0 + v, (7) h0 = n x i=1  −ħ2∇2 i 2m + 1 2m  ω2r2 i + ω2 zz2 i  , (8) v = n x i<j=1 v(|⃗ r i −⃗ r j|), (9) where v is the three-dimensional interaction potential. we compute the n-body density matrix at finite temper- ature using three-dimensional path-integral qmc meth- ods. we thus obtain all the thermodynamic observ- ables [5, 16, 17] for up to n = 106. qmc calculations have clearly demonstrated the presence of a kosterlitz– thouless transition [6] for parameters corresponding to the ens experiment. in the mean-field approximation, one replaces the n- body interaction between atoms in eq. (9) by an effec- tive single-particle potential. the mean-field hamilto- nian writes hmf = h0 + vmf, (10) where vmf = n x i=1 2gn3d(⃗ r i) −g z d⃗ r [n3d(⃗ r )]2 (11) is the mean-field potential energy. from the corre- sponding partition function in the canonical or grand- canonical ensemble, all thermodynamic quantities can be calculated. the three-dimensional density n3d(⃗ r ) in- side the mean-field potential must be determined self- consistently. in all situations treated in the present pa- per, self-consistency is reached through straightforward iteration. mean-field theory leads to an effective schr ̈ odinger equation for the single-particle wavefunction, ψj(⃗ r ), of energy ǫj,  −ħ2∇2 r 2m + 1 2mω2r2 −ħ2∂2 2m∂z2 + 1 2mω2 zz2 + 2gn3d(⃗ r )  ψj(⃗ r ) = ǫjψj(⃗ r ) (12) together with the total density n3d(⃗ r ) = x i ψ∗ j(⃗ r )ψj(⃗ r ) eβ(μ−ǫj) −1 . (13) the exact solution of the mean-field eigenfunctions and eigenvalues for finite systems, is rather involved, but con- siderably simplifies in the local-density approximation. at finite n, in the canonical ensemble, we solve the mean-field equations through a quantum monte carlo 4 simulation with n particles which avoids an explicit cal- culation of all eigenfunctions. in contrast to the usual interaction energy within qmc, consisting, in general, of a pair interaction potential, the mean-field interaction energy is simply given in terms of an anisotropic, single- particle potential proportional to the three-dimensional density profile, n3d(⃗ r ), as in eqs (10) and (11). this interaction potential must be obtained self-consistently as usual in mean-field. once self-consistency in the den- sity is reached, one can compute correlation functions and off-diagonal elements of the reduced one-body den- sity matrix. c. mean-field: local-density approximation mean-field theory simplifies in the quasi-two- dimensional thermodynamic limit, as the local-density approximation then becomes exact. this is because the natural length scale of the system, λt, separates from the macroscopic scale lt of variation of the density (λt/lt →0). the particle numbers inside a region of constant density diverges. the decoupling of length scales implies that the xy-dependence of the single particle wavefunctions in eq. (12) separates in the thermodynamic limit. using scaled variables, eq. (3), this yields  −1 2 d2 d ̃ z2 + 1 2 ̃ z2 + 2 ̃ gt √ 2π ̃ ωz ̃ nmf 3d( ̃ r, ̃ z)  ̃ φν( ̃ r, ̃ z) = ̃ ǫν( ̃ r) ̃ ωz ̃ φν( ̃ r, ̃ z), (14) for the eigenfunctions, ̃ φν( ̃ r, ̃ z), and eigenvalues, ̃ ǫν( ̃ r), in the confined direction, at a given radial distance, ̃ r. the reduced local density ̃ nmf 3d is given by the normalized wavefunctions ̃ φ( ̃ r, ̃ z): ̃ nmf 3d( ̃ r, ̃ z) = x ν ̃ φ2 ν( ̃ r, ̃ z) ̃ nmf ν ( ̃ r) ̃ nmf ν ( ̃ r) = −log [1 −exp ( ̃ μ( ̃ r) − ̃ ǫν( ̃ r)/t)] . (15) the position-dependence in eq. (14) and eq. (15) only enters parametrically through the ̃ r-dependence of the chemical potential, ̃ μ( ̃ r) = ̃ μ − ̃ r2 2 , (16) and the local-density approximation becomes exact in the quasi-two-dimensional thermodynamic limit. within lda, density profiles (as in fig. 1) are directly related to the equation of state ̃ n( ̃ μ) of a quasi-two-dimensional system, which is homogeneous in the xy-plane. the schr ̈ odinger equation of eq. (14) is conveniently written in the basis {ψ0, ψ1, . . . , ψn, . . . } of the one- dimensional harmonic oscillator with ω = m = 1, as it diagonalizes eq. (14) for ̃ g = 0. using ̃ φν( ̃ z) = x μ aμνψμ( ̃ z), (17) (where we have dropped the index corresponding to ̃ r or, equivalently, to ̃ μ), we can write it as a matrix equation  a − ̃ ǫν ̃ ωz  aν = 0 (18) with eigenvalues ̃ ǫν/ ̃ ωz and eigenvectors aν = {a0ν, . . . , anν}, of the (n + 1) × (n + 1) matrix aμν = νδμν + 2 ̃ gt √ 2π ̃ ωz z d ̃ z ψμ( ̃ z) ̃ nmf 3d( ̃ μ, ̃ z)ψν( ̃ z) (19) (where 0 ≤μ, ν ≤n) and the density ̃ nmf 3d( ̃ μ, ̃ z) = −p ν ̃ φ2 ν( ̃ z) log[1 −exp( ̃ μ − ̃ ǫν/t)]. the wavefunctions ψν are easily programmed (see, e.g., [18] sect. 3.1), and the self-consistent mean-field solutions at each value of ̃ μ can be found via iterated matrix diagonalization. this full solution of the lda mean-field equations is analogous to the one in ref.[10]. the mean-field ver- sion used in [11], however, neglects the off-diagonal cou- plings in aνμ with ν ̸= μ. in [6], we used a simplified mean-field potential in order to reach explicit analytical expressions. these different mean-field approximations essentially coincide at all relevant temperatures [11], but ground-state occupations in z slightly differ. further re- placing the coupling constant ̃ g by ̃ g tanh1/2( ̃ ωz/2t) has allowed us, in ref. [6], to improve the agreement with the qmc results close to the transition. this is because mean-field theory overestimates the effect of the inter- actions in the fluctuation regime. in the following we always quantify beyond-mean-field corrections with re- spect to the full lda solution of eq. (19). iii. correlation density and universality comparisons between the qmc and the mean-field density profiles are shown in fig. 1 for the ens parame- ters at reduced temperature t = 0.71, slightly above the kosterlitz–thouless temperature. finite-size effects as well as deviations from mean-field theory are visible for nλ2 t ≳5. in this section, we concentrate on correlation corrections to mean-field theory in the thermodynamic limit and postpone the discussion of finite-size effects to section iv. we analyze the qmc density profiles within the validity of the lda, eq. (16), and compare qmc and mean-field densities at the same local chemical potential [29] which defines the correlation density, ∆ ̃ n, ∆ ̃ n( ̃ μ) = ̃ n( ̃ μ) − ̃ nmf( ̃ μ). (20) as in experiments, the chemical potential is not a control parameter of the qmc calculation, but it can be obtained 5 from a fit of the wings of the density profile with ̃ n ≲1 to the mean-field equation of state. mean-field effects take into account the dominant in- teraction effects which, in particular, determine shape and energies of the ground and excited states in the tightly confined direction. one expects that correla- tion effects do not modify these high-energy modes, but merely affects the low-energy distribution of xy-modes in- side the confining ground state, ̃ φ0( ̃ z). this assumption is supported by a direct comparison of the normalized density distribution in ̃ z between the qmc solution and the mean-field approximation (inset of fig. 1) at differ- ent radial distances, ̃ r. for small ̃ r, the ground state of the confining potential is strongly populated. for larger ̃ r, higher modes of the one-dimensional harmonic oscilla- tor are thermally occupied, and the density distribution broadens. however, the normalized density profile in ̃ z is everywhere well described by mean-field theory, and correlation effects hardly modify the mode structure in the confined direction. 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 density ~ n = nλt 2 position ~ r = rβ1/2 n=100,000 10,000 1000 mf-lda 0 0.5 0 2 n(~ z) (norm.) ~ z=zωz 1/2 ~ r=0 1 2 3 fig. 1: qmc density profile for the ens parameters ( ̃ g = 0.13, ̃ ωz = 0.55) at temperature t = t/t 2d bec = 0.71 for different values of the particle number, compared to the lda mean-field solution of section ii b at the same total number of particles. the inset shows the density distribution n( ̃ z) at different values of the radial distance ̃ r. in the homogeneous two-dimensional gas, corrections to mean-field theory at small ̃ g are described by classical- field theory, and correlation effects in the density pro- file can be expressed in terms of a universal function of β(μ −2gnmf) [8, 12]. in a quasi-two-dimensional geom- etry, the corresponding relevant quantity is given by the local mean-field gap ∆mf between the local ground-state energy in the confining potential and the local chemical potential, ∆mf( ̃ r) = ̃ ǫ0( ̃ r)/t − ̃ μ( ̃ r). (21) within mean-field theory, it fixes the local xy-density in the ground state of the confining potential ̃ nmf 0 ( ̃ r) = −log  1 −e−∆mf( ̃ r) . (22) in the strictly two-dimensional limit, we have ∆mf →β(2 ̃ għ2nmf/m −μ) = ̃ g ̃ nmf/π − ̃ μ, (23) where nmf is the total mean-field density. deviations are noticeable for large densities, as illustrated in fig. 2 (we have absorbed the zero-point energy ̃ ωz/(2t) in the chem- ical potential). within lda, we expect that the correla- tion density ∆ ̃ n coincides to leading order in ̃ g with the classical-field-theory results of the homogeneous strictly two-dimensional system [12], expressed as functions of the mean-field gap ∆mf [30]. -2 -1 0 1 2 3 4 5 6 7 8 -1 0 -∆, density ~ n = nλ2 ~ μ lda density 2d density -gap: ( ~ μ-~ ε0 /t) ~ μ -~ g~ n/π fig. 2: mean-field equation of state and gap ∆mf for ens parameters ̃ g = 0.13, ̃ ωz = 0.55, t = 0.71. the gap differs from the approximation ̃ μ − ̃ g ̃ n/π only at high density. the 2d mean-field curve (see eqs (22) and (23)) illustrates the de- pendence of the equation of state on microscopic parameters. in ref. [12], the critical density ̃ nc and chemical po- tential ̃ μc at the kosterlitz–thouless transition were de- termined to ̃ nc = log ξn ̃ g , ξn = 380 ± 3, (24) ̃ μc = ̃ g π log ξμ ̃ g , ξμ = 13.2 ± 0.4, (25) and the equation of state in the neighborhood of the tran- sition was written as ̃ n − ̃ nc = 2πλ(x), with x = ( ̃ μ − ̃ μc)/ ̃ g. (26) the function λ(x) was tabulated. consistent with the classical-field approximation, we can expand eq. (22) to leading order in ∆mf, ̃ nmf 0 (∆mf) = −log(∆mf), (27) so that we can express x through ∆mf: x(∆mf) = ̃ nmf π −∆mf ̃ g − ̃ μc ̃ g = − ∆mf ̃ g + 1 π log  ξμ ∆mf ̃ g  . (28) 6 thus, we obtain the correlation density as a function of ∆mf: ∆ ̃ n = 2πλ[x(∆mf)] + log (ξn∆mf/ ̃ g) , (29) and a straightforward inversion of eq. (28) allows us to translate the data of [12] in order to obtain the correla- tion density as a function of the mean-field gap. an em- pirical interpolation of the numerical data with ≲10% error inside the fluctuation regime, ∆c mf ≤∆mf ≤∆f mf (∆c mf and ∆f mf are defined below), is given by ∆ ̃ n(∆mf/ ̃ g) ≃1 5  −1 + 1 ∆mf/ ̃ g  1 1 + π∆2 mf/ ̃ g2 (30) where positivity is imposed since, within classical field theory, the correlation density must be positive and of order (∆mf/ ̃ g)−2 for ∆mf/ ̃ g →∞. 0 0.5 1 1.5 2 2.5 3 3.5 0 0.1 0.2 0.3 0.4 correlation density ∆ ~ n ∆mf/~ g mean-field fluctuation region kt class.-field interpolation (ω,t,~ g) = (0.55,0.71,0.13) (1.64,0.57,0.14) (0.82,0.57,0.14 (1.64,0.57,0.07) (0.82,0.57,0.07) fig. 3: correlation density ∆ ̃ n vs. rescaled mean-field gap ∆mf/ ̃ g. qmc data for various interaction strengths and con- finements ̃ ωz are compared with the interpolation eq. (30) of classical-field results [12]. in fig. 3, we plot the classical-field results for the correlation density as a function of the rescaled mean- field gap. we also indicate the onset of the kosterlitz– thouless transition ( ̃ μ = ̃ μc, x = 0 in eq. (29)) at ∆c mf/ ̃ g = 0.0623, which yields ∆ ̃ nc = 3.164. the cor- relation density in the normal phase is thus finite for all interactions, whereas the mean-field density diverges as ̃ nmf = −log ∆c mf ∝−log g for small interactions at tkt. in fig. 3, we furthermore compare the classical-field data for the correlation density with the results of qmc simu- lations of quasi-two-dimensional trapped bose gases with different coupling constants ̃ g and confinement strengths ̃ ωz. the qmc data illustrates that the external trapping and the quasi-two-dimensional geometry preserve univer- sality in the experimental parameter regime. however, the finite coupling constant ̃ g introduces small deviations due to quantum corrections. from fig. 3, we further see that the correlation den- sity is reduced to roughly 10% of its critical value for mean-field gaps ∆f mf ≃ ̃ g/π. thus, only densities with ̃ n ≳ ̃ nf ≈ ̃ nmf(∆mf ≈ ̃ g/π) are significantly affected by correlations, and ̃ nf can be considered as the boundary of the fluctuation regime. in fact, perturbation theory fails inside this regime. for a strictly two-dimensional system, we have ̃ nf ≈log(π/ ̃ g), (31) and the fluctuation regime is reached for densities ̃ n ≳ ̃ nf. outside the fluctuation regime, ̃ n ≲ ̃ nf, mean-field theory is rather accurate, and can be improved pertur- batively, if necessary. to understand this criterion, which is important for the kosterlitz–thouless to bose–einstein cross-over at small n (see section iv), we briefly analyze the per- turbative structure of the two-dimensional single-particle green's function beyond mean-field theory [8]. within classical-field theory, second-order diagrams are ultravi- olet convergent. each additional higher order brings in a factor ħ2 ̃ g/m for the interaction vertex, one integration over two-momenta, a factor t , and two green's functions (the internal lines). dimensional analysis of the integrals involved shows that each vertex insertion adds a factor ̃ g/∆mf. this implies that perturbation theory fails for ̃ g/∆mf ≳1. for lower densities, 1 ≲ ̃ n ≲ ̃ nf, the gas is quantum degenerate, yet it is accurately described by mean-field theory. in contrast to fully three-dimensional gases, the quantum-degenerate regime can be rather broad in two dimensions for gases with ̃ g ≪1. in this regime, the den- sity, yet normal, is no longer given by a thermal gaussian distribution. this was observed in the nist experiment [2] where ̃ nf ≃5. in fig. 4, we illustrate this effect via the approximations for the tails of the distribution ̃ n( ̃ r) with between one and five gaussians, as ̃ n( ̃ r) = π2 6t2 kmax x k=1 kπk exp  −k ̃ r2 2  . (32) where πk is determined by the formal expansion of the logarithm in eq. (15), but also appears as a cycle weight in the path-integral representation of the bosonic den- sity matrix where they can be measured (see the inset of fig. 4). the successive approximations have no free parameters. figure 5 summarizes the density profiles of a strictly two-dimensional bose gas in the limit ̃ g →0 where the classical-field calculations determine the correlation den- sity. at the critical temperature tkt, the density in the center of the trap is critical, ̃ n(0) = ̃ nc. correla- tion effects are important only in the fluctuation regime, ̃ r ≲1.25√ ̃ g, where ̃ n( ̃ r) ≳ ̃ nf. however, the distribution of the correlation density introduces no further qualita- tive features to the mean-field component. the density profile may be integrated using the interpolation formula for the correlation density, eq. (30). the critical tem- perature of the strictly two-dimensional bose gas as a 7 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 density ~ n = nλt 2 position ~ r = rβ1/2 1 gauss. 4 gauss. n=100,000 10,000 1,000 0.001 0.01 0.1 1 5 20 cycle weight πκ k fig. 4: qmc density profile for the nist parameters t = t/t 2d bec = 0.75, ̃ g = 0.02, and ̃ ωz = 0.5, for different n. the data are compared to the expansion of eq. (32), considering the largest (k = 1) and the four largest terms (k = 1, . . . , 4). the inset shows the qmc cycle weights πk for n = 100 000 (see [6]). function of the total number of particles is given by tkt t 2d bec ≃  1 + 3 ̃ g π3 log2 ̃ g 16 + 6 ̃ g 16π2  15 + log ̃ g 16 −1/2 . this expression includes correction of order ̃ g log ̃ g com- pared to the mean-field estimate of refs [6, 8]. since corrections beyond classical-field theory are rather small (see fig. 3), the kosterlitz–thouless temperature of the strictly two-dimensional trapped bose gas is accurately described by this equation even for large coupling con- stants. for general quasi-two-dimensional gases, fig. 5 re- mains qualitatively correct, but the lda mean-field den- sity profile in the quasi-two-dimensional geometry must be used. numerical integration of this ̃ g →0 den- sity profile for the ens parameters with ̃ ωz = 0.55 and ̃ g = 0.13 leads tkt ≃0.71 t 2d bec, in close agree- ment with tkt ≃0.70 t 2d bec determined in ref. [5] di- rectly from qmc calculations using finite-size extrapola- tions. the quasi-two-dimensional transition temperature is smaller than the one of the strictly two-dimensional gas (t 2d kt ≃0.86 t 2d bec for ̃ g = 0.13). for the nist param- eters ( ̃ g = 0.02, ̃ ωz = 0.5), we have tkt ≃0.74 t 2d bec from the integration of the ̃ g →0 density, and qmc data indicate a transition slightly below this value. iv. finite-size effects and bose–einstein cross-over a. central coherence in the normal phase, the off-diagonal elements of the single-particle density matrix remain short-ranged, -log(~ g) -log(~ g)+3 -log(~ g)+6 0 ~ g1/2 2 ~ g1/2 position ~ r ideal gas ~ n mean-field gas ~ nmf ~ nmf+∆~ n 0 1 -log(~ g) ~ n exp(-~ r2/2) ~ r fig. 5: schematic density profile of a strictly two-dimensional trapped bose gas at tkt for ̃ g →0. for ̃ r ≫√ ̃ g, ̃ n coincides with the ideal gas at t 2d bec (inset, the classical boltzmann dis- tribution e− ̃ r2/2 is given for comparison). in the fluctuation regime, for ̃ r ≲1.25√ ̃ g, mean-field and correlation effects be- come important. the density diverges as ∼log(1/ ̃ g), yet the correlation contribution ∆ ̃ n remains finite. so that they can be described locally. from the self- consistent eigenfunctions of the mean-field schr ̈ odinger equation, eq. (12) and eq. (13), we also obtain the off- diagonal reduced single-body density matrix: ̃ n(1) mf (⃗ r ;⃗ r ′) = λ2lz x j ψ∗ j(⃗ r ) ̃ ψj(⃗ r ′) exp ( ̃ μ −βǫj) −1. (33) in the local-density approximation, we can separate the contributions of the different transverse modes, and we obtain ̃ n(1) mf (⃗ r ;⃗ r ′) = x ν ̃ n(1) mf,ν(r; r′) ̃ φν( ̃ z) ̃ φν( ̃ z′) (34) with ̃ n(1) mf,ν(r; r′) = z d2k (2π)2 λ2 teik*(r−r′) eβħ2k2/2m+∆mf( ̃ r) −1. (35) here we have used that within the lda, the density re- mains constant on the scale λt, so that the mean-field gaps at ̃ r and ̃ r′ are the same. at low densities, where the mean-field gap is large, ∆mf ≫1, we can expand the bose function in eq. (35) in powers of exp (−∆mf), and off-diagonal matrix ele- ments rapidly vanish for distances larger than the ther- mal wavelength λt. at higher densities, in the quantum- degenerate regime, ∆mf ≪1, many gaussians con- tribute, and coherence is maintained over larger dis- tances. in the limit ∆mf →0, we can expand the de- nominator in eq. (35), exp  βħ2k2/2m + ∆mf  −1 ≈ βħ2k2/2m + ∆mf, and the off-diagonal density matrix decays exponentially. in this regime, the local mean-field coherence length is given by ξmf = λt/√4π∆mf. 8 in fig. 6 and fig. 7 we compare the normalized off- diagonal coherence function in the center of the trap c(r) = r dz n(1) 3d (r, z; 0, 0) r dz n(1) 3d (0, z; 0, 0) (36) from qmc calculations with lda for the ens and nist conditions. we see that for ̃ n ≲ ̃ nf, as in the case of the density profile, mean-field theory accurately describes the single-particle coherence. however, it is evident that at higher densities, ̃ n ≳ ̃ nf, where correlation effects for the diagonal elements of the density matrix are important, mean-field theory also fails to describe the off-diagonal matrix elements. to characterize the decay of the off-diagonal density matrix in the fluctuation regime, ̃ n ≳ ̃ nf, we consider a simple one-parameter model which neglects the mo- mentum dependence of the self-energies in the ground state of the confining potential. the single parameter of the model, the effective local gap ∆( ̃ r), is chosen such that it reproduces the local density of the qmc data. the density matrix of this "gap"-model, ̃ n(1) ∆(⃗ r ;⃗ r ′) = p ν ̃ n(1) ∆,ν(r; r′) ̃ φν( ̃ z) ̃ φν( ̃ z′), is a straightforward general- ization of mean-field theory, where in eq. (35), we replace ∆ν = ( ∆ for ν = 0 ∆mf otherwise . (37) to fix the gap ∆of this model, we require that the diag- onal elements of the density matrix reproduces the exact density ̃ n(1) ∆,0(r; r) = ̃ nmf 0 ( ̃ r) + ∆ ̃ n( ̃ r). (38) 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 central coherence c(r) position r/λt n=160,000 40,000 10,000 c∆(r) cmf(r) 0 0.2 0.4 0.6 0.8 1 1 2 3 4 r/λt c(r) fig. 6: off-diagonal coherence c(r) for ens parameters with t = 0.71 (main graph, ̃ n > ̃ nf) and t = 0.769 (inset, ̃ n < ̃ nf) compared to the mean-field prediction cmf(r) and the gap model of eq. (37). in the fluctuation regime, finite-size effects for off-diagonal correlations are more pronounced than for the density (see fig. 1). outside the fluctuation regime the gap model reduces to the mean-field limit. inside the fluctuation regime, where a direct comparison of the coherence with mean- field theory is not very useful, the gap model provides the basis to quantify off-diagonal correlations. it cannot describe the build-up of quasi-long-range order at the kosterlitz–thouless transition, but its correlation length ξ∆= λt / p 4π∆( ̃ r) > ξmf bounds from below the true correlation length in the normal phase. in fig. 6, we show that the gap model accounts for the increase of the coherence length inside the fluctuation regime, ̃ n > ̃ nf for the ens parameters. for smaller interactions, as in the nist experiment, finite-size effects qualitatively change the off-diagonal elements of the density matrix (see fig. 7). 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 central coherence c(r) position r/λt n=640,000 160,000 10,000 c∆(r) cmf(r) 0 0.2 0.4 0.6 0.8 1 1 2 3 4 r/λt c(r) fig. 7: off-diagonal coherence c(r) for nist parameters with t = 0.74 (main graph) and t = 0.769 (inset) in comparison with the mean-field prediction, cmf(r), and the gap model, c∆(r), defined in eq. (37). at t = 0.769, the total central density is ̃ n(0) ≃5.1 < ̃ nf, and the system is outside the fluctuation regime. at t = 0.74, ̃ n(0) ≃10.5 > ̃ nf, and the system is close to the kosterlitz–thouless transition, ∆mf/ ̃ g ≃ 0.08. strong finite-size effects are evident in the fluctuation regime. b. density profile finite-size effects in the density profile are less dra- matic than for the coherence (see fig. 1). within mean- field theory, we have compared the density profiles of the finite system directly with those in the thermodynamic limit (lda), using the finite n solution obtained by the adapted qmc calculation described in section ii b. the mean-field analysis indicates that correlation effects are at the origin of the size-effects of the full qmc den- sity profiles in fig. 1, in particular, at small system size, n = 1000. 9 c. bose–einstein cross-over the finite-size effects in the coherence reflect the un- derlying discrete mode structure of level spacing ∼ħω. off-diagonal properties for ξ∆≳lr are cut offby the extension of the unperturbed ground-state wavefunc- tion, lr = (mω/ħ)−1/2, and resemble those of a bose- condensed system with a significant ground-state occu- pation. whereas in the thermodynamic limit, the in- teracting quasi-two-dimensional trapped bose gas under- goes a kosterlitz–thouless phase transition, the cross- over to bose–einstein condensation sets in when ∆≈ βħω. if this happens outside the fluctuation regime, ∆f mf ≲∆mf ≲βħω, the bose condensation will essen- tially have mean-field character. since the temperature scale is given by ħω/t 2d bec = π/ √ 6n, the discrete level spacing is important for small system sizes, n ≲nfs, with nfs(∆mf) ≈1 6 π2 ∆2 mft2 = π2 6 ̃ g2t2(∆mf/ ̃ g)2 . (39) for small ̃ g, close to t 2d bec where ∆mf is of order ̃ g, these finite-size effects trigger bose–einstein condensation for small n. in particular, for systems with n ≲nfs(∆f mf) ≈ π4/(6 ̃ g2), a cross-over to a mean-field-like bose conden- sation occurs[31], whereas for n ≳nfs(∆c mf) ≈400 ̃ g−2 kosterlitz–thouless-like behavior sets in (see inset of fig. 10). we notice that the finite-size scale nfs ∝1/g2 diverges very rapidly with vanishing interactions, which could make the cross-over experimentally observable. for a finite system with n ≲nfs(∆f mf), the condensate wavefunction does not develop immediately a thomas– fermi shape, but remains close to the gaussian ground- state wavefunction of the ideal gas with typical extension lr = (mω/ħ)−1/2. thus, for small condensate fraction n0, deviations of the moment of inertia i of the trapped gas from its classical value, icl = r d2r r2n(r) ∼nl2 t , are negligible, of order (icl −i)/icl ∼n0l2 r/l2 t ∼n −1/2. only for larger condensates with ̃ gn0 ≫2π, the self- interaction energy dominates the kinetic energy, and the condensate wavefunction approaches the thomas–fermi distribution of radius ∼lt , resulting in a non-classical value of the moment of inertia. in this low-temperature regime, the system can be described by a condensate with a temperature-dependent fluctuating phase [13]. there- fore, for small systems, a non-classical moment of iner- tia only occurs at lower temperatures than condensation, roughly, at a condensate fraction n0 ≳ ̃ g. to illustrate the cross-over between the bose–einstein regime at small n and the kosterlitz–thouless regime at large n, we have calculated the condensate fraction and condensate wavefunction for the ens parameter in fig. 8. to determine both quantities in inhomogeneous systems, n(1) 3d (⃗ r ,⃗ r ′) must be explicitly diagonalized, as the eigen- functions of the single-particle density matrix are not fixed by symmetry alone. in quasi-two-dimensional sys- tems, the full resolution of the off-diagonal density ma- trix in the tightly confined z-direction is difficult. it is more appropriate to consider the in-plane density matrix, n(1)(r, r′), where the confined direction is integrated over. n(1)(r, r′) = r dz r dz′ n(1) 3d (r, z; r′, z′). because of rota- tional symmetry, n(1)(r, r′) is block-diagonal in angular- momentum fourier components n(1)(r, r′) = ∞ x n=0 ∞ x l=−∞ nnlφ∗ nl(r′)φnl(r)elα(r,r′) (40) where α(r, r′) denotes the angle between r and r′, and nnl is the occupation number of the normalized eigen- mode φnl. the (in-plane) condensate fraction, n0 = n00/n, corresponds to the largest eigenvalue with l = 0 and condensate wavefunction φ0(r) = φ00(r). projection on the fourier components is convenient for determining the condensate fraction and wavefunction within qmc. for the ens parameters in fig. 8, the central density is already inside the fluctuation regime, and the conden- sate wavefunction differs from the gaussian ground state of an ideal gas. however, for small systems, n ≲103, it still has a gaussian shape indicating that the conden- sate kinetic energy dominates the potential energy. the condensate fraction vanishes as n0 ∼n −1/2. the qmc calculations of [5] demonstrated that the condensate fraction of the quasi-two-dimensional bose gas vanishes in the normal and in the superfluid phase in the thermodynamic limit, n →∞. however, in the low-temperature superfluid phase, the condensate frac- tion approaches zero very slowly with increasing system size, so that an extensive condensate remains for practi- cally all mesoscopic systems. 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 condensate wavefunction r/lr ψ0 n=250 1,000 10,000 40,000 160,000 0 0.05 0.1 0 0.02 0.04 0.06 n0/n n-1/2 fig. 8: condensate fraction n0 (inset) and wavefunction φ0(r) (main graph), computed by qmc for ens parameters with t = 0.71 for various system sizes. the condensate wave- function is modified with respect to the ground-state wave- function ψ0(r) of the unperturbed harmonic oscillator, but the gaussian shape is preserved. the two largest systems are above the scale nfs (see eq. (39)) 10 v. two-particle correlations a. pair-correlation function density–density correlations can be analyzed by con- sidering the three-dimensional pair-correlation function, n(2)(⃗ r ;⃗ r ′). this quantity factorizes within mean-field theory into terms described by the one-particle density matrix, n(1) mf (⃗ r 1;⃗ r 2) (see section iv a): n(2) mf (⃗ r 1;⃗ r 2) = nmf(⃗ r 1)nmf(⃗ r 2) + h n(1) mf (⃗ r 1;⃗ r 2) i2 . (41) for vanishing distances, ⃗ r 1 →⃗ r 2, mean-field theory pre- dicts n(2) mf (⃗ r ,⃗ r ) = 2n2 mf(⃗ r ). for bose-condensed atoms, this bunching effect is absent. in two dimensions, deviations from 2n2 of the pair- correlation function at contact signal beyond-mean-field fluctuations[12, 19, 21]. in ref. [12], the universal char- acter of the contact value was used to define the quasi- condensate density, nqc(r) ≡[2[n(r)]2 −n(2)(r, r)]1/2. this quantity has been studied in ref. [22] for a quasi- two-dimensional trapped gas within classical-field theory. the pair-correlation function of the quasi-two- dimensional gas is obtained by integrating both coordi- nates over the confined direction n(2)(r1; r2) = z dz1 z dz2 n(2)(r1, z1; r2, z2), (42) and mean-field expressions for this quantity follow from eq. (41) together with eqs (34) and (35). inside the fluctuation regime, the gap model (using eq. (37) in the mean-field expressions) again leads to an improved pair- correlation function n(2) ∆. figure 9 illustrates that outside the fluctuation regime mean-field theory describes the pair-correlation function well. in contrast to a strictly two-dimensional gas, the contact value of the pair correlation function is below 2. even in the mean-field regime, the occupation of more than one mode in the confined direction causes a notice- able reduction of the pair-correlation function at contact. the above definition of the quasi-condensate must there- fore be modified in this geometry to maintain its univer- sal character. at short distances r ∼r0, the pair-correlation function depends on the specific form of the interaction. this cannot be reproduced by the single-particle mean-field approximation. however, two-particle scattering proper- ties dominate for small enough distances as, for example, the wavefunction of hard spheres must vanish for overlap- ping particles. this feature can be included in mean-field theory by multiplying its pair-correlation functions by a short-range term χ2d(r), which accounts for two-particle scattering[23]. in two dimensions, χ2d shows a charac- teristic logarithmic behavior for short distances: χ2d(r →0) ≃ " 1 + ̃ g 2π log r πec 2 r λt #2 . 1 1.1 1.2 1.3 1.4 0 0.5 1 1.5 2 g(r) r/λt t = 1.0 g∆(r) χ2d(r) g∆(r) n=100,000 1 1.2 1.4 1.6 0 0.5 1 1.5 t = 0.75 r/λt g(r) fig. 9: central pair correlations, g(r) = n(2)(r, 0)/[n(r)n(0)], of the quasi-two-dimensional trapped bose gas at temperature t = t 2d bec (main figure) and t = 0.75 t 2d bec (inset) for n = 100, 000 atoms (ens parameters), together with the predic- tion of the mean-field gap model, g∆(r) = n(2) ∆(r, 0)/n(r)n(0), and the short-range improved mean-field model, χ2d(r)g∆(r). factorizing out the short-range behavior from the pair- correlation function, the correlation part of the renor- malized pair-correlation function, ̃ n(2) ∆− ̃ n(2)/χ2d, should be dominated by contributions from classical-field the- ory, its contact value is universal, and it might be used to define a quasi-condensate density in quasi-two dimen- sions via ̃ n2 qc = limr→0 h ̃ n(2) ∆(r, 0) − ̃ n(2)(r, 0)/χ2d(r) i (see fig. 10). similar to the correlation density, the quasi-condensate density is universal. at tkt, the clas- sical field result is ̃ nqc ≃7.2, whereas it is around 2.7 at the onset of the fluctuation regime, so that, in the normal phase, nqc/n vanishes as | log ̃ g|−1 for ̃ g →0. b. local-density correlator due to the three-dimensional nature of the underly- ing interaction potential, observables which couple di- rectly to local three-dimensional density fluctuations in- volve the following density correlator k(2) = lim δ→0 √ 2πlz r dz n(2)(r, z; r + δ, z) χ3d(δ) , (43) where χ3d(r ≪λt) ≃(1 −as/r)2 describes the uni- versal short-distance behavior of the three-dimensional two-body wavefunction in terms of the s-wave scattering length as. arguments similar to those in section v a show that the local-density correlator in general differs from the contact value of the quasi-two-dimensional pair- correlation function, further, the integration over the square of the ground state density in z leads to a 11 0 5 10 0 0.1 0.2 0.3 0.4 ~ nqc ∆mf/~ g mean-field fluctuation region kt cft n=100,000 bec kt 16~ g-2 400~ g-2 n ∆mf ∆c mf ∆f mf fig. 10: quasi-condensate density nqc obtained by qmc from the renormalized pair-correlation function at t = 0.71 t 2d bec, and t = 0.75 t 2d bec (see fig. 9) for ens parame- ters, plotted as a function of ∆mf, and compared to classical- field simulations [12]. the inset shows the boundary of the region with strong finite-size effects (see eq. (39)). phase-space-density dependence which destroys the sim- ple mean-field property k(2) ∝2n of strictly two- dimensional bose gases. vi. conclusions in this paper, we have studied the quasi-two- dimensional trapped bose gas in the normal phase above the kosterlitz–thouless temperature for small interac- tions ̃ g < 1. we have discussed the three qualitatively distinct regimes of this gas: for phase-space densities ̃ n ≲1, it is classical. at higher density, 1 ≲ ̃ n ≲ ̃ nf, the gas is in the quantum mean-field regime, and its coher- ence can be maintained over distances much larger than λt. finally, mean-field theory fails in the fluctuation regime ̃ nf ≲ ̃ n ≤ ̃ nc and beyond-mean-field corrections must be taken into account. in the fluctuation regime, for small interactions, the deviations of the density profile with respect to the mean- field profile are universal. mean-field theory thus ac- counts for most microscopic details of the gas (which depend on the interactions and on the trap geometry). we have shown in detail how to extract the correlation density (the difference between the density and the mean- field density at equal chemical potential) from qmc den- sity profiles and the lda mean-field results, and com- pared it to the universal classical field results. quantum corrections to the equation of state, expected of order ̃ g, were demonstrated to be small for current experiments with ̃ g ≲0.2. the smooth behavior of quantum cor- rections, which has been already noticed in qmc cal- culations of the kosterlitz–thouless transition tempera- ture in homogeneous systems [24], strongly differs from the three-dimensional case[25, 26] where quantum correc- tions to universality are non-analytic [27, 28], and where the universal description holds only asymptotically. it would be interesting if these universal deviations from mean-field theory could be observed experimentally. correlation effects in local observables, e.g. the den- sity profile, and local-density correlators, converge rather quickly to their thermodynamic limit value, and corre- lation effects for mesoscopic systems are well described by the local-density approximation. off-diagonal coher- ence properties show much larger finite-size effects, in particular for weak interactions. this introduces quali- tative changes for mesoscopic system sizes and, in par- ticular, the cross-over between bose–einstein physics at small particle number n ≲const/ ̃ g2 and the kosterlitz– thouless physics for larger systems. tuning the inter- action strength via a feshbach resonance might make it possible to observe the cross-over between bose–einstein condensation and kosterlitz–thouless physics in current experiments. acknowledgments w. k. acknowledges the hospitality of aspen center for physics, where part of this work was performed. nb: the computer programs used in this work are avail- able from the authors. [1] z. hadzibabic, p. kr ̈ uger, m. cheneau, b. battelier, and j. dalibard, nature 441, 1118 (2006). [2] p. clade, c. ryu, a. ramanathan, k. helmerson, w.d. phillips phys. rev. lett. 102, 170401 (2009). [3] j. m. kosterlitz and d. j. thouless, j. phys. c 6, 1181 (1973); j. m. kosterlitz, j. phys. c 7, 1046 (1974); v. l. berezinskii, sov. phys. jetp 32, 493 (1971); 34, 610 (1972). [4] p. kr ̈ uger, z. hadzibabic, and j. dalibard, phys. rev. lett. 99, 040402 (2007). [5] m. holzmann and w. krauth, phys. rev. lett. 100 190402 (2008). [6] m. holzmann, m. chevallier, and w. krauth, epl 82 30001 (2008). [7] d.s. fisher and p.c. hohenberg, phys. rev. b 37, 4936 (1988). [8] m. holzmann, g. baym, j.-p. blaizot, and f. lalo ̈ e, proc. nat. acad. sci. 104, 1476 (2007). [9] n. prokof'ev, o. ruebenacker, and b. svistunov, phys. rev. lett. 87, 270402 (2001). [10] z. hadzibabic, p. kr ̈ uger, m. cheneau, s. p. rath, and j. dalibard, new j. phys 10 045006 (2008). 12 [11] r. n. bisset, d. baillie, and p. b. blakie, phys. rev. a 79, 013602 (2009). [12] n. prokof'ev and b. svistunov, phys. rev. a 66, 043608 (2002). [13] d. s. petrov, m. holzmann, and g. v. shlyapnikov, phys. rev. lett. 84, 2551 (2000). [14] d. s. petrov and g. v. shlyapnikov, phys. rev. a 64, 012706 (2001). [15] l.-k. lim, c. m. smith, and h.t.c. stoof, phys. rev. a 78, 013634 (2008). [16] w. krauth, phys. rev. lett. 77, 3695 (1996). [17] m. holzmann, w. krauth, and m. naraschewski, phys. rev. a 59, 2956 (1999). [18] w. krauth, statistical mechanics: algorithms and com- putations, oxford university press (oxford, uk) (2006). [19] yu. kagan, v.a. kashurnikov, a.v. krasavin, n.v. prokof'ev, and b.v. svistunov, phys. rev. a 61, 43608 (2000). [20] m. chevallier and w. krauth, phys. rev. e 76 051109 (2007). [21] l. giorgetti, i. carusotto, and y. castin, phys. rev. a 76, 013613 (2007). [22] r.n. bisset, m.j. davis, t.p. simula, and p.b. blakie, phys. rev. a 79, 033626 (2009). [23] m. holzmann and y. castin, eur. phys. j. d 7, 425 (1999). [24] s. pilati, s. giorgini, and n. prokof'ev, phys. rev. lett. 100, 140405 (2008). [25] g. baym, j.-p. blaizot, m. holzmann, f. lalo ̈ e, and d. vautherin, phys. rev. lett. 83, 1703 (1999); eur. phys. j. b24, 107 (2001). [26] m. holzmann and g. baym, phys. rev. lett. 90, 040402 (2003). [27] m. holzmann, g. baym, j.-p. blaizot, and f. lalo ̈ e, phys. rev. lett. 87, 120403 (2001). [28] p. arnold, g. moore, and b. tom ́ asˆ ık, phys. rev. a 65, 013606 (2002). [29] in general, at equal chemical potential, the mean-field density differs from the exact one, and so does also the total number of particles in the trap. [30] similar to the mean-field gap, one may introduce an effective mean-field coupling constant ̃ gmf = ̃ g r d ̃ z | ̃ φ0( ̃ z)|4/ r d ̃ z | ̃ φ0( ̃ z)|2 which accounts for modifi- cations of the in-plane interactions in the ground state of the confining potential. this leads to small corrections, not visible for the experimental parameters considered in this paper. [31] in contrast to the infinite mean-field gas, these finite mean-field systems undergo a bose–einstein condensa- tion slightly below t 2d bec [8].

0911.1705

simulation-based model selection for dynamical systems in systems and population biology

computer simulations have become an important tool across the biomedical sciences and beyond. for many important problems several different models or hypotheses exist and choosing which one best describes reality or observed data is not straightforward. we therefore require suitable statistical tools that allow us to choose rationally between different mechanistic models of e.g. signal transduction or gene regulation networks. this is particularly challenging in systems biology where only a small number of molecular species can be assayed at any given time and all measurements are subject to measurement uncertainty. here we develop such a model selection framework based on approximate bayesian computation and employing sequential monte carlo sampling. we show that our approach can be applied across a wide range of biological scenarios, and we illustrate its use on real data describing influenza dynamics and the jak-stat signalling pathway. bayesian model selection strikes a balance between the complexity of the simulation models and their ability to describe observed data. the present approach enables us to employ the whole formal apparatus to any system that can be (efficiently) simulated, even when exact likelihoods are computationally intractable.

introduction mathematical models are widely used to describe and analyze complex systems and processes. formulating a model to describe, e.g. a signalling pathway or host par- asite system, requires us to condense our assumptions and knowledge into a single coherent framework [1]. mathematical analysis and computer simulations of such models then allow us to compare model predictions with experimental observations in order to test, and ultimately improve these models. the continuing success, for example of systems biology, relies on the judicious combination of experimental and theoretical lines of argument. because many of the mathematical models in biology (as in many other disciplines) ∗to whom correspondence should be addressed: [email protected], [email protected] 1 arxiv:0911.1705v3 [q-bio.qm] 20 jan 2010 are too complicated to be analyzed in a closed form, computer simulations have become the primary tool in the quantitative analysis of very large or complex bio- logical systems. this, however, can complicate comparisons of different candidate models in light of (frequently sparse and noisy) observed data. whenever proba- bilistic models exist, we can employ standard model selection approaches of either a frequentist, bayesian, or information theoretic nature [2,3]. but if suitable prob- ability models do not exist, or if the evaluation of the likelihood is computationally intractable, then we have to base our assessment on the level of agreement between simulated and observed data. this is particularly challenging when the parameters of simulation models are not known but must be inferred from observed data as well. bayesian model selection side-steps or overcomes this problem by marginal- izing (that is integrating) over model parameters, thereby effectively treating all model parameters as nuisance parameters. for the case of parameter estimation when likelihoods are intractable, approximate bayesian computation (abc) frameworks have been applied successfully [4–9]. in abc the calculation of the likelihood is replaced by a comparison between the ob- served data and simulated data. given the prior distribution p(θ) of parameter θ, the goal is to approximate the posterior distribution, p(θ|d0) ∝f(d0|θ)p(θ), where f(d0|θ) is the likelihood of θ given the data d0. abc methods have the following generic form: 1 sample a candidate parameter vector θ∗from prior distribution p(θ). 2 simulate a data set d∗from the model described by a conditional probability distribution f(d|θ∗). 3 compare the simulated data set, d∗, to the experimental data, d0, using a distance function, d, and tolerance ε; if d(d0, d∗) ≤ε, accept θ∗. the tolerance ε ≥0 is the desired level of agreement between d0 and d∗. the output of an abc algorithm is a sample of parameters from the distribution p(θ|d(d0, d∗) ≤ε). if ε is sufficiently small then this distribution will be a good approximation for the "true" posterior distribution, p(θ|d0). a tutorial on abc methods is available in the suppl. material. such a parameter estimation approach can be used whenever the model is known. however, when several plausible candidate models are available we have a model selection problem, where both the model structure and parameters are unknown. in the bayesian framework, model selection is closely related to parameter estimation, but the focus shifts onto the marginal posterior probability of model m given data d0, p(m|d0) = p(d0|m)p(m) p(d0) where p(d0|m) is the marginal likelihood and p(m) the prior probability of the model [10]. this framework has some conceptual advantages over classical hy- 2 pothesis testing: for example, we can rank an arbitrary number of different non- nested models by their marginal probabilities; and rather than only considering ev- idence against a model the bayesian framework also weights evidence in a model's favour [11]. in practical applications, however, a range of potential pitfalls need con- sidering: model probabilities can show strong dependence on model and parameter priors; and the computational effort needed to evaluate these posterior distributions can make these approaches cumbersome. the computationally expensive step in bayesian model selection is the evaluation of the marginal likelihood, which is obtained by marginalizing over model parameters; i.e. p(d0|m) = r f(d0|m, θ)p(θ|m)dθ, where p(θ|m) is the parameter prior for model m. here we develop a computationally efficient abc model selection formal- ism based on a sequential monte carlo (smc) sampler. we show that our abc smc procedure allows us to employ the whole paraphernalia of the bayesian model selection formalism, and we illustrate the use and scope of our new approach in a range of models: chemical reaction dynamics, gibbs random fields, and real data describing influenza spread and jak-stat signal transduction. 2 abc for model selection our goal is to estimate the marginal posterior distribution of a model, p(m|d0), and in this section we explain two ways in which this problem can be approached. in the joint space based approach we define a joint space of model indicators, m = 1, 2, . . . , |m|, and corresponding model parameters, θ, obtain the joint posterior distribution over the combined space of models and parameters, p(θ, m|d0), and finally marginalize over parameters to obtain p(m|d0). in the second, marginal likelihood based approach, we estimate marginal likelihoods (also called the evidence), p(d0|m), for each given model, and use these to calculate the marginal posterior model distributions through p(m|d0) = p(d0|m)p(m) p m′ p(d0|m′)p(m′). both approaches have been applied under the abc rejection scheme, which is com- putationally prohibitive for models with even an only moderate number of parame- ters [12,13]. here we incorporate ideas from smc to both of the above approaches, making them computationally more efficient. in this section we present only the more powerful approach abc smc model selection on the joint space. we refer the reader to the suppl. material for derivations and details, as well as discussion on the abc smc model selection algorithm based on the marginal likelihood approach. in model selection based on abc rejection we adapt the basic abc procedure (presented in the introduction) to the joint space, where particles (m, θ) consist of a model indicator m and a parameter θ. the abc rejection model selection algorithm on the joint space proceeds as follows [13]: 3 1 draw m∗from the prior p(m). 2 sample θ∗from the prior p(θ|m∗). 3 simulate a candidate data set d∗∼f(d|θ∗, m∗). 4 compute the distance. if d(d0, d∗) ≤ε, accept (m∗, θ∗), otherwise reject it. 5 return to 1. once a sample of n particles has been accepted, the marginal posterior distribution is approximated by p(m = m′|d0) ≈#accepted particles(m′, .) n . in the abc smc model selection algorithm on the joint space, particles (parame- ter vectors) { (m1,θ1), . . . , (mn,θn) } are sampled from the prior distribution, p(m, θ), and propagated through a sequence of intermediate distributions, p(m, θ|d(d0, d∗) ≤ εi), i = 1, . . . , t −1, until they represent a sample from the target distribution, p(m, θ|d(d0, d∗) ≤εt ). the tolerances εi are chosen such that ε1 > . . . > εt ≥0, and the distributions thus gradually evolve towards the target posterior distribution. the algorithm is presented below (and explained in the suppl. tutorial). abc smc model selection algorithm on the joint space ms1 initialize ε1, . . . , εt . set the population indicator t = 1. ms2.0 set the particle indicator i = 1. ms2.1 if t = 1, sample (m∗∗, θ∗∗) from the prior distribution p(m, θ). if t > 1, sample m∗with probability pt−1(m∗) and draw m∗∗∼kmt(m|m∗). sample θ∗from previous population {θ(m∗∗)t−1} with weights wt−1 and draw θ∗∗∼kpt,m∗∗(θ|θ∗). if p(m∗∗, θ∗∗) = 0, return to ms2.1. simulate a candidate data set d(b) ∼f(d|m∗∗, θ∗∗) bt times (b = 1, . . . , bt) and calculate bt(m∗∗, θ∗∗). if bt(m∗∗, θ∗∗) = 0, return to ms2.1. ms2.2 set (m(i) t , θ(i) t ) = (m∗∗, θ∗∗) and calculate the weight of the particle as w(i) t (m(i) t , θ(i) t ) =    bt(m(i) t , θ(i) t ), if t = 1 p(m(i) t , θ(i) t )bt(m(i) t , θ(i) t ) s , if t > 1. 4 where bt(m(i) t , θ(i) t ) = 1 bt bt x b=1 1(d(d0, d∗ b) ≤εt) s = |m| x j=1 pt−1(m(j) t−1)kmt(m(i) t |m(j) t−1) × x k;mt−1=m(i) t w(k) t−1kpt,m(i) t (θ(i) t |θ(k) t−1) pt−1(mt−1 = m(i) t ) if i < n set i = i + 1, go to ms2.1. ms3 normalize the weights wt. sum the particle weights to obtain marginal model probabilities, pt(mt = m) = x i;m(i) t =m w(i) t (m(i) t , θ(i) t ). if t < t, set t = t + 1, go to ms2.0. particles sampled from a previous distribution are denoted by a single asterisk, and after perturbation by a double asterisk. km is a model perturbation kernel which allows us to obtain model m from model m∗and kp is the parameter perturba- tion kernel. bt ≥1 is the number of replicate simulation run for a fixed particle (for deterministic models bt = 1) and |m| denotes the number of candidate models. the output of the algorithm, i.e. the set of particles {(mt , θt )} associated with weights wt , is the approximation of the full posterior distribution on the joint model and parameter space. the approximation of the marginal posterior distribution of the model obtained by marginalization is pt (mt = m) = x i;m(i) t =m w(i) t (m(i) t , θ(i) t ), and we can also straightforwardly obtain the marginalized parameter distributions. the algorithm requires the user to define the prior distribution, distance function, tolerance schedule and perturbation kernels. in all examples presented in the results section we choose uniform prior distributions for all parameters and models; that is all models are a priori equally plausible. such priors are informative in a sense that they define a feasible parameter region (e.g. reaction rates are positive), but they are predominantly non-informative as they do not specify any further preference for particular parameter values. this way the inference will mostly be informed by the information contained in the data. a good tolerance can be found empirically by 5 trying to reach the lowest distance feasible and arrive at the posterior distribution in a computationally efficient way. our perturbation kernels are component-wise truncated uniform or gaussian and are automatically adapted by feeding back in- formation on the obtained parameter ranges from the previous population. distance functions are defined for each model as specified in the results section. the algo- rithm presented in toni et al. [8] is a special case of the above algorithm for discrete uniform km kernel and uniform prior distribution of the model p(m). 3 results in this section we illustrate abc smc for model selection on a simple example of stochastic reaction kinetics. we then compare the computational efficiency of abc smc for stochastic models of gibbs random fields with that of the abc rejection model selection method. finally, we apply the algorithm to several real datasets: first we select between different stochastic models of influenza epidemics (where we can compare our approach with previously published results obtained using exact bayesian model selection), and then apply our approach to choose from among different mechanistic models for the stat5 signaling pathway. 3.1 chemical reaction kinetics we illustrate our algorithm for the stochastic reaction kinetic models x + y k1 − →2y and x k2 − →y . the first is a model of an autocatalytic reaction, where the reaction product y is the catalyst for the reaction. in the second, molecules y do not need to be present for a change from x to y to occur. such models have, for example, been considered in the context of prion replication dynamics [14,15], where x represents a healthy form of a prion protein and y a diseased form. we simulate synthetic datasets of y measured at 20 time points using gillespie algorithm [16] from model 2 with parameter k2 = 30 and initial conditions x0 = 40, y0 = 3 (figure 1(a), suppl. table 1). we apply our abc smc algorithm for model selection, which identifies the correct model with high confidence (figure 1(b)). 3.2 gibbs random fields gibbs random fields have become staple models in machine learning, including appli- cations in computational biology and bioinformatics (see for example [13,17]). here we use two gibbs random field models [18], for which closed form posterior distribu- tions are available. this allows us to compare the abc smc approximated posterior distributions of the models to true posterior distribtuions, and to demonstrate the computational efficiency of our approach when compared to model selection based on abc rejection sampling. both models, m0 and m1, are defined on a sequence of n binary random variables, 6 0.00 0.02 0.04 0.06 0.08 0.10 0 10 20 30 40 number of molecules (a) 0.00 0.02 0.04 0.06 0.08 0.10 0 10 20 30 40 number of molecules (a) 1 2 model 0.0 0.4 0.8 prior 1 2 model 0.0 0.4 0.8 population1 1 2 model 0.0 0.4 0.8 population2 1 2 model 0.0 0.4 0.8 population3 1 2 model 0.0 0.4 0.8 population4 1 2 model 0.0 0.4 0.8 population5 (b) f igure 2: (a) stochastic trajectories of species x (red) and y (blue). model 1 is simulated for k1 = 2 .1 (dark colours), model 2 for k2 = 30 (light colours). data points are represented by circles. (b) w e have repeated the model selection run 20 times; the red sections present 25% and 75% quantiles around the median. prior distribution p(m) is chosen uniform and k1, k2 ? u(0, 100). p erturbation kernels are chosen as follows: k p t(k|k? ) = u(−σ,σ), σ = 2(max{k}t−1 −min{k}t−1) and k m t(m|m? ) = 0 .7 if m = m? and 0.3 otherwise. distance function is mean squared error and epsilon schedule ? = {3000, 1400, 600, 140, 40}. 3 model model 1 2 1 2 1 2 1 2 1 2 1 2 prior population 1 population 2 population 3 population 4 population 5 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 (b) figure 1: 1 figure 1: (a) stochastic trajectories of species x (red) and y (blue). model 1 is simulated for k1 = 2.1 (dashed line), model 2 for k2 = 30 (full line). data points are represented by circles. (b) we have repeated the model selection run 20 times; the red sections present 25% and 75% quantiles around the median. prior distribution p(m) is chosen uniform and k1, k2 ∼u(0, 100). perturbation kernels are chosen as follows: kpt(k|k∗) = k∗+ u(−σ, σ), σ = 2(max{k}t−1 −min{k}t−1) and kmt(m|m∗) = 0.7 if m = m∗and 0.3 otherwise. number of particles n = 1000. bt = 1. distance function is mean squared error and tolerance schedule ε = {3000, 1400, 600, 140, 40}. x = (x1, . . . , xn), xi ∈{0, 1}; m0 is a collection of n iid bernoulli random variables with probability θ0/(1 + exp(θ0)); m1 is equivalent to a standard ising model, i.e. x1 is taken to be a binary random variable and p(xi+1 = xi|xi) = θ1/(1 + exp(θ1)) for i = 2, . . . , xn. the likelihood functions are f0(x|θ0)= eθ0s0(x) (1 + eθ0)n and f1(x|θ1)= eθ1s1(x) 2(1 + eθ1)n−1 , where s0(x) = pn i=1 1(xi = 1) and s1(x) = pn i=2 1(xi = xi−1) are sufficient statis- tics, respectively. we simulate 1000 datasets from both models for different values of parameters θ0 ∼u(−5, 5), θ1 ∼u(0, 6) and n = 100. using abc smc for model selec- tion allows us to estimate posterior model distributions correctly and demonstrate a considerable computational speed-up in abc smc compared to abc rejection (figure 2). 3.3 infuenza infection outbreaks we next apply abc smc for model selection to models of the spread of different strains of the influenza virus. we use data from influenza a (h3n2) outbreaks that occurred in 1977-78 and 1980-81 in tecomseh, michigan [19] (suppl. table 2), and a second dataset of an influenza b infection outbreak in 1975-76 and influenza a (h1n1) infection outbreak in 1978-79 in seattle, washington [20] (suppl. table 3). 7 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 p(m=0) p (m=0) smc (a) 0 500 1000 1500 2000 2500 0.000 0.005 0.010 0.015 0.020 0 5 10 15 0.00 0.01 0.02 0.03 0.04 0.05 frequency n /n smc rej (b) f igure 5: 5 figure 2: (a) true vs. inferred posterior model distribution. in abc smc we use the euclidian distance d(d0, x) = p (s0(d0) −s0(x))2 + (s1(d0) −s1(x))2. n = 500. bt = 1. tolerance schedule: ε = {9, 4, 3, 2, 1, 0}. perturbation kernels: kmt(m|m∗) = 0.75 if m = m∗and 0.25 otherwise; kpt(θ|θ∗) = θ∗+ u(−σ, σ), σ = 0.5(max{θ}t−1 −min{θ}t−1). we have excluded those datasets for which all states are in 0 or 1 (for which p(m = 0) ≈0.3094 is also correctly inferred) from the analysis. (b) comparison of the number of simulation steps needed by abc rejection (nrej) and abc smc (nsmc); abc smc yields an approximately 50-fold speed-up on average. the basic questions to be addressed here are whether (i) different outbreaks of the same strain and (ii) outbreaks of different molecular strains of the influenza virus can be described by the same model of disease spread. we assume that virus can spread from infected to susceptible individuals and distin- guish between spread inside households or across the population at large [20]. let qc denote the probability that a susceptible individual does not get infected from the community and qh the probability that a susceptible individual escapes infection within their household. then wjs, the probability that j out of the s susceptibles in a household become infected, is given by wjs = s j  wjj(qcqj h)s−j, (1) where w0s = qs c, s = 0, 1, 2, . . ., and wjj = 1 −pj−1 i=0 wij. we are interested in inferring the pair of parameters qh and qc of the model (1) using the data from suppl. table 2. these data were obtained from two separate outbreaks of the same strain, h3n2, and the question of interest is whether these are characterized by the same epidemiological parameters (this question was previously considered in [21, 22]). to investigate this issue, we consider two models: one with four pa- rameters, qh1, qc1, qh2, qc2, which describes the hypothesis that each outbreak has its own characteristics; the second models the hypothesis that both outbreaks share the same epidemiological parameter values for qh and qc. prior distributions of all parameters are chosen to be uniform over the range [0, 1]. 8 to apply abc smc, we use a distance function d(d0, d∗) = 1 2(||d1 −d∗(qh1, qc1)||f + ||d2 −d∗(qh2, qc2)||f ), where || ||f denotes the frobenious norm, d0 = d1 ∪d2 with d1 the 1977-78 out- break and d2 the 1980-81 outbreak datasets from suppl. table 2, and d∗is the simulation output from model (1). the results we obtain are sumarized in figure 3(a) - 3(b) and strongly suggest that the two outbreaks appear to have shared the same epidemiological characteristics. figure 3(a) shows the posterior distribution of the four-parameter model. the marginal posterior distributions of qh1 and qc1 are largely overlapping with the marginal posterior distributions of qh2 and qc2 and we therefore, unsurprisingly, get strong evidence in favour of the two-parameter model. figure 3(b) shows the marginal posterior distribution of the model; the posterior probability of model 1 is 0.98 (median over 10 runs), which gives unambiguous sup- port to model 1, meaning that outbreaks of the same strain share the same dynamics. outbreaks due to a different viral strain (suppl. table 3) have different characteris- tics as indicated by the posterior distribution of the four-parameter model presented in figure 3(c). this was confirmed by applying our model selection algorithm; the inferred posterior marginal model probability of a two-parameter model was negli- gible (results not shown). from figure 3(c) we also see that these differences are due to differences in viral spread across the community whereas within-household dynamics are comparable. we thus explore a further model with three parameters, qc1, qc2, qh (model 1), where the two outbreaks share the same within-household characteristics (qh), and compare it against and the four-parameter model (model 2). the obtained bayes factor suggests that there is only very week evidence in favour of model 1 (figure 3(d)), which is in agreement with the result of [21]. in general genetic predisposition, differences in immunity and lifestyle etc. will lead to heterogeneity in susceptibility to viral infection among the host population. such a model can be written as [22] wjs(v) = s−j x i=0 s i  vi(1 −v)s−iwj,s−i. (2) on the basis of the previous results, we combine both outbreak data sets from suppl. table 2, and find some evidence that model (2) explains the data better than model (1), suggesting that the host-virus dynamics are shaped by the molecular nature of the viral strain, as well as by variability in the host population (see suppl. figure 2). 3.4 jak-stat signaling pathway having convinced ourselves that the novel abc smc model selection approach agrees with the analytical model probabilities, and those obtained using conven- 9 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 qh qc (a) 1 2 model 0.0 0.2 0.4 0.6 0.8 1.0 population8 (b) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 qh qc (c) 1 2 model 0.0 0.2 0.4 0.6 0.8 1.0 population7 (d) f igure 2: 2 1 2 1 2 model model approximation of model posterior p(m|d0) approximation of model posterior p(m|d0) figure 3: (a) abc smc posterior distributions for parameters inferred for a four-parameter model from the data in suppl. table 2. marginal posterior distributions of parameters qc1, qh1 (red) and qc2, qh2 (blue). (b) estimation of a posterior marginal distribution p(m|d0). model 1 is a two-parameter and model 2 a four-parameter model (1). all intermediate populations are shown in suppl. figure 1(a). (c) the same as (a) but here the data used is from suppl. table 3. (d) estimation of a posterior marginal distribution. model 1 is a two-parameter and model 2 a three-parameter model (1). all intermediate populations are shown in suppl. figure 1(b). 10 tional bayesian model selection, while outperforming conventional abc rejection model selection approaches, we can now turn our attention to real world scenarios that have not previously been considered from a bayesian (exact or approximate) perspective. here we consider models of signaling though the erythropoietin recep- tor (epor), transduced by stat5 (figure 4(a)) [23, 24]. signaling through this receptor is crucial for proliferation, differentiation, and survival of erythroid progen- itor cells [25]. when the epo hormone binds to the epor receptor, the receptor's cytoplasmic domain is phosporylated, which creates a docking site for signaling molecules, in particular stat5. upon binding to the activated receptor, stat5 first becomes phosphorylated, then dimerizes and translocates to the nucleus, where it acts as a transcription factor. there have been competing hypotheses about what happens with the stat5 in the nucleus. originally it had been suggested that stat5 gets degraded in the nucleus in an ubiquitin-asssociated way [26], but other evidence suggests that they are dephosphorylated in the nucleus and then trafficked back to the cytoplasm [27]. the ambiguity of the shutoffmechanism of stat5 in the nucleus triggered the development of several mathematical models [29,30,32] describing different hypothe- ses. all models assume mass action kinetics and denote the amount of activated epo-receptors by epora, monomeric unphosphorylated and phosporylated stat5 molecules by x1 and x2, respectively, dimeric phosphorylated stat5 in the cyto- plasm by x3 and dimeric phosphorylated stat5 in the nucleus by x4. the most basic model timmer et al. developed, under the assumption that phosphorylated stat5 does not leave the nucleus, consists of the following kinetic equations, ̇ x1 = −k1x1epora (3) ̇ x2 = −k2x2 2 + k1x1epora ̇ x3 = −k3x3 + 1 2k2x2 2 ̇ x4 = k3x3. (4) one can then assume that phosphorylated stat5 dimers dissociate and leave the nucleus; this is modelled by adding appropriate kinetic terms to the equations (3) and (4) of the basic model to obtain ̇ x1 = −k1x1epora + 2k4x4 ̇ x4 = k3x3 −k4x4. the cycling model can be developed further by assuming a delay before stat5 leaves the nucleus: ̇ x1 = −k1x1epora + 2k4x3(t −τ) ̇ x4 = k3x3 −k4x3(t −τ). (5) this model was chosen as the best model in the original analyses [29,30] based on a numerical evaluation of the likelihood, followed by a likelihood ratio test and boot- strap procedure for model selection. the data are partially observed time course 11 arbouzova, n. i. et al. development 2006;133:2605-2616 the canonical model of jak/stat signalling (a) 1 2 3 0.0 0.4 0.8 population1 1 2 3 0.0 0.4 0.8 population2 1 2 3 0.0 0.4 0.8 population3 1 2 3 0.0 0.4 0.8 population4 1 2 3 0.0 0.4 0.8 population5 1 2 3 0.0 0.4 0.8 population6 1 2 3 0.0 0.4 0.8 population7 1 2 3 0.0 0.4 0.8 population8 1 2 3 0.0 0.4 0.8 population9 1 2 3 0.0 0.4 0.8 population10 1 2 3 0.0 0.4 0.8 population11 1 2 3 0.0 0.4 0.8 population12 1 2 3 0.0 0.4 0.8 population13 1 2 3 0.0 0.4 0.8 population14 1 2 3 0.0 0.4 0.8 population15 1 2 3 model 0.0 0.4 0.8 population16 1 2 3 model 0.0 0.4 0.8 population17 1 2 3 model 0.0 0.4 0.8 population18 1 2 3 model 0.0 0.4 0.8 population19 1 2 3 model 0.0 0.4 0.8 population20 population 11 population 11 population 13 population 12 population 14 population 15 population 18 population 19 population 17 population 16 population 20 population 8 population 9 population 10 population 7 population 6 population 2 population 3 population 4 population 5 population 1 (b) figure 1: 1 figure 4: (a) stat5 signaling pathway. adapted from [28]. (b) histograms show pop- ulations of the model parameter m. population 20 represents the approximation of the marginal posterior distribution of m. tolerance schedule: ε = {200, 100, 50, 35, 30, 25, 22, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8}. perturbation kernels: kmt(m|m∗) = 0.6 if m = m∗and 0.2 otherwise; kpt(θ|θ∗) = θ∗+ u(−σ, σ), σ = 0.5(max{θ}t−1 −min{θ}t−1). n = 500. distance function: d(d0, d∗) = s p t  y(1) 0 (t)−y∗(1)(t) σ(1) d0(t)) 2 +  y(2)(t)−y∗(2)(t) σ(2) d0(t) 2 , with d0 = {y(1) 0 , y(2) 0 }, d∗= {y∗(1), y∗(2)} and y(1) the total amount of phosphoryalated stat5 in the cytoplasm and y(2) the total amount of stat5 in the cytoplasm. σ(1) d0 and σ(2) d0 are the associated confidence intervals; reassuringly, other distance functions, e.g. the square root of the sum of squared errors yield identical model selection results (data not shown). 12 measurements of the total amount of stat5 in the cytoplasm, and the amount of phosphorylated stat5 in the cytoplasm; both are only known up to a normalizing factor. we propose a further model with clear physical interpretation where the delay acts on stat5 inside the nucleus (x4) rather than on x3 (in equation (5)), for which a biological interpretation is difficult. instead of x3(t −τ), we propose to model the delay of phosphorylated stat5 x4 in the nucleus directly and obtain [31]: ̇ x1 = −k1x1epora + 2k4x4(t −τ) ̇ x4 = k3x3 −k4x4(t −τ). we perform the abc smc model selection algorithm on the following non-nested models: (1) cycling delay model with x3(t −τ), (2) cycling delay model with x4(t−τ), (3) cycling model without a delay. the model parameter m can therefore take values 1, 2 and 3. for each proposed model and parameter combination we numerically solve the ode equations of the model and add ε ∼n(0, σ) to obtain the simulated time course data. the noise parameter σ can be either fixed or treated as another parameter to be estimated; we consider the latter option, under the assumption that the experi- mental noise is independent and identically distributed for all time points. figure 4(b) shows intermediate populations leading to the abc smc marginal posterior distribution over the model parameters m (population 20). bayes factors can be calculated from the last population and according to the conventional in- terpretation of bayes factors [33], it can be concluded that there is strong evidence in favour of model 3 compared to model 1, positive evidence in favour of model 3 compared to model 2, and positive evidence in favour of model 2 compared to model 1. thus cycling appears to be clearly important and the model that receives the most support is the cycling model without a time-delay. here the flexibility of abc smc has allowed us to perform simultaneous model selection on non-nested models of ordinary and time-delay differential equations. 4 discussion we have developed a novel model selection methodology based on approximate bayesian computation and sequential monte carlo. the results obtained in our ap- plications illustrate the usefulness and wide applicability of our abc smc method, even when experimental data are scarce, there are no measurements for some of the species, temporal data are not measured at equidistant time points, and when parameters such as kinetic rates are unknown. in the context of dynamical systems our method can be applied across all simulation and modelling (including qualitative modelling) frameworks; for jak-stat signal transduction dynamics, for example, 13 we have been able to compare the relative explanatory power of ode and time-delay differential equation models. our model selection procedure is also not confined to dynamical systems; in fact the scope for application is immense and limited only by the availability of efficient simulation approaches. routine application to complex models in systems, computational and population biology with hundreds or thousands of parameters [34] will require further numeri- cal developments due to the high computational cost of repeated simulations. smc based abc methods are, however, highly paralellizable and we believe that future work should exploit this property to make these methods computationally more ef- ficient. further potential improvements might come from (i) regression adjustment techniques that have so far been applied in the parameter estimation abc frame- work [4,35,36] (ii) from automatic generation of the tolerance schedules [37], and (iii) by developing more sophisticated perturbation kernels that exploit inherent prop- erties of biological dynamical systems such as sloppiness [38,39]; here especially we feel that there is substantial room for improvement as the likelihoods of dynamical systems contain information about the qualitative behaviour [40] which can also be exploited in abc frameworks. 5 conclusion we conclude by emphasizing the need for inferential methods which can assess the relative performance and reliability of different models. the need for such reliable model selection procedures can hardly be overstated: with an increasing number of biomedical problems being studied using simulation approaches, there is an obvious and urgent need for statistically sound approaches that allow us to differentiate between different models. if parameters are known or the likelihood is available in a closed form, then the model selection is generally straightforward. however, for many of the most interesting systems biology (and generally, scientific) problems this is not the case and here abc smc can be employed. acknowledgement we are especially grateful to paul kirk for his insightful comments and many valu- able discussions. we furthermore thank the members of theoretical systems biology group for discussions and comments on earlier versions of this paper. funding: this work was supported through a mrc priority studentship (t.t.) and bbsrc grant bb/g009374/1. references [1] may rm. uses and abuses of mathematics in biology. science, 303(5659):790–3, 2004. 14 [2] burnham k and anderson d. model selection and multimodel inference: a practical information-theoretic approach. springer-verlag new york, inc., 2002. [3] vyshemirsky v and girolami ma. bayesian ranking of biochemical system models. bioinfor- matics, 24(6):833–9, 2008. [4] beaumont ma, zhang w and balding dj. approximate bayesian computation in population genetics. genetics, 162(4):2025–2035, 2002. [5] marjoram p, molitor j, plagnol v and tavare s. markov chain monte carlo without likeli- hoods. proc natl acad sci usa, 100(26):15324–8, 2003. [6] sisson sa, fan y and tanaka mm. sequential monte carlo without likelihoods. proc natl acad sci usa, 104(6):1760–5, 2007. [7] ratmann o, jorgensen o, hinkley t, stumpf m, richardson s and wiuf c. using likelihood- free inference to compare evolutionary dynamics of the protein networks of h. pylori and p. falciparum. plos comput biol, 3(11):2266–2278, 2007. [8] toni t, welch d, strelkowa n, ipsen a and stumpf mph. approximate bayesian computation scheme for parameter inference and model selection in dynamical systems. j. r. soc. interface, 6:187–202, 2009. [9] ratmann o, andrieu c, wiuf c and richardson s. model criticism based on likelihood- free inference, with an application to protein network evolution. proc natl acad sci usa, 106(26):10576–81, 2009. [10] gelman a, jb c, stern h and rubin d. bayesian data analysis. chapman & hall/crc, 2nd edition, 2003. [11] jeffreys h. theory of probability. 1st ed. the clarendon press, oxford, 1939. [12] wilkinson rd. bayesian inference of primate divergence times. phd thesis, university of cambridge, 2007. [13] grelaud a, robert cp, marin jm, rodolphe f and taly jf. abc likelihood-free methods for model choice in gibbs random fields. bayesian analysis, 4(2):317–336, 2009. [14] prusiner sb. novel proteinaceous infectious particles cause scrapie. science, 216(4542):136–44, 1982. [15] eigen m. prionics or the kinetic basis of prion diseases. biophys chem, 63(1):a1–18, 1996. [16] gillespie d. exact stochastic simulation of coupled chemical reactions. the journal of physical chemistry, 1977. [17] wei, z. and li h. (2007). a markov random field model for network-based analysis of genomic data. bioinformatics, 23(12), 1537–44. [18] møller j. spatial statistics and computational methods. springer, 2003. [19] addy c, jr il and haber m. a generalized stochastic model for the analysis of infectious disease final size data. biometrics, 961–974, 1991. [20] jr il and koopman j. household and community transmission parameters from final distri- butions of infections in households. biometrics, 115–126, 1982. [21] clancy d and o'neill pd. exact bayesian inference and model selection for stochastic models of epidemics among a community of households. scand j stat, 34(2):259–274, 2007. [22] o'neill p, balding d, becker n, eerola m and mollison d. analyses of infectious disease data from household outbreaks by markov chain monte carlo methods. j roy stat soc c-app, 49:517–542, 2000. [23] darnell je. stats and gene regulation. science, 277(5332):1630–5, 1997. [24] horvath cm. stat proteins and transcriptional responses to extracellular signals. trends biochem sci, 25(10):496–502, 2000. 15 [25] klingmuller u, bergelson s, hsiao jg and lodish hf. multiple tyrosine residues in the cytosolic domain of the erythropoietin receptor promote activation of stat5. proc natl acad sci usa, 93(16):8324–8, 1996. [26] kim tk and maniatis t. regulation of interferon-gamma-activated stat1 by the ubiquitin- proteasome pathway. science, 273(5282):1717–9, 1996. [27] k ̈ oster m and hauser h. dynamic redistribution of stat1 protein in ifn signaling visualized by gfp fusion proteins. eur j biochem, 260(1):137–44, 1999. [28] arbouzova ni and zeidler mp. jak/stat signalling in drosophila: insights into conserved regulatory and cellular functions. development, 133(14):2605–16, 2006. [29] swameye i, muller tg, timmer j, sandra o and klingmuller u. identification of nucleocy- toplasmic cycling as a remote sensor in cellular signaling by databased modeling. proc natl acad sci usa, 100(3):1028–33, 2003. [30] muller tg, faller d, timmer j, swameye i, sandra o and klingm ̈ uller u. tests for cycling in a signalling pathway. journal of the royal statistical society series c, 53(4):557, 2004. [31] zi, z. and klipp, e. (2006) sbml-pet: a systems biology markup language-based parameter estimation tool. bioinformatics, 22(21), 2704–5. [32] timmer j and muller t. modeling the nonlinear dynamics of cellular signal transduction. int j bifurcat chaos, 14:2069–2079, 2004. [33] kass r and raftery a. bayes factors. journal of the american statistical association, 90:773– 795, 1995. [34] chen ww, schoeberl b, jasper pj, niepel m, nielsen ub, lauffenburger da and sorger pk. input-output behavior of erbb signaling pathways as revealed by a mass action model trained against dynamic data. mol syst biol, 5:239, 2009. [35] blum mg and fran ̧ cois o. non-linear regression models for approximate bayesian computa- tion. statistics and computing, in press, 2009. [36] excoffier cldwl. bayesian computation and model selection in population genetics. arxiv:0901.2231v1 [stat.me], 2009. [37] del moral, p., doucet, a., and jasra, a. (2009). an adaptive sequential monte carlo method for approximate bayesian computation. imperial college technical report. [38] gutenkunst r, waterfall j, casey f, brown k, myers c and sethna j. universally sloppy parameter sensitivities in systems biology models. plos comput biol, 3(10):e189, 2007. [39] secrier, m., toni, t., and stumpf, m. p. h. (2009). the abc of reverse engineering biological signalling systems. mol. biosyst, (in press) doi: 10.1039/b908951a. [40] kirk pdw, toni t and stumpf mph. parameter inference for biochemical systems that undergo a hopf bifurcation. biophysical journal, 95(2):540–9, 2008. [41] robert cp and casella g. monte carlo statistical methods. springer, 2004. [42] doucet a, freitas nd and gordon n. sequential monte carlo methods in practice. springer, 2001. [43] moral pd, doucet a and jasra a. sequential monte carlo samplers. j. royal statist. soc. b, 2006. 16 supplementary material a: derivation of abc smc model selection algorithms we start this section by briefly reviewing the building blocks of the abc smc algorithm of toni et al. [8], which is based on sequential importance sampling (sis). the main idea of importance sampling is to sample from the desired target distribution π (which can be impossible or hard to sample from) indirectly through sampling from a proposal distribution η [41]. to get a sample from π, one can instead sample from η and weight the samples by importance weights w(x) = π(x) η(x) . in sis one reaches the target distribution πt through a series of intermediate distributions, πt, t = 1, . . . , t −1 [42, 43]. if it is hard to sample from these distributions one can use the idea of importance sampling described above to sample from a series of proposal distributions ηt and weight the obtained samples by importance weights wt(xt) = πt(xt) ηt(xt) . (6) in sis the proposal distributions are defined as ηt(xt) = z ηt−1(xt−1)κt(xt−1, xt)dxt−1, (7) where ηt−1 is the previous proposal distribution and κt is a markov kernel. to apply sis, we need to define the intermediate and the proposal distributions. in an abc framework [4,5], which is based on comparisons between simulated and experimental datasets, we define the intermediate distributions as [6,8] πt(x) = p(x) bt bt x b=1 1 d(d0, d(b)(x)) ≤εt
, (8) where p(x) denotes the prior distribution and d(1), . . . , d(bt) are bt ≥1 data sets generated for a fixed parameter x, d(b) ∼f(d|x). 1(x) is an indicator function and εt is the tolerance required from particles contributing to the intermediate distribution t. to simplify the notation we define bt(x) = 1 bt pbt b=1 1 d(d0, d(b)(x)) ≤εt
. we define the first proposal distribution to equal the prior distribution, η1(x) = p(x). the proposal distribution at time t (t = 2, . . . , t), ηt, is defined as ηt(xt) = 1 (p(xt) > 0) 1 (bt(xt) > 0) z πt−1(xt−1)kt(xt|xt−1)dxt−1, (9) where kt denotes the perturbation kernel (e.g. random walk around the particle). for details of how this proposal distribution was obtained, see [8]. in the remainder of this section we introduce three different ways in which abc smc ideas pre- sented above can be used in the model selection framework. we start by proposing a simple and naive incorporation of the above building blocks for model selection. we then continue by deriving an abc smc model selection algorithm on the joint model and parameter space, which is presented in the methods section of the paper. in the end we present abc smc algorithm for approximation of the marginal likelihood, which can also be employed for model selection. the only of these three algorithms that we present in the main part of the paper and use in examples is algorithm ii (abc smc model selection on the joint space), since the other two algo- rithms (i and iii) are computationally too expensive and impractical to use. 17 i) abc smcm rejθ model selection algorithm very naively and stragihtforwardly the intermediate distributions can be defined as πt(m) = p(m)bmt(m), where bmt(m) := p θ∼p (θ|m) 1(d(d0, d(θ, m)) < εt) p θ∼p (θ|m) 1(p(θ|m) > 0) . this means that for each model m we calculate bmt(m) as the ratio between the number of accepted particles (where the distance falls below εt) and all sampled particles, where parameters θ of model m are sampled from the prior distribution p(θ|m). if a set of candidate models m of a finite size |m| is being considered, and n denotes the number of particles, then we can write the algorithm as follows: ms1 initialize ε1, . . . , εt . set the population indicator t = 1. ms2 for i = 1, . . . , |m|, calculate the weights as w(i) t (m(i) t ) =    bmt(m(i) t ), if t = 1 p (m(i) t )bmt(m(i) t ) pn j=1 w(j) t−1kmt(m(i) t |m(j) t−1), if t > 1. ms3 normalize the weights. if t < t, set t = t + 1, go to ms2. in this algorithm we estimate the posterior distribution of the model indicator m sequentially (i.e. using ideas from sis), but the integration over model parameters is not sequential; we always sample them from the prior distribution p(θ|m) (i.e. in the rejection sampling manner). this algorithm is therefore computationally very expensive. it would be computationally more efficient to generate θt by exploiting the knowledge about θ that is contained in {θ}t−1. in addition to learning m sequentially, i.e. by exploiting {m}t−1 for generating mt, we would also like to learn θ sequentially. in order to do this, we define ii) abc smc model selection on the joint space let (m, θ) denote a particle from a joint space, where m corresponds to the model indicator and θ are the parameters of model m. we define the intermediate distributions by πt(m, θ) = p(m, θ)bt(m, θ), where bt(m, θ) = 1 bt bt x b=1 1(d(d0, d(b)(m, θ)) ≤εt). in the following equations kmt denotes the perturbation kernel for the model parameter, kpt,m denotes the perturbation kernel for the parameters of model m, and t is the population number. now we derive the sequential importance sampling weights wt(mt, θt) = πt(mt, θt) ηt(mt, θt) . 18 for a particle (mt, θt) from population t, we define the proposal distribution ηt(mt, θt) as ηt(mt, θt) = 1 (p(mt, θt) > 0) 1 (bt(mt, θt) > 0) (10) × z mt−1 πt−1(mt−1)kmt(mt|mt−1)dmt−1 × z θt−1|mt−1=mt πt−1(θt−1)kpt(θt|θt−1)dθt−1 ∝ 1 (p(mt, θt) > 0) 1 (bt(mt, θt) > 0) × |m| x j=1 pt−1(m(j) t−1)kmt(mt|m(j) t−1) × x k;mt−1=mt w(k) t−1 pt−1(mt−1 = mt)kpt,mt(θt|θ(k) t−1), where intermediate marginal model probabilities pt(m) are defined as pt(mt = m) = x mt=m wt(mt, θt). the weights for all accepted particles are (obtained by including (8) and (10) in equation (6)) wt(mt, θt) = p(mt, θt)bt(mt, θt) p|m| j=1 p (j) t−1(m(j) t−1)kmt(mt|m(j) t−1) p k;mt−1=mt w(k) t−1 pt−1(mt−1=mt)kpt,mt(θt|θ(k) t−1) . the resulting abc smc algorithm is presented in the methodology section of the main part of the paper. iii) abc smc approximation of the marginal likelihood p(d0|m) if we can calculate the marginal likelihood p(d0|m) for each of the candidate models that we consider in the model selection problem, then we can calculate the marginal posterior distribution of a model m as p(m|d0) = p(d0|m)p(m) p m′ p(d0|m′)p(m′). (11) we now explain how to calculate p(d0|m) for model m. in the abc rejection-based approach the posterior distribution of the parameters for each model m are estimated independently by employing abc rejection; the marginal likelihood then equals the acceptance rate, p(d0|m) ≈#accepted particles given model m nm , (12) i.e. the ratio between the number of accepted versus the number of proposed particles nm. we can use this marginal likelihood estimate to calculate p(m|d0) using equation (11). this approach has been used in [12]. we now derive how abc smc can be used for estimating the marginal likelihood, which can be then used for model selection. in a usual abc smc setting for drawing samples from the pos- terior parameter distribution p(θ|m, d0) for a given model m, we define intermediate distributions as πt(θ) = p(θ)1(d(d0, d(θ)) ≤εt). (13) the target distribution πt is an unnormalized approximation of the posterior distribution p(θ|m, d0). we are now interested in its normalization constant, i.e. the marginal likelihood, p(d0|m) ≈ z θ πt (θ)dθ. 19 let us call the integrals of πt(θ), r θ πt(θ)dθ, the intermediate marginal likelihoods. in the usual abc smc parameter estimation setting, our goal is to obtain samples from distribu- tion πt (θ), whereas our goal here is to obtain its normalization constant. while this distribution as defined in equation (13) is in general unnormalized, the abc smc parameter estimation algorithm performs normalization of weights at every t and therefore returns its normalized version [8]. so we cannot use the usual output of abc smc directly. instead we proceed as follows. we would like to draw particles from the following target distribution: tt (θ) = p(θ)1[d(d0, d(θ)) ≤εt ] + p(θ)1[d(d0, d(θ)) > εt ], where p(θ) is the prior distribution. to draw samples from tt we can use abc smc, where we define the intermediate distributions as tt(θ) = p(θ)1[d(d0, d(θ)) ≤εt] + p(θ)1[d(d0, d(θ)) > εt] = t 1 t (θ) + t 2 t (θ). in each population we accept n particles, and a particle is only rejected if it falls outside the boundaries of tt. we classify the accepted particles in two sets, θ1 t := {θ; d(d0, d(θ)) ≤εt} and θ2 t := {θ; d(d0, d(θ)) > εt}, depending on the distance reached. in each population t we can then calculate the intermediate marginal likelihoods by z θ t 1 t (θ)dθ = x θ∈θ1 t wt(θ). the target marginal likelihood, r θ t 1 t (θ)dθ, is our approximation of p(d0|m). in an abc rejection setting, where t = 1 and all weights are equal, this result corresponds to (12). after calculating p(d0|m) for each m, we can use equation (11) to calculate the marginal pos- terior distributions for model m, p(m|d0) ≈ p(m) p θ∈θ1 t wt (θ) p(m′) p m′ p θ′∈θ′1 t w′ t (θ′). the model selection algorithm based on approximating the marginal likelihood proceeds as follows: algorithm m1 for model mj, j = 1, . . . , |m| do steps s1 to s4. then go to m2. s1 initialize ε1, . . . , εt . set the population indicator t = 1. s2.0 set the particle indicator i = 1. s2.1 if t = 1, sample θ∗∗independently from p(θ). if t > 1, sample θ∗from the previous population {θ(i) t−1} with weights wt−1 and perturb the particle to obtain θ∗∗∼kt(θ|θ∗), where kt is a perturbation kernel. if p(θ∗∗) = 0, return to s2.1. for a particle θ∗∗simulate a candidate data set d and calculate d(d0, d(θ∗∗)). if d(d0, d(θ∗∗)) ≤εt, add θ∗∗to θ1 t(mj). if d(d0, d(θ∗∗)) > εt, add θ∗∗to θ2 t(mj). s2.1 calculate the weight for particle θ(i) t = θ∗∗: w(i) t (θ(i) t ) = ( 1, if t = 1 p(θ(i) t )/ pn j=1 w(j) t−1kt(θ(i) t |θ(j) t−1)  , if t > 1. if i < n set i = i + 1, go to s2.1. 20 s3 normalize the weights. if t < t, set t = t + 1, go to s2.0. s4 calculate p(d0|mj) ≈ p(mj) p θ∈θ1 t (mj) wt (θ) p m′ p(m′) p θ′∈θ′1 t (m) w′ t (θ′). m2 for each mj calculate p(mj|d0) using equation p(m|d0) = p(d0|m)p(m) p m′ p(d0|m′)p(m′). the computational advantage of this model selection algorithm compared to the marginal likelihood model selection based on abc rejection can be obtained by (i) starting with a small number of particles n in population 1 and increasing it in each subsequent population. this way not much computational effort is spent on simulations in earlier populations, but we nevertheless have a big enough sample set in the last population to obtain a reliable estimate; (ii) exploiting the property that intermediate distributions in the parameter estimation framework should be included in one another, and so range θ1 t ≥range θ1 t+1, t = 1, . . . , t −1. in other words, a proposed particle in population t cannot belong to θ1 t if it cannot be obtained by perturbing any of the particles in θ1 t−1. we can therefore reject some of the proposed particles without simulation. this means a huge saving in computational time, since simulations are the most expensive part of abc based algortihms. however, one of the obvious ways to exploit this property would be to use a truncated perturbation kernel with ranges they cover being smaller than the range of prior distribution. but we find this unsatisfactory and, in the present form, feel that evaluating the marginal model likelihood directly is not practical. supplementary material b: tutorial on abc rejection and abc smc for parameter estimation and model se- lection available in arxiv (reference arxiv:0910.4472v2 [stat.co]). supplementary material c: supplementary figures and datasets available on the bioinformatics webpage. 21

0911.1706

lp convergence with rates of smooth poisson-cauchy type singular operators

in this article we continue the study of smooth poisson-cauchy type singular integral operators on the line regarding their convergence to the unit operator with rates in the lp norm, p greater equal one. the related established inequalities involve the higher order lp modulus of smoothness of the engaged function or its higher order derivative.

introduction the rate of convergence of singular integrals has been studied in [9], [13], [14], [15], [7], [8], [4], [5], [6] and these articles motivate our work. here we study the lp, p ≥1, convergence of smooth poisson- cauchy type singular integral operators over r to the unit operator with rates over smooth functions with higher order derivatives in lp(r). we establish related jackson type inequalities involving the higher lp modulus of smoothness of the engaged function or its higher order derivative. the discussed operators are not in general positive, see [10], [11]. other motivation comes from [1], [2]. 2 results in the next we introduce and deal with the smooth poisson-cauchy type singular integral operators mr,ξ(f; x) defined as follows. for r ∈n and n ∈z+ we set αj =      (−1)r−jr j
j−n, j = 1, . . . , r, 1 − r x j=1 (−1)r−jr j
j−n, j = 0, (1) that is r p j=0 αj = 1. 1 george a. anastassiou, razvan a. mezei 2 let f ∈cn(r) and f (n) ∈lp(r), 1 ≤p < ∞, α ∈n, β > 1 2α, we define for x ∈r, ξ > 0 the lebesgue integral mr,ξ(f; x) = w z ∞ −∞ pr j=0 αjf(x + jt) t2α + ξ2α
β dt, (2) where the constant is defined as w = γ (β) αξ2αβ−1 γ 1 2α
γ β − 1 2α
. note 1. the operators mr,ξ are not, in general, positive. see [10], (18). we notice by w r ∞ −∞ 1 (t2α+ξ2α)β dt = 1, that mr,ξ(c, x) = c, c constant, see also [10], [11], and mr,ξ(f; x) −f(x) = w   r x j=0 αj z ∞ −∞ [f(x + jt) −f(x)] 1 t2α + ξ2α
β dt  . (3) we use also that z ∞ −∞ tk t2α + ξ2α
β dt = (0, k odd, 1 ξ2αβ−k−1α γ( k+1 2α )γ(β−k+1 2α ) γ(β) , k even, with β > k+1 2α , (4) see [16]. we need the rth lp-modulus of smoothness ωr(f (n), h)p := sup |t|≤h ∥∆r tf (n)(x)∥p,x, h > 0, (5) where ∆r tf (n)(x) := r x j=0 (−1)r−j r j  f (n)(x + jt), (6) see [12], p. 44. here we have that ωr(f (n), h)p < ∞, h > 0. we need to introduce δk := r x j=1 αjjk, k = 1, . . . , n ∈n, (7) and denote by ⌊*⌋the integral part. call τ(w, x) := r x j=0 αjjnf (n)(x + jw) −δnf (n)(x). (8) notice also that − r x j=1 (−1)r−j r j  = (−1)r r 0  . (9) according to [3], p. 306, [1], we get τ(w, x) = ∆r wf (n)(x). (10) thus ∥τ(w, x)∥p,x ≤ωr(f (n), |w|)p, w ∈r. (11) george a. anastassiou, razvan a. mezei 3 using taylor's formula, and the appropriate change of variables, one has (see [6]) r x j=0 αj[f(x + jt) −f(x)] = n x k=1 f (k)(x) k! δktk + rn(0, t, x), (12) where rn(0, t, x) := z t 0 (t −w)n−1 (n −1)! τ(w, x)dw, n ∈n. (13) using the above terminology we obtain for β > 2⌊n 2 ⌋+1 2α that ∆(x) := mr,ξ(f; x) −f(x) − ⌊n/2⌋ x m=1 f (2m)(x)δ2m (2m)! γ 2m+1 2α
γ β −2m+1 2α
γ 1 2α
γ β − 1 2α
ξ2m = r∗ n(x), (14) where r∗ n(x) := w z ∞ −∞ rn(0, t, x) 1 t2α + ξ2α
β dt, n ∈n. (15) in ∆(x), see (14), the sum collapses when n = 1. we present our first result. theorem 1. let p, q > 1 such that 1 p + 1 q = 1, n ∈n, α ∈n, β > 1 α  1 p + n + r  and the rest as above. then ∥∆(x)∥p ≤ (2α) 1 p γ (β) γ  qβ 2 − 1 2α  1 q ξnτ 1 p γ  qβ 2  1 q γ 1 2α
1 p γ β − 1 2α
(rp + 1) 1 p [(n −1)!] (q(n −1) + 1)1/q ωr(f (n), ξ)p, (16) where 0 < τ := "z ∞ 0 (1 + u)rp+1 unp−1 (u2α + 1)pβ/2 du − z ∞ 0 unp−1 (u2α + 1)pβ/2 du # < ∞. (17) hence as ξ →0 we obtain ∥∆(x)∥p →0. if additionally f (2m) ∈lp(r), m = 1, 2, . . . ,  n 2  then ∥mr,ξ(f) −f∥p →0, as ξ →0. proof. we observe that |∆(x)|p = w p      z ∞ −∞ rn(0, t, x) 1 t2α + ξ2α
β dt      p ≤w p z ∞ −∞ |rn(0, t, x)| 1 t2α + ξ2α
β dt !p ≤w p z ∞ −∞      z |t| 0 (|t| −w)n−1 (n −1)! |τ(sign(t) * w, x)|dw      1 t2α + ξ2α
β dt !p . (18) hence we have i := z ∞ −∞ |∆(x)|pdx ≤w p z ∞ −∞ z ∞ −∞ γ(t, x) 1 t2α + ξ2α
β dt !p dx ! , (19) where γ(t, x) := z |t| 0 (|t| −w)n−1 (n −1)! |τ(sign(t) * w, x)|dw ≥0. (20) george a. anastassiou, razvan a. mezei 4 therefore by using h ̈ older's inequality suitably we obtain r.h.s.(19) = w p z ∞ −∞ z ∞ −∞ γ(t, x) 1 t2α + ξ2α
β dt !p dx ! = w p *   z ∞ −∞   z ∞ −∞ γ(t, x) 1 t2α + ξ2α
β/2 1 t2α + ξ2α
β/2 dt   p dx   ≤w p *    z ∞ −∞   z ∞ −∞  γ(t, x) 1 t2α + ξ2α
β/2   p dt     z ∞ −∞   1 t2α + ξ2α
β/2   q dt   p q dx    = w p *    z ∞ −∞   z ∞ −∞ γp(t, x) 1 t2α + ξ2α
pβ/2 dt     z ∞ −∞ 1 t2α + ξ2α
qβ/2 dt   p q dx    = w p *   z ∞ −∞   z ∞ −∞ γp(t, x) 1 t2α + ξ2α
pβ/2 dt  dx     γ 1 2α
γ  qβ 2 − 1 2α  γ  qβ 2  αξqαβ−1   p q = ξpαβ−1α [γ (β)]p γ  qβ 2 − 1 2α  p q γ  qβ 2  p q γ 1 2α
γ β − 1 2α
p   z ∞ −∞   z ∞ −∞ γp(t, x) 1 t2α + ξ2α
pβ/2 dt  dx  . (21) again by h ̈ older's inequality we have γp(t, x) ≤ r |t| 0 |τ(sign(t) * w, x)|pdw  ((n −1)!)p |t|np−1 (q(n −1) + 1)p/q . (22) consequently we have r.h.s.(21) ≤ ξpαβ−1α [γ (β)]p γ  qβ 2 − 1 2α  p q γ  qβ 2  p q γ 1 2α
γ β − 1 2α
p *   z ∞ −∞   z ∞ −∞ r |t| 0 |τ(sign(t) * w, x)|pdw  ((n −1)!)p |t|np−1 (q(n −1) + 1)p/q 1 t2α + ξ2α
pβ/2 dt  dx   = : (∗), (calling c1 := ξpαβ−1α [γ (β)]p γ  qβ 2 − 1 2α  p q γ  qβ 2  p q γ 1 2α
γ β − 1 2α
p ((n −1)!)p(q(n −1) + 1)p/q ) (23) george a. anastassiou, razvan a. mezei 5 and (∗) = c1   z ∞ −∞   z ∞ −∞ z |t| 0 |τ(sign(t) * w, x)|pdw ! |t|np−1 1 t2α + ξ2α
pβ/2 dx  dt   = c1   z ∞ −∞   z ∞ −∞ z |t| 0 |∆r sign(t)*wf (n)(x))|pdw ! |t|np−1 1 t2α + ξ2α
pβ/2 dx  dt   = c1   z ∞ −∞ z ∞ −∞ z |t| 0 |∆r sign(t)*wf (n)(x))|pdw ! dx ! |t|np−1 1 t2α + ξ2α
pβ/2 dt   = c1   z ∞ −∞ z |t| 0 z ∞ −∞ |∆r sign(t)*wf (n)(x))|pdx  dw ! |t|np−1 1 t2α + ξ2α
pβ/2 dt   ≤c1   z ∞ −∞ z |t| 0 ωr(f (n), w)p pdw ! |t|np−1 1 t2α + ξ2α
pβ/2 dt  . (24) so far we have proved i ≤c1   z ∞ −∞ z |t| 0 ωr(f (n), w)p pdw ! |t|np−1 1 t2α + ξ2α
pβ/2 dt  . (25) by [12], p. 45 we have (r.h.s.(25)) ≤c1  ωr(f (n), ξ)p p   z ∞ −∞ z |t| 0  1 + w ξ rp dw ! |t|np−1 1 t2α + ξ2α
pβ/2 dt  =: (∗∗). (26) but we see that (∗∗) =  ξc1 rp + 1   ωr(f (n), ξ)p p j , (27) where j = z ∞ −∞  1 + |t| ξ rp+1 −1 ! |t|np−1 1 t2α + ξ2α
pβ/2 dt = 2 z ∞ 0  1 + t ξ rp+1 −1 ! tnp−1 1 t2α + ξ2α
pβ/2 dt. (28) here we find j = 2ξp(n−αβ) z ∞ 0  (1 + u)rp+1 −1  unp−1 1 (u2α + 1)pβ/2 du = 2ξp(n−αβ) "z ∞ 0 (1 + u)rp+1 unp−1 (u2α + 1)pβ/2 du − z ∞ 0 unp−1 (u2α + 1)pβ/2 du # . (29) thus by (17) and (29) we obtain j = 2ξp(n−αβ)τ. (30) george a. anastassiou, razvan a. mezei 6 we notice that 0 < τ < z ∞ 0 (1 + u)rp+1 unp−1 (u2α + 1)pβ/2 du < z ∞ 0 (1 + u)rp+1 (1 + u)np−1 (u2α + 1)pβ/2 du = z ∞ 0 (1 + u)p(n+r) (u2α + 1)pβ/2 du =: i1. also call k := z 1 0 (1 + u)p(n+r) (u2α + 1)pβ/2 du < ∞. then we can write i1 = k + z ∞ 1 (1 + u)p(n+r) (u2α + 1)pβ/2 du < k + 2p(n+r) z ∞ 1 up(n+r) (u2α + 1)pβ/2 du = k + 2p(n+r)i2, where i2 := r ∞ 1 up(n+r) (u2α+1)pβ/2 du. since 1 1+u2α < 1 u2α , we have 1 (1+u2α)pβ/2 < 1 upαβ , for u ∈[1, ∞). so we get i2 < z ∞ 1 up(n+r−αβ)du = lim ε→∞ z ε 1 up(n+r−αβ)du = lim ε→∞  εp(n+r−αβ)+1 −1 p (n + r −αβ) + 1  = −1 p (n + r −αβ) + 1, which is a positive number since β > 1 α  1 p + n + r  . consequently i2 is finite, so is i1, proving τ < ∞. using (27) and (30) we get (∗∗) =  ξc1 rp + 1   ωr(f (n), ξ)p p 2ξp(n−αβ)τ = 2α [γ (β)]p γ  qβ 2 − 1 2α  p q τ (rp + 1) γ 1 2α
γ β − 1 2α
p γ  qβ 2  p q ((n −1)!)p(q(n −1) + 1)p/q ξpn  ωr(f (n), ξ)p p . (31) i.e. we have established that i ≤ 2α [γ (β)]p γ  qβ 2 − 1 2α  p q τ (rp + 1) γ 1 2α
γ β − 1 2α
p γ  qβ 2  p q ((n −1)!)p(q(n −1) + 1)p/q ξpn  ωr(f (n), ξ)p p . (32) that is finishing the proof of the theorem. ■ the counterpart of theorem 1 follows, case of p = 1. theorem 2. let f ∈cn(r) and f (n) ∈l1(r), n ∈n, α ∈n, β > n+r+1 2α . then ∥∆(x)∥1 ≤ 1 (r + 1) (n −1)!γ 1 2α
γ β − 1 2α
(33) * "r+1 x k=1 r + 1 k  γ n + k 2α  γ  β −n + k 2α # ωr(f (n), ξ)1ξn. george a. anastassiou, razvan a. mezei 7 hence as ξ →0 we obtain ∥∆(x)∥1 →0. if additionally f (2m) ∈l1(r), m = 1, 2, . . . ,  n 2  then ∥mr,ξ(f) −f∥1 →0, as ξ →0. proof. it follows |∆(x)| = w      z ∞ −∞ rn(0, t, x) 1 t2α + ξ2α
β dt      ≤w z ∞ −∞ |rn(0, t, x)| 1 t2α + ξ2α
β dt ≤w z ∞ −∞ z |t| 0 (|t| −w)n−1 (n −1)! |τ(sign(t) * w, x)|dw ! 1 t2α + ξ2α
β dt. (34) thus ∥∆(x)∥1 = z ∞ −∞ |∆(x)|dx ≤w * (35) z ∞ −∞ z ∞ −∞ z |t| 0 (|t| −w)n−1 (n −1)! |τ(sign(t) * w, x)|dw ! 1 t2α + ξ2α
β dt ! dx =: (∗) but we see that z |t| 0 (|t| −w)n−1 (n −1)! |τ(sign(t) * w, x)|dw ≤ |t|n−1 (n −1)! z |t| 0 |τ(sign(t) * w, x)|dw. (36) therefore it holds (∗) ≤w z ∞ −∞ z ∞ −∞ |t|n−1 (n −1)! z |t| 0 |τ(sign(t) * w, x)|dw ! 1 t2α + ξ2α
β dt ! dx = w (n −1)! z ∞ −∞ z |t| 0 z ∞ −∞ |τ(sign(t) * w, x)|dx  dw ! |t|n−1 t2α + ξ2α
β dt ! ≤ w (n −1)! z ∞ −∞ z |t| 0 ωr(f (n), w)1dw ! |t|n−1 t2α + ξ2α
β dt ! . (37) i.e. we get ∥∆(x)∥1 ≤ w (n −1)! z ∞ −∞ z |t| 0 ωr(f (n), w)1dw ! |t|n−1 t2α + ξ2α
β dt ! . (38) consequently we have ∥∆(x)∥1 ≤wωr(f (n), ξ)1 (n −1)! z ∞ −∞ z |t| 0  1 + w ξ r dw ! |t|n−1 t2α + ξ2α
β dt ! = 2ξwωr(f (n), ξ)1 (r + 1) (n −1)! z ∞ 0  1 + t ξ r+1 −1 ! tn−1 t2α + ξ2α
β dt ! = 2γ (β) αξ2αβωr(f (n), ξ)1 (r + 1) (n −1)!γ 1 2α
γ β − 1 2α
z ∞ 0  1 + t ξ r+1 −1 ! tn−1 t2α + ξ2α
β dt ! . (39) george a. anastassiou, razvan a. mezei 8 we have gotten so far ∥∆(x)∥1 ≤ 2γ (β) αξ2αβωr(f (n), ξ)1 * λ (r + 1) (n −1)!γ 1 2α
γ β − 1 2α
, (40) where λ := z ∞ 0  1 + t ξ r+1 −1 ! tn−1 t2α + ξ2α
β dt. (41) one easily finds that λ = z ∞ 0 r+1 x k=1 r + 1 k   t ξ k! tn−1 t2α + ξ2α
β dt = ξn−2αβ r+1 x k=1 r + 1 k  z ∞ 0 t n+k−1 (t 2α + 1)β dt = ξn−2αβ r+1 x k=1 r + 1 k  kn+k. (42) where kn+k := z ∞ 0 t n+k−1 (t 2α + 1)β dt = γ n+k 2α
γ β −n+k 2α
γ (β) 2α . (43) ∥∆(x)∥1 ≤ 1 (r + 1) (n −1)!γ 1 2α
γ β − 1 2α
"r+1 x k=1 r + 1 k  γ n + k 2α  γ  β −n + k 2α # ωr(f (n), ξ)1ξn. we have proved (33). ■ the case n = 0 is met next. proposition 1. let p, q > 1 such that 1 p + 1 q = 1, α ∈n, β > 1 α  r + 1 p  and the rest as above. then ∥mr,ξ(f) −f∥p ≤ (2α) 1 p [γ (β)] γ  qβ 2 − 1 2α  1 q θ 1 p γ 1 2α
1 p γ β − 1 2α
γ  qβ 2  1 q ωr(f, ξ)p, (44) where 0 < θ := z ∞ 0 (1 + t)rp 1 (t2α + 1)pβ/2 dt < ∞. (45) hence as ξ →0 we obtain mr,ξ →unit operator i in the lp norm, p > 1. proof. by (3) we notice that, mr,ξ(f; x) −f(x) = w   r x j=0 αj z ∞ −∞ (f(x + jt) −f(x)) 1 t2α + ξ2α
β dt   = w   z ∞ −∞   r x j=0 αj (f(x + jt) −f(x))   1 t2α + ξ2α
β dt   = w   z ∞ −∞   r x j=1 αjf(x + jt) − r x j=1 αjf(x)   1 t2α + ξ2α
β dt   george a. anastassiou, razvan a. mezei 9 = w   z ∞ −∞   r x j=1  (−1)r−j r j  j−n  f(x + jt) − r x j=1  (−1)r−j r j  j−n  f(x)   1 t2α + ξ2α
β dt   = w   z ∞ −∞   r x j=1  (−1)r−j r j  f(x + jt) − r x j=1  (−1)r−j r j  f(x)   1 t2α + ξ2α
β dt   (9) = w   z ∞ −∞   r x j=1  (−1)r−j r j  f(x + jt) +  (−1)r−0 r 0  f(x + 0t)   1 t2α + ξ2α
β dt   = w   z ∞ −∞   r x j=0 (−1)r−j r j  f(x + jt)   1 t2α + ξ2α
β dt   (6) = w z ∞ −∞ (∆r tf) (x) 1 t2α + ξ2α
β dt ! . (46) and then |mr,ξ(f; x) −f(x)| ≤w z ∞ −∞ |∆r tf(x)| 1 t2α + ξ2α
β dt ! . (47) we next estimate z ∞ −∞ |mr,ξ(f; x) −f(x)|pdx ≤ z ∞ −∞ (w)p z ∞ −∞ |∆r tf(x)| 1 t2α + ξ2α
β dt !p dx = (w)p z ∞ −∞   z ∞ −∞  |∆r tf(x)| 1 t2α + ξ2α
β/2     1 t2α + ξ2α
β/2  dt   p dx ≤(w)p z ∞ −∞      z ∞ −∞  |∆r tf(x)| 1 t2α + ξ2α
β/2   p dt   1 p   z ∞ −∞   1 t2α + ξ2α
β/2   q dt   1 q    p dx = (w)p z ∞ −∞   z ∞ −∞ |∆r tf(x)|p 1 t2α + ξ2α
pβ/2 dt     z ∞ −∞ 1 t2α + ξ2α
qβ/2 dt   p q dx = (w)p z ∞ −∞   z ∞ −∞ |∆r tf(x)|p 1 t2α + ξ2α
pβ/2 dt     γ 1 2α
γ  qβ 2 − 1 2α  γ  qβ 2  αξqαβ−1   p q dx = (w)p   γ 1 2α
γ  qβ 2 − 1 2α  γ  qβ 2  αξqαβ−1   p q z ∞ −∞   z ∞ −∞ |∆r tf(x)|p 1 t2α + ξ2α
pβ/2 dt  dx george a. anastassiou, razvan a. mezei 10 = αξαβp−1 [γ (β)]p γ  qβ 2 − 1 2α  p q γ 1 2α
γ β − 1 2α
p γ  qβ 2  p q z ∞ −∞   z ∞ −∞ |∆r tf(x)|p 1 t2α + ξ2α
pβ/2 dx  dt = αξαβp−1 [γ (β)]p γ  qβ 2 − 1 2α  p q γ 1 2α
γ β − 1 2α
p γ  qβ 2  p q z ∞ −∞ z ∞ −∞ |∆r tf(x)|p dx  1 t2α + ξ2α
pβ/2 dt ≤ αξαβp−1 [γ (β)]p γ  qβ 2 − 1 2α  p q γ 1 2α
γ β − 1 2α
p γ  qβ 2  p q z ∞ −∞ ωr(f, |t|)p p 1 t2α + ξ2α
pβ/2 dt = 2αξαβp−1 [γ (β)]p γ  qβ 2 − 1 2α  p q γ 1 2α
γ β − 1 2α
p γ  qβ 2  p q z ∞ 0 ωr(f, t)p p 1 t2α + ξ2α
pβ/2 dt ≤ 2αξαβp−1 [γ (β)]p γ  qβ 2 − 1 2α  p q γ 1 2α
γ β − 1 2α
p γ  qβ 2  p q ωr(f, ξ)p p z ∞ 0  1 + t ξ rp 1 t2α + ξ2α
pβ/2 dt = 2αξαβp−1 [γ (β)]p γ  qβ 2 − 1 2α  p q γ 1 2α
γ β − 1 2α
p γ  qβ 2  p q ωr(f, ξ)p p z ∞ 0 (1 + t )rp 1 (t 2α + 1)pβ/2 ξαpβ ξdt = 2α [γ (β)]p γ  qβ 2 − 1 2α  p q γ 1 2α
γ β − 1 2α
p γ  qβ 2  p q ωr(f, ξ)p p z ∞ 0 (1 + t)rp 1 (t2α + 1)pβ/2 dt. (48) we have established (44). we also notice that θ = z ∞ 0 (1 + t)rp (1 + t2α)pβ/2 dt = z 1 0 (1 + t)rp (1 + t2α)pβ/2 dt + z ∞ 1 (1 + t)rp (1 + t2α)pβ/2 dt < z 1 0 (1 + t)rp (1 + t2α)pβ/2 dt + 2rp z ∞ 1 trp (1 + t2α)pβ/2 dt < z 1 0 (1 + t)rp (1 + t2α)pβ/2 dt + 2rp z ∞ 1 tp(r−αβ)dt = z 1 0 (1 + t)rp (1 + t2α)pβ/2 dt − 2rp p (r −αβ) + 1, the last, since β > 1 α  r + 1 p  , is a finite positive constant. thus 0 < θ < ∞. ■ we also give proposition 2. assume β > r+1 2α . it holds ∥mr,ξf −f∥1 ≤ 2αγ (β) γ 1 2α
γ β − 1 2α
z ∞ 0 (1 + t)r 1 (t2α + 1)β dt ! ωr(f, ξ)1. (49) hence as ξ →0 we get mr,ξ →i in the l1 norm. george a. anastassiou, razvan a. mezei 11 proof. by (47) we have again |mr,ξ(f; x) −f(x)| ≤w z ∞ −∞ |∆r tf(x)| 1 t2α + ξ2α
β dt ! . next we estimate z ∞ −∞ |mr,ξ(f; x) −f(x)| dx ≤w z ∞ −∞ z ∞ −∞ |∆r tf(x)| 1 t2α + ξ2α
β dt ! dx = w z ∞ −∞ z ∞ −∞ |∆r tf(x)| dx  1 t2α + ξ2α
β dt ≤w z ∞ −∞ ωr(f, |t|)1 1 t2α + ξ2α
β dt ≤w2ωr(f, ξ)1 z ∞ 0  1 + t ξ r 1 t2α + ξ2α
β dt = γ (β) ξ2αβ−12α γ 1 2α
γ β − 1 2α
ωr(f, ξ)1 z ∞ 0 ξ (1 + t)r 1 (t2α + 1)β ξ2αβ dt = γ (β) 2α γ 1 2α
γ β − 1 2α
ωr(f, ξ)1 z ∞ 0 (1 + t)r 1 (t2α + 1)β dt. (50) we have proved (49). we also notice that 0 < z ∞ 0 (1 + t)r 1 (t2α + 1)β dt = z 1 0 (1 + t)r (t2α + 1)β dt + z ∞ 1 (1 + t)r (t2α + 1)β dt < z 1 0 (1 + t)r (t2α + 1)β dt + 2r z ∞ 1 tr−2αβdt = z 1 0 (1 + t)r (t2α + 1)β dt − 2r (r −2αβ + 1), which is a positive finite constant. ■ in the next we consider f ∈cn(r) and f (n) ∈lp(r), n = 0 or n ≥2 even, 1 ≤p < ∞and the similar smooth singular operator of symmetric convolution type mξ(f; x) = w z ∞ −∞ f(x + y) 1 y2α + ξ2α
β dy, for all x ∈r, ξ > 0. (51) that is mξ(f; x) = w z ∞ 0 (f(x + y) + f(x −y)) 1 y2α + ξ2α
β dy, for all x ∈r, ξ > 0. notice that m1,ξ = mξ. let the central second order difference ( ̃ ∆2 yf)(x) := f(x + y) + f(x −y) −2f(x). (52) notice that ( ̃ ∆2 −yf)(x) = ( ̃ ∆2 yf)(x). george a. anastassiou, razvan a. mezei 12 when n ≥2 even using taylor's formula with cauchy remainder we eventually find ( ̃ ∆2 yf)(x) = 2 n/2 x ρ=1 f (2ρ)(x) (2ρ)! y2ρ + r1(x), (53) where r1(x) := z y 0 ( ̃ ∆2 tf (n))(x)(y −t)n−1 (n −1)! dt. (54) notice that mξ(f; x) −f(x) = w z ∞ 0 ( ̃ ∆2 yf(x)) 1 y2α + ξ2α
β dy. (55) furthermore by (4), (53) and (55) we easily see that k(x) : = mξ(f; x) −f(x) − n/2 x ρ=1 f (2ρ)(x) (2ρ)! γ 2ρ+1 2α
γ β −2ρ+1 2α
γ 1 2α
γ β − 1 2α
ξ2ρ (56) = w z ∞ 0 z y 0 ( ̃ ∆2 t f (n))(x)(y −t)n−1 (n −1)! dt  1 y2α + ξ2α
β dy, where β > (n+1) 2α . therefore we have |k(x)| ≤w z ∞ 0 z y 0    ̃ ∆2 t f (n)   (x)(y −t)n−1 (n −1)! dt  1 y2α + ξ2α
β dy. (57) here we estimate in lp norm, p ≥1, the error function k(x). notice that we have ω2(f (n), h)p < ∞, h > 0, n = 0 or n ≥2 even. operators mξ are positive operators. the related main lp result here comes next. theorem 3. let p, q > 1 such that 1 p + 1 q = 1, n ≥2 even, α ∈n, β > 1 α  1 p + n + 2  and the rest as above. then ∥k(x)∥p ≤ ̃ τ 1/pα1/pγ  qβ 2 − 1 2α 1/q 2 1 q γ 1 2α
1/p γ β − 1 2α
γ  qβ 2 1/q (q(n −1) + 1)1/q (2p + 1)1/p γ (β) (n −1)!ξnω2(f (n), ξ)p, (58) where 0 < ̃ τ = z ∞ 0  (1 + u)2p+1 −1  upn−1 1 (1 + u2α)pβ/2 du < ∞. (59) hence as ξ →0 we get ∥k(x)∥p →0. if additionally f (2m) ∈lp(r), m = 1, 2, . . . , n 2 then ∥mξ(f) −f∥p →0, as ξ →0. proof. we observe that |k(x)|p ≤w p z ∞ 0 z y 0    ̃ ∆2 tf (n)   (x)(y −t)n−1 (n −1)! dt  1 y2α + ξ2α
β dy !p . (60) call ̃ γ(y, x) := z y 0 | ̃ ∆2 tf (n)(x)|(y −t)n−1 (n −1)! dt ≥0, y ≥0, (61) george a. anastassiou, razvan a. mezei 13 then we have |k(x)|p ≤w p z ∞ 0 ̃ γ(y, x) 1 y2α + ξ2α
β dy !p . (62) consequently λ : = z ∞ −∞ |k(x)|pdx ≤w p z ∞ −∞ z ∞ 0 ̃ γ(y, x) 1 y2α + ξ2α
β dy !p dx = w p z ∞ −∞   z ∞ 0 ̃ γ(y, x) 1 y2α + ξ2α
β/2 1 y2α + ξ2α
β/2 dy   p dx (by h ̈ older's inequality) ≤w p    z ∞ −∞   z ∞ 0 ( ̃ γ(y, x))p 1 y2α + ξ2α
pβ/2 dy     z ∞ 0 1 y2α + ξ2α
qβ/2 dy   p/q dx    = w p   γ 1 2α
γ  qβ 2 − 1 2α  2γ  qβ 2  αξqαβ−1   p/q   z ∞ −∞   z ∞ 0 ( ̃ γ(y, x))p 1 y2α + ξ2α
pβ/2 dy  dx   = [γ (β)]p αξαβp−1γ  qβ 2 − 1 2α p/q 2 p q γ 1 2α
γ β − 1 2α
p γ  qβ 2 p/q   z ∞ −∞   z ∞ 0 ( ̃ γ(y, x))p 1 y2α + ξ2α
pβ/2 dy  dx   = : (∗). (63) by applying again h ̈ older's inequality we see that ̃ γ(y, x) ≤ r y 0 | ̃ ∆2 t f (n)(x)|pdt 1/p (n −1)! y(n−1+ 1 q ) (q(n −1) + 1)1/q . (64) therefore it holds (∗) ≤ [γ (β)]p αξαβp−1γ  qβ 2 − 1 2α p/q 2 p q γ 1 2α
γ β − 1 2α
p γ  qβ 2 p/q [(n −1)!]p (q(n −1) + 1)p/q *   z ∞ −∞   z ∞ 0 z y 0 | ̃ ∆2 tf (n)(x)|pdt  yp(n−1+ 1 q ) 1 y2α + ξ2α
pβ/2 dy  dx   = [γ (β)]p αξαβp−1γ  qβ 2 − 1 2α p/q 2 p q γ 1 2α
γ β − 1 2α
p γ  qβ 2 p/q [(n −1)!]p (q(n −1) + 1)p/q *   z ∞ 0   z ∞ −∞ z y 0 | ̃ ∆2 tf (n)(x)|pdt  yp(n−1+ 1 q ) 1 y2α + ξ2α
pβ/2 dx  dy   = : (∗∗). (65) we call c2 := [γ (β)]p αξαβp−1γ  qβ 2 − 1 2α p/q 2 p q γ 1 2α
γ β − 1 2α
p γ  qβ 2 p/q [(n −1)!]p (q(n −1) + 1)p/q . (66) george a. anastassiou, razvan a. mezei 14 and hence (∗∗) = c2   z ∞ 0 z ∞ −∞ z y 0 | ̃ ∆2 tf (n)(x)|pdt  dx  yp(n−1+ 1 q ) 1 y2α + ξ2α
pβ/2 dy   = c2   z ∞ 0 z y 0 z ∞ −∞ | ̃ ∆2 tf (n)(x)|pdx  dt  ypn−1 1 y2α + ξ2α
pβ/2 dy   = c2   z ∞ 0 z y 0 z ∞ −∞ |∆2 tf (n)(x −t)|pdx  dt  ypn−1 1 y2α + ξ2α
pβ/2 dy   = c2   z ∞ 0 z y 0 z ∞ −∞ |∆2 tf (n)(x)|pdx  dt  ypn−1 1 y2α + ξ2α
pβ/2 dy   ≤c2   z ∞ 0 z y 0 ω2(f (n), t)p pdt  ypn−1 1 y2α + ξ2α
pβ/2 dy   ≤c2ω2(f (n), ξ)p p   z ∞ 0 z y 0  1 + t ξ 2p dt ! ypn−1 1 y2α + ξ2α
pβ/2 dy  . (67) i.e. so far we proved that λ ≤c2ω2(f (n), ξ)p p   z ∞ 0 z y 0  1 + t ξ 2p dt ! ypn−1 1 y2α + ξ2α
pβ/2 dy  . (68) but r.h.s.(68) = c2ξ 2p + 1ω2(f (n), ξ)p p   z ∞ 0  1 + y ξ 2p+1 −1 ! ypn−1 1 y2α + ξ2α
pβ/2 dy  . (69) call m := z ∞ 0  1 + y ξ 2p+1 −1 ! ypn−1 1 y2α + ξ2α
pβ/2 dy, (70) and ̃ τ := z ∞ 0  (1 + u)2p+1 −1  upn−1 1 (1 + u2α)pβ/2 du. (71) that is m = ξp(n−αβ) ̃ τ. (72) therefore it holds λ ≤ ̃ τ [γ (β)]p αξpnγ  qβ 2 − 1 2α p/q ω2(f (n), ξ)p p 2 p q (2p + 1) γ 1 2α
γ β − 1 2α
p γ  qβ 2 p/q [(n −1)!]p (q(n −1) + 1)p/q . (73) we have established (58). ■ the counterpart of theorem 3 follows, p = 1 case. george a. anastassiou, razvan a. mezei 15 theorem 4. let f ∈cn(r) and f (n) ∈l1(r), n ≥2 even, α ∈n, β > n+3 2α . then ∥k(x)∥1 ≤ 1 6γ 1 2α
γ β − 1 2α
(n −1)!  3γ n + 1 2α  γ  β −n + 1 2α  (74) +3γ n + 2 2α  γ  β −n + 2 2α  + γ n + 3 2α  γ  β −n + 3 2α  ω2(f (n), ξ)1ξn. hence as ξ →0 we obtain ∥k(x)∥1 →0. if additionally f (2m) ∈l1(r), m = 1, 2, . . . , n 2 then ∥mξ(f) −f∥1 →0, as ξ →0. proof. notice that ̃ ∆2 tf (n)(x) = ∆2 t f (n)(x −t), (75) all x, t ∈r. also it holds z ∞ −∞ |∆2 tf (n)(x −t)|dx = z ∞ −∞ |∆2 tf (n)(w)|dw ≤ω2(f (n), t)1, all t ∈r+. (76) here we obtain ∥k(x)∥1 = z ∞ −∞ |k(x)|dx (57) ≤w z ∞ −∞ z ∞ 0 z y 0    ̃ ∆2 tf (n)(x)    (y −t)n−1 (n −1)! dt  1 y2α + ξ2α
β dy ! dx ≤w z ∞ −∞ z ∞ 0  yn−1 (n −1)! z y 0    ̃ ∆2 tf (n)(x)    dt  1 y2α + ξ2α
β dy ! dx = w z ∞ 0 z ∞ −∞ z y 0    ̃ ∆2 tf (n)(x)    dt  dx  yn−1 (n −1)! 1 y2α + ξ2α
β ! dy (75) = w z ∞ 0 z ∞ −∞ z y 0   ∆2 tf (n)(x −t)    dt  dx  yn−1 (n −1)! 1 y2α + ξ2α
β ! dy = w z ∞ 0 z y 0 z ∞ −∞   ∆2 tf (n)(x −t)    dx  dt  yn−1 (n −1)! 1 y2α + ξ2α
β ! dy (76) ≤w z ∞ 0 z y 0 ω2(f (n), t)1dt  yn−1 (n −1)! 1 y2α + ξ2α
β ! dy ≤wω2(f (n), ξ)1 z ∞ 0 z y 0  1 + t ξ 2 dt ! yn−1 (n −1)! 1 y2α + ξ2α
β dy ! = wω2(f (n), ξ)1 z ∞ 0  1 + y ξ 3 −1 ! ξ 3 yn−1 (n −1)! 1 y2α + ξ2α
β dy ! = γ (β) αξn γ 1 2α
γ β − 1 2α
(n −1)!3ω2(f (n), ξ)1 z ∞ 0  (1 + y )3 −1  y n−1 1 (y 2α + 1)β dy ! george a. anastassiou, razvan a. mezei 16 = γ (β) αξn γ 1 2α
γ β − 1 2α
(n −1)!3ω2(f (n), ξ)1 z ∞ 0 3y + 3y 2 + y 3
y n−1 1 (y 2α + 1)β dy ! = γ (β) αξn γ 1 2α
γ β − 1 2α
(n −1)!3ω2(f (n), ξ)1 z ∞ 0 3y n + 3y n+1 + y n+2
1 (y 2α + 1)β dy ! = 1 6γ 1 2α
γ β − 1 2α
(n −1)!  3γ n + 1 2α  γ  β −n + 1 2α  +3γ n + 2 2α  γ  β −n + 2 2α  + γ n + 3 2α  γ  β −n + 3 2α  ω2(f (n), ξ)1ξn. (77) we have proved (74). ■ the related case here of n = 0 comes next. proposition 3. let p, q > 1 such that 1 p + 1 q = 1, α ∈n, β > 1 α  2 + 1 p  and the rest as above. then ∥mξ(f) −f∥p ≤ ρ1/pγ (β) α1/pγ  qβ 2 − 1 2α 1/q 2 1 q γ 1 2α
1/p γ β − 1 2α
γ  qβ 2 1/q ω2(f, ξ)p, (78) where 0 < ρ := z ∞ 0 (1 + y)2p 1 (y2α + 1)βp/2 dy < ∞. (79) hence as ξ →0 we obtain mξ →i in the lp norm, p > 1. proof. from (55) we get |mξ(f; x) −f(x)|p ≤w p z ∞ 0    ̃ ∆2 yf(x)    1 y2α + ξ2α
β dy !p . (80) we then estimate z ∞ −∞ |mξ(f; x) −f(x)|pdx ≤w p z ∞ −∞ z ∞ 0    ̃ ∆2 yf(x)    1 y2α + ξ2α
β dy !p dx = w p z ∞ −∞   z ∞ 0    ̃ ∆2 yf(x)    1 y2α + ξ2α
β/2 1 y2α + ξ2α
β/2 dy   p dx ≤w p z ∞ −∞      z ∞ 0      ̃ ∆2 yf(x)    1 y2α + ξ2α
β/2   p dy   1 p   z ∞ 0   1 y2α + ξ2α
β/2   q dy   1 q    p dx = w p z ∞ −∞      z ∞ 0    ̃ ∆2 yf(x)    p 1 y2α + ξ2α
βp/2 dy     z ∞ 0 1 y2α + ξ2α
βq/2 dy   p q   dx = w p   z ∞ 0   z ∞ −∞    ̃ ∆2 yf(x)    p 1 y2α + ξ2α
βp/2 dx  dy     γ 1 2α
γ  qβ 2 − 1 2α  2γ  qβ 2  αξqαβ−1   p q george a. anastassiou, razvan a. mezei 17 = [γ (β)]p αξαβp−1γ  qβ 2 − 1 2α p/q 2 p q γ 1 2α
γ β − 1 2α
p γ  qβ 2 p/q *   z ∞ 0 z ∞ −∞  ∆2 yf(x −y)  p dx  1 y2α + ξ2α
βp/2 dy   ≤ [γ (β)]p αξαβp−1γ  qβ 2 − 1 2α p/q 2 p q γ 1 2α
γ β − 1 2α
p γ  qβ 2 p/q z ∞ 0 ω2(f, y)p p 1 y2α + ξ2α
βp/2 dy ≤ [γ (β)]p αξαβp−1γ  qβ 2 − 1 2α p/q 2 p q γ 1 2α
γ β − 1 2α
p γ  qβ 2 p/q ω2(f, ξ)p p z ∞ 0  1 + y ξ 2p 1 y2α + ξ2α
βp/2 dy = [γ (β)]p αξαβp−1γ  qβ 2 − 1 2α p/q 2 p q γ 1 2α
γ β − 1 2α
p γ  qβ 2 p/q ω2(f, ξ)p p z ∞ 0 (1 + y )2p 1 (y 2α + 1)βp/2 ξαβp ξdy = [γ (β)]p αγ  qβ 2 − 1 2α p/q 2 p q γ 1 2α
γ β − 1 2α
p γ  qβ 2 p/q ω2(f, ξ)p p z ∞ 0 (1 + y)2p 1 (y2α + 1)βp/2 dy. (81) the proof of (78) is now completed. ■ also we give proposition 4. for α ∈n, β > 3 2α, it holds, ∥mξf −f∥1 ≤ " 1 2 + γ 1 α
γ β −1 α
γ 1 2α
γ β − 1 2α
+ γ 3 2α
γ β − 3 2α
2γ 1 2α
γ β − 1 2α
# ω2(f, ξ)1. (82) hence as ξ →0 we get mξ →i in the l1 norm. proof. from (55) we have |mξ(f; x) −f(x)| ≤w z ∞ 0    ̃ ∆2 yf(x)    1 y2α + ξ2α
β dy ! . (83) hence we get z ∞ −∞ |mξ(f; x) −f(x)|dx ≤w z ∞ −∞ z ∞ 0 | ̃ ∆2 yf(x)| 1 y2α + ξ2α
β dy ! dx = w z ∞ 0 z ∞ −∞ | ̃ ∆2 yf(x)|dx  1 y2α + ξ2α
β dy = w z ∞ 0 z ∞ −∞ |∆2 yf(x −y)|dx  1 y2α + ξ2α
β dy = w z ∞ 0 z ∞ −∞ |∆2 yf(x)|dx  1 y2α + ξ2α
β dy ≤w z ∞ 0 ω2(f, y)1 1 y2α + ξ2α
β dy george a. anastassiou, razvan a. mezei 18 ≤wω2(f, ξ)1 z ∞ 0  1 + y ξ 2 1 y2α + ξ2α
β dy = wω2(f, ξ)1 z ∞ 0 (1 + x)2 1 (x2α + 1)β ξ2αβ ξdx = γ (β) α γ 1 2α
γ β − 1 2α
ω2(f, ξ)1 z ∞ 0 (1 + x)2 1 (x2α + 1)β dx = " 1 2 + γ 1 α
γ β −1 α
γ 1 2α
γ β − 1 2α
+ γ 3 2α
γ β − 3 2α
2γ 1 2α
γ β − 1 2α
# ω2(f, ξ)1. (84) we have established (82). ■ references [1] g.a. anastassiou, rate of convergence of non-positive linear convolution type operators. a sharp inequality, j. math. anal. and appl., 142 (1989), 441–451. [2] g.a. anastassiou, sharp inequalities for convolution type operators, journal of approximation theory, 58 (1989), 259–266. [3] g.a. anastassiou, moments in probability and approximation theory, pitman research notes in math., vol. 287, longman sci. & tech., harlow, u.k., 1993. [4] g.a. anastassiou, quantitative approximations, chapman & hall/crc, boca raton, new york, 2001. [5] g.a. anastassiou, basic convergence with rates of smooth picard singular integral operators, j. of computational analysis and applications,vol.8, no.4 (2006), 313-334. [6] g.a. anastassiou, lp convergence with rates of smooth picard singular operators, differential & difference equations and applications, hindawi publ. corp., new york, (2006), 31–45. [7] g.a. anastassiou and s. gal, convergence of generalized singular integrals to the unit, uni- variate case, math. inequalities & applications, 3, no. 4 (2000), 511–518. [8] g.a. anastassiou and s. gal, convergence of generalized singular integrals to the unit, mul- tivariate case, applied math. rev., vol. 1, world sci. publ. co., singapore, 2000, pp. 1–8. [9] g.a. anastassiou and r. mezei, lp convergence with rates of smooth gauss-weierstrass singular operators, nonlinear studies, accepted 2008. [10] g.a. anastassiou and r. a. mezei, global smoothness and uniform convergence of smooth poisson-cauchy type singular operators, submitted 2009. [11] g.a. anastassiou and r. a. mezei, a voronovskaya type theorem for poisson-cauchy type singular operators, submitted 2009. [12] r.a. devore and g.g. lorentz, constructive approximation, springer-verlag, vol. 303, berlin, new york, 1993. [13] s.g. gal, remark on the degree of approximation of continuous functions by singular integrals, math. nachr., 164 (1993), 197–199. george a. anastassiou, razvan a. mezei 19 [14] s.g. gal, degree of approximation of continuous functions by some singular integrals, rev. anal. num ́ er, th ́ eor. approx., (cluj), tome xxvii, no. 2 (1998), 251–261. [15] r.n. mohapatra and r.s. rodriguez, on the rate of convergence of singular integrals for h ̈ older continuous functions, math. nachr. 149 (1990), 117–124. [16] d. zwillinger, crc standard mathematical tables and formulae, 30th edition, chapman & hall/crc, boca raton, 1995.

0911.1707

a dynamic vulnerability map to assess the risk of road network traffic utilization

le havre agglomeration (codah) includes 16 establishments classified seveso with high threshold. in the literature, we construct vulnerability maps to help decision makers assess the risk. such approaches remain static and do take into account the population displacement in the estimation of the vulnerability. we propose a decision making tool based on a dynamic vulnerability map to evaluate the difficulty of evacuation in the different sectors of codah. we use a geographic information system (gis) to visualize the map which evolves with the road traffic state through a detection of communities in large graphs algorithm.

introduction the population of the seine estuary is exposed to several types of natural and industrial hazards. it is included in the drainage basin of the "lézarde" and is also exposed to significant technological risks. the modeling and assessment of the danger is useful when it intersects with the exposed stakes. the most important factor is people. recent events have shown that our agglomerations are vulnerable in front of emergency situations. the examination of impacted populations remains a difficult exercise. in this context, the major risk management direction team (dirm) of le havre agglomeration (codah) has developed a model of spatial and temporal population exposed allocation pret- resse; the scale is the building (bourcier and mallet 2006); thus by distinguishing their day and night occupation. the model was able to locate people during the day both in their workplace and their residence (the unemployed and retirees). although the model was able to locate the diurnal and nocturnal population, it remains static because it does not take into account the daily movement of people and the road network utilization. for a better evacuation of people in the case of a major risk, we need to know the state of road traffic to determine how to allocate the vehicles on the road network and model the movement of these vehicles. in fact, the panic effect of some people can lead to accidents and traffic jams, which may be too grievous with a danger that spreads quickly. the panic generally results from the lack of coordination and dialogue between individuals (provitolo, 2007). in the literature, several models were developed to calculate a score of the vulnerability related to the road network utilization. this score may depend on social, biophysical, demographical or other aspects. most of these models adopt a pessimistic approach to calculate this vulnerability: this case is met when a group of people in a hazardous area decide all to take the same route to evacuate this area, which unfortunately happens quite often in the real world evacuation situations. although it helps decision-makers to estimate the risk by a census vulnerability map, this approach remains static and does not take into account the evolution of the road network traffic. in this paper, we have to simplify the representation of the population displacement, which is a complex phenomena. we also propose a dynamic and pessimistic approach related to the access to the road network. to this end, we model the road network by a dynamic graph (the dynamics is due to the traffic evolution). a simple model based on traffic flow will also be proposed. then, we apply a self-organization algorithm to detect communities on the graph belonging to the collective intelligence algorithms. the algorithm allows us to define the different vulnerable neighborhoods of the agglomeration in the case of an evacuation due to a potential danger, while taking into account the evolution of the road network traffic. the result of this algorithm will be visualized into a gis on a dynamic vulnerability map which categorizes various sectors depending on the difficulty of access to the road network. the map will help decision- makers in a better estimation of risk in the communes of the codah. it will enrich pret-ress static model developed at the codah, taking into account the mobility of the population. 3 directive seveso is an european directive, it lays down to the states to identify potential dangerous site. it intends to prevent major accidents involving dangerous substances and limit their consequences for man and the environment, with a view to ensuring high levels of protection throughout the community. 2 vulnerability assessment approaches traditional methods of conception and evaluation of the population at risk do not sometimes treat the behavioral evacuee's response (e.g. initial response to an evacuation, travel speed, family interactions / group, and so on.); they describe prescriptive rules as the travel distance. these traditional methods are not very sensitive to human behavior for different emergency scenarios. the computerized models offer the potential to evaluate the evacuation of a neighborhood in emergency situations and overcome these limitations (castel 2006). recently, some interesting applications have been developed by including the population dynamics, the models of urban growth patterns and land use. for computer modelers, this integration provides the ability to have computing entities as agents that are linked to real geographical locations. for gis users, it provides the ability to model the emergence of phenomena by various interactions of agents in time and space by using a gis (najlis and north 2004). many researchers have emphasized the need to create models of based vector multi agent systems (mas), which may require topological data structure provided in gis (parker 2004). in this way, one can represent an abm in which may coexist n levels of organizations with several classes of agents (e.g. level 1: individuals or companies, level 2 and 3: agents, cities, communities. . .) (daude 2005). in (cutter et al 2000), the author presents a method to spatially estimate the vulnerability and treats the biophysical and social aspects (access to resources, people with evacuation special needs, people with reduced mobility ...). several layers are created into the gis (a layer by a danger), and all these layers are combined into one composed of intersecting polygons to build a generic vulnerability map. to complete, it was necessary to take into account the infrastructure and various possible emergency exits. so, a new map has been constructed and a new layer has been incorporated. this work has been applied to the george town canton in which we find various natural and industrial risks, and where there are different types of people. in the neighborhood evacuation cases on a micro scale, a number of studies based on micro simulation have been developed. in their paper (church and cova 2000), the authors presented a model to estimate the necessary time to evacuate a neighborhood according to the effective of the population, number of vehicles, roads capacity and number of vehicles per minute. the model is based on the optimization in order to find the critical area around a point at a potential danger in a pessimistic way. this model has been coupled with a gis (arcinfo) to visualize the results (identify evacuation plans) and construct an evacuation vulnerability map for the city (santa barbara). cova and church (1997) opened the way on the study based on geographic information systems to evacuate people. their study identified the communities that may face transport difficulties during an evacuation. research has modeled the population by lane occupation during an evacuation emergency using the city of santa barbara. an optimization based model (graph partitioning problem) was realized to find the neighborhood that causes the highest vulnerability around each node in the graph and a vulnerability map around nodes in the city was constructed. a constructive heuristic has been used to calculate the best cluster around each node. this heuristic was developed in c and the result was displayed on a map (with arcinfo). nevertheless, in this approach, we predefine the maximum number of nodes in a neighborhood, which may not always be realistic and does not take into account the traffic evolution during the calculation of critical neighborhoods. so, the vulnerable neighborhoods don't evolve according to traffic state. in our work, we try to build a dynamic vulnerability map evolving with the traffic dynamics, in which the nodes number of in a critical neighborhood, is not predetermined and can change depending on traffic state. 3 problematic in this paper, the vulnerability is related to the access to the road network. we are persuaded that an accident on the network may cause traffic jams and therefore serious problems. so, it is important to have a pessimistic approach which takes into account the worst behavior of evacuees during a disaster (e.g. all individuals evacuate by taking the same exit route). to address this vulnerability, we have to finely represent the population and the dynamic state of road traffic. in pret- resse model developed within the major risks management team of codah, we have ventilated the day / night population at the buildings. the model was able to locate people during the day both in their workplace and their residence (the unemployed and retirees). it has been estimated that people will be in their residence during the night. a displacement survey was also realized in codah agglomeration and will be included in our work. pret-ress will be enriched by our model to dynamically assess the vulnerability related to the road traffic evolution. 4 dynamic model 4.1 system architecture the system consists of two related modules. the first one is dedicated to simulate the flow and apply the communities detection algorithm on the graph. the second one allows visualizing the result into a gis. the architecture is illustrated in the following figure. figure 1: system architecture the simulation module contains three components: - the dynamic graph extracted from the road network layer and detailed in the following section. - the flow management component consists of vehicles flow simulator applied on the graph. - the communities' detection component, detailed in the section 5. its takes the extracted graph and the current flow as an input and returns the formed communities according to the current state of road traffic. the visualization module consists of the road network layer integrated into the gis. this module communicates with the simulation module: the graph is constructed from this module, which in turn get the simulation result and visualize it as a dynamic vulnerability map. 4.2 environment modeling the road network is integrated as a layer in the geographic information system (gis). from this layer, we extract the data by using the open source java gis toolkit geotools. this toolkit provides several methods to manipulate geospatial data and implements open geospatial consortium (ogc) specifications, so we can read and write esri shapefile format. once the necessary road network data extracted, we use the graphstream tool developed within litis laboratory of le havre to construct a graph. this tool is designed for modeling; processing and visualizing graphs. the data extracted from network layer contains the roads circulation direction, roads id, roads type, their lengths and geometry. the extracted multigraph g (t) = (v (t), e (t)) represents the road network where v (t) is the set of nodes and e (t) the set of arcs. we deal with a multigraph because there was sometimes more than one oriented arc in the same direction between two adjacent nodes due to multiple routes between two points in le havre road network. graphstream facilitates this task because it is adapted to model and visualize multigraphs. in the constructed multigraph: - the nodes represent roads intersections; - the arcs represent the roads taken by vehicles; - the weight on each arc represents the needed time to cross this arc, depending on the current load of the traffic; - dynamic aspect relates to the weights of the arcs, which can evolve in time, according to the evolution of the fluidity of circulation; we have also constructed a voronoi tessellation (thiessen polygon) around nodes and projected the population in buildings on these nodes. the population in buildings is extracted from pret-ress model. 4.3 traffic management for a better evacuation of people in a major risk situation, we need to know the state of the road network to determine how to allocate the vehicles on this network and to model the movement of these vehicles. different types of models can be adopted: - the microscopic model details the behavior of each individual vehicle by representing interactions (modeled by a car following model) with other vehicles and in general by using a spatialization. we can extend the model by adding a regulation model with priorities rules, traffic lights... microscopic models may be applied on crowds movement as in boids collective approaches (reynolds 1987). in our problem, a microscopic model is used on the scale of a sector or a neighborhood. it has the advantage to model vehicle behavior in an evacuation of a neighborhood, people panic, interactions between vehicles, accidents... -the macroscopic model is based on the analogy between vehicular traffic and the fluid flow within a canal. it allows us to visualize the flow on the roads rather than individual vehicles. it is used at many sectors or the entire city scale. the hybrid model allows coupling the two types of dynamics flow models within the same simulation. several works have already borrowed this direction (hennecke et al. 2000, bourrel and henn 2002, magne et al. 2000). however, this approach is relatively new and very few have adopted it to our knowledge (hman et al. 2006). in risk context, the use of a hybrid model is very important especially when dealing with large volume of data: changing the scale from micro to macro in a region where we haven't a crisis situation (everything is normal) allows to economize the computation and the change from macro to micro in a critical situation allows to zoom and detect the behaviors and interactions between entities in danger. in this paper, we used a simple model of macroscopic flow: - a set of flow of cars moving from one arc to another adjacent one. - the arcs are limited in capacity of vehicles. - the flow can be broken and two or more flows can gather on a node. - traffic jams may appear in certain places of the road network; those places will be more vulnerable than others. we have adopted a macroscopic model in which flows circulate normally (no accidents) because the goal now is to establish a dynamic pessimistic vulnerability map which is not always the case in the real world (90% of people takes an exit route and the rest takes another route for example). hence, it is important to have in the near future a micro approach with a change of scale (from micro to macro and vice versa during the simulation) to simulate scenarios of danger in real time (accidents, behavior of drivers, vehicles interactions ...), a study on which we are working actually . 4.4 complex system a complex system is characterized by numerous entities of the same or different nature that interact in a non-trivial way (non-linear, feedback loop ...); the global emergence of new properties not seen in these entities: properties or evolution cannot be predicted by simple calculations. codah can be seen as a complex system in which the environment may influence on evacuation by imposing some rules which may reduce the flow fluidity (existence or not of safe refuge and emergency exits, routes traffic direction, priorities, traffic lights...) and vice versa (a fire or an accident may cause damages and change the environment). this system is in perpetual evolution; it is far from equilibrium dynamics with an absence of any global control. some organizations may appear or disappear according to different interactions. entities as vehicles and pedestrians interact between them and with environment. 5 comunities detection algorithm our aim is to identify communities in graphs, i.e. dense areas strongly linked to each other and more weakly linked to the outside world. the concept of communities in a graph is difficult to explain formally. it can be seen as a set of nodes whose internal connections density is higher than the outside density without defining formal threshold (pons 2005). the goal is to find a partition of nodes in communities according to a certain predefined criteria without fixing the number of such communities or the number of nodes in a community. radicchi (radicchi et al., 2004) proposes two possible definitions to quantify a community definition: • community c in a strong sense: c i i d i d out c out c ∈ ∀ > ), ( ) ( . a node belongs to a strong community if it has more connections within the community than outside. • community c in a weak sense: ∑ ∑ ∈ ∈ ∈ ∀ > c i out c c i out c c i i d i d ), ( ) ( . a community is qualified as weak if the sum of all degrees inside is more important than the sum of degrees towards the rest of the graph. ) (i d out c is the exiting edges number from a node i belonging to the community c towards the nodes of the same community. out c d is the exiting edges number from a node i belonging to the community c towards the nodes of other communities. finding organizations is a new field of research (albert and barabasi, 2002; newman, 2004a). interesting works were developed in the literature on the detection of structure in large communities in graphs (clauset et al. 2004, newman 2004a; b, pons and latapy 2006). in our problem, we look for a self-organization in networks with an evolutionary algorithm close to the detection of communities in large graphs and belonging to collective intelligence algorithms. this algorithm is based on the spread of forces in graphs. figure 2: communities detection in a graph the algorithm principle is to color the graph using pheromones and it uses several colonies of ants, each of a distinct color. each colony will collaborate to colonize zones, whereas colonies compete to maintain their own colored zone (see figure 2). solutions will therefore emerge and be maintained by the ant behavior. the solutions will be the color of each vertex in the graph. indeed, colored pheromones are deposited by ants on edges. the color of a vertex is obtained from the color having the largest proportion of pheromones on all incident edges. we have an interaction between each two local adjacent nodes according to the attraction force that exists between them. this force depends in our case on n/c, where n is the number of vehicles on the arc between 2 nodes neighbors and c represents vehicles capacity of the arc. this report was chosen because, in each community, we will have a large number of vehicles which all decide to exit through the same route in the case of a potential danger; this responds well to one of the purposes listed in beginning to have a pessimistic approach in the calculation of vulnerability. the algorithm has the advantage of not allowing the breaking of a link between 2 adjacent nodes to maintain the structure of the road network. when the traffic evolves, the algorithm detects that and communities can change or disappear as a result of local forces that change between the nodes locally. at each simulation time step, the flow on the arcs change following traffic conditions and the attraction forces may change also. once communities are formed on the graph, the result will be transmitted to the road network layer into the gis to be visualized. 6 conclusion in this paper, we have proposed a decision making tool to assess the danger depending on the road network use by the vehicles. this tool enables decision makers to visualize, on a geographic information system, a dynamic vulnerability map linked to the difficulty of evacuating the various streets in the metropolitan area of le havre agglomeration. we simulated the road network traffic by using a simple model of vehicles flow. a communities detection algorithm in the large graphs was adopted. it enabled us to form communities in a graph thanks to local force propagation rules between adjacent nodes. the communities evolve according to the current state of road network traffic. the result of the evolution of communities is visualized by using a gis. the adopted approach allowed us to estimate the risk due to the use of the road network by vehicles and categorize le havre agglomeration areas by their vulnerability. we will complete our work by using a micro model of traffic with a possibility of change from micro to macro and vice versa when necessary, and this depending on the situation. 7 references [1] aaron clauset, m. e. j. newman and cristopher moore (2004). finding community structure in very large networks. phys. rev. e 70, 066111. [2] a. hennecke, m. treiber and d. helbing (2000). macroscopic simulation of open systems and micro-macro link. in m. schreckenberg d. helbing and h. j. herrmann, editor, traffic and granular flow '99 : social, traffic, and granular dynamics, pages 383–388. springer. [3] castel, c.j.e. (2006). developing a prototype agent-based pedestrian evacuation model to explore the evacuation of king's cross st pancras underground station, centre for advanced spatial analysis (university college london): working paper 108, london. [4] church, r.l. and cova, t.j. (2000). mapping evacuation risk on transportation networks using a spatial optimization model. transportation research part c: emerging technologies, 8(1-6): 321-336. [5] cova, t.j. and church, r.l. (1997) modelling community evacuation vulnerability using gis. international journal of geographical information science, 11(8): 763-784. [6] cutter, s.l., j.t. mitchell and m.s. scott (2000). revealing the vulnerability of people and places: a case study of georgetown county, south carolina. annals of the association of american geographers 90 (4): 713-737. [7] e. bourrel and v. henn (2002). mixing micro and macro representation of traffic flow: a first theoretical step. in proceeding of the 9th meeting of the euro working group on transportation. [8] e. daudé (2005). systèmes multi-agents pour la simulation en géographie : vers une géographie artificielle, chapter 13, pages 355–382. in y. guermont (dir.), modélisation en géographie : déterminismes et complexités, hermès, paris. [9] f. radicchi, c. castellano, f. cecconi, v. loreto and d. parisi. defining and identifying communities in networks. in proceedings of the national academy of sciences, volume 101, pages 2658–2663, 2004. [10] graphstream : un outil de modélisation et de visualisation de graphes dynamiques. distribué sous licence libre (gpl).http://graphstream.sourceforge.net. [11] j.c. bourcier and p. mallet (2006). allocation spatio-temporelle de la population exposée aux risques majeurs. contribution à l'expologie sur le bassin de risques majeurs de l'estuaire de la seine: modèle pret-resse. revue internationale de géomatique, 16(10) : 457-478. [12] l. magne, s. rabut and j. f. gabard (2000). toward an hybrid macro and micro traffic flow simulation model. in informs spring 2000 meeting. [13] m. e. j. newman (2004). detecting community structure in networks, eur. phys. j. b 38, 321–330. [14] m. e. j. newman (2004). fast algorithm for detecting community structure in networks. phys. rev. e 69, 066133. [15] m.s. el hman, h. abouaäissa, d. jolly and a. benasser (2006). simulation hybride de flux de trafic basée sur les systèmes multi-agents. in 6e conférence francophone de modélisation et simulation - mosim'06. [16] najlis, r. and m. j. north (2004). repast for gis. proceedings of agent 2004: social dynamics: interaction, reflexivity and emergence, university of chicago and argonne national laboratory, il, usa. [17] d. c. parker (2004). integration of geographic information systems and agent-based models of land use : challenges and prospects. in maguire, d., j. m. f., goodchild,and m., batty, (eds), gis, spatial analysis and modelling, redlands, ca: esri. press. [18] pascal pons (2005). détection de structures de communautés dans les grands réseaux d'interactions. septièmes rencontres francophones sur les aspects algorithmiques des télécommunications. giens, france. [19] pascal pons and matthieu latapy(2006). computing communities in large networks using random walks. journal of graph algorithms and applications. vol. 10, no. 2, pp. 191-218. [20] d. provitolo (2007). a proposition for a classification of the catastrophe systems based on complexity criteria. in european conference complex systems-epnacs'07, emergent properties in natural and artificial complex systems, dresden, pages 93–106. [21] reynolds c. (1987). flocks, herds, and schools: a distributed behavioral model. computer graphics. siggraph 87 conference, vol. 21(4), 25–34.

0911.1708

different goals in multiscale simulations and how to reach them

in this paper we sum up our works on multiscale programs, mainly simulations. we first start with describing what multiscaling is about, how it helps perceiving signal from a background noise in a ?ow of data for example, for a direct perception by a user or for a further use by another program. we then give three examples of multiscale techniques we used in the past, maintaining a summary, using an environmental marker introducing an history in the data and finally using a knowledge on the behavior of the different scales to really handle them at the same time.

introduction: what this paper is about, and what it's not although we delved into different applications and application domains, the computer science research goals of our team has remained centered on the same subject for years. it can be expressed in different ways that we feel are, if not exactly equivalent, at least closely connected. it can be defined as man- aging multiple scales in a simulation. it also consists in handling emergent structures in a simulation. it can often also be seen as dynamic heuristic clus- tering of dynamic data3. this paper is about this theme, about why we think 3 we will of course later on describe in more details what we mean by all this. 2 pierrick tranouez and antoine dutot it is of interest and what we've done so far in this direction. it is therefore akin to a state of the art kind of article, except more centered on what we did. we will allude to what others have done, but the focus of the article is presenting our techniques and what we're trying to do, like most articles do, and not present an objective description of the whole field, as the different applica- tions examples could make think : we're sticking to the same computer science principles overall. we're taking one step back from our works to contemplate them all, and not the three steps which would be necessary to encompass the whole domain, as it would take us beyond the scope of this book. 2 perception: filtering to make decisions i look at a fluid flow simulation but all i'm interested in is where does the turbulence happen, in a case where i couldn't know before the simulation [tranouez 2005a]. i use a multi-participant communication system in a crisis management piece of software and i would like to know what are the main in- terests of each communicant based on what they are saying [lesage 1999]. i use an individual-based model (ibm) of different fish species but i'm interested in the evolution of the populations, not the individual fish [prevost 2004]. i use a traffic simulation with thousands of cars and a detailed town but what i want to know is where the traffic jams are (coming soon). in all those examples, i use a piece of software which produces huge amounts of data but i'm interested in phenomena of a different scale than the raw basic components. what we aim at is helping the user of the program to reach what he is interested in, be this user a human (clarification of the representation) or another program (automatic decision making). although we're trying to stay general in this part, we focused on our past experience of what we actually managed to do, as described in "some techniques to make these observations in a time scale comparable to the observed", this is not gratuitous philosophy. 2.1 clarification of the representation this first step of our work intends to extract the patterns on the carpet from its threads [tranouez 1984]. furthermore, we want it to be done in "real (pro- gram) time", meaning not a posteriori once the program is ended by examining its traces [servat 1998], and sticking as close as possible to the under layer, the one pumping out dynamic basic data. we don't want the discovery of our structures to be of a greater time scale than a step of the program it works upon. how to detect these structures? for each problem the structure must be analyzed, to understand what makes it stand out for the observer. this im- plies knowing the observer purpose, so as to characterize the structure. the different goals in multiscale simulations 3 answers are problem specific, nevertheless rules seem to appear. in many situations, the structures are groups of more basic entities, which then leads to try to fathom what makes it a group, what is its inside, its outside, its frontier, and what makes them so. quite often in the situation we dealt with, the groups members share some common characteristics. the problem in that case belongs to a subgenre of clustering, where the data changes all the time and the clusters evolve with them, they are not computed from scratch at each change. the other structures we managed to isolate are groups of strongly com- municating entities in object-oriented programs like multiagent simulations. we then endeavored to manage these cliques. in both cases, the detected structures are emphasized in the graphical rep- resentation of the program. this clarification lets the user of the simulation understand what happens in its midst. because modeling, and therefore un- derstanding, is clarifying and simplifying in a chosen direction a multi-sided problem or phenomenon, our change of representation participates to the un- derstanding of the operator. it is therefore also a necessary part of automating the whole understanding, aiming for instance at computing an artificial deci- sion making. 2.2 automatic decision making just like the human user makes something of the emerging phenomena the course of the program made evident, other programs can use the detected organizations. for example in the crisis management communication program, the de- tected favorite subject of interest of each of the communicant will be used as a filter for future incoming communications, favoring the ones on connected subjects. other examples are developed below, but the point is once the struc- tures are detected and clearly identified, the program can use models it may have of them to compute its future trajectory. it must be emphasized that at this point the structures can themselves combine into groups and structures of yet another scale, recursively. we're touching there an important compo- nent of complex system [simon 1996]. we may hope the applications of this principle to be numerous, such as robotics, where perceiving structures in vast amounts of data relatively to a goal, and then being able to act upon these accordingly is a necessity. we're now going to develop these notions in examples coming from our past works. 4 pierrick tranouez and antoine dutot 3 some techniques to make these observations in a time scale comparable to the observed the examples of handling dynamic organization we chose are taken from two main applications, one of a simulation of a fluid flow, the other of the simula- tion of a huge cluster of computed processes, distributed over a dynamic net- work of computing resources, such as computers. the methods titled "main- taining a summary of a simulation" and "reification: behavioral methods" are theories from the hydromechanics simulation, while "traces of the past help understand the present" refers to the computing resources management simulation. we will first describe these two applications, so that an eventual misunderstanding of what they are does not hinder later the clarity of our real purpose, the analysis of multiscale handling methods. in a part of a more general estuarine ecosystem simulation, we developed a simulation of a fluid flow. this flow uses a particle model [leonard 1980], and is described in details in [tranouez 2005a] or [tranouez 2005b]. the basic idea is that each particle is a vorticity carrier, each interacting with all the others following biot-savart laws. as fluid flows tend to organize themselves in vortices, from all spatial scales from a tens of angstrom to the atlantic ocean, this is these vortices we tried to handle as the multiscale characteris- tic of our simulation. the two methods we used are described below. the other application, described in depth in [dutot 2005], is a step toward automatic distribution of computing over computing resources in difficult con- ditions, as: • the resources we want to use can each appear and disappear, increase or decrease in number. • the computing distributed is composed of different object-oriented enti- ties, each probably a thread or a process, like in a multiagent system for example (the system was originally imagined for the ecosystem simula- tion alluded to above, and the entities would have been fish, plants, fluid vortices etc., each acting, moving . . . ) furthermore, we want the distribution to follow two guidelines: • as much of the resources as possible must be used, • communications between the resources must be kept as low as possible, as it should be wished for example if the resources are computers and the communications therefore happen over a network, bandwidth limited if compared to the internal of a computer. this the ultimate goal of this application, but the step we're interested in today consists in a simulation of our communicating processes, and of a program which, at the same time the simulated entities act and communicate, different goals in multiscale simulations 5 fig. 1. studies of water passing obstacles and falling by leonardo da vinci, c. 1508- 9. in codex leicester. advises how they should be regrouped and to which computing resource they should be allocated, so as to satisfy the two guidelines above. 3.1 maintaining a summary of a simulation the first method we would like to describe here relates to the fluid flow sim- ulation. the hydrodynamic model we use is based on an important number of interacting particles. each of these influences all the others, which makes n2 interactions, where n is the number of particles used. this makes a great number of computations. luckily, the intensity of the influence is inversely pro- portional to the square of the distance separating two particles. we therefore use an approximation called fast multipoles method (fmm), which consists in covering the simulation space with grids, of a density proportional to the density of particles (see figure 2-a). the computation of the influence of its colleagues over a given particle is then done exactly for the ones close enough, and averaged on the grid for those further. all this is absolutely monoscale. 6 pierrick tranouez and antoine dutot as the particles are vorticity carriers, it means that the more numerous they are in a region of space, the more agitated the fluid they represent is. we would therefore be interested in the structures built of close, dense par- ticles, surrounded by sparser ones. a side effect of the grids of the fmm, is that they help us do just that. it is not that this clustering is much easier on the grids, it's above all that they are an order of magnitude less numerous, and organized in a tree, which makes the group detection much faster than if the algorithm was ran on the particles themselves. furthermore, the step by step management of the grids is not only cheap (it changes the constant of the complexity of the particles movement method but not the order) but also needed for the fmm. we therefore detect structures on • dynamic data (the particles characteristics) • with little computing added to the simulation, which is what we aimed at. the principle here is that through the grids we maintain a summary of the simulation, upon which we can then run static data algorithm, all this at a cheap computing price. 3.2 traces of the past help understand the present the second method relates to the detection of communication clusters inside a distributed application. the applications we are interested in are composed of a large number of object-oriented entities that execute in parallel, appear, evolve and, sometimes, disappear. aside some very regular applications, often entities tend to communicate more with some than with others. for example in a simulation of an aquatic ecosystem, entities representing a species of fish may stay together, interacting with one another, but flee predators. indeed organizations appear groups of entities form. such simulations are a good ex- ample of applications we intend to handle, where the number of entities is often too large to compute a result in an acceptable time on one unique com- puter. to distribute these applications it would be interesting to both have ap- proximately the same number of entities on each computing resource to bal- ance the load, but also to avoid as much as possible to use the network, that costs significantly more in terms of latency than the internals of a computer. our goal is therefore to balance the load and minimize network communica- tions. sadly, these criteria are conflicting, and we must find a tradeoff. our method consists in the use of an ant metaphor. applications we use are easily seen as a graph, which is a set of connected entities. we can map different goals in multiscale simulations 7 a - each color corresponds to a detected aggregate b - each color corresponds to a computing ressource fig. 2. detection of emergent structures in two applications with distinct methods 8 pierrick tranouez and antoine dutot entities to vertices of the graph, and communications between these entities to the edges of the graph. this graph will follow the evolution of the sim- ulation. when an entity appear, a vertex will appear in the graph, when a communication will be established between two entities, an edge will appear between the two corresponding vertices. we will use such a graph to repre- sent the application, and will try to find clusters of highly communicating entities in this graph by coloring it, assigning a color to each cluster. this will allow to identify clusters as a whole and use this information to assign not entities, but at another scale, clusters to computing resources (figure 2-b). for this, we use numerical ants that crawl the graph as well as their pheromones, olfactory messages they drop, to mark clusters of entities. we use several distinct colonies of ants, each of a distinct color, that drop colored pheromones. each color corresponds to one of the computing resources at our disposal. ants drop colored pheromones on edges of the graph when they cross them. we mark a vertex as being of the color of the dominant pheromone on each of its incident edges. the color indicates the computing resource where the entity should run. to ensure our ants color groups of highly communicating entities of the same color to minimize communications, we use the collaboration between ants: ants are attracted by pheromones of their own color, and attracted by highly communicating edges. to ensure the load is balanced, that is to ensure that the whole graph is not colored only in one color if ten colors are avail- able, we use competition, ants are repulsed by the pheromones of other colors. pheromones in nature being olfactory molecules, they tend to evaporate. ants must maintain them so they do not disappear. consequently, only the interesting areas, zones where ants are attracted, are covered by pheromones and maintained. when a zone becomes less interesting, ants leave it and pheromone disappear. when an area becomes of a great interest, ants col- onize it by laying down pheromones that attract more ants, and the process self-amplifies. we respect the metaphor here since it brings us the very interesting prop- erty of handling the dynamics. indeed, our application continuously changes, the graph that represents it follows this, and we want our method to be able to discover new highly communicating clusters, while abandoning vertices that are no more part of a cluster. as ants continuously crawl through the graph, they maintain the pheromone color on the highly communicating clusters. if entities and communications of the simulation appear or disappear, ants can quickly adapt to the changes. colored pheromones on parts where a cluster disappeared evaporate and ants colonize new clusters in a dynamic way. in- deed, the application never changes completely all the time; it modifies itself smoothly. ants lay down "traces" of pheromones and do not recompute the different goals in multiscale simulations 9 color of each vertex at each time, they reuse the already dropped pheromone therefore continuously giving a distribution advice at a small computing price, and adapting to the reconfigurations of the underlying application. 3.3 reification: behavioral methods this last example of our multiscale handling methods was also developed on the fluid flow simulation (figure 3). once more, we want to detect structures in a dynamic flow of data, without getting rid of the dynamicity by doing a full computation on each step of the simulation. the idea here is doing the full computation only once in a good while, and only relatively to the unknown parts of our simulation. fig. 3. fluid flow around an obstacle. on the left, the initial state. on the right, a part of the flow, some steps later (the ellipses are vortices) we begin with detecting vortices on the basic particles once. vortices will be a rather elliptic set of close particles of the same rotation sense. we then introduce a multiagent system of the vortices (figure 3-right). we have in- deed a general knowledge of the way vortices behave. we know they move like a big particle in our biot-savard model, and we model its structural stabil- ity through social interactions with the surrounding basic particles, the other vortices and the obstacles, through which they can grow, shrink or die (be dis- sipated into particles). the details on this can be found in [tranouez 2005a]. later on, we occasionally make a full-blown vortex detection, but only on the remaining basic particles, as the already detected vortexes are managed by the multiagent system. in this case, we possess knowledge on the structures we want to detect, and we use it to build actually the upper scale level of the simulation, which 10 pierrick tranouez and antoine dutot at the same time lightens ulterior structures detection. we are definitely in the category described in automatic decision making. 4 conclusion our research group works on complex systems and focuses on the computer representation of their hierarchical/holarchical characteristics [koestler 1978], [simon 1996], [kay 2000]. we tried to illustrate that describing a problem at different scales is a well-spread practice at least in the modeling and simulat- ing community. we then presented some methods for handling the different scales, with maintaining a summary, using an environmental marker introduc- ing a history in the data and finally using knowledge on the behavior of the different scales to handle them at the same time. we now believe we start to have sound multiscale methods, and must focus on the realism of the applications, to compare the sacrifice in details we make when we model the upper levels rather than just heavily computing the lower ones. we save time and lose precision, but what is this trade-offworth precisely? references [dutot 2005] dutot, a. (2005) distribution dynamique adaptative ` a l'aide de m ́ ecanismes d'intelligence collective. phd thesis, le havre university. [kay 2000] kay, j. (2000) ecosystems as self-organising holarchic open systems : narratives and the second law of thermodynamics. jorgensen, s.e.; and f. m ̈ uller (eds.), handbook of ecosystems theories and management, lewis publishers. [koestler 1978] koestler, a. (1978) janus. a summing up. vintage books, new york. [leonard 1980] leonard, a. (1980) vortex methods for flow simulation, journal of computational physics, vol. 37, 289-335. [lesage 1999] lesage, f.; cardon, a.; and p. tranouez (1999) a multiagent based prediction of the evolution of knowledge with multiple points of view, kaw'99. [prevost 2004] prevost, g.; tranouez, p.; lerebourg, s.; bertelle, c. and d. olivier (2004) methodology for holarchic ecosystem model based on ontological tool. esm 2004, 164-171. [servat 1998] servat, d.; perrier, e.; treuil, j.-p.; and a. drogoul (1998) when agents emerge from agents: introducing multi-scale viewpoints in multi-agent simulations. mabs 98, 183-198. [simon 1996] simon, h. (1996) the sciences of the artificial (3rd edition). mit press. [tranouez 1984] tranouez, pierre (1984) fascination et narration dans l'oeuvre ro- manesque de barbey d'aurevilly. doctorat d' ́ etat. different goals in multiscale simulations 11 [tranouez 2005a] tranouez, p.; bertelle, c; and d. olivier (2006) changing levels of description in a fluid flow simulation in m.a. aziz-alaoui and c. bertelle (eds), "emergent properties in natural and artificial dynamical systems", un- derstanding complex systems series, 87-99. [tranouez 2005b] tranouez, p. (2005) contribution ` a la mod ́ elisation et ` a la prise en compte informatique de niveaux de descriptions multiples. application aux ́ ecosyst` emes aquatiques (penicillo haere, nam scalas aufero), phd thesis, le havre university.

0911.1709

on a conjecture of v. v. shchigolev

v. v. shchigolev has proven that over any infinite field k of characteristic p>2, the t-space generated by g={x_1^p,x_1^px_2^p,...} is finitely based, which answered a question raised by a. v. grishin. shchigolev went on to conjecture that every infinite subset of g generated a finitely based t-space. in this paper, we prove that shchigolev's conjecture was correct by showing that for any field of characteristic p>2, the t-space generated by any subset {x_1^px_2^p...x_{i_1}^p, x_1^px_2^p...x_{i_2}^p,...}, i_1<i_2<i_3<..., of g has a t-space basis of size at most i_2-i_1+1.

introduction in [2] (and later in [3], the survey paper with v. v. shchigolev), a. v. grishin proved that in the free associative algebra with countably infinite generating set { x1, x2, . . . } over an infinite field of characteristic 2, the t -space generated by the set { x2 1, x2 1x2 2, . . . } is not finitely based, and he raised the question as to whether or not over a field of characteristic p > 2, the t -space generated by { xp 1, xp 1xp 2, . . . } is finitely based. this was resolved by v. v. shchigolev in [4], wherein he proved that over an infinite field of characteristic p > 2, this t - space is finitely based. shchigolev then raised the question in [4] as to whether every infinite subset of { xp 1, xp 1xp 2, . . . } generates a finitely based t -space. in this paper, we prove that over an arbitrary field of characteristic p > 2, every subset of { xp 1, xp 1xp 2, . . . } generates a t -space that can be generated as a t -space by finitely many elements, and we give an upper bound for the size of a minimal generating set. let p be a prime (not necessarily greater than 2) and let k denote an arbitrary field of characteristic p. let x = { x1, x2, . . . } be a countably infinite set, and let k0⟨x⟩denote the free associative k-algebra over the set x. definition 1.1. for any positive integer d, let s(d) = s(d)(x1, x2, . . . , xd) = x σ∈σd d y i=1 xσ(i), where σd is the symmetric group on d letters. then define s(d) 1 = { s(d) }s, the t -space generated by { s(d) }, and for all n ≥1, s(d) n+1 = (s(d) n s(d) 1 )s. 1 let i : i1 < i2 < * * * be a sequence of positive integers (finite or infinite), and then for each n ≥1, let r(d) n,i = pn j=1 s(d) ij . when the sequence i is understood, we shall usually write r(d) n instead of r(d) n,i. finally, let r(d) ∞,i (even if the sequence is finite) denote the t -space generated by { s(d) i | i ∈i }. we shall prove that r(d) ∞,i has a t -space basis of size at most i2 −i1 + 1. definition 1.2. let h1 = { xp 1 }s, and for each n ≥1, let hn+1 = (hnh1)s. then for any positive integer n, let ln,i = pn j=1 hij, and let l∞,i denote the t -space generated by { hi | i ∈i }. we prove that l∞,i is finitely generated as a t -space, with a t -space basis of size at most i2 −i1 + 1. in particular, this proves that shchigolev's conjecture is valid. 2 preliminaries in this section, k denotes an arbitrary field of characteristic an arbitrary prime p, and vi, i ≥1, denotes a sequence of t -spaces of k0⟨x⟩satisfying the following two properties: (i) (vivj)s = vi+j; (ii) for all m ≥1, v2m+1 ⊆vm+1 + v1. lemma 2.1. for any integers r and s with 0 < r < s, vs+t(s−r) ⊆vr + vs for all t ≥0. proof. the proof is by induction on t. there is nothing to show for t = 0. for t = 1, let m = s−r in (ii) to obtain that v2s−2r+1 ⊆vs−r+1 +v1, then multiply by vr−1 to obtain vr−1v2s−2r+1 ⊆vr−1vs−r+1 + vr−1v1 ⊆(vr−1vs−r+1)s + (vr−1v1)s = vs+vr. but then v2s−r = (vr−1v2s−2r+1)s ⊆vs+vr, as required. suppose now that t ≥1 is such that the result holds. then vs+(t+1)(s−r) = (vs+t(s−r)vs−r)s ⊆((vs + vr)vs−r)s = v2s−r + vs ⊆vr + vs + vs = vr + vs. the result follows now by induction. for any increasing sequence i : i1 < i2 < * * * of positive integers, we shall refer to i2 −i1 as the initial gap of i. proposition 2.1. for any increasing sequence i = { ij }j≥1 of positive integers, there exists a set j of size at most i2 −i1 + 1 with entries positive integers such that the following hold: (i) 1, 2 ∈j; (ii) p∞ j=1 vij = p j∈j vij. proof. the proof of the proposition shall be by induction on the initial gap. by lemma 2.1, for a sequence with initial gap 1, we may take j = { i1, i2 } . suppose now that l > 1 is an integer for which the result holds for all increasing 2 sequences with initial gap less than l, and let i1 < i2 < * * * be a sequence with initial gap i2 −i1 = l. if for all j ≥3, vij ⊆vi1 + vi2, then j = { 1, 2 } meets the requirements, so we may suppose that there exists j ≥3 such that vij is not contained in vi1 + vi2. by lemma 2.1, this means that there exists j ≥3 such that ij / ∈{ i2 + ql | q ≥0 }. let r be least such that ir / ∈{ i2 + ql | q ≥0 }, so that there exists t such that i2 + tl < ir < i2 + (t + 1)l. form a sequence i′ from i by first removing all entries of i up to (but not including) ir, then prepend the integer i2 + tl. thus i′ 1, the first entry of i′, is i2 + tl, while for all j ≥2, i′ j = ir+j−2. note that i′ 2 −i′ 1 = ir −(i2 + tl) ≤l −1. by hypothesis, there exists a subset j′ of size at most i′ 2 −i′ 1 + 1 ≤l = i2 −i1 that contains 1 and 2 and is such that p∞ j=1 vi′ j = p j∈j′ vi′ j. set j = { 1, 2 } ∪{ r + j −2 | j ∈j′, j ≥2 }. then |j| = |j′| + 1 ≤i2 −i1 + 1 and vi2+tl + ∞ x j=r vij = ∞ x j=1 vi′ j = x j∈j′ vi′ j = vi2+tl + x j∈j′ j≥2 vi′ j = vi2+tl + x j∈j j≥3 vij and by lemma 2.1, vi2+tl ⊆vi2 + vi2, so vi1 + vi2 + ∞ x j=r vij = vi1 + vi2 + vi2+tl + ∞ x j=r vij = vi1 + vi2 + vi2+tl + x j∈j j≥3 vij = vi1 + vi2 + x j∈j j≥3 vij. finally, the choice of r implies that x j∈j vij = vi1 + vi2 + x j∈j j≥3 vij = vi1 + vi2 + ∞ x j=r vij = ∞ x j=1 vij. this completes the proof of the inductive step. we remark that in proposition 2.1, it is possible to improve the bound from i2 −i1 + 1 to 2(log2(2(i2 −i1)). in the sections to come, we shall examine some important situations of the kind described above. 3 the r(d) n sequence we shall have need of certain results that first appeared in [1]. for completeness, we include them with proofs where necessary. in this section, p denotes an arbitrary prime, k an arbitrary field of characteristic p, and d an arbitrary positive integer. the proof of the first result is immediate. 3 lemma 3.1. let d be a positive integer. then s(d+1)(x1, x2, . . . , xd+1) = d+1 x i=1 s(d)(x1, x2, . . . , ˆ xi, . . . , xd+1)xi (1) = s(d)(x1, x2, . . . , xd)xd+1 + d x i=1 s(d)(x1, x2, . . . , xd+1xi, . . . , xd) (2) = xd+1s(d)(x1, x2, . . . , xd) + d x i=1 s(d)(x1, x2, . . . , xixd+1, . . . , xd). (3) corollary 3.1. let d be any positive integer. then modulo s(d) 1 , s(d+1)(x1, x2, . . . , xd+1) ≡s(d)(x1, . . . , xd)xd+1 ≡xd+1s(d)(x1, . . . , xd). proof. this is immediate from (2) and (3) of lemma 3.1. we remark that corollary 3.1 implies that for every u ∈s(d) 1 and v ∈k0⟨x⟩, [ u, v ] ∈s(d) 1 . while we shall not have need of this fact, we note that in [4], shchigolev proves that if the field is infinite, then for any t -space v , if v ∈v , then [ v, u ] ∈v for any u ∈k0⟨x⟩. the next proposition is a strengthened version of proposition 2.1 of [1]. proposition 3.1. for any u, v ∈k0⟨x⟩, (i) (s(d) 1 uv)s ⊆s(d) 1 + (s(d) 1 u)s + (s(d) 1 v)s; and (ii) (uvs(d) 1 )s ⊆s(d) 1 + (us(d) 1 )s + (vs(d) 1 )s. proof. we shall prove the first statement; the proof of the second is similar and will be omitted. by (1) of lemma 3.1, d x i=1 s(d)(x1, . . . , ˆ xi, . . . , xd+1)xi = s(d+1)(x1, . . . , xd+1) −s(d)(x1, . . . , xd)xd+1 and by (2) of lemma 3.1, s(d+1)(x1, . . . , xd+1) −s(d)(x1, . . . , xd)xd+1 ∈s(d) 1 . let v ∈k0⟨x⟩. then s(d)(x2, . . . , xd+1)x1v+ d x i=2 s(d)(x1, . . . , ˆ xi, . . . , xd+1)xiv = d x i=1 s(d)(x1, x2, . . . , ˆ xi, . . . , xd+1)xiv ∈(s(d) 1 v)s . now for each i = 2, . . . , d, we use two applications of corollary 3.1 to obtain s(d)(x1, . . . , ˆ xi, . . . , xd+1)xiv ≡s(d+1)(x1, . . . , ˆ xi, . . . , xd+1, xiv) ≡s(d)(x2, . . . , ˆ xi, . . . , xd+1, xiv)x1 mod s(d) 1 . 4 thus s(d)(x2, . . . , xd+1)x1v + ( d x i=2 s(d)(x2, . . . , ˆ xi, . . . , xiv)
x1 ∈(s(d) 1 v)s + s(d) 1 . thus for u ∈k0⟨x⟩, we obtain s(d)(x2, . . . , xd+1)uv ∈(s(d) 1 u)s+(s(d) 1 v)s+s(d) 1 , and so (s(d) 1 uv)s ⊆(s(d) 1 u)s + (s(d) 1 v)s + s(d) 1 , as required. corollary 3.2. let d be any positive integer. then the sequence s(d) n , n ≥1, satisfies (i) for all m, n ≥1, (s(d) m s(d) n )s = s(d) m+n; (ii) for all m ≥1, s(d) 2m+1 ⊆s(d) m+1 + s(d) 1 . proof. the first statement follows immediately from definition 1.1 by an ele- mentary induction argument. for the second statement, let m ≥1. then by proposition 3.1, for any u, v ∈s(d) m , (s(d) 1 uv)s ⊆s(d) 1 + (s(d) 1 u)s + (s(d) 1 v)s, which implies that (s(d) 1 s(d) m s(d) m )s ⊆s(d) 1 + (s(d) 1 s(d) m )s. by (i), this yields s(d) 2m+1 ⊆s(d) 1 + s(d) m+1, as required. theorem 3.1. let i denote any increasing sequence of positive integers with initial gap g. then r(d) ∞,i is finitely based, with a t -space basis of size at most g + 1. proof. denote the entries of i in increasing order by ij, j ≥1. by corollary 3.2 and proposition 2.1, there exists a set j of positive integers with |j| ≤i2−i1+1 and r(d) ∞,i = r(d) n,i = p j∈j s(d) ij . since for each i, the t -space s(d) i has a basis consisting of a single element, the result follows. 4 the ln sequence we shall make use of the following well known result. an element u ∈k0⟨x⟩is said to be essential if u is a linear combination of monomials with the property that each variable that appears in any monomial appears in every monomial. lemma 4.1. let v be a t -space and let f ∈v . if f = p fi denotes the decomposition of f into its essential components, then fi ∈v for every i. proof. we induct on the number of essential components, with obvious base case. suppose that n > 1 is an integer such that if f ∈v has fewer than n essential components, then each belongs to v , and let f ∈v have n essential components. since n > 1, there is a variable x that appears in some but not all essential components of f. let zx and fx denote the sum of the essential com- ponents of f in which x appears, respectively, does not appear. then evaluate 5 at x = 0 to obtain that fx = f|x=0∈v , and thus zx = f −fx ∈v as well. by hypothesis, each essential component of fx and of zx belongs to v , and thus every essential component of f belongs to v , as required. corollary 4.1. s(p) 1 ⊆h1. proof. s(p) is one of the essential components of (x1 +x2 +* * *+xp)p, and since (x1 + x2 + * * * + xp)p ∈h1, it follows from lemma 4.1 that s(p) ∈h1. thus s(p) 1 ⊆h1. corollary 4.2. for every m ≥1, s(p) m ⊆hm. proof. the proof is an elementary induction, with corollary 4.1 providing the base case. corollary 4.3. for any u ∈h1 and any v ∈k0⟨x⟩, [ u, v ] ∈h1. proof. it suffices to observe that [ xp, v ] = p x i=0 xi[ x, v ]xp−i = 1 (p −1)!s(p)(x, x, . . . , x, [ x, v ]), which belongs to h1 by virtue of corollary 4.1. we remark again that in [3], shchigolev proves that if k is infinite, then every t -space in k0⟨x⟩is closed under commutator in the sense of corollary 4.3. since we have not required that k be infinite, we have provided this closure result (see also lemma 4.4 below). lemma 4.2. for any m, n ≥1, (hmhn)s = hm+n. proof. the proof is by an elementary induction on n, with definition 1.2 pro- viding the base case. lemma 4.3. for any m ≥1, (s(p) 1 h2m)s ⊆h1 + hm+1 and (h2ms(p) 1 )s ⊆ h1 + hm+1. proof. by proposition 3.1 (i), for any u, v ∈hm, we have s(p) 1 uv ⊆s(p) 1 + (s(p) 1 u)s + (s(p) 1 v)s. by corollary 4.2, this gives s(p) 1 hmhm ⊆h1 + (h1hm)s, and then from lemma 4.2, we obtain s(p) 1 h2m ⊆h1 + hm+1. the proof of the second part is similar. lemma 4.4. let m ≥1. for every u ∈hm and v ∈k0⟨x⟩, [ u, v ] ∈hm. proof. the proof is by induction on m, with corollary 4.3 providing the base case. suppose that m ≥1 is such that the result holds. it suffices to prove that for any v ∈k0⟨x⟩, [ xp 1xp 2 * * * xp mxp m+1, v ] ∈hm+1. we have [ xp 1xp 2 * * * xp mxp m+1, v ] = [ xp 1xp 2 * * * xp m, v ]xp m+1 + xp 1xp 2 * * * xp m[ xp m+1, v ]. 6 by hypothesis, [ xp 1xp 2 * * * xp m, v ] ∈hm, while xp m+1 ∈h1 and thus by corollary 4.3, [ xp m+1, v ] ∈h1 as well. now by definition, [ xp 1xp 2 * * * xp m, v ]xp m+1 ∈hm+1 and xp 1xp 2 * * * xp m[ xp m+1, v ] ∈hm+1, which completes the proof of the inductive step. lemma 4.5. let m ≥1. then his(p)h2m−i ⊆h1 + hm+1 for all i with 1 ≤i ≤2m −1. proof. let m ≥1. we consider two cases: 2m −i ≥m and 2m −i < m. suppose that 2m −i ≥m, and let u ∈hi, w ∈hm−1 and z ∈hm−i+1. then us(p)wz = ([ u, s(p)w ] + s(p)wu)z = [ u, s(p)w ]z + s(p)wuz. since u ∈hi, it follows from lemma 4.4 that [ u, s(p)w ] ∈hi. but then by lemma 4.2, [ u, s(p)w ]z ∈hi+m−i+1 = hm+1. as well, by corollary 4.1 and lemma 4.2, s(p)wuz ∈s(p) 1 hm−1+i+m−i+1 = s(p) 1 h2m, and by lemma 4.3, s(p) 1 h2m ⊆h1+ hm+1. thus us(p)wz ∈h1 + hm+1. this proves that his(p)hm−1hm−i+1 ⊆ h1 + hm+1, and so by lemma 4.2, his(p)h2m−i = his(p)(hm−1hm−i+1)s ⊆ h1 + hm+1. the argument for the case when 2m −i < m is similar and is therefore omitted. proposition 4.1. let p > 2. then for every m ≥1, h2m+1 ⊆h1 + hm+1. proof. first, consider the expansion of (x + y)p for any x, y ∈k0⟨x⟩. it will be convenient to introduce the following notation. let jp = { 1, 2, . . ., p }. for any j ⊆jp, let pj = qp i=1 zi, where for each i, zi = x if i ∈j, otherwise zi = y. as well, for each i with 1 ≤i ≤p −1, we shall let s(p)(x, y; i) = s(p)(x, x, . . . , x | {z } i , y, y, . . ., y | {z } p−i ). observe that s(p)(x, y; i) = i!(p −i)! p j⊆jp |j|=i pj. we have (x + y)p = p x i=0 x j⊆jp |j|=i pj = yp + xp + p−1 x i=1 1 i!(p −i)!s(p)(x, y; i). let u = pp−1 i=1 1 i!(p−i)!s(p)(x, y; i), so that (x + y)p = xp + yp + u, and note that u ∈s(p) 1 . then (x + y)2p = y2p + x2p + 2xpyp + [ yp, xp ] + u2 + (xp + yp)u + u(xp + yp). since (x+y)2p, x2p, y2p, and, by lemma 4.4, [ yp, xp ] all belong to h1, it follows (making use of corollary 4.2 where necessary) that 2xpyp ∈h1 + h1s(p) 1 + s(p) 1 h1. consequently, for any m ≥1, xp 1 m y i=1 (2xp 2ixp 2i+1) ∈h1(h1 + h1s(p) 1 + s(p) 1 h1)m. by corollary 4.1, lemma 4.2, and lemma 4.5, h1(h1 + h1s(p) 1 + s(p) 1 h1)m ⊆ h1 + hm+1, and since p > 2, it follows that q2m+1 i=1 xp i ∈h1 + hm+1. thus h2m+1 ⊆h1 + hm+1, as required. 7 theorem 4.1 (shchigolev's conjecture). let p > 2 be a prime and k a field of characteristic p. for any increasing sequence i = { ij }j≥1, l∞,i is a finitely based t -space of k0⟨x⟩, with a t -space basis of size at most i2 −i1 + 1. proof. by lemma 4.2 and proposition 4.1, the sequence hn of t -spaces of k0⟨x⟩meets the requirements of section 2. thus by proposition 2.1, for any increasing sequence i = { ij }j≥1 of positive integers, there exists a set j of positive integers such that |j| ≤i2 −i1 + 1 and l∞,i = p∞ j=1 hij = p j∈j hij. since for each i, hi has t -space basis { xp 1xp 2 * * * xp i }, it follows that l∞,i has a t -space basis of size |j| ≤i2 −i1 + 1. shchigolev's original result was that for the sequence i+ of all positive in- tegers, l∞,i+ is a finitely-based t -space, with a t -space basis of size at most p. it was then shown in [1], a precursor to this work, that l∞,i+ has in fact a t -space basis of size at most 2 (the bound of theorem 4.1, since i1 = 1 and i2 = 2). it is also interesting to note that the results in this paper apply to finite sequences. of course, if i is a finite increasing sequence of positive integers, then l∞,i has a finite t -space basis, but by the preceding work, we know that it has a t -space basis of size at most i2 −i1 + 1. references [1] c. bekh-ochir and s. a. rankin, on a problem of a. v. grishin, preprint, arxiv:0909.2266. [2] a. v. grishin, t-spaces with an infinite basis over a field of characteristic 2, international conference in algebra and analysis commemorating the hun- dredth anniversary of n. g. chebotarev, proceedings, kazan 5–11, june, 1994, p. 29 (russian). [3] a. v. grishin and v. v. shchigolev, t-spaces and their applications, journal of mathematical sciences, vol 134, no. 1, 2006, 1799–1878 (translated from sovremennaya matematika i-ee prilozheniya, vol. 18, alg` ebra, 2004). [4] v. v. shchigolev, examples of t -spaces with an infinite basis, sbornik math- ematics, vol 191, no. 3, 2000, 459–476. 8

0911.1710

moduli spaces of $j$-holomorphic curves with general jet constraints

in this paper, we prove that the tagent map of the holomorphic $k$- jet evaluation $j^k_{hol}$ from the mapping space to holomorphic $k$-jet bundle, when restricted on the universal moduli space of simple j-holomorphic curves with one marked point, is surjective. from this we derive that for generic $j$, the moduli space of simple $j$-holomorphic curves in class $\beta\in h_2(m)$ with general jet constraints at marked points is a smooth manifold of expected dimension.

introduction let (m, ω) be a symplectic manifold of dimension 2n. denote by jω the set of almost complex structures j on m compatible with ω. let σ be a compact oriented surface without boundary, and (j, u) a pair of complex structure j on σ and a map u : σ →m. we say (j, u) is a j-holomorphic curve if ∂j,ju := 1 2 (du + j ◦du ◦j) = 0. we let m1 (σ, m, j; β) be the standard moduli space of j-holomorphic curves in class β ∈h2 (m, z) with one marked point, and m∗ 1 (σ, m, j; β) be the set of simple (i.e. somewhere injective) j-holomorphic curves in m1 (σ, m; β). since the birth of the theory of j-holomorphic curves, moduli spaces of j- holomorphic curves with constraints at marked points have lead to finer symplectic invarints like gromov-witten invariants and quantumn cohomology. j-holomorphic curves with embedding property also plays important role in low dimesional sym- plectic geometry, like the works of [ht] and [wen]. these constraints all can be viewed as partial differential relations in the 0-jet and 1-jet bundles. in relative gromov-witten theory, contact order of j-holomorphic curves with given symplec- tic hypersurfaces (divisors) was used to define the relevant moduli spaces. in the work of cieliebak-mohke [cm] and oh [oh], the authors studied the moduli space of j-holomorphic curves with prescribed vanishing orders of derivatives at marked points. all these are vanishing conditions in k-jets bundles. it is then natural to ask what properties we can expect for moduli spaces of j-holomorphic curves with general constraints in jet bundles (while all constraints in previous examples are zero sections in various jet bundles). the main purpose of this paper is to confirm that for a wide class of closed partial differential relations in holomorphic jet bundles (definition 1, orginially de- fined in [oh]), the moduli spaces of j-holomorphic curves from σ to m with given constraints at marked points behave well for generic j (theorem 3). namely, they are smooth manifolds of dimension predicted by index theorem, and all elements in date: november 9, 2009. key words and phrases. j-holomorphic curve, jet evaluation map, transversality. 1 2 ke zhu the moduli spaces are fredholm regular. during the proof it appears that holomor- phic jet bundles are the natural framework to put jet constraints for j-holomorphic curves in order to obtain regularity of their moduli spaces. the regularity of j- holomorphic curve moduli spaces fails for general constraints in usual jet bundle (remark 2), but still holds in a special case when the moduli space consists of immersed j-holomorphic curves (theorem 2). the key of the proof is to establish the sujective property of the linearization of k-jet evaluations on the universal moduli spaces of j-holomorphic curves at marked points insider the mapping space, including the parameter j ∈jω(theorem 1). it is important to take the evaluations in holomorphic jet bundles in order to get the surjectivity of the linearization of the k-jet evaluation map. since jω is a huge parameter space to deform j-holomorphic curves, the sujective property here is a reminiscence of the classic thom transversality theorem, which says that the k-jet evaluation on smooth mapping space to the k-jet bundle is transversal to any section there. the framework of the paper is similar to [oh], which in turn is a higher jet generalization of [oz] for 1-jet transversality of j-holomorphic curves. the main steps of the paper are in order: (1) we set up the banach bundle including the finite dimensional holomorphic k-jet subbundle jk hol (σ, m) over the mapping space f1 (σ, m) × jω and define the section υk = ∂, jk hol
, where υk : ((u, j, z0) , j) → ∂j,ju, jk hol (u (z0))
. we inteprete the universal j-holomorphic curve moduli space as m (σ, m) = ∂ −1 (0) = υ−1 k 0, jk hol (σ, m)
. (2) we compute the linearization dυk of the section υk. we express the submersion property of υk as the solvability of a system of equations dυk (ξ, b) = (γ, α) for any (γ, α), where (ξ, b) ∈tuf1 (σ, m) × tjjω, or equivalently, the vanishing of the cokernal element (η, ζ) in the fred- holm alternative system: f ⟨(ξ, b) , (η, ζ)⟩= 0 for all (ξ, b). this is called the cokernal equation. (3) using the abundance of b ∈tjjω we get suppη ⊂{z0}. then we use a structure theorem in distribution to write η as a linear combination of δ function and its derivatives at z0, up to (k −1)-th order derivatives. (4) since suppη ⊂{z0} the cokernal equation is supported at z0. we replace the ξ in the cokernal equation by ξ+h where h = h (z, z) is a suitable polynomial in local coordinates nearby z0, and set b = 0, so that the cokernal equation is reduced to du∂j,jξ, η  = 0 for all ξ. the crucial observation is that to get du∂j,jξ, η  = 0 we do not need so strong conditions of vanishing of 1 ∼k-derivatives of u at z0 as in [oh] and [cm]. this is by exploring the flexibility of h to get rid of redundant terms from the original cokernal equation. (5) then we apply elliptic regularity to conclude η = 0 and consequently ζ = 0. therefore we get the sujectivity of dυk and djk hol. (6) finally, there is an obstruction in step 4 to get h when ζk = 0, where ζk is the k-th component of ζ. but when ζk = 0 the cokernal equation is moduli spaces of j-holomorphic curves with general jet constraints 3 reduced to the (k −1)-jet evaluation setting, so we still get (η, ζ) = (0, 0) by induction on k. acaknowledgement. the author would like to thank yakov eliashberg to suggest the generalization from [oh] to general pde relations. he would also like to thank yong-geun oh on past discussions in holomorphic jet transversality. 2. holomorphic jet bundle we recall the holomorphic jet bundle from [oh]. given σ, m, and (z, x) ∈σ×m, the k-jet with source z and target x is defined as (see [hir]) jk z,x (σ, m) = k y l=0 syml (tzς, txm) , where syml (tzς, txm) is the set of l-multilinear maps from tzς to txm for l ≥1. here for convenience we have set sym0 (tzς, txm) = m. let jk (σ, m) = [ (z,x)∈σ×m jk z,x (σ, m) be the k-jet bundle over σ × m. for the mapping space f1 (σ, m; β) = {((σ, j) , u) |j ∈m (σ) , z ∈σ, u : σ →m, [u] = β} , we consider the map f1 (σ, m; β) →σ × m, (u, j, z) →(z, u (z)) . by this map we can pull back the bundle jk (σ, m) →σ×m to the base f1 (σ, m; β). by abusing notation, we still call the resulted bundle by jk (σ, m) .then jk (σ, m) → f1 (σ, m; β) is a finite dimensional vector bundle over the banach manifold f1 (σ, m; β). we define the k-jet evaluation jk : f1 (σ, m; β) →jk (σ, m) , jk ((u, j) , z) = jk z u ∈jk z,u(z) (σ, m) . then jk is a smooth section. classic thom transversality theorem says that jk is transversal to any section in jk (σ, m). now we turn to the case when σ and m are equipped with (almost) complex structures j and j respectively. the corresponding concept is the holomorphic jet bundle defined in [oh]. with respect to (jz, jx), syml z,x (σ, m) splits into summands indexed by the bigrading (p, q) for p + q = k: syml z,x (σ, m) = sym(l,0) (tzς, txm) ⊕sym(0,l) (tzς, txm) ⊕"mixed parts" let h(l,0) jz,jx (σ, m) = sym(l,0) (tzς, txm) , h(l,0) j,j (σ, m) = [ (z,x)∈σ×m h(l,0) jz,jx (σ, m) . given (j, j), the (j, j)-holomorphic jet bundle jk (j,j)hol (σ, m) is defined as (2.1) jk (j,j)hol (σ, m) = k y l=0 h(l,0) j,j (σ, m) , 4 ke zhu which is a finite dimensional vector bundle over σ × m. we define the bundle jk hol (σ, m) = [ (j,j)∈m(σ)×jω jk (j,j)hol (σ, m) . jk hol (σ, m) →σ × m × m (σ) × jω is a finite dimensional vector bundle over the base banach manifold. using the pull back of the map ev : f1 (σ, m; β) × jω →σ × m × m (σ) × jω, ((u, j) , z, j) →(z, u (z) , j, j) , ev∗jk hol (σ, m)
is a finite dimensional vector bundle over the banach manifold f1 (σ, m; β) × jω. by abusing of notation, we still call ev∗jk hol (σ, m)
by jk hol (σ, m). definition 1. jk hol (σ, m) →f1 (σ, m; β) × jω is called the holomorphic k-jet bundle. let πhol : jk (σ, m) →jk hol (σ, m) be the bundle projection. we define the holomorphic k-jet evaluation jk hol = πhol ◦jk. it is not hard to see jk hol is a smooth section of the banach bundle f1 (σ, m; β) × jω →jk hol (σ, m). according to the summand (2.1), we write jk hol in components jk hol = k y l=0 σl, where the l-th component is σl : f1 (σ, m; β) × jω →h(l,0) j,j (σ, m) , ((u, j) , z, j) →πhol j,j dlu (z)
. we remark that if j is integrable, σl corresponds to the l-th holomorphic derivative ∂l ∂zl u of u at z. the important point is that the holomorphic k-jet bundle and the section jk hol are canonically associated to the pair (σ, j) and (m, j) in the "off-shell level", i.e. on the space of all smooth maps, not only j-holomorphic maps. this enables us to formulate the jet constraints for j-holomorphic maps as some submanifold in the bundle jk hol (σ, m) →f1 (σ, m; β) × jω. 3. fredholm set up the fredholm set up is the same as in [oh], with the simplification that we only need one marked point on σ. the case with more marked points has no essential difference. we introduce the standard bundle h ′′ = [ ((u,j),j) h ′′ ((u,j),j), h ′′ ((u,j),j) = ω(0,1) j,j (u∗t m) and define the section υk : f1 (σ, m; β) × jω →h ′′ × jk (σ, m) as υk ((u, j) , z, j) = ∂(u, j, j) ; jk hol (u, j, j, z)
, where ∂(u, j, j) = ∂j,j (u) = du + j ◦du ◦j 2 . moduli spaces of j-holomorphic curves with general jet constraints 5 given β ∈h2 (m, z), let m1 (σ, m; β) = [ j∈jω m1 (σ, m, j; β) be the universal moduli space of j-holomorphic curves in class β with one marked point. its open subset consisting of somewhere injective j-holomorphic curves is denoted by m∗ 1 (σ, m; β). it is a standard fact in symplectic geometry that m∗ 1 (σ, m; β) is a banach manifold. now we make precise the necessary regularity requirement for the banach man- ifold set-up: (1) to make sense of the evaluation of jku at a point z on σ, we need to take at least w k+1,p-completion with p > 2 of f1 (σ, m; β) so jku ∈w 1,p ֒ →c0. to make the section υk differentiable we need to take w k+2,p completion, since in (4.2) (k + 1)-th derivatives of u are involved. to apply sard-smale theorem, we actually need to take w n,p completion with sufficiently large n = n (β, k). (2) we provide h′′ with topology of a w n,p banach bundle. (3) we also need to provide the banach manifold structure of jω. we can borrow floer's scheme [f, f] for this whose details we refer readers thereto. 4. transversality theorem 1. at every j-holomorphic curve ((u, j) , z, j) ∈m∗ 1 (σ, m; β) ⊂f1 (σ, m; β)× jω, the linearization dυk of the map υk = ∂, jk hol
: f1 (σ, m; β) × jω →h ′′ × jk hol (σ, m) is surjective. especially the linearization djk hol of the holomorphic k-jet evaluation jk hol : f1 (σ, m; β) × jω →jk hol (σ, m) on m∗ 1 (σ, m; β) is sujective. to prove theorem we need to verify that at each ((u, j) , z, j) ∈m∗ 1 (σ, m; β) , the system of equations dj,(j,u)∂(b, (b, ξ)) = γ (4.1) dj,(j,u)jk hol (b, (b, ξ)) (z) + ∇v jk hol (u)
(z) = α (4.2) has a solution (b, (b, ξ) , v) ∈tjjω × tjm (σ) × tuf1 (σ, m; β) × tzς for each given data γ ∈ω(0,1) n−1,p (u∗t m) , ζ = ζ0, ζ1, . . . ζk
∈jk hol tzς, tu(z)m
. it will be enough to consider the triple with b = 0 and v = 0 which we will assume from now on. we compute the dj,(j,u)jk hol (b, (b, ξ)) (z). it is enough to compute dj,(j,u)σl (b, (0, ξ)) (z) for l = 0, 1, * * *k. we have (4.3) dj,(j,u)σl (b, (0, ξ)) (z) = πhol  (∇du)l ξ (z)  +σ0≤s,t≤lb (z)*fst (z) (∇du)s ξ (z) , ∇tu (z)
where fst (z) (*, *) is some vector-valued monomial, and b (z) is a matrix val- ued function, both smoothly depending on z. there is no derivative of b in the above formula, because for any l, σl is the projection of the tensor dlu ∈ 6 ke zhu syml tzς, tu(x)m
to sym(l,0) j,j tzς, tu(x)m
, and the projection only involves j but not its derivatives. since u is (j,j)-holomorphic, it also follows that (4.4) πhol  (∇du)l ξ (z)  =  ∇ ′ du l ξ (z) , where ∇ ′ du = πhol∇du = du∂j,j. there is a formula for du∂j,j and du∂j,j nearby z0 (see [si]): du∂j,jξ = ∂ξ + a (z) ∂ξ + c (z) ξ (4.5) du∂j,jξ = ∂ξ + g (z) ∂ξ + h (z) ξ where a (z) , c (z) , g (z) , h (z) are matrix-valued smooth functions, all vanishing at z0. now we study the solvability of (4.1) and (4.2) by fredholm alternative. we regard ω(0,1) n−1,p (u∗t m) × jk hol tzς, tu(z)m
as a banach space with the norm ∥*∥n−1,p + σk l=1 |*|l where |*|l is any norm induced by an inner product on the 2n-dimensional vector space sym(l,0) j,j tzς, tu(z)m
≃cn. we denote the natural pairing ω(0,1) n−1,p (u∗t m) ×  ω(0,1) n−1,p (u∗t m) ∗ →r by ⟨*, *⟩and the inner product on sym(l,0) j,j tzς, tu(z)m
by (*, *)z. let (η, ζ) ∈  ω(0,1) n−1,p (u∗t m) ∗ × jk hol tzς, tu(z)m
for ζ = ζ1,*** ,ζk
such that (4.6) dj,(j,u)∂(b, (0, ξ)) , η  + σk l=1 dj,(j,u)σl (b, (0, ξ)) (z) , ζl
z = 0 for all ξ ∈ω(0,1) n−1,p (u∗t m) and b ∈tjjω. we want to show (η, ζ) = (0, 0). the idea is to change the above equation into dj,(j,u)∂(b, (0, ξ)) , η  = 0 for all ξ and b by judiciously modifying ξ by a taylor polynomial nearby z, and then use standard techniques in j-holomorphic curve theory to show η = 0, and after that use cauchy integral to show ζ = 0. we first deal with n = k case, and later raise the regularity by ellipticity of cauchy-riemann equation. let ξ = 0, then (4.6) becomes 1 2b ◦du ◦j, η  = 0. using the abundance of b ∈tjjω, and that u is a simple j-holomorphic curve, by standard technique (for example [ms]) we get η = 0 on σ\ {z0}, namely suppη ⊂ {z0}. since η ∈ w k,p
∗, by the structure theorem of distribution with point support (see [gs]), we have (4.7) η = p  ∂ ∂z , ∂ ∂z  δz0 moduli spaces of j-holomorphic curves with general jet constraints 7 where δz0 is the delta function supported at z0, and p is a polynomial in two variables with degree ≤k −1: this is because the evaluation at a point of the k-th derivative of w k,p maps does not define a continuous functional on w k,p. let b = 0. by (4.3) and (4.7), (4.6) becomes (4.8) du∂j,jξ, η  + dj,(j,u)jk hol
ξ (z0) , ζ
z0 = 0. since ξ is arbitrary, we can replace ξ by ξ + χ (z) h (z, z) in the above identity, where h (z, z) is a vector-valued polynomial in z and z, and χ (z) is a smooth cut- offfunction equal to 1 in a coordinate neighborhood of z0 and 0 outside a slightly larger neighborhood, so that χ (z) h (z, z) is a well defined and smooth on whole σ. we want (4.8) becomes du∂j,jξ, η  = 0 after that replacement. for this purpose the h (z, z) should satisfy (4.9)  du∂j,jh, p  ∂ ∂z , ∂ ∂z  δz0  + dj,(j,u)jk hol
h (z0) , ζ
z0 = − dj,(j,u)jk hol
ξ (z0,z0) , ζ
z0 after simplification, the above is a differential equation about h: (4.10) q  ∂ ∂z , ∂ ∂z  h (z0,z0) = w where q (s, t) is a vector-valued polynomial in two variables s, t , q ∂ ∂z, ∂ ∂z
acts on h (z, z) with vector coefficients paired with those of h by inner product, and w := − dj,(j,u)jk hol
ξ (z0) , ζ
z0 is a constant. here comes the crucial observation: when ζk ̸= 0, the highest degree of s in q (s, t) is in the term ζksk. this is because p (s, t) has degree≤k −1 and after integration by parts, ∂ ∂z can fall at du∂j,jh of most (k −1) times, and in (4.5) du∂j,jh = ∂h + a (z) ∂h + c (z) h, where a (z0) = 0. on the other hand, in dj,(j,u)jk hol
h, by (4.3) , (4.4) , (4.5), the highest derivative for ∂ ∂z is ∂ ∂z
k, and is paired with the coefficient ζk in (4.9). when ζk ̸= 0, we take h (z, z) = ζk |ζk|2 1 k! (z −z0)k w, then h solves (4.10). this is because of the following: h is holomorphic nearby z0, so we can ignore all terms in q ∂ ∂z, ∂ ∂z
involving ∂ ∂z; for the remaining terms in q ∂ ∂z, ∂ ∂z
, they must be of the form ∂ ∂z
l with 0 ≤l ≤k, and only ∂ ∂z
k h (z0) ̸= 0. with this h, we reduce the cokernal equation to du∂j,jξ, η  = 0. since η is a weak solution of du∂j,j
∗η = 0 on σ, by ellipticity of the du∂j,j
∗operator, the distribution solution η is smooth on σ (see [ho]). since η = 0 on σ\ {z0} , η = 0 on σ. then it is not hard to conclude ζ = 0 by cauchy integral formula as in [oz] and [oh]. therefore the system of equations (4.1) and (4.2) is solvable for any η ∈w k,p and α ∈jk hol tz0σ, tu(z0)m
. there is one case left: that is when ζk = 0. we still need to show (η, ζ) = (0, 0). if k = 1, then ζ1 = 0 ⇔ζ = 0 so it has been done as above. if k > 1, we notice that the cokernal equation (4.6) now is the cokernal equation for the section dυk−1, since the k-th jet is paired with ζk there, dj,(j,u)jk hol
ξ (z0) , ζ
z0 = dj,(j,u)jk−1 hol
ξ (z0) , ζ
z0 . by induction assumption on k, dυk−1 has trivial cokernal hence (η, ζ) = (0, 0). 8 ke zhu last we raise the regularity from w k+1,p to w n,p, for any n > k. for η ∈ w n−1,p ⊂w k,p, by the above argument we can find a solution ξ ∈w k+1,p in (4.1). by elliptic regularity, the solution ξ ∈w n,p. therefore (4.1) and (4.2) is solvable in w n,p setting. this finishes induction hence the proof of theorem. remark 1. in the above proof, the induction starts from k = 1. in [oz], k = 1 case was treated in the framework of 1-jet transversality at (u, z0) where du (z0) = 0. the above proof includes the k = 1 case as well, but the way of choosing h does not rely on du (z0) = 0 and applies to any z0 on σ. remark 2. it is crucial that we use the holomorphic k-jet bundle instead of the usual k-jet bundle to get the sujective property of dυk. otherwise, as the usual jet evaluation involves mixed derivatives, given ζ(k,0) = 0 we can not reduce the cokernal equation to the (k −1) case by induction, and when k = 1, ζ(1,0) = 0 does not imply ζ = 0. in the case k = 1, we can explicitly see why this submersion property fails in the usual 1-jet bundle: for a j-holomorphic curve u with du (z0) = 0, and γ1 = ∂, jk
: f1 (σ, m; β) × jω →h ′′ × j1 (σ, m), calculations in [oz] yield dγ1 (ξ, b) = du∂j,jξ; du∂j,jξ (z0) , du∂j,jξ (z0)
therefore there is no solution for η, α(0,1),α(1,0)
if η (z0) ̸= α(0,1). however, if du (z0) ̸= 0 then the sujective property still holds in the usual jet bundles. more precisely we have the following theorem 2. at any j-holomrophic curve ((u, j) , z0, j) ∈m∗ 1 (σ, m; β) ⊂f1 (σ, m; β)× jω with du (z0) ̸= 0, the linearization dγk of the section γk = ∂, jk
: f1 (σ, m; β) × jω →h ′′ × jk (σ, m) is a surjective. especially the linearization djk of k-jet evaluation jk : f1 (σ, m; β) × jω →jk (σ, m) at ((u, j) , z0, j) is surjective. proof. it is enough to show that the cokernal equation  du∂j,jξ + 1 2b ◦du ◦j, η  + dj,(j,u)jk
ξ (z0) , ζ
z0 = 0, for all ξ, b only has trivial solution (η, ζ) = (0, 0). to do this, using standard argument in [ms] we again get suppη ⊂{z0}. given ζ ∈jk tz0σ, tu(z0)m
, by taylor polynomial we can construct a smooth ξ supported in arbitrarily small neighborhood of z0 ∈σ, such that dj,(j,u)jk
ξ (z0) = ζ. when du (z0) ̸= 0, by linear algebra (namely the abundance of tjjω) and perturbation method we can construct b ∈tjjω such that du∂j,jξ + 1 2b ◦du ◦j = 0 on σ (see [ms]). so we get from the cokernal equation that 0 + |ζ|2 = 0, i.e. ζ = 0. let b = 0 in the cokernal equation, we get du∂j,jξ, η  = 0 for all ξ. then by elliptic regularity we conclude η = 0 on the whole σ. □ the following theorem is a direct consequence of theorem 1 by applying sard- smale theorem. moduli spaces of j-holomorphic curves with general jet constraints 9 theorem 3. let s be any smooth section of the holomorphic k-jet bundle jk hol (σ, m) → f1 (σ, m; β) × jω. then the section υk is transversal to the section (0, s). the moduli space ms := jk hol
−1 (s) ∩m∗ 1 (σ, m; β) = υ−1 k (0, s) is a banach submanifold of codimension 2kn in m∗ 1 (σ, m; β). under the natural projection π : f1 (σ, m; β) × jω →jω, there exists jreg ⊂jω of second category, such that for any j ∈jreg, the modulis space ms j := ms ∩π−1 (j) is a smooth manifold in m∗ 1 (σ, m, j; β) , with dimension dim ms j = dim m∗ 1 (σ, m, j; β) −2kn, and all the elements in ms j are fredholm regular. remark 3. in [oh], the s is the zero section of the holomorphic k-jet bun- dle jk hol (σ, m), so ms j is the set of j-holomorphic curves with prescribed ram- ification degrees at the marked points. the j-holomorphic curves in our mod- uli space ms j can obey more general constraint s. similar to [oh], the theo- rem also has the version with more than one marked point. also the constaint s need not to be a full section over the base, but only a closed submanifold in jk hol (σ, m) whose tangent space projects onto the horizontal distribution of the bundle jk hol (σ, m) →f1 (σ, m; β) × jω, because the essential part in the proof the theorem is that dυk|(0,s) is surjective. the theorem appears to be a good start of studying moduli spaces of j-holomorphic curves satisfying general jet constraints in the holomorphic jet bundle; for exam- ple, moduli spaces of j-holomorphic curves with self tangency. also in [cm], jet constraints from symplectic hypersurfaces were used to get rid of multicovering bub- bling spheres. this enables them to define genus zero gromov-witten invariants without abstract perturbations. the above theorem tells that the moduli spaces ms j are well-behaved, and the {jt}0≤t≤1 family version of the above theorem tells that they are cobordant to each other by moduli spaces  ms jt 0≤t≤1 for generic path jt ⊂jω. it is interesting to see if the moduli spaces ms j can be used to construct new symplectic invariants. references [cm] cieliebak, k., mohke, k., "symplectic hypersurfaces and transversality in gromov-witten theory", journal of symplectic geometry (2008) [f] floer, a., "the unregularized gradient flow of symplectic action", comm. pure appl. math. 41 (1988), 775-813 [gs] gelfand, i.m., shilov, g.e., "generalized functions", vol 2, academic press, new york and london, 1968 [hir] hirsch, m., "differential topology", gtm 33, springer-verlag, 1976 [ho] h ̈ ormander, l., "the analysis of linera differential operators ii", compre. studies in math. 257, springer-verlag, 1983, berlin [ht] m. hutchings and c. h. taubes, gluing pseudo-holomorphic curves along branched covered cylinders ii, j. symplectic geom., 5 (2007), pp. 43–137; math.sg/0705.2074. [ms] mcduffd., salamon, d., "j-holomorpic curves and symplectic topology", colloquim pub- lications, vol 52, ams, provindence ri, 2004. [oh] oh,y-g, "higher jet evaluation transversality of j-holomorphic curves", math.sg/0904.3573 [oz] oh,y-g., zhu, k., "embedding property of j-holomorphic curves in calabi-yau manifolds for generic j", math.sg/0805.3581, asian j. of math, vol 13, no.3, 2009 10 ke zhu [oz1] oh,y-g., zhu, k., floer trajectories with immersed nodes and scale-dependent gluing, submitted, sg/0711.4187 [si] sikorav, j.c., "some property of holomorphic curves in almost complex manifolds", 165- 189, "holomorphic curves in symplectic geometry", audin, m. and lafontaine, j. ed, birkh ̈ auser, basel, 1994 [wen] c. wendl, automatic transversality and orbifolds of punctured holomorphic curves in di- mension 4, arxiv:0802.3842v1. department of mathematics, the chinese university of hong kong, shatin, hong kong e-mail address: [email protected]

0911.1711

a new class of exact hairy black hole solutions

we present a new class of black hole solutions with minimally coupled scalar field in the presence of a negative cosmological constant. we consider a one-parameter family of self-interaction potentials parametrized by a dimensionless parameter $g$. when $g=0$, we recover the conformally invariant solution of the martinez-troncoso-zanelli (mtz) black hole. a non-vanishing $g$ signals the departure from conformal invariance. all solutions are perturbatively stable for negative black hole mass and they may develop instabilities for positive mass. thermodynamically, there is a critical temperature at vanishing black hole mass, where a higher-order phase transition occurs, as in the case of the mtz black hole. additionally, we obtain a branch of hairy solutions which undergo a first-order phase transition at a second critical temperature which depends on $g$ and it is higher than the mtz critical temperature. as $g\to 0$, this second critical temperature diverges.

introduction four-dimensional black hole solutions of einstein gravity coupled to a scalar field have been an avenue of intense research for many years. questions pertaining to their exis- tence, uniqueness and stability were seeking answers over these years. the kerr-newman solutions of four-dimensional asymptotically flat black holes coupled to an electromagnetic field or in vacuum, imposed very stringent conditions on their existence in the form of "no-hair" theorems. in the case of a minimally coupled scalar field in asymptotically flat spacetime the no-hair theorems were proven imposing conditions on the form of the self- interaction potential [1]. these theorems were also generalized to non-minimally coupled scalar fields [2]. for asymptotically flat spacetime, a four-dimensional black hole coupled to a scalar field with a zero self-interaction potential is known [3]. however, the scalar field diverges on the event horizon and, furthermore, the solution is unstable [4], so there is no violation of the "no-hair" theorems. in the case of a positive cosmological constant with a minimally coupled scalar field with a self-interaction potential, black hole solutions were found in [5] and also a numerical solution was presented in [6], but it was unstable. if the scalar field is non-minimally coupled, a solution exists with a quartic self-interaction potential [7], but it was shown to be unstable [8, 9]. in the case of a negative cosmological constant, stable solutions were found numerically for spherical geometries [10, 11] and an exact solution in asymptotically ads space with hyperbolic geometry was presented in [12] and generalized later to include charge [13]. this solution is perturbatively stable for negative mass and may develop instabilities for positive mass [14]. the thermodynamics of this solution were studied in [12] where it was shown that there is a second order phase transition of the hairy black hole to a pure topological black hole without hair. the analytical and numerical calculation of the quasi-normal modes of scalar, electromagnetic and tensor perturbations of these black holes confirmed this behaviour [15]. recently, a new exact solution of a charged c-metric conformally coupled to a scalar field was presented in [16, 17]. a schwarzschild-ads black hole in five-dimensions coupled to a scalar field was discussed in [18], while dilatonic black hole solutions with a gauss-bonnet term in various dimensions were discussed in [19]. from a known black hole solution coupled to a scalar field other solutions can be gen- erated via conformal mappings [20]. in all black hole solutions in the einstein frame the scalar field is coupled minimally to gravity. applying a conformal transformation to these solutions, other solutions can be obtained in the jordan frame which are not physically equivalent to the untransformed ones [21]. the scalar field in the jordan frame is coupled to gravity non-minimally and this coupling is characterized by a dimensionless parameter ξ. there are strong theoretical, astrophysical and cosmological arguments (for a review see [21]) which fix the value of this conformal coupling to ξ = 1/6. if the scalar poten- tial is zero or quartic in the scalar field, the theory is conformally invariant; otherwise a non-trivial scalar potential introduces a scale in the theory and the conformal invariance is broken. in this work we present a new class of black hole solutions of four-dimensional einstein gravity coupled to a scalar field and to vacuum energy. we analyse the structure and –3– study the properties of these solutions in the einstein frame. in this frame, the scalar self-interaction potential is characterised by a dimensionless parameter g. if this parameter vanishes, then the known solutions of black holes minimally coupled to a scalar field in (a)ds space are obtained [7, 12]. transforming these solutions to the jordan frame, the parameter g can be interpreted as giving the measure of departure from conformal invari- ance. this breakdown of conformal invariance allows the back-scattering of waves of the scalar field offof the background curvature of spacetime, and the creation of "tails" of radiation. this effect may have sizeable observation signatures in cosmology [22]. following [23], we perform a perturbative stability analysis of the solutions. we find that the hairy black hole is stable near the conformal point if the mass is negative and may develop instabilities in the case of positive mass. we also study the thermodynamics of our solutions. calculating the free energy we find that there is a critical temperature above which the hairy black hole loses its hair to a black hole in vacuum. this critical temperature occurs at a point where the black hole mass flips sign, as in the case of the mtz black hole [12]. interestingly, another phase transition occurs at a higher critical temperature which is of first order and involves a different branch of our solution. this new critical temperature diverges as the coupling constant in the potential g →0. these exact hairy black hole solutions may have interesting applications to holographic superconductors [24, 25], where new types of holographic superconductors can be constructed [14, 26]. our discussion is organized as follows. in section 2 we introduce the self-interaction potential and we present the hairy black hole solution. in section 3 we discuss the thermo- dynamics of our solution. in section 4 we perform a stability analysis. finally, in section 5 we summarize our results. 2 black hole with scalar hair to obtain a black hole with scalar hair, we start with the four-dimensional action consisting of the einstein-hilbert action with a negative cosmological constant λ, along with a scalar, i = z d4x√−g r −2λ 16πg −1 2gμν∂μφ∂νφ −v (φ)  , (2.1) where g is newton's constant and r is the ricci scalar. the corresponding field equations are gμν + λgμν = 8πgt matter μν , 2φ = dv dφ , (2.2) where the energy-momentum tensor is given by t matter μν = ∂μφ∂νφ −1 2gμνgαβ∂αφ∂βφ −gμνv (φ) . (2.3) the potential is chosen as v (φ) = λ 4πg sinh2 r 4πg 3 φ –4– + gλ 24πg " 2 √ 3πgφ cosh r 16πg 3 φ ! −9 8 sinh r 16πg 3 φ ! −1 8 sinh  4 √ 3πgφ # (2.4) and it is given in terms of a coupling constant g. setting g = 0 we recover the action that yields the mtz black hole [12]. this particular form of the potential is chosen so that the field equations can be solved analytically. the qualitative nature of our results does not depend on the detailed form of the potential. a similar potential was considered in a different context in [5] (see also [26] for the derivation of a potential that yields analytic solutions in the case of a spherical horizon). if one goes over to the jordan frame, in which the scalar field obeys the klein-gordon equation 2φ −ξrφ −dv dφ = 0 , (2.5) with ξ = 1/6, the scalar potential has the form v (φ) = −2πgλ 9 φ4 − gλ 16πg "r 16πg 3 φ 1 −4πg 3 φ2 + 16πg 9 φ2 1 −4πg 3 φ2 ! −  1 −4πg 3 φ2   1 + 4πg 3 φ2  ln 1 + q 4πg 3 φ 1 − q 4πg 3 φ  . (2.6) evidently, the scalar field is conformally coupled but the conformal invariance is broken by a non-zero value of g. the mass of the scalar field is given by m2 = v ′′(0) = −2 l2 (2.7) where we defined λ = −3/l2. notice that it is independent of g and coincides with the scalar mass that yields the mtz black hole [12]. asymptotically (r →∞), the scalar field behaves as φ ∼r−∆± where ∆± = 3 2 ± q 9 4 + m2l2. in our case ∆+ = 2 and ∆−= 1. both boundary conditions are acceptable as both give normalizable modes. we shall adopt the mixed boundary conditions (as r →∞) φ(r) = α r + cα2 r2 + . . . , c = − r 4πg 3 < 0 . (2.8) this choice of the parameter c coincides with the mtz solution [12]. solutions to the einstein equations with the boundary conditions (2.8) have been found in the case of spherical horizons and shown to be unstable [23]. in that case, for α > 0, it was shown that c < 0 always and the hairy black hole had positive mass. on the other hand, mtz black holes, which have hyperbolic horizons and obey the boundary conditions (2.8) with c < 0, can be stable if they have negative mass [14]. this is impossible with –5– spherical horizons, because they always enclose black holes of positive mass. the numerical value of c is not important (except for the fact that c ̸= 0) and is chosen as in (2.8) for convenience. the field equations admit solutions which are black holes with topology r2 × σ, where σ is a two-dimensional manifold of constant negative curvature. black holes with constant negative curvature are known as topological black holes (tbhs - see, e.g., [27, 28]). the simplest solution for λ = −3/l2 reads ds2 = −ftbh(ρ)dt2 + 1 ftbh(ρ)dρ2 + ρ2dσ2 , ftbh(ρ) = ρ2 l2 −1 −ρ0 ρ , (2.9) where ρ0 is a constant which is proportional to the mass and is bounded from below (ρ0 ≥− 2 3 √ 3l), dσ2 is the line element of the two-dimensional manifold σ which is locally isomorphic to the hyperbolic manifold h2 and of the form σ = h2/γ , γ ⊂o(2, 1) , (2.10) with γ a freely acting discrete subgroup (i.e., without fixed points) of isometries. the line element dσ2 of σ can be written as dσ2 = dθ2 + sinh2 θdφ2 , (2.11) with θ ≥0 and 0 ≤φ < 2π being the coordinates of the hyperbolic space h2 or pseu- dosphere, which is a non-compact two-dimensional space of constant negative curvature. this space becomes a compact space of constant negative curvature with genus g ≥2 by identifying, according to the connection rules of the discrete subgroup γ, the opposite edges of a 4g-sided polygon whose sides are geodesics and is centered at the origin θ = φ = 0 of the pseudosphere. an octagon is the simplest such polygon, yielding a compact surface of genus g = 2 under these identifications. thus, the two-dimensional manifold σ is a compact riemann 2-surface of genus g ≥2. the configuration (2.9) is an asymptotically locally ads spacetime. the horizon structure of (2.9) is determined by the roots of the metric function ftbh(ρ), that is ftbh(ρ) = ρ2 l2 −1 −ρ0 ρ = 0 . (2.12) for − 2 3 √ 3l < ρ0 < 0, this equation has two distinct non-degenerate solutions, corresponding to an inner and to an outer horizon ρ−and ρ+ respectively. for ρ0 ≥0, ftbh(ρ) has just one non-degenerate root and so the black hole (2.9) has one horizon ρ+. the horizons for both cases of ρ0 have the non-trivial topology of the manifold σ. we note that for ρ0 = − 2 3 √ 3l, ftbh(ρ) has a degenerate root, but this horizon does not have an interpretation as a black hole horizon. the boundary has the metric ds2 ∂= −dt2 + l2dσ2 , (2.13) so spatially it is a hyperbolic manifold of radius l (and of curvature −1/l). –6– the action (2.1) with a potential as in (2.4) has a static black hole solution with topology r2 × σ and with scalar hair, and it is given by ds2 = r(r + 2r0) (r + r0)2  −f(r)dt2 + dr2 f(r) + r2dσ2  , (2.14) where f(r) = r2 l2 −gr0 l2 r −1 + gr2 0 l2 −  1 −2gr2 0 l2  r0 r  2 + r0 r  + g r2 2l2 ln  1 + 2r0 r  , (2.15) and the scalar field is φ(r) = r 3 4πg arctanh  r0 r + r0  , (2.16) obeying the boundary conditions (2.8) by design. 3 thermodynamics to study the thermodynamics of our black hole solutions we consider the euclidean continuation (t →iτ) of the action in hamiltonian form i = z h πij ̇ gij + p ̇ φ −nh −nihi i d 3xdt + b, (3.1) where πij and p are the conjugate momenta of the metric and the field respectively; b is a surface term. the solution reads: ds2 = n2(r)f 2(r)dτ 2 + f −2(r)dr2 + r2(r)dσ2 (3.2) where n(r) = r(r + 2r0) (r + r0)2 , f 2(r) = (r + r0)2 r(r + 2r0) f(r) , r2(r) = r3(r + 2r0) (r + r0)2 , (3.3) with a periodic τ whose period is the inverse temperature, β = 1/t. the hamiltonian action becomes i = −β σ z ∞ r+ n(r)h(r)dr + b, (3.4) where σ is the area of σ and h = nr2  1 8πg (f 2)′r ′ r + 2f 2r ′′ r + 1 r 2(1 + f 2) + λ  + 1 2f 2(φ′)2 + v (φ)  . (3.5) the euclidean solution is static and satisfies the equation h = 0. thus, the value of the action in the classical limit is just the surface term b, which should maximize the action within the class of fields considered. –7– we now compute the action when the field equations hold. the condition that the geometries which are permitted should not have conical singularities at the horizon imposes t = f ′(r+) 4π . (3.6) using the grand canonical ensemble (fixing the temperature), the variation of the surface term reads δb ≡δbφ + δbg , where δbg = βσ 8πg h n  rr ′δf 2 −(f 2)′rδr  + 2f 2r  nδr ′ −n′δr i∞ r+ , (3.7) and the contribution from the scalar field equals δbφ = βσnr 2f 2φ′δφ|∞ r+ . (3.8) for the metric, the variation of fields at infinity yields δf 2  ∞ =  2 l2r0 −2(3 + (9 −8g)r2 0/l2) 3r −4r0(1 −4r2 0/l2) r2 + o(r−3)  δr0 , δφ|∞ = r 3 4πg 1 r −2r0 r2 + o(r−3)  δr0 , δr|∞ =  −r0 r + 3r2 0 r2 + o(r−3)  δr0 , (3.9) so δbg|∞ = βσ 8πg 6r0(r −4(1 −2g/9)r0) l2 −2 + o(r−1)  δr0 , δbφ|∞ = βσ 8πg  −6r0(r −4r0) l2 + o(r−1)  δr0 . (3.10) the surface term at infinity is b|∞= −βσ(3 −8gr2 0/l2) 12πg r0 . (3.11) the variation of the surface term at the horizon may be found using the relations δr|r+ = δr(r+) −r ′|r+ δr+ , δf 2  r+ = −(f 2)′  r+ δr+ . we observe that δbφ|r+ vanishes, since f 2(r+) = 0, and δb|r+ = −βσ 16πgn(r+) (f 2)′  r+ δr 2(r+) = −σ 4gδr 2(r+) . –8– thus the surface term at the horizon is b|r+ = −σ 4gr 2(r+) . (3.12) therefore, provided the field equations hold, the euclidean action reads i = −βσ(3 −8gr2 0/l2) 12πg r0 + σ 4gr 2(r+) . (3.13) the euclidean action is related to the free energy through i = −βf. we deduce i = s −βm , (3.14) where m and s are the mass and entropy respectively, m = σ(3 −8gr2 0/l2) 12πg r0 , s = σ 4g r2(r+) = ah 4g (3.15) it is easy to show that the law of thermodynamics dm = tds holds. for g = 0, these expressions reduce to the corresponding quantities for mtz black holes [12]. alternatively, the mass of the black hole can be found by the ashtekar-das method [29]. a straightforward calculation confirms the expression (3.15) for the mass. in the case of the topological black hole (2.12) the temperature, entropy and mass are given by respectively, t = 3 4πl ρ+ l − l 3ρ+  , stbh = σρ2 + 4g , mtbh = σρ+ 8πg ρ2 + l2 −1  , (3.16) and also the law of thermodynamics dm = tds is obeyed. we note that, in the limit r0 →0, f(r) →r2 l2 −1 from eq. (2.15) and the corresponding temperature (3.6) reads t = 1 2πl, which equals the temperature of the topological black hole (3.16) in the limit ρ0 →0 (ρ+ →1). the common limit ds2 ads = − r2 l2 −1  dt2 + r2 l2 −1 −1 dr2 + r2dσ2 (3.17) is a manifold of negative constant curvature possessing an event horizon at r = l. the tbh and our hairy black hole solution match continuously at the critical temperature t0 = 1 2πl , (3.18) which corresponds to mtbh = m = 0, with (3.17) a transient configuration. evidently, at the critical point (3.18) a scaling symmetry emerges owing to the fact that the metric becomes pure ads. at the critical temperature (3.18) a higher-order phase transition occurs as in the case of the mtz black hole (with g = 0). introducing the terms with g ̸= 0 in the potential do not alter this result qualitatively. –9– next we perform a detailed analysis of thermodynamics and examine several values of the coupling constant g. henceforth we shall work with units in which l = 1. we begin with a geometrical characteristic of the hairy black hole, the horizon radius r+ (root of f(r) (eq. (2.15)). in figure 1 we show the r0 dependence of the horizon for representative values of the coupling constant, g = 3 and g = 0.0005. we observe that, for g = 3, the horizon may correspond to more than one value of the parameter r0. for g = 0.0005 we see that, additionally, there is a maximum value of the horizon radius. we note that one may express the radius of the horizon r+ in terms of the dimensionless parameter ξ = r0 r+ (3.19) as r+ = 1 + ξ q 1 + gξ(1 + ξ)(−1 + 2ξ + 2ξ2) + 1 2g ln(1 + 2ξ) . (3.20) the temperature reads t = 1 + ξ(1 + ξ)(4 −g(1 + 2ξ + 2ξ2)) + 1 2g(1 + 2ξ)2 ln(1 + 2ξ) 2π(1 + 2ξ) q 1 + gξ(1 + ξ)(−1 + 2ξ + 2ξ2) + 1 2g ln(1 + 2ξ) , (3.21) or equivalently t = (r+ + r0)(r2 + + 4r0r+ + 4r2 0 −8gr3 0r+ −8gr4 0) 2πr3 + , (3.22) a third order equation in r+, showing that, for a given temperature, there are in general three possible values of ξ. thus we obtain up to three different branches of our hairy black hole solution. we start our analysis with a relatively large value of the coupling constant g, namely g = 3 and calculate the horizon radius, temperature and euclidean action for various values of r0. in figure 2, left panel, we depict r0 versus t and it is clear that there is a t interval for which there are really three corresponding values for r0. outside this interval, there is just one solution. the corresponding graph for the euclidean actions may be seen in the right panel of the same figure. the action for the topological black hole with the common temperature t is represented by a continuous line, while the actions for the hairy black holes are shown in the form of points. we note that equation (2.12) yields for the temperature of the topological black hole t = 1 4π  3ρ+ l2 + 1 ρ+  ⇒ρ+ = 2πt 3 + q 2πt 3
2 −1 3. the largest euclidean action (smallest free energy) will dominate. there are three branches for the hairy black hole, corresponding to the three different values of r0. in particular, for some fixed temperature (e.g., t = 0.16) the algebraically lowest r0 corresponds to the algebraically lowest euclidean action; similarly, the medium and largest r0 parameters correspond to the medium and largest euclidean actions. the medium euclidean action for the hairy black hole is very close to the euclidean action for the topological black hole. in fact, it is slightly smaller than the latter for t < 1 2π ≈0.159 and slightly larger after that value. if it were the only branch present, one would thus conclude that the hairy black hole dominates for small temperatures, while for large temperatures –10– 0.6 0.7 0.8 0.9 1 1.1 1.2 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 rp r0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -5 0 5 10 15 20 25 30 35 rp r0 figure 1: horizon versus parameter r0 for g = 3 (left panel) and g = 0.0005 (right panel). the topological black hole would be preferred. this would be a situation similar to the one of the mtz black hole. however, the two additional branches change completely our conclusions. the upper branch shows that the hairy black hole dominates up to t ≈0.20. when the coupling constant g decreases, equation (3.21) along with the demand that the temperature should be positive, show that the acceptable values of r0 are two rather than three, as may be seen in figure 3, left panel. the lowest branch of the previous corresponding figure 2 shrinks for decreasing g and it finally disappears. an interesting consequence of this is that the temperature has an upper limit. the graph for the euclidean actions (figure 3, right panel) is influenced accordingly. there are just two branches for the hairy black hole, rather than three in figure 2 and the figure ends on its right hand side at t ≈1.25. the continuous line represents the euclidean action for the topological black hole with the same temperature. similar remarks hold as in the previous case, e.g., the phase transition moves to t ≈0.80. in addition, the largest value of r0 corresponds to the upper branch of the hairy black hole. to understand the nature of this phase transition it is instructive to draw a kind of phase diagram, so that we can spot which is the dominant solution for a given pair of g and t. we depict our result in figure 4. the hairy solution dominates below the curve which shows the critical temperature as a function of the coupling constant g. the most striking feature of the graph is that the critical temperature diverges as g →0. thus, it does not converge to the mtz value 1 2π ≈0.159 at g = 0. for even the slightest nonzero values of g the critical temperature gets extremely large values! this appears to put the conformal point (mtz black hole) in a special status within the set of these hairy black holes. in other words, the restoration of conformal invariance is not a smooth process, and the mtz black hole solution cannot be obtained in a continuous way as g →0. in fact, it seems that (even infitesimally) away from the conformal point g = 0 black holes are mostly hairy! 4 stability analysis to perform the stability analysis of the hairy black hole it is more convenient to work in the einstein frame. henceforth, we shall work in units in which the radius of the boundary –11– -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 r0 t 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.1 0.12 0.14 0.16 0.18 0.2 0.22 i t figure 2: parameter r0 (left panel) and euclidean actions versus temperature for g = 3 (right panel). is l = 1. we begin with the hairy black hole line element, ds2 0 = ˆ r(ˆ r + 2r0) (ˆ r + r0)2  −f(ˆ r)dt2 + dˆ r2 f(ˆ r) + ˆ r2dσ2  , (4.1) which can be written in the form ds2 = −f0 h2 0 dt2 + dr2 f0 + r2dσ2 (4.2) using the definitions f0(r) = f(ˆ r)  1 + r2 0 (ˆ r + 2r0)(ˆ r + r0) 2 , h0(r) =  1 + r2 0 (ˆ r + 2r0)(ˆ r + r0)  ˆ r + r0 p ˆ r(ˆ r + 2r0) , (4.3) -5 0 5 10 15 20 25 30 35 0 0.2 0.4 0.6 0.8 1 1.2 1.4 r0 t 0 0.5 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 i t figure 3: parameter r0 (left panel) and euclidean actions versus temperature for g = 0.0005 (right panel). –12– 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 t g figure 4: phase diagram. for points under the curve the hairy solution will be preferred. under the change of coordinates r = ˆ r3/2(ˆ r + 2r0)1/2 ˆ r + r0 . (4.4) the scalar field solution reads φ0(r) = r 3 4πg tanh−1 r0 ˆ r + r0 , (4.5) obeying the boundary conditions (2.8) with α = α0 = r 3 4πg|r0| . (4.6) we are interested in figuring out when the black hole is unstable (losing its hair to turn into a tbh) and discuss the results in the context of thermodynamic considerations. to this end, we apply the perturbation f(r, t) = f0(r) + f1(r)eωt, h(r, t) = h0(r) + h1(r)eωt, φ(r, t) = φ0(r) + φ1(r) r eωt . (4.7) which respects the boundary conditions (2.8) with ω > 0 for an instability to develop. the field equations read: −1 −f −rf ′ + rf h′ h + 8πgr2v (φ) = 0 , (4.8) ̇ f + rf ̇ φφ′ = 0 , (4.9) 2h′ + rh h2 f 2 ̇ φ2 + φ′2  = 0 , (4.10) ̇ h f ̇ φ ! −1 r2  r2f hφ′ ′ + 1 hv ′(φ) = 0 . (4.11) –13– the field equations give a schr ̈ odinger-like wave equation for the scalar perturbation, −d2φ1 dr2 ∗ + vφ1 = −ω2φ1 , (4.12) where we defined the tortoise coordinate dr∗ dr = h0 f0 , (4.13) and the effective potential is given by v = f0 h2 0  −1 2(1 + r2φ′ 0 2)φ′ 0 2f0 + (1 −r2φ′ 0 2)f ′ 0 r + 2rφ′ 0v ′(φ0) + v ′′(φ0)  . (4.14) the explicit form of the schr ̈ odinger-like equation reads: −f(ˆ r) d dˆ r  f(ˆ r)dφ1 dˆ r  + vφ1 = −ω2φ1, (4.15) where the functional form of the function f has been given in equation (2.15) and 1 v = r2 0f(ˆ r) ˆ r2 1 + 2r0 ˆ r
2  1 + 3r0 ˆ r + 3r2 0 ˆ r2 2  5 + 2 + (11g + 54)r2 0 r0ˆ r + 29 + (47g + 189)r2 0 ˆ r2 (4.16) +r0(150 + (−3g + 270)r2 0) ˆ r3 + r2 0(396 + (−351g + 135)r2 0) ˆ r4 + r3 0(612 −873gr2 0) ˆ r5 (4.17) +r4 0(582 −1047gr2 0) ˆ r6 + 324r5 0(1 −2gr2 0) ˆ r7 + 81r6 0(1 −2gr2 0) ˆ r8 (4.18) +g 2  5 + 54r0 ˆ r + 189r2 0 ˆ r2 + 270r3 0 ˆ r3 + 135r4 0 ˆ r4  ln  1 + 2r0 ˆ r  . (4.19) near the horizon the schr ̈ odinger-like equation simplifies to −[f ′(ˆ r+)]2ǫ d dǫ  ǫdφ1 dǫ  = −ω2φ1, ǫ = ˆ r −ˆ r+ (4.20) and its acceptable solution reads φ1 ∼ǫκω, κ = 1 f ′(ˆ r+), ω > 0 . (4.21) regularity of the scalar field at the horizon (r →r+) requires the boundary conditions φ1 = 0 , (r −r+)φ′ 1 = κωφ1 , κ > 0 . (4.22) 1we have set 8πg = 1 . –14– for a given ω > 0, they uniquely determine the wavefunction. at the boundary (ˆ r →∞), the wave equation is approximated by −d2φ1 dr2 ∗ + 5r2 0φ1 = −ω2φ1 , (4.23) with solutions φ1 = e±er∗, e = q ω2 + 5r2 0 , (4.24) where r∗= r dˆ r f(ˆ r) = −1 r + . . . . therefore, for large r, φ1 = a + b r + . . . (4.25) to match the boundary conditions (2.8), we need b a = 2cα0 = −2r0 . (4.26) since the wavefunction has already been determined by the boundary conditions at the horizon and therefore also the ratio b/a, this is a constraint on ω. if (4.26) has a solution, then the black hole is unstable. if it does not, then there is no instability of this type (however, one should be careful with non-perturbative instabilities). in figure 5 (left panel) we show the ratio b/a for the standard mtz black hole (corre- sponding to g = 0) versus ω at a typical value of the mass parameter, namely r0 = −0.10. it is obvious that the value of the ratio lies well below the values 2r0 = +0.20. it is clearly impossible to have a solution to this equation, so the solution is stable. in fact this value of the mass parameter lies in the interesting range for this black hole, since thermodynamics dictates that for negative values of r0 mtz black holes are favored against topological black holes. we find that the mtz black holes turn out to be stable. next we examine the case g = 0.0005, for which we have presented data before in figure 3. as we have explained there, the most interesting branch of the graphs is the upper branch, on the right panel, which dominates the small t part of the graph and corresponds to large values of r0 (typically around 30 on the left panel of the same figure). thus we set g = 0.0005, r0 = +30 and plot b/a versus ω in the right panel of figure 5. it is clear that the curve lies systematically below the quantity −2r0 = −60 and no solution is possible, so the hairy black hole with these parameters is stable. finally we come to the case with g = +3, which has three branches. in the left panel of figure 6 we show the results for r0 = +0.40, which corresponds to the upper branch of figure 2; the curve again lies below the quantity −2r0 = −0.80 and no solution is possible, so this hairy black hole is stable. in the right panel of figure 6 we show the results for r0 = −0.30, which corresponds to the lowest branch of figure 2, which disappears for decreasing g. in this case we find something qualitatively different: the curve cuts the line −2r0 = +0.60 around ω ≈0.40 and a solution is possible, signaling instability. thus, for g = 3 the hairy black hole may be stable or unstable, depending on the value of r0. –15– -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0 0.2 0.4 0.6 0.8 1 -2*mass, b/a omega -350 -300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300 -2*mass, b/a omega figure 5: stability of the standard mtz black hole (left panel) for r0 = −0.10; similarly for the hairy black hole at g = 0.0005, r0 = +30 (right panel). -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 0 2 4 6 8 10 -2*mass, b/a omega -2 0 2 4 6 8 10 12 0 0.5 1 1.5 2 -2*mass, b/a omega figure 6: stability of the hairy black hole for g = 3 and r0 = +0.40 (left panel); similarly for the hairy black hole at r0 = −0.30 (right panel). 5 conclusions we presented a new class of hairy black hole solutions in asymptotically ads space. the scalar field is minimally coupled to gravity with a non-trivial self-interaction potential. a coupling constant g in the potential parametrizes our solutions. if g = 0 the conformal invariant mtz black hole solution, conformally coupled to a scalar field, is obtained. if g ̸= 0 a whole new class of hairy black hole solutions is generated. the scalar field is conformally coupled but the solutions are not conformally invariant. these solutions are perturbative stable near the conformal point for negative mass and they may develop instabilities for positive mass. we studied the thermodynamical properties of the solutions. calculating the free energy we showed that for a general g, apart the phase transition of the mtz black hole at the critical temperature t = 1/2πl, there is another critical temperature, higher than the mtz critical temperature, which depends on g and where a first order phase transition occurs of the vacuum black hole towards a hairy one. the existence of a second critical temperature is a manifestation of the breaking of conformal invariance. as g →0 the second critical –16– temperature diverges, indicating that there is no smooth limit to the mtz solution. the solutions presented and discussed in this work have hyperbolic horizons. there are also hairy black hole solutions with flat or spherical horizons of similar form. however, these solutions are pathological. in the solutions with flat horizons, the scalar field diverges at the horizon, in accordance to the "no-hair" theorems. in the case of spherical horizons, calculating the free energy we find that always the vacuum solution is preferable over the hairy configuration. moreover, studying the asymptotic behaviour of the solutions, we found that they are unstable for any value of the mass. acknowledgments g. s. was supported in part by the us department of energy under grant de-fg05- 91er40627. references [1] j. e. chase, "event horizons in static scalar-vacuum space-times," commun. math. phys. 19, 276 (1970); j. d. bekenstein, "transcendence of the law of baryon-number conservation in black- hole physics," phys. rev. lett. 28, 452 (1972); j. d. bekenstein, "nonexistence of baryon number for static black holes," phys. rev. d 5, 1239 (1972); j. d. bekenstein, "nonexistence of baryon number for black holes. ii," phys. rev. d 5, 2403 (1972); m. heusler, "a mass bound for spherically symmetric black hole spacetimes," class. quant. grav. 12, 779 (1995) [arxiv:gr-qc/9411054]; m. heusler and n. straumann, "scaling arguments for the existence of static, spher- ically symmetric solutions of self-gravitating systems," class. quant. grav. 9, 2177 (1992); d. sudarsky, "a simple proof of a no-hair theorem in einstein-higgs theory," class. quant. grav. 12, 579 (1995); j. d. bekenstein, "novel 'no-scalar-hair' theorem for black holes," phys. rev. d 51, r6608 (1995). [2] a. e. mayo and j. d. bekenstein, "no hair for spherical black holes: charged and nonminimally coupled scalar field with self–interaction," phys. rev. d 54, 5059 (1996) [arxiv:gr-qc/9602057]; j. d. bekenstein, "black hole hair: twenty-five years after," arxiv:gr-qc/9605059. [3] n. bocharova, k. bronnikov and v. melnikov, vestn. mosk. univ. fiz. astron. 6, 706 (1970); j. d. bekenstein, annals phys. 82, 535 (1974); j. d. bekenstein, "black holes with scalar charge," annals phys. 91, 75 (1975). –17– [4] k. a. bronnikov and y. n. kireyev, "instability of black holes with scalar charge," phys. lett. a 67, 95 (1978). [5] k. g. zloshchastiev, "on co-existence of black holes and scalar field," phys. rev. lett. 94, 121101 (2005) [arxiv:hep-th/0408163]. [6] t. torii, k. maeda and m. narita, "no-scalar hair conjecture in asymptotic de sitter spacetime," phys. rev. d 59, 064027 (1999) [arxiv:gr-qc/9809036]. [7] c. martinez, r. troncoso and j. zanelli, "de sitter black hole with a conformally coupled scalar field in four dimensions," phys. rev. d 67, 024008 (2003) [arxiv:hep- th/0205319]. [8] t. j. t. harper, p. a. thomas, e. winstanley and p. m. young, "instability of a four-dimensional de sitter black hole with a conformally coupled scalar field," phys. rev. d 70, 064023 (2004) [arxiv:gr-qc/0312104]. [9] g. dotti, r. j. gleiser and c. martinez, "static black hole solutions with a self interact- ing conformally coupled scalar field," phys. rev. d 77, 104035 (2008) [arxiv:0710.1735 [hep-th]]. [10] t. torii, k. maeda and m. narita, "scalar hair on the black hole in asymptotically anti-de sitter spacetime," phys. rev. d 64, 044007 (2001). [11] e. winstanley, "on the existence of conformally coupled scalar field hair for black holes in (anti-)de sitter space," found. phys. 33, 111 (2003) [arxiv:gr-qc/0205092]. [12] c. martinez, r. troncoso and j. zanelli, "exact black hole solution with a minimally coupled scalar field," phys. rev. d 70, 084035 (2004) [arxiv:hep-th/0406111]. [13] c. martinez, j. p. staforelli and r. troncoso, "topological black holes dressed with a conformally coupled scalar field and electric charge," phys. rev. d 74, 044028 (2006) [arxiv:hep-th/0512022]; c. martinez and r. troncoso, "electrically charged black hole with scalar hair," phys. rev. d 74, 064007 (2006) [arxiv:hep-th/0606130]. [14] g. koutsoumbas, e. papantonopoulos and g. siopsis, "exact gravity dual of a gap- less superconductor," jhep 0907, 026 (2009) [arxiv:0902.0733 [hep-th]]. [15] g. koutsoumbas, s. musiri, e. papantonopoulos and g. siopsis, "quasi-normal modes of electromagnetic perturbations of four-dimensional topological black holes with scalar hair," jhep 0610, 006 (2006) [arxiv:hep-th/0606096]; g. koutsoumbas, e. papantonopoulos and g. siopsis, "phase transitions in charged topological-ads black holes," jhep 0805, 107 (2008) [arxiv:0801.4921 [hep-th]]; g. koutsoumbas, e. papantonopoulos and g. siopsis, "discontinuities in scalar perturbations of topological black holes," class. quant. grav. 26, 105004 (2009) [arxiv:0806.1452 [hep-th]]. –18– [16] c. charmousis, t. kolyvaris and e. papantonopoulos, "charged c-metric with con- formally coupled scalar field," class. quant. grav. 26, 175012 (2009) [arxiv:0906.5568 [gr-qc]]. [17] a. anabalon and h. maeda, "new charged black holes with conformal scalar hair," arxiv:0907.0219 [hep-th]. [18] k. farakos, a. p. kouretsis and p. pasipoularides, "anti de sitter 5d black hole solutions with a self-interacting bulk scalar field: a potential reconstruction approach," phys. rev. d 80, 064020 (2009) [arxiv:0905.1345 [hep-th]]. [19] n. ohta and t. torii, "black holes in the dilatonic einstein-gauss-bonnet theory in various dimensions iv - topological black holes with and without cosmological term," arxiv:0908.3918 [hep-th]. [20] k.-i. maeda, "towards the einstein-hilbert action via conformal transformation," phys. rev. d 39 3159 (1989). [21] v. faraoni, e. gunzig and p. nardone, fund. cosmic phys. 20, 121 (1999) [arxiv:gr- qc/9811047]. [22] v. faraoni and s. sonego, "on the tail problem in cosmology," phys. lett. a 170, 413 (1992) [arxiv:astro-ph/9209004]; t. w. noonan, "huygen's principle in conformally flat spacetimes," class. quant. grav. 12, 1087 (1995). [23] t. hertog and k. maeda, "stability and thermodynamics of ads black holes with scalar hair," phys. rev. d 71, 024001 (2005) [arxiv:hep-th/0409314]. [24] s. a. hartnoll, c. p. herzog and g. t. horowitz, "building a holographic supercon- ductor," phys. rev. lett. 101, 031601 (2008) [arxiv:0803.3295 [hep-th]]. [25] s. a. hartnoll, c. p. herzog and g. t. horowitz, "holographic superconductors," jhep 0812, 015 (2008) [arxiv:0810.1563 [hep-th]]. [26] d. f. zeng, "an exact hairy black hole solution for ads/cft superconductors," arxiv:0903.2620 [hep-th]. [27] r. b. mann, "topological black holes – outside looking in," arxiv:gr-qc/9709039. [28] d. birmingham, "topological black holes in anti-de sitter space," class. quant. grav. 16, 1197 (1999) [arxiv:hep-th/9808032]. [29] a. ashtekar and s. das, "asymptotically anti-de sitter space-times: conserved quan- tities," class. quant. grav. 17, l17 (2000) [arxiv:hep-th/9911230].

0911.1712

spin gaps and spin-flip energies in density-functional theory

energy gaps are crucial aspects of the electronic structure of finite and extended systems. whereas much is known about how to define and calculate charge gaps in density-functional theory (dft), and about the relation between these gaps and derivative discontinuities of the exchange-correlation functional, much less is know about spin gaps. in this paper we give density-functional definitions of spin-conserving gaps, spin-flip gaps and the spin stiffness in terms of many-body energies and in terms of single-particle (kohn-sham) energies. our definitions are as analogous as possible to those commonly made in the charge case, but important differences between spin and charge gaps emerge already on the single-particle level because unlike the fundamental charge gap spin gaps involve excited-state energies. kohn-sham and many-body spin gaps are predicted to differ, and the difference is related to derivative discontinuities that are similar to, but distinct from, those usually considered in the case of charge gaps. both ensemble dft and time-dependent dft (tddft) can be used to calculate these spin discontinuities from a suitable functional. we illustrate our findings by evaluating our definitions for the lithium atom, for which we calculate spin gaps and spin discontinuities by making use of near-exact kohn-sham eigenvalues and, independently, from the single-pole approximation to tddft. the many-body corrections to the kohn-sham spin gaps are found to be negative, i.e., single particle calculations tend to overestimate spin gaps while they underestimate charge gaps.

introduction there is hardly any electronic property of a system that does not depend on whether there is an energy gap for charge excitations, or for particle addition and re- moval. similarly, there is hardly any magnetic property of a system that does not depend in some way on whether there is an energy gap for flipping a spin, or for adding and removing spins from the system. the reliable calculation of charge gaps1 from first prin- ciples is nontrivial and still faces practical problems (rel- evant aspects are reviewed below), but at least con- ceptually it is clear how charge gaps are to be defined and quantified within modern electronic-structure meth- ods, such as density-functional theory (dft).2,3,4 on the other hand, much less is is known about how to calculate, or even define, spin gaps. in the present paper we show how to define and cal- culate the spin gap in spin-dft (sdft), and predict that such calculations will encounter a spin-gap problem similar to the band-gap problem familiar from applica- tions of dft to semiconductors or to strongly correlated systems. section ii of this paper is devoted to charge gaps. in sec. ii a we recapitulate the conceptual difference be- tween fundamental gaps and excitation gaps. in sec. ii b we then recall the quantitative definition of the funda- mental gap and related quantities, such as the single- particle gap, and particle addition and removal energies. section ii c summarizes key aspects of the derivative dis- continuity, while sec. ii d describes the connection be- tween gaps and discontinuities within the framework of ensemble dft. although the final results of these sec- tions are well known, our treatment is different from the usual one in so far as we introduce many-body corrections to the gap and derivative discontinuities in completely independent ways, related only a posteriori via ensemble dft. this way of proceeding is useful for performing the generalization to the spin case. for both the fundamental gap and the optical exci- tation gap, the gapped degree of freedom is related to particles: either particles are added to or removed from the system, or particles are excited to higher energy lev- els within the system under study. in ordinary atoms, molecules and solids, these particles are electrons, and the particle gaps of many-electron systems are a key property in determining the functionality of today's elec- tronic devices. the last decade has witnessed an enormous growth of interest in another type of system, and in devices result- ing from them, in which the key degree of freedom is the spin. in the resulting field of spintronics, and the develop- ment of spintronic devices, one is interested in controlling and manipulating the spin degrees of freedom indepen- dently of, or in addition to, the charge degrees of freedom. here, the issue of the spin gap arises, and a number of questions for electronic-structure and many-body theory appear: what is the energy required to add a spin to the system? what is the energy cost of flipping a spin? how do these concepts differ from the fundamental and optical gaps involving particles? can we calculate spin gaps from spin-density-functional theory, and if yes, what type of exchange-correlation (xc) functional is required? 2 in sec. iii, we answer these questions. in sec. iii a we contrast spin gaps with charge gaps, and in sec. iii b we propose a set of many-body and single-particle definitions for quantities related to the spin gap, such as spin-flip energies and the spin stiffness. we take care to ensure that all quantities appearing in our definitions can, in principle, be calculated from con- ventional sdft or time-dependent sdft (tdsdft), and try to make the definitions in the spin case as analo- gous as possible to the charge case. however, this analogy can only be carried up to a certain point, and important differences between charge gaps and spin gaps emerge al- ready at this level. as a simple example, we consider, in sec. iii c, the lithium atom, for which we confront calculated and experimental spin gaps. in sec. iii d we then use ensembles dft to relate the spin gap to a derivative discontinuity that is simi- lar to, but distinct from, the one usually considered in the charge case. finally, in sec. iii f, we investigate the connection to excitation gaps calculated from tdsdft. equations are given that allow one to extract the var- ious spin gaps and related quantities from noncollinear tdsdft calculations. for illustrative purposes we eval- uate these for the lithium atom, and compare the gaps and discontinuities obtained from time-dependent dft to those obtained in sec. iii c from time-independent considerations. sec. iv contains our conclusions. ii. charge gap a. fundamental gaps vs. excitation gaps to provide the background for this investigation, let us first briefly recapitulate pertinent aspects of charge (or particle1) gaps. while by definition all gaps involve energy differences between a lower-lying state (in practice often the ground state) and a state of higher energy, important differences depend on how the extra energy is added to the system and what degrees of freedom absorb it. therefore, differ- ent notions of gap are appropriate for different purposes. for processes in which particles are added to or removed from the system, which is subsequently allowed to relax to the ground state appropriate to the new particle num- ber, the key quantity is the fundamental gap (sometimes also called the quasiparticle gap) which is calculated from differences of ground-state energies of systems with dif- ferent particle number. as such, it is relevant for instance in transport phenomena and electron-transfer reactions. if energy is added by means of radiation, on the other hand, the particle number does not change, and the rele- vant gap is an excitation energy of the n-particle system. this excitation gap (sometimes also called the optical gap), is relevant, e.g., in spectroscopy. in first-principles electronic-structure calculations, ex- citation gaps are today often calculated from time- dependent density-functional theory (tddft). funda- mental gaps, on the other hand, involve ground-state en- ergies of systems with different particle numbers, and should thus, in principle, be accessible by means of static (ground-state) dft. however, it is well known that com- mon approximations to dft encounter difficulties in this regard. in semiconductors, for example, calculated fun- damental gaps are often greatly underestimated relative to experiment, and in strongly-correlated systems such as transition-metal oxides, gapped materials are frequently incorrectly predicted to be metallic, i.e., to have no gap at all. the resulting band-gap problem of dft has been intensely studied for many decades. a major breakthrough in this field was the discov- ery of the derivative discontinuity of the exact exchange- correlation (xc) functional of dft, which was shown to account for the difference between the gap obtained from solving the single-particle kohn-sham (ks) equations of dft, and the true fundamental gap.5,6,7 the problems occurring in practice for semiconductors and strongly- correlated systems are therefore attributed to the fact that common local and semilocal approximations to the exact xc functional do not have such a discontinuity. the development of dft-based methods allowing to nonem- pirically predict the presence and size of gaps in many- electron systems continues to be a key issue of electronic- structure theory and computational materials science. b. definition of fundamental charge gaps the fundamental charge gap eg is defined as the dif- ference eg(n) = i(n) −a(n), (1) where the electron affinity (energy gained by bringing in a particle from infinity) and ionization energy (energy it costs to remove a particle to infinity) are defined in terms of ground-state energies of the n-particle system, as a(n) = e(n) −e(n + 1) (2) i(n) = e(n −1) −e(n). (3) the order of terms in these differences is the conventional choice. the definition of the fundamental gap is in terms of processes involving addition and removal of charge and spin. the change in the respective quantum numbers is ±1 in n, and ±1/2 in s. in chemistry,3 the average of i and a is identified with the electronegativity of the n-particle system: (i(n) + a(n))/2 = χ(n). the corresponding kohn-sham gap is defined analo- gously as eg,ks(n) = iks(n) −aks(n), (4) where iks(n) = eks(n −1)−eks(n) and aks(n) = eks(n) −eks(n + 1). since the ks total energy is simply the sum of the ks eigenvalues, eks = pn k=1 ǫk, 3 this reduces to iks(n) = −ǫn(n) and aks(n) = −ǫn+1(n), from which one obtains the usual form eg,ks(n) = ǫn+1(n) −ǫn(n), (5) where ǫn(n) and ǫn+1(n) are the highest occupied and the lowest unoccupied state of the n-particle system, re- spectively. the fundamental gap can also be written in terms of ks eigenvalues by means of the ionization-potential theo- rem (sometimes known as koopmans' theorem of dft), which states i(n) = −ǫn(n) (6) a(n) = i(n + 1) = −ǫn+1(n + 1), (7) so that i(n) ≡iks(n), and eg(n) = ǫn+1(n + 1) −ǫn(n). (8) note that in contrast with the ks gap (5) these eigen- values pertain to different systems. the relation between both gaps is established by rewriting the fundamental gap as eg = eg,ks + ∆xc, (9) which defines ∆xc as the xc correction to the single- particle gap. by making use of the previous relations we can cast ∆xc as8,9,10,11 ∆xc = ǫn+1(n +1)−ǫn+1(n) = aks(n)−a(n). (10) the important thing to notice in these expressions is that, due to protection by koopmans' theorem, the ion- ization energy does not contribute to the xc correction ∆xc, so that the correction of the affinity and of the fun- damental gap are one and the same quantity. also, note that all of these definitions can be made without any recourse to ensemble dft and without any mention of derivative discontinuities. c. nonuniqueness and derivative discontinuities the basic euler equations of dft is2,3,4 δe[n] δn(r) = μ. (11) since e[n] = f[n] + r d3r vext(r)n(r) and eks[n] = ts[n] + r d3r vs(r)n(r), this implies δf[n] δn(r) = μ −vext(r) (12) and δts[n] δn(r) = μ −vs(r), (13) where ts[n] is the noninteracting kinetic energy func- tional, f[n] = t [n] + u[n] is the internal energy func- tional, expressed in terms of the interacting kinetic en- ergy t [n] and the interaction energy u[n], μ is the chem- ical potential, vext(r) is the external potential and vs(r) the effective ks potential. both the effective and the external potential are only defined up to a constant, which does not change the form of the eigenfunctions. consider now a gapped open sys- tem, connected to a particle reservoir with fixed chemical potential initially in the gap, and gradually change the constant. as long as the change is sufficiently small, the chemical potential remains in the gap, the density n(r) does not change, and the derivatives on the left-hand side of eqs. (12) and (13) change continuously however, once the change in the constant is large enough to affect the number of occupied levels, the sit- uation changes: as soon as a new level falls below the chemical potential, or emerges above it, the number of particles in the system changes discontinuously by an in- teger, and the chemical potential adjusts itself to the new total particle number. for later convenience we call the two values of μ on the left and the right of integer particle number μ−and μ+, respectively. when the right-hand side of eqs. (12) and (13) changes discontinuously, the left-hand side must also change discontinuously. this means that the functional derivatives of f[n] and ts[n] change discontinuously for variations δn(r) such that n passes through an integer, and are not defined precisely at the integer. we can also argue conversely that if the functional derivatives existed at all n(r) they would determine the potentials uniquely. since the potentials are unique only up to a constant, the functional derivatives cannot exist for the density varia- tions δn arising from changing the potential by a con- stant. in a gapped system, these are the δn integrating to an integer. either way, we see that the indeterminacy of the po- tentials with respect to a constant implies that the func- tionals f[n] and ts[n] display derivative discontinuities for certain directions in density space along which the total particle number changes by an integer. this is the famous integer discontinuity of dft.5,6,7 d. connection of discontinuities and gaps: ensemble dft up to this point we have defined ∆xc as a many-body correction to the single-particle gap, and deduced the existence of derivative discontinuities from noting the nonuniqueness of the external potentials with respect to a constant. these two conceptually distinct phenomena are related by ensemble dft for systems with fractional particle number, describing open systems in contact with a particle reservoir.5,6,7 for such systems ensemble dft guarantees that the ground-state energy as a function of particle number, e(n), is a set of straight lines connect- 4 ing values at integer particle numbers. for straight lines, the derivative at any n can be ob- tained from the values at the endpoints: −a = e(n+1)−e(n) = ∂e ∂n     n+δn = μ+ = δe δn(r)     n+δn (14) and −i = e(n)−e(n−1) = ∂e ∂n     n−δn = μ−= δe δn(r)     n−δn . (15) the many-body fundamental gap is thus the derivative discontinuity of the total energy across densities integrat- ing to an integer: eg = i −a = δe[n] δn(r)     n+δn −δe[n] δn(r)     n−δn . (16) this energy functional is commonly written as e = ts + v + eh + exc, where the external potential energy v and the hartree energy eh are manifestly continuous functionals of the density. hence, the energy gap reduces to the sum of the discontinuity of the noninteracting ki- netic energy ts and that of the xc energy exc. the entire argument up to this point can be repeated for a noninteracting system in external potential vs. the energy of this system is eks = ts+vs, of which only the first term can be discontinuous. hence the fundamental gap of the ks system is given by the discontinuity of ts eg,ks = δts[n] δn(r)     n+δn −δts[n] δn(r)     n−δn . (17) returning now to the many-body gap, written as the sum of the discontinuities of ts and exc, we arrive at eg = eg,ks + δexc[n] δn(r)     n+δn −δexc[n] δn(r)     n−δn , (18) or, by means of eq. (9), ∆xc = δexc[n] δn(r)     n+δn −δexc[n] δn(r)     n−δn . (19) this identifies the xc correction to the single-particle gap as the derivative discontinuity arising from the nonuniqueness of the potentials with respect to an ad- ditive constant.5,6,7 importantly, this connection is not required to define the xc corrections and neither is its existence enough to conclude that these corrections are nonzero. many-body corrections to the single-particle gap can be defined inde- pendently of any particular property of the density func- tional (or even without using any density-functional the- ory), and whether for a given system these corrections are nonzero or not depends on the electronic structure of that particular system, and does not follow from the for- mal possibility of a derivative discontinuity, because this discontinuity itself might be zero. thus, the question of the existence and size of xc corrections to the charge gap must be asked for each system anew. as we will see in the next section, the same is true for the spin gap. iii. spin gap we have provided the above rather detailed summary of the definition of the fundamental charge gap and its connection to nonuniqueness and to derivative disconti- nuities to prepare the ground for the following discussion of the spin gap. in order to arrive at a consistent dft definition of spin gaps, we follow the steps outlined in the charge case: (i) define appropriate gaps and their xc corrections, (ii) use the nonuniqueness of the sdft potentials to show the existence of spin derivative dis- continuities, and (iii) identify a suitable spin ensemble to connect the two. a. spin gap vs. charge gap to introduce a spin gap or a spin-flip energy (see be- low for precise definitions) we consider processes in which only the total spin of the system is changed, while the particle number remains the same. there cannot be any definition in terms of particle addition and removal en- ergies, since in these processes the charge changes, too, which is not what one wants the spin gap to describe. in other words, the change of quantum numbers related to a spin flip is ±1 for the spin and 0 for charge. note that this is an excitation energy, where the excitation takes place under the constraints of constant particle number and change of total spin by one unit. this is the key difference to the previous section, from which all other differences follow. b. definition of spin gaps: spin-flip energies and spin stiffness first, we define the spin up-flip energy and the spin down-flip energy in terms of many-body energies as esf+(n) = e(n, s + 1) −e(n, s) (20) esf−(n) = e(n, s −1) −e(n, s). (21) here e(n, s) is the lowest energy in the n-particle spin- s subspace, where s is the eigenvalue of the z-component of the total spin, and we assumed that spin-up and spin- down are good quantum numbers. this implies, in par- ticular, that spin-orbit coupling is excluded from our analysis. (of course these definitions only apply if the respective flips are actually possible; in other words, if s does not yet have the maximal or minimal value for a given n.) the differences esf+(n) and esf−(n) are similar to the concepts of affinity and ionization energy, eqs. (2) and (3). however, affinities and ionization energies are always defined with the smaller value (of n) as the first term in the differences, whereas spin-flip energies are con- ventionally defined as final state minus initial state, i.e. 5 dos   sc ks g e , f  n  n ks g sc ks g e e , ,    sc ks g e ,  f dos 0 , ,    ks g sc ks g e e  sf ks e  sf ks e  sf ks e  sf ks e b a. b b.  n  n fig. 1: left: spin-resolved single-particle (ks) density-of- states of a spin-polarized insulator. two spin-flip energies and two spin-conserving gaps can be defined. right: the half-metallic ferromagnet is a special case in which the gap in one spin channel (say spin up) is zero. in this case, there is only one spin-conserving gap, equal to the sum of both spin- flip energies, esf− ks + esf+ ks = esc s,ks, and the ks charge gap is zero, due to the presence of the gapless spin down channel. figure courtesy of daniel vieira. both spin-flip energies measure an energy cost. there- fore, the down-flip is the spin counterpart to the ioniza- tion energy, while the up-flip is the spin counterpart to minus the electron affinity. a more important difference is that the spin-flip en- ergies involve excited-state energies e(n, s + 1) and e(n, s −1) of the n-particle system, instead of ground- state energies, and in this sense are more similar to the optical gap in the charge case than to the quantities used in evaluating the fundamental gap. alternatively, these energies can also be considered ground-state energies for sectors of hilbert space restricted to a given total s, but we will not make use of this alternative interpretation in the following. ks spin-flip energies are related analogously to single- particle eigenvalues, according to esf+ ks = ǫl(↑) −ǫh(↓) (22) esf− ks = ǫl(↓) −ǫh(↑), (23) where all energies are calculated at the same n, s. here l(σ) means the lowest unoccupied spin σ state, and h(σ) means the highest occupied spin σ state. similarly, l(σ) and h(σ) denote the lowest occupied spin σ state and h(σ) the highest unoccupied spin σ state, respectively. this notation is nonstandard, but helpful, and further illustrated in fig. 1. in the same way, we can also define the spin conserving (sc) single-particle gaps in each spin channel, as esc,↑ g,ks = ǫl(↑) −ǫh(↑) (24) esc,↓ g,ks = ǫl(↓) −ǫh(↓). (25) the spin-conserving gaps and the spin-flip energies are necessarily related by esc↑ g,ks + esc↓ g,ks = esf− ks + esf+ ks . a look at fig. 1 clarifies these definitions. if the system is non spin polarized, both spin-flip energies and both spin- conserving gaps become equal to the ordinary ks charge gap, which in our present notation reads ǫl −ǫh. in the same way as for charge gaps, we can now also consider the sum and the difference of the spin-flip ener- gies. the sum es = esf−+ esf+ (26) of the energies it costs to flip a spin up and a spin down is formally analogous to the fundamental gap (1), but with the important difference that es involves excited- state energies. the unusual sign (sum instead of differ- ence) arises simply because both spin-flip energies mea- sure costs, whereas the affinity featured in eq. (1) mea- sures an energy gain. the formal analogy to eq. (1) suggests that the quan- tity defined in eq. (26) be called the fundamental spin gap. in practice, however, the name spin gap is more appropriately applied to the individual spin-flip energies. the physical interpretation of their sum, eq. (26), is re- vealed by expressing it in terms of the many-body ener- gies by means of eqs. (20) and (21): es = e(n, s + 1) + e(n, s −1) −2e(n, s). (27) this is of the form of a discretized second derivative ∂2e(n, s)/∂s2, which identifies es as the discretized spin stiffness [we anticipated this interpretation when at- taching a subscript s for stiffness to the sum in eq. (26)]. we note that half of i−a is known in quantum chemistry as chemical hardness, which conveys a very similar idea as stiffness. generically, we refer to all three quantities esf−, esf+ and es as spin gaps. the spin electronegativity can be defined as half of the difference of the spin-flip energies, χs = (esf−−esf+)/2. this quantity has the following interpretation: if χs > 0, it costs less energy to flip a spin up than to flip a spin down, whereas if χs < 0 the down flip is energetically cheaper. the ks spin stiffness is defined as the sum of ks spin- flip energies, es,ks = esf− ks + esf+ ks , (28) or, with eqs. (22) and (23), es,ks = [ǫl(↓) −ǫh(↑)] + [ǫl(↑) −ǫh(↓)]. (29) this is analogous to (5), except that in spin flips nothing is removed to infinity or brought in from infinity. thus, differently from the ks ionization energy and electron affinity, the spin-flip energies require two single-particle energies for their definition instead of one, and in contrast with the ks charge gap the ks spin stiffness requires four single-particle energies instead of two. this missing analogy is physically meaningful: con- ventional gaps are defined in terms of particle addition and removal processes and are ground-state properties. 6 table i: kohn-sham energy eigenvalues (in ev) for the lithium atom. the 1s ↑, 2s ↑and 1s ↓levels are oc- cupied. ks: energy eigenvalues obtained by inversion from quasi-exact densities. xx denotes exact exchange,12 and kli is the krieger-li-iafrate approximation.14 ksa ksb xxa kli-xx lsda −ǫ1s↑ 55.97 58.64 55.94 56.64 51.02 −ǫ2s↑ 5.39 5.39 5.34 5.34 3.16 −ǫ2p↑ 3.54 - 3.48 3.50 1.34 −ǫ1s↓ 64.41 64.41 67.18 67.14 50.81 −ǫ2s↓ 8.16 5.87 8.25 8.23 2.09 aref. 12 bref. 13 to define pure spin gaps (i.e., spin-flip energies and spin stiffness) in which the charge does not change, we cannot make use of particle addition and removal processes but have to use spin flip processes instead. however, spin flips are excitation energies, and we must specify both initial and final states to define them properly. we also note that the many-body spin stiffness has no simple expression in terms of eigenvalues which would be analogous to eq. (8). such an expression would require the spin counterpart to koopmans' theorem i(n) ≡ iks(n), which is not available for spin-flip energies. hence, in general both spin-flip energies esf−and esf+ may be individually different from their ks counterparts esf− ks and esf+ ks : esf−= esf− ks + ∆sf− xc (30) esf+ = esf+ ks + ∆sf+ xc . (31) we can, moreover, establish a relation between the many- body spin stiffness and the ks spin stiffness by rewriting the former as es = es,ks + ∆s xc = esf− ks + esf+ ks + ∆s xc, (32) which defines ∆s xc as the xc correction to the ks spin stiffness. the important thing to notice in eq. (32) is that there is no reason to attribute ∆s xc only to the up- flip energy. this is a key difference to the charge case, where i(n) ≡iks(n) and the xc correction could thus be attributed only to the electron affinity. rather, the spin-flip corrections are connected by ∆s xc = ∆sf− xc + ∆sf+ xc . (33) c. example: the li atom to give an explicit example of the quantities intro- duced in the previous section, we now consider the li atom. for this system, ks eigenvalues ǫh↑= ǫ2s↑, ǫl↑= ǫ2p↑, ǫl↓= ǫ2s↓and ǫh↓= ǫ1s↓have been obtained table ii: single-particle spin-flip energies (34) and (35) and spin stiffness (29), their experimental (exp) counterparts, (20), (21) and (26), and the resulting xc corrections defined in (30), (31) and (32), for the lithium atom. in the columns labelled ks we employ ks eigenvalues obtained from near- exact densities, while in the columns labelled xx, kli-xx and lsda we use approximate eigenvalues obtained from standard sdft calculations. the experimental values were obtained using spectroscopic data for the lowest quartet state 4p 0 from ref. 15 as well as accurate wave-function based the- ory from ref. 16. all values are in ev. ksa ksb xxa kli-xx lsda exp esf+ 60.87 60.87 63.70 63.64 49.47 57.41 esf− −2.77 −0.48 −2.91 −2.89 1.07 0 es 58.10 60.39 60.79 60.75 50.54 57.41 ∆sf+ xc −3.46 −3.46 −6.29 −6.23 7.94 ∆sf− xc 2.77 0.48 2.91 2.89 −1.07 ∆s xc −0.69 −2.98 −3.38 −3.34 6.87 aref. 12 bref. 13, taking −ǫ2p↑= 3.54 ev from ref. 12 by numerical inversion of the ks equation starting from near-exact densities (see table i).12,13 the ks spin-flip energies are obtained as esf+ ks = ǫ2p↑−ǫ1s↓ (34) esf− ks = ǫ2s↓−ǫ2s↑. (35) they are given in table ii, together with the spin stiff- ness es, see eq. (26). table ii also presents the cor- responding experimental many-body energy differences for the li atom, which were obtained using spectroscopic data for the lowest quartet state 4p 0 and accurate wave- function based theory.15,16 relativistic effects and other small corrections included in the experimental data are ignored since they are too small on the scale of energies we are interested in. table ii also gives the xc corrections to the single- particle spin flip energies, see eqs. (30) and (31), and the xc correction to the spin stiffness, ∆s xc, see eq. (32). as a consistency test we verified that relation (33), which connects the xc corrections of the spin-flip energies to the xc corrections of the spin stiffness, is satisfied. we also carried out calculations using the exact- exchange (xx) eigenvalues of ref. 12 in order to sepa- rately assess the size of exchange and correlation effects. the resulting value of ∆s x = −3.38 ev indicates a larger (more negative) correction than in the calculation includ- ing correlation. an approximate kli-xx calculation14 yields very similar results, while the lsda data are com- pletely different and do not even reproduce the correct sign. three of the required "exact" ks single-particle eigen- values are also reported in ref. 13 (we use the result of ref. 12 for the missing value of ǫ2p↑). the value of ǫ2s↓is 7 quite different than the value reported in ref. 12 (−5.87 ev versus −8.16 ev), and consequently we obtain a rather different value of ∆s xc (−2.98 ev versus −0.69 ev). nev- ertheless, both sets of data sustain our main conclusions in this section: (i) simple lsda calculations give rise to serious qual- itative errors. as can be seen from table ii, one obtains spin-flip energies that are drastically to small (esf+) or have the wrong sign (esf−). the resulting xc corrections also suffer from having the wrong sign. these shortcom- ings of the lsda are hardly surprising in view of its well-established failure to describe the charge gap. (ii) even the precise ks eigenvalues do not predict the exact spin flip energies and spin stiffness, i.e. the xc cor- rections introduced in sec. iii b on purely formal grounds are indeed nonzero. the absolute size of these corrections implies that a simple ks eigenvalue calculation of spin gaps can be seriously in error. (iii) exchange-only calculations overestimate (in mod- ulus) the size of the gap corrections. this implies that there is substantial cancellation between the exchange and the correlation contribution to the full correction. this is the same trend known for charge gaps. (iv) the xc corrections to both the up-flip energy and the spin stiffness turn out to be negative; in other words the ks calculation overestimates these quantities. this is the opposite of what occurs in the case of the funda- mental charge gap, which is underestimated by the ks calculation. we note that hints of an underestimation of the experimental spin-flip energies by ks eigenvalue differences have also been observed for half-metallic fer- romagnets. in the case of cro2, for example, ref. 17 reports experimental spin-flip energies in the range 0.06 to 0.25 ev and compiles sdft predictions that range from 0.2 to 0.7 ev (and in one case even 1.7 ev). d. nonuniqueness and derivative discontinuities in sdft above we pointed out that the effective and external potentials of dft are determined by the ground-state density up to an additive constant. however, this state- ment only holds when one formulates dft exclusively in terms of the charge density, as we have done in dis- cussing charge gaps. it does not hold when one works with spin densities, as in sdft, or current densities, as in current-dft (cdft). in these cases the densities still determine the wave function, but they do not uniquely determine the corre- sponding potentials. a first example of this nonunique- ness problem of generalized dfts was already encoun- tered in early work on sdft, for the single-particle ks hamiltonian.18 later, this observation was extended to the sdft many-body hamiltonian,19,20 and further ex- amples were obtained in cdft21 and dft on lattices.22 nonuniqueness is a generic feature of generalized (mul- tidensity) dfts, consequences of which are still under investigation.23,24,25,26,27,28 in particular, refs. 19 and 20 already point out that the nonuniqueness of the po- tentials of sdft implies that the sdft functionals can have additional derivative discontinuities, because, if the functional derivatives of f and ts in multi-density dfts such as sdft and cdft existed for all densities, they would determine the corresponding potentials uniquely. very recently, g ́ al and collaborators28 pointed out that one-sided derivatives may still exist, and explored con- sequences of this for the dft description of chemical reactivity indices. just as in the charge case, derivative discontinuities re- sult from the nonuniqueness of the spin-dependent poten- tials, while corrections to single-particle gaps result from the auxiliary nature of kohn-sham eigenvalues. in the charge case, both distinct phenomena could be connected by means of ensemble dft for systems of fractional par- ticle number. the question then arises if a similar con- nection can also be established in the spin case. this requires an investigation of spin-ensembles. e. spin ensembles consequences of the nonuniqueness of the potentials of sdft for the calculation of spin gaps were already hinted at in refs. 19 and 20, where it was pointed out that there may be a spin-gap problem in sdft similarly to the well known band-gap problem of dft. to make these hints more precise, we first recall, from the above, that the quantity usually called the spin gap is actually what we here called the spin-flip gap, and is analogous to the ionization energy or the electron affin- ity in the charge case, not to the fundamental particle gap. the spin-dependent quantity that is most analo- gous to the fundamental particle gap is the discretized spin stiffness of eqs. (26) and (27). however, regardless of whether one focuses on the spin-flip energies or on the spin stiffness, the spin situation is not completely anal- ogous to the charge situation because both the spin-flip gaps and the spin stiffness are defined in terms of excited states of an n-particle hamiltonian, while charge gaps are defined in terms of ground-state energies of hamilto- nians with different particle numbers. to identify a suitable ensemble, we write the energy associated with a generic ensemble of two systems, a and b, as ew = (1 −w)ea + web, (36) where 0 ≤w ≤1 is the ensemble weight. if a and b have different particle numbers, na and nb = na ± 1, this becomes the usual fractional-particle number ensemble, which is unsuitable for our present investigation where the involved systems differ in the spin but not the charge quantum numbers. a spin-dependent ensemble was recently constructed by yang and collaborators29,30 in order to understand the static correlation error of common density functionals. in 8 this spin ensemble, a and b have different (possibly frac- tional) spin, but are degenerate in energy. the constancy condition, whose importance and utility is stressed in refs. 29,30, arises directly from the restriction of the en- semble to degenerate states. while useful for the pur- poses of analyzing the static correlation error, this spin ensemble is too restrictive for our purposes, as it excludes the excited states involved in the definition of spin-flip gaps and of the spin stiffness. ensembles involving excited states have been employed in dft in connection with the calculation of excitation energies.31,32,33 here a and b differ in energy but stem from the same hamiltonian, with fixed particle number. excited-state ensemble theory leads to a simple expres- sion relating the first excitation energy to a ks eigenvalue difference32 eb −ea = ǫw m+1 −ǫw m + ∂ew xc[n] ∂w     n=nw , (37) where eb and ea are the energies of first excited and the ground state of the many-body system, respectively, ǫw m+1 and ǫw m are the highest occupied and lowest unoc- cupied ks eigenvalues, and ew xc is the ensemble xc func- tional. equation (37) holds for ensemble weights in the range 0 ≤w ≤1/2. levy showed34 that the last term in this equation is related to a derivative discontinuity according to ∂ew xc[n] ∂w     n=nw = δew=0 xc [n] δn(r)     n=nw=0 −δew xc[n] δn(r)     n=nw (38) for w →0. here nw = (1 −w)na + wnb is the ensemble density, and the discontinuity arises because even in the w →0 limit the ensemble density does contain an admix- ture of the state b with energy eb > ea and thus decays differently from n0 as r →∞.34 levy developed his ar- gument explicitly only for the spin-unpolarized case, but already pointed out in the original paper that the results carry over to spin-polarized situations. in our case, we take a to be the ground state and b to be the lowest-lying state differing from it by a spin flip. to be specific, let us assume that the spin is flipped up. in this case we obtain from eqs. (37) and (38) in the limit w →0, and using our present notation, esf+(n) = e(n, s + 1) −e(n, s) (39) = ǫw l(↑) −ǫw h(↓) + ∂ew xc[n↑, n↓] ∂w     n↑=nw ↑ n↓=nw ↓ (40) = ǫw l(↑) −ǫw h(↓) + δew=0 xc [n↑, n↓] δn↓(r)     n↑=nw=0 ↑ n↓=nw=0 ↓ −δew xc[n↑, n↓] δn↓(r)     n↑=nw ↑ n↓=nw ↓ (41) = esf+ w,ks(n) + ∆sf+ w,xc (42) for w →0. equation (42), which is the ensemble ver- sion of our eq. (31), illustrates that ks spin-flip excita- tions, too, acquire a many-body correction arising from a derivative discontinuity. in the particular case in which the spin flip costs no energy in the many-body and in the ks system, the pre- ceding equation reduces to ∆sf+ w,xc = 0, which is the con- stancy condition derived in refs. 29,30 for spin ensembles of degenerate states. we note that the ks eigenvalues and the discontinu- ity in eqs. (40) to (42) must be evaluated by taking the w →0 limit of the w-dependent quantities, while the quantities in eq. (31) have no ensemble dependence. this complicates the evaluation of spin-flip energies and their discontinuities, as defined in sec. iii b, from en- semble dft. therefore, we turn to still another density- functional approach to excited states in order to evaluate these quantities: tddft. f. connection to tddft tddft has established itself as the method of choice for calculating excitation energies in atomic and molec- ular systems, and is making rapid progress in nanoscale systems and solids as well.35,36 in this section we will make a connection between the preceding discussion and tddft, which will allow us to derive simple approxima- tions for the xc corrections to the single-particle spin-flip excitation energies and the spin stiffness. to calculate the spin-conserving and the spin-flip exci- tation energies, it is necessary to use a noncollinear spin- density response theory, even if the system under study has a ground state with collinear spins (i.e., spin-up and -down with respect to the z axis are good quantum num- bers). in this way the spin-up and spin-down density responses can become coupled, and the description of spin-flip excitations (for instance, due to a transverse magnetic perturbation) becomes possible. in tddft, the spin-conserving and the spin-flip excitation energies can be obtained from the following eigenvalue equations, which are a generalization of the widely used casida equations37 for systems with noncollinear spin:38 x σσ′ x i′a′ nh δi′iδa′aδσαδσ′α′ωa′σ′i′σ + kαα′,σσ′ iαaα′,i′σa′σ′ i xi′σa′σ′ + kαα′,σ′σ iαaα′,i′σa′σ′yi′σ,a′σ′ o = −ωxiα,aα′ (43) 9 x σσ′ x i′a′ n kα′α,σσ′ iαaα′,i′σa′σ′xi′σa′σ′ + h δa′aδi′iδσ′α′δσαωa′σ′i′σ + kα′α,σ′σ iαaα′,i′σa′σ′ i yi′σ,a′σ′ o = ωyiα,aα′ , (44) where we use the standard convention that i, i and a, a′ are indices of occupied and unoccupied ks orbitals, re- spectively, and α′α′, σσ′ are spin indices, and ωa′σ′i′σ = ǫa′σ′ −ǫi′σ. choosing the ks orbitals to be real, without loss of generality, we have kαα′,σσ′ iαaα′,i′σa′σ′(ω) = z d3r z d3r′ ψiα(r)ψaα′(r) × f hxc αα′,σσ′(r, r′, ω)ψi′σ(r′)ψa′σ′(r′) . (45) here, the subscript indices of the matrix elements k refer to the ks orbitals in the integrand, and the superscript spin indices refer to the hartree-xc kernel f hxc α,α′,σσ′(r, r′, ω) = δαα′δσσ′ |r −r′| + f xc αα′,σσ′(r, r′, ω) , (46) where the frequency-dependent xc kernel is defined as the fourier transform of the time-dependent xc kernel f xc αα′,σσ′(r, t, r′, t′) = δvxc αα′(r, t) δnσσ′(r′, t′)     n(r,t)=n0(r) . (47) here, n(r, t) and n0(r) are the time-dependent and the ground-state 2×2 spin-density matrix, which follow from the dft formalism for noncollinear spins.38,39,40,41,42 eqs. (43) and (44) give, in principle, the exact spin- conserving and spin-flip excitation energies of the system, provided the exact ks orbitals and energy eigenvalues are known, as well as the exact functional form of f xc αα′,σσ′. we will now consider a simplified solution known as the single-pole approximation.43,44 it is obtained from the full system of equations (43), (44) by making the tamm- dancoffapproximation (i.e., neglecting the off-diagonals) and focusing only on the h(σ) →l(σ′) excitations. in other words, we need to solve the 4 × 4 problem x σσ′ h δσ′α′δσαωlσ′hσ + kα′α,σ′σ hαlα′,hσlσ′ i yhσ,lσ′ = ωyhα,lα′ . (48) for ground states with collinear spins, the only nonva- nishing elements of the hartree-xc kernel are f hxc ↑↑,↑↑, f hxc ↓↓,↓↓, f hxc ↑↑,↓↓, f hxc ↓↓,↑↑, f xc ↑↓,↑↓, f xc ↓↑,↓↑ (49) (notice that there is no hartree term in the spin-flip channel), and the spin-conserving and spin-flip excita- tion channels decouple into two separate 2 × 2 problems. for the spin-conserving case, we have det      ω↑↑−ωsc + m↑↑,↑↑ m↑↑,↓,↓ m↓↓,↑↑ ω↓↓−ωsc + m↓↓,↓↓      = 0 , (50) where we abbreviate mαα′,σσ′ = kα′α,σ′σ hαlα′,hσlσ′(ω) and ωσ′σ = ωlσ′,hσ = ǫlσ′ −ǫhσ. from this, we get the two spin-conserving excitation energies as esc↑,↓= ω↑↑+ ω↓↓+ m↑↑,↑↑+ m↓↓,↓↓ 2 ± h m↑↑,↓↓m↓↓,↑↑ + 1 4(ω↑↑−ω↓↓+ m↑↑,↑↑−m↓↓,↓↓)2i1/2 , (51) with the spin-conserving kohn-sham single-particle gaps esc↑(↓) g,ks = ω↑↑(↓↓). the two spin-flip excitations follow immediately as esf+ = ω↑↓+ m↑↓,↑↓ (52) esf−= ω↓↑+ m↓↑,↓↑, (53) where esf+ ks = ω↑↓and esf− ks = ω↓↑. this gives a simple approximation for the xc correction to the spin stiffness: ∆s xc = m↑↓,↑↓+ m↓↑,↓↑. (54) explicit expressions for f xc αα′,σσ′ can be obtained from the local spin-density approximation (lsda), and we list them here for completeness (see also wang and ziegler38): f xc ↑↑,↑↑= ∂2(neh xc) ∂n2 + 2(1 −ζ)∂2eh xc ∂n∂ζ + (1 −ζ)2 n ∂2eh xc ∂ζ2 f xc ↓↓,↓↓= ∂2(neh xc) ∂n2 −2(1 + ζ)∂2eh xc ∂n∂ζ + (1 + ζ)2 n ∂2eh xc ∂ζ2 f xc ↑↑,↓↓= ∂2(neh xc) ∂n2 −2ζ ∂2eh xc ∂n∂ζ −(1 −ζ2) n ∂2eh xc ∂ζ2 f xc ↑↓,↑↓= 2 nζ ∂eh xc(n, ζ) ∂ζ , (55) where f xc ↓↓,↑↑= f xc ↑↑,↓↓and f xc ↓↑,↓↑= f xc ↑↓,↑↓, and it is under- stood that all expressions are multiplied by δ(r −r′) and evaluated at the local ground-state density and spin po- larization, n0(r) = n0↑(r) + n0↓(r) and ζ0(r) = [n0↑(r) − n0↓(r)]/n0(r). for the xc energy density of the spin- polarized homogeneous electron gas we take the standard interpolation formula eh xc(n, ζ) = eh xc(n, 0) + (1 + ζ)4/3 + (1 −ζ)4/3 −2 24/3 −2 × [eh xc(n, 1) −eh xc(n, 0)]. (56) the case of exact exchange (xx) in linear response can be treated exactly, though with considerable technical and numerical effort.45,46 a simplified expression of the xx xc kernel was developed by petersilka et al.43, and we have generalized their expression for the linear response 10 table iii: top part: spin-conserving and spin-flip excita- tion energies, calculated with lsda and kli-xx using dif- ferences of ks eigenvalues and tddft in the single-pole ap- proximation (51)–(53). bottom part: tddft xc corrections to the ks spin-flip excitation energies, from eqs. (52), (53), and to the ks spin gap, eq. (54). all numbers are in ev. lsda kli-xx exact ks tddft ks tddft ksa expb esc↑ 1.83 2.00 1.84 2.01 1.85 1.85 esc↓ 48.72 48.89 58.90 59.31 56.25 56.36 esf+ 49.47 48.23 63.64 62.12 60.87 57.41 esf− 1.07 0.99 −2.89 −2.97 −2.77 0.0 ∆sf+ xc −1.24 −1.52 −3.46 ∆sf− xc −0.08 −0.07 +2.77 ∆s xc −1.32 −1.59 −0.69 aevaluated from the ks eigenvalues of ref. 12 bspectroscopic data from ref. 47 (esc↑,↓) and refs. 15,16 (esf+) of the spin-density matrix. we obtain (details will be published elsewhere): f x ↑↑,↑↑(r, r′) = − n↑ x i,k ψk↑(r)ψ∗ k↑(r′)ψ∗ i↑(r)ψi↑(r′) |r −r′|n↑(r)n↑(r′) (57) and similarly for f x ↓↓,↓↓(r, r′), and f x ↑↓,↑↓(r, r′) = − n↑,n↓ x i,k ψk↑(r)ψ∗ k↑(r′)ψ∗ i↓(r)ψi↓(r′) |r −r′| p n↑(r)n↓(r)n↑(r′)n↓(r′) = f x ↓↑,↓↑(r, r′). (58) here, n↑and n↓are the number of occupied spin-up and spin-down orbitals. we have evaluated eqs. (51)–(54) for the spin-con- serving and spin-flip excitation energies of the lithium atom involving the h(σ) and l(σ′) orbitals. the lsda and kli-xx orbital eigenvalues that are needed as input are given in table i. the associated excitation energies are shown in table iii, where we compare ks excitations, i.e., differences of ks eigenvalues, with tddft excitations obtained using the single-pole approximation described above. all in all, the tddft excitation energies are not much improved compared to the ks orbital eigenvalue differences. the main reason is that the lsda and kli-xx ks energy eigenvalues are not particularly close to the exact ks energy eigenvalues, and furthermore that the single-pole approximation is too simplistic for this open-shell atom. however, we observe that the xc correction ∆s xc to the spin stiffness es, when directly calculated within lsda or kli-xx using the tddft formula (54), is reasonably close to the exact value, and has the correct sign. this tells us that, even though the ks spin gap itself may be not very good, the simple tddft expression (54) gives a reasonable approximation for the xc correction to it. iv. conclusion the calculation of spin gaps and related quantities is important for phenomena like spin-flip excitations in fi- nite systems,38 the magnetic and transport properties of extended systems such as half-metallic ferromagnets17 and, quite generally, in the emerging field of spintronics and spin-dependent transport. our aim in this paper was to show how to define and calculate spin gaps and related quantities from density- functional theory. the proper definition of spin gaps in sdft is by no means obvious, and the straightforward extrapolation of concepts and properties from the charge case to the spin case is fraught with dangers. there- fore, we started our investigation by disentangling two aspects of the gap problem that in the charge case are usually treated together: the derivative discontinuity and the many-body correction to single-particle gaps. on this background, we then provided a set of dft- based definitions of quantities that are related to spin gaps, such as spin-conserving gaps, spin-flip gaps and the spin stiffness, pointing out in each case where pos- sible analogies to the charge case exist, and when these analogies break down. in particular, spin-flips involve excitations, while particle addition and removal involves ground-state energies. as a consequence, single-particle spin-flip energies involve two eigenvalues (and not one) and single-particle spin gaps involve four (and not two). moreover, each spin-flip energy may have its own xc cor- rection (there is no koopmans' theorem for spin flips). an evaluation of our definitions for the lithium atom, making use of highly precise kohn-sham eigenvalues and spectroscopic data, shows that the many-body correction to spin gaps can indeed be nonzero. in fact, unlike what is common in the charge case, this correction turns out to be negative, i.e. the single-particle calculation overes- timates the spin gap while it underestimates the charge gap. while this result for a single atom is consistent with available data on half-metallic ferromagnets,17 sim- ilar calculations must be performed for other systems be- fore broad trends can be identified. next, we connected the many-body corrections to the spin gap and related quantities to ensemble dft and to tddft. the former connection makes use of a suitable excited-state spin ensemble (different from the degenerate-state spin ensemble recently proposed by yang and collaborators29,30) and depends on a crucial in- sight of levy34 regarding excited-state derivative discon- tinuities. the latter connection employs a noncollinear version of the casida equations,38 which we evaluate, again for the lithium atom, within the single-pole ap- proximation, in lsda and for exact exchange. the development of approximate density functionals and computational methodologies that permit the reli- able calculation of spin gaps and related quantities, in- cluding their many-body (xc) corrections, remains a chal- lenge for the future. acknowledgments k.c. thanks the physics depart- 11 ment of the university of missouri-columbia, where part of this work was done, for generous hospitality, and d. vieira for preparing and discussing fig. 1. k.c. is supported by brazilian funding agencies fapesp and cnpq. c.a.u. acknowledges support from nsf grant no. dmr-0553485. c.a.u. would also like to thank the kitp santa barbara for its hospitality and partial support under nsf grant no. phy05-51164. g.v. ac- knowledges support from nsf grant no. dmr-0705460. 1 the expression 'charge gap' is actually a misnomer, as what is added, removed or excited is a particle (with charge and spin) not just charge. charge gap is the common ex- pression, however, and we will use it interchangeably with the more correct 'particle gap'. in one dimensional systems, on the other hand, it is important to distinguish charge and particle gaps, due to the possibility of spin-charge separa- tion. 2 r. m. dreizler and e. k. u. gross, density functional theory (springer, berlin, 1990). 3 r. g. parr and w. yang, density-functional theory of atoms and molecules (oxford university press, oxford, 1989). 4 g. f. giuliani and g. vignale, quantum theory of the electron liquid (cambridge university press, cambridge, 2005). 5 j. p. perdew, r. g. parr, m. levy, and j. l. balduz, phys. rev. lett. 49, 1691 (1982). 6 j. p. perdew and m. levy, phys. rev. lett. 51, 1884 (1983). 7 l. j. sham and m. schl ̈ uter, phys. rev. lett. 51, 1888 (1983). 8 l. j. sham and m. schl ̈ uter, phys. rev. b 32, 3883 (1985). 9 a. seidl, a. g ̈ orling, p. vogl, j. a. majewski, and m. levy, phys. rev. b 53, 3764 (1996). 10 k. capelle, m. borgh, k. karkkainen, and s.m. reimann phys. rev. lett. 99, 010402 (2007). 11 f. p. rosselli, a. b. f. da silva, and k. capelle, arxiv:physics/0611180v2 (2006). 12 j. chen, j. b. krieger, r. o. esquivel, m. j. stott, and g. j. iafrate, phys. rev. a 54, 1910 (1996). 13 o. v. gritsenko and e. j. baerends, j. chem. phys. 120, 8364 (2004). 14 j. b. krieger, y. li, and g. j. iafrate, phys. rev. a 45, 101 (1992). 15 s. mannervik and h. cederquist, phys. scr. 27, 175 (1983). 16 c. f. bunge and a. v. bunge, phys. rev. a 17, 816 (1978); c. f. bunge, j. phys. b: at. mol. opt. phys. 14, 1 (1981); j.-j. hsu, k. t. chung, and k.-n. huang, phys. rev. a 44, 5485 (1991). 17 j. m. d. coey and m. venkatesan, j. appl. phys. 91, 8345 (2002). 18 u. von barth and l. hedin, j. phys. c 5, 1629 (1972). 19 h. eschrig and w. e. pickett, solid state commun. 118, 123 (2001). 20 k. capelle and g. vignale, phys. rev. lett. 86, 5546 (2001). 21 k. capelle and g. vignale, phys. rev. b 65, 113106 (2002). 22 c. a. ullrich, phys. rev. b 72, 073102 (2005). 23 o. gritsenko and e. j. baerends, j. chem. phys. 120, 8364 (2004). 24 n. argaman and g. makov, phys. rev. b 66, 052413 (2002). 25 n.. i. gidopoulos, phys. rev. b 75, 134408 (2007). 26 w. kohn, a. savin, and c. a. ullrich, int. j. quantum chem. 100, 20 (2004). 27 k. capelle, c. a. ullrich, and g. vignale, phys. rev. a 76, 012508 (2007). 28 t. g ́ al, p. w. ayers, f. de proft, and p. geerlings, j. chem. phys. 131, 154114 (2009). 29 a. j. cohen, p. mori-s ́ anchez, and w. yang, science 321, 792 (2008). 30 p. mori-s ́ anchez, a. j. cohen, and w. yang, phys. rev. lett. 102, 066403 (2009). 31 a. k. theophilou, j. phys. c 12, 5419 (1979). 32 e. k. u. gross, l. n. oliveira, and w. kohn, phys. rev. a 37, 2805 (1988); ibid 2809; ibid 2821. 33 a. nagy, phys. rev. a 49, 3074 (1994); ibid 42, 4388 (1990). 34 m. levy, phys. rev. a 52, r4313 (1995). 35 time-dependent density functional theory, edited by m. a. l. marques, c. a. ullrich, f. nogueira, a. rubio, k. burke, and e. k. u. gross, lecture notes in physics 706 (springer, berlin, 2006). 36 p. elliott, k. burke, and f. furche, in recent advances in density functional methods 26, edited by k. b. lipkowitz and t. r. cundari (wiley, hoboken, nj, 2009), p. 91. 37 m. e. casida, in recent advances in density functional methods, edited by d. e. chong (world scientific, singa- pore, 1995), vol. 1, p. 155. 38 f. wang and t. ziegler, j. chem. phys. 121, 12191 (2004); j. chem. phys. 122, 074109 (2005); int. j. quant. chem. 106, 2545 (2006). 39 j. sticht, k. h. h ̈ ock, and j. k ̈ ubler, j. phys.: condens. matter 1, 8155 (1989). 40 l. m. sandratskii, adv. phys. 47, 91 (1998). 41 o. heinonen, j. m. kinaret, and m. d. johnson, phys. rev. b 59, 8073 (1999). 42 c. a. ullrich and m. e. flatt ́ e, phys. rev. b 66, 205305 (2002) and phys. rev. b 68, 235310 (2003). 43 m. petersilka, u. j. gossmann, and e. k. u. gross, phys. rev. lett. 76, 1212 (1996). 44 h. appel, e. k. u. gross, and k. burke, phys. rev. lett. 90, 043005 (2003). 45 y.-h. kim and a. g ̈ orling, phys. rev. lett. 89, 096402 (2002). 46 m. hellgren and u. von barth, phys. rev. b 78, 115107 (2008). 47 j. e. sansonetti, w. c. martin, and s. l. young, handbook of basic atomic spectroscopic data (national institute of standards and technology, gaithersburg, 2005).

0911.1714

adhesion of surfaces via particle adsorption: exact results for a lattice of fluid columns

we present here exact results for a one-dimensional gas, or fluid, of hard-sphere particles with attractive boundaries. the particles, which can exchange with a bulk reservoir, mediate an interaction between the boundaries. a two-dimensional lattice of such one-dimensional gas `columns' represents a discrete approximation of a three-dimensional gas of particles between two surfaces. the effective particle-mediated interaction potential of the boundaries, or surfaces, is calculated from the grand-canonical partition function of the one-dimensional gas of particles, which is an extension of the well-studied tonks gas. the effective interaction potential exhibits two minima. the first minimum at boundary contact reflects depletion interactions, while the second minimum at separations close to the particle diameter results from a single adsorbed particle that crosslinks the two boundaries. the second minimum is the global minimum for sufficiently large binding energies of the particles. interestingly, the effective adhesion energy corresponding to this minimum is maximal at intermediate concentrations of the particles.

introduction the interactions of surfaces are often affected by nanoparticles or macro- molecules in the surrounding medium. non-adhesive particles cause attrac- tive depletion interactions between the surfaces, since the excluded volume of the molecules depends on the surface separation [2, 3, 4]. adhesive parti- cles, on the other hand, can directly bind two surfaces together if the surface 1 separation is close to the particle diameter [5, 6, 7]. in a recent letter [8], we have presented a general, statistical-mechanical model for two surfaces in contact with adhesive particles. in this model, the space between the sur- faces is discretized into columns of the same diameter d as the particles. the approximation implied by this discretization is valid for small bulk volume fractions of the particles, since three-dimensional packing effects relevant at larger volume fractions are neglected. for short-ranged particle-surface in- teractions, the gas of particles between the surfaces is as dilute as in the bulk for large surface separations, except for the single adsorption layers of particles at the surfaces. in this article, we present an exact solution of the one-dimensional gas of hard-sphere particles in a single column between two 'surfaces'. our aim here is two-fold. first, the exact solution presented here corroborates our previ- ous, approximate solution for this one-dimensional gas obtained from a virial expansion in the particle concentration [8]. second, the exactly solvable, one-dimensional model considered here is a simple toy model to study the interplay of surface adhesion and particle adsorption. exactly solvable, one- dimensional models have played an important role in statistical mechanics [9, 10]. one example is the kac-baker model [11, 12, 13], which has shed light on the statistical origin of phase transitions of the classical van der waals type. more recent examples are models for one-dimensional interfaces, or strings, which have revealed the relevance of entropy and steric interactions in membrane unbinding and wetting transitions [14, 15, 16]. other examples are the tonks model [1] and its various generalizations [17, 18, 19, 20], which have influenced our understanding of the relations between short-ranged par- ticle interactions, thermodynamics, and statistical correlations in simple flu- ids. the tonks model has been exploited also in soft-matter physics to investigate structures of confined fluids [21, 22], depletion phenomena in two- component mixtures [23], thermal properties of columnar liquid crystals [24] and the phase behavior of polydisperse wormlike micelles [25]. a recent bio- physical modification of the tonks model addresses the wrapping of dna around histone proteins [26]. the model considered here is a novel extension of the tonks model. in our model, a one-dimensional gas of hard-sphere particles is attracted to the system boundaries, or 'surfaces', by short-ranged interactions. we calculate the effective, particle-mediated interaction potential between the surfaces, v , by explicit integration over the particles' degrees of freedom in the partition function. the potential v is a function of the surface separa- tion land exhibits a minimum at surface contact, which reflects depletion interactions, and a second minimum at separations close to the diameter of the adhesive particles. the effective, particle-mediated adhesion energy of 2 bulk b) a) l d r l r l d figure 1: a) a one-dimensional gas, or fluid, of hard-sphere particles in a column of the same diameter d as the particles. the interaction between the particles and the boundaries, or 'surfaces', is described by a square-well potential of depth u and range lr < d/2. a particle thus gains the binding energy u if its center is located at a distance smaller than lr+d/2 from one of the surfaces. the length lof the column corresponds to the separation of the surfaces. we consider the grand-canonical ensemble in which the particles in the column exchange with a bulk reservoir. - b) for small bulk volume fractions, a two-dimensional lattice of such columns represents a discrete approximation of a three-dimensional gas of particles between two surfaces [8]. since the particle-surface interactions are short-ranged in our model, the particle gas between the surfaces is as dilute as in the bulk for large surfaces separations, except for the adsorption layer of particles at each of the surfaces. the surfaces, w, can be determined from the interaction potential v . the adhesion energy is the minimal work that has to be performed to bring the surfaces apart from the equilibrium state corresponding to the deepest well of the potential v (l). interestingly, the adhesion energy w attains a maximum value at an optimal particle concentration in the bulk, and is considerably smaller both for lower and higher particle bulk concentrations. this article is organized as follows. in section 2, we introduce our model and define the thermodynamic quantities of interest. in section 3, we cal- culate the particle-mediated interaction potential v (l) of the surfaces. the global minimum of this potential is determined in section 4, and the effective adhesion energy of the surfaces in section 5. in section 6, we show that the interaction potential v (l) exhibits a barrier at surface separations slightly larger than the particle diameter, because a particle bound to one of the sur- 3 faces 'blocks' the binding of a second particle to the apposing surface. the particle binding probability is calculated and analyzed in section 7. 2 model and definitions we consider a one-dimensional gas of particles with attractive boundaries, see figure 1. the particles are modeled as hard spheres, and the attractive in- teraction between the particles and the boundaries, or 'surfaces', is described by a square-well potential with depth u and range lr. the length lof the gas 'column' corresponds to the separation of the surfaces and the width of the column is chosen to be equal to the particle diameter d. the particles in the column exchange with a bulk reservoir of particles. the position of the center of mass of particle k is denoted by xk, and its momentum by pk. for the system of n hard particles confined in the column of length l> nd, one has d/2 < x1, x1 < x2−d, x2 < x3−d, . . . xn < l−d/2. we assume that the 1-st and n-th particle interact with the surfaces, i.e. with the bases of the columns, via the square-well potential vn {xk} = −uθ d 2 + lr −x1  −uθ  xn −l+ lr + d 2  , (1) where u > 0 and lr > 0 are the potential depth and range, respectively. we also assume that lr < d/2. here and below, θ denotes the heaviside step function with θ(x) = 1 for x > 0 and θ(x) = 0 for x < 0. the configuration energy for the system of n particles in the column is hn {xk, pk} = vn {xk} + n x k=1 p2 k 2m (2) and the corresponding canonical partition function can be written as zn = 1 λn z l−d 2 (n−1 2)d dxn z xn−d (n−3 2)d dxn−1 . . . z x3−d 3 2d dx2 z x2−d 1 2 d dx1e−vn{xk}/t (3) after integration over the momenta of the particles, see, e.g., [1, 21, 26]. here, λ = h/(2πmt)1/2 is the thermal de broglie wavelength, and t denotes the temperature times the boltzmann constant. in other words, t is the basic energy scale. since the particles can exchange with the bulk solution, the number n of particles in the column is not constant. such a system is described by the grand-canonical ensemble in which the temperature, the column length 4 l, and the particle chemical potential μ are fixed. the corresponding grand- canonical partition function z = 1 + ⌊l/d⌋ x n=1 znenμ/t (4) is a sum of a finite number of elements, where ⌊l/d⌋denotes the largest integer less than or equal l/d. the upper limit of the sum on the right hand side of equation (4) is the largest number of hard particles with diameter d that can be accommodated in a column of length l. the partition function z given by equation (4) determines the grand potential fgc = −t ln z, (5) the bulk density of the grand potential fgc = lim l→∞ fgc d l (6) and, hence, the surface contribution to the grand potential f (s) gc = fgc − fgc l/d. the effective interaction potential of the surfaces 1, v = f (s) gc d2 = fgc d −fgc l d3 , (7) is defined as the density of the surface contribution to the grand potential fgc. for consistency with our previous model for particle-mediated surface interactions [8], the column bases are chosen here to be squares of side length d. thus the d2 in the denominator of equation (7) is the column base area. the surface potential v defined by equation (7) is the main quantity of interest here and will be determined in the next section. 3 effective surface interaction potential equations (4) - (7) imply that exp  −v d2 t  =  1 + ⌊l/d⌋ x n=1 znen μ/t  exp fgc l t d  . (8) 1the surface interaction potential v was called the 'effective adhesion potential' in reference [8]. 5 to determine the surface interaction potential v , we thus have to calculate the canonical partition function zn and the bulk density of the grand po- tential, fgc, defined in equation (6). the n-particle partition function zn is defined in equation (3). the change of variables yk = xk − k −1 2
d in equation (3), with k = 1, 2 . . ., n, leads to zn = 1 λn z l−nd 0 dyneuθ(yn−l+nd+lr)/t z yn 0 dyn−1 . . . z y3 0 dy2 z y2 0 dy1euθ(lr−y1)/t, (9) where u is the binding energy of the particles. the integral (9) can be evaluated, see a, and after some computation we arrive at zn = eu/t −1
2 φn (l−2lr) −2eu/t eu/t −1
φn (l−lr) + e2u/t φn (l) , (10) with φn (l0) = 1 n! l0 −nd λ n θ (l0 −nd) (11) for any length l0. for u = 0, equation (10), reduces to the partition function zn = 1 λnn! (l−nd)n θ (l−nd) (12) of the classical tonks gas [1]. the grand potential density fgc can be derived from this exact result as shown in the following subsection. 3.1 thermodynamic potentials in the bulk the canonical and grand-canonical ensembles are equivalent in the thermo- dynamic limit, i.e. for infinite surface separation l. in this limit, the grand potential density fgc therefore can be obtained from the canonical poten- tial density via legendre transformation. first, we define the canonical free energy fca = −t ln zn and the free energy density in the bulk, fca = lim ∞ fca d l , (13) where lim∞denotes the thermodynamic limit in which both the column length land the particle number n go to infinity while the particle 'volume fraction' φ = n d l (14) remains constant. the particle volume fraction in one dimension defined by equation (14) attains values 0 ≤φ ≤1, and could also be called a 'length 6 fraction'. in the one-dimensional model considered here, close packing cor- responds to φ = 1. the free energy density fca defined by equation (13) is an intensive quan- tity in the thermodynamic limit and, therefore, does not depend on the boundary conditions. in particular, the free energy density fca and its deriva- tives do not depend on the boundary potential (1), which is characterized by the binding energy u and range lr. with the exact expression for the canon- ical partition function zn given in equation (10) and the sterling formula ln(n!) ≈n ln n −n + 1 2 ln (2πn) , (15) we get fca = tφ  −1 + ln λ d φ 1 −φ  . (16) as expected, the particle-surface interactions characterized by the binding energy u and range lr do not affect the free energy density fca in the ther- modynamic limit. from the canonical free energy density fca, we obtain the chemical potential μ = ∂fca ∂φ  t (17) for the particles in the bulk. equations (16) and (17) lead to μ = t ln λ d φ 1 −φ  + t φ 1 −φ (18) which can be rewritten as eμ/t = λ d φ 1 −φ exp  φ 1 −φ  . (19) finally, the grand potential density fgc follows from the legendre transfor- mation fgc = fca −μ φ. (20) equations (16), (18) and (20) lead to fgc = −tφ 1 −φ, (21) with the dependence between the chemical potential μ and the particle bulk volume fraction φ given in equation (18). basic thermodynamics implies that the gas pressure in the bulk is p = −fgc/d3. this leads to the pressure p = tφ/[(1 −φ)d3] here, which is the correct equation of state for the tonks gas [1]. 7 3.2 potential profile by combining equations (8), (19) and (21), the effective, particle-mediated interaction potential v of the surfaces can be expressed as a function of the particle bulk volume fraction φ and the separation lbetween the surfaces: v = −t d2 ln  1 + ⌊l/d⌋ x n=1 zn λ d φ 1 −φ n exp  n φ 1 −φ  + lt d3 φ 1 −φ, (22) with the exact expression for the canonical partition function zn given in equations (10) and (11), the potential v can be evaluated numerically for any finite surface separation l, see figures 2 and 3. for large bulk volume fractions φ (dashed curve in figure 2), the potential v exhibits oscillations up to surface separations lof the order of several particle diameters before approaching a constant, asymptotic value v∞for large surface separations. the oscillations are related to successive layers of particles formed in the space between the surfaces. for small bulk volume fractions φ, in contrast, the interaction potential v attains an approximately constant value for surface separations l> 2(d + lr), see solid curve in figure 2. 3.3 potential asymptote since the interactions between the particles and the surfaces are short-ranged, the potential v has a horizontal asymptote, i.e. v (l) approaches a constant value v∞for large surface separation l, see figures 2 and 3. in this subsection, we calculate the position of the asymptote. to simplify the notation, we first introduce the auxiliary variable ζ = φ 1 −φ, (23) and the function g l d, l0 d , ζ  = ⌊l/d⌋ x n=1 ζn n! exp  −l−nd d ζ  l−l0 d −n n θ l−l0 d −n  (24) defined for an arbitrary length l0. by combining equations (8), (10), (19) and (21), we then express the potential v as e−v (l) d2/t = e−ζl/d + (eu/t −1)2 g l d, 2lr d , ζ  −2eu/t(eu/t −1) g l d, lr d , ζ  + e2u/t g l d, 0, ζ  . (25) 8 1 2 3 4 5 6 -8 -6 -4 -2 0 vd t 2 l/d φ = 0.05 φ = 0.5 figure 2: rescaled surface interaction potential v d2/t, given by equation (22), as a function of the rescaled surface separation l/d where d is the par- ticle diameter and t denotes the temperature in energy units. the particle binding energy here is u = 5 t, the binding range is lr = 0.3 d, and the particle bulk volume fraction is φ = 0.05 (solid line) and φ = 0.5 (dashed line), respectively. for large surface separations l≫d, the potential v (l) attains an approximately constant value v∞. according to equation (28), the asymptotic values are v∞≈−6.6446 t/d2 for φ = 0.5 (dashed line) and v∞≈−2.3422 t/d2 for φ = 0.05 (solid line), in excellent agreement with numerical values obtained from equation (22). using the saddle-point approximation and stirling's formula (15), one can prove that lim l→∞g l d, l0 d , ζ  = e−ζ l0/d 1 + ζ , (26) see b. we now apply this result to equation (25) and arrive at lim l→∞exp  −v d2 t  = eu/t −(eu/t −1)e−ζlr/d
2 1 1 + ζ . (27) hence, the asymptotic value of the potential v (l) is given by the following exact expression: v∞= −t d2 ln (1 −φ) −2 t d2 ln  eu/t −(eu/t −1) exp  −lr d φ 1 −φ  . (28) for non-adhesive particles with u = 0 or lr = 0, the asymptotic value of the potential v (l) is v∞= −(t/d2) ln(1 −φ). for small bulk volume fractions 9 φ ≪1 of the particles and large binding energy u with eu/t ≫1, we obtain v∞≈−2 t d2 ln  1 + φ eu/t lr d  . (29) we will use equations (28) and (29) in section 5 to calculate the effective adhesion energy of surfaces. in the following section 4, we determine the global minimum of the potential v (l). 4 global minimum of the surface interaction potential in the present calculation we have chosen the free energy reference state in such a way that the effective surface interaction potential vanishes at surface contact l= 0, i.e. v (l= 0) = 0, see equation (7). for surface separations 0 < l< d, the potential v (l) increases linearly with l, i.e. v (l) = ltφ/[(1 −φ)d3], since fgc = 0 for these separations. the potential v (l) decreases for separations d < l< d+lr, and increases again for d+lr < l< 2d. the potential thus attains a minimum at l= d + lr, see figures 2 and 3. the value vmin = v (l= lr + d) at this minimum can be calculated again from equation (22). for l= d + lr, equation (10) reduces to z1 = e2u/tlr/λ and zn = 0 for n ≥2 since only a single particle fits into the column. insertion of these results into equation (22) leads to vmin = −t d2 ln  1 + lr d φ 1 −φ exp 2u t + φ 1 −φ  + t d2 lr d + 1  φ 1 −φ. (30) the potential v (l) thus has two local minima, one located at l= 0 with v (l= 0) = 0 and the other at l= lr + d with v (l= lr + d) = vmin. the two minima result from the interplay of depletion interactions and adhesive interactions. the global minimum of v (l) is located at l= d+lr if vmin < 0, i.e. for e−ζ + lr d ζ e2u/t −eζ lr/d > 0 (31) with ζ = φ/(1 −φ), see equation (23). the inequality (31) is fulfilled for sufficiently large particle binding energies u. for small binding energies u, in contrast, the potential v (l) has its global minimum at surface contact l= 0, see figure 3. in the experimentally relevant case of small particle bulk volume fractions φ ≪1 and large binding energy u with eu/t ≫1, equation (30) reduces to vmin ≈−t d2 ln  1 + φ e2u/t lr d  (32) 10 1 2 3 4 5 -0.05 0 0.05 0.1 l/d vd t 2 u = 0.6 t u = 0.9 t figure 3: rescaled surface interaction potential v d2/t, given by equation (22), versus the rescaled surface separation l/d. the bulk volume fraction of the particles here is φ = 0.1, the binding range is lr = 0.3 d, and the binding energy is u = 0.9 t (dashed line) and u = 0.6 t (solid line). the global minimum of the potential v (l) is located at the separation l= d + lr for u = 0.9 t (dashed line) and at surface contact l= 0 for u = 0.6 t (solid line). and the inequality (31) simplifies to u > t 2 ln  1 + d lr  . (33) and, thus, to a relation that is independent of the particle bulk volume frac- tion φ. 5 adhesion energy in this section, we assume that the binding energy u of the particles is sufficiently large so that the inequality (31) is fulfilled. the global minimum of the interaction potential v (l) then is located at l= d + lr. the minimum value vmin = v (l= d + lr) is given by equation (30). for large surface separations l≫d, the potential v (l) attains a constant value v∞given by equation (28). the difference between the asymptotic and the minimum value of the potential v (l) is the effective adhesion energy w = v∞−vmin (34) 11 of the surfaces. the effective adhesion energy is the minimal work that has to performed to bring the two surfaces far apart from the separation l= d + lr. from equations (28), (30), and (34), we obtain the exact result w = t d2 ln  1 + lr d φ 1 −φ exp 2u t + φ 1 −φ  −t d2 lr d + 1  φ 1 −φ −t d2 ln (1 −φ) −2 t d2 ln  eu/t −(eu/t −1) exp  −lr d φ 1 −φ  . (35) in figure 4, the adhesion energy w is plotted as a function of the particle bulk volume fraction φ. interestingly, the adhesion energy w exhibits a maximum at an optimal bulk volume fraction φ⋆of the particles. 0.001 0.002 0.005 0.01 0.02 0.05 0.1 2 2.25 2.5 2.75 3 3.25 3.5 φ t 2 wd figure 4: rescaled adhesion energy wd2/t as a function of the particle bulk volume fraction φ. the solid line corresponds to the exact result (35), and the dashed line to the approximation (36). the particle binding energy here is u = 5 t and the binding range is lr = 0.3 d. the adhesion energy has a maximum at φ = φ⋆with φ⋆≈e−u/td/lr. for small bulk volume fractions φ ≪1 and large binding energy u with eu/t ≫1, the asymptotic value and minimum value of v (l) are approxi- mately given by equations (29) and (32), respectively. the adhesion energy w = v∞−vmin then simplifies to w ≈t d2 ln 1 + φ e2u/t lr/d (1 + φ eu/t lr/d)2. (36) this expression is identical with our previous result obtained from a virial expansion in φ up to second order terms [8]. for φ ≪1 and eu/t ≫1, the 12 adhesion energy (36) is a good approximation of the exact result (35), see figures 4 and 5. from equation (36), we obtain the approximate expression φ⋆≈d lr e−u/t (37) for the optimum bulk volume fraction φ⋆at which the adhesion energy w attains its maximum value. the adhesion energy (36) can be understood as the difference of two lang- muir adsorption free energies per fluid column, or pair of apposing binding sites [8]: (i) the adsorption free energy (t/d2) ln 1 + q φ e2u/t
for small surface separations at which a particle binds both surfaces with total bind- ing energy 2u, and (ii) the adsorption free energy (t/d2) ln 1 + q φ eu/t
for large surface separations, counted twice in (36) because we have two sur- faces. these langmuir adsorption free energies result from a simple two-state model in which a particle is either absent (boltzmann weight 1) or present (boltzmann weights q φ e2u/t and q φ eu/t, respectively) at a given binding site, see e.g. [20]. the factor q depends on the degrees of freedom of a single adsorbed particle. in our model, we obtain q = lr/d. to assess the quality of approximate expression (36), we analyze its rela- tive error in reference to the exact result (35). the relative error is the mag- nitude of the difference between the exact result (35) and the approximate expression (36) divided by the magnitude of the exact result (35). figure 5 shows parameter regions in which the relative error of the expression (36) is smaller or larger than 1%, 2% and 5%, respectively. in this example, the binding range is lr = 0.3. for intermediate and large binding energies with u > 6 t, we find that the relative error of the approximate expression (36) is smaller than 1% in a broad range of volume fractions φ. 6 potential barrier for large binding energies u with eu/t ≫1, the effective interaction potential has a barrier at surface separations d + 2lr < l< 2(d + lr), see figure 2. at these separations, only a single particle fits between the surfaces, but this particle can just bind one of the surfaces. the particle thus 'blocks' the binding site at the apposing surface. the potential barrier attains its maximum value vba = v (l= 2d) at the separation l= 2d, see figure 2. from equation (25), we obtain vba = −t d2 ln  1 +  2lr d eu/t −1
+ 1  φ 1 −φ exp  φ 1 −φ  + 2 t d2 φ 1 −φ (38) 13 0 0.05 0.10 0.15 0.20 0.25 1 2 3 4 5 6 φ t u 1 % 2 % 5 % figure 5: relative error of the approximate expression (36) for the binding range lr = 0.3/, d of the particles. in the parameter region above the dotted line, the relative error is smaller than 1%. below this line, the relative error is larger than 1%. the relative error is smaller than 2% above the dashed line, and smaller than 5% above the solid line. for binding energies u > 6 t, the simple expression (36) approximates the exact result (35) very well since the relative error is smaller than 1% for a broad range of volume fractions φ. for φ ≪1 and eu/t ≫1, we get vba ≈−t d2 ln  1 + 2φ eu/t lr d  (39) the barrier height uba = vba −v∞then is uba ≈t d2 ln 1 + φ eu/t lr/d
2 1 + 2φ eu/t lr/d (40) since the asymptotic value v∞is given by equation (29) in this limiting case. the width of the barrier is approximately lba ≈d, see figure 2. equation (40) is again identical with our previous result obtained from a virial expansion in φ up to second order terms [8]. 14 7 binding probability another quantity of interest here is the binding probability ns = 1 2⟨θ (lr −x1 + d/2)⟩+ 1 2⟨θ (xn −l+ lr + d/2)⟩ (41) defined as the probability that the separation of the closest particle from a column base is smaller than the binding range lr. in other words, the binding probability ns is the probability of finding a particle bound to one of the bases. the binding probability corresponds to the surface coverage in the case of a three-dimensional gas of particles between two parallel attractive surfaces. equations (1) - (5) imply that the binding probability can be calculated by differentiation of the grand potential fgc with respect to binding energy u, i.e. ns = −1 2  ∂fgc ∂u  . since the grand potential density fgc given by equation (21) does not depend on the binding energy u, the binding probability can also be obtained from the effective surface interaction potential via ns = −d2 2 ∂v ∂u  . (42) with the exact expression (25) for the interaction potential v , the binding probability ns can be determined numerically for any finite separation l. in figure 6, the binding probability ns is plotted as a function of surface separation lfor three different volume fractions φ around the optimal volume fraction φ⋆at which the adhesion energy w is maximal. in the vicinity of φ⋆, the binding probability at large separations l> 2(d + lr) is sensitive to small variations of φ, while the binding probability at separations lin the surface binding range d < l< d + 2lr remains practically constant at almost 100% . for small bulk volume fractions φ of the particles and large particle bind- ing energies u, the asymptotic and minimum value of the interaction po- tential v (l) are given by equations (29) and (32), respectively. from these equations and relation (42), we obtain the approximate expressions ns, ∞≈ φ φ + φ⋆ (43) for the binding probability at large surface separation land ns, min ≈ φ φ + φ⋆e−u/t , (44) 15 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 φ = 0.007 n l/d φ = 0.003 φ = 0.001 s figure 6: binding probability ns, calculated numerically from equations (42) and (25), as a function of rescaled surface separation l/d where d is the diameter of the adhesive particles. the binding energy here is u = 7 t, the binding range is lr = 0.3 d and the particle bulk volume fraction is φ = 0.001 < φ⋆(dashed line), φ = 0.003 ≈φ⋆(solid line) and φ = 0.007 > φ⋆ (dotted line). the optimal volume fraction φ⋆at which the adhesion energy becomes maximal is given by equation (37). for the particle binding probability at the binding separation l= d+lr of the surfaces, with the optimum bulk volume fraction φ⋆given in equation (37). these expressions correspond to the well-known langmuir adsorption equa- tion [20]. at the optimal volume fraction φ⋆, the particle binding probability for unbound and bound surfaces is ns, ∞= 1/2 and ns, min ≈1, respectively. bringing the surfaces from large separations l> 2(d + lr) within binding separations d < l< d + 2lr thus does not require desorption or adsorption of particles at φ = φ⋆. 8 conclusions we have considered one-dimensional gas of hard-sphere particles with at- tractive boundaries, a novel extension of the tonks model [1]. we have solved this model analytically in the whole range of parameters by explicit integration over the particles' degrees of freedom in the partition function. in contrast to other studies on one-dimensional models for hard spheres [18, 19, 20, 21, 22, 23, 24, 25, 26], we have focused on the boundary con- 16 tribution to the free energy of the system, which corresponds to the effective, particle-mediated interaction potential between the boundaries, or surfaces, see figures 2 and 3. the effective adhesion energy obtained from the inter- action potential depends non-monotonically on the volume fraction φ of the particles in the bulk, see figure 4. the adhesion energy exhibits a maximum at an optimum volume fraction, which can lead to reentrant transitions in which the surfaces first bind with increasing volume fraction φ, and unbind again when the volume fraction φ is increased beyond its optimum value. a lattice of such one-dimensional gas columns represents a discrete ap- proximation of a three-dimensional gas of particles between two adsorbing surfaces, see figure 1 and reference [8]. for small volume fractions φ and short-ranged particle-surface interactions considered here, the gas of parti- cles between two well-separated surfaces is as dilute as in the bulk, except for the single adsorption layers of particles at the surfaces. at larger volume frac- tions, three-dimensional packing effects become relevant. these effects are not captured correctly in the one-dimensional model. however, it has been pointed out [22] that approximations based on one-dimensional models do well in comparison to density functional theories for three-dimensional hard sphere fluids confined in planar, non-adsorbing pores [27]. in principle, the quality of the one-dimensional approximation can be tested by monte carlo or molecular dynamics simulations, which have been used to study various three-dimensional systems of hard spheres confined between non-adsorbing surfaces [28, 29, 30, 31, 32]. for simplicity, and for consistency with our previous publication [8], we have considered here a square lattice of columns between the surfaces. in particular, the factor d2 in the denominator of equation (7) is the column base area in the square lattice. for a hexagonal lattice of columns, the corresponding area is ( √ 3/2) d2, and the corresponding effective interaction potential of the surfaces is thus obtained by multiplying the right hand side of equation (22) with a factor 2/ √ 3 ≈1.1547. the adhesion energy of the surfaces has to be rescaled with the same factor in the case of hexagonal lattice of columns, but its functional dependence on the bulk volume fraction φ, binding energy u, and binding range lr remains unchanged. we have considered an equilibrium situation in which the particles ex- change with a bulk solution. for polymers between surfaces, such an equi- librium has been termed 'full equilibrium'. in a 'restricted equilibrium', in contrast, the polymers are trapped between the surfaces [33, 34, 35], which is less likely for the spherical particles considered here. 17 a canonical partition function in this section we calculate the n-particle partition function zn as given by equation (9) for l> nd. first, if one notices that eu θ(lr−y1)/t = 1 + eu/t −1
θ(lr −y1) (45) and eu θ(yn−l+nd+lr)/t = 1 + eu/t −1
θ(yn −l+ nd + lr), (46) the integral (9) can be written as a sum of four terms zn = 1 λni1 + 1 λn eu/t −1
i2 + 1 λn eu/t −1
i3 + 1 λn eu/t −1
2 i4, (47) with i1 = z l−nd 0 dyn z yn 0 dyn−1 . . . z y3 0 dy2 z y2 0 dy1, (48) i2 = z l−nd 0 dynθ (yn −l+ nd + lr) z yn 0 dyn−1 . . . z y3 0 dy2 z y2 0 dy1, (49) i3 = z l−nd 0 dyn z yn 0 dyn−1 . . . z y3 0 dy2 z y2 0 dy1θ (lr −y1) , (50) i4 = z l−nd 0 dynθ (yn −l+ nd + lr) z yn 0 dyn−1 . . . z y2 0 dy1θ (lr −y1) . (51) the first integral i1 = 1 (n −1)! z l−nd 0 yn−1 n dyn = 1 n! (l−nd)n (52) and second integral i2 = 1 (n −1)! z l−nd 0 yn−1 n θ (yn −l+ nd + lr) dyn = 1 n! ((l−nd)n −(l−nd −lr)n θ (l−nd −lr)) (53) can be easily calculated. to calculate the third integral, we start from z y2 0 θ (lr −y1) dy1 = min [y2, lr] = y2 −(y2 −lr) θ (y2 −lr) . (54) 18 in the next steps, we find z y3 0 (y2 −(y2 −lr) θ (y2 −lr)) dy2 = 1 2y2 3 −1 2 (y3 −lr)2 θ (y3 −lr) (55) and z y4 0 1 2y2 3 −1 2 (y3 −lr)2 θ (y3 −lr)  dy3 = 1 6y3 4 −1 6 (y4 −lr)3 θ (y4 −lr) (56) iterating these results leads to i3 = 1 (n −1)! z l−nd 0  yn−1 n −(yn −lr)n−1 θ (yn −lr)  dyn. (57) the integral i3 can now be evaluated as i3 = 1 n! (l−nd)n − 1 (n −1)! z l−nd 0 (yn −lr)n−1 θ (yn −lr) dyn = 1 n! (l−nd)n −1 n! (l−nd −lr)n θ (l−nd −lr) . (58) note that i2 = i3. the fourth integral, i4, can be brought to the form i4 = 1 (n −1)! z l−nd 0 θ (yn −l+ nd + lr)  yn−1 n −(yn −lr)n−1 θ (yn −lr)  dyn (59) if one uses again (54) and iterates the integration as in (55) and (56). the first term on the right hand side of equation (59) is equal to i2, see equation (53). thus i4 = i2 −i5, where i5 = 1 (n −1)! z l−nd 0 θ (yn −l+ nd + lr) θ (yn −lr) (yn −lr)n−1 dyn (60) to determine the integral i5, one has to distinguish three cases: (i) for l> nd + 2lr, we obtain i5 = 1 n! ((l−nd −lr)n −(l−nd −2lr)n) , (61) (ii) for nd + lr < l< nd + 2lr, we obtain i5 = 1 n! (l−nd −lr)n (62) 19 and (iii) for l< nd + lr one gets i5 = 0. in summary i4 = i2 −1 n! (l−nd −lr)n θ (l−nd −lr) + 1 n! (l−nd −2lr)n θ (l−nd −2lr) (63) if one now gathers the results (52), (53), (58), (63) and returns to equation (47), one obtains the partition function zn = 1 λnn!  j1 + 2(eu/t −1)(j1 −j2) + (eu/t −1)2(j1 −2j2 + j3)  (64) with j1 = (l−nd)n θ (l−nd) , (65) j2 = (l−nd −lr)n θ (l−nd −lr) , (66) j3 = (l−nd −2lr)n θ (l−nd −2lr) . (67) this result can be written as zn = 1 λnn!  eu/t −1
2 (l−nd −2lr)n θ (l−nd −2lr) −2eu/t eu/t −1
(l−nd −lr)n θ (l−nd −lr) +e2u/t (l−nd)n θ (l−nd)  (68) which simplifies to equation (10). b proof of equality (26) here, we explore the asymptotics of the function g(l/d, l0/d, ζ) defined in equation (24) and, hence, prove the equality (26). to simplify the notation, let l/d = n, where n is a large integer number, and l0/d = λ. then g (n, λ, ζ) = n x n=1 ζn n! e−(n−n)ζ (n −n −λ)n θ (n −n −λ) . (69) in the next step, we introduce the auxiliary function χ(n) = −(n −n)ζ + n ln ζ + n ln(n −n −λ) −n ln n + n −1 2 ln(2πn) (70) 20 and rewrite the function g(n, λ, ζ) given by equation (69) in the form g = n x n=1 eχ(n) θ (n −n −λ) , (71) using stirling's formula (15). for large n, one can replace the sum on the right hand side of equation (71) by an integral and write g ≈ z n−λ 1 eχ(n) dn. (72) the function χ(n) has a global maximum at n = n0 with n0 ≈(n −λ) ζ 1 + ζ (73) for large n. note that for large numbers n, the location n0 ≈(n −λ) φ of the global minimum scales linearly with n. if we now expand the function χ(n) around the point n0 up to second order terms and apply the saddle-point approximation, we get g ≈eχ(n0) z n−λ 1 exp 1 2χ ′′(n0) (n −n0)2  dn. (74) a simple change of variables m = n −n0 leads to g ≈eχ(n0) z (n−λ)(1−φ) −(n−λ)φ+1 eχ ′′(n0) m2/2 dm (75) where we have used n0 ≈(n −λ) φ, which follows from equation (73) and relation (23) between variables ζ and φ. in the limit of large n, we thus obtain g ≈eχ(n0) z ∞ −∞ eχ ′′(n0) m2/2 dm (76) because 0 < φ < 1. now, we can calculate the gaussian integral in (76) to get g ≈eχ(n0) s 2π −χ ′′(n0) . (77) from the definition (70) of the auxiliary function χ(n) and equation (73) for the point n = n0 at which function χ(n) has its global maximum, we get χ(n0) ≈−λ ζ −1 2 ln(2πn0) (78) 21 and χ ′′(n0) ≈−(1 + ζ)2 n0 (79) note that χ ′′(n0) < 0, and that the function χ(n) has indeed a maximum at n = n0. combining equations (77), (78) and (79) leads to g ≈e−λ ζ 1 + ζ (80) for large n values and, thus, to equation (26) quod erat demonstrandum. 22 references [1] tonks l, 1936 phys. rev. 50 955 [2] asakura s and oosawa f, 1954 j. chem. phys. 22 1255 [3] dinsmore a d, yodh a g and pine d j, 1996 nature 383 239 [4] anderson v j and lekkerkerker h n w, 2002 nature 416 811 [5] baksh m m, jaros m and groves j t, 2004 nature 427 139 [6] winter e m and groves j t, 2006 anal. chem. 78 174 [7] hu y, doudevski i, wood d, moscarello m, husted c, genain c, za- sadzinski j a and israelachvili j, 2004 proc. natl. acad. sci. usa 101 13466 [8] r ́ o ̇ zycki b, lipowsky r and weikl t r, 2008 epl 84 26004 [9] lieb e h and mattis d c, mathematical physics in one dimension (london, academic press, 1966) [10] baxter r j, exactly solved models in statistical mechanics (london, academic press, 1982) [11] kac m, 1959 phys. fluids 2 8 [12] baker g, 1961 phys. rev. 126 1477 [13] kac m, uhlenbeck g and hemmer p, 1963 j. math. phys. 4 216 [14] j ̈ ulicher f, lipowsky r and m ̈ uller-krumbhaar h, 1990 europhys. lett. 11 657 [15] lipowsky r, 1995 z. phys. b 97 193 [16] r ́ o ̇ zycki b and napi ́ orkowski m, 2003 j. phys. a: math. gen. 36 4551 r ́ o ̇ zycki b and napi ́ orkowski m, 2004 europhys. lett. 66 25 [17] gursey f, 1950 proc. cambridge phil. soc. 46 182 [18] salsburg z, zwanzig r and kirkwood j, 1953 j. chem. phys. 21 1098 [19] baur m and nosanow l, 1962 j. chem. phys. 37 153 23 [20] davis h t, statistical mechanics of phases, interfaces, and thin films (vch publishers, new york, 1996) [21] davis h t, 1990 j. chem. phys. 93 4339 [22] henderson t, 2007 mol. phys. 105 2345 [23] lekkerkerker h n w and widom b, 2000 physica a 285 483 [24] wensink h h, 2004 phys. rev. lett. 93 157801 [25] van der schoot p, 1996 j. chem. phys. 104 1130 [26] chou t, 2003 europhys. lett. 62 753 [27] pizio o, partykiejew a and soko lowki s, 2001 mol. phys. 99 57 [28] chu x l, nikolov a d and wasan d t, 1994 langmuir 10 4403 [29] schmidt m and l ̈ owen h, 1997 phys. rev. e 55 7228 [30] schoen m, gruhn t and diestler d j, 1998 j. chem. phys. 109 301 [31] fortini a and gijkstra m, 2006 j. phys.: condens. matter 18 l371 [32] mittal j, truskett t m, errington j r and hummer g, 2008 phys. rev. lett. 100 145901 [33] ennis j and j ̈ onsson b j, 1999 phys. chem. b 103, 2248 [34] bleha t and cifra p, 2004 langmuir 20, 764 [35] leermakers f a m and butt h j, 2005 phys. rev. e 72, 021807 24

0911.1715

doppler-tuned bragg spectroscopy of excited levels in he-like uranium: a discussion of the uncertainty contributions

we present the uncertainty discussion of a recent experiment performed at the gsi storage ring esr for the accurate energy measurement of the he-like uranium 1s2p3p2- 1s2s3s1 intra-shell transition. for this propose we used a johann-type bragg spectrometer that enables to obtain a relative energy measurement between the he-like uranium transition, about 4.51 kev, and a calibration x-ray source. as reference, we used the ka fluorescence lines of zinc and the li-like uranium 1s22p2p3/2 - 1 s22s 2s1/2 intra-shell transition from fast ions stored in the esr. a comparison of the two different references, i.e., stationary and moving x-ray source, and a discussion of the experimental uncertainties is presented.

introduction we present the uncertainty discussion of a recent experiment performed at the gsi (darmstadt, germany) for the accurate energy measurement of the he-like uranium 1s2p 3p2 →1s2s 3s1 intra-shell transition. this measurement allows, for the first time, to test two-photons quantum electrodynamics in he-like heavy ions. in this article we describe the techniques adopted in the measurement where x rays emitted from fast ions in a storage ring have been detected with a bragg spectrometer. in particular, we study the contribution of the different uncertainties related to the relativistic velocity of the ions when a stationary or moving calibration x-ray source is considered. additional details of such an experiment can be found in ref. [1]. 2. description of the set-up the experiment was performed at the gsi experimental storage ring esr [2] in august 2007. here, a h-like uranium beam with up to 108 ions was stored, cooled, and decelerated to an energy of 43.57 mev/u. excited he-like ions were formed by electron capture during the interaction of the ion beam with a supersonic nitrogen gas-jet target. at the selected velocity, electrons zn kα2 kα1 0 200 400 600 800 1000 x-axis (channels) -150 -100 -50 0 50 100 y-axis (channels) 0 10 20 30 40 50 60 70 counts per pixel -150 -100 -50 0 50 100 0 200 400 600 800 1000 y-axis (channels) x-axis (channels) li-like u 1s22p 3p2 - 1s22s 3s1 figure 1. reflection of the zinc kα (left) and li-like uranium intra-shell 1s22p 2p3/2 → 1s22s 2s1/2 transition (right) on the bragg spectrometer ccd. the transition energy increases with the increasing of x-position. the slightly negative slope of the line is due to the relativistic velocity of the li-like ions. are primarily captured into shells with principal quantum number of n ≤20, which efficiently populate the n = 2 3p2 state via cascade feeding. this state decays to the n = 2 3s1 state via an electric dipole (e1) intra-shell transition (branching ration 30%) with the emission of photons of an energy close to 4.51 kev detected by a bragg spectrometer. the crystal spectrometer [3] was mounted in the johann geometry in a fixed angle configuration allowing for the detection of x rays with a bragg angle θ around 46.0◦. the spectrometer was equipped with a ge(220) crystal cylindrically bent, with a radius of curvature r = 800 mm, and a newly fitted x-ray ccd camera (andor do420) as position sensitive detector. the imaging properties of the curved crystal were used to resolve spectral lines from fast x-ray sources nearly as well as for stationary sources [4]. for this purpose, it was necessary to place the rowland-circle plane of the spectrometer perpendicular to the ion–beam direction. for a minimizing the systematic effects due to the ion velocity and alignment uncertainties, the observation angle θ = 90◦was chosen. the value of the ion velocity v was selected such that the photon energy, e in the ion frame, was doppler-shifted to the value elab = 4.3 kev in the laboratory frame, where elab = e/[γ(1 −β cos θ)], with β = v/c (c is speed of light) and γ = 1/ p 1 −β2. this value of elab was chosen to have the he-like uranium spectral line position on the ccd close to the position of the 8.6 kev kα1,2 lines of zinc, which were observed in second order diffraction. the zinc lines were used for calibration and they were produced by a commercial x-ray tube and a removable zinc plate between the target chamber and the crystal. an image of the zinc kα lines from the bragg spectrometer is presented in fig. 1 (left side). as an alternative method to the measurement of the he-like uranium intra-shell transition, we used a calibration line originating from fast ions, rather than one from the stationary source. for this purpose the 1s22p 2p3/2 →1s22s 2s1/2 transition in li-like u at 4459.37 ± 0.21 ev [5, 6] was chosen. at the esr, the li-like ions were obtained by electron capture into he-like uranium ions. to match the energy of the he-like transition, an energy of 32.63 mev/u was used to doppler-shift the li-like transition. an image of the li-like uranium transition in the bragg spectrometer is presented in fig. 1 (right side). starting from the bragg's law in differential form, ∆e ≈−e ∆θ/ tan θ, one obtains an approximate dispersion formula that is valid for small bragg angle differences ∆θ. taking into account the relativistic doppler effect, the measured value of the he-like u transition is given by e = e0 n n0 1 −δ(e0)/ sin2 θ0
1 −δ(e)/ sin2 θ
γ(1 −β cos θ) γ0(1 −β0 cos θ)  1 + ∆x tan θ0 d  , (1) where n0 and n are the diffraction order of the he-like u and reference lines, respectively, θ0 and θ = θ0 + ∆θ the correspondent bragg angle, ∆x is the position difference of the spectral lines on the ccd along the dispersion direction and d is the crystal–ccd distance. δ(e) is the deviation of the index of refraction nr(e) = 1 + δ(e) of the crystal material from the unity, which depends on the energy e of the reflected x ray (δ ∼10−5 −10−6 typically). in the case n = n0, the corrections due to the refraction index and other energy dependent corrections for curved crystals [7, 8] are negligible. in the case of a stationary calibration source, γ0 = 1 and β0 = 0. the measured value of the he-like uranium transition energy and additional information can be found in ref. [1]. in the following section we present in detail the analysis of the systematic uncertainties. 3. evaluation of the experimental uncertainties one of the main sources of uncertainty in the present experiment is the low amount of collected data (see fig. 1, right). this limits the accuracy of the he- and li-like uranium line position, i.e., the accuracy of ∆x, which is proportional to the energy uncertainty, (δe)stat ∝δ(∆x) (see eq. (1)). in our specific experiment, characterized by the parameters listed in table 1, numerical values of (δe)stat are presented in table 2. due to the high statistics in the stationary calibration source measurement, the zn kα spectrum, (δe)stat is ∼ √ 2 smaller than when the moving li- like ion emission is used as a reference. in the case of systematic uncertainties, three major sources dominate: the accuracy of the reference energy e0, of the ion velocity and of the observation angle θ. similarly to the statistical uncertainty, the contribution of δe0 is much smaller when zn lines are used for calibration instead of the li-like u transition (see table 2). this is due to the high accuracy of the zinc kα transition energy, which in the case of kα1 is 8638.906 ± 0.073 ev [9], compared to the li-like u transition accuracy of 0.21 ev [5, 6]. if on one hand, doppler tuning of the photon energy in the laboratory frame produce two important systematic uncertainty contributions, due to the ions velocity and the observation angle; on the other hand, it allows for detecting the different spectral lines in the same narrow spatial region of the ccd detector, i.e., ∆x/d ≪1. this results in a drastic reduction of other systematic effects such as the influence of uncertainty of the crystal–ccd distance d, the accuracy of the ccd pixel size [10], the accuracy of the inter-plane distance of the crystal and effects from the optical aberrations in the johann geometry set-up. systematic uncertainties related to the relativistic velocity of the ions are treated in detail in the following subsections. 3.1. ion velocity uncertainty the ion velocity in the storage ring is imposed by the velocity of the electrons in the electron cooler [2]. this is related to the cooler voltage v by the simple relations γ = 1 + ev mec2, β = r 2 ev mec2 +  ev mec2 2 1 + ev mec2 ≃ r 2 ev mec2 , (2) where me and e are the mass and charge of the electron, respectively. the factor ev/(mec2) is in general very small, of the order of 4 × 10−2 in our specific case. the cooler voltage uncertainty δv propagates to the energy uncertainty via the parameters γ and β in eq. (1). more precisely, (δγ)v = e/(mec2) δv and (δβ)v ≃[e/(2mec2v )]1/2 δv . we note that δγ/δβ = o( p ev/(mec2)). for this reason an observation angle of 90◦, where the effect of δβ is minimal, was chosen. in the following formulas we will consider only the case θ = 90◦. the uncertainty of the cooler voltage has two principal sources: the accuracy of the absolute value and its linearity. the relation between the real voltage value vreal and the set value v can be written as v = a + b vreal, where the single uncertainties on the factors a and b has to be considered for our analysis. with this notation, for an observation angle of θ = 90◦, the uncertainty propagation to the energy value is, for the case of a stationary calibration source, (δe)a e = 1 γ     ∂γ ∂a     δa = e mec2 1 γ δa, (δe)b e = 1 γ     ∂γ ∂b     δb = ev mec2 1 γ δb, (3) and (δe)a e = γ0 γ     ∂ ∂a  γ γ0     δa = e2 m2 ec4 |v −v0| γ γ0 δa, (δe)b e = γ0 γ     ∂ ∂b  γ γ0     δb = e mec2 |v −v0| γ γ0 δb, (4) in the case of a moving calibration source, where v0 and γ0 are the corresponding parameters and where the approximation v ≈vreal has been applied. a reduction of the uncertainties is obtained when the moving calibration source is used. the uncertainty (δe)a due to the offset error of v is drastically decreased, by a factor e|v −v0|/(mec2γ0) ≪1, here as the uncertainty due to the linearity is also reduced, but only by a factor |v −v0|/(v γ0). numerical values for our experiments are given in table 1 and 2. 3.2. observation angle uncertainty if the effect of the uncertainty due to the ion velocity is minimal when θ = 90◦, the effect of the uncertainty δθ of the observation angle itself is maximal. in the case θ = 90◦, starting from eq. (1), for a stationary calibration source we have (δe)θ e = 1 1 −β cos θ     d dθ(1 −β cos θ)     θ=90◦ δθ = βδθ, (5) and (δe)θ e = 1 −β0 cos θ 1 −β cos θ     d dθ  1 −β cos θ 1 −β0 cos θ     θ=90◦ δθ = |β −β0|δθ. (6) table 1. principal parameters of the ion beams for the he- and li-like transition measurement (see text). he-like u li-like u β 0.295578 0.257944 γ 1.046771 1.035026 v (volt) 23900 17898 θ 90.00◦± 0.38◦ a(volt) 0 ± 10 b 1. ± 2 × 10−4 table 2. different uncertainty contributions (in ev) when li-like uranium and zinc transitions are used as the reference. li-like u zn kα (δe)stat 0.43 0.30 (δe)e0 0.21 0.04 (δe)a 1 × 10−3 0.08 (δe)b 0.01 0.04 (δe)θ 0.11 0.88 (δe)tot 0.50 0.93 for a moving calibration source. again, analogously to the calculation of the preceding subsection, a reduction of a factor |β −β0|/β in the uncertainty is obtained when x rays from fast ions are used for the calibration. in our experimental set-up, the value of δθ is principally due to the accuracy of the position of the gas-jet target with respect to the main axis of the spectrometer (±0.5 mm). numerical values for our experiments are in table 1 and 2. a direct evaluation of deviation ∆θ from 90◦can be obtained via eq. (6) with the measurement of the li-like uranium energy 4460.12 ± 0.31 ev (statistical uncertainty only) using the zinc kα lines as reference, and its comparison with the literature value 4459.37±0.21 ev [5, 6]. we estimated ∆θ = −0.37◦± 0.18◦in agreement with the expected deviation (see table 1). 4. conclusions we present the uncertainty discussion of an accurate energy measurement on he-like uranium intra-shell transition obtained via bragg spectroscopy of doppler-tuned x rays emitted from fast ions. we have evaluated and compared the systematic uncertainties when a stationary or moving calibration source is used. in particular in our experiment, the use of the x-ray emission of fast li-like uranium ions as reference enables to reduce systematics uncertainties by a factor of about 4. references [1] trassinelli m, kumar a, beyer h, indelicato p, m ̈ artin r, reuschl r, kozhedub y, brandau c, br ̈ auning h, geyer s, gumberidze a, hess s, jagodzinski p, kozhuharov c, trotsenko s, weber g and st ̈ ohlker t 2008 submitted to phys. rev. lett. [2] franzke b 1987 nucl. instrum. meth. phys. res. b 24-25 18–25 [3] beyer h f, indelicato p, finlayson k d, liesen d and deslattes r d 1991 phys. rev. a 43 223 [4] beyer h f and liesen d 1988 nucl. instrum. meth. phys. res. a 272 895–905 [5] beiersdorfer p, knapp d, marrs r e, elliott s r and chen m h 1993 phys. rev. lett. 71 3939 [6] beiersdorfer p 1995 nucl. instrum. meth. phys. res. b 99 114–116 [7] cembali f, fabbri r, servidori m, zani a, basile g, cavagnero g, bergamin a and zosi g 1992 j. appl. crystallogr. 3 424–31 [8] chukhovskii f n, h ̈ olzer g, wehrhan o and f ̈ orster e 1996 j. appl. crystallogr. 29 438–445 [9] deslattes r d, kessler jr e g, indelicato p, de billy l, lindroth e and anton j 2003 rev. mod. phys. 75 35–99 [10] indelicato p, le bigot e o, trassinelli m, gotta d, hennebach m, nelms n, david c and simons l m 2006 rev. sci. inst. 77 043107 (pages 10)

0911.1717

optical and near-ir spectroscopy of candidate red galaxies in two z~2.5 proto-clusters

we present a spectroscopic campaign to follow-up red colour-selected candidate massive galaxies in two high redshift proto-clusters surrounding radio galaxies. we observed a total of 57 galaxies in the field of mrc0943-242 (z=2.93) and 33 in the field of pks1138-262 (z=2.16) with a mix of optical and near-infrared multi-object spectroscopy. we confirm two red galaxies in the field of pks1138-262 at the redshift of the radio galaxy. based on an analysis of their spectral energy distributions, and their derived star formation rates from the h-alpha and 24um flux, one object belongs to the class of dust-obscured star-forming red galaxies, while the other is evolved with little ongoing star formation. this result represents the first red and mainly passively evolving galaxy to be confirmed as companion galaxies in a z>2 proto-cluster. both red galaxies in pks1138-262 are massive, of the order of 4-6x10^11 m_sol. they lie along a colour-magnitude relation which implies that they formed the bulk of their stellar population around z=4. in the mrc0943-242 field we find no red galaxies at the redshift of the radio galaxy but we do confirm the effectiveness of our jhk_s selection of galaxies at 2.3<z<3.1, finding that 10 out of 18 (56%) of jhk_s-selected galaxies whose redshifts could be measured fall within this redshift range. we also serendipitously identify an interesting foreground structure of 6 galaxies at z=2.6 in the field of mrc0943-242. this may be a proto-cluster itself, but complicates any interpretation of the red sequence build-up in mrc0943-242 until more redshifts can be measured.

introduction at every epoch, galaxies with the reddest colours are known to trace the most massive objects (tanaka et al. 2005). the study of massive galaxies at high redshift therefore places important constraints on the mechanisms and physics of galaxy formation (e.g., hatch et al. 2009). studies of clusters at both low and high redshift also yield valuable insights into the assembly and evo- lution of large-scale structure, which itself is a sensitive cosmo- logical probe (e.g., eke et al. 1996). the most distant spectroscopically confirmed cluster lies at z = 1.45 and has 17 confirmed members, a large fraction of which are consistent with massive, passively evolving el- lipticals (stanford et al. 2006; hilton et al. 2007). high redshift proto-clusters have been inferred through narrow-band imaging ⋆based in part on data collected at subaru telescope, which is oper- ated by the national astronomical observatory of japan, and in part on data collected with eso's very large telescope ut1/antu, under pro- gram id 080.a-0463(b). and eso's new technology telescope under program id 076.a-0670(b) searches for ly-α emitters in the vicinity of radio galaxies (e.g., kurk et al. 2004a; venemans et al. 2005). such searches have been highly successful, with follow-up spectroscopy confirm- ing 15 −35 ly-α emitters per proto-cluster. however, the ly-α emitters are small, faint, blue star-forming galaxies, and likely constitute a small fraction of both the number of cluster galaxies and the total mass budget in the cluster. massive elliptical galaxies are seen to dominate cluster cores up to a redshift of z ≈1 (e.g., van dokkum et al. 2001), with the red sequence firmly in place by that epoch (ellis et al. 2006; blakeslee et al. 2003) and in fact showing very little evolution even out to z ∼1.4 (e.g., lidman et al. 2008). what remains uncertain is whether this holds true at yet higher redshift. the fundamental questions are (1) at what epoch did massive, rich clusters first begin to assemble, and (2) from what point do they undergo mostly passive evolution? there is certainly evidence that the constituents of high redshift clusters are very differ- ent from that of rich clusters in the local universe, with the distant clusters containing a higher fraction of both blue spi- rals (e.g., ellis et al. 2006) and active galaxies (galametz et al. 2 m. doherty et al.: spectroscopy of red galaxies in z ∼2.5 proto-clusters 2009). van dokkum & stanford (2003) found early-type galax- ies with signatures of recent star formation in a z = 1.27 clus- ter, perhaps symptomatic of a recently formed (or still forming) cluster. locating massive galaxies in higher redshift clusters will therefore yield valuable insight into the earliest stages of cluster formation and the build-up of complex structures which are ob- served in the local universe. in two previous papers, we found over-densities of red colour-selected galaxies in several proto-clusters associated with high redshift radio galaxies (kajisawa et al. 2006; kodama et al. 2007). follow up studies of distant red galaxies (drgs) have shown that this colour selection criterion contains two pop- ulations: (1) dusty star-forming galaxies and (2) passively evolved galaxies. the exact fraction between these two pop- ulations is still a matter of debate (f ̈ orster schreiber et al. 2004; grazian et al. 2006, 2007). with our new jhk colour- selection, we hope to be more efficient in identifying the evolved galaxy population. we have now embarked upon a spectroscopic follow-up investigation with the aim of confirming cluster mem- bers and determining their masses and recent star formation his- tories. such a goal is ambitious as redshifts of these red galax- ies are generally very difficult to obtain, especially amongst the population with no on-going star formation (i.e., lacking emis- sion lines). here we present the first results from optical and near-infrared spectroscopy of targets in two of the best studied proto-clusters, mrc 1138−262 (z = 2.16) and mrc 0943−242 (z = 2.93). in section 2 we explain the target selection and in section 3 outline the observations and data reduction steps. section 4 presents the redshifts identified and in section 5 we briefly anal- yse the properties of the two galaxies discovered at the redshift of mrc 1138−262,including their relative spatial location to the rg, ages, masses and star formation rates from spectral energy distribution fits and star formation rates from the hα emission line flux. in section 6 we summarise the results and draw some wider conclusions. throughout this paper we assume a cosmology of h0=71 km s−1, ωm = 0.27, ωλ=0.73 (spergel et al. 2003) and magnitudes are on the vega system. 2. target selection 2.1. proto-clusters we have concentrated the initial stages of our spectroscopic campaign on mrc 1138−262 and mrc 0943−242 for several reasons. first, these two proto-clusters fields were both observed in the space infrared by a spitzer survey of 70 high redshift radio galaxies (rgs; seymour et al. 2007). they are also amongst the eight z > 2 radio galaxies observed in the broad and narrow-band imaging survey of venemans et al. (2007). both targets therefore are among the best studied high-redshift radio galaxies and have a large quantity of broadband data available from the u-band to the spitzer bands, over a wide field of view. their redshifts span the 2 < z < 3 region in which it has been shown that the red sequence most likely starts to build up kodama et al. (2007), and they therefore probe an interesting and poorly understood redshift regime for understanding the build-up of red, massive galaxies in proto-clusters. our group previously obtained deep, wide-field near-infrared imaging of mrc 1138−262 and mrc 0943−242 with the multi-object infrared camera and spectrograph (moircs; suzuki et al. 2008; ichikawa et al. 2006), the relatively new 4′ × 7′ imager on subaru (kodama et al. 2007). the data were ob- tained in exceptional conditions, with seeing ranging from 0. ′′5 to 0. ′′7. we use our previously published photometric catalogues, and quote the total magnitudes (magauto in sextractor) for individual band photometry, and 1.5′′ diameter aperture magni- tudes for all colours. we refer to (kodama et al. 2007) for fur- ther details on the photometry. we also used an h−band image of mrc 1138−262 obtained with the son of isaac (sofi) on the new technology telescope (ntt) on ut 2006 march 23. the total exposure time was 9720 s. we reduced the data using the standard procedures in iraf. using this data set, we selected galaxies expected to lie at the same redshifts as the radio galaxies on the basis of their j − ks or jhks colours. these deep data indicate a clear excess of the near-infrared-selected galaxies clustered towards the radio galaxies (e.g., figures 3–5 in kodama et al. 2007). the excess is a factor of ≈2 −3 relative to the goods-s field. 2.2. mrc 1138–262 at z = 2.16 the red galaxies in mrc 1138−262 were selected according to the 'classic' drg criterion of j −ks > 2.3 (van dokkum et al. 2004). we note that in kodama et al. (2007) the zero points of the photometry in this field were incorrect by 0.25 and 0.30 mag- nitudes in j and ks bands, respectively, with j′ = j −0.25, k′ s = ks −0.30 for this field, where the primed photometry represents the corrected values. since the corrections for both bands are similar, there is little change in the sample of drgs identified, which was the primary thrust of that paper. we have adjusted the magnitudes and colours before revising the tar- get selection for the current paper. we also include three near- infrared spectroscopic targets in the field of mrc 1138−262 from the hubble space telescope nicmos imaging reported by zirm et al. (2008), two of which fulfill the drg criterion and one of which is slightly bluer. we targeted 33 red galaxies in this field, down to a ks magnitude of 21.7 . 2.3. mrc 0943–242 at z = 2.93 for the redshift z ∼3 proto-cluster mrc 0943−242, we de- fine two classes of colour-selected objects: blue jhks galaxies (bjhks) defined to have: j −ks > 2(h −ks) + 0.5 and j −ks > 1.5, and red jhks galaxies (rjhks) defined to have: j −ks > 2(h −ks) + 0.5 and j −ks > 2.3. figure 1 shows these colour cuts and the objects which were targeted for spectroscopy. note that although we have not used h−band data in the selection of targets for mrc 1138−262, we include a jhks diagram for that field as a comparison to mrc 0943−242 - it is obvious from these diagrams that the two colour cuts select distinct populations of objects. figure 2 demonstrates how our jhks selection works to se- lect candidate proto-cluster members around mrc 0943−242. the solid curves represent the evolutionary tracks of galaxies over 0 < z < 4 with different star formation histories (passive, in- termediate, and active) formed at zform = 5 (kodama et al. 1999). the dashed line connects the model points at z = 3, and indicates the region where we expect to find galaxies at a similar redshift to the radio galaxy. we therefore apply the colour-cut (shown by the dot-dashed lines) to exclusively search for galaxies associ- ated with the radio galaxies at 2.3 < z < 3.1. this technique was originally used by kajisawa et al. (2006) and has a great advan- tage over the drg selection used at lower redshift since we also select the bluer populations within the redshift range of interest. m. doherty et al.: spectroscopy of red galaxies in z ∼2.5 proto-clusters 3 0.0 0.5 1.0 1.5 2.0 h−ks 1 2 3 4 j−ks mrc 0943−242 z=2.93 16 18 20 22 ks 0 1 2 3 4 5 j−ks zform=5 4 mrc 0943−242 z=2.93 (a) (b) 0.0 0.5 1.0 1.5 2.0 h−ks 1 2 3 4 j−ks mrc 1138−262 z=2.16 16 18 20 22 ks 0 1 2 3 4 5 j−ks zform=5 4 mrc 1138−262 z=2.16 (c) (d) fig. 1. spectroscopic targets shown in colour-colour (left) and colour-magnitude (right) space for mrc 0943−242 (top) and mrc 1138−262 (bottom). objects observed with optical spec- troscopy by either fors2 or focas are indicated by large circles; objects observed with near-infrared spectroscopy by moircs are indicated by squares. with respect to the radio galaxy, filled blue symbols are confirmed foreground objects and red filled symbols are background galaxies. green symbols show galaxies confirmed to lie at the redshift of the radio galaxy. we targeted 38 jhks selected galaxies (including the rg itself) down to a magnitude ks = 22. in section 4.1, we will show that we could measure redshifts of half of these sources with a suc- cess rate of our selection technique of ∼55%. in the remaining spectroscopic slits, we included 22 additional "filler" targets (see table 2), mostly those with colours just outside our selection cri- terion, but also two distant red galaxies (j−k > 2.3) and three lyα emitters. where two or more objects clashed in position, we priori- tised objects with 24μm detections from the multiband imaging photometer for spitzer (mips; rieke et al. 2004). for the op- tical spectroscopy, we also prioritised objects with brighter i- band magnitudes. after selecting primary targets according to the above colour criteria, a couple of additional 'filler' targets were selected depending on the geometry of the mask. these filler objects were drawn from a list of candidate ly-α emit- ters which do not yet have confirmed redshifts (venemans et al. 2007). figure 1 shows the distribution of selected targets in colour-colour and colour-magnitude space. whilst by no means complete sample, we believe our sample to be representative of the candidate objects. 3. observations and data reduction we obtained optical multi-object spectra of targeted galaxies in the two proto-clusters using the faint object camera and 0.0 0.5 1.0 1.5 2.0 h−ks 0 1 2 3 4 5 j−ks z=0 z=1 z=2 z=3 z=4 bulge (e) b+d (50% + 50%) disk (t=5gyr) e(b−v)=0.2 fig. 2. colour selection technique designed to select galaxies at 2.3 < z < 3.1. the solid curves represent the evolutionary tracks of galaxies over 0 < z < 4 with different star formation histories (passive, intermediate, and active, from top to bottom, respec- tively) formed at zform = 5 (kodama et al. 1999). the dashed line connects the model points at z = 3. objects targeted spectroscop- ically are represented by circles; filled circles indicate where a redshift identification has been made. blue circles indicate ob- jects at z < 2.3 and red circles indicate sources at z > 3.1 - i.e., outside of the targeted redshift range. cyan circles show objects whose redshifts fall within the targeted range. spectrograph (focas; kashikawa et al. 2000) on subaru, and the focal reducer and low dispersion spectrograph-2 (fors2; appenzeller & rupprecht 1992) on the vlt1. we obtained near- infrared multi-object spectra of targeted galaxies using moircs on subaru. for one galaxy, we also obtained a spectrum with x- shooter (d'odorico et al. 2006) on the vlt during a com- missioning run. a summary of the observations is presented in table 1. 3.1. optical spectra we were scheduled two nights, ut 2007 march 14-15, of focas spectroscopy on subaru. however, 1.5 out of the two nights were lost to bad weather. in the first half of the second night, we obtained three hours integration on mrc 0943−242 in variable seeing. we used the 300b grism with the l600 filter and binned the readout 3 × 2 (spatial × spectral). this set-up gives a dispersion of 1.34 å pix−1 and a spectral resolution of 9 å full width at half maximum (fwhm; as measured from sky lines). there were 22 slits on the mask, but the spectra for three objects fell on bad ccd defects so we treat these as unobserved. two ly-α emitter candidates were also placed on the mask to fill in space where there were no suitable near-infrared selected galax- ies. the data were reduced in a standard manner using iraf and custom software. although the night was not photometric, spec- tra were flux calibrated with standard stars observed on the same night, as it is important to calibrate at least the relative spectral response given that we are interested in breaks across the spec- tra. we obtained fors2/mxu spectra with ut2/kueyen on ut 2008 march 11-12. the nights were clear with variable seeing averaging around 1′′. the 300v grating was used, providing a 1 eso program 080.a-0463(b). 4 m. doherty et al.: spectroscopy of red galaxies in z ∼2.5 proto-clusters dispersion of 3.36 å pix−1 and a resolution of 12 å (fwhm, measured from sky lines). slitlets were 1′′ wide and 6−10′′ long. we used 2′′ nods to shift objects along the slitlets, thereby cre- ating a-b pairs to improve sky subtraction. we first created time-contiguous pairs of subtracted frames, and then averaged all of the a−b pairs together in order to improve the result- ing sky subtraction and to reject cosmic rays (in both the pos- itive and negative images). these combined frames were then processed using the fors2 pipeline (ver. 3.1.7) with standard inputs. the extracted one-dimensional spectra were boxcar ex- tracted over a 1′′ aperture and calibrated with flux standards taken on the same night. in total, we obtained 10 hours of in- tegration on mrc 0943−242 and five hours of integration on mrc 1138−262. 3.2. near-infrared spectra we obtained moircs spectra for mrc 1138−262 over the lat- ter halves of ut 2008 january 11-12. both nights were photo- metric, and we obtained a total of five hours of integration. since chip 1 of moircs was the engineering grade chip at that time, we gained no useful data on that chip and essentially lost half of the field coverage. this detector suffered a loss of sensitivity compared to chip 2. more problematic was a prominent large ring-like structure with significantly high (and variable) dark noise, as well as several smaller and patchy structures, which we were unable to calibrate out successfully. we obtained k-band spectra with the medium resolution (r1300) grating, yielding a dispersion of 3.88 å pix−1 and a resolution of 26 å (fwhm, measured from sky lines). the objects were nodded along the slitlets and a-b pairs combined to remove the sky background. the frames were then flat-fielded and corrected for optical dis- tortion using the mscgeocorr task in iraf. they were then ro- tated to correct for the fact that the slits are tilted on the detector. we extracted spectra using a box-car summation over the width of the continuum, or, in the case of no continuum, over the spa- tial extent of any detected line emission. we obtained moircs data for mrc 0943−242 on ut 2009 january 10, with four hours total integration time over the latter half of the night. the low resolution hk500 grating was used, providing a dispersion of 7.72 å pix−1 and a resolution of 42 å (fwhm, measured from sky lines). 3.3. uv/optical/nir spectrum we also obtained data for a single target in the field of mrc 0943−242 with x-shooter during a commissioning run on 2009 june 5 under variable seeing conditions (between 0.5′′ and 2′′ during the exposure). slit widths of 1.0′′, 0.9′′ and 0.9′′ were used in the uv–blue, visual–red and near–ir arm, re- spectively, resulting in resolutions of ruvb=5100, rvis=8800 and rnir=5100. the data were reduced using a beta version of the pipeline developed by the x-shooter consortium (goldoni et al. 2006). the visual-red arm spectrum (550nm– 1000nm) presented below is a combination of three 1200s expo- sures. the near-ir arm spectrum (1000nm-2500nm) is a combi- nation of two 1300s exposures in a nodding sequence. 4. results/redshift identifications redshifts were identified by visual inspection of both the pro- cessed, stacked two-dimensional data and the extracted, one- dimensional spectra. we based redshifts on both emission and absorption line features, as well as possible continuum breaks and assigned confidence flags. table 2 shows the sources observed with optical spec- troscopy and the corresponding redshift identifications and qual- ity flags. quality 1 indicates confirmed, confident redshifts; qual- ity 2 indicates doubtful redshifts; quality 3 indicates that we de- tected continuum emission but no identification was made, typi- cally due to the low signal-noise ratio (s/n) of the data; and qual- ity 4 indicated that no emission was detected. in some cases we can infer an upper limit to the redshift if the continuum extends to the blue wavelength cutoffof the optical spectrum. this is useful for distinguishing foreground objects. in practice, due to the lower efficiency of the grisms, the spectra become too noisy to distinguish the continuum below 3900 å for fors and 4100 å for focas, corresponding to ly-α at z=2.3 and z=2.4, re- spectively. 4.1. mrc 0943−242 out of 32 objects observed with fors2 in mrc 0943−242, 16 redshifts have been identified (fig. 3). nineteen objects were ob- served with focas, including 17 near-infrared-selected galax- ies and two ly-α candidates observed as filler objects (fig. 4). we obtained five redshifts, only one of which lies close to the redshift of the radio galaxy, lae#381 at z = 2.935 (quality 1) and this is not one of our red galaxy candidate members but was one of the lyman-alpha emitter candidates. the low success rate with focas is most likely due to the short exposure time since that observing run was largely weathered out. we identified a further 8 redshifts with moircs (out of 21 targeted sources), giving a total of 27 sources in the field of mrc 0943−242 with spectroscopic redshifts, out of 57 targeted (fig. 5). we also observed one object (#792) with x-shooter. the visual-red arm spectrum (fig. 6) shows one line at 8985.7å. the near-ir arm spectrum (fig. 7) shows two lines at 15815.7å and 15868.2å. we identify those lines as [oii]λ3727, hα and [nii]λ6583 at z=1.410. this redshift is consistent with the one determined independently from the spectrum obtained with fors. from the remaining 30 sources, we obtained five redshift up- per limits from the optical spectroscopy, excluding these sources from being members of the proto-cluster. the remaining 25 ob- jects cannot be excluded as either foreground or background ob- jects, and an unknown fraction of these may even be at the red- shift of the radio galaxy. however, we found an extended fore- ground structure in this field at redshift z ∼2.6. the redshift distribution is shown in figure 8. this foreground structure com- plicates any interpretation of the red sequence, or lack thereof, in this proto-cluster, as a large number of galaxies along the 'red sequence' in mrc 0943−242 may be members of this interven- ing structure. however, it is in itself very interesting as it lies at a high redshift and may also be a young, forming cluster. we find six galaxies in this structure, with a mean redshift < z >= 2.65 and ∆z = 0.03. several of these sources may contain agn fea- tures (civ, heii lines), though the s/n of our spectra is insuffi- cient to make a clear statement about this. for one object, #420, we obtained two different redshifts from fors2 and moircs. upon closer inspection of the i- band and k−band images, we found out that this is actually a pair of galaxies: (i) a blue one at z =0.458 detected in the i−band image , and used to centre our fors2 spectroscopic slit (fig. 3), and (ii) a red one at z = 2.174 detected in the k−band image and m. doherty et al.: spectroscopy of red galaxies in z ∼2.5 proto-clusters 5 fig. 3. fors2 spectra of objects with identified redshifts in the field of mrc 0943−242. objects are ordered by redshift from low to high. the vertical green line indicates the region of atmospheric absorption at 7600 å. the horizontal dashed line indicates zero 6 m. doherty et al.: spectroscopy of red galaxies in z ∼2.5 proto-clusters fig. 3. cont. m. doherty et al.: spectroscopy of red galaxies in z ∼2.5 proto-clusters 7 fig. 3. cont. 8 m. doherty et al.: spectroscopy of red galaxies in z ∼2.5 proto-clusters fig. 4. focas spectra of objects with identified spectra in the field mrc 0943−242. m. doherty et al.: spectroscopy of red galaxies in z ∼2.5 proto-clusters 9 fig. 5. moircs spectra of objects with identified redshifts in the field mrc 0943−242. sky lines are shown as a dotted green spectrum beneath each object spectrum. 10 m. doherty et al.: spectroscopy of red galaxies in z ∼2.5 proto-clusters table 1. summary of observations. proto-cluster instrument fov filter grating spectral coveragea (μm) ut date obs. total exp. mrc 1138−262 moircs 4′ × 3. ′5 k r1300 2.0 −2.4 11-12 jan 2008 5 hr mrc 1138−262 fors2 7′ × 7′ – 300v 0.33 −0.9 11-12 mar 2008 5 hr mrc 0943−242 fors2 7′ × 7′ – 300v 0.33 −0.9 11-12 mar 2008 10 hr mrc 0943−242 focas 6′ diameter l600 300b 0.37 −0.6 15 mar 2007 3 hr mrc 0943−242 moircs 4′ × 7′ oc hk hk500 1.3 −2.5 10 jan 2009 4 hr mrc 0943−242 x-shooter 1′′ × 11′′ – – 0.3 −2.5 5 jun 2009 1 hr a exact spectral coverage for each object depends on the position of the slit on the detector. fig. 6. x-shooter visual–red arm spectrum of object #792 in the field of mrc 0943−242. used to centre the moircs slit (fig. 5). these two galaxies are separated by only 1. ′′1. a hint of the blue galaxy is seen in the k−band image, and vice versa, a hint of the red galaxy is seen in the i−band image, but none of these can be properly deblended. the red galaxy at z = 2.174 is the one selected by our colour criterion. more than a half (10/18, or 56%) of the jhks selected galax- ies whose redshifts were successfully measured turn out to be actually located in the redshift range 2.3 < z < 3.1. of the red sequence candidates (i.e., rjhk galaxies), 6 out of 10 are within the range 2.3 < z < 3.1. these numbers are summarised in table 4. the majority of redshift identifications were based on emission lines, though in several cases the fors2 data yielded absorption line identifications where the continuum is strong enough (e.g., see objects #749 and #760 in figure 3). 4.2. mrc 1138−262 we observed 27 objects (25 red galaxies and 2 lα emitters) in the mrc 1138−262 field with optical spectroscopy, but could identify only two redshifts, both of which are background ly-α emitters at z > 3 (figure 9). no redshifts were confirmed at the fig. 7. x-shooter nir arm spectrum of object #792 in the field of mrc 0943−242. redshift of the radio galaxy from optical spectroscopy. although the total exposure time is half of that spent on mrc 0943−242, given the 50% confirmation rate in that field we may naively ex- pect a higher success rate here given the lower redshift of this radio galaxy. however, about twice as many sources (25 out of 57) in the mrc 0943−242 field have i < 24.5, compared to the mrc 1138−262 field (7 out of 33), so the lower success rate in mrc 1138−262 may be partially due to the sources being fainter. we are confident that the astrometry in both masks is ac- curate - confirmed by several stars and bright galaxies (includ- ing the radio galaxy itself) placed on the slitmasks. the astrom- etry was internally consistent as the slit positions were derived from preimaging obtained with fors2. however, the efficiency of the 300v grism drops offquickly at the blue end, and since the sources observed in this field have such faint magnitudes, it is likely not possible to detect lyman breaks at the redshift of the radio galaxy; we estimate a minimum redshift of z = 2.2 to detect ly-α with fors2, and the sources were too faint to identify redshifts from absorption lines. furthermore, given the difference in colour cuts from mrc 0943−242 which selects in- trinsically different populations (figure 1), we do not expect to find many higher redshift interlopers. m. doherty et al.: spectroscopy of red galaxies in z ∼2.5 proto-clusters 11 table 2. results of optical and near-ir spectroscopy in the mrc 0943−242 field. id typea ra(j2000) dec.(j2000) vtotal itotal jtotal ks,total j −ks (1. ′′5) instr. z qualityb 897 bjhk 09 45 25.34 −24 25 40.3 >26.8 >25.6 23.69 21.65 2.15 fors2 ... 4 420 rjhk 09 45 25.57 −24 28 59.8 ...c ...c 23.65 21.35 2.65 focas ... 3 moircs 2.174 2 413 rjhk 09 45 25.74 −24 30 28.8 >26.8 >25.6 21.85 19.79 2.41 moircs ... 3 410 bjhk 09 45 26.07 −24 30 20.2 23.73 22.85 21.78 20.36 1.52 moircs 1.420 2 402 rjhk 09 45 26.15 −24 29 38.2 >26.8 >25.6 23.38 20.34 2.96 moircs 2.466 2 873 bjhk 09 45 26.16 −24 26 48.2 24.54 23.27 21.64 19.99 1.59 focas 0.272 1 860 filler 09 45 26.55 −24 28 14.8 27.2 24.99 21.28 19.01 2.25 fors2 ... 4 387 bjhk 09 45 26.90 −24 31 08.2 >26.8 >25.6 23.29 20.02 2.28 fors2 ... 4 845 rjhk 09 45 27.07 −24 28 44.5 25.3 24.01 22.24 19.60 2.72 fors2 2.335 2 moircs ... 3 373 filler 09 45 27.27 −24 30 37.7 24.70 23.52 21.63 20.06 1.59 fors2 <2.3 3 830 rjhk 09 45 27.39 −24 26 45.9 >26.8 >25.6 22.94 20.39 2.93 fors2 ... 4 moircs ... 4 368 filler 09 45 27.46 −24 30 46.8 24.68 22.79 20.95 18.93 2.18 focas ... 4 803 bjhk 09 45 28.25 −24 26 03.8 24.31 23.82 22.89 21.10 1.66 fors2 2.68 1 792 bjhk 09 45 28.52 −24 26 16.8 24.74 23.19 21.43 19.67 1.78 fors2 1.408 1 focas <2.4 3 moircs ... 3 xshoot 1.410 1 341 bjhk 09 45 28.76 −24 29 10.9 25.20 23.12 21.53 19.44 2.07 focas <2.4 3 775 filler 09 45 29.12 −24 28 23.9 26.47 24.60 22.75 20.35 2.14 fors2 ... 4 760 rjhk 09 45 29.58 −24 28 02.5 >26.8 >25.6 23.70 21.25 2.33 fors2 2.65 2 749 rjhk 09 45 29.91 −24 27 24.7 >26.8 >25.6 22.01 19.71 2.53 fors2 2.635 1 moircs ... 3 722 filler 09 45 30.36 −24 25 26.8 23.96 22.40 22.18 20.44 1.62 fors2 ... 3 moircs 1.507 1 700 bjhk 09 45 31.14 −24 27 01.3 24.58 23.91 22.75 20.43 2.26 focas 2.386 1 267 rjhk 09 45 31.31 −24 29 34.8 24.96 24.60 23.24 20.83 2.43 fors2 <2.3 3 focas ... 4 moircs 2.419 1 694 filler 09 45 31.38 −24 27 39.1 26.81 24.43 23.01 21.21 1.93 fors2 ... 4 690 filler 09 45 31.50 −24 27 17.8 24.31 23.78 22.48 20.99 1.68 focas <2.4 3 686 rjhk 09 45 31.58 −24 26 41.0 >26.8 >25.6 23.14 20.60 2.83 moircs ... 3 250 filler 09 45 31.77 −24 29 24.4 >26.8 >25.6 22.34 20.59 1.71 moircs ... 3 675 rjhk 09 45 31.99 −24 28 14.8 24.96 24.51 22.89 20.33 2.47 focas 2.520 1 241 bjhk 09 45 32.28 −24 28 50.6 >26.8 >25.6 22.88 20.96 1.89 fors2 1.16 2 660 rjhk 09 45 32.53 −24 26 38.0 25.06 24.26 22.87 21.07 2.63 fors2 1.12 2 hzrg bjhk 09 45 32.76 −24 28 49.3 ... ... 20.65 19.12 1.70 moircs 2.923 1 221 rjhk 09 45 32.81 −24 29 28.2 >26.8 >25.6 22.26 20.38 2.39 moircs ... 4 646 bjhk 09 45 32.97 −24 27 59.6 25.73 24.26 23.35 21.34 1.93 focas ... 4 623 rjhk 09 45 33.66 −24 27 13.0 26.45 25.04 24.85 22.54 2.74 fors2 ... 4 175 filler 09 45 35.06 −24 29 24.3 25.02 24.09 22.23 20.15 1.89 fors2 <2.8 3 174 rjhk 09 45 35.11 −24 31 34.6 25.46 25.37 22.51 19.82 2.75 focas ... 4 167 drg 09 45 35.32 −24 29 46.8 >26.8 >25.6 22.24 20.59 3.02 fors2 0.464 2 153 rjhk 09 45 35.72 −24 31 38.1 26.51 23.66 23.00 20.62 2.59 fors2 3.94 2 143 bjhk 09 45 36.10 −24 30 34.4 24.98 24.00 23.33 21.25 1.96 focas ... 3 139 rjhk 09 45 36.34 −24 29 56.9 >26.8 >25.6 21.99 19.80 2.52 moircs ... 3 138 filler 09 45 36.42 −24 30 25.6 24.29 23.65 22.27 20.95 1.37 fors2 1.34 1 544 bjhk 09 45 36.52 −24 26 11.8 24.59 23.75 23.14 21.72 1.58 focas ... 3 130 rjhk 09 45 36.82 −24 29 51.8 >26.8 25.33 23.53 19.90 3.63 moircs ... 3 123 filler 09 45 37.03 −24 31 18.1 >26.8 >25.6 21.60 19.78 1.93 fors2 ... 3 116 filler 09 45 37.23 −24 31 48.5 26.45 23.78 23.11 21.75 1.34 fors2 2.58 2 114 rjhk 09 45 37.23 −24 29 59.2 >26.8 >25.6 24.91 21.10 3.38 fors2 ... 4 110 filler 09 45 37.26 −24 30 20.8 >26.8 >25.6 22.22 20.39 1.87 fors2 2.65 1 103 bjhk 09 45 37.41 −24 30 12.1 24.81 24.23 23.61 21.36 2.10 fors2 2.62 1 focas 2.625 1 522 bjhk 09 45 37.44 −24 27 50.4 27.5 25.23 23.15 21.42 1.78 fors2 ... 4 100 rjhk 09 45 37.47 −24 29 09.5 24.61 23.73 23.10 20.16 2.55 moircs 2.649 2 511 rjhk 09 45 37.84 −24 28 43.9 25.7 24.91 22.75 20.04 2.88 moircs ... 3 498 drg 09 45 38.26 −24 28 08.6 26.4 24.59 22.34 19.99 2.39 focas ... 4 72 rjhk 09 45 38.30 −24 30 46.0 >26.8 >25.6 22.86 20.81 2.35 fors2 ... 4 495 filler 09 45 38.33 −24 27 03.3 >26.8 >25.6 22.20 20.44 1.73 fors2 ... 3 64 filler 09 45 38.82 −24 30 39.0 26.0 25.07 22.86 20.83 2.12 focas ... 4 39 filler 09 45 39.72 −24 32 02.6 25.4 24.13 22.22 20.57 1.59 fors2 3.76 2 34 rjhk 09 45 39.92 −24 29 19.7 27.9 25.75 22.60 19.99 2.60 moircs ... 3 23 rjhk 09 45 40.40 −24 29 00.6 >26.8 >25.6 24.31 21.00 3.23 moircs ... 4 16 filler 09 45 40.57 −24 31 24.7 24.16 23.48 22.34 20.93 1.47 fors2 2.44 1 focas <2.7 2 lae1141 lae 09 45 29 518 24 29 15 72 l itt did t moircs 3 495 2 12 m. doherty et al.: spectroscopy of red galaxies in z ∼2.5 proto-clusters table 3. results of optical and near-ir spectroscopy in the mrc 1138−262 field. id ra(j2000) dec.(j2000) btotal itotal jtotal ks,total j −ks (1. ′′5) instr. z qualitya 493 11 40 33.42 −26 29 14.6 >27 >26.2 22.89 20.55 2.56 fors2 ... 4 867 11 40 34.09 −26 30 38.7 >27 25.32 >23.3 21.30 2.49 fors2 ... 4 156 11 40 34.82 −26 27 46.4 >27 >26.2 >23.3 21.15 4.23 fors2 ... 4 174 11 40 35.45 −26 27 51.8 25.35 25.06 >23.3 >22 2.51 fors2 ... 3 676 11 40 36.96 −26 29 55.6 >27 >26.2 22.57 19.79 3.07 fors2 ... 4 808 11 40 38.19 −26 30 24.3 >27 24.87 22.91 20.44 2.56 fors2 ... 4 552 11 40 38.28 −26 29 25.1 >27 25.37 22.67 20.47 2.41 moircs ... 4 364 11 40 39.76 −26 28 45.4 24.17 22.87 21.24 19.03 2.21 moircs ... 3 877 11 40 40.24 −26 30 41.2 25.96 24.91 23.44 21.30 2.35 fors2 ... 3 341 11 40 41.47 −26 28 37.2 >27 24.81 23.63 20.48 3.06 fors2 3.263 1 147 11 40 43.48 −26 27 44.1 >27 >26.2 21.98 20.04 2.43 fors2 ... 3 369 11 40 43.46 −26 28 45.2 >27 >26.2 >23.3 21.30 3.83 moircs ... 4 905 11 40 44.01 −26 30 47.0 >27 >26.2 23.45 20.86 2.69 moircs ... 4 456 11 40 44.28 −26 29 07.7 >27 24.71 21.28 18.93 2.45 fors2 ... 3 moircs 2.172 2 464 11 40 46.09 −26 29 11.5 >27 >26.2 21.54 19.06 2.41 moircs 2.149 2 558 11 40 46.52 −26 29 27.1 >27 24.71 21.64 19.03 2.55 moircs ... 4 476 11 40 46.68 −26 29 10.4 >27 24.90 22.30 19.86 2.46 fors2 ... 3 605 11 40 47.58 −26 29 38.2 >27 >26.2 >23.3 21.07 3.51 fors2 ... 4 moircs ... 4 830 11 40 48.36 −26 30 30.6 >27 20.48 20.67 18.86 2.20 moircs ... 3 492 11 40 49.59 −26 29 07.7 >27 25.60 23.09 20.02 2.84 fors2 ... 4 392 11 40 50.35 −26 28 49.6 >27 >26.2 23.47 20.56 2.47 fors2 ... 4 573 11 40 50.75 −26 29 32.5 >27 24.68 21.98 20.04 2.06 moircs ... 4 601 11 40 51.30 −26 29 38.5 >27 23.45 21.27 19.05 2.35 fors2 ... 3 506 11 40 53.13 −26 29 18.1 >27 23.55 21.33 19.01 2.38 fors2 ... 3 165 11 40 54.01 −26 27 48.1 >27 >26.2 23.35 20.85 2.59 fors2 ... 4 241 11 40 54.77 −26 28 03.6 >27 25.44 23.16 20.78 2.76 fors2 ... 3 223 11 40 55.99 −26 28 02.9 26.13 24.30 >23.3 21.36 2.37 fors2 3.455 1 268 11 40 57.02 −26 28 17.5 >27 23.02 20.57 18.35 2.64 fors2 ... 3 206 11 40 57.87 −26 27 59.3 >27 24.52 23.51 20.28 2.40 fors2 ... 3 565 11 41 00.10 −26 29 28.1 >27 >26.2 22.72 20.31 2.57 fors2 ... 4 451 11 41 00.84 −26 29 04.7 >27 >26.2 >23.3 21.26 3.11 fors2 ... 4 90 11 41 01.54 −26 27 31.2 >27 23.55 21.38 18.89 2.51 fors2 ... 3 728 11 41 02.69 −26 30 07.8 >27 >26.2 21.03 18.89 2.56 fors2 ... 4 ... 11 40 43.85 −26 31 26.4 ly-α emitter candidate fors2 ... 3 ... 11 40 50.68 −26 31 00.0 ly-α emitter candidate fors2 ... 4 aquality flags are the same as in table 2. limiting magnitudes are 5σ. table 4. spectroscopy summary for near-ir selected candidates. selection criterion # candidates # targeted # confirmed redshifts # @ 2.3 < z < 3.1 # @ zrg mrc 0943−242 rjhk 62 23 10 6 0 bjhk 70 15 8 4 1a total 132 38 18 10 1a mrc 1138−262 drg 97 30 15 4 2 a this is the radio galaxy itself, which is also selected as a bjhk candidate. from the moircs spectra, we detected two emission line objects (out of 12 targets), which, if identified as hα, places the objects at the same redshift as the radio galaxy. one of these is a clear detection, the other is marginal (3σ). both the two- dimensional and extracted one-dimensional spectra are shown in figures 10 and 11. the target information for these two objects are shown in table 3. the small number of confirmed redshifts in this field is in fact not surprising given that detecting interstellar absorp- tion lines and the lyman-break in the optical was unsuccessful. as we seem to detect only a surprisingly low number of star- forming red galaxies in our colour-selection, the remaining frac- tion of passively evolving red sequence galaxies will not have strong emission lines, and hence will be harder to confirm red- shifts of. our current results show that the most effective strategy would in fact be to observe much deeper in the near-ir so that we can detect the 4000å continuum break feature, if present (e.g., kriek et al. 2006). in fact, there are a total of 13 objects in which we detect continuum in our near-ir spectra, but in general it is too weak and/or diffuse to identify any features with the current data. m. doherty et al.: spectroscopy of red galaxies in z ∼2.5 proto-clusters 13 fig. 9. fors2 spectra of background ly-α emitters identified in the field of mrc 1138−262. 5. properties of the two confirmed red galaxies in mrc 1138−262 these are the first examples of red galaxies confirmed as mem- bers in a proto-cluster above z > 2. here we compare their physi- cal properties, insofar as possible with the information available, with the other confirmed proto-cluster members which are gen- erally small, blue, star-forming, ly-α and hα emitters. we make use of the multi-wavelength broad-band data to fit spectral energy distributions (seds) and deduce ages, masses and star formation rates (sfrs) for these two galaxies. we also calculate the sfrs from the hα line fluxes in the moircs spectra and from spitzer 24 μm imaging of the mrc 1138−262 field. finally, given the projected spatial location with respect to the rg, and the calculated stellar masses, we attempt to in- fer whether or not these galaxies will eventually merge with mrc 1138−262 . 5.1. stellar masses using broad-band ubrijhks[3.6][4.5][5.8][8.0] photometry, we fit the seds of the two galaxies confirmed to lie at the red- shift of the mrc 1138−262 to model templates with two aims. the first is to test if the photometric redshifts produced are reli- able, which is useful for future studies. the second is to derive estimates of the physical properties of these galaxies - in par- ticular, their stellar masses. tanaka et al. (in prep.) describes the details of the model fitting, but we briefly outline the procedure here. the model templates were generated using the updated ver- sion of bruzual & charlot (2003) population synthesis code (charlot & bruzual in prep), which takes into account the effects of thermally pulsating agb stars. we adopt the salpeter (1955) initial mass function (imf) and solar and sub-solar metallicities (z = 0.02 and 0.008). we generated model templates assuming an exponentially decaying sfr with time scale τ, dust extinc- tion, and age. we implement effects of the intergalactic extinc- tion following furusawa et al. (2000), who used the recipe by madau (1995), as we are exploring the z > 2 universe. we use conventional χ2 minimizing statistics to fit the models to the ob- served data. firstly, we generate model templates at various redshifts and perform the sed fit. errors of 0.1 magnitudes are added in quadrature to all bands to ensure that systematic zero point er- rors do not dominate the overall error budget. the resulting pho- tometric redshift of the object #456 is 2.25+0.06 −0.09, which is consis- tent with the spectroscopic redshift (zspec = 2.1719). we then fix the redshift of the templates at the spectroscopic redshift and fit the sed again. we impose the logical constraint that the model galaxies must be younger than the age of the universe. the derived properties of the galaxy #456 are an age of 1.6+1.1 −0.7 gyr, an e-folding time scale τ = 0.1+0.4 −0.1 gyr, and dust extinction of τv = 0.4+1.4 −0.4, where τv is the optical depth in the v- band. it is a relatively old galaxy with a modest amount of dust, 14 m. doherty et al.: spectroscopy of red galaxies in z ∼2.5 proto-clusters fig. 8. redshift distribution for confirmed sources in the field of mrc 0943−242 at z = 2.93 (redshift indicated by an arrow). the proportion of sources in each bin with quality flag 1 (i.e. reliable redshifts) is shown with red hatching. the blue hatching shows sources with quality 3 (i.e., representing an upper limit rather than a confirmed redshift). no filling (i.e., simply the black out- line) shows all sources with redshifts (quality flags 1 and 2). fig. 10. (bottom) 2d and (top) 1d spectra for the object #464 with z=2.149. the oh skylines are shown in red underneath the 1d spectrum. the 2d and 1d spectra are on the same wavelength scale. although the error on the extinction is large. the apparent red color of the galaxy is probably due to its old stellar populations. we find a high photometric stellar mass of 2.8+1.5 −1.0 × 1011 m⊙ for this galaxy. the short time scale derived (τ of only 0.1 gyr) suggests that the galaxy formed in an intense burst of star for- mation. the low current star formation rate of 0.0+1.0 −0.0 m⊙yr−1 derived from the sed fit indeed confirms that the high star for- fig. 11. bottom: 2d spectrum of object #456 showing h-α detec- tion, middle: 2d spectrum rebinned in 2x2 pixels to emphasize the detection and top: 1d spectrum extracted from the rebinned 2d spectrum. mation phase has ended and the galaxy is now in a quiescent phase. for the galaxy #464, the best fit photometric redshift is zphot = 2.05+0.26 −0.13, consistent with the spectroscopic redshift (zspec = 2.149). the best fit at the spectroscopic redshift yields an age of 2.4+0.4 −1.0 gyr, an e-folding time τ = 0.2+0.1 −0.2 gyr, dust extinction of τv = 0.7+0.7 −0.6, a stellar mass of 5.1+1.5 −2.0 × 1011 m⊙ and a sfr of 0.0+1.0 −0.0 m⊙yr−1. the best fitting seds for each galaxy are shown in figures 13 and 12. the broad-band magnitudes of the two galax- ies are summarized in table 5. a more extensive analysis of the properties of galaxies in the mrc 1138−262 field will be pre- sented in tanaka et al. (in prep.). several earlier papers presented measurements of different classes of objects in the field of mrc 1138−262. kurk et al. (2004b) converted k-band magnitudes to masses, for the ly-α and hα emitters, estimating masses of 0.3 −3 × 1010m⊙for the ly-α emitters (except for one object with a mass of 7.5×1010m⊙) and 0.3 −11 × 1010m⊙for the hα emitters, the latter on average being clearly more massive than the ly-α emitters. furthermore hatch et al. (2009) find stellar masses between 4 × 108 and 3×1010 for candidate ly-α emitting companions within 150 kpc of the central rg. our red galaxies are thus an order of magni- m. doherty et al.: spectroscopy of red galaxies in z ∼2.5 proto-clusters 15 table 5. total magnitudes of the two hα detected galaxies. in this table, the magnitudes are on the ab system for the ease of fitting seds. magnitude limits are 3σ limits. id u b r i z j h ks 3.6μm 4.5μm 5.8μm 8.0μm 456 > 26.6 > 26.0 > 25.2 25.15 ± 0.41 > 24.2 22.19 ± 0.08 21.27 ± 0.04 20.72 ± 0.06 20.11 ± 0.03 20.00 ± 0.04 19.69 ± 0.13 20.24 ± 0.25 464 > 26.6 > 26.0 > 25.2 > 26.1 > 24.2 22.45 ± 0.11 21.39 ± 0.04 20.86 ± 0.09 19.96 ± 0.03 19.80 ± 0.03 19.63 ± 0.12 20.05 ± 0.21 fig. 12. sed fit for object #456, with the fainter hα emission line detection. fig. 13. sed fit for object #464, with the brighter hα emission line detection. tude more massive than the most massive ly-α emitters and 2–3 times more massive than the most massive hα emitters discov- ered through nb imaging searches in this field. 5.2. star formation rates we now use the line fluxes for the two hα detections to infer a lower limit to the instantaneous sfr. we obtained a rough flux calibration of the spectra by scaling the total flux in the spec- trum to the observed ks magnitude of the two objects, and as- suming a flat spectral throughput in the ks -band. as the ob- served lines are near the centre of the ks -band, this approx- imation does not significantly effect the derived hα flux, and we assume a calibration uncertainty of ≈20%. both line de- tections lie on the edge of weak telluric absorption features be- tween 20400-20800 å and also very close to sky lines (espe- cially #458), increasing the measurement uncertainties. the hα flux of object #464 thus derived is 1.3 ± 0.3 × 10−16 erg s−1 cm−2 and the fainter object, #456, has an hα flux of 6.2 ± 2 × 10−17 erg s−1 cm−2. the measured line flux in #464 is fully consistent with the published f(hα)=1.35 × 10−16 erg s−1 cm−2 derived from narrow-band imaging (object 229 of kurk et al. 2004b). the previously published near-ir spectroscopy of kurk et al. (2004a) found f(hα)=7.1±1.9 × 10−17 erg s−1 cm−2, which is somewhat lower than our flux, suggesting possible slit losses. in the same narrow-band image, object #458 does not show any excess flux compared to the full k−band image, suggesting that our line flux may be overestimated due to incomplete skyline subtraction. using the kennicutt (1998) relation, sfr (m⊙yr−1) = 7.9 × 10−42 l(hα) erg s−1, which assumes solar metallicities and a salpeter (1955) imf, this leads to sfrs of 35±8 and 17±6 m⊙yr−1, respectively. the sfrs derived from hα are higher than those from the sed fits (figs 12 and 13), which give sfr up to 4m⊙yr−1 at 2σ, i.e. a factor of 5–10 lower. an additional independent esti- mate for the sfr can be obtained from the mips 24μm imag- ing of the field (seymour et al. 2007). object #464 is detected at s(24μm)=470±30μjy, while object #458 remains undetected at at the 2σ<60 μjy level. we first convert the 24μm flux to the total ir flux using the relation of reddy et al. (2006), and con- verted the latter to a sfr using the formula of kennicutt (1998). the derived sfr are 34 and <4 m⊙yr−1 for objects #464 and #458, respectively. for #464, this is fully consistent with the the sfr derived from hα, but for #458, the value derived from hα seems strongly over-estimated. this may be either be due to the low s/n and incomplete skyline removal in our near-ir spec- troscopy, or it may indicate that the hα may have a contribution by an agn. to confirm the latter hypothesis, we need to obtain other emission lines such [oiii], but the possible agn contri- bution may not be surprising given the higher fraction of agns observed in proto-clusters (e.g. galametz et al. 2009). in summary, we find that object #458 most likely belongs to the class of passively evolving galaxies, while #464 belongs to the class of dusty star-forming red galaxies. 5.3. merging timescales the spatial locations of the two red galaxies identified at the red- shift of mrc 1138−262 are shown in figure 14. they both lie along the east-west axis, where most of the confirmed agn in this field were found (croft et al. 2005; pentericci et al. 2002) and where the brighter drgs in kodama et al. (2007) also trace a filament. the two galaxies are relatively central, being located within 1′ (500 kpc, physical) of the radio galaxy. this is signif- icant given that no obvious density gradient has been observed for the lyman-α emitters (e.g., pentericci et al. 2000; kurk et al. 2003), yet kurk et al. (2004b) argue there is an indication that 16 m. doherty et al.: spectroscopy of red galaxies in z ∼2.5 proto-clusters the hα emitters and extremely red objects have a higher density near the rg than further out. our new results and these former results are consistent with the picture that accelerated evolution takes place in high density environments. to properly assess the possibility of a future merger be- tween the rg and one or both of the red companion galax- ies, numerical simulations are needed. however, we can at least gain an idea of the relative timescales involved as follows. the crossing time can be approximated by tcross = (r3/gmtotal)1/2 (binney & tremaine 2008) where r is the distance between the two galaxies and m is the total mass (dark matter + stellar) of the central galaxy, approximately 1013m⊙in this case (hatch et al. 2009). for the closer galaxy to the rg, #464, the distance is r ∼300 kpc (physical) and the crossing time is therefore roughly 0.7 gyr. the second galaxy has a crossing time between 2–3 times longer, let us say approximately 2 gyr. for a major merger (unless it is a high speed encounter in which case the galaxies would pass through each other with little disruption) the galaxies should merge into a single system within a few cross- ing times (binney & tremaine 2008). there is therefore enough time for the two red galaxies to merge with the central rg before redshift z = 0, increasing its stellar mass of 1012m⊙(hatch et al. 2009) by a factor of at least 50% . fig. 14. relative spatial distribution of the two confirmed mas- sive galaxies in mrc 1138−262. the central radio galaxy is marked by a large green asterisk. the red galaxy candidates are marked as red points; those targeted spectroscopically are marked with circles. the two filled red circles are the galaxies confirmed to lie at the redshift of the radio galaxy. confirmed lyman-α emitters are marked as blue triangles (pentericci et al. 2000), and confirmed agn are marked as the smaller green squares (croft et al. 2005; pentericci et al. 2002). objects which have been excluded as background objects are marked with a plus sign. 6. conclusions we observed a total of 57 near-infrared selected galaxies in mrc 0943−242 and 33 in mrc 1138−262 with a mix of op- tical and near-infrared multi-object spectroscopy to attempt the confirmation of massive galaxies in the vicinity of the cen- tral radio galaxy in these candidate forming proto-clusters. we have determined that the jhks selection criteria presented in kajisawa et al. (2006) select mostly 2.3 < z < 3.1 galaxies, thus confirming the validity of this colour selection technique (10 out of 18 confirmed redshifts of jhks-selected galaxies are in this redshift range). of the 57 sources in the field of the radio galaxy mrc 0943−242 we identify 27 spectroscopic redshifts and for the remaining 30 sources we obtained five upper limits on the redshifts, excluding them to be part of the proto-cluster. the re- maining 25 objects could not be excluded as either foreground or background. an unknown fraction of these may be at the red- shift of the radio galaxy. however, we also pinpoint a foreground (but still quite distant) large scale structure at redshift z ≈2.65 in this field, so it is likely that some of the remaining galaxies are also associated with this structure. of the 33 galaxies observed in mrc 1138−262 we exclude two background objects and confirm two objects at the redshift of the radio galaxy (all on the basis of emission lines). we can- not exclude the remaining 29 objects as being potential proto- cluster members - we require deep near-infrared spectroscopy to locate the 4000å break. on the basis of sed fits, and the sfr derived from both the hα line flux and the 24 μm flux, we deduce that one of the two red galaxies confirmed to be at the redshift of the proto-cluster belongs to the class of dusty star-forming (but still massive) red galaxies, while the other is evolved and massive, and formed rapidly in intense bursts of star formation. this represents the first red galaxies to be confirmed as a mem- ber of a proto-cluster above z > 2. the only other galaxies of the same class are evolved galaxies found in an overdensity at z = 1.6 in the galaxy mass assembly ultra-deep spectroscopic survey (kurk et al. 2009), and red massive galaxies in the multi- wavelength survey by yale-chile (musyc) sample at z ∼2.3 (e.g. kriek et al. 2006). since their phase of high star forma- tion has ended (the current sfr is of order 1 m⊙yr−1), we only just detect them through their emission line fluxes. however, this shows that even in these galaxies where we do not expect strong emission lines, it is still sometimes possible to derive redshifts from the hα line. it is highly likely that there are more galaxies on the red sequence in this field which have completely ceased star formation and are thus very difficult to identify spectroscop- ically. the next stage of our spectroscopic campaign is to obtain deep near-infrared spectroscopy in the j- and h-bands to search for objects with 4000å continuum breaks. given the inferred sfrs and stellar masses of the two con- firmed z ∼2.15 galaxies, they have to have formed quite early, which fits into the down-sizing scenario (cowie et al. 1996). although this conclusion is drawn from limited numbers, the low fraction of sources with hα emission lines in our sample suggests that most of the sources probably don't have much on- going star formation (sfr< 0.5 m⊙yr−1) and so their formation epoch might be quite high. this would be contrary to the idea of the giant elliptical assembly epoch being z ∼2 −3 (e.g., van dokkum et al. 2008; kriek et al. 2008), but is consistent with results from sed age fitting of stellar populations, which point to zform > 3 (e.g., eisenhardt et al. 2008). the discrepancy may be resolved if subclumps form early and merge without in- ducing much star formation (i.e. dry merging). alternatively, the fact that these galaxies are located in over–densities may predis- pose us to structures that formed early. finally, the proximity of these two massive galaxies to the rg implies that they will have an important impact on its future evolution. given the crossing times, compared with the time re- maining until z = 0, it is plausible that one or both of the galaxies may eventually merge with the rg. m. doherty et al.: spectroscopy of red galaxies in z ∼2.5 proto-clusters 17 acknowledgements. we thank the referee g. zamorani for a very constructive referee report, which has substantially improved this paper. this work was fi- nancially supported in part by the grant-in-aid for scientific research (no.s 18684004 and 21340045) by the japanese ministry of education, culture, sports and science. cl is supported by nasa grant nnx08aw14h through their graduate student researcher program (gsrp). we thank dr. bruzual and dr. charlot for kindly providing us with their latest population synthesis code. md thanks andy bunker and rob sharp for useful discussions on manipulating moircs data. the work of ds was carried out at jet propulsion laboratory, california institute of technology, under a contract with nasa. jk acknowl- edges financial support from dfg grant sfb 439. the authors wish to respect- fully acknowledge the significant cultural role and reverence that the summit of mauna kea has always had within the indigenous hawaiian community. we are fortunate to have the opportunity to conduct scientific observations from this mountain. references appenzeller, i. & rupprecht, g. 1992, the messenger, 67, 18 binney, j. & tremaine, s. 2008, galactic dynamics: second edition (princeton university press) blakeslee, j. p., franx, m., postman, m., et al. 2003, apj, 596, l143 bruzual, g. & charlot, s. 2003, mnras, 344, 1000 cowie, l. l., songaila, a., hu, e. m., & cohen, j. g. 1996, aj, 112, 839 croft, s., kurk, j., van breugel, w., et al. 2005, aj, 130, 867 d'odorico, s., dekker, h., mazzoleni, r., et al. 2006, in society of photo- optical instrumentation engineers (spie) conference series, vol. 6269, society of photo-optical instrumentation engineers (spie) conference series eisenhardt, p. r. m., brodwin, m., gonzalez, a. h., et al. 2008, apj, 684, 905 eke, v. r., cole, s., & frenk, c. s. 1996, mnras, 282, 263 ellis, s. c., jones, l. r., donovan, d., ebeling, h., & khosroshahi, h. g. 2006, mnras, 368, 769 f ̈ orster schreiber, n. m., van dokkum, p. g., franx, m., et al. 2004, apj, 616, 40 furusawa, h., shimasaku, k., doi, m., & okamura, s. 2000, apj, 534, 624 galametz, a., stern, d., eisenhardt, p. r. m., et al. 2009, apj, 694, 1309 goldoni, p., royer, f., franc ̧ois, p., et al. 2006, in society of photo-optical instrumentation engineers (spie) conference series, vol. 6269, society of photo-optical instrumentation engineers (spie) conference series grazian, a., fontana, a., moscardini, l., et al. 2006, a&a, 453, 507 grazian, a., salimbeni, s., pentericci, l., et al. 2007, a&a, 465, 393 hatch, n. a., overzier, r. a., kurk, j. d., et al. 2009, mnras, 395, 114 hilton, m., collins, c. a., stanford, s. a., et al. 2007, apj, 670, 1000 ichikawa, t., suzuki, r., tokoku, c., et al. 2006, in presented at the society of photo-optical instrumentation engineers (spie) conference, vol. 6269, society of photo-optical instrumentation engineers (spie) conference series kajisawa, m., kodama, t., tanaka, i., yamada, t., & bower, r. 2006, mnras, 371, 577 kashikawa, n., inata, m., iye, m., et al. 2000, in society of photo-optical instrumentation engineers (spie) conference series, vol. 4008, society of photo-optical instrumentation engineers (spie) conference series, ed. m. iye & a. f. moorwood, 104–113 kennicutt, r. c. 1998, ara&a, 36, 189 kodama, t., bell, e. f., & bower, r. g. 1999, mnras, 302, 152 kodama, t., tanaka, i., kajisawa, m., et al. 2007, mnras, 377, 1717 kriek, m., van der wel, a., van dokkum, p. g., franx, m., & illingworth, g. d. 2008, apj, 682, 896 kriek, m., van dokkum, p. g., franx, m., et al. 2006, apj, 645, 44 kurk, j., cimatti, a., zamorani, g., et al. 2009, arxiv e-prints kurk, j., r ̈ ottgering, h., pentericci, l., miley, g., & overzier, r. 2003, new astronomy review, 47, 339 kurk, j. d., pentericci, l., overzier, r. a., r ̈ ottgering, h. j. a., & miley, g. k. 2004a, a&a, 428, 817 kurk, j. d., pentericci, l., r ̈ ottgering, h. j. a., & miley, g. k. 2004b, a&a, 428, 793 lidman, c., rosati, p., tanaka, m., et al. 2008, a&a, 489, 981 madau, p. 1995, apj, 441, 18 pentericci, l., kurk, j. d., carilli, c. l., et al. 2002, a&a, 396, 109 pentericci, l., kurk, j. d., r ̈ ottgering, h. j. a., et al. 2000, a&a, 361, l25 reddy, n. a., steidel, c. c., fadda, d., et al. 2006, apj, 644, 792 rieke, g. h., young, e. t., engelbracht, c. w., et al. 2004, apjs, 154, 25 salpeter, e. e. 1955, apj, 121, 161 seymour, n., stern, d., de breuck, c., et al. 2007, apjs, 171, 353 spergel, d. n., verde, l., peiris, h. v., et al. 2003, apjs, 148, 175 stanford, s. a., romer, a. k., sabirli, k., et al. 2006, apj, 646, l13 suzuki, r., tokoku, c., ichikawa, t., et al. 2008, pasj, 60, 1347 tanaka, m., kodama, t., arimoto, n., et al. 2005, mnras, 362, 268 van dokkum, p. g., franx, m., f ̈ orster schreiber, n. m., et al. 2004, apj, 611, 703 van dokkum, p. g., franx, m., kriek, m., et al. 2008, apj, 677, l5 van dokkum, p. g. & stanford, s. a. 2003, apj, 585, 78 van dokkum, p. g., stanford, s. a., holden, b. p., et al. 2001, apj, 552, l101 venemans, b. p., r ̈ ottgering, h. j. a., miley, g. k., et al. 2005, a&a, 431, 793 venemans, b. p., r ̈ ottgering, h. j. a., miley, g. k., et al. 2007, a&a, 461, 823 zirm, a. w., stanford, s. a., postman, m., et al. 2008, apj, 680, 224

0911.1718

time-dependent quantum transport with superconducting leads: a discrete basis kohn-sham formulation and propagation scheme

in this work we put forward an exact one-particle framework to study nano-scale josephson junctions out of equilibrium and propose a propagation scheme to calculate the time-dependent current in response to an external applied bias. using a discrete basis set and peierls phases for the electromagnetic field we prove that the current and pairing densities in a superconducting system of interacting electrons can be reproduced in a non-interacting kohn-sham (ks) system under the influence of different peierls phases {\em and} of a pairing field. an extended keldysh formalism for the non-equilibrium nambu-green's function (negf) is then introduced to calculate the short- and long-time response of the ks system. the equivalence between the negf approach and a combination of the static and time-dependent bogoliubov-degennes (bdg) equations is shown. for systems consisting of a finite region coupled to ${\cal n}$ superconducting semi-infinite leads we numerically solve the static bdg equations with a generalized wave-guide approach and their time-dependent version with an embedded crank-nicholson scheme. to demonstrate the feasibility of the propagation scheme we study two paradigmatic models, the single-level quantum dot and a tight-binding chain, under dc, ac and pulse biases. we provide a time-dependent picture of single and multiple andreev reflections, show that andreev bound states can be exploited to generate a zero-bias ac current of tunable frequency, and find a long-living resonant effect induced by microwave irradiation of appropriate frequency.

introduction in the last two decades superconducting nanoelectron- ics has emerged as an interdisciplinary field bridging dif- ferent areas of physics like superconductivity, quantum transport and quantum computation.1–3 for practical ap- plications the reduction of heat losses in superconducting circuits constitutes a major advantage over semiconduc- tor electronics where a molecular junction is more subject to thermal instabilities.4–7 the idea of exploiting atomic-size quantum point con- tacts or quantum dots coupled to superconducting leads as quantum bits (qubit) has received significant atten- tion both theoretically and experimentally.8–11 the state of a qubit evolves in time according to the schr ̈ odinger equation for open quantum systems and can be manip- ulated using electromagnetic pulses of the duration of few nano-seconds or even faster. due to the reduced dimensionality and the high speed of the pulses these systems can be classified as ultrafast josephson nano- junctions (uf-jnj). the microscopic description of the out-of-equilibrium properties of an uf-jnj is not only of importance for their potential applications in future electronics but also of considerable fundamental interest. the quantum nature of the nanoscale device leads to a sub-harmonic gap structure,12–16 ac characteristics,17,18 current-phase relation,19,20 etc. that differ substantially from those of a macroscopic josephson junction. further- more, there are regimes in which the electron-electron scattering inside the device plays an important role.21–25 we here focus on a different relevant aspect of uf- jnj, namely the ab initio description of their short time responses. considerable theoretical progresses have been made to construct a first-principle scheme of electron transport through molecules placed between normal met- als. on the contrary, despite the recent experimental advances in fabricating superconducting quantum point contacts, a first-principle approach to superconducting nanoelectronics is still missing. furthermore, time- dependent (td) properties like the switch on/offtime of the current or the response to time-dependent ac fields or train pulses has remained largely unexplored. there are several difficulties related to the construction of a feasible time-dependent approach already at a mean-field level. the system is open, the electronic energy scales are 2-3 orders of magnitude larger than a typical supercon- ducting gap, the problem is intrinsically time-dependent (even for dc biases), and the possible formation of an- dreev bound states (abs) give rise to persistent oscilla- tions in the density and current. the time-evolution of lo- calized wave-packets scattering across a superconductor- normal interface was explored long ago.26–28 more re- cently the analysis has been extended to scattering states in superconductor-device-normal (s-d-n) junctions us- ing the wide-band-limit (wbl) approximation29 and in superconductor-device-superconductor (s-d-s) junctions by approximating the leads with finite size reservoirs.30 2 however, there has been no attempt to calculate the re- sponse of s-d-s junctions to td applied voltages using truly semi-infinite leads. in this work we propose a one-particle framework to study td quantum transport in uf-jnj, construct a suitable propagation scheme and apply it to study gen- uine td properties like the switch on/offof the current, the onset of a josephson regime, abs oscillations, ac transport and the time-evolution of multiple andreev re- flections. the one-particle framework, described in section ii a and ii b, is an extension of td superconducting density functional theory31 to systems with a discrete basis and is built on the mapping from densities to potentials pro- posed by van leeuwen32 and vignale.33 it is shown that under reasonable assumptions the current density and pairing density of an interacting system perturbed by a td electromagnetic field can be reproduced in a kohn- sham system of non-interacting electrons perturbed by a td electromagnetic and pairing fields, and that these fields are unique. in the special case of normal systems such result provides a formulation of td current density functional theory in tight-binding models. an extended keldysh formalism for the non- equilibrium nambu-green's function is introduced in section ii c and used to calculate the time-dependent current, density and pairing density of the kohn-sham hamiltonian. by adding a vertical imaginary track to the original keldysh contour34–36 we are able to extract the response of the system just after the application of the bias (transient regime) and to describe the onset of the josephson regime. we also show the equivalence be- tween the equations of motion for the nambu-green's function on the extended contour and the combination of the static and td bogoliubov-degennes equations. in section iii we illustrate a procedure for the calcula- tion of the one-particle eigenstates of a system consisting of n semi-infinite superconducting leads coupled to a fi- nite region c. these states are then propagated in time according to the td bogoliubov-degennes equations us- ing an embedded crank-nicholson algorithm which re- duces to that of refs. 37,38 in the case of normal leads. the propagation scheme is unitary (norm conserving) and incorporates exactly the transparent boundary con- ditions. the feasibility of the method is demonstrated in sec- tion iv where we calculate the td current, density and pairing density of s-d-s junctions under dc, ac and pulse biases. the paradigmatic model with a single atomic level connected to a left and right superconducting leads is investigated in detail. we provide a time-dependent picture of single and multiple andreev reflections and of the consequent formation of cooper pairs at the inter- face. we show that the smaller is the bias the longer and the more complex is the transient regime. we also study how the system relaxes after the bias is switched off. due to the presence of abs a tiny difference in the switch-off time can cause a large difference in the relaxation be- havior with persistent oscillations of tunable frequency. abs also play a crucial role in microwave ac transport. tuning the frequency of the microwave field according to the abs energy difference one produces a long-living transient resonant effect in which the amplitude of the ac current is about an order of magnitude larger than that of the current out of resonance. finally we consider one-dimensional atomic chains coupled to superconduct- ing leads. we calculate the td current density pattern along the chain for dc (ac) biases and show a clear-cut transient scenario of the multiple (photon-assisted) an- dreev reflections. a summary of the main findings and an outlook on future perspectives are drawn in section v. ii. general formulation a. hamiltonian of the system the hamiltonian of a system of interacting electrons can be written in terms of the field operators ˆ ψσ(r) ( ˆ ψ† σ(r)) which destroy (create) an electron of spin σ in position r. we expand the field operators in some suitable basis of localized orbitals φm(r) as ˆ ψσ(r) = p m ˆ cmσφm(r). assuming, for simplicity, that the φm's are orthonormal the ˆ c's operators obey the anticommu- tation relations {ˆ cmσ, ˆ c† nσ′} = δσσ′δnm. (1) in the presence of an external static electromagnetic and pairing field the hamiltonian has the general form ˆ h0 = ˆ k0 + ˆ ∆0 + ˆ ∆† 0 + ˆ hint. (2) the first term is the free-electron part and reads ˆ k0 = x σ x mn tmneiγmnˆ c† mσˆ cnσ (3) with real symmetric hopping parameters tmn = tnm and real antisymmetric phases γmn = −γnm. the phases ac- count for the presence of an external vector potential a(r), in accordance with the peierls prescription. if we use a grid basis for the expansion of the field operators with grid points rm then γmn = 1 c r rm rn dl*a(r). the sec- ond term in eq. (2) represents the pairing field operator which couples the pairing density operator to an external field and reads ˆ ∆0 = x m ∆mˆ c† m↑ˆ c† m↓. (4) we notice that the pairing field ∆m is local in the chosen basis. this term is usually set to zero since the transition to a superconducting state is caused by the interaction part. our motivation to include it at this stage will soon become clear. the interaction part of the hamiltonian 3 ˆ hint contains terms more than quadratic in the ˆ c's op- erators. we do not specify the form of ˆ hint which can be any. we, however, require that it commutes with the density operator ˆ nmσ ≡ˆ c† mσˆ cmσ [ ˆ hint, ˆ nmσ] = 0, ∀m, σ. (5) the above condition is fulfilled on a grid basis as well as in tight-binding models with hubbard-like interactions. we are interested in the dynamics of the system when an extra time-dependent electromagnetic field and pair- ing potential is switched on at t = 0. the pairing po- tential must here be considered as an independent ex- ternal field. since the time-dependent part of the scalar potential can always be gauged away we restrict to time- dependent hamiltonians of the form ˆ h(t) = ˆ k(t) + ˆ ∆(t) + ˆ ∆†(t) + ˆ hint, (6) where ˆ k(t) = x σ x mn tmneiγmn(t)ˆ c† mσˆ cnσ (7) and ˆ ∆(t) = x m ∆m(t)ˆ c† m↑ˆ c† m↓. (8) in 1994 wacker, k ̈ ummel and gross31 put forward a rigorous framework, known as td density functional theory for superconductors (scdft), to study the dy- namics of a superconducting system in the continuum case. the continuum hamiltonian can be obtained from the hamiltonian in eq. (6) with the φm's a grid ba- sis in the limit of zero spacing. they proved that given an initial many-body state |φ0⟩the current and pair- ing densities evolving under the influence of two different vector potentials a and a′ and/or two different pair- ing potentials ∆and ∆′ are always different. this re- sult renders all observable quantities functionals of the current and pairing densities, which can therefore be cal- culated in a one-particle manner.31 the original formu- lation relies on the assumption that the time-dependent current and pairing densities of the interacting hamil- tonian can be reproduced in a non-interacting hamil- tonian under the influence of another vector and pair- ing potential, i.e., that the interacting a-∆densities are also non-interacting a-∆representable. the interact- ing versus non-interacting representability assumption is present also in the original formulation of td density functional theory (dft) by runge and gross39 and td current density functional theory (cdft) by ghosh and dhara.40 the representability problem in tddft was solved by van leeuwen who proved that the td den- sity of a system with interaction ˆ hint under the influence of a td scalar potential v can be reproduced in another system with interaction ˆ h′ int under the influence of a td scalar potential v ′ and that v ′ is unique.32 we will re- fer to such result as the van leeuwen theorem. taking ˆ h′ int = 0 the van leeuwen theorem implies that the td interacting density can be reproduced in a system of non- interacting electrons. later vignale extended the van leeuwen theorem to solve the representability problem in tdcdft.33 in the next section we show that the results by van leeuwen and vignale can be further extended to solve the representability problem in tdscdft. the theory is formulated on a discete basis and it is not lim- ited to pure states, implying that we also have access to the finite-temperature domain. b. the one-particle kohn-sham scheme of tdscdft let ˆ ρ(t) be the density matrix at time t of the system described by the hamiltonian in eq. (6). we denote by o(t) ≡tr {ˆ ρ(t) ˆ o(t)} the time-dependent ensemble average of a generic operator ˆ o(t), where the "tr " sym- bol signifies the trace over a complete set of many-body states. the average o(t) obeys the equation of motion d dto(t) = ∂ ∂to(t) + itr {ˆ ρ(t)[ ˆ h(t), ˆ o(t)]}. (9) it is easy to verify that when ˆ o(t) is the density operator ˆ nm ≡p σ ˆ c† mσˆ cmσ, eq. (9) yields d dtnm(t) = x n jmn(t) −4im  ∆∗ m(t)pm(t)e−2itmmt , (10) where jmn(t) and pm(t) are the expectation value of the bond-current operator ˆ jmn(t) ≡1 i x σ  tmneiγmn(t)ˆ c† mσˆ cnσ −h.c.  (11) and pairing density operator ˆ pm(t) ≡ˆ cm↓ˆ cm↑e2i r t 0 dt′tmm = ˆ cm↓ˆ cm↑e2itmmt. (12) equation (10) is the proper extension of the continuity equation to systems exposed to a pairing field. the term ˆ ∆(t) + ˆ ∆†(t) acts as if there were td sources and sinks. notice that under the gauge transformation ˆ cnσ → eiβn(t)ˆ cnσ (with βn(0) = 0) the on-site energies change as tmm →tmm −dβm(t)/dt while the phases and the pair- ing field change according to γmn(t) →γmn(t) + βm(t) − βn(t) and ∆m(t) →∆m(t) exp[2iβm(t)]. therefore the bond-current operator ˆ jmn and pairing density operator ˆ pm are gauge invariant. in a grid basis representation with grid points rm the phases βm(t) are the discretized values of the scalar function λ(rm, t) which defines the gauge-transformed vector potential a and scalar poten- tial v : a →a + c∇λ and v →v −∂λ/∂t. the equation of motion for the bond-current jmn(t) can be cast as follows d dtjmn(t) = kmn(t) d dtγmn(t) + fmn(t). (13) 4 the first term in the r.h.s. is exactly ∂jmn(t)/∂t; the operator ˆ kmn(t) ≡p σ tmneiγmn(t)ˆ c† mσˆ cnσ + h.c.
is the energy density of the bond m-n. the second term in the r.h.s. is, therefore, the average of ˆ fmn(t) ≡ i[ ˆ h(t), ˆ jmn(t)], see eq. (9). the derivation of the equation of motion for the pairing density pm(t) is also straightforward and leads to  d dt −2itmm  pm(t) = i∆m(t)[nm(t) −1]e2itmmt + igm(t)e2itmmt, (14) with ˆ gm(t) ≡[ ˆ k(t) + ˆ hint, ˆ cm↓ˆ cm↑]. we now ask the question whether the densities jmn(t) for all bonds m-n with tmn ̸= 0 and pm(t) can be re- produced in a system with a different interaction hamil- tonian ˆ h′ int under the influence of td phases γ′(t) and pairing potential ∆′(t) starting from an initial density matrix ˆ ρ′(0). for the densities to be the same at time t = 0 we have to choose ˆ ρ′(0) and γ′(0) in such a way that tr {ˆ ρ′(0) ˆ j′ mn(0)} = tr {ˆ ρ(0) ˆ jmn(0)}, (15) tr {ˆ ρ′(0) ˆ pm(0)} = tr {ˆ ρ(0) ˆ pm(0)}. (16) notice that in the primed system the bond-current op- erator ˆ j′ mn is different from ˆ jmn since the phases γ′ are generally different from γ. on the contrary the pairing density operator is the same in the two systems. equa- tions (15,16) define the compatible initial configurations of the primed system. we answer the above question affirmatively by showing that given a compatible initial configuration [ˆ ρ′(0), γ′(0)] and under reasonable conditions there exist γ′(t) and ∆′(t) for which the bond-current and pairing density of the original and primed system are the same at all times. the formal statement is enunciated in the following theorem : given a compatible initial configuration [ˆ ρ′(0), γ′(0)] such that k′ mn(0) = tr {ˆ ρ′(0) x σ (tmneiγ′ mn(0)ˆ c† mσˆ cnσ + h.c.)} ̸= 0 (17) for all bonds m-n with tmn ̸= 0, and n′ m(0) = tr {ˆ ρ′(0)ˆ nm} ̸= 1, (18) which implies that at time t = 0 none of the orbitals φm are half filled in the primed system, there exist a unique set of continuous phases γ′(t) and pairing potential ∆′(t) that reproduce in the primed system the densities jmn(t) and pm(t) of the original system. remarks : before presenting the proof of the the- orem we discuss few relevant implications. (1) if the original system is a superconducting system with an at- tractive interaction ˆ hint and a vanishing pairing field, i.e., ˆ ∆= 0, the theorem implies that the bond-currents and pairing densities can be reproduced in a system of non-interacting electrons, i.e., ˆ h′ int = 0 perturbed by td phases γ′ and pairing field ∆′. in the following we will refer to such non-interacting system as the kohn- sham (ks) system and to the td perturbation as the ks phases and ks pairing potential. in section iii we describe how to perform the time-evolution of such ks systems for geometries relevant to quantum trans- port. (2) for interacting systems with ∆= 0 and ini- tially in equilibrium in the absence of electromagnetic fields the phases γ(0) = 0 and hence jmn(0) = 0 for all bonds. in the ks system a possible compatible initial configuration is therefore γ′(0) = 0 and ˆ ρ′(0) such that the expectation value of the one-particle density matrix n′ mn(0) = p σ tr {ˆ ρ′(0)ˆ c† mσˆ cnσ} is real. for such initial configurations the condition (17) becomes n′ mn(0) ̸= 0 for all bonds m-n with tmn ̸= 0. (3) if we ask the ques- tion whether only the bond-currents jmn(t) of a system with hamiltonian (6) and zero pairing field, i.e., ∆= 0, can be reproduced in a system with zero pairing field, i.e., ∆′ = 0, and different interactions ˆ h′ int under the in- fluence of different phases γ′ starting from some initial density matrix ˆ ρ′(0), the answer is affirmative provided that ˆ ρ′(0) and γ′(0) fulfill eqs. (15,17). this corollary ex- tends tdcdft to tight-binding models using the peierls phases as the basic ks fields and lays down the basis for a density functional td theory in discrete systems.41 we conclude this section with the proof of the theo- rem. proof : the current and pairing densities of the primed system obey the equations of motion (13,14) with kmn(t) →k′ mn(t), fmn(t) →f ′ mn(t) and nm(t) → n′ m(t), gm(t) →g′ m(t). therefore, for a generic time t the densities of the two systems are the same provided that k′ mn(t) d dtγ′ mn(t) = kmn(t) d dtγmn(t) + fmn(t) −f ′ mn(t), (19) [n′ m(t) −1]∆′ m(t) = [nm(t) −1]∆m(t) + gm(t) −g′ m(t). (20) a discussion on the existence and uniqueness of the so- lution for the coupled eqs. (19-20) is rather complicated since the dependence on the phases γ′ and potentials ∆′ in f ′ and g′ enters implicitly via the td density matrix ˆ ρ′(t). to proceed further we then follow the approach of vignale and assume that the time-dependent phases and pairing potentials and hence all expectation values are analytic functions of time around t = 0.33 expand- ing all quantities in eqs. (19-20) in their taylor series and equating the coefficients with the same power of t 5 we obtain (l + 1)k′(0) mn γ′(l+1) mn = − l−1 x k=0 (k + 1)k′(l−k) mn γ′(k+1) mn + l x k=0 (k + 1)k(l−k) mn γ(k+1) mn + f ′(l) mn −f (l) mn, (21) [n′(0) m −1]∆′(l) m = − l−1 x k=0 n′(l−k) m ∆′(k) m + l x k=0 n(l−k) m ∆(k) m −∆(l) m + g′(l) m −g(l) m , (22) where for a generic analytic function f(t) we defined f (l) as the l-th coefficient of the taylor expansion. we now show that eqs. (21-22) constitute a set of recursive re- lations to calculate all γ′(l) and ∆′(l) once all γ′(k) and ∆′(k) are known for k < l. we first observe that the l-th derivative of the density matrix ˆ ρ′(t) in t = 0 de- pends at most on the (l −1) derivative of γ′ and ∆′ since i d dt ˆ ρ′(t) = [ ˆ h′(t), ˆ ρ′(t)]. the quantity f ′ mn depends on (γ′, ∆′) implicitly through ˆ ρ′(t) and explicitly through the commutator [ ˆ h′(t), ˆ j′ mn(t)]. since the l-th derivative of the commutator depends on all (γ′(k), ∆′(k)) with k ≤l the quantity f ′(l) mn is a function of (γ′(k), ∆′(k)) with k ≤l. on the contrary, the quantities k′, g′ depend implicitly on (γ′, ∆′) through ˆ ρ′(t) but they explicitly depend only on γ′, i.e., there is no explicit dependence on the pair- ing potential ∆′. we therefore conclude that k′(l) and g′(l) depend on the γ′(k) with k ≤l and on ∆′(k) with k < l. finally, from eq. (10) we see that the l-th deriva- tive of the density n′ m(t) depends at most on the l −1 derivative of γ′ and ∆′. the table below summarizes the dependency of the various quantities on the order of the derivatives of γ′ and ∆′ f ′(l) k′(l) g′(l) n′(l) {γ′(k)} k ≤l k ≤l k ≤l k < l {∆′(k)} k ≤l k < l k < l k < l (23) from the above considerations it follows that eq. (22) with l = 0 can be used to determine ∆′(0) since the r.h.s. depends only on γ′(0) = γ′(0) and from eq. (18) the pref- actor [n′(0) m −1] ̸= 0. having ∆′(0) we can easily calculate γ′(1) from eq. (21) with l = 0 since the r.h.s. depends only on γ′(0) and ∆′(0) and from eq. (17) k′(0) mn ̸= 0. with γ′(1), γ′(0) and ∆′(0) we can use eq. (22) with l = 1 to extract ∆′(1), then eq. (21) with l = 1 to extract γ′(2) and so on and so forth. c. keldysh-green's function in the nambu space 1. keldysh contour we now specialize to interacting systems which are ini- tially in equilibrium at temperature t = 1/β and chem- ical potential μ; such initial configurations are the rele- vant ones in quantum transport experiments, see section ii d.42 from static scdft43 we can choose the initial density matrix of the ks system as the thermal density matrix of a system described by the equilibrium hamilto- nian (2) with ˆ hint = 0 and ks phases γ and pairing po- tentials ∆, and from the results of the previous section we know that such ks system can reproduce the td bond- currents and pairing densities of the interacting system if perturbed by td ks phases γ(t) and pairing potentials ∆(t). denoting by ˆ hs(t) = ˆ k(t) + ˆ ∆(t) + ˆ ∆†(t) the td hamiltonian and by ˆ ρs(t) the td density matrix of the ks system we then have ˆ ρs(t) = 1 z ˆ ss(t)e−β( ˆ hs−μ ˆ n) ˆ s† s(t) (24) where z = tr {e−β( ˆ hs−μ ˆ n)} is the partition function and ˆ ss(t) is the ks evolution operator to be determined from i d dt ˆ ss(t) = ˆ hs(t) ˆ ss(t) with boundary condition ˆ ss(0) = 1. the hamiltonian ˆ hs = ˆ hs(0) is the equi- librium ks hamiltonian while ˆ n is the total number of particles operator. it is worth to notice that in general [ ˆ hs, ˆ n] ̸= 0 due to the presence of the pairing field. the td expectation value os(t) of a generic operator ˆ o(t) is in the ks system given by34–36,44 os(t) = tr {ˆ ρs(t) ˆ o(t)} ≡⟨tk n ˆ o(z = t±) o ⟩ (25) where we have introduced the short hand notation ⟨tk{. . .}⟩= tr h tk n e−i r γk d ̄ z ˆ hμ,s( ̄ z) . . . oi tr h tk n e−i r γk d ̄ z ˆ hμ,s( ̄ z)oi . (26) in the above equation γk is the keldysh contour45 il- lustrated in fig. 1 which is an oriented contour com- posed by an upper branch going from 0 to ∞, a lower branch going from ∞to 0 and a purely imaginary (ther- mal) segment going from 0 to −iβ. the operator tk is the contour ordering operator and move operators with later contour variable to the left (an extra minus sign has to be included for odd permutations of fermion fields). finally ˆ hμ,s( ̄ z = ̄ t±) = ˆ hs( ̄ t) where the contour points ̄ t−/ ̄ t+ lie on the upper/lower branch at a distance ̄ t from the origin while for ̄ z on the thermal segment ˆ hμ,s( ̄ z = −iτ) = ˆ hs −μ ˆ n. thus, the denominator in eq. (26) is simply the partition function z. in eq. (25) the variable z on the contour can be taken either on the upper (t−) or lower (t+) branch at a distance t from the origin. 6 fig. 1: the keldysh contour γk described in the main text. the contour variable z = t−/t+ denotes a point on the up- per/lower branch at a distance t from the origin while z = −iτ denotes a point on the imaginary track at a distance τ from the origin. in the figure we also illustrate the points 0−(ear- liest point on γk), 0+ and −iβ (latest point on γk). 2. keldysh-nambu-green's function the ks expectation value os(t) of an operator ˆ o(t) is in general different from the expectation value o(t) produced by the original system. however if ˆ o(t) is the ks bond-current operator or the pairing density operator the average over the ks system yields exactly the bond- current and pairing density of the original system. it is therefore convenient to introduce the non-equilibrium nambu-green's functions (negf) from which the ex- pectation value of any one-particle operator can be ex- tracted. a further reason for us to introduce the negf is that the equilibrium and time-dependent bogoliubov- degennes equations can be elegantly derived from them, thus illustrating the equivalence between the negf and the bogoliubov-degennes formalisms. the normal and anomalous components of the negf are defined accord- ing to46 gσ,mn(z; z′) = 1 i ⟨tk  ˆ cmσ(z)ˆ c† nσ(z′) ⟩, (27) fmn(z; z′) = 1 i ⟨tk {ˆ cm↓(z)ˆ cn↑(z′)}⟩, (28) fmn(z; z′) = −1 i ⟨tk n ˆ c† n↑(z′)ˆ c† m↓(z) o ⟩, (29) where z, z′ run on the keldysh contour γk.34,35,44,47 the ˆ c operators carry a dependence on the z variable; such de- pendence simply specifies their position along the contour so to have a well defined action of tk.44 the td bond- current and pairing density can be expressed in terms of gσ(z; z′) and f(z; z′) as jmn(t) = − x σ  tmneiγmn(t)gσ,nm(t−; t+) + h.c.  , (30) pm(t) = ifmm(t+; t−)e2itmmt. (31) 3. equations of motion the negf of the ks system obey the following equa- tions of motion ( i − → d dz 1 −hμ(z) ) g(z; z′) = 1δ(z −z′), (32) g(z; z′) ( −i ← − d dz′ 1 −hμ(z′) ) = 1δ(z −z′), (33) where all underlined quantities are 2 × 2 matrices in the nambu space with matrix elements 1mn =  δmn 0 0 δmn  and gmn(z; z′) =   g↑,mn(z; z′) −fnm(z′; z) fmn(z; z′) −g↓,nm(z′; z)  , (34) hμ,mn(z) =   kμ,mn(z) δmn∆m(z) δmn∆∗ m(z) −kμ,nm(z)  . (35) the matrix elements of hμ(z) are  kμ,mn(t±) = tmneiγmn(t) ∆m(t±) = ∆m(t) (36) for z = t± on the horizontal branches and  kμ,mn(−iτ) = tmneiγmn −μδmn ∆m(−iτ) = ∆m (37) for z = −iτ on the imaginary track. since hμ(−iτ) is independent of τ we write hμ(−iτ) = h0 −μσ with σmn = σz1mn and σz the third pauli matrix. in the next section we show that the solution of the equations of motion is equivalent to first solve the static bogoliubov-degennes (bdg) equations and then their td version. 4. keldysh components and bogoliubov-degennes equations we introduce the left and right contour evolution ma- trices sr/l(z) which satisfy i d dz sr(z) = hμ(z)sr(z), (38) −i d dz′ sl(z′) = sl(z′)hμ(z′), (39) with boundary conditions sr/l(0−) = 1. the most gen- eral solution of the equations of motion (32,33) can then be written as g(z; z′) = sr(z)  θ(z; z′)g> + θ(z′; z)g< sl(z′), (40) 7 with g>−g< = −i1 and the contour heaviside function θ(z; z′) = 1 if z is later than z′ and zero otherwise. equa- tion (40) is a solution for all matrices g> = −i1 + g<. in order to determine g> or g< we use the boundary conditions g(0−; z′) = −g(−iβ; z′), (41) g(z; 0−) = −g(z; −iβ), (42) which follow directly from the definitions (27-29) of the negf. using eq. (40) one finds g(0−; z′) = g<sl(z′) and g(−iβ; z′) = sr(−iβ)g>sl(z′) from which we con- clude that g< = −sr(−iβ)g>. (43) similarly, from eq. (42) one finds g> = −g<sl(−iβ). (44) exploiting the fact that hμ(−iτ) = h0 −μσ is con- stant along the imaginary track one readily realizes that sr/l(−iβ) = exp[±β(h0 −μσ)] and hence g< = i 1 + exp[β(h0 −μσ)]. (45) from the exact solution (40) we can extract any observ- able quantity at times t ≥0 and not only its limiting behavior at t →∞. below we calculate the different components of the negf. we introduce the eigenstates ψq, with eigenenergies eq, of the matrix h0 −μσ. the vector ψq = [uq, vq] is a two-dimensional vector in the nambu space and, by definition, satisfies the eigenvalue problem x n tmneiγmnuq(n) + ∆mvq(m) = (eq + μ)uq(m), (46) − x n tnmeiγnmvq(n)+∆∗ muq(m) = (eq−μ)vq(m). (47) due to the presence of the pairing field the components uq and vq are coupled and the eigenstates ψq are a mixture of one-particle spin-up electron states and spin- down hole states. we will refer to the eigenstates ψq as bogolons. the above equations have the structure of the static bdg equations which follow from the bcs approximation.48,49 in our case eqs. (46,47) follow from scdft43 and therefore yield the exact equilibrium bond- current and pairing density provided that the exact ks phases and pairing fields are used. inserting the complete set of eigenstates in eq. (40) and taking into account eq. (45) we find the following expansion for the negf g(z; z′) = i x q sr(z)ψq  θ(z; z′)f >(eq) +θ(z′; z)f <(eq)  ψ† q sl(z′), (48) where f <(ω) = 1/[1 + exp(βω)] is the fermi function and f >(ω) = f <(ω)−1. taking z and z′ on the real axis but on different branches of the keldysh contour, we can extract the lesser and greater component of the negf. we first notice that for z = t± the contour evolution operators reduce to the standard evolution operators, i.e., sr(t±) = s(t) and sl(t±) = s†(t) with i d dts(t) = h(t)s(t), s(0) = 1 , (49) and h(t) = hμ(t±), see eq. (36). then, in terms of the evolved states ψq(t) = s(t)ψq with components ψq(t) = [uq(t), vq(t)] we find g≶(t; t′) ≡g(t∓; t′ ±)=    g≶ ↑(t; t′) −f≷,t (t′; t) f ≶(t; t′) −g≷,t ↓ (t′; t)    = i x q f ≶(eq)   uq(t)u† q(t′) uq(t)v† q(t′) vq(t)u† q(t′) vq(t)v† q(t′)  ,(50) where the superscript t in f≷,t and g≷,t ↓ denotes the transpose of the matrix, see also eq. (34). the func- tions uq(t) and vq(t) can be determined by solving a cou- pled system of first-order differential equations. from eq. (49) it follows that i d dtuq(m, t) = x n tmneiγmn(t)uq(n, t) + ∆m(t)vq(m, t), (51) i d dtvq(m, t) = − x n tnmeiγnm(t)vq(n, t) + ∆∗ m(t)uq(m, t), (52) which have the structure of the td bdg equations.26,50 as in the static case, however, the solution of eqs. (51- 52) yields the exact densities and not their bcs approx- imation. we notice that for the ks system to reproduce the time-independent densities of an interacting system in equilibrium it must be ∆m(t) = e−2iμt∆m (53) for which one finds the solutions uq(t) = e−i(eq+μ)tuq and vq(t) = e−i(eq−μ)tvq. the above time-dependence of the pairing field is the same as in the bcs approximation. using eq. (50) the retarded (r) and advanced (a) negf are gr/a(t; t′) ≡±θ(±t ∓t′)  g>(t; t′) −g<(t; t′)  = ∓iθ(±t ∓t′)s(t)s†(t′), (54) with components gr/a mn (t; t′) =    gr/a ↑,mn(t; t′) −fa/r nm (t′; t) f r/a mn (t; t′) −ga/r ↓,nm(t′; t)   . (55) 8 it follows that g≶(t; t′) can also be written as g≶(t; t′) = gr(t; 0) g≶(0; 0) ga(0; t′). (56) d. application to quantum transport we here apply the above formalism to systems de- scribed by α = 1, . . . , n bulk superconducting leads in contact with a central region c which can be, e.g., a quantum dot, a molecule or a nanostructure. assuming no direct coupling between the leads the hamiltonian hμ is written in terms of its projections on different sub- spaces as hμ = n x α=1 hμ,αα + hμ,cc + n x α=1 (hμ,αc + hμ,cα), (57) where hμ,αα describes the α-th lead, hμ,cc the nanos- tructure c and hμ,αc + hμ,cα the coupling between lead α and c. we assume region c to be a constric- tion so small that the bulk equilibrium of the leads is not altered by the coupling to c. furthermore we con- sider time-dependent perturbations which correspond to the switching on of a longitudinal electric field in lead α. the time to screen the external electric field in the leads is in the plasmon time-scale region. if we are in- terested in external fields which vary on a much longer time-scale it is reasonable to expect that the leads remain in local equilibrium. therefore the coarse-grained time evolution of the system can be described by the following td hamiltonian hμ(t±) = h(t) hαα(t) = exp (−iμtσz) hαα(0) exp (iμtσz) , (58) hαc(t) = exp  i z t 0 d ̄ t uα( ̄ t)σz  hαc(0), (59) hcα(t) = [hcα(t)]†. (60) we do not specify the time dependence of hcc(t) since it can be any, see below. the td field uα(t) is the sum of the external and hartree field and is homogeneous, i.e., it does not carry any dependence on the internal struc- ture of the leads, in accordance with the above discus- sion. it has been shown that for macroscopic leads the assumption of homogeneity is verified with rather high accuracy.51 as for the case of normal leads the equations of motion for the keldysh-green's function can be solved by an em- bedding procedure. we define the uncontacted green's function g which obeys the equations of motion (32,33) with hμ,αc = hμ,cα = 0 and the same boundary con- ditions as g. then, the equation of motion for gcc projected onto regions cc takes the form ( i − → d dz 1cc −hμ,cc(z) ) gcc(z; z′) = 1ccδ(z −z′) + z d ̄ z σ(z; ̄ z) gcc( ̄ z; z′),(61) where the embedding self-energy is expressed in terms of g as σ( ̄ z; ̄ z′) = n x α=1 σα( ̄ z; ̄ z′) = n x α=1 hμ,cα( ̄ z) gαα( ̄ z; ̄ z′) hμ,αc( ̄ z′). (62) the above equation of motion is defined on the keldysh contour of fig. 1. converting eq. (61) in equations for real times results in a set of coupled equations known as kadanoff-baym equations34,52–56 recently implemented to study transient responses of interacting electrons in model molecular junctions.51,57 the use of the kadanoff- baym equations to address transient and relaxation ef- fects in other contexts has been pioneered by sch ̈ afer,58 bonitz et al.,59 and binder et al..60 the importance of using an uncontacted green's func- tion g with boundary conditions (41,42) for a proper de- scription of g≶(t; t′) at finite times has been discussed elsewhere in the context of transient regimes36,51 and it has been shown that it leads to coupled equations be- tween the keldysh-green's function with two real times and those with one real and one imaginary time. in the next section we propose a wave-function based propagation scheme to solve eq. (61) for td hamiltoni- ans of the form (58-60). iii. numerical algorithm we consider semi-infinite periodic leads with a super- cell of dimension n α cell for lead α. the projected hamil- tonian h0,αα = hαα(0) can then be organized as follows h0,αα =     hα tα 0α . . . t† α hα tα . . . 0α t† α hα . . . . . . . . . . . . . . .    , (63) where hα is the 2n α cell × 2n α cell nambu hamiltonian of the supercell with matrix structure hα =  ǫα ∆α ∆∗ α −ǫt α  , (64) while tα describes the contact between two nearest neigh- bor supercells. since the pairing field is local the off- diagonal terms of tα are zero and therefore the general structure of the hopping matrix is tα =  tα 0α 0α −tt α  . (65) the matrices ǫα, ∆α and tα in hα and tα have the di- mension of the unit cell, i.e., n α cell × n α cell. in particular ∆α is a diagonal matrix. 9 a. calculation of initial states given the above structure of the leads hamiltonian the eigenstates of h0 −μσ can be grouped in scattering states with incoming bogolons from lead α = 1, . . . , n and andreev bound states (abs). 1. scattering states the lead α is characterized by energy bands eα ν (p) with ν = 1, . . . , 2n α cell and p ∈(0, π). for a given p the energies eα ν (p) are the solutions of the eigenvalue problem hα + tαeip + t† αe−ip −μσα
u α νp = eα ν (p)u α νp (66) with u α νp the nambu-bloch eigenvectors. we write the index of the localized orbital φm as m = s, j, α; here s labels the orbital within the supercell, j the supercell and α the lead. the index s runs between 1 and n α cell while the supercell index j = 0, . . . , ∞. the scattering state for an incoming bogolon from lead α has the general form ψα νp(m) =                u α νp(s)e−ipj + p ρ rαα νp,ρ w αα νp,ρ(s) eiqαα νp,ρj m = s, j, α ψα νp,c(m) m ∈c p ρ t αβ νp,ρ w αβ νp,ρ(s)eiqαβ νp,ρj m = s, j, β ̸= α (67) with reflection coefficients r and transmission coefficients t . the momenta qαβ νp,ρ (for all leads β including β = α) are associated to states with energy e = eα ν (p) and can therefore be obtained from the roots of det[hβ + tβeiq + t† βe−iq −μσβ −e1β] = 0. (68) the above equation admits, in general, complex solu- tions for q. in eq. (67) the sums over ρ run over real solutions q for which the sign of the fermi velocity vβ ρ (q) = ∂eβ ρ (q)/∂q is opposite to the sign of the fermi velocity vα ν (p) of the incoming bogolon and over all com- plex solutions q for which im[q] > 0 (evanescent states). once the qαβ νp,ρ are known the bloch state w αβ νp,ρ is sim- ply the eigenvector with zero eigenvalue of the matrix hβ + tβeiqαβ νp,ρ + t† βe−iqαβ νp,ρ −μσβ −e1β. for the cal- culation of the reflection and transmission coefficients as well as of the amplitude ψα νp,c(m) in the central region we extended a recently proposed wave-guide approach.61 the method is based on projecting the schr ̈ odinger equa- tion (h0 −μσ)ψ = eψ onto the central region and onto all the supercells in contact with the central region, i.e., with j = 0. the projection onto a j = 0 supercell leads to an equation which couples the amplitude of ψ in j = 0 with that in j = 1. exploiting the analytic form of the eigenstate in eq. (67) the amplitude in the leads can entirely be expressed in terms of the unknown r's and t 's for all j. in this way the equations can be closed and the problem is mapped into a simple linear system of equations for the unknown rαβ νp,ρ, t αβ νp,ρ and ψα νp,c(m). 2. andreev bound states the presence of a gap in the spectrum of the supercon- ducting leads may lead to the formation of localized abs within the gap. the procedure to calculate the abs is slightly different from the one previously presented since the abs energy is not an input parameter and the abs state is normalized to 1 over the whole system. the en- ergy eb of an abs ψb is outside the lead continua. pro- jecting the schr ̈ odinger equation (h0 −μσ)ψb = ebψb onto different regions and solving for the projection ψb,c in region c one finds (heff 0,cc(eb)−μσcc)ψb,c = ebψb,c where heff 0,cc(e)=h0,cc+ x α h0,cα 1 e−(h0,αα−μσαα)h0,αc. (69) the abs energies eb can then be extracted from the roots of det[heff 0,cc(e) −μσcc −e1cc] = 0 and the eigenvector with zero eigenvalue of heff 0,cc(eb)−μσcc − eb1cc is proportional to the projection ψb,c of the abs in region c. we call cb the unknown constant of propor- tionality. as for the scattering states we can construct the abs everywhere in the system according to ψb(m) =      p ρ bα b,ρw α b,ρ(s)eiqα b,ρj m = s, j, α ψb,c(m) m ∈c . (70) the momenta qα b,ρ and bloch states w α b,ρ are calculated in the same way as for the scattering states. by definition all momenta have a finite imaginary part and the sum in 10 eq. (70) runs over those with a positive imaginary part. the constants bα b,ρ can be simply obtained by project- ing the schr ̈ odinger equation (h0 −μσ)ψb = ebψb onto the supercells in contact with region c, i.e., with j = 0. the resulting equation couples the amplitude of ψb in j = 0 with that in j = 1 and with the known amplitude cbψb,c(m). exploiting the analytic form of ψb in the leads the amplitude in j = 1 can entirely be expressed in terms of the constants cbbα b,ρ thus yielding a linear system of equations for each lead. once the cbbα b,ρ are known the constant of proportionality cb is fixed by im- posing that the abs is normalized to 1. this can be easily done since the sums over j are geometrical series. b. embedded crank-nicholson propagation scheme to propagate the generic eigenstate ψ of h0 −μσ we extend the embedded crank-nicholson37,38 scheme to su- perconducting leads. the equations of motion (51,52) can be written in a compact form as i d dtψ(t) = h(t)ψ(t), ψ(0) = ψ (71) where the components of the td hamiltonian are given in eqs. (58-60). we first perform the gauge transfor- mation ψα(t) = exp[−iμσααt]φα(t) for the projection of the state ψ onto lead α and ψc(t) = φc(t) for region c. the state φ(t) obeys the equation i d dtφ(t) = ̃ h(t)φ(t), φ(0) = ψ (72) with ̃ hαα(t) = hαα(0) −μσαα (73) ̃ hαc(t) = exp  i  μt + z t 0 d ̄ t uα( ̄ t)  σαα  hαc(0) (74) and ̃ hcc(t) = hcc(t). the advantage of the gauge transformed equations is that the lead hamiltonian is now independent of time. we discretize the time as tm = 2mδ and define φ(m) = φ(tm) and ̃ h (m) = 1 2 h ̃ h(tm+1) + ̃ h(tm) i . the differential operator in eq. (72) is then approximated by the cayley propagator  1 + iδ ̃ h (m) φ(m+1) =  1 −iδ ̃ h (m) φ(m). (75) the above propagation scheme is known as crank- nicholson algorithm and it is norm-conserving and ac- curate up to second order in δ. as the matrix ̃ h is in- finite dimensional the direct implementation of eq. (75) is not possible. a significant progress can be done using an embedding procedure which, as we shall see, entails perfect transparent boundary conditions at the interfaces between region c and leads α. projecting eq. (75) onto lead α and iterating one finds φ(m+1) α = gm+1 αα φ(0) α − iδ 1αα + iδ ̃ hαα m x j=0 gj αα ̃ h (m−j) αc ×  φ(m+1−j) c + φ(m−j) c  ,(76) where we have defined the propagator gαα = 1αα −iδ ̃ hαα 1αα + iδ ̃ hαα , (77) and made use of the fact that ̃ hαα(t) ≡ ̃ hαα is time- independent. the time-dependence of the contacting hamiltonian can be easily extracted from eq. (74) and reads ̃ h (m) αc = exp  iμ(m+1) α σαα  + exp  iμ(m) α σαα  2 ̃ hαc(0), (78) where we have defined μ(m) α = μtm + z tm 0 d ̄ t uα( ̄ t). (79) at this point comes a crucial observation which allows for extending the propagation scheme of refs. 37,38 to the superconducting case. since the pairing field is local in the chosen basis the off-diagonal part of the contacting hamiltonian is zero and hence ̃ hcασαα = σcc ̃ hcα. it follows that eq. (78) can also be rewritten as ̃ h (m) αc = ̃ hαc(0) exp  iμ(m+1) α σcc  + exp  iμ(m) α σcc  2 ≡ ̃ hαc(0) ̄ z(m) α , (80) which implicitly define the matrices ̄ z(m) α = (z(m) α )∗. next we project eq. (75) onto region c and use eq. (76) to express the φα at a given time step in terms of the φc at all previous time steps. the resulting equation is  1cc + iδ ̃ h (m) eff  φ(m+1) c =  1cc −iδ ̃ h (m) eff  φ(m) c + x α  s(m) α + m (m) α  (81) and contains only quantities with the dimension of region c. we emphasize that eq. (81) is an exact reformulation of the original eq. (75) but it has the advantage of being implementable. indeed, exploiting the result in eq. (80) the boundary term s(m) α and memory term m (m) α read s(m) α = −iδz(m) α ̃ hcα(0) gm αα  1αα + gαα  φ(0) α , (82) 11 m (m) α = −δ2 m−1 x j=0 z(m) α  q(j+1) α + q(j) α  ̄ z(m−1−j) α ×  φ(m−j) c + φ(m−1−j) c  , (83) while the effective hamiltonian is given by ̃ h (m) eff = ̃ h (m) cc −iδ x α z(m) α q(0) α ̄ z(m) α , (84) where the embedding matrices q(m) α have twice the di- mension of region c and are defined according to q(m) α = ̃ hcα(0)  1αα −iδ ̃ hαα m  1αα + iδ ̃ hαα m+1 ̃ hαc(0). (85) in appendix a we describe a recursive scheme to calcu- late the embedding matrices. in appendix b we further show that the boundary term s(m) α can be expressed in terms of the qα's thus rendering eq. (81) a well defined equation for time propagations. in the next section we apply the numerical scheme to uf-jnj model systems and obtain results for the td densities and currents. iv. real-time simulations of s-d-s junctions due to the vast phenomenology of s-d-s junctions it is not possible to address these systems in a single work. furthermore the analysis of the time-dependent regime is generally more complex than that in the josephson regime and it is therefore advisable to first gain some insight by investigating simple cases. our intention in this section is to demonstrate the feasibility of the prop- agation scheme and to present genuine td properties of simple model systems. we consider a tight-binding chain (region c) with nearest neighbor hopping tc and on-site energy ǫc con- nected to a left (l) and right (r) wide-band leads. the α = l, r lead is described by a semi-infinite tight-binding chain with nearest neighbor hopping tα and a constant pairing field ∆α, and is coupled to the α end-point of the central chain through its surface site with a hop- ping tcα = tαc. the system is initially in equilibrium at temperature t = 0 and chemical potential μ = 0 and driven out of equilibrium by a td bias voltage uα(t) applied to lead α at positive times. from sec- tion ii d, the hamiltonian for this kind of systems read ˆ h(t) = p α( ˆ hαα(t) + ˆ hαc(t) + ˆ hcα(t)) + ˆ hcc where ˆ hαα(t) = tα ∞ x j=0 x σ (ˆ c† j+1σαˆ cjσα + h.c.) + (e−2iμt∆αˆ c† j↑αˆ c† j↓α + h.c.) (86) describes the lead α = l, r, ˆ hlc(t) = tlcei r t 0 dt′ul(t′) x σ ˆ c† 0σlˆ c0σ + h.c. (87) ˆ hrc(t) = trcei r t 0 dt′ur(t′) x σ ˆ c† 0σrˆ cnσ + h.c. (88) accounts for the coupling between region c and the leads, and ˆ hcc = tc n−1 x m=0 x σ (ˆ c† m+1σˆ cmσ+h.c.)+ǫc n x m=0 x σ ˆ c† mσˆ cmσ (89) is the hamiltonian of the chain with n + 1 atomic sites. the currents jl(t) ≡j0l,0(t) and jr(t) ≡jn,0r(t) through the bonds connecting the chain to the left and right leads are obtained from eq. (30) and eq. (50) and read jl(t) = −itlceiγlc(t) "x q f <(eq)uq(0l, t)u∗ q(0, t) − x q f >(eq)vq(0, t)v∗ q(0l, t) # + h.c., (90) jr(t) = −it∗ rce−iγrc(t) "x q f <(eq)uq(0, t)u∗ q(0r, t) − x q f >(eq)vq(0r, t)v∗ q(0, t) # + h.c., (91) where γαc(t) = i r t 0 dt′uα(t′) and the sum over q runs over all abs and scattering states. similarly, the pairing density pm(t) on an arbitrary site of the chain is obtained from eq. (31) and eq. (50) and reads pm(t) = x q f <(eq)uq(m, t)v∗ q(m, t)e2iǫct. (92) we will write the pairing field as ∆α = ξαeiχα∆and measure energies in units of ∆, times in units of ħ/∆ and currents in units of |e|∆/ħ, with |e| the absolute charge of the carriers. since we consider wide-band leads with tα ≫tαc, tc and the chemical potential is set to zero the results depend only on the ratio γα ≡2t2 αc/tα (tunneling rate) and not on tαc and tα separately. in the following we therefore specify the value of γα only. in practical calculations the longitudinal vector p ∈(0, π) of the scattering states, see eq. (67), is discretized with np mesh points and only states with energy within the range (μ−λ, μ+ λ) are propagated in time. we will call np,α the number of scattering states from lead α that are propagated. the cutoffλ is chosen about an order of magnitude larger than the typical energy scales of the problem, i.e., uα, γα, ∆α, tc, ǫc. 12 a. the single-level quantum dot model the single-level quantum dot (qd) model corresponds to a central chain with only one atomic site (n = 0). for ∆l = ∆r = 0 (n-qd-n) the td response of this system has been investigated by several authors and an analytic formula for the td current is also available.36,62,63 scarce attention, however, has been devoted to the system with one superconducting lead29 (n-qd-s) and to the best of our knowledge the only available results when both leads are superconducting (s-qd-s) have been published in ref. 30. 1. n-qd-s model under dc bias we first consider the n-qd-s case schematically illus- trated in fig. 2(a). to highlight the different scattering mechanisms we shift the central level by ǫc = 0.5, choose weak couplings to the leads γl = γr = 0.2, and drive the system out of equilibrium by applying four different biases ul = 0.3, 0.6, 0.9, 1.2 to the left normal lead. for biases in the subgap region, i.e., ul < ∆r = 1, transport is dominated by andreev reflections (ar). in fig. 2(b) we show the currents jl(t) and jr(t) of eqs. (90,91). for ul = 0.3 < ǫc the ar are strongly suppressed since electrons at the left electrochemical potential μl = ul have just enough energy to enter the resonant window (ǫc −2γ, ǫc + 2γ), where 2γ = γl + γr. resonant ar can occur for ul > ǫc and constitute the dominant mech- anism for electron tunneling. this is clearly visible in the second panel of fig. 2(b) where the steady-state values of jr for ul = 0.6 and ul = 0.9 are approximatively the same. at larger biases ul = 1.2 > ∆r electrons can also tunnel via standard quasi-particle scattering and the steady-state current increases. this interpretation is con- firmed by the behavior of the pairing density p0(t) on the qd, third panel of fig. 2(b). for times up to ∼5 the pairing density decreases since pre-existent cooper pairs in lead r move away from the qd. however, while |p0(t)| remains below its equilibrium value at ul = 0.3, for all other biases, ul > ǫc, |p0(t)| increases after t ∼5, mean- ing that a cooper pair is forming at the interface. we also notice that the values of |p0(t →∞)| for ul = 0.9 and ul = 1.2 are very close while the corresponding cur- rents jr differ appreciably. this is again in agreement with the fact that electrons with energy larger than ∆r do not undergo ar and thus no extra cooper pairs are formed. finally we observe that the transient regime is longer in the n-qd-s case than in the n-qd-n case, see inset in panel 2 and 3 of fig. 2(b), as also pointed out in ref. 29. 2. s-qd-s model under dc bias we now turn to the more interesting case in which the qd is connected to a left and right superconduct- 0 20 40 60 0 0.05 0.1 ul = 0.3 ul = 0.6 ul = 0.9 ul = 1.2 0 0.1 0.2 0 0.03 0.06 0 10 20 30 0 0.1 0.2 0 10 20 30 0 0.08 0.16 t jl jr |p0| (b) jl t jr t n-qd-n n-qd-n fig. 2: (a) schematic of the transport set up. a single level qd with on-site energy εc = 0.5 is weakly connected (γl = γr = 0.2) to a left normal lead and a right supercon- ducting lead. in equilibrium both temperature t and chemi- cal potential μ are zero. the system is driven out of equilib- rium by a step-like voltage bias ul = 0.3, 0.6, 0.9, 1.2 in the normal lead. for ul < ∆r the dominant scattering mecha- nism is the ar in which an electron is reflected as a hole and a cooper pair is formed in lead r. (b) time-dependent cur- rent at the left interface (first panel), right interface (second panel) and absolute value of the pairing density on the qd (third panel). the insets show the td current for the same parameters but ∆r = 0, i.e., for a normal r lead. the results are obtained with a time-step δ = 0.05, cutoffλ = 6 and a number of scattering states np,l = 1070, np,r = 1056. ing lead (s-qd-s), see fig. 3(a). we focus on sym- metric couplings γl = γr = γ = 1 and on pairing fields ∆l = ∆reiχ = eiχ with the same magnitude but different phase. this system always support two andreev bound states (abs) in the gap. their en- ergy can be obtained analytically from the solution of det[heff 0,cc(e)−μσcc −e1cc] = 0 (see section iii a 2) which, in terms of the dimensionless variables x = e/∆, 13 0 50 100 150 200 250 0 0.8 1.6 0 10 20 30 40 0 0.6 1.2 (b) (c) ul = 0.2 ul = 0.3 ul = 0.4 ul = 0.5 ul = 1.0 ul = 2.0 ul = 3.0 t t jl jl fig. 3: (a) schematic of the s-qd-s model with γl = γr = 1.0, ∆l = ∆r = 1, and ǫc = 0. this system admits two abs in the gap. the abs energy depends on the superconducting phase difference χ as illustrated in the inset. (b-c) time- dependent current jl(t) at the left interface as a function of time for (b) ul = 3.0, 2.0, 1.0 [the curves corresponding to bias ul = n.0 are shifted upward by 0.3(n−1)] and (c) ul = 0.5, 0.4, 0.3, 0.2 [the curves corresponding to bias ul = 0.n are shifted upward by 0.6(n −2)]. the results are obtained with a time-step δ = 0.05, cutoffλ = 12.1, and a number of scattering states np,l = np,r = 768 for panel (b) and δ = 0.05, λ = 4, np,l = np,r = 788 for panel (c). γ = γ/∆and e = (ǫc −μ)/∆, reads x2(1 + γ √ 1 −x2 )2 −e2 −α2γ2 1 −x2 = 0, (93) where α = q 1+cos χ 2 and varies in the range (0, 1). in fig. 3(a) we plot the solutions of eq. (93) as a function of χ for ǫc = μ = 0. in equilibrium and at zero temperature one abs is fully occupied and the other is empty. at time t = 0 a constant bias ul is applied to the left lead. in fig. 3(b) we display the td current at the left interface jl(t) for χ = 0 and ul = 3, 2, 1. after a transient the current oscillates in time with period tj = 2π/(2ul), as expected. for ul > 2 the s-qd-s system behaves similarly to a macroscopic josephson junction with an almost pure monochromatic response, albeit the average value jdc of the current over a period is different from zero. for ul = 1 < 2∆, i.e., in the subgap region, the transient regime becomes much longer and jl(t) deviates 0 1 2 3 4 ω 0 0.5 1 1.5 |jl(ω)| (arb. units) ul = 0.3 ul = 0.4 ul = 0.5 ul = 0.6 ul = 1.0 0.3 0.4 0.5 0.6 ul 0 0.05 0.1 jdc 0 50 100 150 200 t 0 1 2 3 jl,abs (a) (b) (c) ul = 0.3 ul = 0.4 ul = 0.5 ul = 0.6 ul = 0.2 2∆/5 2∆/4 fig. 4: (a) discrete fourier transform of jl(t) in arbitrary units [the curves corresponding to bias ul = 0.n are shifted upward by 0.7(n −3) while that corresponding to bias ul = 1.0 is shifted upward by 2.8. (b) values of the average current for biases in the subgap region. (c) abs contribution to the current jl(t) for biases ul = 0.2, 0.3, 0.4, 0.5, 0.6 [the curves corresponding to bias ul = 0.n are shifted upward by 0.8(n− 2)]. the numerical parameters are the same as in fig. 3. from a perfect monochromatic function. at ul = 1 the dominant scattering mechanism is the single ar. as discussed in ref. 15 the presence of the resonant level modifies substantially the jdc −v (v = ul −ur) characteristic and for γ = 1 the subharmonic gap struc- ture is almost entirely washed out. however, a very rich structure is observed in the td current. in fig. 3(c) we display jl(t) for biases ul = 0.5, 0.4, 0.3, 0.2. the charge carriers undergo multiple ar (mar) before ac- quiring enough energy and escaping from the qd. the dwelling time increases with decreasing bias and the tran- sient current has a highly non-trivial behavior before the josephson regime sets in. from the simulations in fig. 3(c) at bias ul = 0.2 the propagation time t = 250 is not sufficient for the development of the josephson oscil- lations. we also observe that the smaller is the bias the larger is the contribution of high-order harmonics, which is in contrast with one would naively expect from linear response theory. in fig. 4(a) we display the fourier transform of jl(t)−jdc in the josephson regime. replica of the main josephson frequency ωj = 2ul are clearly visible for ul < ∆. the values of jdc as obtained from time propa- gation are reported in fig. 4(b) and are consistent with a smeared sub-harmonic gap structure. from the curves jl(t) it is not evident how to estimate the duration of the transient time. we found useful to look at the contribution of the abs, jl,abs, to the total current jl, since jl,abs(t →∞) = 0. this quantity is evaluated from eq. (90) by restricting the sum over q to the abs and is shown in fig. 4(c). abs play a crucial 14 0 0.5 1 1.5 1 2 3 0 15 30 45 60 0 0.5 1 t jr n0 |p0| 0 π/4 π/2 3π/4 π π 3π/4 π/2 π/4 0 π 3π/4 π/2 π/4 0 fig. 5: time-dependent current at the right interface jr (first panel) as well as the density n0 (second panel) and pair- ing density |p0| (third panel) on the qd. the curves from bot- tom to top corresponds to a switch-offtime t(n) off= 5π + nπ/8, with n = 0, 1, 2, 3, 4. since the bias is ul = 1 the ac- cumulated phase difference χ(n) at the end of the pulse is χ(n) = 2t(n) off= nπ/4. for the switch-offtime t(n) offthe curves of jr are shifted upward by 0.3n, those of n0 by 0.5n and those of |p0| by 0.2n. the results are obtained with a time- step δ = 0.05, cutoffλ = 12.1, and a number of scattering states np,l = np,r = 768. role in the relaxation mechanism as we shall see in the next section. 3. s-qd-s model under dc pulses as mentioned in the introduction the possibility of em- ploying uf-jnj in future electronics rely on our under- standing of their td properties. in the previous section we studied the transient behavior of a s-qd-s system under the sudden switch-on of an applied bias. equally important is to study how the system responds when the bias is switched off. we therefore consider the same s- qd-s model as before with γl = γr = 1, ǫc = 0, ∆l = ∆r = 1 initially in equilibrium at zero temper- ature and chemical potential. at time t = 0 a constant bias ul = 1 is applied to lead l until the time toffat which the bias is switched off. how does the system re- lax? in fig. 5 we show the current jr at the right interface as well as the density n0 and pairing density |p0| on the qd for switch-offtimes t(n) off = 5π + nπ/8 with n = 0, 1, 2, 3, 4. despite the fact that the switch- offtimes are all very close [t(0) off∼15.71 and t(4) off∼17.28] the system reacts in different ways and actually relaxes only in one case. the strong dependence on toffis due to the two abs in the gap. similarly to what happens in normal systems64 the asymptotic (t →∞) form of the density on the qd is n0(t) −n0,cont ∼ x ij fij cos((ǫ(i) abs −ǫ(j) abs)t), (94) where ǫ(i) abs, i = 1, 2, are the abs eigenenergies of the hamiltonian after the bias has been switched offand n0,cont is the contribution of the continuum states to the density. the coefficients fij = fji are matrix ele- ments of the fermi function f( ˆ h(0)) calculated at the equilibrium hamiltonian and depend on the history of the applied bias.65,66 contrary to the normal case, how- ever, the energy of the abs depends on when the bias is switched offsince after a time toffthe phase difference χ changes from zero to 2ultoff. this fact together with eq. (94) explains the persistent oscillations at different frequencies. indeed χ(n) = 2ult(n) off= nπ/4 and from fig. 3(a) we see that [ǫ(1) abs(χ(n)) −ǫ(2) abs(χ(n))] varies from ∼1.08 to zero when n varies from zero to 4. the amplitude of the oscillations as well as the average value of the density n0, however, do not depend only on χ but also on the history of the applied bias. two different biases ul(t) and u ′ l(t) yielding the same phase differ- ence χ = 2 r toff 0 dτul(τ) = 2 r toff 0 dτu ′ l(τ) give rise to different persistent oscillations, albeit with the same fre- quency. from the results of this section we conclude that for devices coupled to superconducting leads a small differ- ence in the switch-offtime of the bias can cause a large difference in the relaxation time of the device. this prop- erty may be exploited to generate zero bias ac currents of tunable frequency. 4. s-qd-s model under ac bias the time-propagation approach has the merit of not being limited to step-like biases as it can deal with any td bias at the same computational cost. of spe- cial importance is the case of ac biases where a mi- crowave radiation ur sin(ωrt) is superimposed to a dc sig- nal v = ul −ur. the study of uf-jnj in the presence of microwave radiation started with the work of cuevas et al.67 who predicted the occurrence of subharmonic shapiro spikes in the jdc −v characteristic of supercon- ducting point contacts. later on zhu et al.68 extended the analysis to the s-qd-s model and discuss how the abs modify the jdc −v characteristic. the replicas of the shapiro spikes have been experimentally observed69 and can be explained in terms of photon-assisted mul- tiple andreev reflections. using a generalized floquet 15 0 0.4 0.8 -0.2 0 0.2 jl,abs jl,cont 0 130 260 -0.2 0 0.2 re[p0] im[p0] |p0| (a) (b) (c) ωr = 0.5 ωr = 1.08 ωr = 1.5 jl t fig. 6: (a) td current at the left interface for ul = 0, ur = 0.05ωr with ωr = 0.5, 1.08 [the curve is shifted upward by 0.4], and 1.5 [the curve is shifted upward by 0.8]. (b) abs and continuum contribution to the total current in the resonant case ωr = 1.08, ur = 0.05ωr and ul = 0. (c) pairing potential on the qd for the same parameters as in panel (b). the results are obtained with a time-step δ = 0.05, cut-off λ = 4, and a number of scattering states np,l = np,r = 788. formalism one can show that in the long-time limit67 jl(t) = x mn jn m(v, γ, ωr)ei(mωj+nωr)t (95) where γ = ur/ωr and ωj = 2v is the josephson fre- quency. the calculation of jn m is, in general, rather com- plicated and to the best of our knowledge the full td profile of jl(t) as well as the duration of the transient time before the photon-assisted josephson regime sets in have not been addressed before. we here consider the s-qd-s model with γl = γr = 1, εc = 0, ∆l = ∆r = 1 under a dc bias and in the presence of a superimposed microwave radiation ul(t) = ul + ur sin(ωrt) and ur = 0. in fig. 6(a) we display the td current at the left interface for fixed γ = ur/ωr = 0.05 and different values of the frequency ωr = 0.5, 1.08, 1.5. the first striking feature is the occurrence of a transient resonant effect at ωr = 1.08 ∼ωabs ≡ǫ(1) abs −ǫ(2) abs. at the resonant frequency the amplitude of the oscillations increases linearly in time till a maximum value ∼0.3. the fourier decomposition (not shown) reveals that the peak at ω = 1.08 splits into two peaks, one above and one below 1.08, which is consistent with the observed beat- ing. the effect is absent at larger (ωr = 1.5) and smaller (ωr = 0.5) frequencies for which the amplitude of the oscillations remains below 0.05 and two main harmonics, one at ωr and the other at ωabs, are visible in the fourier decomposition (not shown). the peak at ω = ωabs is due to a transient excitation with a long life-time and cannot be described using floquet based approaches. the abs play a crucial role in determining the td profile of jl at the resonant frequency. the total current jl(t) = jl,cont(t) + jl,abs(t) is the sum of the current jl,cont coming from the evolution of the continuum states and the abs current jl,abs(t). these two currents are shown in fig. 6(b) from which it is evident that abs carry an important amount of current not only in the dc josephson effect30,70 but also in the transient regime. in fig. 6(c) we show the pairing density on the qd for the resonant frequency ωr = 1.08. in the presence of an external bias the abs contribute to the current only in the transient regime. the duration of the transient is investigated in fig. 7 where we show jr,abs for dc biases with a superimposed microwave radiation described by ul(t) = ul + ur sin(ωrt), with ur = 0.05ωr, ωr = 1.08, and ul = 0.0, 0.03, 0.1, 0.3. the interplay between the ac josephson effect and the resonant microwave driving leads to complicated td pat- terns for small ul. increasing ul the life-time of the quasi abs decreases resulting in a fast damping of the oscillations, see fig. 7 with ul = 0.3. b. long atomic chains we consider a chain of n + 1 = 21 atomic sites with onsite energy ǫc = 0 and nearest neighbor hopping tc = 1, see eq. (89), symmetrically coupled, γl = γr = γ, to superconducting electrodes with |∆l| = |∆r| = ∆. in the limit of long chains one can prove that the current phase relation (at zero bias) is linear if tc = γ/2.30,70 this is the so called ishii's sawtooth behavior71 and is due to perfect ar. to better visualize the mar in the transient regime we therefore choose tc = γ/2. in equilibrium there are 16 abs in the gap. at time t = 0 the system is driven out of equilibrium by a dc bias ul applied to lead l. in fig. 8 we display the contour plot of the cur- rents jn,n+1(t) along the bond (n, n + 1) of region c as a function of time for different values of ul = 2∆/4, 2∆/3, 2∆/2. the mar pattern is illustrated with black arrows. there is a clear-cut transient sce- nario during which electrons undergo n ar before the ac josephson regime sets in, with n = ul/2∆. at every ar the current increases since the electrons are mainly reflected as holes and holes as electrons. the same nu- merical simulation in a normal system would have given a current in region 1ar smaller than the current in region 0ar. for the same system parameters we also considered a dc bias ul = 0.8 for which the dominant scattering mech- anism is the 3-rd order ar. the contour plot of the bond 16 0 250 500 750 1000 t 0 1 2 3 jr,abs ul = 0.0 ul = 0.03 ul = 0.1 ul = 0.3 fig. 7: abs contribution to the current at the right interface for dc biases with a superimposed microwave radiation described by ul(t) = ul + ur sin(ωrt), with ur = 0.05ωr, ωr = 1.08 and ul = 0.0, 0.03, 0.1, 0.3. the system is the same as in fig. 6 with ∆l = ∆r = 1, γl = γr = 1 and ǫc = 0. the time-step is δ = 0.05. fig. 8: td picture of mar. a chain of 21 atomic sites is symmetrically connected with γl = γr = 2tc = 2 to two identical superconducting leads with ∆l = ∆r = 1. a dc bias ul = 2∆/n, n = 4, 3, 2, is applied to lead l at time t = 0. the panels show the contour plots of the bond-current jn,n+1(t) across the atomic bonds of region c. the results are obtained with a time-step δ = 0.05, cut-offλ = 4 and a number of scattering states np,l = np,r = 1232. current is displayed in the top-left panel of fig. 9 and is similar to the case ul = 2∆/3 of fig. 8. a new scat- tering channel does, however, open if a microwave radia- tion of appropriate frequency is superimposed to ul. we therefore applied an ac bias ur(t) = ur sin(ωrt) to lead r and choose ωr to fulfill 2ul + ωr = 2∆, i.e., ωr = 0.4. in fig. 9 we report the contour plot of the bond-current for different values of ur = 0.0, 0.1, 0.3, 0.5. at ur ̸= 0 the right-going wave-front reduces its intensity just af- ter crossing the bond 10 due to scattering against the left-going wave-front from lead r, see the characteris- tic λ-shape in the bottom-right panel. when the right- going wave-front hits the right interface the bond cur- rent sharply increases. furthermore, the larger is ur the shorter is the transient regime. this can be explained as follows. at large ur the dominant scattering mechanism is the one in which an electron from lead l and energy ul is reflected as a hole and at the same time absorbs a photon of energy ωr. the energy of the reflected hole is 2ul + ωr = 2∆, no extra ar are needed for charge transfer and the photon-assisted josephson regime sets in. v. conclusions and outlooks in this paper we proposed a one-particle framework and a propagation scheme to study the td response of uf-jnj. by projecting the continuum hamiltonian onto a suitable set of localized states we reduced the problem to the solution of a discrete system in which the electro- magnetic field is described in terms of peierls phases. the latter provide the basic quantities to construct a density functional theory of superconducting (and as a special case normal) systems. we proved that under reasonable conditions the td bond current and pairing density of an interacting system driven out of equilibrium by peierls phases γ(t) can be reproduced in a system of noninter- acting ks electrons under the influence of peierls phases γ′(t) and pairing field ∆′(t) and that γ′(t) and ∆′(t) are unique. we considered the ks system initially in equilib- rium at given temperature and chemical potential when at time t = 0 an external electromagnetic field is switched on. to calculate the response of the system at times t > 0 we used a non-equilibrium formalism in which the normal and anomalous propagators are defined on an extended keldysh contour that includes a purely imaginary (ther- 17 fig. 9: photon-assisted mar in a chain of 21 atomic sites. the equilibrium parameters are the same as in fig. 8. an ac bias ur = ur sin(ωrt) in lead r is superimposed to a dc bias ul = 0.8 in lead l. the panels show the contour plots of the bond-current jn,n+1(t) across the atomic bonds of region c for different values of ur = 0.0, 0.1, 0.3, 0.5 and ωr = 0.4. the results are obtained with a time-step δ = 0.05, cut-off λ = 4 and a number of scattering states np,l = np,r = 1232. mal) path going from 0 to −iβ. we showed that the solution of the equations of motion for the negf are equivalent to first solve the static bdg equations and then the td bdg equations. it is worth emphasizing that in tdscdft the bdg equations do not follow from the bcs approximation and that their solution yields the exact bond-current and pairing density of an interacting system provided that the exact ks peierls phases and pairing field are used. for systems consisting of n superconducting leads in contact with a finite region c and driven out of equilib- rium by a longitudinal electric field a numerical algorithm is proposed. the initial eigenstates are obtained from a recent generalized wave-guide approach properly adapted to the superconducting case.61 the initial states are prop- agated in time using an embedded crank-nicholson al- gorithm which is norm-conserving, accurate up to sec- ond order in the time-step and that exactly incorpo- rates transparent boundary conditions. the propagation scheme reduces to the one of refs. 37,38 in the case of normal leads. the method described in this work allows for obtain- ing the td current across an uf-jnj and hence to fol- low the time evolution of several ar until the josephson regime sets in. as a first calculation of these kind we explored in detail the popular single-level qd model in the weak and intermediate coupling regime. we demon- strated that the transient time increases with decreasing bias and provided a quantitative picture of the mar. the rich structure of the transient regime is due to the abs which play a crucial role in the relaxation process. for dc pulses we showed that abs can be exploited to generate zero bias ac currents of tunable frequency. furthermore, irradiating the biased system with a microwave field of appropriate frequency the abs give rise to a long-living transient resonant effect. the transient regime increases also with the length of the junction. we considered one- dimensional atomic chains coupled to superconducting leads under dc and ac biases. here we showed that in conditions of perfect ar there exists a clear-cut transient scenario for mar. for biases ul = 2∆/n the dominant scattering channel is the n-th order ar and the transient regime lasts for about nn/vc where n is the length of the chain and vc the electron velocity at the fermi level. similar considerations apply to photon assisted mar. a more careful analysis of the transient regime is beyond the scope of the present paper. however such analysis is of utmost importance if the ultimate goal of supercon- ducting nanoelectronics is to use these devices for ultra- fast operations. the td properties presented in this work have been obtained using rather simple, yet so far unexplored, mod- els. a more sophisticated description of the hamilto- nian is, however, needed for a quantitative parameter- free comparison with experiments. theoretical advances also involve the development of approximate function- als for the self-consistent calculation of the td pairing potential and peierls phases. self-consistent calculations have so far been restricted to equilibrium s-d-s mod- els with a point-like attractive interaction treated in the bcs approximation.72–75 for biased systems, however, the pairing potential and peierls phases must be treated on equal footing and a first step in this direction would be the bcs approximation for the pairing field and the hartree-fock approximation for the peierls phases. more difficult is the study of uf-jnj in the coulomb blockade regime for which electron correlations beyond hartree- fock must be incorporated. finally, the approach presented in this work is not lim- ited to two terminal systems. the coupling of the cen- tral region to a third normal lead, or gate, allows for controlling the josephson current by varying the gate voltage.25,76,77 these systems can be potentially used for fast switches and transistors,78,79 and a microscopic un- derstanding of their ultrafast properties is therefore nec- essary to optimize their functionalities. appendix a: calculation of the embedding matrices without loss of generality we include few layers of each lead in the explicitly propagated region c. then, the embedding matrix q(m) α is zero everywhere except in the block of dimension 2n α cell × 2n α cell which is connected to the α lead. denoting with q(m) α such non-vanishing block 18 in q(m) α we have q(m) α = tα     1αα −iδ ̃ hαα m  1αα + iδ ̃ hαα m+1    0.0 t† α , (a1) where the subscript (0, 0) denotes the first diagonal block (supercell with j = 0) of the matrix in the square brack- ets. we notice that from eq. (73) the matrix ̃ hαα is the same as the matrix hαα(0) in eq. (63) but with renor- malized diagonal blocks ̃ hα = hα −μσα. in order to compute the q(m) α 's we introduce the generating matrix function qα(x, y) ≡tα  1 x1αα + iyδ ̃ hαα  0,0 t† α, (a2) which can also be expressed in terms of continued matrix fractions qα(x, y) = tα 1 x1α + iyδ ̃ hα + y2δ2tα 1 x1α + iyδ ̃ hα + y2δ2tα 1 ...... t† α t† α t† α = tα 1 x1α + iyδ ̃ hα + y2δ2qα(x, y) t† α ≡tαpα(x, y)t† α, (a3) where the last step is an implicit definition of pα(x, y). the q(m) α 's are obtained from the generating matrix func- tion as q(m) α = tα 1 m!  −∂ ∂x + ∂ ∂y m pα(x, y)     x=y=1 t† α = tαp(m) α t† α. (a4) using the identity 1 m![−∂ ∂x + ∂ ∂y]mp−1 α (x, y)pα(x, y) = 0, we derive the following recursive scheme (1α+ iδ ̃ hα)p(m) α = (1α−iδ ̃ hα)p(m−1) α −δ2 m x k=0 (q(k) α + 2q(k−1) α + q(k−2) α )p(m−k) α (a5) with p(m) α = q(m) α = 0 for m < 0. the above relation can be used to calculate q(m) α provided that all p(k) α are known for k < m. to obtain p(0) α we can use eq. (a3) with x = y = 1 in which the continued fraction is truncated after a number nlevel of levels. convergence can be easily checked by increasing nlevel. appendix b: calculation of the boundary term from eq. (81) we see that in order to propagate an eigenstate of h0 −μσ we need to know the boundary term defined in eq. (82). the state φ(0) can be either a scattering state or an abs. as shown in section iii a the projection onto lead α of a generic eigenstate with energy e can be written as a linear combination of states of the form φα k(m = s, j, α) = zα k (s)eikj, (b1) where the amplitudes zα k satisfies the eigenvalue equation hα + tαeik + t† αe−ik −μσα
zα k = ezα k . (b2) in the following we show how to compute the action of the operator ̃ hcα(0)gm αα  1αα + gαα  on φα k. we define the nambu vector in region c φα(m) c,k ≡ ̃ hcα(0)gm αα  1αα + gαα  φα k = 2 ̃ hcα(0)  1αα −iδ ̃ hαα m  1αα + iδ ̃ hαα m+1 φα k , (b3) from which the boundary term can easily be extracted by taking the appropriate linear combination of the φα(m) c,k and then multiplying by −iδz(m) α , see eq. (82). since re- gion c includes few layers of the leads the vector φα(m) c,k is zero everywhere except for the components correspond- ing to orbitals in contact with lead α. if we call φα(m) c,k the vector with such components from eq. (b3) we can write φα(m) c,k = 2tα     1αα −iδ ̃ hαα m  1αα + iδ ̃ hαα m+1 φα k    j=0 ≡2tαv α(m) k , (b4) where the subscript j = 0 in the square brackets denotes the vector of dimension 2n α cell with components given by the projection of the full vector onto the first (j = 0) supercell. as for the embedding matrices we introduce the generating function v α k (x, y) =  1 x1αα + iyδ ̃ hαα φα k  j=0 (b5) 19 from which the v α(m) k are obtained via multiple deriva- tives v α(m) k = 1 m!  −∂ ∂x + ∂ ∂y m v α k (x, y)     x=y=1 . (b6) the generating function can be obtained as follows. tak- ing φα k as in eq. (b1) and exploiting the property in eq. (b2) it is easy to realize that h ̃ hααφα k i j = (e −δj,0e−ikt† α) [φα k]j , (b7) where the subscript j denotes the vector of dimension 2n α cell with components given by the projection of the full vector onto the j-th supercell. then, multiplying the dyson identity 1 x1αα + iδy ̃ hαα = 1 x −iyδ x 1 x1αα + iyδ ̃ hαα ̃ hαα (b8) on the right by φα k , using eq. (b7) and solving for v α k (x, y) we obtain the following result v α k (x, y) = 1 + iyδe−ikpα(x, y)t† α x + iyδe zα k , (b9) where pα(x, y) is the generating function defined in eq. (a3). the quantity v α(m) k can now be obtained from eq. (b6) and reads v α(m) k = (1 −iδe)m (1 + iδe)m+1 zα k + iδe−ik m x n=0 (1 −iδe)m−n (1 + iδe)m−n+1 ×  p(n) α + p(n−1) α  t† αzα k . (b10) this conclude the calculation of the boundary term. 1 c. j. lambert and r. raimondi, j. phys.: condens. mat- ter 10, 901 (1998). 2 y. makhlin, g. sch ̈ on, and a. shnirman, rev. mod. phys. 73, 357 (2001). 3 k. k. likharev superconductor devices for ultrafast com- puting in applications of superconductivity, ed. h. wein- stock, kluwer (1999). 4 r. s. sorbello, solid state phys. 51, 159 (1997). 5 t. n. todorov, j. hoekstra, and a. p. sutton, phys. rev. lett. 86, 3606 (2001). 6 m. di ventra, s. t. pantelides, and n. d. lang, phys. rev. lett. 88, 046801 (2002). 7 c. verdozzi, g. stefanucci, and c.-o. almbladh, phys. rev. lett. 97, 046603 (2006). 8 j. clarke and f. k. wilhelm, nature 453, 1031 (2008). 9 a. zazunov, v. s. shumeiko, e. n. bratus, j. lantz, and g. wendin, phys. rev. lett. 90, 087003 (2003). 10 g. wendin and v. s. shumeiko, low temp. phys. 33, 724 (2007). 11 c. buizert, a. oiwa, k. shibata, k. hirakawa, and s. tarucha, phys. rev. lett. 99, 136806 (2007). 12 d. c. ralph, c. t. black, and m. tinkham, phys. rev. lett. 74, 3241 (1995). 13 e. n. bratus, v. s. shumeiko, and g. wendin, phys. rev. lett. 74, 2110 (1995). 14 m. c. koops, g. v. van duyneveldt, and r. de bruyn ouboter, phys. rev. lett. 77, 2542 (1995). 15 a. l. yeyati, j. c. cuevas, a. lopez-davalos, and a. martin-rodero, phys. rev. b 55, r6137 (1997). 16 g. johansson, e. n. bratus, v. s. shumeiko, and g. wendin, phys. rev. b 60, 1382 (1999). 17 j. c. cuevas, a. martin-rodero, and a. l. yeyati, phys. rev. b 54, 7366 (1996). 18 q. f. sun, h. guo, and j. wang, phys. rev. b 65, 075315 (2002). 19 c. w. j. beenakker, phys. rev. lett. 67, 3836 (1991). 20 u. gunsenheimer, u. sch ̈ ussler, and r. k ̈ ummel, phys. rev. b 49, 6111 (1994). 21 r. fazio and r. raimondi, phys. rev. lett. 80, 2913 (1998). 22 a. a. clerk and v. ambegaokar, phys. rev. b 61, 9109 (2000). 23 y. avishai, a. golub, and a. d. zaikin, phys. rev. b 67, 041301 (2003). 24 e. vecino, a. martin-rodero, and a. l. yeyati, phys. rev. b 68, 035105 (2003). 25 m. governale, m. g. pala, and j. k ̈ onig, phys. rev. b 77, 134513 (2008). 26 r. k ̈ ummel, z. phys. 218, 472 (1969). 27 h. d. raedt, k. michielsen, and t. m. klapwijk, phys. rev. b 50, 631 (1994). 28 a. jacobs and r. k ̈ ummel, phys. rev. b 64, 104515 (2001). 29 y. xing, q. f. sun, and j. wang, phys. rev. b 75, 125308 (2007). 30 e. perfetto, g. stefanucci, and m. cini, phys. rev. b 80, 205408 (2009). 31 o.-j. wacker, r. k ̈ ummel, and e. k. u. gross, phys. rev. lett. 73, 2915 (1994). 32 r. van leeuwen, phys. rev. lett. 82, 3863 (1999). 33 g. vignale, phys. rev. b 70, 201102(r) (2004). 34 p. danielewicz, ann. phys. (n.y.) 152, 239 (1984). 35 m. wagner, phys. rev. b 44, 6104 (1991). 36 g. stefanucci and c.-o. almbladh, phys. rev. b 69, 195318 (2004). 37 s. kurth, g. stefanucci, c.-o. almbladh, a. rubio, and e. k. u. gross, phys. rev. b 72, 035308 (2005). 38 g. stefanucci, e. perfetto, and m. cini, phys. rev. b 78, 075425 (2008). 39 e. runge and e. k. u. gross, phys. rev. lett. 52, 997 (1984). 40 s. k. ghosh and a. k. dhara, phys. rev. a 38, 1149 (1988). 41 at present there is no rigorous formulation of standard tddft in discrete systems. the difficulty here consists in finding simple criteria for the existence of a ks system, see c. verdozzi, phys. rev. lett. 101, 166401 (2008) and 20 references therein. 42 for a discussion on general initial configurations the reader is referred to refs. 34,35 and d. semkat, d. kremp and m. bonitz, phys. rev. e 59, 1557 (1998); k. morawetz, m. bonitz, v. g. morozov, g. r ̈ opke, d. kremp, phys. rev. e 63, 020102 (2001). 43 n. n. oliveira, e. k. u. gross, and w. kohn, phys. rev. lett. 60, 2430 (1988). 44 r. van leeuwen, n. e. dahlen, g. stefanucci, c. o. alm- bladh, and u. von barth, lect. notes phys. 706, 33 (2006). 45 l. v. keldysh, jetp 20, 1018 (1965). 46 y. nambu, phys. rev. 117, 648 (1960). 47 j. rammer and h. smith, rev. mod. phys. 58, 323 (1986). 48 a. f. andreev, sov. phys. jetp 19, 1228 (1964). 49 p. g. de gennes, superconductivity of metals and alloys (benjamin, new york, 1966). 50 r. k ̈ ummel, in physics and applications of mesoscopic josephson junctions, edited by h. ohta and c. ishii (the physical society of japan, tokyo, 1999), p. 19. 51 p. my ̈ oh ̈ anen, a. stan, g. stefanucci, and r. van leeuwen, phys. rev. b 80, 115107 (2009). 52 l. p. kadanoffand g. baym, quantum statistical mechan- ics (benjamin, new york, 1962). 53 n.-h. kwong and m. bonitz, phys. rev. lett. 84, 1768 (2000). 54 n. e. dahlen and r. van leeuwen, phys. rev. lett. 98, 153004 (2007). 55 a. stan, n. e. dahlen, and r. van leeuwen, j. chem. phys. 130, 224101 (2009). 56 m. p. von friesen, c. verdozzi, and c.-o. almbladh, phys. rev. lett. 103, 176404 (2009). 57 p. my ̈ oh ̈ anen, a. stan, g. stefanucci, and r. van leeuwen, europhys. lett. 84, 67001 (2008). 58 w. sch ̈ afer, j. opt. soc. am. b 13, 1291 (1996). 59 m. bonitz, d. kremp, d. c. scott, r. binder, w. d. kraeft, and h. s. k ̈ hler, j. phys.: condens. matter 8, 6057 (1996). 60 r. binder, h. s. k ̈ ohler, m. bonitz, and n. kwong, phys. rev. b 55, 5110 (1997). 61 g. stefanucci, e. perfetto, s. bellucci, and m. cini, phys. rev. b 79, 073406 (2009). 62 a.-p. jauho, n. s. wingreen, and y. meir, phys. rev. b 50, 5528 (1994). 63 g. schaller, p. zedler, and t. brandes, phys. rev. a 79, 032110 (2009). 64 g. stefanucci, phys. rev. b 75, 195115 (2007). 65 e. khosravi, s. kurth, g. stefanucci, and e. k. u. gross, applies phys. a 93, 355 (2008). 66 e. khosravi, g. stefanucci, s. kurth, and e. k. u. gross, phys. chem. chem. phys. 11, 4535 (2009). 67 j. c. cuevas, j. heurich, a. martin-rodero, a. l. yeyati, and g. sch ̈ on, phys. rev. lett. 88, 157001 (2002). 68 y. zhu, w. li, z. s ma, and t. h lin, phys. rev. b 69, 024518 (2004). 69 m. chauvin, p. vom stein, h. pothier, p. joyez, m. e. huber, d. esteve, and c. urbina, phys. rev. lett. 97, 067006 (2006). 70 i. affleck, j. s. caux, and a. m. zagoskin, phys. rev. b 62, 1433 (2000). 71 c. ishii, prog. theor. phys. 44, 1525 (1970). 72 a. martin-rodero, f. j. garcia-vidal, and a. l. yeyati, phys. rev. lett. 72, 554 (1994). 73 a. spuntarelli, p. pieri and g. c. strinati, con- mat/0911.4026. 74 a. m. martin and j. f. annett, phys. rev. b 57, 8709 (1998). 75 j. j. hogan-o'neill, a. m. martin, and j. f. annett, phys. rev. b 60, 3568 (1999). 76 f. k. wilhelm, g. sch ̈ on, and a. d. zaikin, phys. rev. lett. 81, 1682 (1998). 77 p. samuelsson, j. lantz, v. s. shumeiko, and g. wendin, phys. rev. b 62, 1319 (2000). 78 t. akazaki, h. takayanagi, and j. nitta, appl. phys. lett. 68, 418 (1996). 79 j. j. a. baselmans, a. f. morpurgo, b. j. van wees, and t. m. klapwijk, nature 397, 43 (1999).

0911.1719

bosonic colored group field theory

bosonic colored group field theory is considered. focusing first on dimension four, namely the colored ooguri group field model, the main properties of feynman graphs are studied. this leads to a theorem on optimal perturbative bounds of feynman amplitudes in the "ultraspin" (large spin) limit. the results are generalized in any dimension. finally integrating out two colors we write a new representation which could be useful for the constructive analysis of this type of models.

introduction group field theories (gft's) or quantum field theories over group manifolds were introduced in the beginning of the 90's [1, 2] as generalizations of matrix and tensor field theories [3, 4]. they are currently under active investigation as they provide one of the most complete definition for spin foam models, themselves acknowledged as good candidates for a background independent quantum theory of gravity [5, 6]. also they are more and more studied per se as a full theory of quantum gravity [4, 7, 8]. gft's are gauge invariant theories characterized by a non-local interaction which pairs the field arguments in a dual way to the gluing of simplicial complexes. hence, the feynman diagrams of a gft are fat graphs, and their duals are triangulations of (pseudo)manifolds made of vertices, edges, faces and higher dimensional simplices discretizing a particular spacetime. the first and simplest gft's [1, 2] have feynman amplitudes which are products of delta functions on the holonomies of a group connection associated to the faces of the feynman diagram. these amplitudes are therefore discretizations of the topological bf theory. but the gft is more than simply its perturbative feynman amplitudes. a quantum field theory formulated by a functional integral also assigns a precise weight to each feynman graph, and should resum all such graphs. in particular, it sums over different spacetime topologies1 thus providing a so-called "third quantization". however a difficulty occurs, shared by other discrete approaches to quantum gravity: since a gft sums over arbitrary spacetime topologies, it is not clear how the low energy description of such a theory would lead to a smooth and large manifold, namely our classical spacetime (see [7] for more details). in order to overcome this difficulty we propose to include the requirement of renormaliz- ability as a guide [8]. the declared goal would be to find, using a novel scale analysis, which of the gft models leads to a physically relevant situation after passing through the renormaliza- tion group analysis. let us mention that a number of non-local noncommutative quantum field theories have been successfully renormalized in the recent years (see [9] and references therein). even though the gft's graphs are more complicated than the ribbon graphs of noncommutative field theories, one can hope to extend to gft's some of the tools forged to renormalize these noncommutative quantum field theories. recent works have addressed the first steps of a renormalization program for gft's [7, 8] and for spin foam models [10, 11] combining topological and asymptotic large spin (also called "ultraspin") analysis of the amplitudes. a first systematic analysis of the boulatov model was started in [7]. the authors have identified a specific class of graphs called "type i" for which a complete procedure of contraction is possible. the exact power counting for these graphs has been established. they also formulated the conjecture that these graphs dominate in the ultraspin regime. the boulatov model was also considered in [8] as well as its freidel-louapre constructive regularization [12]. here, feynman amplitudes have been studied in the large cutoff limit and their optimal bounds have been found (at least for graphs without generalized tadpoles, see appendix below). these bounds show that the freidel-louapre model is perturbatively more divergent than the ordinary one. at the constructive field theory level, borel summability of the connected functions in the coupling constant has been established via a convergent "cactus" expansion, together with the correct scaling of the borel radius. these two seminal works on the general scaling properties in gft [7, 8] were restricted to three dimensions. the purpose of this paper is to extend the perturbative bounds of [8] to any dimension. a difficulty (which was overlooked in [8], see the appendix of this paper) comes from the fact that the power counting of the most general topological models is governed by "generalized tadpoles". 1 in this sense, gft's are quantum field theories of the spacetime and not solely on a spacetime [4]. 1 in the mean time, a fermionic colored gft model possessing a su(d + 1) symmetry in dimension d has been introduced by gurau [13], together with the homology theory of the corresponding colored graphs. in contradistinction with the general bosonic theory, the "bubbles" of this theory can be easily identified and tadpoles simply do not occur. apart from the su(d +1) symmetry, these nice features are shared by the bosonic version of this colored model, which is the one considered in this paper. we prove that a vacuum graph of such a theory is bounded in dimension 4 by knλ9n/2, where λ is the "ultraspin" cutoffand n the number of vertices of the graph. for any dimension d, similar bounds are also derived. we also take the first steps towards the constructive analysis of the model. the paper's organization is as follows. section 2 is devoted to the definition of the colored models, the statement of some properties of their feynman graphs, and the perturbative bounds which can be proved in any dimension. section 3 further analyses the model by integrating out two particular colors. this leads to a matthews-salam formulation which reveals an interesting hidden positivity of the model encouraging for a constructive analysis. a conclusion is provided in section 4 and an appendix discusses the not yet understood case of graphs with tadpoles in general gft's. 2 perturbative bounds of colored models in this section, we introduce the colored ooguri model or colored su(2) bf theory in four dimensions2. some useful (feynman) graphical properties are stated and allow us to bound a general feynman amplitude. these bounds are then generalized to any dimension. 2.1 the colored ooguri model the dynamical variables of a d dimensional gft are fields, defined over d copies of a group g. for the moment, let us specialize to d = 4 and g = su(2), hence to ooguri-type models. in the colored bosonic model, these fields are themselves d + 1 complex valued functions φl, l= 1, 2, . . . , d+1 = 5. the upper index ldenotes the color index of the field φ = (φ1, . . . , φ5). the fields are required to be invariant under the "diagonal" action of the group g φl(g1h, g2h, g3h, g4h) = φl(g1, g2, g3, g4), h ∈g, ̄ φl(g1h, g2h, g3h, g4h) = ̄ φl(g1, g2, g3, g4), h ∈g, (1) but are not symmetric under any permutation of their arguments. we will use the shorthand notation φl α1,α2,α3,α4 := φl(gα1, gα2, gα3, gα4). the dynamics is traditionally written in terms of an action s[φ] := z 4 y i=1 dgi 5 x l=1 ̄ φl 1,2,3,4 φl 4,3,2,1 + λ1 z 10 y i=1 dgi φ1 1,2,3,4 φ5 4,5,6,7 φ4 7,3,8,9 φ3 9,6,2,10 φ2 10,8,5,1 +λ2 z 10 y i=1 dgi ̄ φ1 1,2,3,4 ̄ φ5 4,5,6,7 ̄ φ4 7,3,8,9 ̄ φ3 9,6,2,10 ̄ φ2 10,8,5,1 (2) supplemented by the gauge invariance constraints (1), ̄ φlφlis a quadratic mass term. integrations are performed over copies of g using products of invariant haar measures dgi of this group and λ1,2 are coupling constants. 2su(2) is chosen here for simplicity. the so(4) or so(3, 1) bf theories could be treated along the same lines. supplemented with plebanski constraints they are a starting point for four dimensional quantum gravity. 2 a2 a3 1 a a4 a a 2 a 3 1a a 4 a a2 a3 1 a a4 a a 2 a 3 1a a 4 a covariance figure 1: the propagator or covariance of the colored model. 15 14 13 12 1 43 42 41 5 4 4 21 23 2 24 25 35 34 32 31 3 52 53 54 51 5 51 52 53 54 5 32 31 35 34 3 21 25 24 23 2 45 41 42 43 4 15 14 13 12 1 pentachore i pentachore ii figure 2: the vertices of the colored model: pentachores i and ii are associated with interactions of the form φ5 and ̄ φ5, respectively. the initial model of this type was fermionic [13], hence the fields were complex grassmann variables ψland ̄ ψl. the corresponding monomials ψ1ψ2ψ3ψ4ψ5 and ̄ ψ1 ̄ ψ2 ̄ ψ3 ̄ ψ4 ̄ ψ5 are su(5) invariant. in the bosonic model, this invariance is lost as the monomials are only invariant under transformations with permanent 1, which do not form a group. more rigorously one should consider the gauge invariance constraints as part of the prop- agator of the group field theory. the partition function of this bosonic colored model is then rewritten as, z(λ1, λ2) = z dμc[ ̄ φ, φ] e−λ1t1[φ]−λ2t2[ ̄ φ], (3) where t1,2 stand for the interaction parts in the action (2) associated with the φ's and ̄ φ's respectively, and dμc[ ̄ φ, φ] denotes the degenerate gaussian measure (see a concise appendix in [8]) which, implicitly, combines the (ordinary not well defined) lebesgue measure of fields d[ ̄ φ, φ] = q ld ̄ φldφl, the gauge invariance constraint (1) and the mass term. hence dμc[ ̄ φ, φ] is associated with the covariance (or propagator) c given by z ̄ φl 1,2,3,4 φl′ 4′,3′,2′,1′ dμc[ ̄ φ, φ] = cll′(g1, g2, g3, g4; g′ 4, g′ 3, g′ 2, g′ 1) = δll′ z dh 4 y i=1 δ(gih(g′ i)−1). (4) as in the ordinary gft situation, the covariance c, a bilinear form, can also be considered as an operator, which is a projector (satisfying c2 = c) and acts on the fields as follows: [cφ]l 1,2,3,4 = x l′ z dg′ i cll′(g1, g2, g3, g4; g′ 4, g′ 3, g′ 2, g′ 1) φl′ 4′,3′,2′,1′ = z dh φl(g4h, g3h, g2h, g1h). (5) 3 2.2 properties of feynman diagrams in a given dimension d, feynman graphs of a gft are dual to d dimensional simplicial complexes triangulating a topological spacetime. in this subsection, we illustrate the particular features of this duality in the above colored model. associating the field φlto a tetrahedron or 3-simplex with its group element arguments gi representing its faces (triangles), then it is well known that the order of arguments of the fields in the quintic interactions (2) follows the pattern of the gluing of the five tetrahedra l= 1, ..., 5, along one of their faces in order to build a 4-simplex or pentachore (see fig.2). besides, the propagator can be seen as the gluing rule for two tetrahedra belonging to two neighboring pentachores. for the bf theory this gluing is made so that each face is flat. in the present situation, the model (2) adds new features in the theory: the fields are complex valued and colored. consequently, we can represent the complex nature of the fields by a specific orientation of the propagators (big arrows in fig.1) and the color feature of the fields is reflected by a specific numbering (from 1 to 5, see fig.2) of the legs of the vertex. hence, the only admissible propagation should be between a field ̄ φland φlof the same color index, and "physically" it means that only tetrahedra of the same color belonging to two neighboring pentachores can be glued together. finally, given a propagator with color l= 1, 2, ..., 5, it has itself sub-colored lines (called sometimes "strands") that we write cyclically lj, and the gluing also respects these subcolors (because our model does not include any permutation symmetry of the strands). let us enumerate some useful properties of the colored graphs. the following lemmas hold. lemma 2.1 given a n-point graph with n internal vertices, if one color is missing on the external legs, then (i) n is a even number; (ii) n is also even and external legs have colors which appear in pair. proof. let us consider a n-point graph with n internal vertices. consequently, one has n fields for each color. since one color is missing on the external legs, and since by parity, any contraction creating an internal line consumes two fields of the same color, the full contraction process for that missing color consumes all the n fields of that color and an even number of fields. hence n must be even. this proves the point (i). we prove now the point (ii). we know that the number n of internal vertices is even. now, if a color on the external legs appears an odd number of times, the complete internal contraction for that color would involve an odd number of fields hence would be impossible. □ an interesting corollary is that for n < d + 1 the conclusions of lemma 2.1 must hold. in particular the colored theory in dimension d has no odd point functions with less than d arguments. for example, the colored ooguri model in four dimensions has no one and three point functions. this property is reminiscent of ordinary even field theories like the φ4 4 model which has also neither one nor three point functions. it may simplify considerably the future analysis of renormalizable models of this type. we recall that in a colored gft model, (i) a face (or closed cycle) is bi-colored with an even number of lines; (ii) a chain (opened cycle) of length > 1 is bi-colored. 4 t23 t41 t14 = t 32 figure 3: generalized tadpoles. this is in fact the definition of a face in [13] but can also be easily deduced from the fact that each strand in a line l(0) a joining a vertex v (0) to a vertex v (1) possesses a double label that we denote by ab (see fig. 2): a is the color index of l(0) a and b denotes the color of the line l(1) b after the vertex v1, where the strand ab will propagate. in short, a vertex connects in a unique way, the strand ab of the line la to the strand ba of the line lb. the point is that, in return, the strand ba belonging to l(1) b can only be connected to a strand of the form ab in a line l(2) a through another vertex v (2). hence only chains of the bi-colored form abbaabba . . . ba (for closed cycles) and abbaabba . . . ab (for open chains) could be obtained. closed cycles include clearly an even number of lines. □ definition 2.1 a generalized tadpole is a (n < 5)-point graph with only one external vertex (see fig.3). in the case of a generalized tadpole with one external leg, that external leg must contract to a single vertex, creating a new tadpole but with four external legs. theorem 2.1 there is no generalized tadpole in the colored ooguri group field model. proof. by definition, a generalized tadpole is a (n < 5)-point graph, meaning that at least one of the colors is missing on the external legs. then, lemma 2.1 tells us that n should be even (n = 2 or 4) and the external colors appear in pairs. having in mind that, still by definition, an ordinary vertex of the colored theory has no common color in its fields, the external vertex of a generalized tadpole having colors appearing in pairs cannot be an ordinary colored vertex. □ hence, the colored ooguri model has no generalized tadpole and so, a fortiori, no tadpole. this property does not actually depend on the dimension. the same strategy as in [8] is used in order to determine the types of vertex operators from which the feynman amplitude of a general vacuum graph will be bounded in the next subsection. we first start by some definitions. definition 2.2 (i) a set a of vertices of a graph g is called connected if the subgraph made of these vertices and all their inner lines (that is all the lines starting and ending at a vertex of a) is connected. (ii) an (a, b)-cut of a two-point connected graph g with two external vertices va and vb is a partition of the vertices of g into two subsets a and b such that va ∈a, vb ∈b and a and b are still connected. (iii) a line joining a vertex of a to a vertex of b in the graph is called a frontier line for the (a, b)-cut. 5 (iv) a vertex of b is called a frontier vertex with respect to the cut if there is a frontier line attached to that vertex. (v) an exhausting sequence of cuts for a connected two-point graph g of order n is a sequence a0 = ∅& a1 & a2 & * * * & an−1 & an = g such that (ap, bp := g \ ap) is a cut of g for any p = 1, * * * , n −1. given a graph g, an exhausting sequence of cuts is a kind of total ordering of the vertices of g, such that each vertex can be 'pulled' successively through a 'frontier' from one part b to another part a, and this without disconnecting a or b. lemma 2.2 let g be a colored connected two-point graph. there exists an exhausting sequence of cuts for g. the proof can be worked out by induction along the lines of a similar lemma in [8]. the main ingredient for this proof in the colored model is the absence of generalized tadpoles as estab- lished by theorem 2.1. we reproduce here this proof in detail in the four dimensional case for completeness, but the result holds in any dimension. proof of lemma 2.2. let us consider a two-point graph g with n vertices and assume that a sequence a0 = ∅& a1 & a2 & * * * ap and its corresponding sequence {bj=0,1,...,p}, have been defined for 0 ≤p < n −1, such that for all j = 0, 1, . . . , p, (aj, bj) is a cut for g. then another frontier vertex vp+1 has to be determined such that ap+1 = ap ∪{vp+1} and bp+1 = g \ ap+1 define again a cut for g. let us consider a tree tp with a fixed root vb which spans the remainder set of vertices bp and give them a partial ordering. the set bp being finite, we can single out a maximal frontier vertex vmax with respect to that ordering, namely a frontier vertex such that there is no other frontier vertex in the "branch above vmax" in tp. we prove first that vmax ̸= vb. the proposition vmax = vb would imply that vb is the only frontier vertex left in bp. the vertex vb has four internal lines and therefore two cases may occur: (1) all these are frontier lines then this means that {vb} = bp which contradicts the fact that p < n −1; (2) not all lines are frontier lines which implies that some of these lines span a generalized tadpole. this possibility does not occur by theorem 2.1. let us then assume that vmax ̸= vb and choose vp+1 = vmax. we would want to prove that the set bp+1 = bp \ {vp+1} is still connected through internal lines. cutting lp+1, the unique link between vp+1 and vb in tp splits the tree into two connected components. we call rp the part containing vb, and sp the other one. note that sp becomes a rooted tree with root and only frontier vertex vp+1 (remember that vp+1 is maximal). from the property of vp+1 to be a frontier vertex, one deduces that it has at most four lines in bp (and so at least one frontier line) and hence there are at most three lines from vp+1 to bp+1 distinct from lp+1. since the tree rp does not have any line hooked to vp+1, its lines remain inner lines of bp+1 confining all of its vertices to a single connected component of bp+1. assume that bp+1 is not connected, this would imply first, that sp contains other vertices than vp+1. second, removing the root vp+1 from sp and the at most three lines hooked to it, one, two, or three connected components would be obtained. these component are made of the vertices of sp \ {vp+1} plus their inner lines, which no longer hook to rp through inner lines of bp+1. due to the fact that these components have no frontier vertices hence no other frontier lines, it would mean that they must have been hooked to the total graph g through at most three lines from vp+1 to bp+1 distinct from lp+1, hence they would have been a generalized tadpole which is not admissible. □ 6 14 o o 23 figure 4: the vertex operators. b b b b b b b b b b b b b b b b b b b b b b g23 n = g14 n = figure 5: the chains of graphs g23 n and g14 n with 2n vertices. hi 1 hi 2 hi 3 hi 4 h 1 i+1 h 3 i+1 4 hi+1 2 hi+1 hi 0 figure 6: elements in the chain g14 n . 2.3 perturbative bounds to begin with the study of perturbative bounds of the feynman graphs, let us introduce a cutoff of the theory in order to define a regularized version of the partition function z(λ1, λ2) (3). we then truncate the peter-weyl field expansion as φl 1,2,3,4 = λ x j1,j2,j3,j4 tr φj1,j2,j3,j4dj1(g1)dj2(g2)dj3(g3)dj4(g4)
, (6) where the summation is over the spin indices j1,...,4, up to λ, dj(g) denotes the (2j+1)-dimensional matrix representing g and φj1,j2,j3,j4 are the corresponding modes. on the group, the delta function with cutoffis of the form δλ(h) = pλ j (2j + 1) tr dj(h). this function behaves as usual as r dg δλ(hg−1)δλ(gk−1) = δλ(hk−1), r dgδ(gh) = 1, and diverges as pλ j j2 ∼λ3. we mention that the bounds obtained hereafter are only valid for vacuum graphs. for general graphs with external legs, a bound can be easily obtained by cauchy-schwarz inequalities by taking the product of the l2-norms of these external legs (seen as test functions) times the amplitude of a vacuum graph. 7 we now prove the following theorem theorem 2.2 there exists a constant k such that for any connected colored vacuum graph g of the ooguri model with n internal vertices, we have |ag| ≤knλ9n/2+9. (7) this bound is optimal in the sense that there exists a graph gn with n internal vertices such that |ag| ≃knλ9n/2+9. proof. lemma 2.2 shows that the vertices can be "pulled out" one by one from the connected graph g. from this procedure, we build two kinds of vertex operators which compose the graph g. these are the operators o14 and o23 (see fig.4) and their adjoint acting as o14 : hλ 0 − → hλ 0 ⊗hλ 0 ⊗hλ 0 ⊗hλ 0 o23 : hλ 0 ⊗hλ 0 − → hλ 0 ⊗hλ 0 ⊗hλ 0 (8) where hλ 0 := hλ ∩im c is a subspace of su(2)-right invariant function belonging also to hλ ⊂ l2(su(2)4) the subspace of l2 integrable functions with λ-truncated peter-weyl expansion. the norm of these operators can be computed according to the formula ||h|| = lim n→∞  tr[h†h]n1/2n , (9) where h† denotes the adjoint operator associated with h. we start by the calculations of tr(o14o41)n using the formula [7] tr(o14o41)n = z y l∈lg14 n dhl y f∈fg14 n δλ  ⃗ y l∈∂fhl  (10) with g14 n (see fig.5 and fig.6) the vacuum graph obtained from tr(o14o41)n, lg14 n its set of lines, fg14 n its set of faces (closed cycles of strands). the oriented product in the argument of the delta function is to be performed on the hl belonging to each oriented line of each oriented face. the rule is that if the orientation of the line and the face coincide, one takes hl in the product; if the orientations disagree one takes the value h−1 l . we get after a reduction (we omit henceforth the subscript λ in the notation of the truncated delta functions) tr(o14o41)n = z n y i=1 4 y j=0 dhi j ( δ( n y i=1 hi 1hi 0)δ(hn 0 h1 1)δ( n y i=1 hi 2hi 0)δ(hn 0 h1 2)δ( n y i=1 hi 3hi 0)δ(hn 0 h1 3)δ( n y i=1 hi 4hi 0)δ(hn 0 h1 4) n y i=1 δ(hi 1(hi 2)−1)δ(hi 1(hi 3)−1)δ(hi 1(hi 4)−1)δ(hi 2(hi 3)−1)δ(hi 2(hi 4)−1)δ(hi 3(hi 4)−1) ) . (11) after integration, it is simple to deduce that tr(o14o41)n ≤λ9n+9 and the operator norm is bounded by ||o14|| ≤λ9/2. (12) a similar calculation allows us to write tr(o23o32)2n ≤λ6n+15 (see graph g23 n in fig.5) such that we get the bound ||o23|| ≤λ3/2. □ (13) the meaning of the norms of the operators o14 and o23 is the following: each vertex in a graph gn diverges at most as λ9/2 which is the bound of ||o14||. roughly speaking the amplitude of a colored graph is bounded by knλ9n/2 where n is its number of vertices. 8 2 a a1 d a 2 a a1 d a (d+1) 1 (d+1) d (d+1) 2 d+1 d d+1 d 1 d 2 d 21 23 2(d+1) 2d 2 12 1d 1(d+1) 1 vertex propagator figure 7: propagator and vertex φd+1 in d dimensional gft. 2.4 d dimensional colored gft we treat in this subsection, in a streamlined analysis, how to extend the above perturbative bounds to any dimension d. the d dimensional gft model. in dimension d, the action (2) finds the following extension sd[φ] := z d y i=1 dgi d+1 x l=1 ̄ φl 1,2,...,d φl d,...,2,1 +λ1 z y dgij φ1 12,13,...,1(d+1) φ(d+1) (d+1)1,(d+1)2,(d+1)3,...,(d+1)d φd dd+1,d1,d2,...,dd−1 . . . . . . . . . φ3 34,35,...,3d+1,31,32 φ2 23,24,...,2d+1,21 d+1 y j̸=i δ(gij(gji)−1) +λ2 z y dgij ̄ φ1 12,13,...,1(d+1) ̄ φd+1 (d+1)1,(d+1)2,(d+1)3,...,(d+1)d ̄ φd dd+1,d1,d2,...,dd−1 . . . . . . . . . ̄ φ3 34,35,...,3d+1,31,32 ̄ φ2 23,24,...,2d+1,21 d+1 y j̸=i δ(gij(gji)−1) (14) where the complex colored fields φl: gd →c will be denoted by φl(gli, glj, . . . , glk) = φl li,lj,...,lk, and glk is a group element materializing the link in the vertex of two colors land k (the general propagator and vertex φd+1 are pictured in fig.7; the vertex for ̄ φd+1 can be easily found by conjugation). perturbative bounds. an extension of theorem 2.1 can be easily realized here in any dimension using similar conditions as in lemma 2.1 for any n-point function (the argument on the parity of the number of lines in complete contraction procedure will still hold here). since there is no generalized tadpole again in the d dimensional theory, we are able to still provide a exhaustive sequence of cuts for any colored graph. from this point, we can formulate the following statement: theorem 2.3 there exists a constant k such that for any connected colored vacuum graph g of the d dimensional gft model with n internal vertices, we have |ag| ≤knλ3(d−1)(d−2)n/4+3(d−1). (15) 9 d+1−p p figure 8: the operator o(d + 1 −p, p). this bound is optimal in the sense that there exists a graph gn with n internal vertices such that |ag| ≃knλ3(d−1)(d−2)n/4+3(d−1). proof. given 0 < p ≤d, the generalized vertex operator is an operator with d + 1 −p legs in a part a and remaining p legs in a part b, for a (a, b)-cut for g (see fig.8). from the generalized formula for tr(od+1−p,pop,d+1−p)2n = λ3[((d−p−1)(d−p)+(p−1)(p−2))n+(d+1−p)p−1], (16) we can determine the bound on the norm of the operator od+1−p,p as ||od+1−p,p|| ≤λ 3(d−1)(d−2)−2(p−1)(d−p) 4 . (17) then the maximum of the bound occurs for p = 1, giving an operator od,1. for this operator we have ||od,1|| ≤(λ3/2) (d−1)(d−2) 2 . □ (18) setting d = 4 and p = 1, (18) recovers the bounds of o14 (12). in the specific d = 3 dimensional case, (18) reduces to λ3/2 bound of the vertex operators of boulatov's model as established in [8]. 3 integration of two fields: first steps of constructive analysis a main interesting feature of the colored theory, is the possibility of an explicit integration of two fields leading to a determinant as in the matthews-salam formalism [14]. we explore this possibility in four dimensions in this subsection, but the result again is general. let us start by writing the vertex terms in the form (setting henceforth λ1 = λ2 = λ) sv = λ (z 4 y i=1 dgidg′ i φ1 1,2,3,4  δ(g1(g′ 1)−1) z dg5dg6dg7 φ5 4,2′,5,6 φ4 6,3,3′,7 φ3 7,5,2,4′  φ2 4′,3′,2′,1′ + z 4 y i=1 dgidg′ i ̄ φ1 4′,3′,2′,1′  δ(g′ 4(g4)−1) z dg5dg6dg7 ̄ φ5 1′,3,5,6 ̄ φ4 6,2,2′,7 ̄ φ3 7,5,3′,1  ̄ φ2 1,2,3,4 ) ,(19) such that the following operators h(g1, g2, g3, g4; g′ 4, g′ 3, g′ 2, g′ 1) = δ(g1(g′ 1)−1) z dg5dg6dg7 φ5 4,2′,5,6 φ4 6,3,3′,7 φ3 7,5,2,4′ , (20) h∗(g1, g2, g3, g4; g′ 4, g′ 3, g′ 2, g′ 1) = δ(g4(g′ 4)−1) z dg5dg6dg7 ̄ φ5 1′,3,5,6 ̄ φ4 6,2,2′,7 ̄ φ3 7,5,3′,1 (21) 10 allow us to express the partition function (3) as z(λ) = z dμc[ ̄ φ, φ] exp[−λ[ z 4 y i=1 dgidg′ i φ1 1,2,3,4 h(g1, g2, g3, g4; g′ 4, g′ 3, g′ 2, g′ 1) φ2 4′,3′,2′,1′ + ̄ φ2 1,2,3,4 h∗(g1, g2, g3, g4; g′ 4, g′ 3, g′ 2, g′ 1) ̄ φ1 4′,3′,2′,1′]]. (22) note the important property of h∗to be the adjoint of h. indeed, a quick inspection shows that h∗= ( ̄ h)t which can be checked by a complex conjugation of the fields and a symmetry of arguments such that 1 →4 and 2 →3. the integration of this function follows standard techniques in quantum field theory. we can introduce the vector v = (reφ1, imφ1, reφ2, imφ2) and its transpose vt and such that φ1hφ2 and ̄ φ1 ̄ h ̄ φ2 and the mass terms ̄ φl=1,2 1,2,3,4φl=1,2 4,3,2,1 can be expressed as in the matrix form z(λ) = z dμc[ ̄ φ, φ] e−λvt a v, (23) where the matrix operator a can be expressed as a =     0 0 h ih 0 0 ih −h h∗ −ih∗ 0 0 −ih∗ −h∗ 0 0    . (24) after integration over the colors 1 and 2, using the normalized gaussian measure dμc[ ̄ φ, φ] = dμ′ c′[ ̄ φ1,2; φ1,2]dμ′′ c′′[ ̄ φ3,4,5, φ3,4,5], we get z(λ) = z dμ′′ c′′[ ̄ φ3,4,5, φ3,4,5] k[det(1 + λca)]−1 = z dμ′′ c′′[ ̄ φ3,4,5, φ3,4,5] ke−tr log(1+λca), (25) where k is an unessential normalization constant that we omit in the sequel. mainly the operator product ca can be calculated by composing ch and ch∗. we obtain for ch and ch∗, respectively, the following operators defined by their kernel h(g1, g2, g3, g4; g′ 4, g′ 3, g′ 2, g′ 1) = z 4 y i=1 dg′′ i c(g1, g2, g3, g4; g′′ 4, g′′ 3, g′′ 2, g′′ 1)h(g′′ 1, g′′ 2, g′′ 3, g′′ 4; g′ 4, g′ 3, g′ 2, g′ 1) (26) = z dg5dg6dg7 z dh δ(g1h(g′ 1)−1) φ5(g4h, g′ 2, g5, g6) φ4(g6, g3h, g′ 3, g7) φ3(g7, g5, g2h, g′ 4) , h∗(g1, g2, g3, g4; g′ 4, g′ 3, g′ 2, g′ 1) = z 4 y i=1 dg′′ i c(g1, g2, g3, g4; g′′ 4, g′′ 3, g′′ 2, g′′ 1)h∗(g′′ 1, g′′ 2, g′′ 3, g′′ 4; g′ 4, g′ 3, g′ 2, g′ 1) (27) = z dg5dg6dg7 z dh δ(g4h(g′ 4)−1) φ5(g′ 1, g3h, g5, g6) φ4(g6, g′ 2, g2h, g7) φ3(g7, g5, g′ 3, g1h). it is convenient to see ca as a sum of two matrices h + h∗which are defined by the offblock diagonal matrices of ca involving at each matrix element h or h∗. then the determinant integrant of (25) can be expressed, up to a constant, more simply by e−tr log(1+λ(h+h∗)) = e+tr p∞ n=1 (−λ)n n (h+h∗)n 11 = e+tr p∞ p=1 λ2p 2p ([h+h∗)2])p = e 1 2tr p∞ p=1 λ2p p (q)p = e−1 2tr log(1−λ2q) (28) where we have used the fact that tr((h+h∗)2p+1) = 0 for all p, and q := (h+h∗)2 = hh∗+h∗h, since h2 = 0 = (h∗)2. the operator q is hermitian and is given as q =     2hh∗ −2ihh∗ 0 0 2ihh∗ 2hh∗ 0 0 0 0 2h∗h 2ih∗h 0 0 −2ih∗h 2h∗h    . (29) thus, q has positive real eigenvalues and −λ2q is positive for λ = ic, c ∈r. the next purpose is to investigate a bound on the radius of the series f(λ) = log z(λ) denoting the free energy and computing the amplitude sum of connected feynman graphs. for this, we will use a cactus expansion [17] and the brydges-kennedy forest formula (see [18] and references therein; for a short pedagogical approach see [8]) on the partition function z(λ). expanding e−1 2 tr log(1−λ2q) in terms of vλ = −1 2tr log(1 −λ2q), using the replica trick, and then applying the brydges-kennedy formula, one comes to z(λ) = z dνc[ ̄ φ5,4,3, φ5,4,3] e−1 2tr log(1−λ2q) = ∞ x n=0 1 n! x f ∈fn y l∈f z 1 0 dhl ! y l∈f ∂ ∂hl ! z dνhf n (σ1, . . . , σn) n y v=1 vλ(σv), (30) where we have changed the notation dνc = dμ′′ c′′ and each of the σi represent an independent copy of the six fields ( ̄ φ5,4,3, φ5,4,3)i; the second sum is over the set fn of forests built on n points or vertices; the product is over lines l in a given forest f; hf is a n(n −1)/2-tuple with element hf l = minp hp, where p take values in the unique path in f connecting the source s(l) and target t(l) of a line l ∈f, if such a path exists, otherwise hf l = 0. it is well known that the summand factorizes along connected components of each forest. therefore log z(λ) is given by the same series in terms of trees (connected forests) as f(λ) = ∞ x n=1 1 n! x t∈tn y l∈t z 1 0 dhl ! y l∈t ∂ ∂hl ! z dνht n (σ1, . . . , σn) n y v=1 vλ(σv). (31) the trees t join the new vertices vλ(σv)'s often called "loop vertices". the covariance of dνht n (σ1, . . . , σn) can be expressed by, cht ij; ab(gi; gj) =    1 if i = j and a = b δab ht l c(gi; gj) if i ̸= j 0 otherwise (32) where l = {ij} denotes the line with source s(l) = i and target t(l) = j. we use also the notation (gi, gj) = ((gi)k=1,2,3,4, (gj)k=1,2,3,4) ∈g4×4, a and b are color indices such that formally, we have dνht n (σ1, . . . , σn) = e r dgdg′ pn i,j=1 p5 a,b=1 δ δ ̄ φa i (gi) cht ij;ab(gi,gj) δ δφb j(gj ) . (33) the partial derivative ∂/∂hl in (31) acts on the measure dνht n (σ1, . . . , σn) and one obtains f(λ) = ∞ x n=1 1 n! x t∈tn y l∈t z 1 0 dhl ! z dνht n (σ1, . . . , σn) (34) y l∈t z d4gs(l)d4gt(l) c(gs(l); gt(l)) δ2 δ ̄ φa(l) s(l)(gs(l))δφa(l) t(l) (gt(l)) n y v=1 vλ(σv), 12 with the color index a(l) denoting the color of the line l. let us denote by kv the coordination number of a given loop vertex vλ(σv) hooked to the tree t by half lines lv = 1, . . . , kv such that s(lv) = v or t(lv) = v. since each half line corresponds to a derivative, we can decompose the kv derivatives in two parts pv + qv = kv such that, up to inessential constants t (λ; kv) = pv y l1 v=1 δ δ ̄ φa(l1 v) v (gl1 v) qv y l2 v=1 δ δφa(l2 v) v (gl2 v)  −1 2tr log(1 −λ2q[ ̄ φ5,4,3 v ; φ5,4,3 v ])  (35) = tr x r pr+qr>0 p pr=pv;p qr=qv r y j=1      δpj+qj λ2q qpj l(1;j) v =1 δ ̄ φa(l(1;j) v ) v (gl(1;j) v ) qqj l(2;j) v =1 δφa(l(2;j) v ) v (gl(2;j) v ) 1 1 −λ2q      . note that, as previously mentionned the coupling constant λ = ic is a pure imaginary complex number so that the denominator 1 + c2q is positive. a rigorous bound for (35) is complicated to determine. however an encouraging remark is to consider the constant field modes or constant fields themselves (even if the physical relevance of these "background" modes is not clear at this stage). if we restrict to these constant modes, then (35) can be bounded as follows. noting that q is a polynomial in the fields φa of degree six, the product of derivatives acting on it behaves like t (λ; kv) ≤ x r pr+qr>0 p pr=pv;p qr=qv r y j=1 |λ1/3φ|6−pj−qj(λ1/3)pj+qj 1 + |λ1/3φ|6 ≤ x r pr+qr>0 p pr=pv;p qr=qv r y j=1 (λ1/3)pj+qj. (36) note that similar formulas exist in dimension d with q a polynomial of degree 2(d −1). hence this method opens a constructive perspective for colored gft in any dimension. this perspective is distinct or could be a complement to the freidel-louapre approach. 4 conclusion perturbative bounds of a general vacuum graph of colored gft have been obtained in any dimension. we have shown that, in dimension d, the scaling of the amplitude of any vacuum graph, in the "ultraspin" cutoffλ, behaves like knλ3(d−1)(d−2)n/4+3(d−1) where n is the number of vertices. in addition, it has been revealed that, by an integration of two colors, the model is positive if the coupling constant is purely imaginary. we did not need further regularization procedure, such as an inclusion of a freidel-louapre interaction term which in return may be more divergent than the ordinary theory. using a loop vertex expansion, we have reached the property that for at least the constant modes of the fields the forest-tree formula leads to a convergent series (which should be the borel-le roy sum of appropriate order of the ordinary perturbation series). these results are encouraging for a constructive program. the bounds obtained in this paper for colored gft models should now be completed into a more precise power counting and scaling analysis. they should also be extended to the physically more interesting models in particular the so called eprl-fk model [16, 15]. we recall that the most simple divergencies of the eprl-fk model have been studied in [10]. this work analyzes the elementary "bubble" (built out of two vertices) corresponding to the one-loop self-energy correction and the "ball" (built out of five vertices), corresponding to the one-loop vertex correction. the authors prove that putting the external legs at zero spin, the degree of 13 figure 9: nonplanar tadpoles with their additional divergence (dashed line) when inserted in more involved graphs. divergence in the cutoffof the ball could be logarithmic. this result sounds very promising from the renormalization point of view. the first step in this program is to establish the power counting of a simplified colored ooguri model with commutative group, which we call "linearized" colored ooguri model. we checked that power counting in this case is given by a homology formula [13, 19]. then we intend to perform a similar analysis of the colored linearized eprl-fk model, and find out the analogs of the multiscale renormalization group analysis in this context. this should help for the more complicated study of the "non-linear" models in which homotopy rather than homology should ultimately govern the power counting. acknowledgments the work of j.b.g. is supported by the laboratoire de physique th ́ eorique d'orsay (lpto, uni- versit ́ e paris sud xi) and the association pour la promotion scientifique de l'afrique (apsa). discussions with r. gurau are gratefully acknowledged, which in particular lead to the represen- tation of section 3. 5 appendix: the case of tadpoles in this appendix, we give some precisions on the scaling properties of amplitudes of graphs containing generalized tadpoles as occur in the non colored gft. we will restrict the discussion to the case of dimension 3 [8, 7] but similar properties definitely appear in greater dimension. referring to [8], the vertex operators of boulatov's model are bounded by λ3/2 such that any connected vacuum graph without generalized tadpoles is bounded by knλ3n/2 (omitting the overall trace factor), n being as usual the number of vertices. however, nothing more can be said in the case of graphs with generalized tadpoles. for instance, there exist non-planar tadpoles (see fig.9) which contribute more than λ3/2 per vertex. indeed, in the typical situation of fig.9, these tadpoles cost a factor λ3/2+3/2 per vertex, and violate the ordinary vertex bound. finally, one concludes that the bounds of [8] are correct for vacuum graphs without gen- eralized tadpoles, such as the vacuum graphs that occur in the colored models, but that, when tadpoles are present, they are wrong for the most general model. in any non-colored case and any dimension, we can expect similar features. 14 references [1] d. v. boulatov, mod. phys. lett. a7 (1992) 1629; eprint hep-th/9202074. [2] h. ooguri, mod. phys. lett. a7 (1992) 2799; eprint hep-th/9205090. [3] l. freidel, int. j. theor. phys. 44 (2005) 1769; eprint hep-th/0505016. [4] d. oriti, "the group field theory approach to quantum gravity," eprint gr-qc/0607032 (2006). [5] c. rovelli, quantum gravity (cambridge university press, cambridge, 2004). [6] t. thiemann, modern canonical quantum general relativity (cambridge university press, cambridge 2007). [7] l. freidel, r. gurau and d. oriti, phys. rev. d80 (2009) 044007; eprint 0905.3772[hep-th]. [8] j. magnen, k. noui, v. rivasseau and m. smerlak, "scaling behaviour of three dimensional group field theory," eprint 0906.5477[hep-th] (2009). [9] v. rivasseau, noncommutative renormalization, poincar ́ e seminar x, "espaces quantiques", ed. b. duplantier et al, (2007) 15-95; eprint 0705.0705[hep-th]. [10] c. perini, c. rovelli and s. speziale, "self-energy and vertex radiative corrections in lqg," eprint 0810.1714[gr-qc] (2008). [11] j. w. barrett, r. j. dowdall, w. j. fairbairn, h. gomes and f. hellmann, "asymptotic analysis of the eprl four-simplex amplitude," eprint 0902.1170[gr-qc] (2009). [12] l. freidel and d. louapre, phys. rev. d68 (2003) 104004; eprint hep-th/0211026. [13] r. gurau, "colored group field theory," eprint 0907.2582[hep-th] (2009). [14] p. t. matthews and a. salam, nuovo cimento 12 (1954) 563; ibid 2 (1955) 120. [15] l. freidel and k. krasnov, class. quant. grav. 25 (2008) 125018; eprint 0708.1595[gr-qc]. [16] j. engle, e. livine, r. pereira and c. rovelli, nucl. phys. b799 (2008) 136; eprint 0711.0146[gr-qc]. [17] v. rivasseau, jhep 9 (2007) 008; eprint 0706.1224[hep-th]. [18] v. rivasseau, from perturbative to constructive renormalization (princeton university press, princeton, 1991); a. abdesselam and v. rivasseau, trees, forests and jungles: a botanical garden for cluster expansions, in constructive physics, ed. v. rivasseau, lecture notes in physics 446, springer verlag, 1995. [19] j. ben geloun, j. magnen and v. rivasseau, in progress. 15

0911.1720

evolutionary game theory: temporal and spatial effects beyond replicator dynamics

evolutionary game dynamics is one of the most fruitful frameworks for studying evolution in different disciplines, from biology to economics. within this context, the approach of choice for many researchers is the so-called replicator equation, that describes mathematically the idea that those individuals performing better have more offspring and thus their frequency in the population grows. while very many interesting results have been obtained with this equation in the three decades elapsed since it was first proposed, it is important to realize the limits of its applicability. one particularly relevant issue in this respect is that of non-mean-field effects, that may arise from temporal fluctuations or from spatial correlations, both neglected in the replicator equation. this review discusses these temporal and spatial effects focusing on the non-trivial modifications they induce when compared to the outcome of replicator dynamics. alongside this question, the hypothesis of linearity and its relation to the choice of the rule for strategy update is also analyzed. the discussion is presented in terms of the emergence of cooperation, as one of the current key problems in biology and in other disciplines.

introduction 2 2 basic concepts and results of evolutionary game theory 4 2.1 equilibria and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 replicator dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 the problem of the emergence of cooperation . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 the effect of different time scales 8 3.1 time scales in the ultimatum game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 time scales in symmetric binary games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2.1 slow selection limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2.2 fast selection limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 email addresses: [email protected] (carlos p. roca), [email protected] (jos ́ e a. cuesta), [email protected] (angel s ́ anchez) preprint submitted to elsevier november 9, 2009 arxiv:0911.1720v1 [q-bio.pe] 9 nov 2009 4 structured populations 21 4.1 network models and update rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 spatial structure and homogeneous networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3 synchronous vs asynchronous update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4 heterogeneous networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.5 best response update rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.6 weak selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5 conclusion and future prospects 41 a characterization of birth-death processes 47 b absorption probability in the hypergeometric case 47 1. introduction the importance of evolution can hardly be overstated, in so far as it permeates all sciences. indeed, in the 150 years that have passed since the publication of on the origin of the species [1], the original idea of darwin that evolution takes place through descent with modification acted upon by natural selection has become a key concept in many sciences. thus, nowadays one can speak of course of evolutionary biology, but there are also evolutionary disciplines in economics, psychology, linguistics, or computer science, to name a few. darwin's theory of evolution was based on the idea of natural selection. natural selection is the process through which favorable heritable traits become more common in successive generations of a population of reproducing organisms, displacing unfavorable traits in the struggle for resources. in order to cast this process in a mathematically precise form, j. b. s. haldane and sewall wright introduced, in the so-called modern evolutionary synthesis of the 1920's, the concept of fitness. they applied theoretical population ideas to the description of evolution and, in that context, they defined fitness as the expected number of offspring of an individual that reach adulthood. in this way they were able to come up with a well-defined measure of the adaptation of individuals and species to their environment. the simplest mathematical theory of evolution one can think of arises when one assumes that the fitness of a species does not depend on the distribution of frequencies of the different species in the population, i.e., it only depends on factors that are intrinsic to the species under consideration or on environmental influences. sewall wright formalized this idea in terms of fitness landscapes ca. 1930, and in that context r. fisher proved his celebrated theorem, that states that the mean fitness of a population is a non-decreasing function of time, which increases proportionally to variability. since then, a lot of work has been done on this kind of models; we refer the reader to [2, 3, 4, 5] for reviews. the approach in terms of fitness landscapes is, however, too simple and, in general, it is clear that the fitness of a species will depend on the composition of the population and will therefore change accordingly as the population evolves. if one wants to describe evolution at this level, the tool of reference is evolutionary game theory. brought into biology by maynard smith [6] as an "exaptation"1 of the game theory developed originally for economics [8], it has since become a unifying framework for other disciplines, such as sociology or anthropology [9]. the key feature of this mathematical apparatus is that it allows to deal with evolution on a frequency-dependent fitness landscape or, in other words, with strategic interactions between entities, these being individuals, groups, species, etc. evolutionary game theory is thus the generic approach to evolutionary dynamics [10] and contains as a special case constant, or fitness landscape, selection. in its thirty year history, a great deal of research in evolutionary game theory has focused on the properties and applications of the replicator equation [11]. the replicator equation was introduced in 1978 by taylor and jonker [12] and describes the evolution of the frequencies of population types taking into account their 1borrowing the term introduced by gould and vrba in evolutionary theory, see [7]. 2 mutual influence on their fitness. this important property allows the replicator equation to capture the essence of selection and, among other key results, it provides a connection between the biological concept of evolutionarily stable strategies [6] with the economical concept of nash equilibrium [13]. as we will see below, the replicator equation is derived in a specific framework that involves a number of assumptions, beginning with that of an infinite, well-mixed population with no mutations. by well-mixed population it is understood that every individual either interacts with every other one or at least has the same probability to interact with any other individual in the population. this hypothesis implies that any individual effectively interacts with a player which uses the average strategy within the population (an approach that has been traditionally used in physics under the name of mean-field approximation). deviations from the well-mixed population scenario affect strongly and non-trivially the outcome of the evolution, in a way which is difficult to apprehend in principle. such deviations can arise when one considers, for instance, finite size populations, alternative learning/reproduction dynamics, or some kind of structure (spatial or temporal) in the interactions between individuals. in this review we will focus on this last point, and discuss the consequences of relaxing the hypothesis that every player interacts or can interact with every other one. we will address both spatial and temporal limitations in this paper, and refer the reader to refs. [10, 11] for discussions of other perturbations. for the sake of definiteness, we will consider those effects, that go beyond replicator dynamics, in the specific context of the emergence of cooperation, a problem of paramount importance with implications at all levels, from molecular biology to societies and ecosystems [14]; many other applications of evolutionary dynamics have also been proposed but it would be too lengthy to discuss all of them here (the interested reader should see, e.g., [10]). cooperation, understood as a fitness-decreasing behavior that increases others' fitness, is an evolutionary puzzle, and many researchers have considered alternative approaches to the replicator equation as possible explanations of its ubiquity in human (and many animal) societies. as it turns out, human behavior is unique in nature. indeed, altruism or cooperative behavior exists in other species, but it can be understood in terms of genetic relatedness (kin selection, introduced by hamilton [15, 16]) or of repeated interactions (as proposed by trivers [17]). nevertheless, human cooperation extends to genetically unrelated individuals and to large groups, characteristics that cannot be understood within those schemes. subsequently, a number of theories based on group and/or cultural evolution have been put forward in order to explain altruism (see [18] for a review). evolutionary game theory is also being intensively used for this research, its main virtue being that it allows to pose the dilemmas involved in cooperation in a simple, mathematically tractable manner. to date, however, there is not a generally accepted solution to this puzzle [19]. considering temporal and spatial effects means, in the language of physics, going beyond mean-field to include fluctuations and correlations. therefore, a first step is to understand what are the basic mean field results. to this end, in section 2 we briefly summarize the main features of replicator equations and introduce the concepts we will refer to afterwards. subsequently, section 3 discusses how fluctuations can be taken into account in evolutionary game theory, and specifically we will consider that, generically, interactions and dynamics (evolution) need not occur at the same pace. we will show that the existence of different time scales leads to quite unexpected results, such as the survival and predominance of individuals that would be the less fit in the replicator description. for games in finite populations with two types of individuals or strategies, the problem can be understood in terms of markov processes and the games can be classified according to the influence of the time scales on their equilibrium structure. other situations can be treated by means of numerical simulations with similarly non-trivial results. section 4 deals with spatial effects. the inclusion of population structure in evolutionary game theory has been the subject of intense research in the last 15 years, and a complete review would be beyond our purpose (see e.g. [20]). the existence of a network describing the possible interactions in the population has been identified as one of the factors that may promote cooperation among selfish individuals [19]. we will discuss the results available to date and show how they can be reconciled by realizing the role played by different networks, different update rules for the evolution of strategies and the equilibrium structure of the games. as a result, we will be able to provide a clear-cut picture of the extent as to which population structure promote cooperation in two strategy games. finally, in section 5 we discuss the implications of the reviewed results on a more general context. our 3 major conclusion will be the lack of generality of models in evolutionary game theory, where details of the dynamics and the interaction modify qualitatively the results. a behavior like that is not intuitive to physicists, used to disregard those details as unimportant. therefore, until we are able to discern what is and what is not relevant, when dealing with problems in other sciences, modeling properly and accurately specific problems is of utmost importance. we will also indicate a few directions of research that arise from the presently available knowledge and that we believe will be most appealing in the near future. 2. basic concepts and results of evolutionary game theory in this section, we summarize the main facts about evolutionary game theory that we are going to need in the remainder of the paper. the focus is on the stability of strategies and on the replicator equation, as an equivalent to the dynamical description of a mean field approach which we will be comparing with. this summary is by no means intended to be comprehensive and we encourage the reader to consult the review [21] or, for full details, the books [6, 9, 11]. 2.1. equilibria and stability the simplest type of game has only two players and, as this will be the one we will be dealing with, we will not dwell into further complications. player i is endowed with a finite number ni of strategies. a game is defined by listing the strategies available to the players and the payoffs they yield: when a player, using strategy si, meets another, who in turn uses strategy sj, the former receives a payoffwij whereas the latter receives a payoffzij. we will restrict ourselves to symmetric games, in which the roles of both players are exchangeable (except in the example considered in section 3.1); mathematically, this means that the set of strategies are the same for both players and that w = zt . matrix w is then called the payoffmatrix of the normal form of the game. in the original economic formulation [8] payoffs were understood as utilities, but maynard smith [6] reinterpreted them in terms of fitness, i.e. in terms of reproductive success of the involved individuals. the fundamental step to "solving" the game or, in other words, to find what strategies will be played, was put forward by john nash [13] by introducing the concept of equilibrium. in 2 × 2 games, a pair of strategies (si, sj) is a nash equilibrium if no unilateral change of strategy allows any player to improve her payoff. when we restrict ourselves to symmetric games, one can say simply, by an abuse of language [21], that a strategy si is a nash equilibrium if it is a best reply to itself: wii ≥wij, ∀sj (a strict nash equilibrium if the inequality is strict). this in turn implies that if both players are playing strategy si, none of them has any incentive to deviate unilaterally by choosing other strategy. as an example, let us consider the famous prisoner's dilemma game, which we will be discussing throughout the review. prisoner's dilemma was introduced by rapoport and chammah [22] as a model of the implications of nuclear deterrence during the cold war, and is given by the following payoffmatrix (we use the traditional convention that the matrix indicates payoffs to the row player) c d c d  3 5 0 1  . (1) the strategies are named c and d for cooperating and defecting, respectively. this game is referred to as prisoner's dilemma because it is usually posed in terms of two persons that are arrested accused of a crime. the police separates them and makes the following offer to them: if one confesses and incriminates the other, she will receive a large reduction in the sentence, but if both confess they will only get a minor reduction; and if nobody confesses then the police is left only with circumstancial evidence, enough to imprison them for a short period. the amounts of the sentence reductions are given by the payoffs in (1). it is clear from it that d is a strict nash equilibrium: to begin with, it is a dominant strategy, because no matter what the column player chooses to do, the row player is always better offby defecting; and when both players defect, none will improve her situation by cooperating. in terms of the prisoners, this translates into the fact that both will confess if they behave rationally. the dilemma arises when one realizes that both players would be better offcooperating, i.e. not confessing, but rationality leads them unavoidable to confess. 4 the above discussion concerns nash equilibria in pure strategies. however, players can also use the so-called mixed strategies, defined by a vector with as many entries as available strategies, every entry indicating the probability of using that strategy. the notation changes then accordingly: we use vectors x = (x1 x2 . . . xn)t , which are elements of the simplex sn spanned by the vectors ei of the standard unit base (vectors ei are then identified with the n pure strategies).the definition of a nash equilibrium in mixed strategies is identical to the previous one: the strategy profile x is a nash equilibrium if it is a best reply to itself in terms of the expected payoffs, i.e. if xt wx ≥xt wy, ∀y ∈sn. once mixed strategies have been introduced, one can prove, following nash [13] that every normal form game has at least one nash equilibrium, albeit it need not necessarily be a nash equilibrium in pure strategies. an example we will also be discussing below is given by the hawk-dove game (also called snowdrift or chicken in the literature [23]), introduced by maynard smith and price to describe animal conflicts [24] (strategies are labeled h and d for hawk and dove, respectively) d h d h  3 5 1 0  . (2) in this case, neither h nor d are nash equilibria, but there is indeed one nash equilibrium in mixed strategies, that can be shown [6] to be given by playing d with probability 1/3. this makes sense in terms of the meaning of the game, which is an anti-coordination game, i.e. the best thing to do is the opposite of the other player. indeed, in the snowdrift interpretation, two people are trapped by a snowdrift at the two ends of a road. for every one of them, the best option is not to shovel snow offto free the road and let the other person do it; however, if the other person does not shovel, then the best option is to shovel oneself. there is, hence, a temptation to defect that creates a dilemmatic situation (in which mutual defection leads to the worst possible outcome). in the same way as he reinterpreted monetary payoffs in terms of reproductive success, maynard smith reinterpreted mixed strategies as population frequencies. this allowed to leave behind the economic con- cept of rational individual and move forward to biological applications (as well as in other fields). as a consequence, the economic evolutionary idea in terms of learning new strategies gives way to a genetic transmission of behavioral strategies to offspring. therefore, maynard smith's interpretation of the above result is that a population consisting of one third of individuals that always use the d strategy and two thirds of h-strategists is a stable genetic polymorphism. at the core of this concept is his notion of evolutionarily stable strategy. maynard smith defined a strategy as evolutionarily stable if the following two conditions are satisfied xt wx ≥ xt wy, ∀y ∈sn, (3) if x ̸= y and xt wy = xt wx, then xt wy > yt wy. (4) the rationale behind this definition is again of a population theoretical type: these are the conditions that must be fulfilled for a population of x-strategists to be non-invadable by any y-mutant. indeed, either x performs better against itself than y or, if they perform equally, x performs better against y than y itself. these two conditions guarantee non-invasibility of the population. on the other hand, comparing the definitions of evolutionarily stable strategy and nash equilibrium one can immediately see that a strict nash equilibrium is an evolutionarily stable strategy and that an evolutionarily stable strategy is a nash equilibrium. 2.2. replicator dynamics after nash proposed his definition of equilibrium, the main criticism that the concept has received relates to how equilibria are reached. in other words, nash provided a rule to decide which are the strategies that rational players should play in a game, but how do people involved in actual game-theoretical settings but without knowledge of game theory find the nash equilibrium? furthermore, in case there is more than one nash equilibrium, which one should be played, i.e., which one is the true "solution" of the game? these questions started out a great number of works dealing with learning and with refinements of the concept 5 that allowed to distinguish among equilibria, particularly within the field of economics. this literature is out of the scope of the present review and the reader is referred to [25] for an in-depth discussion. one of the answers to the above criticism arises as a bonus from the ideas of maynard smith. the notion of evolutionarily stable strategy has implicit some kind of dynamics when we speak of invasibility by mutants; a population is stable if when a small proportion of it mutates it eventually evolves back to the original state. one could therefore expect that, starting from some random initial condition, populations would evolve to an evolutionarily stable strategy, which, as already stated, is nothing but a nash equilibrium. thus, we would have solved the question as to how the population "learns" to play the nash equilibrium and perhaps the problem of selecting among different nash equilibria. however, so far we have only spoken of an abstract dynamics; nothing is specified as to what is the evolution of the population or the strategies that it contains. the replicator equation, due to taylor and jonker [12], was the first and most successful proposal of an evolutionary game dynamics. within the population dynamics framework, the state of the population, i.e. the distribution of strategy frequencies, is given by x as above. a first key point is that we assume that the xi are differentiable functions of time t: this requires in turn assuming that the population is infinitely large (or that xi are expected values for an ensemble of populations). within this hypothesis, we can now postulate a law of motion for x(t). assuming further that individuals meet randomly, engaging in a game with payoffmatrix w, then (wx)i is the expected payofffor an individual using strategy si, and xt wx is the average payoffin the population state x. if we, consistently with our interpretation of payoffas fitness, postulate that the per capita rate of growth of the subpopulation using strategy si is proportional to its payoff, we arrive at the replicator equation (the name was first proposed in [26]) ̇ xi = xi[(wx)i −xt wx], (5) where the term xt wx arises to ensure the constraint p i xi = 1 ( ̇ xi denotes the time derivative of xi). this equation translates into mathematical terms the elementary principle of natural selection: strategies, or individuals using a given strategy, that reproduce more efficiently spread, displacing those with smaller fitness. note also that states with xi = 1, xj = 0, ∀j ̸= i are solutions of eq. (5) and, in fact, they are absorbing states, playing a relevant role in the dynamics of the system in the absence of mutation. once an equation has been proposed, one can resort to the tools of dynamical systems theory to de- rive its most important consequences. in this regard, it is interesting to note that the replicator equation can be transformed by an appropriate change of variable in a system of lotka-volterra type [11]. for our present purposes, we will focus only on the relation of the replicator dynamics with the two equilib- rium concepts discussed in the preceding subsection. the rest points of the replicator equation are those frequency distributions x that make the rhs of eq. (5) vanish, i.e. those that verify either xi = 0 or (wx)i = xt wx, ∀i = 1, . . . , n. the solutions of this system of equations are all the mixed strategy nash equilibria of the game [9]. furthermore, it is not difficult to show (see e.g. [11]) that strict nash equilibria are asymptotically stable, and that stable rest points are nash equilibria. we thus see that the replicator equation provides us with an evolutionary mechanism through which the players, or the population, can arrive at a nash equilibrium or, equivalently, to an evolutionarily stable strategy. the different basins of attraction of the different equilibria further explain which of them is selected in case there are more than one. for our present purposes, it is important to stress the hypothesis involved (explicitly or implicitly) in the derivation of the replicator equation: 1. the population is infinitely large. 2. individuals meet randomly or play against every other one, such that the payoffof strategy si is proportional to the payoffaveraged over the current population state x. 3. there are no mutations, i.e. strategies increase or decrease in frequency only due to reproduction. 4. the variation of the population is linear in the payoffdifference. assumptions 1 and 2 are, as we stated above, crucial to derive the replicator equation in order to replace the fitness of a given strategy by its mean value when the population is described in terms of frequencies. of 6 course, finite populations deviate from the values of frequencies corresponding to infinite ones. in a series of recent works, traulsen and co-workers have considered this problem [27, 28, 29]. they have identified different microscopic stochastic processes that lead to the standard or the adjusted replicator dynamics, showing that differences on the individual level can lead to qualitatively different dynamics in asymmetric conflicts and, depending on the population size, can even invert the direction of the evolutionary process. their analytical framework, which they have extended to include an arbitrary number of strategies, provides good approximations to simulation results for very small sizes. for a recent review of these and related issues, see [30]. on the other hand, there has also been some work showing that evolutionarily stable strategies in infinite populations may lose their stable character when the population is small (a result not totally unrelated to those we will discuss in section 3). for examples of this in the context of hawk-dove games, see [31, 32]. assumption 3 does not pose any severe problem. in fact, mutations (or migrations among physically separated groups, whose mathematical description is equivalent) can be included, yielding the so-called replicator-mutator equation [33]. this is in turn equivalent to the price equation [34], in which a term involving the covariance of fitness and strategies appears explicitly. mutations have been also included in the framework of finite size populations [29] mentioned above. we refer the reader to references [33, 35] for further analysis of this issue. assumption 4 is actually the core of the definition of replicator dynamics. in section 4 below we will come back to this point, when we discuss the relation of replicator dynamics to the rules used for the update of strategies in agent-based models. work beyond the hypothesis of linearity can proceed otherwise in different directions, by considering generalized replicator equations of the form ̇ xi = xi[wi(x) −xt w(x)]. (6) the precise choice for the functions wi(x) depends of course on the particular situation one is trying to model. a number of the results on replicator equation carry on for several such choices. this topic is well summarized in [21] and the interested reader can proceed from there to the relevant references. assumption 2 is the one to which this review is devoted to and, once again, there are very many different possibilities in which it may not hold. we will discuss in depth below the case in which the time scale of selection is faster than that of interaction, leading to the impossibility that a given player can interact with all others. interactions may be also physically limited, either for geographical reasons (individuals interact only with those in their surroundings), for social reasons (individuals interact only with those with whom they are acquainted) or otherwise. as in previous cases, these variations prevents one from using the expected value of fitness of a strategy in the population as a good approximation for its growth rate. we will see the consequences this has in the following sections. 2.3. the problem of the emergence of cooperation one of the most important problems to which evolutionary game theory is being applied is the under- standing of the emergence of cooperation in human (albeit non-exclusively) societies [14]. as we stated in the introduction, this is an evolutionary puzzle that can be accurately expressed within the formalism of game theory. one of the games that has been most often used in connection with this problem is the prisoner's dilemma introduced above, eq. (1). as we have seen, rational players should unavoidably defect and never cooperate, thus leading to a very bad outcome for both players. on the other hand, it is evident that if both players had cooperated they would have been much better off. this is a prototypical example of a social dilemma [36] which is, in fact, (partially) solved in societies. indeed, the very existence of human society, with its highly specialized labor division, is a proof that cooperation is possible. in more biological terms, the question can be phrased using again the concept of fitness. why should an individual help other achieve more fitness, implying more reproductive success and a chance that the helper is eventually displaced? it is important to realize that such a cooperative effort is at the roots of very many biological phenomena, from mutualism to the appearance of multicellular organisms [37]. when one considers this problem in the framework of replicator equation, the conclusion is immediate and disappointing: cooperation is simply not possible. as defection is the only nash equilibrium of prisoner's 7 dilemma, for any initial condition with a positive fraction of defectors, replicator dynamics will inexorably take the population to a final state in which they all are defectors. therefore, one needs to understand how the replicator equation framework can be supplemented or superseded for evolutionary game theory to become closer to what is observed in the real world (note that there is no hope for classical game theory in this respect as it is based in the perfect rationality of the players). relaxing the above discussed assumptions leads, in some cases, to possible solutions to this puzzle, and our aim here is to summarize and review what has been done along these lines with assumption 2. 3. the effect of different time scales evolution is generally supposed to occur at a slow pace: many generations may be needed for a noticeable change to arise in a species. this is indeed how darwin understood the effect of natural selection, and he always referred to its cumulative effects over very many years. however, this needs not be the case and, in fact, selection may occur faster than the interaction between individuals (or of the individuals with their environment). thus, recent experimental studies have reported observations of fast selection [38, 39, 40]. it is also conceivable that in man-monitored or laboratory processes one might make selection be the rapid influence rather than interaction. another context where these ideas apply naturally is that of cultural evolution or social learning, where the time scale of selection is much closer to the time scale of interaction. therefore, it is natural to ask about the consequences of the above assumption and the effect of relaxing it. this issue has already been considered from an economic viewpoint in the context of equilibrium selection (but see an early biological example breaking the assumption of purely random matching in [41], which considered hawk-dove games where strategists are more likely to encounter individuals using their same strategy). this refers to a situation in which for a game there is more than one equilibrium, like in the stag hunt game, given e.g. by the following payoffmatrix c d c d  6 5 1 2  . (7) this game was already posed as a metaphor by rousseau [42], which reads as follows: two people go out hunting for stag, because two of them are necessary to hunt down such a big animal. however, any one of them can cheat the other by hunting hare, which one can do alone, leaving the other one in the impossibility of getting the stag. therefore, we have a coordination game, in which the best option is to do as the other: hunt stag together or both hunting hare separately. in game theory, this translates into the fact that both c and d are nash equilibria, and in principle one is not able to determine which one would be selected by the players, i.e., which one is the solution of the game. one rationale to choose was proposed by harsanyi and selten2 [43], who classified c as the pareto-efficient equilibrium (hunting stag is more profitable than hunting hare), because that is the most beneficial for both players, and d as the risk-dominant equilibrium, because it is the strategy that is better in case the other player chooses d (one can hunt hare alone). here the tension arises then from the risk involved in cooperation, rather than from the temptation to defect of snowdrift games [44] (note that both tensions are present in prisoner's dilemma). kandori et al [45] showed that the risk-dominant equilibrium is selected when using a stochastic evo- lutionary game dynamics, proposed by foster and young [46], that considers that every player interacts with every other one (implying slow selection). however, fast selection leads to another result. indeed, robson and vega-redondo [47] considered the situation in which every player is matched to another one and therefore they only play one game before selection acts. in that case, they showed that the outcome changed and that the pareto-efficient equilibrium is selected. this result was qualified later by miekisz [48], who showed that the selected equilibrium depended on the population size and the mutation level of the dynamics. recently, this issue has also been considered in [49], which compares the situation where the 2harsanyi and senten received the nobel prize in economics for this contribution, along with nash, in 1994. 8 contribution of the game to the fitness is small (weak selection, see section 4.6 below) to the one where the game is the main source of the fitness, finding that in the former the results are equivalent to the well-mixed population, but not in the latter, where the conclusions of [47] are recovered. it is also worth noticing in this regards the works by boylan [50, 51], where he studied the types of random matching that can still be approximated by continuous equations. in any case, even if the above are not general results and their application is mainly in economics, we have already a hint that time scales may play a non-trivial role in evolutionary games. in fact, as we will show below, rapid selection affects evolutionary dynamics in such a dramatic way that for some games it even changes the stability of equilibria. we will begin our discussion by briefly summarizing results on a model for the emergence of altruistic behavior, in which the dynamics is not replicator-like, but that illustrates nicely the very important effects of fast selection. we will subsequently proceed to present a general theory for symmetric 2 × 2 games. there, in order to make explicit the relation between selection and interaction time scales, we use a discrete-time dynamics that produces results equivalent to the replicator dynamics when selection is slow. we will then show that the pace at which selection acts on the population is crucial for the appearance and stability of cooperation. even in non- dilemma games such as the harmony game [52], where cooperation is the only possible rational outcome, defectors may be selected for if population renewal is very rapid. 3.1. time scales in the ultimatum game as a first illustration of the importance of time scales in evolutionary game dynamics, we begin by dealing with this problem in the context of a specific set of such experiments, related to the ultimatum game [53, 54]. in this game, under conditions of anonymity, two players are shown a sum of money. one of the players, the "proposer", is instructed to offer any amount to the other, the "responder". the proposer can make only one offer, which the responder can accept or reject. if the offer is accepted, the money is shared accordingly; if rejected, both players receive nothing. note that the ultimatum game is not symmetric, in so far as proposer and responder have clearly different roles and are therefore not exchangeable. this will be our only such an example, and the remainder of the paper will only deal with symmetric games. since the game is played only once (no repeated interactions) and anonymously (no reputation gain; for more on explanations of altruism relying on reputation see [55]), a self-interested responder will accept any amount of money offered. therefore, self-interested proposers will offer the minimum possible amount, which will be accepted. the above prediction, based on the rational character of the players, contrasts clearly with the results of actual ultimatum game experiments with human subjects, in which average offers do not even approximate the self-interested prediction. generally speaking, proposers offer respondents very substantial amounts (50% being a typical modal offer) and respondents frequently reject offers below 30% [56, 57]. most of the experiments have been carried out with university students in western countries, showing a large degree of individual variability but a striking uniformity between groups in average behavior. a large study in 15 small-scale societies [54] found that, in all cases, respondents or proposers behave in such a reciprocal manner. furthermore, the behavioral variability across groups was much larger than previously observed: while mean offers in the case of university students are in the range 43%-48%, in the cross-cultural study they ranged from 26% to 58%. how does this fit in our focus topic, namely the emergence of cooperation? the fact that indirect reciprocity is excluded by the anonymity condition and that interactions are one-shot (repeated interaction, the mechanism proposed by axelrod to foster cooperation [58, 59], does not apply) allows one to interpret rejections in terms of the so-called strong reciprocity [60, 61]. this amounts to considering that these behaviors are truly altruistic, i.e. that they are costly for the individual performing them in so far as they do not result in direct or indirect benefit. as a consequence, we return to our evolutionary puzzle: the negative effects of altruistic acts must decrease the altruist's fitness as compared to that of the recipients of the benefit, ultimately leading to the extinction of altruists. indeed, standard evolutionary game theory arguments applied to the ultimatum game lead to the expectation that, in a well-mixed population, punishers (individuals who reject low offers) have less chance to survive than rational players (individuals who accept 9 figure 1: left: mean acceptance threshold as a function of simulation time. initial condition is that all agents have ti = 1. right: acceptance threshold distribution after 108 games (note that this distribution, for small s, is not stationary). initial condition is that all agents have uniformly distributed, random ti. in both cases, s is as indicated from the plot. any offer) and eventually disappear. we will now show that this conclusion depends on the dynamics, and that different dynamics may lead to the survival of punishers through fluctuations. consider a population of n agents playing the ultimatum game, with a fixed sum of money m per game. random pairs of players are chosen, of which one is the proposer and another one is the respondent. in its simplest version, we will assume that players are capable of other-regarding behavior (empathy); consequently, in order to optimize their gain, proposers offer the minimum amount of money that they would accept. every agent has her own, fixed acceptance threshold, 1 ≤ti ≤m (ti are always integer numbers for simplicity). agents have only one strategy: respondents reject any offer smaller than their own acceptance threshold, and accept offers otherwise. money shared as a consequence of accepted offers accumulates to the capital of each player, and is subsequently interpreted as fitness as usual. after s games, the agent with the overall minimum fitness is removed (randomly picked if there are several) and a new agent is introduced by duplicating that with the maximum fitness, i.e. with the same threshold and the same fitness (again randomly picked if there are several). mutation is introduced in the duplication process by allowing changes of ±1 in the acceptance threshold of the newly generated player with probability 1/3 each. agents have no memory (interactions are one-shot) and no information about other agents (no reputation gains are possible). we note that the dynamics of this model is not equivalent to the replicator equation, and therefore the results do not apply directly in that context. in fact, such an extremal dynamics leads to an amplification of the effect of fluctuations that allows to observe more clearly the influence of time scales. this is the reason why we believe it will help make our main point. fig. 1 shows the typical outcome of simulations of our model for a population of n = 1000 individuals. an important point to note is that we are not plotting averages but a single realization for each value of s; the realizations we plot are not specially chosen but rather are representative of the typical simulation results. we have chosen to plot single realizations instead of averages to make clear for the reader the large fluctuations arising for small s, which are the key to understand the results and which we discuss below. as we can see, the mean acceptance threshold rapidly evolves towards values around 40%, while the whole distribution of thresholds converges to a peaked function, with the range of acceptance thresholds for the agents covering about a 10% of the available ones. these are values compatible with the experimental results discussed above. the mean acceptance threshold fluctuates during the length of the simulation, never reaching a stationary value for the durations we have explored. the width of the peak fluctuates as well, but in a much smaller scale than the position. the fluctuations are larger for smaller values of s, and when s becomes of the order of n or larger, the evolution of the mean acceptance threshold is very smooth. as is clear from fig. 1, for very small values of s, the differences in payoffarising from the fact that only some players play are amplified by our extreme dynamics, resulting in a very noisy behavior of the mean threshold. this is a crucial point and will be discussed in more detail below. importantly, the 10 typical evolution we are describing does not depend on the initial condition. in particular, a population consisting solely of self-interested agents, i.e. all initial thresholds set to ti = 1, evolves in the same fashion. indeed, the distributions shown in the left panel of fig. 1 (which again correspond to single realizations) have been obtained with such an initial condition, and it can be clearly observed that self-interested agents disappear in the early stages of the evolution. the number of players and the value m of the capital at stake in every game are not important either, and increasing m only leads to a higher resolution of the threshold distribution function, whereas smaller mutation rates simply change the pace of evolution. to realize the effect of time scales, it is important to recall previous studies of the ultimatum game by page and nowak [62, 63]. the model introduced in those works has a dynamics completely different from ours: following standard evolutionary game theory, every player plays with every other one in both roles (proponent and respondent), and afterwards players reproduce with probability proportional to their payoff (which is fitness in the reproductive sense). simulations and adaptive dynamics equations show that the population ends up composed by players with fair (50%) thresholds. note that this is not what one would expect on a rational basis, but page and nowak traced this result back to empathy, i.e. the fact that the model is constrained to offer what one would accept. in any event, what we want to stress here is that their findings are also different from our observations: we only reach an equilibrium for large s. the reason for this difference is that the page-nowak model dynamics describes the s/n →∞limit of our model, in which between death-reproduction events the time average gain obtained by all players is with high accuracy a constant o(n) times the mean payoff. we thus see that our model is more general because it has one free parameter, s, that allows selecting different regimes whereas the page-nowak dynamics is only one limiting case. those different regimes are what we have described as fluctuation dominated (when s/n is finite and not too large) and the regime analyzed by page and nowak (when s/n →∞). this amounts to saying that by varying s we can study regimes far from the standard evolutionary game theory limit. as a result, we find a variability of outcomes for the acceptance threshold consistent with the observations in real human societies [54, 57]. furthermore, if one considers that the acceptance threshold and the offer can be set independently, the results differ even more [64]: while in the model of page and nowak both magnitudes evolve to take very low values, close to zero, in the model presented here the results, when s is small, are very similar to the one-threshold version, leading again to values compatible with the experimental observations. this in turn implies that rapid selection may be an alternative to empathy as an explanation of human behavior in this game. the main message to be taken from this example is that fluctuations due to the finite number of games s are very important. among the results summarized above, the evolution of a population entirely formed by self-interested players into a diversified population with a large majority of altruists is the most relevant and surprising one. one can argue that the underlying reason for this is precisely the presence of fluctuations in our model. for the sake of definiteness, let us consider the case s = 1 (agent replacement takes place after every game) although the discussion applies to larger (but finite) values of s as well. after one or more games, a mutation event will take place and a "weak altruistic punisher" (an agent with ti = 2) will appear in the population, with a fitness inherited from its ancestor. for this new agent to be removed at the next iteration, our model rules imply that this agent has to have the lowest fitness, and also that it does not play as a proposer in the next game (if playing as a responder the agent will earn nothing because of her threshold). in any other event this altruistic punisher will survive at least one cycle, in which an additional one can appear by mutation. it is thus clear that fluctuations indeed help altruists to take over: as soon as a few altruists are present in the population, it is easy to see analytically that they will survive and proliferate even in the limit s/n →∞. 3.2. time scales in symmetric binary games the example in the previous subsection suggests that there certainly is an issue of relative time scales in evolutionary game theory that can have serious implications. in order to gain insight into this question, it is important to consider a general framework, and therefore we will now look at the general problem of symmetric 2 × 2 games. asymmetric games can be treated similarly, albeit in a more cumbersome manner, and their classification involves many more types; we feel, therefore, that the symmetric case is a much 11 clearer illustration of the effect of time scales. in what follows, we review and extend previous results of us [65, 66], emphasizing the consequences of the existence of different time scales. let us consider a population of n individuals, each of whom plays with a fixed strategy, that can be either c or d (for "cooperate" and "defect" respectively, as in section 2). we denote the payoffthat an x-strategist gets when confronted to a y -strategist (x and y are c or d) by the matrix element wxy . for a certain time individuals interact with other individuals in pairs randomly chosen from the population. during these interactions individuals collect payoffs. we shall refer to the interval between two interaction events as the interaction time. once the interaction period has finished reproduction occurs, and in steady state selection acts immediately afterwards restoring the population size to the maximum allowed by the environment. the time between two of these reproduction/selection events will be referred to as the evolution time. reproduction and selection can be implemented in at least two different ways. the first one is through the fisher-wright process [5] in which each individual generates a number of offspring proportional to her payoff. selection acts by randomly killing individuals of the new generation until restoring the size of the population back to n individuals. the second option for the evolution is the moran process [5, 67]. it amounts to randomly choosing an individual for reproduction proportionally to payoffs, whose single offspring replaces another randomly chosen individual, in this case with a probability 1/n equal for all. in this manner populations always remains constant. the fisher-wright process is an appropriate model for species which produce a large number of offspring in the next generation but only a few of them survive, and the next generation replaces the previous one (like insects or some fishes). the moran process is a better description for species which give rise to few offspring and reproduce in continuous time, because individuals neither reproduce nor die simultaneously, and death occurs at a constant rate. the original process was generalized to the frequency-dependent fitness context of evolutionary game theory by taylor et al. [68], and used to study the conditions for selection favoring the invasion and/or fixation of new phenotypes. the results were found to depend on whether the population was infinite or finite, leading to a classification of the process in three or eight scenarios, respectively. both the fisher-wright and moran processes define markov chains [69, 70] on the population, charac- terized by the number of its c-strategists n ∈{0, 1, . . . , n}, because in both cases it is assumed that the composition of the next generation is determined solely by the composition of the current generation. each process defines a stochastic matrix p with elements pn,m = p(m|n), the probability that the next generation has m c-strategists provided the current one has n. while for the fisher-wright process all the elements of p may be nonzero, for the moran process the only nonzero elements are those for which m = n or m = n±1. hence moran is, in the jargon of markov chains, a birth-death process with two absorbing states, n = 0 and n = n [69, 70]. such a process is mathematically simpler, and for this reason it will be the one we will choose for our discussion on the effect of time scales. to introduce explicitly time scales we will implement the moran process in the following way, generalizing the proposal by taylor et al. [68]. during s time steps pairs of individuals will be chosen to play, one pair every time step. after that the above described reproduction/selection process will act according to the payoffs collected by players during the s interaction steps. then, the payoffs of all players are set to zero and a new cycle starts. notice that in general players will play a different number of times -some not at all- and this will reflect in the collected payoffs. if s is too small most players will not have the opportunity to play and chance will have a more prominent role in driving the evolution of the population. quantifying this effect requires that we first compute the probability that, in a population of n individ- uals of which n are c-strategists, an x-strategist is chosen to reproduce after the s interaction steps. let nxy denote the number of pairs of x- and y -strategists that are chosen to play. the probability of forming a given pair, denoted pxy , will be pcc = n(n −1) n(n −1), pcd = 2 n(n −n) n(n −1), pdd = (n −n)(n −n −1) n(n −1) . (8) 12 then the probability of a given set of nxy is dictated by the multinomial distribution m({nxy }; s) =    s!pncc cc ncc! pncd cd ncd! pndd dd ndd!, if ncc + ncd + ndd = s, 0, otherwise. (9) for a given set of variables nxy , the payoffs collected by c- and d-strategists are wc = 2nccwcc + ncdwcd, wd = ncdwdc + 2nddwdd. (10) then the probabilities of choosing a c- or d-strategist for reproduction are pc(n) = em  wc wc + wd  , pd(n) = em  wd wc + wd  , (11) where the expectations em [*] are taken over the probability distribution m (9). notice that we have to guarantee wx ≥0 for the above expressions to define a true probability. this forces us to choose all payoffs wxy ≥0. in addition, we have studied the effect of adding a baseline fitness to every player, which is equivalent to a translation of the payoffmatrix w, obtaining the same qualitative results (see below). once these probabilities are obtained the moran process accounts for the transition probabilities from a state with n c-strategists to another with n±1 c-strategists. for n →n+1 a c-strategist must be selected for reproduction (probability pc(n)) and a d-strategist for being replaced (probability (n −n)/n). thus pn,n+1 = p(n + 1|n) = n −n n pc(n). (12) for n →n −1 a d-strategist must be selected for reproduction (probability pd(n)) and a c-strategist for being replaced (probability n/n). thus pn,n−1 = p(n −1|n) = n n pd(n). (13) finally, the transition probabilities are completed by pn,n = 1 −pn,n−1 −pn,n+1. (14) 3.2.1. slow selection limit let us assume that s →∞, i.e. the evolution time is much longer than the interaction time. then the distribution (9) will be peaked at the values nxy = spxy , the larger s the sharper the peak. therefore in this limit pc(n) → wc(n) wc(n) + wd(n), pd(n) → wd(n) wc(n) + wd(n), (15) where wc(n) = n n  n −1 n −1(wcc −wcd) + wcd  , wd(n) = n −n n  n n −1(wdc −wdd) + wdd  . (16) in general, for a given population size n we have to resort to a numerical evaluation of the various quantities that characterize a birth-death process, according to the formulas in appendix a. however, for large n the transition probabilities can be expressed in terms of the fraction of c-strategists x = n/n as pn,n+1 = x(1 −x) wc(x) xwc(x) + (1 −x)wd(x), (17) pn,n−1 = x(1 −x) wd(x) xwc(x) + (1 −x)wd(x), (18) 13 where wc(x) = x(wcc −wcd) + wcd, wd(x) = x(wdc −wdd) + wdd. (19) the terms wc and wd are, respectively, the expected payoffof a cooperator and a defector in this case of large s and n. the factor x(1 −x) in front of pn,n+1 and pn,n−1 arises as a consequence of n = 0 and n = n being absorbing states of the process. there is another equilibrium x∗where pn,n±1 = pn±1,n, i.e. wc(x∗) = wd(x∗), with x∗given by x∗= wcd −wdd wdc −wcc + wcd −wdd . (20) for x∗to be a valid equilibrium 0 < x∗< 1 we must have (wdc −wcc)(wcd −wdd) > 0. (21) this equilibrium is stable3 as long as the function wc(x) −wd(x) is decreasing at x∗, for then if x < x∗ pn,n+1 > pn+1,n and if x > x∗pn,n−1 > pn−1,n, i.e. the process will tend to restore the equilibrium, whereas if the function is increasing the process will be led out of x∗by any fluctuation. in terms of (19) this implies wdc −wcc > wdd −wcd. (22) notice that the two conditions wc(x∗) = wd(x∗), w′ c(x∗) < w′ d(x∗), (23) are precisely the conditions arising from the replicator dynamics for x∗to be a stable equilibrium [10, 11], albeit expressed in a different manner than in section 2 (w′ x represents the derivative of wx with respect to x). out of the classic dilemmas, condition (21) holds for stag hunt and snowdrift games, but condition (22) only holds for the latter. thus, as we have already seen, only snowdrift has a dynamically stable mixed population. this analysis leads us to conclude that the standard setting of evolutionary games as advanced above, in which the time scale for reproduction/selection is implicitly (if not explicitly) assumed to be much longer than the interaction time scale, automatically yields the distribution of equilibria dictated by the replicator dynamics for that game. we have explicitly shown this to be true for binary games, but it can be extended to games with an arbitrary number of strategies. in the next section we will analyze what happens if this assumption on the time scales does not hold. 3.2.2. fast selection limit when s is finite, considering all the possible pairings and their payoffs, we arrive at pc(n) = s x j=0 s−j x k=0 2s−j−k s!ns−k(n −1)j(n −n)s−j(n −n −1)k j!k!(s −j −k)!n s(n −1)s × 2jwcc + (s −j −k)wcd 2jwcc + 2kwdd + (s −j −k)(wcd + wdc), (24) and pd(n) = 1 −pc(n). we have not been able to write this formula in a simpler way, so we have to evaluate it numerically for every choice of the payoffmatrix. however, in order to have a glimpse at the effect of reducing the number of interactions between successive reproduction/selection events, we can examine analytically the extreme case s = 1, for which pn,n+1 = n(n −n) n(n −1)  2wcd wdc + wcd + n n wdc −wcd wdc + wcd −1 n  , (25) pn,n−1 = n(n −n) n(n −1)  1 + n n wdc −wcd wdc + wcd −1 n  . (26) 3here the notion of stability implies that the process will remain near x∗for an extremely long time, because as long as n is finite, no matter how large, the process will eventually end up in x = 0 or x = 1, the absorbing states. 14 from these equations we find that pn,n−1 pn,n+1 = dn + s(n −1) d(n + 1) + s(n −1) −d(n + 1), d = wdc −wcd, s = wdc + wcd, (27) and this particular dependence on n allows us to find the following closed-form expression for cn, the probability that starting with n cooperators the population ends up with all cooperators (see appendix b) cn = rn rn , rn =      n y j=1 s(n −1) + dj s(n −1) −d(n + 1 −j) −1, if d ̸= 0, n, if d = 0. (28) the first thing worth noticing in this expression is that it only depends on the two off-diagonal elements of the payoffmatrix (through their sum, s, and difference, d). this means that in an extreme situation in which the evolution time is so short that it only allows a single pair of players to interact, the outcome of the game only depends on what happens when two players with different strategies play. the reason is obvious: only those two players that have been chosen to play will have a chance to reproduce. if both players have strategy x, an x-strategist will be chosen to reproduce with probability 1. only if each player uses a different strategy the choice of the player that reproduces will depend on the payoffs, and in this case they are precisely wcd and wdc. of course, as s increases this effect crosses over to recover the outcome for the case s →∞. we can extend our analysis further for the case of large populations. if we denote x = n/n and c(x) = cn, then we can write, as n →∞, c(x) ∼enφ(x) −1 enφ(1) −1, φ(x) = z x 0 [ln(s + dt) −ln(s + d(t −1))] dt. (29) then φ′(x) = ln  s + dx s + d(x −1)  , (30) which has the same sign as d, and hence φ(x) is increasing for d > 0 and decreasing for d < 0. thus if d > 0, because of the factor n in the argument of the exponentials and the fact that φ(x) > 0 for x > 0, the exponential will increase sharply with x. then, expanding around x = 1, φ(x) ≈φ(1) −(1 −x)φ′(1), (31) so c(x) ∼exp{−n ln(1 + d/s)(1 −x)}. (32) the outcome for this case is that absorption will take place at n = 0 for almost any initial condition, except if we start very close to the absorbing state n = n, namely for n ≳n −1/ ln(1 + d/s). on the contrary, if d < 0 then φ(x) < 0 for x > 0 and the exponential will be peaked at 0. so expanding around x = 0, φ(x) ≈xφ′(0) (33) and c(x) ∼1 −exp{−n ln(1 −d/s)x}. (34) the outcome in this case is therefore symmetrical with respect to the case d > 0, because now the probability of ending up absorbed into n = n is 1 for nearly all initial conditions except for a small range near n = 0 determined by n ≲1/ ln(1 −d/s). in both cases the range of exceptional initial conditions increases with decreasing |d|, and in particular when d = 0 the evolution becomes neutral,4 as it is reflected in the fact that in that special case cn = n/n (cf. eq. (28)) [5]. 4notice that if d = 0 then wdc = wcd and therefore the evolution does not favor any of the two strategies. 15 (a) (b) (c) figure 2: absorption probability cn to state n = n starting from initial state n, for a harmony game (payoffs wcc = 1, wcd = 0.25, wdc = 0.75 and wdd = 0.01), population sizes n = 10 (a), n = 100 (b) and n = 1000 (c), and for values of s = 1, 2, 3, 10 and 100. the values for s = 100 are indistinguishable from the results of replicator dynamics. in order to illustrate the effect of a finite s, even in the case when s > 1, we will consider all possible symmetric 2 × 2 games. these were classified by rapoport and guyer [71] in 12 non-equivalent classes which, according to their nash equilibria and their dynamical behavior under replicator dynamics, fall into three different categories: (i) six games have wcc > wdc and wcd > wdd, or wcc < wdc and wcd < wdd. for them, their unique nash equilibrium corresponds to the dominant strategy (c in the first case and d in the second case). this equilibrium is the global attractor of the replicator dynamics. (ii) three games have wcc > wdc and wcd < wdd. they have several nash equilibria, one of them with a mixed strategy, which is an unstable equilibrium of the replicator dynamics and therefore acts as a separator of the basins of attractions of two nash equilibria in pure strategies, which are the attractors. (iii) the remaining three games have wcc < wdc and wcd > wdd. they also have several nash equilibria, one of them with a mixed strategy, but in this case this is the global attractor of the replicator dynamics. examples of the first category are the harmony and prisoner's dilemma games. category (ii) includes the stag hunt game, whereas the snowdrift game belongs to category (iii). we will begin by considering one example of category (i): the harmony game. to that aim we will choose the parameters wcc = 1, wcd = 0.25, wdc = 0.75 and wdd = 0.01. the name of this game refers to the fact that it represents no conflict, in the sense that all players get the maximum payoffby following strategy c. the values of cn obtained for different populations n and several values of s are plotted in fig. 2. the curves for large s illustrate the no-conflicting character of this game as the probability cn is almost 1 for every starting initial fraction of c-strategists. the results for small s also illustrate the effect of fast selection, as the inefficient strategy, d, is selected for almost any initial fraction of c-strategists. the effect is more pronounced the larger the population. the crossover between the two regimes takes place at s = 2 or 3, but it depends on the choice of payoffs. a look at fig. 2 reveals that the crossing over to the s →∞ regime as s increases has no connection whatsoever with n, because it occurs nearly at the same values for any population size n. it does depend, however, on the precise values of the payoffs. as a further check, in fig. 3 we plot the results for s = 1 for different population sizes n and compare with the asymptotic prediction (32), showing its great accuracy for values of n = 100 and higher; even for n = 10 the deviation from the exact results is not large. let us now move to category (ii), well represented by the stag hunt game, discussed in the preceding subsection. we will choose for this game the payoffs wcc = 1, wcd = 0.01, wdc = 0.8 and wdd = 0.2. the values of cn obtained for different populations n and several values of s are plotted in fig. 4. the panel (c) for s = 100 reveals the behavior of the system according to the replicator dynamics: both strategies are attractors, and the crossover fraction of c-strategists separating the two basins of attraction (given by 16 figure 3: same as in fig. 2 plotted against n −n, for s = 1 and n = 10, 100 and 1000. the solid line is the asymptotic prediction (32). (a) (b) (c) figure 4: absorption probability cn to state n = n starting from initial state n, for a stag hunt game (payoffs wcc = 1, wcd = 0.01, wdc = 0.8 and wdd = 0.2), population sizes n = 10 (a), n = 100 (b) and n = 1000 (c), and for values of s = 1, 3, 5, 10 and 100. results from replicator dynamics are also plotted for comparison. eq. (20)) is, for this case, x∗≈0.49. we can see that the effect of decreasing s amounts to shifting this crossover towards 1, thus increasing the basins of attraction of the risk-dominated strategy. in the extreme case s = 1 this strategy is the only attractor. of course, for small population sizes (fig. 4(a)) all these effects (the existence of the threshold and its shifting with decreasing s) are strongly softened, although still noticeable. an interesting feature of this game is that the effect of a finite s is more persistent compared to what happens to the harmony game. whereas in the latter the replicator dynamics was practically recovered, for values of s ≥10 we have to go up to s = 100 to find the same in stag hunt. finally, a representative of category (iii) is the snowdrift game, for which we will choose the payoffs wcc = 1, wcd = 0.2, wdc = 1.8 and wdd = 0.01. for these values, the replicator dynamics predicts that both strategies coexist with fractions of population given by x∗in (20), which for these parameters takes the value x∗≈0.19. however, a birth-death process for finite n always ends up in absorption into one of the absorbing states. in fact, for any s and n and this choice of payoffs, the population always ends up absorbed into the n = 0 state -except when it starts very close to n = n. but this case has a peculiarity that makes it entirely different from the previous ones. whereas for the former cases the absorption time (50) is τ = o(n) regardless of the value of s, for snowdrift the absorption time is o(n) for s = 1 but grows very fast with s towards an asymptotic value τ∞(see fig. 5(a)) and τ∞grows exponentially with n (see fig. 5(b)). this means that, while for s = 1 the process behaves as in previous cases, being absorbed into the n = 0 state, as s increases there is a crossover to a regime in which the transient states become more 17 (a) (b) figure 5: absorption time starting from the state n/n = 0.5 for a snowdrift game (payoffs wcc = 1, wcd = 0.2, wdc = 1.8 and wdd = 0.01), as a function of s for population size n = 100 (a) and as a function of n in the limit s →∞(b). note the logarithmic scale for the absorption time. relevant than the absorbing state because the population spends an extremely long time in them. in fact, the process oscillates around the mixed equilibrium predicted by the replicator dynamics. this is illustrated by the distribution of visits to states 0 < n < n before absorption (48), shown in fig. 6. thus the effect of fast selection on snowdrift games amounts to a qualitative change from the mixed equilibrium to the pure equilibrium at n = 0. having illustrated the effect of fast selection in these three representative games, we can now present the general picture. similar results arise in the remaining 2 × 2 games, fast selection favoring in all cases the strategy with the highest payoffagainst the opposite strategy. for the remaining five games of category (i) this means favoring the dominant strategy (prisoner's dilemma is a prominent example of it). the other two cases of category (ii) also experience a change in the basins of attraction of the two equilibria. finally, the remaining two games of category (iii) experience the suppression of the coexistence state in favor of one of the two strategies. the conclusion of all this is that fast selection changes completely the outcome of replicator dynamics. in terms of cooperation, as the terms in the off-diagonal of social dilemmas verify wdc > wcd, this change in outcome has a negative influence on cooperation, as we have seen in all the games considered. even for some payoffmatrices of a non-dilemma game such as harmony, it can make defectors invade the population. two final remarks are in order. first, these results do not change qualitatively with the population size. in fact, eqs. (32) and (34) and fig. 3 very clearly illustrate this. second, there might be some concern about this analysis which the extreme s = 1 case puts forward: all this might just be an effect of the fact that most players do not play and therefore have no chance to be selected for reproduction. in order to sort this out we have made a similar analysis but introducing a baseline fitness for all players, so that even if a player does not play she can still be selected for reproduction. the probability will be, of course, smaller than the one of the players who do play; however, we should bear in mind that when s is very small, the overwhelming majority of players are of this type and this compensates for the smaller probability. thus, let fb be the total baseline fitness that all players share per round, so that sfb/n is the baseline fitness every player has at the time reproduction/selection occurs. this choice implies that if fb = 1 the overall baseline fitness and that arising from the game are similar, regardless of s and n. if fb is very small (fb ≲0.1), the result is basically the same as that for fb = 0. the effect for fb = 1 is illustrated in fig. 7 for harmony and stag hunt games. note also that at very large baseline fitness (fb ≳10) the evolution is almost neutral, although the small deviations induced by the game -which are determinant for the ultimate fate of the population- still follow the same pattern (see fig. 8). interestingly, traulsen et al. [72] arrive at similar 18 figure 6: distribution of visits to state n before absorption for population n = 100, initial number of cooperators n = 50 and several values of s. the game is the same snowdrift game of fig. 5. the curve for s = 100 is indistinguishable from the one for s →∞(labeled 'replicator'). results by using a fermi like rule (see sec. 4.1 below) to introduce noise (temperature) in the selection process, and a interaction probability q of interactions between individuals leading to heterogeneity in the payoffs, i.e., in the same words as above, to fluctuations, that in turn reduce the intensity of selection as is the case when we introduce a very large baseline fitness. 19 (a) (b) figure 7: absorption probability starting from state n for the harmony game of fig. 2 (a) and the stag hunt game of fig. 4 (b) when n = 100 and baseline fitness fb = 1. (a) (b) figure 8: same as fig. 7 for fb = 10. 20 4. structured populations having seen the highly non-trivial effects of considering temporal fluctuations in evolutionary games, in this section we are going to consider the effect of relaxing the well-mixed hypothesis by allowing the existence of spatial correlations in the population. recall from section 2.2 that a well-mixed population presupposes that every individual interacts with equal probability with every other one in the population, or equivalently that each individual interacts with the "average" individual. it is not clear, however, that this hypothesis holds in many practical situations. territorial or physical constraints may limit the interactions between individuals, for example. on the other hand, an all-to-all network of relationships does not seem plausible in large societies; other key phenomena in social life, such as segregation or group formation, challenge the idea of a mean player that everyone interacts with. it is adequate, therefore, to take into consideration the existence of a certain network of relationships in the population, which determines who interacts with whom. this network of relationships is what we will call from now on the structure of the population. consistently, a well-mixed population will be labeled as unstructured and will be represented by a complete graph. games on many different types of networks have been investigated, examples of which include regular lattices [73, 74], scale-free networks [75], real social networks [76], etc. this section is not intended to be an exhaustive review of all this existent work and we refer the reader to [20] for such a detailed account. we rather want to give a panoramic and a more personal and idiosyncratic view of the field, based on the main available results and our own research. it is at least reasonable to expect that the existence of structure in a population could give rise to the appearance of correlations and that they would have an impact on the evolutionary outcome. for more than fifteen years investigation into this phenomena has been a hot topic of research, as the seminal result by nowak and may [73], which reported an impressive fostering of cooperation in prisoner's dilemma on spatial lattices, triggered a wealth of work focused on the extension of this effect to other games, networks and strategy spaces. on the other hand, the impossibility in most cases of analytical approaches and the complexity of the corresponding numerical agent-based models have made any attempt of exhaustive approach very demanding. hence most studies have concentrated on concrete settings with a particular kind of game, which in most cases has been the prisoner's dilemma [73, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91]. other games has been much less studied in what concerns the influence of population structure, as show the comparatively much smaller number of works about snowdrift or hawk-dove games [92, 93, 94, 95, 96, 97], or stag hunt games [98, 99, 100]. moreover, comprehensive studies in the space of 2 × 2 games are very scarce [74, 75]. as a result, many interesting features of population structure and its influence on evolutionary games have been reported in the literature, but the scope of these conclusions is rather limited to particular models, so a general understanding of these issues, in the broader context of 2 × 2 games and different update rules, is generally missing. however, the availability and performance of computational resources in recent years have allowed us to undertake a systematic and exhaustive simulation program [101, 102] on these evolutionary models. as a result of this study we have reached a number of conclusions that are obviously in relation with previous research and that we will discuss in the following. in some cases, these are generalizations of known results to wider sets of games and update rules, as for example for the issue of the synchrony of the updating of strategies [73, 77, 78, 95, 96, 100, 103] or the effect of small-world networks vs regular lattices [84, 96, 104, 105]. in other cases, the more general view of our analysis has allowed us to integrate apparently contradictory results in the literature, as the cooperation on prisoner's dilemma vs. snowdrift games [73, 92, 93, 94, 96], or the importance of clustering in spatial lattices [85, 89, 96]. other conclusions of ours, however, refute what seems to be established opinions in the field, as the alleged robustness of the positive influence of spatial structure on prisoner's dilemma [73, 74, 77]. and finally, we have reached novel conclusions that have not been highlighted by previous research, as the robustness of the influence of spatial structure on coordination games, or the asymmetry between the effects on games with mixed equilibria (coordination and anti-coordination games) and how it varies with the intensity of selection. it is important to make clear from the beginning that evolutionary games on networks may be sensitive to another source of variation with respect to replicator dynamics besides the introduction of spatial cor- relations. this source is the update rule, i.e. the rule that defines the evolution dynamics of individuals' 21 strategies, whose influence seems to have been overlooked [74]. strictly speaking, only when the model implements the so-called replicator rule (see below) one is considering the effect of the restriction of rela- tionships that the population structure implies, in comparison with standard replicator dynamics. when using a different update rule, however, we are adding a second dimension of variability, which amounts to relax another assumption of replicator dynamics, namely number 4, which posits a population variation linear in the difference of payoffs (see section 2). we will show extensively that this issue may have a huge impact on the evolutionary outcome. in fact, we will see that there is not a general influence of population structure on evolutionary games. even for a particular type of network, its influence on cooperation depends largely on the kind of game and the specific update rule. all one can do is to identify relevant topological characteristics that have a consistent effect on a broad range of games and update rules, and explain this influence in terms of the same basic principles. to this end, we will be looking at the asymptotic states for different values of the game parameters, and not at how the system behaves when the parameters are varied, which would be an approach of a more statistical mechanics character. in this respect, it is worth pointing out that some studies did use this perspective: thus, it has been shown that the extinction transitions when the temptation parameter varies within the prisoner's dilemma game and the evolutionary dynamics is stochastic fall in the directed percolation universality class, in agreement with a well known conjecture [106]. in particular, some of the pioneering works in using a physics viewpoint on evolutionary games [82, 107] have verified this result for specific models. the behavior changes under deterministic rules such as unconditional imitation (see below), for which this extinction transition is discontinuous. although our ultimate interest may be the effect on the evolution of cooperation, measuring to which extent cooperation is enforced or inhibited is not enough to clarify this effect. as in previous sections, our basic observables will be the dynamical equilibria of the model, in comparison with the equilibria of our reference model with standard replicator dynamics –which, as we have explained in section 2, are closely related to those of the basic game–. the understanding of how the population structure modifies qualitatively and quantitatively these equilibria will give us a much clearer view on the behavior and properties of the model under study, and hence on its influence on cooperation. 4.1. network models and update rules many kinds of networks have been considered as models for population structure (for recent reviews on networks, see [108, 109]). a first class includes networks that introduce a spatial arrangement of relationships, which can represent territorial or physical constraints in the interactions between individuals. typical examples of this group are regular lattices, with different degrees of neighborhood. other important group is that of synthetic networks that try to reproduce important properties that have been found in real networks, such as the small-world or scale-free properties. prominent examples among these are watts- strogatz small-world networks [110] and barab ́ asi-albert scale-free networks [111]. finally, "real" social networks that come directly from experimental data have also been studied, as for example in [112, 113]. as was stated before, one crucial component of the evolutionary models that we are discussing in this section is the update rule, which determines how the strategy of individuals evolves in time. there is a very large variety of update rules that have been used in the literature, each one arising from different backgrounds. the most important for our purposes is the replicator rule, also known as the proportional imitation rule, which is inspired on replicator dynamics and we describe in the following.5 let i = 1 . . . n label the individuals in the population. let si be the strategy of player i, wi her payoffand ni her neighborhood, with ki neighbors. one neighbor j of player i is chosen at random, j ∈ni. the probability of player i adopting the strategy of player j is given by pt ij ≡p {st j →st+1 i } =  (w t j −w t i )/φ , w t j > w t i , 0 , w t j ≤w t i , (35) with φ appropriately chosen as a function of the payoffs to ensure p {*} ∈[0, 1]. 5to our knowledge, helbing was the first to show that a macroscopic population evolution following replicator dynamics 22 figure 9: asymptotic density of cooperators x∗with the replicator rule on a complete network, when the initial density of cooperators is x0 = 1/3 (left, a), x0 = 1/2 (middle, b) and x0 = 2/3 (right, c). this is the standard outcome for a well-mixed population with replicator dynamics, and thus constitutes the reference to assess the influence of a given population structure (see main text for further details). the reason for the name of this rule is the fact that the equation of evolution with this update rule, for large sizes of the population, is equal, up to a time scale factor, to that of replicator dynamics [9, 11]. therefore, the complete network with the replicator rule constitutes the finite-size, discrete-time version of replicator dynamics on an infinite, well-mixed population in continuous time. fig. 9 shows the evolutionary outcome of this model, in the same type of plot as subsequent results in this section. each panel of this figure displays the asymptotic density of cooperators x∗for a different initial density x0, in a grid of points in the st-plane of games. the payoffmatrix of each game is given by c d c d  1 s t 0  . (36) we will consider the generality of this choice of parameters at the end of this section, after introducing the other evolutionary rules. note that, in the notation of section 3, we have taken wcc = 1, wcd = s, wdc = t, wdd = 0; note also that for these payoffs, the normalizing factor in the replicator rule can be chosen as φ = max(ki, kj)(max(1, t)−min(0, s)). in this manner, we visualize the space of symmetric 2×2 games as a plane of co-ordinates s and t –for sucker's and temptation–, which are the respective payoffs of a cooperator and a defector when confronting each other. the four quadrants represented correspond to the following games: harmony (upper left), stag hunt (lower left), snowdrift or hawk-dove (upper right) and prisoner's dilemma (lower right). as expected, these results reflect the close relationship between the equilibria of replicator dynamics and the equilibria of the basic game. thus, all harmony games end up in full cooperation and all prisoner's dilemmas in full defection, regardless of the initial condition. snowdrift games reach a mixed strategy equilibrium, with density of cooperators xe = s/(s +t −1). stag hunt games are the only ones whose outcome depends on the initial condition, because of their bistable character with an unstable equilibrium also given by xe. to allow a quantitative comparison of the degree of cooperation in each game, we have introduced a quantitative index, the average cooperation over the region corresponding to each game, which appears beside each quadrant. the results in fig. 9 constitute the reference against which the effect of population structure will be assessed in the following. one interesting variation of the replicator rule is the multiple replicator rule, whose difference consists on checking simultaneously all the neighborhood and thus making more probable a strategy change. with could be induced by a microscopic imitative update rule [114, 115]. schlag proved later the optimality of such a rule under certain information constraints, and named it proportional imitation [116]. 23 this rule the probability that player i maintains her strategy is p {st i →st+1 i } = y j∈ni (1 −pt ij), (37) with pt ij given by (35). in case the strategy update takes place, the neighbor j whose strategy is adopted by player i is selected with probability proportional to pt ij. a different option is the following moran-like rule, also called death-birth rule, inspired on the moran dynamics, described in section 3. with this rule a player chooses the strategy of one of her neighbors, or herself's, with a probability proportional to the payoffs p {st j →st+1 i } = w t j −ψ p k∈n∗ i (w t k −ψ), (38) with n ∗ i = ni ∪{i}. because payoffs may be negative in prisoner's dilemma and stag hunt games, the constant ψ = maxj∈n ∗ i (kj) min(0, s) is subtracted from them. note that with this rule a player can adopt, with low probability, the strategy of a neighbor that has done worse than herself. the three update rules presented so far are imitative rules. another important example of this kind is the unconditional imitation rule, also known as the "follow the best" rule [73]. with this rule each player chooses the strategy of the individual with largest payoffin her neighborhood, provided this payoffis greater than her own. a crucial difference with the previous rules is that this one is a deterministic rule. another rule that has received a lot of attention in the literature, specially in economics, is the best response rule. in this case, instead of some kind of imitation of neighbor's strategies based on payoffscoring, the player has enough cognitive abilities to realize whether she is playing an optimum strategy (i.e. a best response) given the current configuration of her neighbors. if it is not the case, she adopts with probability p that optimum strategy. it is clear that this rule is innovative, as it is able to introduce strategies not present in the population, in contrast with the previous purely imitative rules. finally, an update rule that has been widely used in the literature, because of being analytically tractable, is the fermi rule, based on the fermi distribution function [82, 117, 118]. with this rule, a neighbor j of player i is selected at random (as with the replicator rule) and the probability of player i acquiring the strategy of j is given by p {st j →st+1 i } = 1 1 + exp −β (w t j −w t i )
. (39) the parameter β controls the intensity of selection, and can be understood as the inverse of temperature or noise in the update rule. low β represents high temperature or noise and, correspondingly, weak selection pressure. whereas this rule has been employed to study resonance-like behavior in evolutionary games on lattices [119], we use it in this work to deal with the issue of the intensity of selection (see subsection 4.6). having introduced the evolutionary rules we will consider, it is important to recall our choice for the payoff matrix (36), and discuss its generality. most of the rules (namely the replicator, the multiple replicator, the unconditional imitation and the best response rules) are invariant on homogeneous networks6 under translation and (positive) scaling of the payoffmatrix. among the remaining rules, the dynamics changes upon translation for the moran rule and upon scaling for the fermi rule. the corresponding changes in these last two cases amount to a modification of the intensity of selection, which we also treat in this work. therefore, we consider that the parameterization of (36) is general enough for our purposes. it is also important to realize that for a complete network, i.e. for a well-mixed or unstructured population, the differences between update rules may be not relevant, as far as they do not change in general the evolutionary outcome [121]. these differences, however, become crucial when the population has some structure, as we will point out in the following. 6the invariance under translations of the payoffmatrix does not hold if the network is heterogenous. in this case, players with higher degrees receive comparatively more (less) payoffunder positive (negative) translations. only very recently has this issue been studied in the literature [120]. 24 figure 10: asymptotic density of cooperators x∗in a square lattice with degree k = 8 and initial density of cooperators x0 = 0.5, when the game is the prisoner's dilemma as given by (40), proposed by nowak and may [73]. note that the outcome with replicator dynamics on a well-mixed population is x∗= 0 for all the displayed range of the temptation parameter t. notice also the singularity at t = 1.4 with unconditional imitation. the surrounding points are located at t = 1.3999 and t = 1.4001. the results displayed in fig. 9 have been obtained analytically, but the remaining results of this section come from the simulation of agent-based models. in all cases, the population size is n = 104 and the allowed time for convergence is 104 time steps, which we have checked it is enough to reach a stationary state. one time step represents one update event for every individual in the population, exactly in the case of synchronous update and on average in the asynchronous case, so it could be considered as one generation. the asymptotic density of cooperators is obtained averaging over the last 103 time steps, and the values presented in the plots are the result of averaging over 100 realizations. cooperators and defectors are randomly located at the beginning of evolution and, when applicable, networks have been built with periodic boundary conditions. see [101] for further details. 4.2. spatial structure and homogeneous networks in 1992 nowak and may published a very influential paper [73], where they showed the dramatic effect that the spatial distribution of a population could have on the evolution of cooperation. this has become the prototypical example of the promotion of cooperation favored by the structure of a population, also known as network reciprocity [19]. they considered the following prisoner's dilemma: c d c d  1 0 t ε  , (40) with 1 ≤t ≤2 and ε ≲0. note that this one-dimensional parameterization corresponds in the st-plane to a line very near the boundary with snowdrift games. fig. 10 shows the great fostering of cooperation reported by [73]. the authors explained this influence in terms of the formation of clusters of cooperators, which give cooperators enough payoffto survive even when surrounded by some defectors. this model has a crucial detail, whose importance we will stress later: the update rule used is unconditional imitation. since the publication of this work many studies have investigated related models with different games and networks, reporting qualitatively consistent results [20]. however, hauert and doebeli published in 2004 another important result [93], which casted a shadow of doubt on the generality of the positive influence of spatial structure on cooperation. they studied the following parameterization of snowdrift games: c d c d  1 2 −t t 0  , (41) 25 figure 11: asymptotic density of cooperators x∗in a square lattice with degree k = 8 and initial density of cooperators x0 = 0.5, when the game is the snowdrift game as given by (41), proposed by hauert and doebeli [93]. the result for a well-mixed population is displayed as a reference as a dashed line. with 1 ≤t ≤2 again. the unexpected result obtained by the authors is displayed in fig. 11. only for low t there is some improvement of cooperation, whereas for medium and high t cooperation is inhibited. this is a surprising result, because the basic game, the snowdrift, is in principle more favorable to cooperation. as we have seen above, its only stable equilibrium is a mixed strategy population with some density of cooperators, whereas the unique equilibrium in prisoner's dilemma is full defection (see fig. 9). in fact, a previous paper by killingback and doebeli [92] on the hawk-dove game, a game equivalent to the snowdrift game, had reported an effect of spatial structure equivalent to a promotion of cooperation. hauert and doebeli explained their result in terms of the hindrance to cluster formation and growth, at the microscopic level, caused by the payoffstructure of the snowdrift game. notwithstanding the different cluster dynamics in both games, as observed by the authors, a hidden contradiction looms in their argument, because it implies some kind of discontinuity in the microscopic dynamics in the crossover between prisoner's dilemma and snowdrift games (s = 0, 1 ≤t ≤2). however, the equilibrium structure of both basic games, which drives this microscopic dynamics, is not discontinuous at this boundary, because for both games the only stable equilibrium is full defection. so, where does this change in the cluster dynamics come from? the fact is that there is not such a difference in the cluster dynamics between prisoner's dilemma and snowdrift games, but different update rules in the models. nowak and may [73], and killingback and doebeli [92], used the unconditional imitation rule, whereas hauert and doebeli [93] employed the replicator rule. the crucial role of the update rule becomes clear in fig. 12, where results in prisoner's dilemma and snowdrift are depicted separately for each update rule. it shows that, if the update rule used in the model is the same, the influence on both games, in terms of promotion or inhibition of cooperation, has a similar dependence on t. for both update rules, cooperation is fostered in prisoner's dilemma and snowdrift at low values of t, and cooperation is inhibited at high t. note that with unconditional imitation the crossover between both behaviors takes place at t ≈1.7, whereas with the replicator rule it occurs at a much lower value of t ≈1.15. the logic behind this influence is better explained in the context of the full st-plane, as we will show later. the fact that this apparent contradiction has been resolved considering the role of the update rule is a good example of its importance. this conclusion is in agreement with those of [96], which performed an exhaustive study on snowdrift games with different network models and update rules, but refutes those of [74], which defended that the effect of spatial lattices was almost independent of the update rule. in consequence, the influence of the network models that we consider in the following is presented separately for each kind of update rule, highlighting the differences in results when appropriate. apart from this, to assess and explain the influence of spatial structure, we need to consider it along with games that have different equilibrium structures, not only a particular game, in order to draw sufficiently general conclusions. 26 (a) (b) figure 12: asymptotic density of cooperators x∗in square lattices with degree k = 8 and initial density of cooperators x0 = 0.5, for both prisoner's dilemma (40) and snowdrift games (41), displayed separately according to the update rule: (a) unconditional imitation (nowak and may's model [73]), (b) replicator rule (hauert and doebeli's model [93]). the result for snowdrift in a well-mixed population is displayed as a reference as a dashed line. it is clear the similar influence of regular lattices on both games, when the key role of the update rule is taken into account (see main text for details). one way to do it is to study their effect on the space of 2 × 2 games described by the parameters s and t (36). a first attempt was done by hauert [74], but some problems in this study make it inconclusive (see [101] for details on this issue). apart from lattices of different degrees (4, 6 and 8), we have also considered homogeneous random networks, i.e. random networks where each node has exactly the same number of neighbors. the aim of comparing with this kind of networks is to isolate the effect of the spatial distribution of individuals from that of the mere limitation of the number of neighbors and the context preservation [85] of a degree-homogeneous random network. the well-mixed population hypothesis implies that every player plays with the "average" player in the population. from the point of view of the replicator rule this means that every player samples successively the population in each evolution step. it is not unreasonable to think that if the number of neighbors is sufficiently restricted the result of this random sampling will differ from the population average, thus introducing changes in the evolutionary outcome. fig. 13 shows the results for the replicator rule with random and spatial networks of different degrees. first, it is clear that the influence of these networks is negligible on harmony games and minimal on prisoner's dilemmas, given the reduced range of parameters where it is noticeable. there is, however, a clear influence on stag hunt and snowdrift games, which is always of opposite sign: an enhancement of cooperation in stag hunt and an inhibition in snowdrift. second, it is illuminating to consider the effect of increasing the degree. for the random network, it means that its weak influence vanishes. the spatial lattice, however, whose result is very similar to that of the random one for the lowest degree (k = 4), displays remarkable differences for the greater degrees (k = 6 and 8). these differences are a clear promotion of cooperation in stag hunt games and a lesser, but measurable, inhibition in snowdrift games, specially for low s. the relevant topological feature that underlies this effect is the existence of clustering in the network, understood as the presence of triangles or, equivalently, common neighbors [108, 109]. in regular lattices, for k = 4 there is no clustering, but there is for k = 6 and 8. this point explains the difference between the conclusions of cohen et al. [85] and those of ifti et al. [89] and tomassini et al. [96], regarding the role of network clustering in the effect of spatial populations. in [85], rectangular lattices of degree k = 4 were considered, which have strictly zero clustering because there are not closed triangles in the network, hence finding no differences in outcome between the spatial and the random topology. in the latter case, on the contrary, both studies employed rectangular lattices of degree k = 8, which do have clustering, and thus they identified it as a key feature of the network, for particular parameterizations of the games they were 27 figure 13: asymptotic density of cooperators x∗in homogeneous random networks (upper row, a to c) compared to regular lattices (lower row, d to f), with degrees k = 4 (a, d), 6 (b, e) and 8 (c, f). the update rule is the replicator rule and the initial density of cooperators is x0 = 0.5. the plots show that the main influence occurs in stag hunt and snowdrift games, specially for regular lattices with large clustering coefficient, k = 6 and 8 (see main text). studying, namely prisoner's dilemma [89] and snowdrift [96]. an additional evidence for this conclusion is the fact that small-world networks, which include random links to reduce the average path between nodes while maintaining the clustering, produce almost indistin- guishable results from those of fig. 13 d-f. this conclusion is in agreement with existent theoretical work about small-world networks, on prisoner's dilemma [84, 104, 105] and its extensions [122, 123], on snowdrift games [96], and also with experimental studies on coordination games [124]. the difference between the effect of regular lattices and small-world networks consists, in general, in a greater efficiency of the latter in reaching the stationary state (see [101] for a further discussion on this comparison). the mechanism that explains this effect is the formation and growth of clusters of cooperators, as fig. 14 displays for a particular realization. the outcome of the population is then totally determined by the stability and growth of these clusters, which in turn depend on the dynamics of clusters interfaces. this means that the result is no longer determined by the global population densities but by the local densities that the players at the cluster interfaces see in their neighborhood. in fact, the primary effect that the network clustering causes is to favor, i.e. to maintain or to increase, the high local densities that were present in the population from the random beginning. this favoring produces opposite effects in stag hunt and snowdrift games. as an illustrating example, consider that the global density is precisely that of the mixed equilibrium of the game. in stag hunt games, as this equilibrium is unstable, a higher local density induces the conversion of nearby defectors to cooperators, thus making the cluster grow. in snowdrift games, on the contrary, as the equilibrium is stable, it causes the conversion of cooperators to defectors. see [101] for a full discussion on this mechanism. in view of this, recalling that these are the results for the replicator rule, and that therefore they corre- spond to the correct update rule to study the influence of population structure on replicator dynamics, we 28 figure 14: snapshots of the evolution of a population on a regular lattice of degree k = 8, playing a stag hunt game (s = −0.65 and t = 0.65). cooperators are displayed in red and defectors in blue. the update rule is the replicator rule and the initial density of cooperators is x0 = 0.5. the upper left label shows the time step t. during the initial steps, cooperators with low local density of cooperators in their neighborhood disappear, whereas those with high local density grow into the clusters that eventually take up the complete population. 29 figure 15: asymptotic density of cooperators x∗in homogeneous random networks (upper row, a to c) compared to regular lattices (lower row, d to f), with degrees k = 4 (a, d), 6 (b, e) and 8 (c, f). the update rule is unconditional imitation and the initial density of cooperators is x0 = 0.5. again as in fig. 13, spatial lattices have greater influence than random networks when the clustering coefficient is high (k = 6 and 8). in this case, however, the beneficial effect for cooperation goes well into snowdrift and prisoner's dilemma quadrants. can state that the presence of clustering (triangles, common neighbors) in a network is a relevant topological feature for the evolution of cooperation. its main effects are, on the one hand, a promotion of cooperation in stag hunt games, and, on the other hand, an inhibition (of lower magnitude) in snowdrift games. we note, however, that clustering may not be the only relevant factor governing the game asymptotics: one can devise peculiar graphs, not representing proper spatial structure, where other influences prove relevant. this is the case of networks consisting of a complete subgraphs connected to each other by a few connections [119], a system whose behavior, in spite of the high clustering coefficient, is similar to those observed on the traditional square lattice where the clustering coefficient is zero. this was subsequently related [125] to the existence of overlapping triangles that support the spreading of cooperation. we thus see that our claim about the outcome of evolutionary games on networks with clustering is anything but general and depends on the translational invariance of the network. other stochastic non-innovative rules, such as the multiple replicator and moran rules, yield similar results, without qualitative differences [101]. unconditional imitation, on the contrary, has a very different influence, as can be seen in fig. 15. in the first place, homogenous random networks themselves have a marked influence, that increases with network degree for stag hunt and snowdrift games, but decreases for prisoner's dilemmas. secondly, there are again no important differences between random and spatial networks if there is no clustering in the network (note how the transitions between the different regions in the results are the same). there are, however, stark differences when there is clustering in the network. interestingly, these are the cases with an important promotion of cooperation in snowdrift and prisoner's dilemma games. in this case, the dynamical mechanism is the formation and growth of clusters of cooperators as well, 30 figure 16: snapshots of the evolution of a population on a regular lattice of degree k = 8, playing a stag hunt game (s = −0.65 and t = 0.65). cooperators are displayed in red and defectors in blue. the update rule is unconditional imitation and the initial density of cooperators is x0 = 1/3 (this lower value than that of fig. 14 has been used to make the evolution longer and thus more easily observable). the upper left label shows the time step t. as with the replicator rule (see fig. 14), during the initial time steps clusters emerge from cooperators with high local density of cooperators in their neighborhood. in this case, the interfaces advance deterministically at each time step, thus giving a special significance to flat interfaces and producing a much faster evolution than with the replicator rule (compare time labels with those of fig. 14) 31 figure 17: asymptotic density of cooperators x∗in regular lattices of degree k = 8, for different initial densities of cooperators x0 = 1/3 (a, d), 1/2 (b, e) and 2/3 (c, f). the update rules are the replicator rule (upper row, a to c) and unconditional imitation (lower row, d to f). with the replicator rule, the evolutionary outcome in stag hunt games depends on the initial condition, as is revealed by the displacement of the transition line between full cooperation and full defection. however, with unconditional imitation this transition line remains in the same position, thus showing the insensitivity to the initial condition. in this case, the outcome is determined by the presence of small clusters of cooperators in the initial random population, which takes place for a large range of values of the initial density of cooperators x0. and the fate of the population is again determined by the dynamics of cluster interfaces. with unconditional imitation, however, given its deterministic nature, interfaces advance one link every time step. this makes very easy the calculation of the conditions for their advancement, because these conditions come down to those of a flat interface between cooperators and defectors [101]. see fig. 16 for a typical example of evolution. an interesting consequence of the predominant role of flat interfaces with unconditional imitation is that, as long as there is in the initial population a flat interface (i.e. a cluster with it, as for example a 3 × 2 cluster in a 8-neighbor lattice), the cluster will grow and eventually extend to the entire population. this feature corresponds to the 3×3 cluster rule proposed by hauert [74], which relates the outcome of the entire population to that of a cluster of this size. this property makes the evolutionary outcome quite independent of the initial density of cooperators, because even for a low initial density the probability that a suitable small cluster exists will be high for sufficiently large populations; see fig. 17 d-f about the differences in initial conditions. nevertheless, it is important to realize that this rule is based on the dynamics of flat interfaces and, therefore, it is only valid for unconditional imitation. other update rules that also give rise to clusters, as replicator rule for example, develop interfaces with different shapes, rendering the particular case of flat interfaces irrelevant. as a consequence, the evolution outcome becomes dependent on the initial condition, as fig. 17 a-c displays. in summary, the relevant topological feature of these homogeneous networks, for the games and update rules considered so far, is the clustering of the network. its effect depends largely on the update rule, and the most that can be said in general is that, besides not affecting harmony games, it consistently promotes 32 figure 18: asymptotic density of cooperators x∗in regular lattices of degree k = 8, with synchronous update (left, a and c) compared to asynchronous (right, b and d). the update rules are the replicator rule (upper row) and unconditional imitation (lower row). the initial density of cooperators is x0 = 0.5. for the replicator rule, the results are virtually identical, showing the lack of influence of the synchronicity of update on the evolutionary outcome. in the case of unconditional imitation the results are very similar, but there are differences for some points, specially snowdrift games with s ≲0.3 and t > 5/3 ≈1.67. the particular game studied by huberman and glance [103], which reported a suppression of cooperation due to asynchronous update, belongs to this region. cooperation in stag hunt games. 4.3. synchronous vs asynchronous update huberman and glance [103] questioned the generality of the results reported by nowak and may [73], in terms of the synchronicity of the update of strategies. nowak and may used synchronous update, which means that every player is updated at the same time, so the population evolves in successive generations. huberman and glance, in contrast, employed asynchronous update (also called random sequential update), in which individuals are updated independently one by one, hence the neighborhood of each player always remains the same while her strategy is being updated. they showed that, for a particular game, the asymptotic cooperation obtained with synchronous update disappeared. this has become since then one of the most well-known and cited examples of the importance of synchronicity in the update of strategies in evolutionary models. subsequent works have, in turn, critizised the importance of this issue, showing that the conclusions of [73] are robust [77, 126], or restricting the effect reported by [103] to particular instances of prisoner's dilemma [78] or to the short memory of players [100]. other works, however, in the different context of snowdrift games [95, 96] have found that the influence on cooperation can be positive or negative, in the asynchronous case compared with the synchronous one. we have thoroughly investigated this issue, finding that the effect of synchronicity in the update of strategies is the exception rather than the rule. with the replicator rule, for example, the evolutionary outcome in both cases is virtually identical, as fig. 18 a-b shows. moreover, in this case, the time evolution is also very similar (see fig.19 a-b). with unconditional imitation there are important differences only in 33 figure 19: time evolution of the density of cooperators x in regular lattices of degree k = 8, for typical realizations of stag hunt (left, a and c) and snowdrift games (right, b and d), with synchronous (continuous lines) or asynchronous (dashed lines) update. the update rules are the replicator rule (upper row) and unconditional imitation (lower row). the stag hunt games for the replicator rule (a) are: a, s = −0.4, t = 0.4; b, s = −0.5, t = 0.5; c, s = −0.6, t = 0.6; d, s = −0.7, t = 0.7; e, s = −0.8, t = 0.8. for unconditional imitation the stag hunt games (c) are: a, s = −0.6, t = 0.6; b, s = −0.7, t = 0.7; c, s = −0.8, t = 0.8; d, s = −0.9, t = 0.9; e, s = −1.0, t = 1.0. the snowdrift games are, for both update rules (b, d): a, s = 0.9, t = 1.1; b, s = 0.7, t = 1.3; c, s = 0.5, t = 1.5; d, s = 0.3, t = 1.7; e, s = 0.1, t = 1.9. the initial density of cooperators is x0 = 0.5. the time scale of the asynchronous realizations has been re-scaled by the size of the population, so t hat for both kinds of update a time step represents the same number of update events in the population. figures a and b show that, in the case of the replicator rule, not only the outcome but also the time evolution is independent of the update synchronicity. with unconditional imitation the results are also very similar for stag hunt (c), but somehow different in snowdrift (d) for large t, displaying the influence of synchronicity in this subregion. note that in all cases unconditional imitation yields a much faster evolution than the replicator rule. one particular subregion of the space of parameters, corresponding mostly to snowdrift games, to which the specific game studied by huberman and glance belongs (see fig. 18 c-d and 19 c-d). 34 figure 20: asymptotic density of cooperators x∗with the replicator update rule, for model networks with different degree heterogeneity: homogeneous random networks (left, a), erd ̋ os-r ́ enyi random networks (middle, b) and barab ́ asi-albert scale- free networks (right, c). in all cases the average degree is ̄ k = 8 and the initial density of cooperators is x0 = 0.5. as degree heterogeneity grows, from left to right, cooperation in snowdrift games is clearly enhanced. figure 21: asymptotic density of cooperators x∗with unconditional imitation as update rule, for model networks with different degree heterogeneity: homogeneous random networks (left, a), erd ̋ os-r ́ enyi random networks (middle, b) and barab ́ asi-albert scale-free networks (right, c). in all cases the average degree is ̄ k = 8 and the initial density of cooperators is x0 = 0.5. as degree heterogeneity grows, from left to right, cooperation in snowdrift games is enhanced again. in this case, however, cooperation is inhibited in stag hunt games and reaches a maximum in prisoner's dilemmas for erd ̋ os-r ́ enyi random networks. 4.4. heterogeneous networks the other important topological feature for evolutionary games was introduced by santos and co-workers [75, 127, 128], who studied the effect of degree heterogeneity, in particular scale-free networks. their main result is shown in fig. 20, which displays the variation in the evolutionary outcome induced by increasing the variance of the degree distribution in the population, from zero (homogeneous random networks) to a finite value (erd ̋ os-r ́ enyi random networks), and then to infinity (scale-free networks). the enhancement of cooperation as degree heterogeneity increases is very clear, specially in the region of snowdrift games. the effect is not so strong, however, in stag hunt or prisoner's dilemma games. similar conclusions are obtained with other scale-free topologies, as for example with klemm-egu ́ ıluz scale-free networks [129]. very recently, it has been shown [130] that much as we discussed above for the case of spatial structures, clustering is also a factor improving the cooperative behavior in scale-free networks. the positive influence on snowdrift games is quite robust against changes in network degree and the use of other update rules. on the other hand, the influence on stag hunt and prisoner's dilemma games is quite restricted and very dependent on the update rule, as fig. 21 reveals. in fact, with unconditional imi- tation cooperation is inhibited in stag hunt games as the network becomes more heterogeneous, whereas in prisoner's dilemmas it seems to have a maximum at networks with finite variance in the degree distribution. 35 a very interesting insight from the comparison between the effects of network clustering and degree het- erogeneity is that they mostly affect games with one equilibrium in mixed strategies, and that in addition the effects on these games are different. this highlights the fact that they are different fundamental topological properties, which induce mechanisms of different nature. in the case of network clustering we have seen the formation and growth of clusters of cooperators. for network heterogeneity the phenomena is the bias and stabilization of the strategy oscillations in snowdrift games towards the cooperative strategy [131, 132], as we explain in the following. the asymptotic state of snowdrift games in homogeneous networks consists of a mixed strategy population, where every individual oscillates permanently between cooperation and defec- tion. network heterogeneity tends to prevent this oscillation, making players in more connected sites more prone to be cooperators. at first, having more neighbors makes any individual receive more payoff, despite her strategy, and hence she has an evolutionary advantage. for a defector, this is a short-lived advantage, because it triggers the change of her neighbors to defectors, thus loosing payoff. a high payoffcooperator, on the contrary, will cause the conversion of her neighbors to cooperators, increasing even more her own payoff. these highly connected cooperators constitute the hubs that drive the population, fully or partially, to cooperation. it is clear that this mechanism takes place when cooperators collect more payofffrom a greater neighborhood, independently of their neighbors' strategies. this only happens when s > 0, which is the reason why the positive effect on cooperation of degree heterogeneity is mainly restricted to snowdrift games. 36 figure 22: asymptotic density of cooperators x∗in a square lattice with degree k = 8 and best response as update rule, in the model with snowdrift (41) studied by sysi-aho and co-workers [94]. the result for a well-mixed population is displayed as a reference. note how the promotion or inhibition of cooperation does not follow the same variation as a function of t than in the case with the replicator rule studied by hauert and doebeli [93] (fig. 11). 4.5. best response update rule so far, we have dealt with imitative update rules, which are non-innovative. here we present the results for an innovative rule, namely best response. with this rule each player chooses, with certain probability p, the strategy that is the best response for her current neighborhood. this rule is also referred to as myopic best response, because the player only takes into account the last evolution step to decide the optimum strategy for the next one. compared to the rules presented previously, this one assumes more powerful cognitive abilities on the individual, as she is able to discern the payoffs she can obtain depending on her strategy and those of her neighbors, in order to chose the best response. from this point of view, it constitutes a next step in the sophistication of update rules. an important result of the influence of this rule for evolutionary games was published in 2005 by sysi- aho and co-workers [94]. they studied the combined influence of this rule with regular lattices, in the same one-dimensional parameterization of snowdrift games (41) that was employed by hauert and doebeli [93]. they reported a modification in the cooperator density at equilibrium, with an increase for some subrange of the parameter t and a decrease for the other, as fig. 22 shows. at the moment, it was intriguing that regular lattices had opposite effects (promotion or inhibition of cooperation) in some ranges of the parameter t, depending on the update rule used in the model. very recently we have carried out a thorough investigation of the influence of this update rule on a wide range of networks [102], focusing on the key topological properties of network clustering and degree heterogeneity. the main conclusion of this study is that, with only one relevant exception, the best response rule suppresses the effect of population structure on evolutionary games. fig. 23 shows a summary of these results. in all cases the outcome is very similar to that of replicator dynamics on well-mixed populations (fig. 9), despite the fact that the networks studied explore different options of network clustering and degree heterogeneity. the steps in the equilibrium density of snowdrift games, as those reported in [94], show up in all cases, with slight variations which depend mostly on the mean degree of the network. the exception to the absence of network influence is the case of regular lattices, and consists of a modification of the unstable equilibrium in stag hunt games, in the sense that it produces a promotion of cooperation for initial densities lower than 0.5 and a corresponding symmetric inhibition for greater densities. an example of this effect is given in fig. 24, where the outcome should be compared to that of well-mixed populations in fig. 9 a. the reason for this effect is that the lattice creates special conditions for the advancement (or receding) of the interfaces of clusters of cooperators. we refer the interested reader to [102] for a detailed description of this phenomena. very remarkably, in this case network clustering is not relevant, because the effect also takes place for degree k = 4, at which there is no clustering in the network. 37 figure 23: asymptotic density of cooperators x∗in random (left, a and d), regular (middle, b and e), and scale-free networks (right, c and f) with average degrees ̄ k = 4 (upper row, a to c) and 8 (lower row, d to f). the update rule is best response with p = 0.1 and the initial density of cooperators is x0 = 0.5. differences are negligible in all cases; note, however, that the steps appearing in the snowdrift quadrant are slightly different. figure 24: asymptotic density of cooperators x∗in regular lattices with initial density of cooperators x0 = 1/3. the degrees are k = 4 (left, a), k = 6 (middle, b) and k = 8 (right, c). the update rule is best response with p = 0.1. comparing with fig. 9 a, there is a clear displacement of the boundary between full defection and full cooperation in stag hung games, which amounts to a promotion of cooperation. the widening of the border in panel c is a finite size effect, which disappears for larger populations. see main text for further details. 38 4.6. weak selection this far, we have considered the influence of population structure in the case of strong selection pressure, which means that the fitness of individuals is totally determined by the payoffs resulting from the game. in general this may not be the case, and then to relax this restriction the fitness can be expressed as f = 1 −w + wπ [133]. the parameter w represents the intensity of selection and can vary between w = 1 (strong selection limit) and w ≳0 (weak selection limit). with a different parameterization, this implements the same idea as the baseline fitness discussed in section 3. we note that another interpretation has been recently proposed [134] for this limit, namely δ-weak selection, which assumes that the game means much to the determination of reproductive success, but that selection is weak because mutant and wild-type strategies are very similar. this second interpretation leads to different results [134] and we do not deal with it here, but rather we stick with the first one, which is by far the most generally used. the weak selection limit has the nice property of been tractable analytically. for instance, ohtsuki and nowak have studied evolutionary games on homogeneous random networks using this approach [135], finding an interesting relation with replicator dynamics on well-mixed populations. using our normalization of the game (36), their main result can be written as the following payoffmatrix  1 s + ∆ t −∆ 0  . (42) this means that the evolution in a population structured according to a random homogeneous network, in the weak selection limit, is the same as that of a well-mixed population with a game defined by this modified payoffmatrix. the effect of the network thus reduces to the term ∆, which depends on the game, the update rule and the degree k of the network. with respect to the influence on cooperation it admits a very straightforward interpretation: if both the original and the modified payoffmatrices correspond to a harmony or prisoner's dilemma game, then there is logically no influence, because the population ends up equally in full cooperation or full defection; otherwise, cooperation is enhanced if ∆> 0, and inhibited if ∆< 0. the actual values of ∆, for the update rules pairwise comparison (pc), imitation (im) and death-birth (db) (see [135] for full details), are ∆p c = s −(t −1) k −2 (43) ∆im = k + s −(t −1) (k + 1)(k −2) (44) ∆db = k + 3(s −(t −1)) (k + 3)(k −2) , (45) k being the degree of the network. a very remarkable feature of these expressions is that for every pair of games with parameters (s1, t1) and (s2, t2), if s1 −t1 = s2 −t2 then ∆1 = ∆2. hence the influence on cooperation for such a pair of games, even if one is a stag hunt and the other is a snowdrift, will be the same. this stands in stark contrast to all the reported results with strong selection, which generally exhibit different, and in many cases opposite, effects on both games. besides this, as the term s −(t −1) is negative in all prisoner's dilemmas and half the cases of stag hunt and snowdrift games, the beneficial influence on cooperation is quite reduced for degrees k as those considered above [101]. another way to investigate the influence of the intensity of selection if to employ the fermi update rule, presented above, which allows to study numerically the effect of varying the intensity of selection on any network model. figs. 25 and 26 display the results obtained, for different intensities of selection, on networks that are prototypical examples of strong influence on evolutionary games, namely regular lattices with high clustering and scale-free networks, with large degree heterogeneity. in both cases, as the intensity of selection is reduced, the effect of the network becomes weaker and more symmetrical between stag hunt and snowdrift games. therefore, these results show that the strong and weak selection limits are not comparable from the viewpoint of the evolutionary outcome, and that weak selection largely inhibits the influence of population structure. 39 figure 25: asymptotic density of cooperators x∗in regular lattices of degree k = 8, for the fermi update rule with β equal to 10 (a), 1 (b), 0.1 (c) and 0.01 (d). the initial density of cooperators is x0 = 0.5. for high β the result is quite similar to that obtained with the replicator rule (fig. 13 f). as β decreases, or equivalently for weaker intensities of selection, the influence becomes smaller and more symmetrical between stag hunt and snowdrift games. figure 26: asymptotic density of cooperators x∗in barab ́ asi-albert scale-free networks of average degree ̄ k = 8, for the fermi update rule with β equal to 10 (a), 1 (b), 0.1 (c) and 0.01 (d). the initial density of cooperators is x0 = 0.5. as in fig. 25, for high β the result is quite similar to that obtained with the replicator rule (fig. 20 c), and analogously, as β decreases the influence of the network becomes smaller and more symmetrical between stag hunt and snowdrift games. 40 5. conclusion and future prospects in this review, we have discussed non-mean-field effects on evolutionary game dynamics. our reference framework for comparison has been the replicator equation, a pillar of modern evolutionary game theory that has produced many interesting and fruitful insights on different fields. our purpose here has been to show that, in spite of its many successes, the replicator equation is only a part of the story, much in the same manner as mean-field theories have been very important in physics but they cannot (nor are they intended to) describe all possible phenomena. the main issues we have discussed are the influence of fluctuations, by considering the existence of more than one time scale, and of spatial correlations, through the constraints on interaction arising from an underlying network structure. in doing so, we have shown a wealth of evidence supporting our first general conclusion: deviations with respect to the hypothesis of a well-mixed population (including nonlinear dependencies of the fitness on the payoffor not) have a large influence on the outcome of the evolutionary process and in a majority of cases do change the equilibria structure, stability and/or basins of attraction. the specific question of the existence of different time scales was discussed in section 3. this is a problem that has received some attention in economics but otherwise it has been largely ignored in biological contexts. in spite of this, we have shown that considering fast evolution in the case of ultimatum game may lead to a non-trivial, unexpected conclusion: that individual selection may be enough to explain the experimental evidence that people do not behave rationally. this is an important point in so far as, to date, simple individual selection was believed not to provide an understanding of the phenomena of altruistic punishment reported in many experiments [56]. we thus see that the effect of different time scales might be determinant and therefore must be considered among the relevant factors with an influence on evolutionary phenomena. this conclusion is reinforced by our general study of symmetric 2 × 2 symmetric games, that shows that the equilibria of about half of the possible games change when considering fast evolution. changes are particularly surprising in the case of the harmony game, in which it turns out that when evolution is fast, the selected strategy is the "wrong" one, meaning that it is the less profitable for the individual and for the population. such a result implies that one has to be careful when speaking of adaptation through natural selection, because in this example we have a situation in which selection leads to a bad outcome through the influence of fluctuations. it is clear that similar instances may arise in many other problems. on the other hand, as for the particular question of the emergence of cooperation, our results imply that in the framework of the classical 2 × 2 social dilemmas, fast evolution is generally bad for the appearance of cooperative behavior. the results reported here concerning the effect of time scales on evolution are only the first ones in this direction and, clearly, much remains to be done. in this respect, we believe that it would be important to work out the case of asymmetric 2 × 2 games, trying to reveal possible general conclusions that apply to families of them. the work on the ultimatum game [64] is just a first example, but no systematic analysis of asymmetric games has been carried out. a subsequent extension to games with more strategies would also be desirable; indeed, the richness of the structures arising in those games (such as, e.g., the rock-scissors-papers game [11]) suggests that considering fast evolution may lead to quite unexpected results. this has been very recently considered in the framework of the evolutionary minority game [136] (where many strategies are possible, not just two or three) once again from an economics perspective [137]; the conclusion of this paper, namely that there is a phase transition as a function of the time scale parameter that can be observed in the predictability of market behavior is a further hint of the interest of this problem. in section 4 we have presented a global view of the influence of population structure on evolutionary games. we have seen a rich variety of results, of unquestionable interest, but that on the downside reflect the non-generality of this kind of evolutionary models. almost every detail in the model matters on the outcome, and some of them dramatically. we have provided evidence that population structure does not necessarily promote cooperation in evolu- tionary game theory, showing instances in which population structure enhances or inhibits it. nonetheless, we have identified two topological properties, network clustering and degree heterogeneity, as those that allow a more unified approach to the characterization and understanding of the influence of population 41 structure on evolutionary games. for certain subset of update rules, and for some subregion in the space of games, they induce consistent modifications in the outcome. in summary, network clustering has a positive impact on cooperation in stag hunt games and degree heterogeneity in snowdrift games. therefore, it would be reasonable to expect similar effects in other networks which share these key topological properties. in fact, there is another topological feature of networks that conditions evolutionary games, albeit of a different type: the community structure [108, 109]. communities are subgraphs of densely interconnected nodes, and they represent some kind of mesoscopic organization. a recent study [76] has pointed out that communities may have their own effects on the game asymptotics in otherwise similar graphs, but more work is needed to assess this influence. on the other hand, the importance of the update rules cannot be overstated. we have seen that for the best response and fermi rules even these "robust" effects of population structure are greatly reduced. it is very remarkable from a application point of view that the influence of population structure is inhibited so greatly when update rules more sophisticated than merely imitative ones are considered, or when the selection pressure is reduced. it is evident that a sound justification of several aspects of the models is mandatory for applications. crucial details, as the payoffstructure of the game, the characteristics of the update rule or the main topological features of the network are critical for obtaining significant results. for the same reasons, unchecked generalizations of the conclusions obtained from a particular model, which go beyond the kind of game, the basic topology of the network or the characteristics of the updated rule, are very risky in this field of research. very easily the evolutionary outcome of the model could change dramatically, making such generalizations invalid. this conclusion has led a number of researchers to address the issue from a further evolutionary viewpoint: perhaps, among the plethora of possible networks one can think of, only some of them (or some values of their magnitudes) are really important, because the rest are not found in actual situations. this means that networks themselves may be subject to natural selection, i.e., they may co-evolve along with the game under consideration. this promising idea has already been proposed [138, 139, 140, 141, 142, 143, 144, 145] and a number of interesting results, which would deserve a separate review on their own right7, have been obtained regarding the emergence of cooperation. in this respect, it has been observed that co-evolution seems to favor the stabilization of cooperative behavior, more so if the network is not rewired from a preexisting one but rather grows upon arrival of new players [147]. a related approach, in which the dynamics of the interaction network results from the mobility of players over a spatial substrate, has been the focus of recent works [148, 149]. albeit these lines of research are appealing and natural when one thinks of possible applications, we believe the same caveat applies: it is still too early to draw general conclusions and it might be that details would be again important. nevertheless, work along these lines is needed to assess the potential applicability of these types of models. interestingly, the same approach is also being introduced to understand which strategy update rules should be used, once again as a manner to discriminate among the very many possibilities. this was pioneered by harley [150] (see also the book by maynard smith [6], where the paper by harley is presented as a chapter) and a few works have appeared in the last few years [151, 152, 153, 154, 155]; although the available results are too specific to allow for a glimpse of any general feature, they suggest that continuing this research may render fruitful results. we thus reach our main conclusion: the outcome of evolutionary game theory depends to a large extent on the details, a result that has very important implications for the use of evolutionary game theory to model actual biological, sociological or economical systems. indeed, in view of this lack of generality, one has to look carefully at the main factors involved in the situation to be modeled because they need to be included as close as necessary to reality to produce conclusions relevant for the case of interest. note that this does not mean that it is not possible to study evolutionary games from a more general viewpoint; as we have seen above, general conclusions can be drawn, e.g., about the beneficial effects of clustering for cooperation or the key role of hubs in highly heterogeneous networks. however, what we do mean is that one should not take such general conclusions for granted when thinking of a specific problem or phenomenon, because it might well be that some of its specifics render these abstract ideas unapplicable. on the other hand, it might be 7for a first attempt, see sec. 5 of [146]. 42 possible that we are not looking at the problem in the right manner; there may be other magnitudes we have not identified yet that allow for a classification of the different games and settings into something similar to universality classes. whichever the case, it seems clear to us that much research is yet to be done along the lines presented here. we hope that this review encourages others to walk offthe beaten path in order to make substantial contributions to the field. acknowledgements this work has been supported by projects mosaico, from the spanish ministerio de educaci ́ on y ciencia, and mossnoho and simumat, from the comunidad aut ́ onoma de madrid. references [1] c. darwin, on the origin of species by means of natural selection, or the preservation of favoured races in the struggle for life, 1st edition, john murray, london, 1859. [2] j. roughgarden, theory of population genetics and evolutionary ecology: an introduction, macmillan publishing co., new york, 1979. [3] l. peliti, fitness landscapes and evolution, in: t. riste, d. sherrington (eds.), physics of biomaterials: fluctuations, self-assembly and evolution, kluwer academic, dordrecht, 1996, pp. 287–308. [4] b. drossel, biological evolution and statistical physics, adv. phys 50 (2001) 209–295. [5] w. j. ewens, mathematical population genetics, 2nd edition, springer, new york, 2004. [6] j. maynard smith, evolution and the theory of games, cambridge university press, cambridge, 1982. [7] s. j. gould, e. s. vrba, exaptation - a missing term in the science of form, paleobiology 8 (1982) 4–15. [8] j. von neumann, o. morgenstern, the theory of games and economic behavior, princeton university press, princeton, 1947. [9] h. gintis, game theory evolving, princeton university press, princeton, 2000. [10] m. a. nowak, evolutionary dynamics: exploring the equations of life, the belknap press of harvard university press, cambridge, 2006. [11] j. hofbauer, k. sigmund, evolutionary games and population dynamics, cambridge university press, cambridge, 1998. [12] p. d. taylor, l. jonker, evolutionarily stable strategies and game dynamics, math. biosci. 40 (1978) 145–156. [13] j. nash, equilibrium points in n-person games, proc. natl. acad. sci. usa 36 (1950) 48–49. [14] e. pennisi, how did cooperative behavior evolve?, science 309 (2005) 93. [15] w. d. hamilton, the genetical evolution of social behaviour i, j. theor. biol. 7 (1964) 1–16. [16] w. d. hamilton, the genetical evolution of social behaviour ii, j. theor. biol. 7 (1964) 17–52. [17] r. l. trivers, the evolution of reciprocal altruism, q. rev. biol. 46 (1971) 35–57. [18] p. hammerstein (ed.), genetic and cultural evolution of cooperation (dahlem workshop report 90), mit press, cambridge, 2003. [19] m. a. nowak, five rules for the evolution of cooperation, science 314 (2006) 1560–1563. [20] g. szab ́ o, g. f ́ ath, evolutionary games on graphs, phys. rep. 446 (2007) 97–216. [21] j. hofbauer, k. sigmund, evolutionary game dynamics, bull. am. math. soc. 40 (2003) 479–519. [22] a. rapoport, a. m. chammah, prisoner's dilemma, university of michigan press, ann arbor, 1965. [23] r. sugden, the economics of rights, co-operation and welfare, basil blackwell,, oxford, 1986. [24] j. maynard smith, g. price, the logic of animal conflict, nature 246 (1973) 15–18. [25] f. vega-redondo, economics and the theory of games, cambridge university press, cambridge, 2003. [26] p. schuster, k. sigmund, replicator dynamics, j. theor. biol. 100 (1983) 533–538. [27] a. traulsen, j. c. claussen, c. hauert, coevolutionary dynamics: from finite to infinite populations, phys. rev. lett. 95 (2005) 238701. [28] j. c. claussen, a. traulsen, non-gaussian fluctuations arising from finite populations: exact results for the evolutionary moran process, phys. rev. e 71 (2005) 25101. [29] a. traulsen, j. c. claussen, c. hauert, coevolutionary dynamics in large, but finite populations, phys. rev. e 74 (2006) 011901. [30] j. c. claussen, discrete stochastic processes, replicator and fokker-planck equations of coevolutionary dynamics in finite and infinite populations, banach center publications 80 (2008) 17–31. [31] d. b. fogel, g. b. fogel, p. c. andrews, on the instability of evolutionary stable strategies, biosys. 44 (1997) 135–152. [32] g. b. fogel, p. c. andrews, d. b. fogel, on the instability of evolutionary stable strategies in small populations, ecol. modelling 109 (1998) 283–294. [33] k. m. page, m. a. nowak, unifying evolutionary dynamics, j. theor. biol. 219 (2002) 93–98. [34] g. r. price, selection and covariance?, nature 227 (1970) 520–521. [35] s. a. frank, george price's contributions to evolutionary genetics, j. theor. biol. 175 (1995) 373–388. [36] p. kollock, social dilemmas: the anatomy of cooperation, annu. rev. sociol. 24 (1998) 183–214. [37] j. maynard smith, e. szathm ́ ary, the major transitions in evolution, oxford university press, oxford, 1995. 43 [38] a. p. hendry, m. t. kinnison, perspective: the pace of modern life: measuring rates of contemporary microevolution, evolution 53 (1999) 1637–1653. [39] a. p. hendry, j. k. wenburg, p. bentzen, e. c. volk, t. p. quinn, rapid evolution of reproductive isolation in the wild: evidence from introduced salmon, science 290 (2000) 516–518. [40] t. yoshida, l. e. jones, s. p. ellner, g. f. fussmann, n. g. h. jr., rapid evolution drives ecological dynamics in a predator-prey system, nature 424 (2003) 303–306. [41] r. m. fagen, when doves conspire: evolution of non-damaging fighting tactics in a non-random encounter animal conflict model, am. nat. 115 (1980) 858–869. [42] b. skyrms, the stag hunt and evolution of social structure, cambridge university press, cambridge, 2003. [43] j. c. harsanyi, r. selten, a general theory of equilibrium selection in games, mit, cambridge, massachussets, 1988. [44] m. w. macy, a. flache, learning dynamics in social dilemmas, proc. nat. acad. sci. usa 99 (2002) 7229–7236. [45] m. kandori, g. j. mailath, r. roy, learning, mutation and long-run equilibria in games, econometrica 61 (1993) 29–56. [46] d. foster, p. h. young, stochastic evolutionary game dynamics, theor. popul. biol. 38 (1990) 219–232. [47] a. j. robson, f. vega-redondo, efficient equilibrium selection in evolutionary games with random matching, j. econ. theory 70 (1996) 65–92. [48] j. miekisz, equilibrium selection in evolutionary games with random matching of players, j. theor. biol. 232 (2005) 47–53. [49] b. w ̈ olfing, a. traulsen, stochastic sampling of interaction partners versus deterministic payoffassignment, j. theor. biol. 257 (2009) 689–695. [50] r. t. boylan, laws of large numbers for dynamical systems with randomly matched individuals, j. econ. theory 57 (1992) 473–504. [51] r. t. boylan, continuous approximation of dynamical systems with randomly matched individuals, j. econ. theory 66 (1995) 615–625. [52] a. n. licht, games commissions play: 2x2 games of international securities regulation, yale j. int. law 24 (1999) 61–125. [53] w. g ̈ uth, r. schmittberger, b. schwarze, an experimental analysis of ultimate bargaining, j. econ. behav. org. 3 (1982) 367–388. [54] j. henrich, r. boyd, s. bowles, c. camerer, e. fehr, h. gintis (eds.), foundations of human sociality: economic experiments and ethnographic evidence from fifteen small-scale societies, oxford university press, oxford, 2004. [55] m. a. nowak, k. sigmund, evolution of indirect reciprocity by image scoring, nature 393 (1998) 573–577. [56] c. f. camerer, behavioral game theory, princeton university press, princeton, 2003. [57] e. fehr, u. fischbacher, the nature of human altruism, nature 425 (2003) 785–791. [58] r. axelrod, w. d. hamilton, the evolution of cooperation, science 211 (1981) 1390–1396. [59] r. axelrod, the evolution of cooperation, basic books, new york, 1984. [60] h. gintis, strong reciprocity and human sociality, j. theor. biol. 206 (2000) 169–179. [61] e. fehr, u. fischbacher, s. g ̈ achter, strong reciprocity, human cooperation and the enforcement of social norms, hum. nat. 13 (2002) 1–25. [62] k. page, m. nowak, a generalized adaptive dynamics framework can describe the evolutionary ultimatum game, j. theor. biol. 209 (2000) 173–179. [63] k. page, m. nowak, empathy leads to fairness, bull. math. biol. 64 (2002) 1101–1116. [64] a. s ́ anchez, j. a. cuesta, altruism may arise from individual selection, j. theor. biol. 235 (2005) 233–240. [65] c. p. roca, j. a. cuesta, a. s ́ anchez, time scales in evolutionary dynamics, phys. rev. lett. 97 (2006) 158701. [66] c. p. roca, j. a. cuesta, a. s ́ anchez, the importance of selection rate in the evolution of cooperation, eur. phys. j. special topics 143 (2007) 51–58. [67] p. a. p. moran, the statistical processes of evolutionary theory, clarendon press, oxford, 1962. [68] c. taylor, d. fudenberg, a. sasaki, m. a. nowak, evolutionary game dynamics in finite populations, bull. math. biol. 66 (2004) 1621–1644. [69] s. karlin, h. m. taylor, a first course in stochastic processes, 2nd edition, academic press, new york, 1975. [70] c. m. grinstead, j. l. snell, introduction to probability, 2nd edition, american mathematical society, providence, 1997. [71] a. rapoport, m. guyer, a taxonomy of 2 × 2 games, general systems 11 (1966) 203–214. [72] a. traulsen, m. a. nowak, j. m. pacheco, stochastic payoffevaluation increases the temperature of selection, j. theor. biol. 244 (2007) 349–356. [73] m. a. nowak, r. m. may, evolutionary games and spatial chaos, nature 359 (1992) 826–829. [74] c. hauert, effects of space in 2 × 2 games, int. j. bifurc. chaos 12 (2002) 1531–1548. [75] f. c. santos, j. m. pacheco, t. lenaerts, evolutionary dynamics of social dilemmas in structured heterogeneous popu- lations, proc. natl. acad. sci. usa 103 (2006) 3490–3494. [76] s. lozano, a. arenas, a. s ́ anchez, mesoscopic structure conditions the emergence of cooperation on social networks, plos one 3 (2008) e1892. [77] m. a. nowak, s. bonhoeffer, r. m. may, spatial games and the maintenance of cooperation, proc. natl. acad. sci. usa 91 (1994) 4877–4881. [78] k. lindgren, m. g. nordahl, evolutionary dynamics of spatial games, physica d 75 (1994) 292–309. [79] v. hutson, g. vickers, the spatial struggle of tit-for-tat and defect, phil. trans. r. soc. lond. b 348 (1995) 393–404. [80] p. grim, spatialization and greater generosity in the stochastic prisoner's dilemma, biosys. 37 (1996) 3–17. [81] m. nakamaru, h. matsuda, y. iwasa, the evolution of cooperation in a lattice-structured population, j. theor. biol. 184 (1997) 65–81. [82] g. szab ́ o, c. t ̋ oke, evolutionary prisoner's dilemma game on a square lattice, phys. rev. e 58 (1998) 69–73. 44 [83] k. brauchli, t. killingback, m. doebeli, evolution of cooperation in spatially structured populations, j. theor. biol. 200 (1999) 405–417. [84] g. abramson, m. kuperman, social games in a social network, phys. rev. e 63 (2001) 030901(r). [85] m. d. cohen, r. l. riolo, r. axelrod, the role of social structure in the maintenance of cooperative regimes, rationality and society 13 (2001) 5–32. [86] m. h. vainstein, j. j. arenzon, disordered environments in spatial games, phys. rev. e 64 (2001) 051905. [87] y. lim, k. chen, c. jayaprakash, scale-invariant behavior in a spatial game of prisoners' dilemma, phys. rev. e 65 (2002) 026134. [88] f. schweitzer, l. behera, h. m ̈ uhlenbein, evolution of cooperation in a spatial prisoner's dilemma, adv. complex sys. 5 (2002) 269–299. [89] m. ifti, t. killingback, m. doebeli, effect of neighbourhood size and connectivity on the spatial continuous prisoner's dilemma, j. theor. biol. 231 (2004) 97–106. [90] c.-l. tang, w.-x. wang, x. wu, b.-h. wang, effects of average degree on cooperation in networked evolutionary game, eur. phys. j. b 53 (2006) 411–415. [91] m. perc, a. szolnoki, social diversity and promotion of cooperation in the spatial prisoner's dilemma, phys. rev. e 77 (2008) 011904. [92] t. killingback, m. doebeli, spatial evolutionary game theory: hawks and doves revisited, proc. r. soc. lond. b 263 (1996) 1135–1144. [93] c. hauert, m. doebeli, spatial structure often inhibits the evolution of cooperation in the snowdrift game, nature 428 (2004) 643–646. [94] m. sysi-aho, j. saram ̈ aki, j. kert ́ esz, k. kaski, spatial snowdrift game with myopic agents, eur. phys. j. b 44 (2005) 129–135. [95] a. kun, g. boza, i. scheuring, asynchronous snowdrift game with synergistic effect as a model of cooperation, behav. ecol. 17 (2006) 633–641. [96] m. tomassini, l. luthi, m. giacobini, hawks and doves on small-world networks, phys. rev. e 73 (2006) 016132. [97] l.-x. zhong, d.-f. zheng, b. zheng, c. xu, p. hui, networking effects on cooperation in evolutionary snowdrift game, europhys. lett. 76 (2006) 724–730. [98] l. e. blume, the statistical mechanics of strategic interaction, games econ. behav. 5 (1993) 387–424. [99] g. ellison, learning, local interaction, and coordination, econometrica 61 (1993) 1047–1071. [100] o. kirchkamp, spatial evolution of automata in the prisoners' dilemma, j. econ. behav. org. 43 (2000) 239–262. [101] c. p. roca, j. cuesta, a. s ́ anchez, effect of spatial structure on the evolution of cooperation, phys. rev. e, in press. [102] c. p. roca, j. cuesta, a. s ́ anchez, promotion of cooperation on networks? the myopic best response case, eur. phys. j. b, in press. [103] b. a. huberman, n. s. glance, evolutionary games and computer simulations, proc. natl. acad. sci. usa 90 (1993) 7716–7718. [104] n. masuda, k. aihara, spatial prisoner's dilemma optimally played in small-world networks, phys. lett. a 313 (2003) 55–61. [105] m. tomochi, defectors' niches: prisoner's dilemma game on disordered networks, soc. netw. 26 (2004) 309–321. [106] h. hinrichsen, non-equilibrium critical phenomena and phase transitions into absorbing states, adv. phys. 49 (2000) 815–958. [107] j. r. n. chiappin, m. j. de oliveira, emergence of cooperation among interacting individuals, phys. rev. e 59 (1999) 6419–6421. [108] m. newman, the structure and function of complex networks, siam review 45 (2003) 167–256. [109] s. bocaletti, v. latora, y. moreno, m. chavez, d. u. hwang, complex networks: structure and dynamics, phys. rep. 424 (2006) 175–308. [110] d. j. watts, s. h. strogatz, collective dynamics of 'small-world' networks, nature 393 (1998) 440–442. [111] r. albert, a.-l. barab ́ asi, statistical mechanics of complex networks, rev. mod. phys. 74 (2002) 47–97. [112] p. holme, a. trusina, b. j. kim, p. minnhagen, prisoners dilemma in real-world acquaintance networks: spikes and quasi-equilibria induced by the interplay between structure and dynamics, phys. rev. e 68 (2003) 030901. [113] r. guimer` a, l. danon, a. d ́ ıaz-guilera, f. giralt, a. arenas, self-similar community structure in a network of human interactions, phys. rev. e 68 (2003) 065103. [114] d. helbing, interrelations between stochastic equations for systems with pair interactions, physica a 181 (1992) 29–52. [115] d. helbing, a mathematical model for behavioral changes through pair interactions, in: g. hagg, u. mueller, k. g. troitzsch (eds.), economic evolution and demographic change, springer–verlag, berlin, 1992, pp. 330–348. [116] k. h. schlag, why imitate, and if so, how? a boundedly rational approach to multi-armed bandits, j. econ. theory 78 (1998) 130–156. [117] l. blume, now noise matters, games econ. behav. 44 (2003) 251–271. [118] a. traulsen, m. a. nowak, j. m. pacheco, stochastic dynamics of invasion and fixation, phys. rev. e 74 (2006) 011909. [119] g. szab ́ o, j. vukov, a. szolnoki, phase diagrams for an evolutionary prisoners dilemma game on two-dimensional lattices, phys. rev. e 72 (2005) 047107. [120] l. luthi, m. tomassini, e. pestelacci, evolutionary games on networks and payoffinvariance under replicator dynamics, biosys. 96 (2009) 213–222. [121] c. p. roca, j. cuesta, a. s ́ anchez, in preparation. [122] z.-x. wu, x.-j. xu, y. chen, y.-h. wang, spatial prisoner's dilemma game with volunteering in newman-watts small- world networks, phys. rev. e 71 (2005) 037103. 45 [123] z.-x. wu, x.-j. xu, y.-h. wang, prisoner's dilemma game with heterogeneous influential effect on regular small-world networks, chin. phys. lett. 23 (2006) 531–534. [124] a. cassar, coordination and cooperation in local, random and small world networks: experimental evidence, games econ. behav. 58 (2007) 209–230. [125] j. vukov, g. szab ́ o, a. szolnoki, cooperation in the noisy case: prisoner's dilemma game on two types of regular random graphs, phys. rev. e 73 (2006) 067103. [126] c. hauert, g. szab ́ o, game theory and physics, am. j. phys. 73 (2005) 405. [127] f. c. santos, j. m. pacheco, scale-free networks provide a unifying framework for the emergence of cooperation, phys. rev. lett. 95 (2005) 98104. [128] f. c. santos, j. m. pacheco, t. lenaerts, graph topology plays a determinant role in the evolution of cooperation, proc. roy. soc. b 273 (2006) 51–56. [129] k. klemm, v. m. egu ́ ıluz, growing scale-free networks with small-world behavior, phys. rev. e 65 (2002) 057102. [130] s. assenza, j. g ́ omez-garde ̃ nes, v. latora, enhancement of cooperation in highly clustered scale-free networks, phys. rev. e 78 (2009) 017101. [131] j. g ́ omez-garde ̃ nes, m. campillo, l. m. flor ́ ıa, y. moreno, dynamical organization of cooperation in complex networks, phys. rev. lett. 98 (2007) 108103. [132] j. poncela, j. g ́ omez-garde ̃ nes, l. m. flor ́ ıa, y. moreno, robustness of cooperation in the prisoner's dilemma in complex networks, new j. phys. 9 (2007) 184. [133] m. a. nowak, a. sasaki, c. taylor, d. fudenberg, emergence of cooperation and evolutionary stability in finite popu- lations, nature 428 (2004) 646–650. [134] g. wild, a. traulsen, the different limits of weak selection and the evolutionary dynamics of finite populations, j. theor. biol. 247 (2007) 382–390. [135] h. ohtsuki, m. a. nowak, the replicator equation on graphs, j. theor. biol. 243 (2006) 86–97. [136] d. challet, y. c. zhang, emergence of cooperation and organization in an evolutionary game, physica a 246 (1997) 407–418. [137] l.-x. zhong, t. qiu, b.-h. chen, c.-f. liu, effects of dynamic response time in an evolving market, physica a 388 (2009) 673–681. [138] m. g. zimmermann, v. m. egu ́ ıluz, m. san miguel, coevolution of dynamical states and interactions in dynamic networks, phys. rev. e 69 (2004) 65102. [139] m. marsili, f. vega-redondo, f. slanina, the rise and fall of a networked society: a formal model, proc. nat. acad. sci. usa 101 (2004) 1439–1442. [140] v. egu ́ ıluz, m. zimmermann, m. g. cela-conde, m. san miguel, cooperation and the emergence of role differentiation in the dynamics of social networks, am. j. sociol. 110 (2005) 977–1008. [141] f. c. santos, j. m. pacheco, t. lenaerts, cooperation prevails when individuals adjust their social ties, plos comput. biol. 2 (2006) e140. [142] j. m. pacheco, a. traulsen, m. a. nowak, coevolution of strategy and structure in complex networks with dynamical linking, phys. rev. lett. 97 (2006) 258103. [143] h. ohtsuki, m. a. nowak, j. m. pacheco, breaking the symmetry between interaction and replacement in evolutionary dynamics on graphs, phys. rev. lett. 98 (2007) 108106. [144] f. mengel, c. fosco, cooperation through imitation and exclusion in networks, mpra paper 5258, university library of munich, germany (oct. 2007). url http://ideas.repec.org/p/pra/mprapa/5258.html [145] j. m. pacheco, a. traulsen, h. ohtsuki, m. a. nowak, repeated games and direct reciprocity under active linking, j. theor. biol. 250 (2008) 723–734. [146] t. gross, b. blasius, adaptive coevolutionary networks: a review, j. r. soc. interface 5 (2008) 259–271. [147] j. poncela, j. g ́ omez-garde ̃ nes, l. m. flor ́ ıa, a. s ́ anchez, y. moreno, complex cooperative networks from evolutionary preferential attachment, plos one 3 (2008) e2449. [148] e. a. sicardi, h. fort, m. h. vainstein, j. j. arenzon, random mobility and spatial structure often enhance cooperation, j. theor. biol. 256 (2009) 240–246. [149] d. helbing, w. yu, the outbreak of cooperation among success-driven individuals under noisy conditions, proc. natl. acad. sci. usa 106 (2009) 3680–3685. [150] c. b. harley, learning the evolutionarily stable strategy., j. theor. biol. 89 (1981) 611–633. [151] o. kirchkamp, simultaneous evolution of learning rules and strategies., j. econ. behav. org. 40 (1999) 295–312. [152] l. g. moyano, a. s ́ anchez, evolving learning rules and emergence of cooperation in spatial prisoner's dilemma, j. theor. biol. 259 (2009) 84–95. [153] g. szab ́ o, a. szolnoki, j. vukov, application of darwinian selection to dynamical rules in spatial prisoners dilemma games, preprint. url \unhbox\voidb@x\hbox{http://www.mfa.kfki.hu/$\sim$szabo/szaborms.html} [154] a. szolnoki, m. perc, coevolution of teaching activity promotes cooperation, new j. phys 10 (2008) 043036. [155] a. szolnoki, m. perc, promoting cooperation in social dilemmas via simple coevolutionary rules, eur. phys. j. b 67 (2008) 337–342. 46 a. characterization of birth-death processes one of the relevant quantities to determine in a birth-death process is the probability cn that, starting from state n, the process ends eventually absorbed into the absorbing state n = n. there is a simple relationship between cn and the stochastic matrix p, namely cn = pn,n−1cn−1 + pn,ncn + pn,n+1cn+1, 0 < n < n, (46) with the obvious boundary conditions c0 = 0 and cn = 1. the solution to this equation is [69] cn = qn qn , qn = n−1 x j=0 qj, q0 = 1, qj = j y i=1 pi,i−1 pi,i+1 (j > 0). (47) another relevant quantity is vk,n, the expected number of visits that, starting from state k, the process pays to site n before it enters one absorbing state. if v = (vk,n), with 0 < k, n < n, then v = i + r + r2 + * * * = (i −r)−1, (48) where i is the identity matrix and r is the submatrix of p corresponding to the transient (non-absorbing) states. the series converges because r is substochastic [70]. thus v fulfills the equation v = v r+i, which amounts to an equation similar to (46) for every row of v , namely vk,n = vk,n−1pn−1,n + vk,npn,n + vk,n+1pn+1,n + δk,n, 0 < k, n < n, (49) where δk,n = 1 if k = n and 0 otherwise. contrary to what happens with eq. (46), this equation has no simple solution and it is better solved as in (48). finally, τk, the number of steps before absorption occurs into any absorbing state, when starting at state k, is obtained as τk = n−1 x n=1 vk,n. (50) b. absorption probability in the hypergeometric case for the special case in which pn,n−1 pn,n+1 = αn + β α(n + 1) + γ (51) the absorption probability into state n = n, cn, can be obtained in closed form. according to (47) the sequence qj fulfills the hypergeometric relation qj qj−1 = αj + β α(j + 1) + γ , (52) from which (α(j + 1) + γ)qj = (αj + β)qj−1. (53) adding this equation up for j = 1, . . . , n −1 we get α n−1 x j=1 (j + 1)qj + γ(qn −1) = α n−2 x j=0 (j + 1)qj + β(qn −qn−1), (54) and therefore (γ −β)qn = γ + α −(β + αn)qn−1. (55) 47 thus, provided γ ̸= β, we obtain qn = γ + α γ −β  1 − n y j=1 αj + β αj + γ  . (56) if α = 0 this has the simple form qn = γ γ −β  1 −(β/γ)n . (57) if α ̸= 0, then we can rewrite qn = γ + α γ −β  1 −γ(β/α + n + 1)γ(γ/α + 1) γ(γ/α + n + 1)γ(β/α + 1)  . (58) the case γ = β can be obtained from (55) or as the limit of expression (58) when γ →β. both ways yield qn = n x j=1 α + γ αj + γ . (59) 48

0911.1721

phase diagram of the lattice su(2) higgs model

we perform a detailed study of the phase diagram of the lattice higgs su(2) model with fixed higgs field length. consistently with previsions based on the fradkin shenker theorem we find a first order transition line with an endpoint whose position we determined. the diagram also shows cross-over lines: the cross-over corresponding to the pure su(2) bulk is also present at nonzero coupling with the higgs field and merges with the one that continues the line of first order transition beyond the critical endpoint. at high temperature the first order line becomes a crossover, whose position moves by varying the temperature.

introduction in this work we present a detailed study of the phase diagram of the lattice higgs su(2) model with higgs field in the fundamental representation and of fixed length1. the model in which higgs length is allowed to change has received quite a lot of attention in the past for its possible phenomenological implications (see e.g. [2]), so that its phase diagram is known with good precision. no systematic study exists instead for the case in which the higgs length is frozen. however this last model is often used, mainly because of its computational simplicity in numerical simulations, as the prototype of a non-abelian gauge theory coupled with matter in the fundamental representation. in particular some work has been done to study confinement in this model ([3] and [4]). in those works some properties of the phase diagram are usually taken for granted, like the existence of a line of first order transitions for β ≳2.3, but they have never really been tested in simulations. as a first step towards a complete understanding of this model, we thus started to systematically investigate its phase diagram, in order to obtain precise estimates of the location of its critical points. the action of lattice higgs su(2) model we adopt is s = β x x,μ<ν  1 −1 2retrpμν(x)  −κ 2 x x,μ>0 tr[φ†(x)uμ(x + ˆ μ)φ(x + ˆ μ)] (1) where the first term is the standard wilson action and the higgs field φ (which transforms in the fundamental representation) is rewritten as an su(2) matrix (see e.g. [5], [6]). since the action is linear in the fields at each point, standard heatbath ([7], [8]) and overrelaxation ([9]) algorithms can be used for the monte carlo update. if the higgs length is allowed to change this is no longer true, since 1a first report of this study was presented at the lattice 2008 conference, [1]. 1 in this case a quartic term is needed, which has the form λ{ 1 2tr[φ†(x)φ(x)]−1}2 and destroys linearity. in the limit β →∞the theory reduces to an o(4) non-linear sigma model, which is known to have a mean field phase transition for κ ≈0.6 (see e.g. [10]); in [11] it was shown that this second order transition becomes a first order ` a la coleman-weinberg when the gauge field is introduced as a small perturbation (β large). the authors of [11], using the cluster expansion developed in [12], were also able to prove the existence of a region of parameter space near the axis β = 0 and for κ →∞where every local observable is analytic. this statement is often referred to as the fradkin-shenker (fs) theorem. it is important to notice that, in this context, an observable is defined as local when its support is contained in a compact set in the thermodynamic limit; observables not satisfying this requirement can have non-analytic behaviour (see e.g. [13, 14]). this two results suggest a phase diagram like that shown in fig. 1: the analyticity region is indicated by ar and is limited by the dotted line, the thick line is the line of first order transitions and the two dots are its second order endpoints. as long as we consider the model at zero temperature, the ǫ-expansion predicts the endpoints to be in the mean field universality class. s s β 0 ∞ κ ∞ ar figure 1: phase diagram of the higgs su(2) model as predicted in [11]. a similar phase diagram was observed in the works [2], [15] for the lattice higgs su(2) model with fourth-order scalar coupling λ ≤0.5, while in the model considered here λ →∞. because of the supposed triviality of the φ4 model in four dimensions, λ is expected to be a marginally irrelevant parameter and therefore the phase diagram not to change qualitatively for λ →∞(see e.g. [16]); however it was observed since long time that the first order transition gets weaker as λ is increased, so that the phase diagram of fig. 1 has not really been checked at large values of λ. after the seminal work ref. [17] on very small lattices, in ref. [18] the observation of a double peak structure was reported at β = 2.3 on a 164 lattice, which however was probably only a consequence of the poor statistics, since in a later study, ref. [19], no double peak was found at β = 2.3. there it was stated that "the system exhibits a transient behaviour up to l = 24 along which the order of the transition cannot be discerned". in this paper we produce the first clear evidence of the line of first order transition and we obtain an estimate of the endpoint position. 2 2 numerical results the obvious observables to look at for this system are • the gauge-higgs coupling, g = 1 2⟨tr[φ†(x)uμ(x + ˆ μ)φ(x + ˆ μ)]⟩ • the plaquette, p = 1 2⟨trpμν⟩ • the energy density e = 6βp + 4κg besides these natural ones, we monitored also the following observables: • the polyakov loop pl(⃗ x) = 1 2tr hqlt−1 t=0 u0(t, ⃗ x) i , pl = 1 v ⟨p ⃗ x pl(⃗ x)⟩ • the z2 monopoles density, m = 1 − 1 nc p c σc, where c stand for the elementary cube and σc = q pμν∈∂c sign trpμν the polyakov loop behaviour is used as an indicator of confinement. the study of the z2 monopoles is motivated by the similarity of the first order transition with the bulk transition of the su(2) pure gauge theory, which is driven by lattice artefacts such as the z2 monopoles. both these points will be discussed more accurately in the following. data were analyzed by using the optimized histogram method ([20]) and the statistical errors were estimated by using the moving block bootstrap method (see e.g. [21]). the presentation of the simulation results will be divided in several parts 1. we will show that at β = 2.5 there is no signal of a phase transition and data are consistent with a smooth cross-over 2. we will show that for β ≥2.775 the scaling is consistent with a first order transition and we will present evidence of a double peak structure 3. two independent estimates of the endpoint will be obtained 4. we will give hints that the above transitions are not related to confinement 5. we will investigate the relation between the line of first order transition, which becomes a smooth cross-over beyond the endpoint, and the pure su(2) bulk transition 6. finally we will present some exploratory results at t ̸= 0 all results in the first four parts have been obtained by fixing the value of β and by looking for transitions in κ. 2.1 cross-over region in fig. 2 the maxima of the susceptibilities of g, p and m are plotted for various lattice sizes and β = 2.5. for lattices up to l ≈18 the maxima of susceptibilities are well described by a function of the form a + bl4, so that they seem to scale linearly with volume as expected for a first order transition. however on larger lattices all susceptibilities saturate and no singularity seems to develop in the thermodynamical limit. this means that the system has a 3 10 20 30 40 l 0.5 1 g susceptibility 10 15 20 25 30 35 40 l 0.03 0.035 0.04 0.045 0.05 p susceptibility 5 10 15 20 25 30 35 40 45 l 1.4 1.5 1.6 1.7 1.8 m susceptibility figure 2: top: g susceptibility peak heights at β = 2.5 (the continuous line is a fit to a + bx4). center: p susceptibility peak heights at β = 2.5. bottom: m susceptibility peak heights at β = 2.5. clearly they all saturate at large l. 4 correlation length of order ≈10 lattice spacing, so that the increase of the susceptibilities with volume, that in previous studies was interpreted as due to a first order transition, is just a signal that the lattices were too small. to look for transitions, besides susceptibilities we also check the behaviour of the energy binder cumulant, which is defined as b = 1 −⟨e4⟩/(3⟨e2⟩2). it can be shown (see e.g. [22]) that near a transition b develops minima whose depth scales as b|min = 2 3 −1 12 e+ e− −e− e+ 2 +o(l−4) = 2 3 −1 3 ∆ ǫ 2 +o(∆3)+o(l−4) (2) where e± = limβ→β± c ⟨e⟩, ∆= e+ −e−and ǫ = 1 2(e+ + e−). in particular the thermodynamical limit of b|min is less than 2/3 if and only if a latent heat is present. the scaling of the energy binder cumulant minima at β = 2.5 is shown in fig. 3. also here two different behaviours are clearly visible: on small lattices there is scaling consistent with a first order transition. by increasing the volume b →2/3, indicating a smooth cross-over or a second order transition. 0 5e-05 0.0001 0.00015 0.0002 0.00025 v -1 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 2/3-b figure 3: binder cumulant minima of e at β = 2.5 (v = l4). these results indicate that at β = 2.5 there is no transition; this is in sharp contrast with all previous studies of this model, which concluded that for β ≥2.3 the system undergoes a first order transition. this wrong conclusion was based on the analysis of lattices of size up to 254, which we have just shown to be too small. 2.2 first order region at β = 2.775, β = 2.79 and β = 2.8 the scaling of the susceptibilities and of the energy binder cumulant remains consistent with first order also for the 5 0 2e+06 4e+06 v 0.6 0.8 1 1.2 g susceptibility 0 2e+06 4e+06 v 0.0055 0.006 0.0065 0.007 p susceptibility 0 2e+06 4e+06 v 0.11 0.12 0.13 0.14 m susceptibility 0 2e+06 4e+06 v 10 12 14 16 18 e susceptibility 0 1e-06 2e-06 3e-06 v -1 0 5e-08 1e-07 1.5e-07 2e-07 2.5e-07 2/3-b figure 4: scaling of the maxima of the susceptibilities and of the e binder cumulant minima for β = 2.775 (v = l4 and the larger v values correspond to lattice sizes l = 30, 35, 40, 45). 6 0.219 0.22 0.221 0 0.01 0.02 0.03 0.04 0.05 l=45 l=50 figure 5: histogram of the observable g for β = 2.8 on the two largest lattices. larger lattices, as shown for example in fig. 4 for β = 2.775. in this range of β values the transition gets stronger as β increases. however we know that the transition at β →∞has to become of second order, so that going at β high enough the transition has to get weaker. we could not reach this regime in our simulations since when increasing the β value it is also necessary to use larger lattices. to have results free of spurious finite size transitions the lattice must be large enough for the corresponding pure su(2) gauge theory to be confined and exponentially large lattices in β are needed. although all observables scale consistently with a first order transition, a clear signal of metastability was revealed only at β = 2.8, where the transition is stronger, and only in the two largest lattices, namely l = 45 and l = 50; the histograms for the observable g on these two lattices are shown in fig. 5, where the formation of a double peak structure is visible. 2.3 endpoint having three sets of data with first order scaling and knowing the universality class of the endpoint, we can estimate its position βc assuming that we are close enough to it. we can do this in two independent ways: 1. from the second form of equation 2 we know the dependence of b on the discontinuity ∆and for a mean-field transition we have ∆∝t1/2, where t is the reduced temperature, t = (t −tc)/tc. in our case, near the critical 7 2.725 2.75 2.775 2.8 β -5e-09 0 5e-09 1e-08 1.5e-08 2e-08 2/3 - b 2.72 2.74 2.76 2.78 2.8 β 0 1e-06 2e-06 3e-06 ∆ 2 figure 6: top: determination of the critical point using the method (1). bot- tom: determination of the critical point using the method (2). 8 point, we can use t ∝(β −βc) and therefore equation 2 can be written as2 lim l→∞bl|min = 2 3 −γ1(β −βc) + o(β −βc)2 (3) 2. for a first order transition the susceptibilities scale for large volume as χ = v (⟨o2⟩−⟨o⟩2) ≈const + v ∆2 (4) so that fitting the χ maxima to the relation χ = const + v ∆2 and using again ∆∝t1/2, we obtain ∆2 = γ2(β −βc) + o(β −βc)2 (5) the fit in the planes (β, b) and (β, ∆2) are shown in fig. 6 (we use the g susceptibility); in both cases the fit is very good and the estimates for the critical point position are β(1) c = 2.7266(16) β(2) c = 2.7299(36) (6) which agree with each other within errors. 2.4 relation to confinement the next question is to understand if the two "phases" found at large β have different confinement properties. we recall that they cannot be considered as different thermodynamical phases as a consequence of fs theorem. in this model wilson loops never obey the area law because of the presence of the higgs field, which destroys the center symmetry of the pure gauge theory; as a consequence the polyakov loop is always non-zero (a direct check of this is shown in fig.8) and cannot be an order parameter. nevertheless polyakov loop is commonly used as a confinement tracker also in theories where it is not an order parameter, e.g. in qcd, since it abruptly jumps at the deconfinement transition. for the lattice su(2) higgs model the polyakov loop doesn't seem to be influenced in any way by the transition: for small lattices it slightly increases, however for larger ones, the transition gets stronger but pl gets flatter in the transition region, as can be seen in fig. 8 (the transition is at κ = 0.704675(30)). also polyakov loop correlators, measured by using the multilevel algorithm of ref. [24], do not show any significant change across the transition, as shown in the bottom of fig. 8. an alternative possibility is to use the order parameter introduced in [25], which we will denote by of m. this is defined by the limit for r →∞of the quantity of m(r), which is constructed as the square of the mean value of a staple-shaped parallel transport connecting two higgs fields divided by a wilson loop; the size of the staple is related to that of the wilson loop as shown in fig. 7, where higgs fields are represented by dots. to our knowledge of m 2a subtlety has to be considered here: since in this model the two relevant operators at the mean field critical point are not related to any symmetry of the system, there is in general mixing between the magnetic and thermal relevant operators (see e.g. [23]). however, since the ∆'s are measured along the coexistence curve, near the critical endpoint the mixing is negligible and the relation between ∆and t is as usual ∆∝tβ (here "β" is the β critical exponent). 9 s s ⟨ ⟩ 2 r 6 ? 2r -  2r 6 ? 2r -  ⟨ ⟩ of m(r) = figure 7: of m(r) definition. has never been measured in a monte carlo simulation, so we have no previous results to compare with; its strong coupling limit was computed in [26]. of m(r) measures the overlap between a higgs-higgs dipole of linear dimension r and the vacuum; if asymptotic colored states exist they are orthogonal to the vacuum, so of m(r) →0 for r →∞, otherwise we must have of m(r) →α > 0. the results obtained by measuring of m(r) on a 404 lattice at β = 2.775 for κ slightly above and slightly below the transition points are shown in fig. 9, where the two lines are fits to the function3 f(x) = a + b 1 xc (symmetrized because of periodic boundary condition). we have for the parameter a, c the estimates: a(0.7046) = 4.9(1) * 10−5 c(0.7046) = 2.98(5) (7) a(0.7048) = 10.5(5) * 10−5 c(0.7048) = 3.30(23) (8) so on both sides of the transition charge is screened and according to the inter- pretation of [25] there is color confinement. 2.5 the pure gauge su(2) bulk another interesting point to study is what relation (if any) the first order has with the bulk transition of the su(2) pure gauge (we call this "transition" for the sake of simplicity, although it is only a rapid but smooth cross-over). an hint on the existence of such a relation can be obtained by noting that the value β ≈2.3 previously thought to be the first order line endpoint position is the value of the bulk su(2) transition. since some test simulations indicated that the position of the su(2) bulk is very stable for small values of κ (so that in the plane (β, κ) this crossover follows a line perpendicular to the β axis), it seems more convenient to hold κ fixed and vary the β value; the results obtained on a 304 lattice for the plaquette susceptibilities for several κ values are shown in fig. 10. from these data a structure of cross-over lines emerges as shown in fig. 11. for κ < 0.6 the bump in β of the plaquette susceptibility which corresponds to the su(2) bulk transition is independent on κ both in position and in shape (vertical dotted line in fig. 11). for κ larger than 0.6 two peaks in β appear, the bulk one and the first order transition studied above. the latter peak persists also when κ is greater than the critical value corresponding to the endpoint 3this ansatz is motivated by the observation that for r big enough such that the wilson loop scales with perimeter, the exponentials in numerator and denominator are the same, leaving a power law. 10 25 30 35 40 45 l 0 0.0005 0.001 0.0015 pl 0.7042 0.7044 0.7046 0.7048 0.705 0.7052 0.7054 κ 0.0001 0.001 pl l=25 l=30 l=35 l=40 l=45 0 5 10 15 20 y 1e-09 1e-08 <pl(x)pl(x+y)> κ=0.70460 κ=0.70480 figure 8: top: polyakov loop for β = 2.775, κ = 0.70464 and various l, the line is a fit to a + b exp(−cl) and the result for the asymptotic value is a = 1.7(1)*10−4. center): polyakov loop for β = 2.775 and various l. bottom: polyakov loop correlators measured on a 404 lattice for β = 2.775 and κ slightly below (0.7046) and above (0.7048) the critical value. 11 0 2 4 6 8 10 r 0.0001 0.001 0.01 0.1 ofm κ=0.7046 κ=0.7048 figure 9: results for of m(r) at β = 2.775 on a 404 lattices slightly above and below the transition point. andthere is no real transition, although it is smaller that the su(2) bulk signal (see fig. 10, κ = 0.75). by increasing the κ value the cross-over remnant of the first order transition moves towards smaller β values, until it intersects the su(2) bulk in the point indicated by a in fig. 11. in the neighborhood of the point a the "first order continuation" peak gets stronger and the su(2) bulk disappears. this increase could have been misinterpreted as the first order transition line endpoint in previous works. a direct check to ensure that in this region there is no transition is shown in fig. 12. for still larger κ values only one maximum is present in susceptibilities, which gets weaker as κ →∞. this interplay between the first order transition line and the bulk su(2) suggests that the first order transition could be in some way related to the same lattice artifacts that drive the bulk su(2) transition, namely the z2 monopoles. the density of z2 monopoles seems indeed to have a jump across the transition (see fig. 4). 2.6 finite temperature motivated by the last observation we try to investigate if the first order transi- tion line can itself be thought as a lattice artefact, like the pure gauge so(3) first order bulk transition. to answer this question the simplest method is to consider the system at finite temperature: bulk transitions are insensitive to the temporal extension lt of the lattice, while physical transitions scale by varying lt. for each lt value we simulate the system with several spatial extents ls, in 12 1.8 2 2.2 2.4 2.6 β 0.02 0.04 0.06 0.08 0.1 0.12 0.14 χ(β,κ) κ=0.75 κ=0.775 κ=0.8 κ=0.85 κ=0.9 κ=0.95 1.8 2 2.2 2.4 2.6 β -0.02 0 0.02 0.04 0.06 0.08 0.1 χ(β,κ)−χ(β,0) κ=0.75 κ=0.775 κ=0.8 κ=0.85 κ=0.9 κ=0.95 figure 10: up: plaquette susceptibility. down: plaquette susceptibility with subtracted the value at κ = 0. 13 s s β 0 ∞ κ ∞ a κ ≈0.6 figure 11: phase diagram of the higgs su(2) with rapid crossovers indicated by dotted lines. order to perform a finite size analysis. some of the results obtained are shown in fig. 13. it is clear that the position of the maxima of the g susceptibility moves by varying lt, so that the line of first order transition seen at zero temperature cannot be a lattice artifact. moreover for all the values of lt considered the maxima saturate by increasing ls, indicating the absence of a phase transition. we expect that for sufficiently small temperature (i.e. large enough lt) the first order transition is restored, however to observe this behaviour numerically we should use lattices with lt > 20 and ls ≫lt, typically ls ≳3lt. instead we have to restrict ourselves to l3 slt ≲4 × 106 because of computer capability. in any case we see the cross-over gets stronger by increasing the lattice temporal extension. 3 conclusions this paper is a study of the phase diagram of the su(2) higgs gauge theory with the higgs in the fundamental representation and fixed length. we find that to investigate the phase structure of this system it is necessary to use much larger lattices than the ones adopted in the past due to the large correlation length. we present the first clear evidence of the presence of a line of first order transition and we estimate its endpoint position to be at βc = 2.7266(16), thus confirming an expectation based on the fradkin-shenker theorem. we give indications that this transition is not a deconfinement transition. we attempt an exploration of the system at finite temperature; the first order transition becames then a cross-over whose position moves by varying the temperature. 4 acknowledgments simulation were performed using the italian grid infrastructure and one of us (c. b.) wishes to thank tommaso boccali for his technical assistance in its use. we thank a. d'alessandro for substantial help in the early stages of this work. 14 2.15 2.16 2.17 2.18 2.19 2.2 β 0.06 0.08 0.1 0.12 0.14 0.16 0.18 p susceptibility l=30 l=40 figure 12: plaquette susceptibility for κ = 0.85. a data in this appendix we give some details of our numerical results. the notation is as follows • κpc is the location in κ of the g susceptibility maximum • χo is the peak value of the susceptibility of the o observable • b is the value at minimum of the energy binder cumulant β = 2.725 l κpc χg χp χm χe 2/3-b 20 0.7090(2) 0.615(18) 0.005575(88) 0.1147(19) 10.85(30) 6.36(18)e-07 25 0.7089(2) 0.692(16) 0.005831(71) 0.1194(15) 11.89(26) 2.839(62)e-07 30 0.7088(1) 0.858(20) 0.006485(78) 0.1319(15) 14.51(32) 1.679(38)e-07 35 0.70873(5) 1.018(24) 0.007082(89) 0.1431(17) 16.64(37) 1.039(23)e-07 40 0.70866(5) 1.022(24) 0.00704(12) 0.1426(18) 16.85(37) 6.17(13)e-08 45 0.70886(5) 0.994(31) 0.00725(23) 0.1469(25) 16.25(47) 3.71(10)e-08 15 0.704 0.706 0.708 0.71 0.712 κ 0.3 0.4 0.5 0.6 0.7 g susceptibility ls=30 lt=20 ls=40 lt=20 ls=50 lt=20 ls=30 lt=16 ls=40 lt=16 ls=30 lt=12 ls=40 lt=12 figure 13: g susceptibility for β = 2.775; the vertical line indicates the place where on symmetric lattices the transition is observed. β = 2.775 l κpc χg χp χm χe 2/3-b 25 0.7049(1) 0.5740(65) 0.005459(31) 0.10807(62) 9.460(98) 2.156(22)e-07 27 0.70480(8) 0.6211(82) 0.005528(40) 0.10884(74) 10.16(12) 1.702(21)e-07 30 0.70476(5) 0.734(13) 0.005769(52) 0.1135(10) 11.82(19) 1.299(21)e-07 35 0.70466(5) 0.812(16) 0.006044(64) 0.1182(12) 12.88(24) 7.64(14)e-08 40 0.70469(3) 0.956(17) 0.006573(77) 0.1285(12) 14.66(25) 5.099(88)e-08 45 0.704675(30) 1.114(24) 0.00698(10) 0.1359(16) 16.72(34) 3.631(75)e-08 β = 2.79 l κpc χg χp χm χe 2/3-b 25 0.70380(10) 0.5512(97) 0.005301(48) 0.10362(93) 9.05(14) 2.030(32)e-07 30 0.70360(5) 0.646(12) 0.005481(49) 0.10632(95) 10.39(17) 1.125(19)e-07 40 0.70356(3) 0.912(19) 0.006187(78) 0.1200(13) 14.01(27) 4.799(93)e-08 45 0.703525(30) 1.140(23) 0.006901(97) 0.1318(15) 17.01(33) 3.639(70)e-08 β = 2.8 l κpc χg χp χm χe 2/3-b 25 0.7032(2) 0.683(22) 0.005080(80) 0.0978(15) 11.25(34) 2.500(75)e-07 30 0.70298(8) 0.7212(85) 0.005279(38) 0.10181(60) 11.63(13) 1.246(13)e-07 35 0.70286(5) 0.760(16) 0.005685(62) 0.1093(11) 11.64(22) 6.74(13)e-08 40 0.70282(4) 0.946(22) 0.006239(75) 0.1192(13) 14.36(31) 4.87(11)e-08 45 0.70279(3) 1.219(27) 0.007033(89) 0.1339(16) 17.75(37) 3.759(78)e-08 16 references [1] c. bonati, g. cossu, a. d'alessandro, m. d'elia, a. di giacomo on the phase diagram of the higgs su(2) model. pos(lattice2008)252 (arxiv:0901.4429). [2] z. fodor, j. hein, k. jansen, a. jaster, i. montvay simulating the elec- troweak phase transition in the su(2) higgs model. nucl. phys. b 439, 147 (1995) (arxiv:hep-lat/9409017). [3] j. greensite, ˇ s. olejn ́ ık vortices, symmetry breaking, and temporary con- finement in su(2) gauge-higgs theory. phys. rev. d 74, 014502 (2006) (arxiv:hep-lat/0603024). [4] w. caudy, j. greensite on the ambiguity of spontaneously broken gauge symmetry, phys. rev. d 78, 025018 (2008) (arxiv:0712.0999 [hep-lat]). [5] i. montvay correlations and static energies in the standard higgs model. nucl. phys. b 269, 170 (1986). [6] i. montvay, g. m ̈ unster quantum fields on a lattice. cambridge university press (1994). [7] m. creutz monte carlo study of quantized su(2) gauge theory. phys. rev. d 21, 2308 (1980). [8] a. d. kennedy, b. j. pendleton improved heatbath method for monte carlo calculations in lattice gauge theories. phys. lett. b 156, 393 (1985). [9] m. creutz overrelaxation and monte carlo simulation. phys. rev. d 36, 515 (1987). [10] a. hasenfratz, k. jansen, j. jers ́ ak, b.b. lang, t. neuhaus, h. yoneyama study of the four component φ4 model. nucl. phys. b 317, 81 (1987). [11] e. fradkin, s. h. shenker phase diagrams of lattice gauge theories with higgs fields. phys. rev. d 19, 3682 (1979). [12] k. osterwalder, e. seiler gauge field theories on a lattice. ann. phys. 110, 440 (1978). [13] r. bertle, m. faber, j. greensite, ˇ s. olejn ́ ık center dominance in su(2) gauge-higgs theory. phys. rev. d 69, 014007 (2004) (arxiv:hep-lat/0310057). [14] m. grady reconsidering gauge-higgs continuity. phys. lett. b 626, 161 (2005) (arxiv:hep-lat/0507037). [15] w. bock, h. g. evertz, j. jers ́ ak, d. p. landau, t. neuhaus, j. l. xu search for critical points in the su(2) higgs model. phys. rev. d 41, 2573 (1990). [16] d. j. e. callaway triviality pursuit: can elementary scalar particles exist? phys. rep. 167, 241 (1988). 17 [17] c.b. lang, c. rebbi, m.virasoro the phase structure of a non-abelian gauge higgs field system. phys. lett. b 104, 294 (1981). [18] w. langguth, i. montvay two-state signal at the deconfinement-higgs phase transition in the standard su(2) higgs model. phys. lett. b 165, 135 (1985). [19] i. campos on the su(2)-higgs phase transition. nucl. phys. b 514, 336 (1998) (arxiv:hep-lat/9706020). [20] a. m. ferrenberg, r. h. swendsen optimized monte carlo data analysis. phys. rev. lett. 63, 1195 (1989). [21] s. mignani, r. rosa the moving block bootstrap to asses the accuracy of statistical estimates in ising model simulations. comp. phys. comm. 92, 203 (1995). [22] j. lee, j. m. kosterlitz finite-size scaling and monte carlo simulations of first-order phase transitions. phys. rev. b 43, 3265 (1991). [23] n. b. wilding simulation studies of fluid critical behaviour. j. phys.: con- dens. matter 9, 585 (1997) (arxiv:cond-mat/9610133). [24] m. luscher, p. weisz locality and exponential error reduction in numerical lattice gauge theory. jhep 09, 010 (2001) (arxiv:hep-lat/0108014). [25] k. fredenhagen, m. marcu confinement criterion for qcd with dynam- ical quarks. phys. rev. lett. 56, 223 (1986). [26] k. holland, p. minkowki, m. pepe, u.-j. wiese exceptional con- finement in g(2) gauge theory. nucl. phys. b 668, 207 (2003) (arxiv:hep-lat/0302023). 18

0911.1722

universality of brunnian ($n$-body borromean) four and five-body systems

we compute binding energies and root mean square radii for weakly bound systems of $n=4$ and $5$ identical bosons. ground and first excited states of an $n$-body system appear below the threshold for binding the system with $n-1$ particles. their root mean square radii approach constants in the limit of weak binding. their probability distributions are on average located in non-classical regions of space which result in universal structures. radii decrease with increasing particle number. the ground states for more than five particles are probably non-universal whereas excited states may be universal.

introduction. the efimov effect was predicted in [1] as the appearance of a series of intimately related excited three-body states when at least two scattering lengths are infinitely large. these states can appear at all length scales and their properties are independent of the de- tails of the potentials. this effect has in recent years been studied intensively and extended to a wider group of physical phenomena, beginning to be known as efimov physics. in general we define efimov physics as quantum physics where universality and scale invariance apply. universality means independence of the shape of the in- terparticle potential. scale invariance means indepen- dence of the length scale of the system. these conditions are rather restrictive but a number of systems are known to exist within this window [2–7]. the great advantage is that one theory is sufficient to explain properties without any detailed knowledge of the interactions [8]. further- more, properties in different subfields of physics are de- scribed as manifestations of the same underlying theory. our physical definition of scale invariance originates from the halo physics first realized and discussed in nu- clei [9], but quickly observed as applicable also to small molecules like the helium trimers [5]. this original defi- nition of scale invariance, that the concept applies to any length scale is obviously continuous as exemplified by nu- clei, atoms and molecules. often the notion of scale in- variance is used in a different mathematical sense where the spatial extension of the structures in one given sys- tem repeats itself in discrete steps like the factor 22.7 for identical particles [1]. this is a result of the independence of potential details and here precisely defining our mean- ing with the notion of universality. as the concepts can be defined in different ways we will use throughout the paper this original physical meaning of scale invariance. together, these two concepts constitute our meaning of efimov physics which to the best of our knowledge has been left undefined in all previous publications. the range of validity for such a global theory is only well described for two and three particles [2, 5, 10]. for n = 4 two states were found in the zero-range, inherently universal, effective field model [11]. these states also ap- peared as universal in finite-range models in connection with each efimov state [12]. this is in contrast to [13] where the disentanglement of the scales used to regularize the three and four-body zero-range faddeev-yakubovsky equations gives rise to a dependence of the four-body ground state on interaction details. then a four-body scale is needed in analogy to the three-body scale ap- pearing independently on top of the two-body proper- ties. this apparent discrepancy between refs. [11, 12] and [13] is not yet resolved. recently, three experiments evidenced two four-body bound states connected to an efimov trimer [14–16] in accordance with the theoretical predictions of ref. [12]. in two of these experiments were also observed deviations from universality [15, 16]. surprisingly, the greatest devi- ation were observed for large scattering lengths (→±∞) - exactly at the region where universality should apply [16]. this requires a theoretical explanation where some- thing should be added in the universal model. very little is known for five particles with complete so- lutions containing all correlations as dictated by the in- teraction. with specific assumptions about only s-waves and essentially no correlations it was concluded in [5, 17] that ground state halos cannot exist for n > 3. these as- sumptions are rather extreme and could be wrong or only partly correct. however, if halos exist they have universal structures as the n = 4 states obtained in [11, 12]. these results can only be reconciled by wrong assumptions in the halo discussion or by impermissible comparison be- tween halo ground states and excited states. it was concluded in [18] that efimov states do not exist for n > 3 and furthermore for three particles exist only for dimensions between 2.3 and 3.8 [19]. however, by 2 restricting to two-body correlations within the n-body system, a series of (highly) excited n-body states were found with the characteristic efimov scaling of energies and radii [20, 21]. whether they maintain their identity and the universal character, when more correlations are allowed in the solutions, remains to be seen. two limits to the universality are apparent. the first appears for large binding energy where the resulting small radii locate the system within the range of the po- tentials and sensitivity to details must appear. the less strict second limit is for excitation energies above the threshold for binding subsystems with fewer particles. structures with such energies are necessarily continuum states which may, or more often may not, be classified as universal states depending on their structures and the final states reached after the decay. even for four particles where universality is found [11, 12], a number of questions are still unanswered. for five and more particles the information becomes very scarce. a novel study claiming universality for ground states of a van der waals potential has appeared for par- ticle number less than 40 [22]. the critical mass is found as a substitute for the critical strength, but the computed radii at threshold cannot be reliably extracted. the purpose of the present paper is to explore the win- dow for efimov physics. we shall investigate the bound- aries for universality preferentially leading to general con- clusions applicable to systems of n-particles. we first discuss qualitative features and basic properties, then ex- tract numerical results for 4 and 5 particles very close to thresholds of binding, and relate to classically allowed regions. we only investigate brunnian (n-body bor- romean) systems [23] where no subsystem is bound. qualitative considerations. for two particles the in- finite scattering length corresponds to a bound state at zero energy. variation of 1/a around zero produces ei- ther a bound state of spatial extension a or a continuum state corresponding to spatial configurations correlated over the radius a. for three particles the efimov effect appears, i.e. for the same interaction, a = ±∞(1/a = 0), infinitely many three-body bound states emerge with progressively smaller binding and correspondingly larger radii [1]. the ratios of two and three-body threshold strengths for sev- eral potentials were derived in [22, 24, 25]. these thresh- olds for binding one state can be characterized by a value of 1/a [10, 12]. infinitely many bound three-body states appear one by one as 1/a is changed from the three- body threshold for binding to the threshold for two- body binding 1/a = 0. moving opposite by decreas- ing the attraction these states one by one cease to be bound. they move into the continuum and continue as resonances [26]. for asymmetric systems with a bound two-body subsystem the three-body bound state passes through the particle-dimer threshold becoming a virtual state [27]. this behavior holds even for particles with different masses [28]. all three-body s-wave states from a certain energy and up are universal. however, this is not an a priori obvious conclusion but nevertheless true because two effects work together, i.e., for 1/a = 0 the system is large for the ex- cited efimov states and for finite 1/a the binding is weak and the radius diverges with binding [5]. both efimov states and weakly bound states are much larger than the range of the interaction. the continuous connection of these bound states and resonances is therefore also in the universal region. the recent results for four particles were that each three-body state has two four-body states attached with larger binding energy [11, 12]. these four-body states are both described as having universal features unam- biguously related to the corresponding three-body states for interactions of both positive and negative scattering lengths. detailed information of structure, correlations, and posssible limits to universality are not available. the one-to-one correspondence between the two four- body states and one three-body state can perhaps be extended such that two weakly bound n-body states ap- pear below the ground state of the (n −1)-body state. this seems to be rather systematic for n-body efimov states obtained with only two-body correlations [20, 21]. if these n-body efimov states remain after extension of the hilbert space to allow all correlations, we can ex- pect these sequences to be continued to the thresholds for binding by decreasing the attraction. however, ground and lowest excited states may be outside the universal region but the sequences may still exist. in any case the scaling properties are different for the n-body efimov states in [20, 21] and the universal four-body states in [11, 12]. the basic reason for the difficulties in finding detailed and general answers is related to the fact that the thresh- olds for binding are moving monotonously towards less attraction with n [24, 25]. for n = 2 weak binding and large scattering length is synonymous. already for n = 3 this connection is broken but the weak binding still causes the size to diverge [5]. the indications are that for n > 3 the size remains finite even in the limit of zero binding. basic properties. we consider a system of n identical bosons each of mass m. they are confined by a harmonic trap of frequency ωt corresponding to a length parameter b2 t = ħ/(mωt). the particles interact pairwise through a potential v of short range b ≪bt. we shall use the gaus- sian shape v = v0 exp(−r2/b2). the chosen values of v0, b, and m lead to a two-body scattering length a and an effective range re. the solution to the schr ̈ odinger equation is approximately found by the stochastic vari- ational method [29]. the results are energies and root mean square radii. for two-body systems we know that the n′th radial moment only diverge at threshold of binding when the 3 angular momentum l ≤(n + 1)/2, see [5]. the equality sign implies a logarithmic divergence with binding b2 in contrast to the normal power law bl−n/2−1/2 2 . for the mean square radius this implies divergence for l ≤3/2. for an n-body system with all contributions entirely from s-waves we can generalize these rigorous results from two-body systems [5]. the number of degrees of freedom is f = 3(n −1) and the generalized centrifugal barrier is obtained with an effective angular momentum l∗= (f −3)/2. divergent root mean square radius is then expected when l∗≤3/2 or equivalently when f ≤6 or n ≤3. if this result holds, four-body systems should have finite root mean square radii even at the threshold of binding. the size of the system is measured by the square root of the mean square radius, ⟨r2/b2 ⟩, which is expressed in units of the "natural" size of the systems, i.e. the range of the binding potential. the dimensionless unit of the binding energy bn of the system is ̄ b = mb2|bn|/ħ2. both universality and scale invariance is therefore de- tected by inspection of these quantities as functions of parameters and shape of the potentials. in regions where the curves are proportional, we conclude that the prop- erties are universal and scale invariant. results for dif- ferent potential shapes can be expressed in terms of a standard potential by scaling the range. then the in- dividual curves would fall on top of each other in the universal regime. clasical allowed region. universal properties can in- tuitively only appear when the structures are outside the potentials because otherwise any small modification would have an effect on the wavefunction. consequently the property would be dependent on these details in con- flict with the assumption of universality. for two-body systems the relative wavefunction is therefore universal only if the largest probability is found outside the poten- tial. this means that this classically forbidden region is occupied. the system is extremely quantum mechanical and very far from obeying the laws of classical physics. to investigate the relation between universality and the classical forbidden regions for n particles we need to compare features of universality with occupation of classical forbidden regions. for two-body systems this is straightforward since the coordinate of the wavefunction and the potential is the same. the probability of finding the system where the energy is smaller than the potential energy is then easy to compute as a simple spatial integral over absolute square of the wavefunction. for more than two particles the problem is well de- fined but the classically forbidden regions (total energy is smaller than the potential energy) themselves are diffi- cult to locate. we attempt a crude estimate which at best can only be valid on average. the energy is computed by adding kinetic and potential energy, i.e. −bn = n⟨t1⟩+ 1 2n(n −1)⟨v12⟩, (1) where we choose an arbitrary particle 1 to get the kinetic part and a set of particles 1 and 2 to get the potential energy. the classical region is defined by having positive kinetic energy. for a two-body gaussian potential we then obtain an estimate of an average, rcl, for the classical radius from ⟨v12⟩> − 2bn n(n −1) = v0 exp(−r2 cl/b2) . (2) if the distance between two particles is larger than rcl we should be in the universal region. this value can then be compared to the size obtained from the average distance between two particles, ⟨r2 12⟩, computed in the n-body system from the mean square radius [20], i.e. ⟨r2 b2 ⟩= n −1 2n ⟨r2 12 b2 ⟩. (3) thus in the classical forbidden region r2 cl/b2 from eq.(2) should be smaller than ⟨r2 12/b2⟩from eq.(3). the four-body system. we show size versus binding energy for n = 4 in fig.1. the variation arises by change of the strength, v0, of the attractive gaussian. the sys- tem is for numerical convenience confined by an external one-body field. however, we are only interested in struc- tures independent of that field, i.e. intrinsic properties of the four-body system. we therefore increase the trap size until the states are converged and located at dis- tances much smaller than the confining walls. we now know that this happens for four particles in contrast to the three-body system where the size diverges when the binding energy approaches zero. in fig.1 we show results for two trap sizes deviating by an order of magnitude and larger than the interaction range b = 11.65a0 (a0 is the bohr radius) by a factor of 20 and 200, respectively. for large binding in the lower right corner the results for the ground state is independent of trap size. when the probability extends by about a factor of 2 further out than b the effect of the small trap can be seen. the tail of the distribution then extends out to 20b even though the mean square is 10 times smaller. in the limit of very small binding energy the radius approaches a constant independent of the binding. the trap size has to be increased to 200b before the trap has no influence which implies that the probability distribu- tion is entirely within that distance when the threshold for zero binding is reached. the converged size is about 2.7b for the ground state. somewhat surprisingly also the first excited state, which also is below the energy of the three-body state, has converged to a value, 7.6b, independent of the trap size. a shape different from a gaussian would again lead to constants related through specific properties of the potentials, but the ratio would remain unchanged. this is precisely as found in two di- mensions for three particles [30]. both states are at the threshold on average very much smaller than both traps. 4 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 1 10 100 4< 4 * < <r 2 / b 2 > b 4 * > 5 * 4> 5 fig. 1: (color online) the mean square radius as function of the four and five-body binding energies, all in dimensionless units. the trap sizes for four particles are bt = 230.942a0 (red lines with 4<), 2630.956a0 (black lines with 4>), and bt = 372.073a0 (blue lines with 5) for five particles. here a0 is the bohr radius. we show ground (solid) and excited states (long-dashed with particle numbers tagged with an *), and "classical" two-body radius (dotted (4) and dot-dashed (5)) translated by eq.(3). nevertheless the smallest trap would still influence the tail of the distribution. in fig.1 we also show the estimated classical average distance between pairs of particles within the n-body system. this curve is above the ground state radius for large binding. here the probability is mostly found in the classical region within the potential, i.e. in the non- universal region. another potential shape would then move these curves. the classical and root mean square radius cross each other when the size is slightly larger than the range b. this limit for universality is similar to the halo condition for universality established in [5, 17]. at smaller binding energy the classical radius becomes less than the size of the system and the probability is on average located outside the potential in the non-classical, universal region. for the extremely small binding energies close to the threshold our estimate of the classical radius diverges log- arithmically with binding energy. thus at some point it has to exceed the size of the system which we concluded converge to a finite value for zero binding. this is sim- ply due to the character of the gaussian potential which approaches zero for large radii. zero energy must then be matched by an infinite radius. however, this gaus- sian tail is too small to obstruct the convergence of the probability distribution to a finite size. this cannot de- stroy universality because the tail has no influence on the wavefunction in this region far outside the range of the potential. for universality only the binding energy is decisive as one can see explicitly for the two-body sys- tem. for n-body systems the same result follows from the asymptotic large distance behavior of the wavefunc- tion expressed in hyperspherical coordinates [31]. thus the classical average radius argument fails for these ex- treme energies when the probability has settled outside the range of the short-range potential. five-body system. in fig.1 we also show results for n = 5 where convergence is reached for the trap size of bt = 372a0. we found two pentamers with energies -0.0281 ħ2 mb2 and -0.0113 ħ2 mb2 below the four-body thresh- old (-0.0103 ħ2 mb2 ) for infinite scattering length. the sizes for both ground and excited states increase again with decreasing binding energy b5 and approach finite values when b5 = 0. these limiting radii of about 1.94b and 4.0b are substantially smaller than corresponding values for four particles. still the largest probability is found out- side the potential providing the binding. this strongly indicates that also these structures are in the universal region. again their ratio is anticipated to be essentially independent of potential shape. this conclusion is sup- ported by the comparison in fig.1 to the classical ra- dius which always is smaller than the radius of the ex- cited state and comparable to the radius of the ground state. as argued for four particles the largest binding for the ground state corresponds to non-universal structure. when the binding energy is about 0.3 in the dimension- less units on the figure the universal structure appears. this happens at about the same energy as for four parti- cles. in both cases the probability is pushed outside the potential and universality is expected for smaller binding energies. conclusions. we have investigated the behavior of brunnian systems near threshold for binding. ground and first excited state for four and five identical bosons appear below the threshold for binding three and four particles, respectively. their radii are for small binding energies larger than the range of the potential holding them together. the largest part of the probability is found in non-classical regions resulting in universal struc- tures. for six and more particles the ground states would be located inside the potential and thus of non-universal structures. excited states are larger and may still be universal but already for seven or eight particles also the first excited state is expected to be non-universal. the numerical results are obtained for a two-body gaussian potential but the features originating from wavefunctions in non-classical regions of space are expected to be inde- pendent of the potential shape. acknowledgement. we are greatful for helpful discus- sions with m. thøgersen. mty thanks fapesp and cnpq for partial support. 5 [1] v. efimov, phys.lett. b 33, 563 (1970); nature physics 5, 533 (2009). [2] k. riisager, a.s. jensen and p. møller, nucl. phys. a548, 393 (1992). [3] e. nielsen, d.v. fedorov and a.s. jensen, j. phys. b31, 4085 (1998). [4] d.v. fedorov, a.s. jensen and k. riisager, phys. rev. c49, 201 (1994); phys. rev. c50, 2372 (1994). [5] a.s. jensen, k. riisager, d.v. fedorov and e. garrido, rev. mod. phys. 76 215 (2004). [6] m.t. yamashita, t. frederico, l.tomio and a. delfino, phys. rev. a 68, 033406 (2003). [7] m.t. yamashita, r.s. marques de carvalho, l. tomio and t. frederico, phys. rev. a68, 012506 (2003). [8] a.e.a. amorim, t. frederico, l. tomio, phys.rev. c 56 r2378 (1997). [9] p.g. hansen and b. jonson, europhys. lett. 4, 409 (1987). [10] e. braaten and h.-w. hammer, phys. rep. 428, 259 (2006). [11] h.w. hammer and l. platter, eur. phys. j. a32, 113 (2007). [12] j. von stecher, p. d'incao, c.h. greene, nature phys. 5, 417 (2009). [13] m.t. yamashita, l. tomio, a. delfino, and t. frederico, europhys. lett. 75, 555 (2006). [14] f. ferlaino et al., phys. rev. lett. 102, 140401 (2009). [15] m. zaccanti et al., nature phys. 5, 586 (2009). [16] s. e. pollack, d. dries and r. g. hulet, science 326, 1683 (2009). [17] k. riisager, d.v. fedorov and a.s. jensen europhys. lett. 49, 547 (2000). [18] r.d. amado and j. v. noble, phys. lett. b35, 25 (1971); r.d. amado and f.c. greenwood phys. rev. d7, 2517 (1973). [19] e. nielsen, d.v. fedorov, a.s. jensen and e. garrido, phys. rep. 347, 373 (2001). [20] o. sørensen, d.v. fedorov and a.s. jensen, phys. rev. lett. 89, 173002 (2002). [21] m. thøgersen, d.v.fedorov, and a.s. jensen, europhys. lett. 83 30012 (2008). [22] g.j. hanna and d. blume, phys. rev. a74, 063604 (2006). [23] h. brunn, ̈ uber verkettung, s.-b.math.-phys.kl. bayer akad. wiss. 22, 77 (1892). [24] j.-m. richard and s. fleck, phys. rev. lett. 73 1464 (1994). [25] j. goy, j.-m. richard, and s. fleck, phys.rev. a52 3511 (1995). [26] f. bringas, m.t yamashita and t. frederico, phys. rev. a 69, 040702(r) (2004). [27] m.t. yamashita, t. frederico, a. delfino and l. tomio, phys. rev. a 66, 052702 (2002). [28] m.t. yamashita, t. frederico and l. tomio, phys. rev. lett. 99, 269201 (2007); phys. lett. b660, 339 (2008). [29] y. suzuki and k. varga, stochastic variational approach to quantum-mechanical few-body problems, springer (1998). [30] e. nielsen, d.v. fedorov, and a.s. jensen, phys. rev. a56, 3287 (1997). [31] s.p. merkuriev, c. gignoux, a. laverne, ann phys. 99, 30 (1976).

0911.1723

the last breath of the young gigahertz-peaked spectrum radio source pks 1518+047

we present the results from multi-frequency vlba observations from 327 mhz to 8.4 ghz of the gigahertz-peaked spectrum radio source pks 1518+047 (4c 04.51) aimed at studying the spectral index distribution across the source. further multi-frequency archival vla data were analysed to constrain the spectral shape of the whole source. the pc-scale resolution provided by the vlba data allows us to resolve the source structure in several sub-components. the analysis of their synchrotron spectra showed that the source components have steep spectral indices, suggesting that no supply/re-acceleration of fresh particles is currently taking place in any region of the source. by assuming the equipartition magnetic field of 4 mg, we found that only electrons with $\gamma$ < 600, are still contributing to the radio spectrum, while electrons with higher energies have been almost completed depleted. the source radiative lifetime we derived is 2700+/-600 years. considering the best fit to the overall spectrum, we find that the time in which the nucleus has not been active represents almost 20% of the whole source lifetime, indicating that the source was 2150+/-500 years old when the radio emission switched off.

introduction the radio emission of extragalactic powerful radio sources is due to synchrotron radiation from relativistic particles with a power-law energy distribution. they are produced in the "central engine", namely the active galactic nucleus (agn), and reaccelerated in the hot spot, that is the region where the particles, channelled through the jets, interact with the external medium. the energy distribution of the plasma is described by a power-law: n(e) = noe−δ that results in a power-law radio spectrum sν ∝ν−α, with a moderate steepening sν ∝ν−α+0.5 caused by energy losses in the form of radio emission. the spectral steepening occurs at the break frequency, νb, which is related to the age of relativistic electrons (see e.g. pacholczyk 1970). this model, known as the continuous injection (ci) model, requires a continuous injection of power-law distributed electrons into a volume permeated by a constant magnetic field. the model has been applied to the interpretation of the total spectra of the radio sources (see e.g. murgia et al. ⋆e-mail: [email protected] 1999), assuming that the emission of the lobes (supposed to have a constant and uniform magnetic field) dominates over that of core, jets and/or hot spots. it is nowadays clear that powerful radio sources represent a small fraction of the population of the active galactic nuclei associated with elliptical galaxies, implying that the radio activity is a transient phase in the life of these systems. the onset of radio activity is currently thought to be related to merger or accretion events occurring in the host galaxy. the evolutionary stage of each powerful radio source is indicated by its linear size. intrinsically compact radio sources with linear size ls ⩽1 kpc, are therefore inter- preted as young radio objects in which the radio emission originated a few thousand years ago (e.g. fanti et al. 1995; snellen et al. 2000). these sources are characterised by a rising synchrotron spectrum peaking at frequencies around a few gigahertz and are known as gigahertz-peaked spec- trum (gps) objects. when imaged with high resolution, they often display a two-sided morphology which looks like a scaled-down version of the classical and large (up to a few mpc) double radio sources (phillips & mutel 1982). for these sources, kinematic (polatidis & conway 2003) 2 m. orienti et al. and radiative (murgia 2003) studies provided ages of the order of 103 and 104 years, supporting the idea that they are young radio sources whose fate is probably to become classical doubles. however, it has been claimed (alexander 2000; marecki et al. 2006) that a fraction of young radio sources would die in an early stage, never becoming the classical extended radio galaxies with linear sizes of a few hundred kpc and ages of the order of 107 – 108 yr. support for this scenario comes from a statistical study of gps sources by gugliucci et al. (2005): they showed that the age distribution of the small radio sources they considered has a peak around 500 years. so far, only a few objects have been recognised as dy- ing radio sources. they are difficult to find owing to their extremely steep radio spectrum which makes them under-represented in flux-limited catalogues. the ob- jects 0809+404 (kunert-bajraszewska et al. 2006) and 1542+323 (kunert-bajraszewska et al. 2005) are possible examples of young radio sources that are fading. these sources, with an estimated age of 104 – 105 years, have been suggested as "faders" due to the lack of active structures, like cores and hot-spots, although their optically-thin radio spectra do not display the typical form expected for a fader. in this paper, we present results from multi- frequency vlba and vla data of the gps radio source pks 1518+047 (ra=15h 21m 14s and dec= 04◦30 ′ 21 ′′, j2000), identified with a quasar at redshift z = 1.296 (stickel & kuhr 1996). it is a powerful radio source (log p1.4 ghz (w/hz) = 28.53), with a linear size of 1.28 kpc. this source was selected from the gps sample of stanghellini et al. (1998) on the basis of its steep spectrum α8.4ghz 1.4ghz = 1.2, uncommon in young radio sources. the goal of this paper is to determine by means of the analysis of the radio spectrum and its slope in the optically thin region, whether this source is in a phase where its radio activity has, perhaps temporarily, switched off. throughout this paper, we assume the following cosmology: h0 = 71 km s−1 mpc−1, ωm = 0.27 and ωλ = 0.73, in a flat universe. at the redshift of the target 1 ′′ = 8.436 kpc. the spectral index is defined as s(ν) ∝ν−α. 2 radio data the target source was observed on march 5, 2008 (project code bd129) with the vlba at 0.327, 0.611 (p band), 1.4 and 1.6 ghz (l band) with a recording band width of 16 mhz at 256 mbps for 20 min in l band and 1 hr in p band. the correlation was performed at the vlba correlator in socorro. these observations have been complemented with archival vlba data at 4.98 (c band) and 8.4 ghz (x band) carried out on march 28, 2001 (project code bs085). the data reduction was carried out with the nrao aips package. during the observations, the gain corrections were found to have variations within 5% in l, c and x bands, and 10% in p band respectively. in p band, the antennas located in kitt peak, mauna kea and north liberty had erratic system temperatures, probably due to local rfi, and their freq. tota nb sb ghz mjy mjy mjy 0.32 2278±38 1136±117 512±58 0.61 4050±200c 1778±182 897±98 0.96 4720±240d - - 1.4 3937±120 2468±123 1555±78 1.6 3376±101 2008±101 1311±56 3.9 1460±50d - - 4.5 1164±30 - - 4.9 1013±30 392±19 479±24 5.0 994±52e - - 8.1 486±15 - - 8.4 441±13 148±7 232±11 11.1 305±30d - - 15 165±1 - - 22 80±5 - - table 1. multi-frequency flux density of pks 1518+047 and of its northern and southern components. a = archival vla data; b = vlba data; c = wsrt data from stanghellini et al. (1998); d = ratan data from stanghellini et al. (1998); e = wsrt data from xiang et al. (2006). data had to be flagged out completely. the resulting loss of both resolution and sensitivity did not affect the source structure and flux density. to constrain the spectral shape of the whole source, we analysed archival vla data at 0.317, 1.365, 1.665, 4.535, 4.985, 8.085, 8.465, 14.940 and 22.460 ghz, obtained on august 22, 1998 (project code as637). the data reduction was carried out with the standard procedure implemented in the nrao aips package. uncertainties in the deter- mination of the flux density are dominated by amplitude calibration errors, which are found to be around 3% at all frequencies. vla and vlba final images were obtained after a number of phase-only self-calibration iterations. at 4.9 and 8.4 ghz, besides the full resolution vlba image, we also produced a lower resolution image using the same uv-range, image sampling and restoring beam of the 1.6 ghz, in order to produce spectral index maps of the source. images at the various frequencies were aligned using the task lgeom by comparing the position of the compact and bright components at each frequency. flux density and deconvolved angular sizes were measured by means of the task jmfit which performs a gaussian fit to the source components on the image plane, or, in the case of extended components, by tvstat which performs an aperture integration on a selected region on the image plane. the parameters derived in this way are reported in table 1. about 8% and 20% of the total flux density is missing in our vlba images at 4.9 and 8.4 ghz respectively if we consider the vla measurements. it is likely that such a missing flux density is from a steep-spectrum and diffuse emission which is not sampled by the vlba data due to the lack of appropriate short spacings. the last breath of the young gps spectrum radio source pks 1518+047 3 3 results 3.1 source structure multi-frequency vlba observations with parsec-scale res- olution allow us to resolve the structure of pks 1518+047 into two main source components (fig. 1), separated by 135 mas (1.1 kpc), in agreement with previous images by dallacasa et al. (1998), xiang et al. (2002), and xiang et al. (2006). both the northern and southern components have angular size of 23×15 mas2. at the highest frequencies, where we achieve the best resolution, the northern component is resolved into two sub-structures (labelled n1 and n2) separated by 11 mas (90 pc) and position angle of 11◦. n1 accounts for 295 and 119 mjy at 4.9 and 8.4 ghz, respectively, with angular size of 7.7×6.8 mas2. component n2 is fainter than n1 and it accounts for 97 and 29 mjy at 4.9 and 8.4 ghz, respectively, with angular size of 7.4 ×5 mas2. the southern complex is resolved into 4 compact regions located at 3, 7 and 16 mas (25, 60 and 130 pc for s2, s3 and s4 respectively) with respect to s1, all with almost the same position angle of 40◦. the resolution, not adequate to properly fit the several components of the southern lobe, does not allow us to reliably determine the observational parameters of each single sub-component. the spectral index in both the northern and southern complexes, computed considering the integrated component flux density, is steep, with α8.4 1.4 = 1.5 ± 0.1 and = 1.0 ± 0.1, respectively. errors on the spectral indices were calculated following the error propagation theory. the analysis of the spectral index distribution in the southern complex (fig. 2) shows a steepening of the spectral index going from s1 inwards to s4, suggesting that s1 is likely the last place where relativistic electrons coming from the source core were re-accelerated. in the northern lobe, the spectral index between 1.6 and 4.9 ghz is almost constant across the component, showing a small gradient going from n1 towards n2, while in the spectral index maps at higher frequencies it steepens from n1 to n2 (fig. 2). the steep spectral indices showed by the northern and southern components suggest that no current particle acceleration across the source is taking place, indicating that active regions, like conventional jet-knots and hot spots, are no longer present. at 327 and 611 mhz, i.e. in the optically-thick part of the spectra, the low spatial resolution does not allow us to separate the individual sub-components. however, their spectral indices, obtained considering the integrated flux density, are α611 327 ∼-0.6±0.2 and -0.9±0.2 for the northern and southern component respectively, that is much flatter than the value expected in the presence of "classical" syn- chrotron self-absorption from a homogeneous component, indicating that what we see is the superposition of the spectra of the various sub-components, each characterised by its own spectral peak occurring in a range of frequencies and then causing the broadening of the overall spectrum (i.e. the northern and southern components are far from be- ing homogeneously filled by magnetised relativistic plasma). 3.2 the spectral shape to understand the physical processes taking place in this source we fit the optically-thin part of the overall spectrum, as well as the spectra of the northern and southern com- ponents, assuming two different models. the first model assumes that fresh relativistic particles are continuously injected in the source (ci model), while in the second model the continuous supply of particles is over and the radio source is already in the relic phase (ci off model). the ci synchrotron model is described by three param- eters: i) αinj, the injection spectral index; ii) νb, the break frequency; iii) norm, the flux normalization. as the injection of fresh particles stops (ci off model), a second break, νb2, appears at high frequencies, and beyond that the spectrum cuts offexponentially. this second break is related to the first according to: νb2 = νb  ts toff 2 (1) where ts is the total source's age and toff is the time elapsed since the injection switched off (see e.g. komissarov & gubanov 1994; slee et al. 2001; parma et al. 2007). indeed, compared to the basic ci model, the ci off model is characterized by one more free parameter: i) αinj, the injection spectral index; ii) νb, the lowest break frequency; iii) norm, the flux normalization; iv) toff/ts, i.e. the relic to total source age ratio. the spectral shapes of both models cannot be described by analytic equations and must be computed numerically (see e.g. murgia et al. 1999 and slee et al. 2001 for further details). in the study of the overall spectrum, in addition to the archival vla data analysed in this paper, we con- sider also ratan observations at 0.96, 3.9, and 11.1 ghz (stanghellini et al. 1998), and wsrt observations at 5 ghz (xiang et al. 2006), to have a better frequency sam- pling. due to the presence of another radio source located at about 1 arcminute from the target (fig. 3), we do not consider observations with resolution worse than ∼1 ′. during the fitting procedure, a particular care was taken in choosing the most accurate injection spectral index. the peculiar shape of the radio spectrum of this source does not allow us to directly derive the injection spectral index from the optically-thin emission below the break, because it falls below the peak frequency where the spectrum is absorbed. for this reason, we fit the spectrum assuming various in- jection spectral indices, choosing the one that provides the best reduced chi-square (fig. 4c). the best fit we obtain with the ci model implies a very steep injection spectral index αinj = 1.1, that is reflected in an uncommon electron energy distribution δ = 3.2, (fig. 4a). on the other hand, the ci off model provides a more common injection spectral index αinj = 0.7 (fig. 4b). fur- thermore, the comparison between the reduced chi-square of 4 m. orienti et al. center at ra 15 21 14.4193420 dec 04 30 21.660520 cont: j1521+04 ipol 326.482 mhz 1521 4 92.icl001.1 plot file version 1 created 25-aug-2008 13:03:49 cont peak flux = 7.7505e-01 jy/beam levs = 2.840e-02 * (-1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512) milliarc sec milliarc sec 150 100 50 0 -50 -100 -150 -200 150 100 50 0 -50 -100 -150 -200 -250 312 mhz peak= 775.0; f.c.= 28.4 (mjy/beam) s n j1521+0340 312 mhz 1518+047 312 mhz center at ra 15 21 14.4193420 dec 04 30 21.660520 cont: j1521+04 ipol 610.982 mhz 1521 3 50ma.icl001.1 plot file version 2 created 13-jul-2009 16:44:56 cont peak flux = 1.5853e+00 jy/beam levs = 4.104e-02 * (-1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512) milliarc sec milliarc sec 100 50 0 -50 -100 -150 50 0 -50 -100 -150 n s j1521+0340 611 mhz peak= 1585.3; f.c.= 41.0 (mjy/beam) 1518+047 611 mhz center at ra 15 21 14.4193420 dec 04 30 21.660520 cont: j1521+04 ipol 1643.392 mhz 1521 if2.icl001.1 plot file version 2 created 13-jul-2009 17:00:33 cont peak flux = 8.8571e-01 jy/beam levs = 1.050e-02 * (-1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048) milliarc sec milliarc sec 60 40 20 0 -20 -40 -60 -80 -100 40 20 0 -20 -40 -60 -80 -100 -120 -140 n2 n1 s4 s j1521+0340 1.6 ghz peak= 885.7; f.c.= 10.5 (mjy/beam) 1518+047 1.6 ghz center at ra 15 21 14.3960180 dec 04 30 21.367740 cont: 1518+047 ipol 4975.459 mhz 1518 c3.icl001.1 plot file version 2 created 13-jul-2009 16:54:12 cont peak flux = 2.5670e-01 jy/beam levs = 3.600e-04 * (-1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512) milliarc sec milliarc sec 120 100 80 60 40 20 0 -20 -40 140 120 100 80 60 40 20 0 -20 5.0 ghz 5.0 ghz n2 s2 s3 s4 n1 s1 peak= 256.7; f.c.= 0.4 (mjy/beam) j1521+0340 5.0 ghz 1518+047 5.0 ghz 1518+047 4.9 ghz center at ra 15 21 14.3960180 dec 04 30 21.367740 cont: 1518+047 ipol 8409.459 mhz 1518 x4.icl001.1 plot file version 2 created 14-jul-2009 13:17:49 cont peak flux = 1.1479e-01 jy/beam levs = 4.590e-04 * (-1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512) milliarc sec milliarc sec 80 60 40 20 0 -20 120 100 80 60 40 20 0 peak= 114.8; f.c.=0.4 (mjy/beam) j1521+0340 8.4 ghz n2 n1 s2 s3 s1 1518+047 8.4 ghz figure 1. vlba images at 0.327, 0.611, 1.6, 4.9 and 8.4 ghz of pks 1518+047. on the images we show the observing frequency, the peak flux density and the first contour intensity (f.c.) which is 3 times the 1σ noise level measured on the image plane. contours increase of a factor 2. the beam is plotted on the bottom left corner. the different models (fig. 4c) shows that the ci off model is more accurate in fitting the data. in the context of the ci off model, we derive a break frequency νbr = 2.4 ghz, and toff/ts = 0.2. a similar result is found for the spectrum of the southern component. as in the overall spectrum, both ci and ci off models well reproduce the spectral shape (fig. 5), but the former implies an uncommonly steep injec- tion spectral index. a different result is found in the analysis of the northern lobe. in this case, we find that the ci off model with αinj = 0.7 well reproduces the spectral shape, providing a break frequency νbr = 0.8 ghz and toff/ts ∼0.27 (fig. 6b), while the ci model provides a worse chi-square even considering a steep injection spectral index (fig. 6a). the optically-thick part of the spectra is well modelled by an absorbed spectrum with αthick = −1.2 ± 0.1, that is different from the canonical -2.5 expected in the pres- ence of synchrotron self-absorption from a homogeneous component. this result, together with the source structure resolved in several sub-components, indicates that in the optically-thick regime the observed spectra are the super- position of the spectra of many components that cannot be resolved due to resolution limitation. the best fit parameters obtained with the ci and ci off models, and assuming synchrotron self-absorption (ssa) are reported in table 2. 3.3 physical parameters physical parameters for the radio source components were computed assuming equipartition condition and using stan- dard formulae (pacholczyk 1970). we considered particles with energy between γmin =100 and γmax =600. the high energy cut-offcorresponds to the value for which the break frequency in the observer frame occurs at 2.4 ghz, as ob- tained from the best fit to the model (section 3.2). pro- ton and electron energy densities are assumed to be equal, and the spectral index is α = 0.7 (see section 3.2). we as- sume that the volume v of the emitting regions is a prolate spheroid: v = π 6 ab2φ the last breath of the young gps spectrum radio source pks 1518+047 5 center at ra 15 21 14.3960180 dec 04 30 21.367740 cont: 1518+047 ipol 4975.459 mhz 1521 c v l.icl001.1 grey: j1521+04 spix 1525.392 mhz 1521 l tap.spix.1 plot file version 3 created 28-aug-2008 13:33:18 grey scale flux range= -2.000 -0.500 sp index cont peak flux = 4.0254e-01 jy/beam levs = 5.520e-04 * (-1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512) -2.0 -1.5 -1.0 -0.5 milliarc sec milliarc sec 100 80 60 40 20 0 -20 -40 140 120 100 80 60 40 20 0 -20 2 1.5 1.0 0.5 contours: 4.9 ghz spix:1.6 − 4.9 ghz 1.2 1.3 0.9 1.3 a peak= 402.5; f.c.= 0.5 (mjy/beam) center at ra 15 21 14.3960180 dec 04 30 21.367740 cont: 1518+047 ipol 8409.459 mhz 1521 xvl.lgeom.1 grey: 1518+047 spix 4975.459 mhz 1521 cx.spix.1 plot file version 4 created 03-nov-2009 15:40:44 grey scale flux range= -3.000 0.000 sp index cont peak flux = 1.9741e-01 jy/beam levs = 7.537e-04 * (-1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024) -3 -2 -1 0 milliarc sec milliarc sec 100 80 60 40 20 0 -20 -40 140 120 100 80 60 40 20 0 -20 1 2 3 0 contours: 8.4 ghz spix: 4.9 − 8.4 ghz 1.7 1.3 peak= 197.4; f.c.= 0.7 (mjy/beam) b 1.1 1.5 center at ra 15 21 14.3960180 dec 04 30 21.367740 cont: 1518+047 ipol 8409.459 mhz 1521 xvl.lgeom.1 grey: j1521+04 spix 1525.392 mhz 1521 l x.spix.1 plot file version 1 created 28-aug-2008 13:48:59 grey scale flux range= -2.000 -0.500 sp index cont peak flux = 1.9741e-01 jy/beam levs = 7.920e-04 * (-1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512) -2.0 -1.5 -1.0 -0.5 milliarc sec milliarc sec 100 80 60 40 20 0 -20 -40 140 120 100 80 60 40 20 0 -20 2.0 1.5 1.0 0.5 contours: 8.4 ghz spix: 1.6 − 8.4 ghz 1.1 1.3 1.8 peak= 197.4; f.c.= 0.7 (mjy/beam) c figure 2. vlba spectral index images of pks 1518+047 between 1.6 and 4.9 ghz (panel a), between 4.9 and 8.4 ghz (panel b), and between 1.6 and 8.4 ghz (panel c). on the images we show the observing frequency of the contours, the peak flux density and the first contour intensity (f.c.) which is 3 times the 1σ noise level measured on the image plane. contours increase of a factor 2. the beam is plotted on the bottom left corner. the grey scale is shown by the wedge at the top of each image. where a and b are the spheroid major and minor axes, and φ is the filling factor. in the case of the entire radio source, we consider a =140 mas, b =25 mas, and φ = 1. with these assumptions, we obtain an equipartition mag- netic field heq = 4 mg, and a minimum energy density umin = 1.4× 10−6 erg/cm3. in the case of component n and s we assume a = 23 mas, b =15 mas and φ=1, and we infer a magnetic field of 7 mg, and a minimum energy density of =5×10−6 erg/cm3, in agreement with the values derived for the entire source structure. 4 discussion in general the shape of the synchrotron spectrum of active radio sources is the result of the interplay between freshly injected relativistic particles and energy losses that cause a depletion of high-energy particles which results into a steep- ening of the spectrum at high frequencies. the discovery of a gps radio source, considered to be a young object where the radio emission started a few thousand years ago, but dis- playing a steep spectral index (α8.4 1.6 > 1.0) in all its compo- nents is somewhat surprising. indeed, the strong steepening found across the components of pks 1518+047 is incompat- ible with the moderate high-frequency steepening predicted by a continuous injection of particles, suggesting that no in- jection/acceleration of fresh relativistic particles is currently occurring in any region of the source. furthermore, when the injection of fresh particles is over, the strong adiabatic losses cause a fast shift of the spectral turnover towards lower fre- quencies and a decrement of the peak flux density. therefore, detections of dying radio sources with a spectral peak still occurring in the ghz regime are extremely rare. a possible cont: 1518+047 ipol 1652.500 mhz 1521 1.4vla.icln.1 plot file version 1 created 21-jul-2009 15:23:26 cont peak flux = 3.5540e+00 jy/beam levs = 6.060e-03 * (1, 2, 4, 8, 16, 32, 64, 128, 256, 512) declination (j2000) right ascension (j2000) 15 21 20 18 16 14 12 04 31 30 00 30 30 00 29 30 00 1518+047 1.6 ghz vla peak= 3554.0; f.c.= 6.1 (mjy/beam) target 1518+047b figure 3. vla image at 1.6 ghz of pks 1518+047. another source, 1518+047b, is located at ∼1 arcminute in the south-east direction from the target. explanation may arise from the presence of a dense ambient medium enshrouding the radio source which may confine the relativistic particles reducing the adiabatic losses. this is probably the case of fading radio sources found in galaxy clusters where the intracluster medium (icm) is likely limit- ing adiabatic cooling (slee et al. 2001). however, in the case of pks 1518+047 the source is surrounded by the interstel- lar medium (ism) of the host galaxy, but its confinement cannot be related to neutral hydrogen, because observations searching for hi absorption did not provide evidence of a 6 m. orienti et al. comp. model αinj νp νb toff/ts χ2 red mhz mhz n ci 1.1 1300+80 −80 870+630 −440 - 12.0 n ci off 0.7 1620+380 −280 800+1230 −410 0.27 5.4 s ci 1.1 1400+880 −190 <35000 - 0.7 s ci off 0.7 1720+880 −540 1260+15870 −1110 >0.01 0.57 tot ci 1.1 1050+46 −40 7130+1740 −1470 - 5.6 tot ci off 0.7 1000+110 −74 2380+1130 −950 0.21 0.46 table 2. best fit parameters obtained with the ci and ci off models + ssa for the source pks 1518+047 and for its northern and southern components. dense environment (gupta et al. 2006). from the analysis of the optically-thin spectrum of pks 1518+047, we find that the steepening may be ex- plained assuming that the supply of relativistic plasma is still taking place, but the energy distribution of the injected particles is uncommonly steep. another explanation is that the steep spectral shape is due to the absence of freshly in- jected/reaccelerated particles and the energy losses are driv- ing the spectrum evolution. support to this scenario comes from the best fit to the spectrum of the northern compo- nent, where the continuous injection model fails in repro- ducing the spectral shape, even assuming that relativistic particles are injected with a steep energy distribution. this steep spectrum can be best reproduced by a synchrotron model in which no particle supply is taking place. further- more, the time spent by the source in its "fader" phase is about 20% of the whole source age. we estimate the break energy γb of the electron spectrum and the electron radiative time by: γb ∼487ν1/2 b h−1/2(1 + z)1/2 (2) and ts = 5.03 × 104h−3/2ν−1/2 b (1 + z)−1/2 (yr) (3) where h is in mg and νb in ghz. if in eqs. 2 and 3 we consider the equipartition magnetic field and νb derived from the fits, we find that the energy of the electrons responsible for the break should be γ ∼ 400 – 600 and the source radiative lifetime is ts = 2700 ± 600 yr. as a consequence, electrons with γ > 600 (i.e. with shorter radiative lifetime) have already depleted their energy and they contribute only to the high-frequency tail of the spectrum. if we consider the toff/ts ∼0.2, as derived from the fit to the overall spectrum, we find that toff = 550 ± 100 years, and this represents the time elapsed since the last supply/acceleration of relativistic particles. these values indicate that the radio source was 2150±500 years old when the radio emission switched off. the small toff derived suggests that adiabatic losses had not enough time to shift the spectral peak of pks 1518+047 far from the ghz regime. in the presence of adiabatic expansion, and considering a magnetic field frozen in the plasma, the spectral peak is shifted towards lower frequencies: νp,1 = νp,0 t0 t1 4 (4) where νp,0 and νp,1 are the peak frequency at the time t0 and t1 respectively (orienti & dallacasa 2008). as the time elapsed after the switch offincreases, the peak moves to lower and lower frequencies, making the source unrecognisable as a young gps. for example, when toff will represent more than 40% of the total source lifetime, the peak should be below 300 mhz, so it will be difficult to identify as a dying young radio source. future instruments like lofar may unveil a population of such fading radio sources. 5 conclusions we presented results from multi-frequency vlba and vla observations of the gps radio source pks 1518+047. the analysis of the spectral index distribution across the whole source structure showed that all the source components have very steep optically-thin synchrotron spectra. the radio spectra are well explained by energy losses of rel- ativistic particles after the cessation of the injection of new plasma in the radio lobes. this result, together with the lack of the source core and active non-steep spectrum components like hot spots and knots in the jets, suggests that no injection/acceleration of fresh particle is currently occurring in any region of the source. for this reason, pks 1518+047 can be considered a fading young radio source, in which the radio emission switched offshortly after its onset. when the supply of energy is switched off, given the high magnetic fields in the plasma, the spectral turnover moves rapidly towards low frequencies, making the source undetectable at the frequencies commonly used for radio surveys. however, in pks 1518+047 the time elapsed since the last particle acceleration is of the order of a few hundred years, suggesting that pks 1518+047 still has a gps spectrum because adiabatic losses have not had enough time to affect the source spectrum, shifting the peak far from the ghz regime. if the interruption of the radio activity is a temporary phase and the radio emission from the central engine will re-start soon, it is possible that the source will appear again as a gps without the severe steep- ening at high frequencies. if this does not happen, the fate of this radio source is to emit at lower and lower frequencies, the last breath of the young gps spectrum radio source pks 1518+047 7 a b c figure 4. the best fit to the overall spectrum of pks 1518+047 using the ci (left) and the ci off models (center), and the reduced chi-square versus the injection spectral index (right). error bars are rather small, and often result within the symbol. a b figure 5. the best fit to the spectrum of the southern component of pks 1518+047 using the ci (top) and the ci off models (bottom). a b figure 6. the best fit to the spectrum of the northern component of pks 1518+047 using the ci (top) and the ci off models (bottom). 8 m. orienti et al. until it disappears at frequencies well below the mhz regime. acknowledgemnet we thank the anonymous referee for carefully reading the manuscript and valuable suggestions. the vlba is oper- ated by the us national radio astronomy observatory which is a facility of the national science foundation operated under a cooperative agreement by associated university, inc. this work 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. references alexander, a., 2000, mnras, 319, 8 dallacasa, d., bondi, m., alef, w., mantovani, f., 1998, a&as, 129, 219 fanti, c., fanti, r., dallacasa, d., schilizzi, r. t., spencer, r. e., stanghellini, c., 1995, a&a, 302, 317 gugliucci, n.e., taylor, g.b., peck, a.b., giroletti, m., 2005, apj,622, 136 gupta, n., salter, c.j., saikia, d.j., ghosh, t., jeyaku- mar, s., 2006, mnras, 373, 972 komissarov, s.s., gubanov, a.g., 1994, a&a, 285, 27 kunert-bajraszewska, m., marecki, a., thomasson, p., spencer, r. e., 2005, a&a, 440, 93 kunert-bajraszewska, m., marecki, a., thomasson, p., 2006, a&a, 450, 945 marecki, a., kunert-bajraszewska, m., spencer, r.e., 2006, a&a, 449, 985 murgia, m., fanti, c., fanti, r., gregorini, l., klein, u., mack, k.-h., vigotti, m., 1999, a&a, 345, 769 murgia, m., 2003, pasa, 20, 19 orienti, m., dallacasa, d., 2008, a&a, 477, 807 pacholczyk, a.g., 1970, radio astrophysics, (san fran- cisco: freeman & co.) parma, p., murgia, m., de ruiter, h.r., fanti, r., mack, k.-h., govoni, f., 2007, a&a, 470, 875 phillips, r.b., mutel, r.l., 1982, a&a, 106, 21 polatidis, a.g., & conway, j.e., 2003, pasa, 20, 69 slee, o.b., roy, a.l., murgia, m., andernach, h., ehle, m., 2001, aj, 122, 1172 snellen, i. a. g., schilizzi, r. t., miley, g. k., de bruyn, a. g., bremer, m. n., r ̈ ottgering, h. j. a., 2000, mnras, 319, 445 stanghellini, c., o'dea, c.p., dallacasa, d., baum, s.a., fanti, r., fanti, c., 1998, a&as, 131, 303 stickel, m., kuhr, h., 1996, a&as, 115, 11 xiang, l., stanghellini, c., dallacasa, d., haiyan, z., 2002, a&a, 385, 768 xiang, l., reynolds, c., strom, r.g., dallcasa, d., 2006, a&a, 454, 729

0911.1724

on the approach to thermal equilibrium of macroscopic quantum systems

we consider an isolated, macroscopic quantum system. let h be a micro-canonical "energy shell," i.e., a subspace of the system's hilbert space spanned by the (finitely) many energy eigenstates with energies between e and e + delta e. the thermal equilibrium macro-state at energy e corresponds to a subspace h_{eq} of h such that dim h_{eq}/dim h is close to 1. we say that a system with state vector psi in h is in thermal equilibrium if psi is "close" to h_{eq}. we show that for "typical" hamiltonians with given eigenvalues, all initial state vectors psi_0 evolve in such a way that psi_t is in thermal equilibrium for most times t. this result is closely related to von neumann's quantum ergodic theorem of 1929.

introduction if a hot brick is brought in contact with a cold brick, and the two bricks are otherwise isolated, then energy will flow from the hot to the cold brick until their temperatures ∗departments of mathematics and physics, rutgers university, 110 frelinghuysen road, piscataway, nj 08854-8019, usa. †e-mail: [email protected] ‡e-mail: [email protected] §dipartimento di fisica dell'universit` a di genova and infn sezione di genova, via dodecaneso 33, 16146 genova, italy. ¶e-mail: [email protected] ∥department of mathematics, rutgers university, 110 frelinghuysen road, piscataway, nj 08854- 8019, usa. ∗∗e-mail: [email protected] ††e-mail: [email protected] 1 become equal, i.e., the system equilibrates. since the bricks ultimately consist of elec- trons and nuclei, they form a quantum system with a huge number (> 1020) of particles; this is an example of an isolated, macroscopic quantum system. from a microscopic point of view the state of the system at time t is described by a vector ψ(t) = e−ihtψ(0) (1) in the system's hilbert space or a density matrix ρ(t) = e−ihtρ(0)eiht , (2) where h is the hamiltonian of the isolated system and we have set ħ= 1. in this paper we prove a theorem asserting that for a sufficiently large quantum system with a "typical" hamiltonian and an arbitrary initial state ψ(0), the system's state ψ(t) spends most of the time, in the long run, in thermal equilibrium. (of course, before the system even reaches thermal equilibrium there could be a waiting time longer than the present age of the universe.) this implies the same behavior for an arbitrary ρ(0). this behavior of isolated, macroscopic quantum systems is an instance of a phe- nomenon we call normal typicality [5], a version of which is expressed in von neumann's quantum ergodic theorem [17]. however, our result falls outside the scope of von neu- mann's theorem, because of the technical assumptions made in that theorem. our result also differs from the related results in [4, 15, 16, 12, 13, 10], which use different notions of when a system is in an equilibrium state. in particular they do not regard the ther- mal equilibrium of an isolated macroscopic system as corresponding to its wave function being close to a subspace heq of hilbert space. see section 6 for further discussion. the rest of this paper is organized as follows. in the remainder of section 1, we define more precisely what we mean by thermal equilibrium. in section 2 we outline the problem and our result, theorem 1. in section 3 we prove the key estimate for the proof of theorem 1. in section 4 we describe examples of exceptional hamiltonians, illustrating how a system can fail to ever approach thermal equilibrium. in section 5 we compare our result to the situation with classical systems. in section 6 we discuss related works. 1.1 the equilibrium subspace let htotal be the hilbert space of a macroscopic system in a box λ, and let h be its hamiltonian. let {φα} be an orthonormal basis of htotal consisting of eigenvectors of h with eigenvalues eα. consider an energy interval [e, e + δe], where δe is small on the macroscopic scale but large enough for the interval [e, e + δe] to contain very many eigenvalues. let h ⊆htotal be the corresponding subspace, h = span  φα : eα ∈[e, e + δe] . (3) a subspace such as h is often called a micro-canonical energy shell. let d be the dimension of h , i.e., the number of energy levels, including multiplicities, between e 2 and e + δe. in the following we consider only quantum states ψ that lie in h , i.e., of the form ψ = x α cα φα (4) with cα ̸= 0 only for α such that eα ∈[e, e + δe]. according to the analysis of von neumann [17, 18] and others (cf. [6]), the macro- scopic (coarse-grained) observables in a macroscopic quantum system can be naturally "rounded" to form a set of commuting operators,  mi i=1,...,k . (5) the operators are defined on htotal, but since we can take them to include (and thus commute with) a coarse-grained hamiltonian, we can (and will) take them to commute with the projection to h , and thus to map h to itself. we write ν = (m1, . . . , mk) for a list of eigenvalues mi of the restriction of mi to h , and hν for the joint eigenspace. such a set of operators generates an orthogonal decomposition of the hilbert space h = m ν hν , (6) where each hν, called a macro-space, represents a macro-state of the system. the dimension of hν is denoted by dν; note that p ν dν = d. if any hν has dimension 0, we remove it from the family {hν}. in practice, dν ≫1, since we are considering a macroscopic system with coarse-grained observables. it can be shown in many cases, and is expected to be true generally, that among the macro-spaces hν there is a particular macro-space heq, the one corresponding to thermal equilibrium, such that deq/d ≈1 , (7) indeed with the difference 1 −deq/d exponentially small in the number of particles.1 this implies, in particular, that each of the macro-observables mi is "nearly constant" on the energy shell h in the sense that one of its eigenvalues has multiplicity at least deq ≈d. we say that a system with quantum state ψ (with ∥ψ∥= 1) is in thermal equilibrium if ψ is very close (in the hilbert space norm) to heq, i.e., if ⟨ψ|peq|ψ⟩≈1 , (8) where peq is the projection operator to heq. the corresponding relation for density matrices is tr(peqρ) ≈1 . (9) 1this dominance of the equilibrium state can be expressed in terms of the (boltzmann) entropy sν of a macroscopic system in the macro-state ν, be it the equilibrium state or some other (see [9]), defined as sν = kb log dν, where kb is the boltzmann constant: deq/d being close to 1 just expresses the fact that the entropy of the equilibrium state is close to the micro-canonical entropy smc, i.e., seq = kb log deq ≈kb log d = smc. 3 the condition (8) implies that a quantum measurement of the macroscopic observable mi on a system with wave function ψ will yield, with probability close to 1, the "equilibrium" value of mi. likewise, a joint measurement of m1, . . . , mk will yield, with probability close to 1, their equilibrium values. let μ(dψ) be the uniform measure on the unit sphere in h [14, 19]. it follows from (7) that most ψ relative to μ are in thermal equilibrium. indeed, z ⟨ψ|peq|ψ⟩μ(dψ) = 1 dtr peq = deq d ≈1 . (10) since the quantity ⟨ψ|peq|ψ⟩is bounded from above by 1, most ψ must satisfy (8).2 1.2 examples of equilibrium subspaces to illustrate the decomposition into macro-states, we describe two examples. as exam- ple 1, consider a system composed of two identical subsystems designated 1 and 2, e.g., the bricks mentioned in the beginning of this paper, with hilbert space htotal = h1⊗h2. the hamiltonian of the total system is h = h1 + h2 + λv , (11) where h1 and h2 are the hamiltonians of subsystems 1 and 2 respectively, and λv is a small interaction between the two subsystems. we assume that h1, h2, and h are positive operators. let h be spanned by the eigenfunctions of h with energies between e and e + δe. in this example, we consider just a single macro-observable m, which is a projected and coarse-grained version of h1/e, i.e., of the fraction of the energy that is contained in subsystem 1 alone. we cannot take m to simply equal h1/e because h1 is defined on htotal, not h , and will generically not map h to itself, while we would like m to be an operator on h . to obtain an operator on h , let p be the projection htotal →h and set h′ 1 = ph1p (12) (more precisely, h′ 1 is ph1 restricted to h ). note that h′ 1 is a positive operator, but might have eigenvalues greater than e. now define3 m = f(h′ 1/e) (13) with the coarse-graining function f(x) =          0 if x < 0.01, 0.02 if x ∈[0.01, 0.03), 0.04 if x ∈[0.03, 0.05), etc. . . . (14) 2it should in fact be true for a large class of observables a on h that, for most ψ relative to μ, ⟨ψ|a|ψ⟩≈tr(ρmca), where ρmc is the micro-canonical density matrix, i.e., 1/d times the identity on h . this is relevant to the various results on thermalization described in section 6. 3recall that the application of a function f to a self-adjoint matrix a is defined to be f(a) = p f(aα)|φα⟩⟨φα| if the spectral decomposition of a reads a = p aα|φα⟩⟨φα|. 4 the hν are the eigenspaces of m; clearly, ⊕νhν = h . if, as we assume, λv is small, then we expect h0.5 = heq to have the overwhelming majority of dimensions. in a thorough treatment we would need to prove this claim, as well as that h′ 1 is not too different from h1, but we do not give such a treatment here. as example 2, consider n bosons (fermions) in a box λ = [0, l]3 ⊆r3; i.e., htotal consists of the square-integrable (anti-)symmetric functions on λn. let the hamiltonian be h = −1 2m n x i=1 ∇2 i + x i<j v |qi −qj|
, (15) where the laplacian ∇2 i has dirichlet boundary conditions, v(r) is a given pair potential, and qi is the triple of position coordinates of the i-th particle. let h again be spanned by the eigenfunctions with energies between e and e + δe. in this example, we consider again a single macro-observable m, based on the oper- ator nleft for the number of particles in the left half of the box λ: nleftψ(q1, . . . , qn) = #  i : qi ∈[0, l/2] × [0, l]2 ψ(q1, . . . , qn) . (16) note that the spectrum of nleft consists of the n + 1 eigenvalues 0, 1, 2, . . ., n. to obtain an operator on h , let p be the projection htotal →h and set n′ left = pnleftp. note that the spectrum of n′ left is still contained in [0, n]. now define m = f(n′ left/n) with the coarse-graining function (14). we expect that for large n, the eigenspace with eigenvalue 0.5, heq = h0.5, has the overwhelming majority of dimensions (and that n′ left ≈nleft). 2 formulation of problem and results our goal is to show that, for typical macroscopic quantum systems, ⟨ψ(t)|peq|ψ(t)⟩≈1 for most t . (17) to see this, we compute the time average of ⟨ψ(t)|peq|ψ(t)⟩. we denote the time average of a time-dependent quantity f(t) by a bar, f(t) = lim t→∞ 1 t z t 0 dt f(t) . (18) since ⟨ψ(t)|peq|ψ(t)⟩is always a real number between 0 and 1, it follows that if its time average is close to 1 then it must be close to 1 most of the time. moreover, for μ-most ψ(0), where μ is the uniform measure on the unit sphere of h , ψ(t) is in thermal equilibrium most of the time. this result follows from fubini's theorem (which implies that taking the μ-average commutes with taking the time average) and the unitary invariance of μ: z ⟨ψ(t)|peq|ψ(t)⟩μ(dψ) = z ⟨ψ|eihtpeqe−iht|ψ⟩μ(dψ) = z ⟨ψ|peq|ψ⟩μ(dψ) ≈1 . (19) 5 that is, the ensemble average of the time average is near 1, so, for μ-most ψ(0), the time average must be near 1, which implies our claim above. so the interesting question is about the behavior of exceptional ψ(0), e.g., of systems which are not in thermal equilibrium at t = 0. do they ever go to thermal equilibrium? as we will show, for many hamiltonians the statement (17) holds in fact for all ψ(0) ∈h . from now on, let h denote the restriction of the hamiltonian to h , and let φ1, . . . , φd be an orthonormal basis of h consisting of eigenvectors of the hamiltonian h with eigenvalues e1, . . . , ed. if ψ(0) = d x α=1 cα φα , cα = ⟨φα|ψ(0)⟩ (20) then ψ(t) = d x α=1 e−ieαtcα φα . (21) thus, ⟨ψ(t)|peq|ψ(t)⟩= d x α,β=1 ei(eα−eβ)t c∗ α cβ⟨φα|peq|φβ⟩. (22) if h is non-degenerate (which is the generic case) then eα −eβ vanishes only for α = β, so the time averaged exponential is δαβ, and ⟨ψ(t)|peq|ψ(t)⟩= d x α=1  cα  2⟨φα|peq|φα⟩. (23) thus, for the system to be in thermal equilibrium most of the time it is necessary and sufficient that the right hand side of (23) is close to 1. now if an energy eigenstate φα is not itself in thermal equilibrium then when ψ(0) = φα the system is never in thermal equilibrium, since this state is stationary. conversely, if we have that ⟨φα|peq|φα⟩≈1 for all α , (24) then the system will be in thermal equilibrium most of the time for all ψ(0). this follows directly from (23) since the right hand side of (23) is an average of the ⟨φα|peq|φα⟩. we show below that (24) is true of "most" hamiltonians, and thus, for "most" hamiltonians it is the case that every wave function spends most of the time in thermal equilibrium. 2.1 main result the measure of "most" we use is the following: for any given d (distinct) energy values e1, . . . , ed, we consider the uniform distribution μham over all hamiltonians with these eigenvalues. choosing h at random with distribution μham is equivalent 6 to choosing the eigenbasis {φα} according to the uniform distribution μonb over all orthonormal bases of h , and setting h = p α eα|φα⟩⟨φα|. the measure μonb can be defined as follows: choosing a random basis according to μonb amounts to choosing φ1 according to the uniform distribution over the unit sphere in h , then φ2 according to the uniform distribution over the unit sphere in the orthogonal complement of φ1, etc. alternatively, μonb can be defined in terms of the haar measure μu(d) on the group u(d) of unitary d × d matrices: any given orthonormal basis {χα} of h defines a one-to-one correspondence between u(d) and the set of all orthonormal bases of h , associating with the matrix u = (uαβ) ∈u(d) the basis φα = d x β=1 uαβχβ ; (25) the image of the haar measure under this correspondence is in fact independent of the choice of {χβ} (because of the invariance of the haar measure under right multiplication), and is μonb. put differently, the ensemble μham of hamiltonians can be obtained by starting from a given hamiltonian h0 on h (with distinct eigenvalues e1, . . ., ed) and setting h = uh0u−1 (26) with u a random unitary matrix chosen according to the haar measure. note that, while considering different possible hamiltonians h in h , we keep heq fixed, although in practice it would often be reasonable to select heq in a way that depends on h (as we did in the examples of section 1.2). for our purpose it is convenient to choose the basis {χα} in such a way that the first deq basis vectors lie in heq and the other ones are orthogonal to heq. then, we have that ⟨φα|peq|φα⟩= deq x β=1 |uαβ|2 (27) with uαβ the unitary matrix satisfying (25). we will show first, in lemma 1, that for every 0 < ε < 1, if d is sufficiently large and deq/d sufficiently close to 1, then most orthonormal bases {φα} are such that ⟨φα|peq|φα⟩> 1 −ε for all α . (28) this inequality is a precise version of (24). how close to 1 should deq/d be? the fact that the average of ⟨ψ|peq|ψ⟩over all wave functions ψ on the unit sphere of h equals deq/d, mentioned already in (10), implies that (28) cannot be true of most orthonormal bases if deq/d ≤1 −ε. to have enough wiggling room, we require that deq d > 1 −ε 2 . (29) 7 we will show then, in theorem 1, that for every (arbitrarily small) 0 < η < 1 and for sufficiently large d, most h are such that for every initial wave function ψ(0) ∈h with ∥ψ(0)∥= 1, the system will spend most of the time in thermal equilibrium with accuracy 1 −η, where we say that a system with wave function ψ is in thermal equilibrium with accuracy 1 −η if ⟨ψ|peq|ψ⟩> 1 −η . (30) this inequality is a precise version of (8). in order to have no more exceptions in time than the fraction 0 < δ′ < 1, we need to set the ε in (28) and (29) equal to ηδ′. lemma 1. let μu(d) denote the haar measure on u(d), and sε := n u ∈u(d)   ∀α : deq x β=1 |uαβ|2 > 1 −ε o . (31) then for all 0 < ε < 1 and 0 < δ < 1, there exists d0 = d0(ε, δ) > 0 such that if d > d0 and deq > (1 −ε/2)d then μu(d)(sε) ≥1 −δ . (32) the proof of lemma 1 is given in section 3. it also shows that d0 can for example be chosen to be d0(ε, δ) = max  103ε−2 log(4/δ), 106ε−4 . (33) from (27), we obtain: theorem 1. for all η, δ, δ′ ∈(0, 1), all integers d > d0(ηδ′, δ) and all integers deq > (1−ηδ′/2)d the following is true: let h be a hilbert space of dimension d; let heq be a subspace of dimension deq; let peq denote the projection to heq; let e1, . . ., ed be pairwise distinct but otherwise arbitrary; choose a hamiltonian at random with eigenvalues eα and an eigenbasis φα that is uniformly distributed. then, with probability at least 1 −δ, every initial quantum state will spend (1 −δ′)-most of the time in thermal equilibrium as defined in (30), i.e., lim inf t→∞ 1 t     0 < t < t : ⟨ψ(t)|peq|ψ(t)⟩> 1 −η    ≥1 −δ′ , (34) where |m| denotes the size (lebesgue measure) of the set m. proof. it follows from lemma 1 that under the hypotheses of theorem 1, ⟨ψ(t)|peq|ψ(t)⟩≥1 −ηδ′ with probability at least 1 −δ. thus, since ηδ′ ≥1 −⟨ψ(t)|peq|ψ(t)⟩≥η ̃ δ, where ̃ δ is the lim supt→∞of the fraction of the time in (0, t) for which ⟨ψ(t)|peq|ψ(t)⟩≤1 −η, it follows that ̃ δ ≤δ′. 8 2.2 remarks normal typicality. theorem 1 can be strengthened; with the same sense of "most" as in theorem 1, we have that for most hamiltonians and for all ψ(0) ⟨ψ(t)|pν|ψ(t)⟩≈dim hν dim h , for all ν (35) for most t. for ν = eq, this implies that ⟨ψ(t)|peq|ψ(t)⟩≈1. this stronger statement we have called normal typicality [5]. a version of normal typicality was proven by von neumann [17]. however, because of the technical assumptions he made, von neumann's result, while much more difficult, does not quite cover the simple result of this paper. typicality and probability. when we express that something is true for most h or most ψ relative to some normalized measure μ, it is often convenient to use the language of probability theory and speak of a random h or ψ chosen with distribution μ. however, by this we do not mean to imply that the actual h or ψ in a concrete physical situation is random, nor that one would obtain, in repetitions of the experiment or in a class of similar experiments, different h's or ψ's whose empirical distribution is close to μ. that would be a misinterpretation of the measure μ, one that suggests the question whether perhaps the actual distribution in reality could be non-uniform. this question misses the point, as there need not be any actual distribution in reality. rather, theorem 1 means that the set of "bad" hamiltonians has very small measure μham. consequences for example 2. from lemma 1 it follows for example 2 that typical hamiltonians of the form (26) with h0 given by the right hand side of (15) are such that all eigenfunctions are close to h0.5; this fact in turn strongly suggests (though we have not proved this) that the eigenfunctions are essentially concentrated on those configurations that have approximately 50% of the particles in the left half and 50% in the right half of the box. equilibrium statistical mechanics. theorem 1 implies that, for typical h, every ψ(0) ∈h is such that for most t, ⟨ψ(t)|mi|ψ(t)⟩≈tr(ρmcmi) , (36) where ρmc is the standard micro-canonical density matrix (i.e., 1/d times the projection htotal →h ), for all macro-observables mi as described in section 1.1. this justifies replacing |ψ(t)⟩⟨ψ(t)| by ρmc as far as macro-observables in equilibrium are concerned. however, this does not, by itself, justify the use of ρmc for observables a not among the {mi}. for example, consider a microscopic observable a that is not "nearly constant" on the energy shell h . then, standard equilibrium statistical mechanics tells us to use ρmc for the expected value of a in equilibrium. we believe that this is in fact correct for most such observables, but it is not covered by theorem 1. results concerning many such observables are described in section 6. these results, according to which, in an appropriate sense, ⟨ψ(t)|a|ψ(t)⟩≈tr(ρmca) (37) 9 for suitable a and ψ(0), are valid only in quantum mechanics. the justification of the broad use of ρmc in classical statistical mechanics relies on rather different sorts of results requiring different kinds of considerations. 3 proof of lemma 1 proof. let us write p for the haar measure μu(d), and let p := p  d \ α=1 n deq x β=1 |uαβ|2 > 1 −ε o . (38) observe that p = 1 −p  d [ α=1 n deq x β=1 |uαβ|2 ≤1 −ε o (39) ≥1 −d max α p n deq x β=1 |uαβ|2 ≤1 −ε o . (40) since u = (uαβ) is a random unitary matrix with haar distribution, its α-th column is a random unit vector ⃗ u := (uαβ)β whose distribution is uniform over the unit sphere of cd (i.e., the distribution is, up to a normalizing constant, the surface area measure). therefore, the probability in the last line does not, in fact, depend on α, and so the step of taking the maximum over α can be omitted. a random unit vector such as ⃗ u can be thought of as arising from a random gaussian vector ⃗ g by normalization: let gβ for β = 1, . . . , d be independent complex gaussian random variables with mean 0 and variance e|gβ|2 = 1/d; i.e., re gβ and im gβ are independent real gaussian random variables with mean 0 and variance 1/2d. then the distribution of ⃗ g = (g1, . . . , gd) is symmetric under rotations from u(d), and thus ⃗ g ∥⃗ g∥ = ⃗ u in distribution, with ∥⃗ g∥2 = d x β=1 |gβ|2 . (41) we thus have that p ≥1 −d p n deq x β=1 |gβ|2 ∥⃗ g∥2 ≤1 −ε o . (42) to estimate the probability on the right hand side of (42), we introduce three different 10 events: a(η′) := n ∥⃗ g∥2 −1   < η′o , (43) b(η′′) := n (1 −η′′)deq d < deq x β=1 |gβ|2 < (1 + η′′)deq d o , (44) c(η′′′) := n (1 −η′′′)deq d < deq x β=1 |gβ|2 ∥⃗ g∥2 < (1 + η′′′)deq d o . (45) let us now assume that deq d > 1 −ε 2 . (46) we then have that (1 −ε/2)deq d > 1 −ε + ε2 4 > 1 −ε , (47) so that c(ε/2) ⊆ n (1 −ε/2)deq d < deq x β=1 |gβ|2 ∥⃗ g∥2 o ⊆ n 1 −ε < deq x β=1 |gβ|2 ∥⃗ g∥2 o (48) and thus p ≥1 −d p(cc(ε/2)) , (49) where the superscript c means complement. our goal is to find a good upper bound for p(cc(ε/2)). if the event a(η′) occurs for 0 < η′ < 1 2 then 1 −η′ < 1 ∥⃗ g∥2 < 1 + 2η′ , (50) and consequently, if a(η′) ∩b(η′′) occurs then deq d (1 −η′)(1 −η′′) < deq p β=1 |gβ|2 ∥⃗ g∥2 < deq d (1 + 2η′)(1 + η′′) . (51) it is now easy to see that a(η′)∩b(η′′) ⊆c(2η′+η′′+2η′η′′), so if we choose η′ = η′′ = ε/8 we obtain that a( ε 8) ∩b( ε 8) ⊆c( 3 8ε + 1 32ε2) ⊆c(ε/2) for 0 < ε < 1 . (52) we thus have the following upper bound: p(cc(ε/2)) ≤p(ac(ε/8)) + p(bc(ε/8)) . (53) 11 to find an estimate of p(a(ε/8)) and p(b(ε/8)) we use the large deviations prin- ciple. it is convenient to use a slightly stronger version of this principle than usual, see section 2.2.1 of [3], which states that for a sequence of n i.i.d. random variables xi, p   n x i=1 xi n −e(x1)   > δ  ≤2e−ni(e(x1)+δ) (54) where i(x) is the rate function [3] associated with the distribution of the xi, defined to be i(x) = sup θ>0 (θx −log eeθxi) . (55) in our case, where xi will be the square of a standard normal random variable, the rate function is i(x) = 1 2(x −1 −log x) ∀x > 1 , (56) as a simple calculation shows. to estimate p(a(ε/8)), set n = 2d , xβ = 2d(re gβ)2 , xd+β = 2d(im gβ)2 for β = 1, . . . , d . (57) thus, for i = 1, . . . , 2d, the xi are i.i.d. variables with mean exi = 2d e(re gi)2 = 1; we thus obtain p(ac(ε/8)) = p n ∥⃗ g∥2 −1   > ε/8 o = (58) = p n  d x β=1 |gβ|2 −1   > ε/8 o (59) = p n  2d x i=1 xi 2d −1   > ε/8 o (60) ≤2e−2d i(1+ε/8) (61) = 2e−d(ε/8−log(1+ε/8)) (62) ≤2 exp  −dε2 192  . (63) in the last step we have used that log(1 + x) ≤x −x2/3 for 0 < x < 1/2. we use a completely analogous argument for b, setting n = 2deq , xβ = 2d(re gβ)2 , xd+β = 2d(im gβ)2 , for β = 1, . . . , deq , (64) 12 and obtain that p(bc(ε/8)) = p n   deq x β=1 |gβ|2 −deq d   /deq d > ε/8 o (65) = p n  2deq x i=1 xi 2deq −1   > ε/8 o (66) ≤2 exp  −deqε2 192  . (67) from (53), (63), and (67) it follows that p(cc(ε/2)) ≤2 exp  −deqε2 192  + 2 exp  −dε2 192  ≤4 exp  −dε2 384  , (68) where we have used that deq > d/2. therefore, by (49), p ≥1 −4d exp  −dε2 384  . (69) the last term converges to 0 as d →∞, so there exists a d0 > 0 such that for all d > d0, p ≥1 −δ , (70) which is what we wanted to show. in order to check this for the d0 specified in (33) right after lemma 1, note that the desired relation 4d exp  −dε2 384  ≤δ (71) is equivalent to d  ε2 384 −log d d  ≥log(4/δ) . (72) thus, it suffices that d > 103ε−2 log(4/δ) and log d d < 10−3ε2 . (73) since log d < √ d for all positive numbers d, condition (73) will be satisfied if √ d > 103ε−2, i.e., if d > 106ε−4. 4 examples of systems that do not approach ther- mal equilibrium we shall now present examples of atypical behavior, namely examples of "bad" hamilto- nians, i.e., hamiltonians for which not all wave functions approach thermal equilibrium 13 (or, equivalently, for which (24) is not satisfied). according to theorem 1, bad hamil- tonians form a very small subset of the set of all hamiltonians. of course, to establish that (24) holds for a particular hamiltonian can be a formidable challenge. moreover, the small subset might include all standard many-body hamiltonians (e.g., all those which are a sum of kinetic and potential energies). but there is no a priori reason to believe that this should be the case. the first example consists of two non-interacting subsystems. this can be expressed in the framework provided by example 1 in section 1.2 with the hamiltonian h = h1 + h2 + λv by setting λ = 0. let {φ1 i } be an orthonormal basis of h1 consisting of eigenvectors of h1 with eigenvalues e1 i , and {φ2 j} one of h2 consisting of eigenvectors of h2 with eigenvalues e2 j . clearly, for λ = 0 not every wave function will approach thermal equilibrium. after all, in this case, the φ1 i ⊗φ2 j form an eigenbasis of h, while h = span  φ1 i ⊗φ2 j : e1 i + e2 j ∈[e, e + δe] (74) and heq = span  φ1 i ⊗φ2 j : e1 i ∈[0.49e, 0.51e) and e1 i + e2 j ∈[e, e + δe] . (75) thus, any φ1 i ⊗φ2 j such that e1 i + e2 j ∈[e, e + δe] but, say, e1 i < 0.49e, will be an example of an element of h that is orthogonal to heq and, as it is an eigenfunction of h, forever remains orthogonal to heq. as another example, we conjecture that some wave functions will fail to approach thermal equilibrium also when λ is nonzero but sufficiently small. we prove this now for a slightly simplified setting, corresponding to the following modification of example 1 of section 1.2. for the usual energy interval [e, e + δe], let h be, independently of λ, given by (74), and, instead of h1 + h2 + λv , let h be given by h = h(λ) = p(h1 + h2 + λv )p , (76) where p is the projection to h . then h defines a time evolution on h that depends on λ. (note that h is still an "energy shell" for all sufficiently small λ, as all nonzero eigenvalues of h(λ) are still contained in an interval just slightly larger than [e, e+δe], and the corresponding eigenvectors lie in h .) let heq for λ ̸= 0 also be given by (75). again, choose one particular φ1 i and one particular φ2 j (independently of λ) so that e1 i + e2 j ∈[e, e + δe] and e1 i < 0.49e, and consider as the initial state of the system again ψ(t = 0) = φ1 i ⊗φ2 j , (77) which evolves to ψ(λ, t) = e−ih(λ)tφ1 i ⊗φ2 j . (78) suppose for simplicity that h(λ = 0) = h1 +h2 is non-degenerate.4 then, according to standard results of perturbation theory [7], also h(λ), regarded as an operator on h , 4since this requires that no eigenvalue difference of h1, e1 i −e1 i′, coincides with an eigenvalue difference of h2, e2 j −e2 j′, we need to relax our earlier assumption that system 1 and system 2 be identical; so, let them be almost identical, with slightly different eigenvalues, and let h1 and h2 each be non-degenerate. 14 is non-degenerate for all λ ∈(−λ0, λ0) for some λ0 > 0; moreover, its eigenvalues e(λ) depend continuously (even analytically) on λ, and so do the eigenspaces. in particular, it is possible to choose for every λ ∈(−λ0, λ0) a normalized eigenstate φ(λ) ∈h of h(λ) with eigenvalue e(λ) in such a way that φ(λ) and e(λ) depend continuously on λ, and φ(λ = 0) = φ1 i ⊗φ2 j. we are now ready to show that for sufficiently small λ > 0, ⟨ψ(λ, t)|peq|ψ(λ, t)⟩≈0 (79) for all t; that is, ψ(λ, t) is nearly orthogonal to heq for all t, and thus is never in thermal equilibrium. to see this, note first that since φ(0) ≈φ(λ) for sufficiently small λ, and since e−ih(λ)t is unitary, also e−ih(λ)tφ(0) ≈e−ih(λ)tφ(λ) (80) (with error independent of t). since the right hand side equals e−ie(λ)tφ(λ) ≈e−ie(λ)tφ(0) , (81) we have that ⟨e−ih(λ)tφ(0)|peq|e−ih(λ)tφ(0)⟩≈⟨φ(0)|peq|φ(0)⟩= 0 . (82) this proves (79) with an error bound independent of t that tends to 0 as λ →0. another example of "bad" hamiltonians is provided by the phenomenon of anderson localization (see in particular [1, 11]): certain physically relevant hamiltonians possess some eigenfunctions φα that have a spatial energy density function that is macroscopi- cally non-uniform whereas wave functions in heq should have macroscopically uniform energy density over the entire available volume. thus, some eigenfunctions are not close to heq, violating (24). 5 comparison with classical mechanics in classical mechanics, one would expect as well that a macroscopic system spends most of the time in the long run in thermal equilibrium. let us define what thermal equilibrium means in classical mechanics. (we defined it for quantum systems in (8).) we denote a point in phase space by x = (q1, . . . , qn, p1, . . . , pn). instead of the orthogonal decomposition of h into subspaces hν we consider a partition of an energy shell γ in phase space, γ = {x : e ≤h(x) ≤e + δe}, into regions γν corresponding to different macro-states ν, i.e., if the micro-state x of the system is in γν then the macro-state of the system is ν. it has been shown [8] for realistic systems with large n that one of the regions γν, corresponding to the macro-state of thermal equilibrium and denoted γeq, is such that, in terms of the (uniform or liouville) phase space volume measure μ on γ, μ(γeq) μ(γ) ≈1 . (83) 15 though the subspaces hν play a role roughly analogous to the regions γν, a basic difference between the classical and the quantum cases is that while every classical phase point in γ belongs to one and only one γν, and thus is in one macro-state, a quantum state ψ need not lie in any one hν, but can be a non-trivial superposition of vectors in different macro-states. (indeed, almost all ψ do not lie in any one hν. that is why we defined being in thermal equilibrium in terms of ψ lying in a neighborhood of heq, rather than lying in heq itself.) the time evolution of the micro-state x is given by the solution of the hamiltonian equations of motion, which sends x (at time 0) to xt (at time t), t ∈r. we expect that for realistic systems with a sufficiently large number n of constituents and for every macro-state ν, most initial phase points x ∈γν will be such that xt spends most of the time in the set γeq. this statement follows if the system is ergodic,5 but in fact is much weaker than ergodicity. theorem 1 is parallel to this statement in that it implies, for typical hamiltonians, that initial states (here, ψ(0)) out of thermal equilibrium will spend most of the time in thermal equilibrium; it is different in that it applies, for typical hamiltonians, to all, rather than most, initial states ψ(0). 6 comparison with the literature von neumann [17] proved, as his "quantum ergodic theorem," a precise version of normal typicality (defined in section 2.2); his proof requires much more effort, and more refined methods, than our proof of theorem 1. however, his theorem assumes that the dimension dν of each macro-space hν is much smaller than the full dimension d, and thus does not apply to the situation considered in this paper, in which one of the macro-spaces, heq, has the majority of dimensions. the reason von neumann treated the more difficult case of small dν but left out the easier and particularly interesting case of the thermal equilibrium macrostate is that he had in mind a notion of thermal equilibrium different from ours. he thought of a thermal equilibrium wave function ψ, not as one in (or close to) a particular hν, but as one with ∥pνψ∥2 ≈dν/d for every ν, i.e., one for which |ψ⟩⟨ψ| ≈ρmc in a suitable coarse-grained sense. because of this different focus, he did not consider the situation presented here. we also note that von neumann's quantum ergodic theorem makes an assumption on h that we do not need in our theorem 1; this assumption, known as a "no resonances" [6, 16] or "non-degenerate energy gaps" [10] condition, asserts that eα −eβ ̸= eα′ −eβ′ unless ( either α = α′, β = β′ or α = β, α′ = β′ . (84) the schnirelman theorem [2] states that, in the semi-classical limit and under suit- able hypotheses, the wigner distribution corresponding to an eigenstate φα becomes the 5a classical system is ergodic if and only if the time evolved micro-state xt spends, in the long run, a fraction of time in each (measurable) set b ⊆γ that is equal to μ(b)/μ(γ) for μ-almost all x. 16 micro-canonical measure. that is, the φα have a property resembling thermal equilib- rium, similar to our condition (24) expressing that all eigenstates are in thermal equilib- rium. srednicki [15] observed other thermal equilibrium properties in energy eigenstates of example systems, a phenomenon he referred to as "eigenstate thermalization." the results of [16, 12, 10] also concern conditions under which a quantum system will spend most of the time in "thermal equilibrium." for the sake of comparison, their results, as well as ours, can be described in a unified way as follows. let us say that a system with initial wave function ψ(0) equilibrates relative to a class a of observables if for most times τ, ⟨ψ(τ)|a|ψ(τ)⟩≈tr  |ψ(t)⟩⟨ψ(t)|a  for all a ∈a . (85) we then say that the system thermalizes relative to a if it equilibrates and, moreover, tr  |ψ(t)⟩⟨ψ(t)|a  ≈tr ρmca
for all a ∈a , (86) with ρmc the micro-canonical density matrix (in our notation, 1/d times the projection p to h ). with these definitions, the results of [16, 12, 10] can be formulated by saying that, under suitable hypotheses on h and ψ(0) and for large enough d, a system will equilibrate, or even thermalize, relative to a suitable class a . our result is also of this form. we have just one operator in a , namely peq. we establish thermalization for arbitrary ψ(0) assuming h is non-degenerate and satisfies ⟨φα|peq|φα⟩≈1 for all α, which (we show) is typically true. von neumann's quantum ergodic theorem [17] establishes thermalization for a fam- ily a of commuting observables; a is the algebra generated by {m1, . . . , mk} in the notation of section 1.1. he assumes that the dimensions of the joint eigenspaces hν are not too small and not too large; that h obeys (84); he makes an assumption about the relation between h and the subspaces hν that he shows is typically true; and he admits arbitrary ψ(0). see [5] for further discussion. rigol, dunjko, and olshanii [13] numerically simulated an example system and concluded that it thermalizes relative to a certain class a consisting of commuting observables. tasaki [16] as well as linden, popescu, short, and winter [10] consider a system coupled to a heat bath, htotal = hsys ⊗hbath, and take a to contain all operators of the form asys ⊗1bath. tasaki considers a rather special class of hamiltonians and establishes thermalization assuming that max α |⟨φα|ψ(0)⟩|2 ≪1 , (87) a condition that implies that many eigenstates of h contribute to ψ(0) appreciably and that can (more or less) equivalently be rewritten as x α  ⟨φα|ψ(0)⟩  4 ≪1 . (88) 17 under the assumption (88) on ψ(0), linden et al. establish equilibration for h satisfying (84). they also establish a result in the direction of thermalization under the additional hypothesis that the dimension of the energy shell of the bath is much greater than dim hsys. reimann's mathematical result [12] can be described in the above scheme as follows. let a be the set of all observables a with (possibly degenerate) eigenvalues between 0 and 1 such that the absolute difference between any two eigenvalues is at least (say) 10−1000. he establishes equilibration for h satisfying (84), assuming that ψ(0) satisfies (88). acknowledgements. we thank matthias birkner (lmu m ̈ unchen), peter reimann (biele- feld), anthony short (cambridge), avraham soffer (rutgers), and eugene speer (rut- gers) for helpful discussions. s. goldstein was supported in part by national science foundation [grant dms-0504504]. n. zangh` ı is supported in part by istituto nazionale di fisica nucleare. j. l. lebowitz and c. mastrodonato are supported in part by nsf [grant dmr 08-02120] and by afosr [grant af-fa 09550-07]. references [1] p. w. anderson: absence of diffusion in certain random lattices. phys. rev. 109, 1492–1505, 1958. [2] y. colin de verdi` ere: ergodicit ́ e et fonctions propres du laplacien. commun. math. phys. 102, 497–502, 1985. [3] a. dembo, o. zeitouni: large deviations techniques and applications, 2nd ed. springer, new york, 1998. [4] j. m. deutsch: quantum statistical mechanics in a closed system. phys. rev. a 43, 2046–2049, 1991. [5] s. goldstein, j. l. lebowitz, c. mastrodonato, r. tumulka, n. zangh` ı: normal typicality and von neumann's quantum ergodic theorem. http://arxiv.org/abs/0907.0108, 2009. [6] r. jancel: foundations of classical and quantum statistical mechanics. oxford: pergamon, 1969. translation by w. e. jones of les fondements de la m ́ ecanique statistique classique e quantique. paris: gauthier-villars, 1963. [7] t. kato: a short introduction to perturbation theory for linear operators. new york: springer-verlag, 1982. [8] o. e. lanford: entropy and equilibrium states in classical statistical mechanics. in a. lenard (ed.), lecture notes in physics 2, 1–113, springer-verlag, 1973. 18 [9] j. l. lebowitz: from time-symmetric microscopic dynamics to time-asymmetric macroscopic behavior: an overview. in g. gallavotti , w. l. reiter, j. yngva- son (editors), boltzmann's legacy, 63–88. european mathematical society (2007). http://arxiv.org/abs/0709.0724 [10] n. linden, s. popescu, a. j. short, a. winter: quantum mechani- cal evolution towards thermal equilibrium. phys. rev e 79, 061103, 2009. http://arxiv.org/abs/0812.2385 [11] v. oganesyan, d. a. huse: localization of interacting fermions at high tempera- ture. phys. rev. b 75, 155111, 2007. [12] p. reimann: foundation of statistical mechanics under experimentally realistic conditions. phys. rev. lett. 101, 190403, 2008. [13] m. rigol, v. dunjko, m. olshanii: thermalization and its mechanism for generic isolated quantum systems. nature 452, 854–858, 2008. [14] e. schr ̈ odinger: statistical thermodynamics. second edition, cambridge university press, 1952. [15] m. srednicki: chaos and quantum thermalization. phys. rev. e 50, 888, 1994. [16] h. tasaki: from quantum dynamics to the canonical distribution: general pic- ture and a rigorous example. phys. rev. lett. 80, 1373-1376, 1998. [17] j. von neumann: beweis des ergodensatzes und des h-theorems in der neuen mechanik. z. physik 57, 30, 1929. [18] j. von neumann: mathematical foundation of quantum mechanics. princeton university press, 1955. translation of mathematische grundlagen der quanten- mechanik. springer-verlag, berlin, 1932. [19] j. d. walecka: fundamentals of statistical mechanics. manuscript and notes of felix bloch. stanford university press, stanford, ca, 1989. 19

0911.1726

higher-order phase transitions with line-tension effect

the behavior of energy minimizers at the boundary of the domain is of great importance in the van de waals-cahn-hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. this problem, also known as the liquid-drop problem, was studied by modica in [21], and in a different form by alberti, bouchitte, and seppecher in [2] for a first-order perturbation model. this work shows that using a second-order perturbation cahn-hilliard-type model, the boundary layer is intrinsically connected with the transition layer in the interior of the domain. precisely, considering the energies $$ \mathcal{f}_{\varepsilon}(u) := \varepsilon^{3} \int_{\omega} |d^{2}u|^{2} + \frac{1}{\varepsilon} \int_{\omega} w (u) + \lambda_{\varepsilon} \int_{\partial \omega} v(tu), $$ where $u$ is a scalar density function and $w$ and $v$ are double-well potentials, the exact scaling law is identified in the critical regime, when $\varepsilon \lambda_{\varepsilon}^{{2/3}} \sim 1$.

introduction in this paper we seek to estimate the asymptotic behavior of the family of energies ε3 z ω |d2u|2 dx + 1 ε z ω w(u) dx + λε z ∂ω v (tu) dhn−1, where u ∈h2(ω), ωis a bounded open set in rn of class c2, tu is the trace of u on ∂ω, w and v are continuous and non-negative double-well potentials with quadratic growth at infinity, and lim ε→0+ λε = ∞. it is known that the transition layer in the interior of the domain has width of order ε (see [22], [20], [21], [2], [13], [9], [15]). to formally find the order of the width of the transition layer on the boundary, it suffices to study the case n = 2. therefore, by focusing on a neighborhood of a point on the boundary (assuming the boundary is flat), consider a 2 −d energy in the half ball of radius δ centered at that point x0 of the boundary, and changing variables to a fixed domain, e.g. the unit ball, we obtain ε3 δ2 zz b+ |d2u|2 dx dy + δ2 ε zz b+ w(u) dx dy + λεδ z e v (tu) dh1. equi-partition of energy between the first and last terms leads to δ ≈ελ −1 3 ε which, in turn, yields δ2 ε ≈ελ −2 3 ε , which vanishes with ε, which seems to indicate that the middle term will not contribute for the transition on the boundary. one also concludes that on the boundary, the energy will scale as ε3 δ2 ≈λεδ ≈ελ 2 3 ε . hence there are three essential regimes for this energy depending on how the quantity ελ 2 3 ε behaves as ε →0+. in this paper we study the case in which ελ 2 3 ε converges to a finite and strictly positive value. the other two regimes will be treated in a forthcoming paper. consider the functional fε(u) :=    ε3 z ω |d2u|2 dx + 1 ε z ω w(u) dx + λε z ∂ω v (tu) dhn−1 if u ∈h2(ω), ∞ otherwise. (1.1) 1 arxiv:0911.1726v1 [math.ap] 9 nov 2009 2 b. galv ̃ ao-sousa theorem 1.1 (compactness). let ω⊂rn be a bounded open set of class c2 and let w : r →[0, ∞) be such that (hw 1 ) w is continuous and w −1({0}) = {a, b} for some a, b ∈r, a < b; (hw 2 ) w(z) ⩾c|z|2 −1 c for all z ∈r and for some c > 0. let v : r →[0, ∞) be such that (hv 1 ) v is continuous and v −1({0}) = {α, β} for some α, β ∈r, α < β; (hv 2 ) v (z) ⩾c|z|2 −1 c for all z ∈r and for some c > 0; (hv 3 ) v (z) ⩾1 c min  |z −β|, |z −α| 2 for all z ∈(α −ρ, α + ρ) ∪(β −ρ, β + ρ) and for some c, ρ > 0. assume that ελ 2 3 ε →l ∈(0, ∞) as ε →0+ and consider a sequence {uε} ⊂h2(ω) such that supε>0 fε(uε) < ∞. then there exist a subsequence {uε} (not relabeled), u ∈bv ω; {a, b}
, and v ∈bv ∂ω; {α, β}
such that uε →u in l2(ω) and tuε →v in l2(∂ω). the next theorem concerns the critical regime where ε and λε are "balanced", i.e. ελ 2 3 ε ∼1, and all terms play an important role. here λε is large enough to render the energy sensitive to the transition that occurs on the boundary, but not too big as to force the value on the boundary to converge to a constant. we define (i) ea := {x ∈ω: u(x) = a} for all u ∈bv ω; {a, b}
; (ii) m is the energy density per unit area on the transition interfaces between the interior potential wells, precisely, m := inf z r −r w(f(t)) + |f ′′(t)|2
dt : f ∈h2 loc(r), f(−t) = a, f(t) = b for all t ⩾r, r > 0  ; (1.2) (iii) σ is the interaction energy on the transition interface between bulk wells and boundary wells, i.e., σ(z, ξ) := inf z r 0 w(f(t)) + |f ′′(t)|2
dt : f ∈h2 loc (0, ∞)
, f(0) = ξ, f(t) = z for all t ⩾r, r > 0  ; (1.3) (iv) fα := {x ∈∂ω: v(x) = α} for all v ∈bv ∂ω; {α, β}
; (v) c is a lower bound to the energy on a transition interface between the wells of the boundary potential, c := inf ( 1 8 z r −r z r −r |f ′(x) −f ′(y)|2 |x −y|2 dx dy + z r −r v f(x)
dx : f ∈h 3 2 loc(r), f ′ ∈h 1 2 (r), f(−t) = α, f(t) = β for all t ⩾r, r > 0 o ; (1.4) (vi) c is an upper bound to the energy on a transition interface between the wells of the boundary potential, c := inf  7 16 z ∞ −∞ z ∞ −∞ |f ′(x) −f ′(y)|2 |x −y|2 dx dy + z ∞ −∞ v f(x)
dx : f ∈h 3 2 loc(r), f(−t) = α, f(t) = β for all t ⩾r, r > 0 o . (1.5) theorem 1.2 (critical case). under the same hypotheses of theorem 1.1 the following statements hold: (i) (lower bound) for every u ∈bv (ω; {a, b}) and v ∈bv (∂ω; {α, β}) and for every sequence {uε} ⊂h2(ω) such that uε →u in l2(ω), tuε →v in l2(∂ω), we have lim inf ε→0+ fε(uε) ⩾mperω(ea) + x z=a,b x ξ=α,β σ(z, ξ)hn−1{tu = z} ∩{v = ξ}
+ clper∂ω(fα); higher-order phase transitions with line-tension effect 3 (ii) (upper bound) for every u ∈bv (ω; {a, b}) and v ∈bv (∂ω; {α, β}), there exists a sequence {uε} ⊂h2(ω) such that uε →u in l2(ω), tuε →v in l2(∂ω), and lim sup ε→0+ fε(uε) ⩽mperω(ea) + x z=a,b x ξ=α,β σ(z, ξ)hn−1{tu = z} ∩{v = ξ}
+ clper∂ω(fα). the main results, theorems 1.1 and 1.2, imply, in particular, that min a<- r ωu dx<b α<- r ∂ωv dhn−1<β fε = o(1) as ε →0+, where we impose a mass constraint to avoid trivial solutions which yield no energy. note that these conditions pose no difficulties to the γ-convergence due to the strong convergence of uε and tuε. thus we identify the precise scaling law for the minimum energy in the parameter regime ελ 2 3 ε ∼1. observe that, although theorem 1.2 does not prove that the sequence {fε}ε>0 γ-converges as ε →0+, since the constants of the lower and upper bounds for the last transition term do not match, we can apply theorem 8.5 from [18] to prove that there exists a subsequence εn →0+ such that the corresponding subsequence of functionals γ-converges. hence theorem 1.2 shows that the limiting functional concentrates on the three different kinds of transition layers: an interior transition layer of dimension n −1, where the limiting value of u makes the transition between a and b; the boundary of the domain, also of dimension n −1, where there is the transition between the interior phases a and b and the boundary phases α and β; and a transition interface on the boundary, of dimension n −2, where the limiting value of the trace tu makes the transition between α and β. the difficulties in proving a γ-convergence result arise mainly from the nature of the functional under consideration. on one hand, the energy involves second-order derivatives, which prevents us from following the usual techniques in phase transitions, such as truncation and rearrangement arguments to obtain monotonically increasing test functions for the constant c. in [2], these techniques are crucial to find a test function that matches both the lifting constant and the optimal profile problem for the boundary wells. on the other hand, for the boundary term, the functionals are also nonlocal. thus the estimates for the recovery sequence have to be sharper, since the nonlocality extends its contribution beyond the characteristic length of the phase transition. the usual methods for localization make use of truncation arguments, which do not apply in this setting due to the fact that the fractional seminorm is of higher-order. similar difficulties can also be found in the papers [6,7,8,5] where, similarly, the γ-convergence is not established. the difference between the constants c and c arises from two factors. first, from proposition 2.9 it does not follow that the lifting constant is independent of the value of the trace g. and second, when estimating the upper bound for the recovery sequence, the transition between α and β is accomplished on a layer of thickness δε = o(ε). so we rescale the integrals by δε, but because of the non-locality of the fractional energy, it obtains a contribution from a layer of thickness ε, which after rescaling becomes of thickness ε/δε →∞. this accounts for the fact that the integration limits of the constant c extend to infinity, while for c they are bounded. the proofs of theorems 1.1 and 1.2 are divided through the next sections. we begin by studying two auxiliary one-dimensional problems. more precisely, let i, j ⊂r be two open intervals and define the following functionals fε(u; i) :=    ε3 z i |u′′(x)|2 dx + 1 ε z i w u(x)
dx if u ∈h2(i), ∞ otherwise, (1.6) and gε(v; j) :=      ε3 8 z j z j  v′(x) −v′(y)  2 |x −y|2 dx dy + λε z j v v(x)
dx if v ∈h 3 2 (j), ∞ otherwise. (1.7) in sections 4.1 and 4.2 we prove a compactness result and a lower bound for fε which follows the techniques developed in [13]. in section 4.3 we will prove a compactness result for gε, while in section 4.4 we will prove a lower bound by finding "good points" x± i such that most of the transition energy is concentrated between x− i and x+ i and we modify 4 b. galv ̃ ao-sousa the original sequence {un} on a small set to be admissible for c. in section 5.1 we will prove theorem 1.1 in the critical regime using a slicing argument to reduce the compactness in the interior to the auxiliary problem studied in section 4.1, and analogously, we reduce the compactness on the boundary to the one-dimensional problem for gε studied in section 4.3. in section 5.2 we prove the lower bound result for theorem 1.2 using the fact that the energy concentrates in different mutually singular sets. finally, in section 5.3 we prove the upper bound for theorem 1.2. from theorem 1.2, we deduce the following corollary. corollary 1.3. under the same hypotheses of theorem 1.1, and assuming that α = β, then the sequence {fε}ε>0 γ-converges as ε →0+ to f0(u) :=      mperω(ea) + x z=a,b σ(z, α)hn−1{tu = z}
if u ∈bv (ω; {a, b}), ∞ otherwise, where m is defined as in (1.2) and σ is defined as in (1.3). from the result of theorem 1.2, we know that the γ-limit of the functionals fε as ε →0+ will concentrate its energy on three surfaces: the discontinuity surface of u, the boundary ∂ω, and the discontinuity surface of v. moreover, we know the precise energy of the first two terms. for the last term, we expect it to be the product of the perimeter of the surface times the value c of the transition between the two boundary preferred phases α and β. since the fractional norm on the boundary is non-local, the definition of c should span the whole real line and the lifting constant should be independent of the function g, as in the first-order case (see [2]). we offer the following conjecture. conjecture. under the same hypotheses of theorem 1.1, then the sequence {fε}ε>0 γ-converges as ε →0+ to f0(u, v) :=      mperω(ea) + x z=a,b x ξ=α,β σ(z, ξ)hn−1{tu = z} ∩{v = ξ}
+ clper∂ω(fα) if (u, v) ∈v, ∞ otherwise, where v := bv (ω; {a, b}) × bv (∂ω; {α, β}), m is defined as in (1.2), σ is defined as in (1.3), and c is defined by c := inf  ζ z ∞ −∞ z ∞ −∞ |f ′(x) −f ′(y)|2 |x −y|2 dx dy + z ∞ −∞ v f(x)
dx : f ∈h 3 2 loc(r), lim x→∞f(−x) = α, lim x→∞f(x) = β  , (1.8) and ζ is defined by ζ := inf      rr r×r+  d2u(x, y)  2 dx dy r r r r  g′(x)−g′(y)  2 |x−y|2 dx dy : u ∈h2(r × r+), tu(*, 0) = g in r      , (1.9) which is independent of g ∈h 3 2 loc(r) such that lim g(−x) = α as x →∞and lim g(x) = β as x →∞. 2. preliminaries 2.1. slicing we now show a slicing argument introduced by [2] and improved in [13]. first we fix some notation. given a bounded open set a ⊂rn, a unit vector e in rn, and a function u : a →r, we denote by m the orthogonal complement of e, ae the projection of a onto m, ay e := {t ∈r : y + te ∈a}, for all y ∈ae, uy e the trace of u on ay e, i.e., uy e(t) := tu(y + te), for all y ∈ae. definition 2.1. for every δ > 0, two sequences {vε}, {wε} ⊂l1(e) are said to be δ−close if for every ε > 0 ∥vε −wε∥l1(e) < δ. higher-order phase transitions with line-tension effect 5 proposition 2.2. assume that e is a lipschitz, bounded and open subset of rn−1. if {wε} ⊂l1(e) is equi-integrable and if there are n −1 linearly independent unit vectors ei such that for every δ > 0 and for every fixed i = 1, . . . , n −1, there exist a sequence {vε} (depending on i) that is δ−close to {wε} with {vy ε} precompact in l1(ey ei) for hn−2-a.e. y ∈eei, then {wε} is precompact in l1(e). 2.2. fractional order sobolev spaces we will use the norms and seminorms of several fractional order spaces, introduced by besov and nikol'skii and summarized in [1] and [27]. consider the following norms and seminorms for the space w 3 2 ,2(j) where j ⊂r is an open interval. |u|2 h 1 2 (j) := z j z j  u(x) −u(y)  2 |x −y|2 dx dy, |u|2 h 3 2 (j) := z j z j  u(x) −2u x+y 2
+ u(y)  2 |x −y|4 dx dy, ∥u∥2 w 3 2 ,2(j) := ∥u∥2 h1(j) + |u′|2 h 1 2 (j), ∥u∥2 h 3 2 (j) := ∥u∥2 l2(j) + |u|2 h 3 2 (j). we will need to compare the two seminorms and for that we invoke an auxiliary result (see [12,25]). proposition 2.3. let r > 1 and let u : (a, b) − →[0, ∞] be a borel function. then z b a 1 (x −a)r z x a u(y) dy  dx ⩽ 1 r −1 z b a u(x) (x −a)r−1 dx. lemma 2.4. let j ⊂r be an open interval and let u ∈h 3 2 (j). then |u|2 h 3 2 (j) ⩽1 8|u′|2 h 1 2 (j). proposition 2.5 (gagliardo-nirenberg-type inequality). let j ⊂r be an open interval. then there exists c = c(j) > 0 such that ∥u∥h1(j) ⩽c  ∥u∥ 1 3 l2(j)|u′| 2 3 h 1 2 (j) + ∥u∥l2(j)  for all u ∈h 3 2 (j). we recall two inequalities due to gagliardo and nirenberg (see [14,24]). proposition 2.6. let ω⊂rn be a bounded open set satisfying the cone property. if u ∈l2(ω) and ∇2u ∈l2(ω), then u ∈h2(ω) and ∥∇u∥l2(ω) ⩽cln(ω)  ∥u∥ 1 2 l2(ω)∥∇2u∥ 1 2 l2(ω;rn×n) + ∥u∥l2(ω)  , where c > 0 is independent of u and ω. proposition 2.7. let j ⊂r be an open bounded interval. if u ∈l1(j) and u′′ ∈l2(j) then u ∈h2(j) and ∥u′∥l 4 3 (j) ⩽c  ∥u∥ 1 2 l1(j)∥u′′∥ 1 2 l2(j) + ∥u∥l1(j)  , for some constant c > 0. 6 b. galv ̃ ao-sousa 2.3. lifting inequalities we need to relate the l2 norm of the hessian with its equivalent on the boundary, i.e., the h 1 2 fractional seminorm of the derivative of the trace. in this section, we estimate the ratio between these two seminorms. we start with an auxiliary lemma from [10]. lemma 2.8. let 1 ⩽p < ∞, let e ⊂rn and f ⊂rm be measurable sets and let u ∈lp(e × f). then z f z e |u(x, y)| dx p dy  1 p ⩽ z e z f |u(x, y)|p dy  1 p dx. proposition 2.9. let g ∈h 3 2 (0, r) and consider the triangle t + r := {(x, y) ∈r2 : 0 < y < r 2 , y < x < r −y}. then, 1 8 ⩽ζr,g := inf      rr t + r  d2u(x, y)  2 dx dy r r 0 r r 0  g′(x)−g′(y)  2 |x−y|2 dx dy : u ∈h2(t + r ), tu(*, 0) = g in (0, r)      ⩽7 16. (2.1) proof. we divide the proof in two steps. step 1: upper bound. define the diamond tr :=  (x, y) ∈r2 : 0 ⩽x ⩽r, |y| ⩽min{x, r −x} . (2.2) given a function g ∈h 3 2 (0, r), we lift it to the diamond tr by u(x, y) := 1 2y z x+y x−y g(t) dt. we are only interested in the lifting on the positive part of the diamond, i.e., on the triangle t + r , but observe that u(x, *) is even, and we will take advantage of that fact for some estimates. since g is continuous, one deduces immediately that u is continuous and tu′(x, 0) = lim y→0+ ∂u ∂x(x, y) = lim y→0+ g(x + y) −g(x −y) 2y = g′(x). moreover, ∂2u ∂x2 (x, y) = g′(x + y) −g′(x −y) 2y , ∂2u ∂x∂y (x, y) = g′(x + y) + g′(x −y) 2y −g(x + y) −g(x −y) 2y2 , ∂2u ∂y2 (x, y) = g′(x + y) −g′(x −y) 2y −g(x + y) + g(x −y) y2 + 1 y3 z x+y x−y g(t) dt. we can easily deduce that ∂2u ∂x2 2 l2(t + r ) = 1 4|g′|2 h 1 2 (0,r), and note that ∂2u ∂x∂y (x, y) = 1 2y2 z y 0 (g′(x + y) −g′(s + x) + g′(x −y) −g′(s + x −y)) ds. use hardy's inequality from proposition 2.3 to obtain ∂2u ∂x∂y 2 l2(t + r ) ⩽1 16|g′|2 h 1 2 (0,r). finally, notice that ∂2u ∂y2 (x, y) = 1 y3 z y 0 f2(r; x, y) dr, higher-order phase transitions with line-tension effect 7 where f2(r; x, y) := z x+y r+x (g′(x + y) −g′(s)) ds + z r+(x−y) x−y (g′(s) −g′(x −y)) ds. using hardy's inequality in propo- sition 2.3 again, we deduce that ∂2u ∂y2 2 l2(t + r ) ⩽1 16|g′|2 h 1 2 (0,r). we finally put the three estimates for the partial derivatives of u of second order together to obtain zz t + r |∇2u|2 dx dy ⩽7 16|g′|2 h 1 2 (0,r). step 2: lower bound in (2.1) case 1: assume that v ∈l1(t + r ; r2) ∩c∞(t + r ; r2) is such that ∇v ∈l2(t + r ; r2×2). first it is easy to prove that     v(x + y, 0) −v(x −y, 0) 2y     2 ⩽1 2 z 1 0  ∇v(x + y −ty, ty)   dt + z 1 0  ∇v(x −y + ty, ty)   dt 2 . by estimating the right-hand side using lemma 2.8 and minkowski inequality, we obtain |v(*, 0)|2 h 1 2 (0,r) ⩽8∥∇v∥2 l2(t + r ). case 2: assume that v ∈l1(t + r ; r2) is such that ∇v ∈l2(t + r ; r2×2). first by reflection, extend the function to v ∈l1(tr; r2) with ∇v ∈l2(tr; r2×2). let φε be the standard mollifiers and consider vε := v ⋆φε defined in t ε r :=  (x, y) ∈tr : d (x, y), ∂tr
> ε . then vε →u in l1 loc(a; r2), ∇vε →∇v in l2(a; r2×2) and vε(*, 0) →tv in l1a ∩(r × {0}); r2
for any open set a ⋐tr. we can find a subsequence (not relabeled) such that vε(x, 0) →tv(x) for l1-a.e. x ∈a ∩(r × {0}). then by case 1, we have z a∩(r×{0}) z a∩(r×{0})     tv(x) −tv(y) x −y     2 dx dy ⩽lim inf ε→0+ z a∩(r×{0}) z a∩(r×{0})     vε(x, 0) −vε(y, 0) x −y     2 dx dy ⩽8 lim ε→0+ zz a∩t + r |∇vε|2 dx dy = 8 zz a∩t + r |∇v|2 dx dy. let an ⊂an+1 ⋐tr be such that tr = s an. then one deduces that z r 0 z r 0     tv(x) −tv(y) x −y     2 dx dy ⩽8 zz t + r |∇v|2 dx dy. apply this result to v := ∇u to deduce z r 0 z r 0     g(x) −g(y) x −y     2 dx dy ⩽8 zz t + r |∇2u|2 dx dy, which proves the lower bound in (2.1). 2.4. slicing on bv we use here the same notation as in section 2.1. theorem 2.10 (slicing of bv functions). let u ∈l1(ω). then u ∈bv (ω) if and only if there exist n linearly independent unit vectors ei such that uy ei ∈bv (ωy ei) for ln−1-a.e. y ∈ωei and z ωy ei |duy ei|(ωy ei) dy < ∞ for all i = 1, . . . , n. we state an immediate corollary of theorem 1.24 from [16]. 8 b. galv ̃ ao-sousa proposition 2.11. let ω⊂rn be a bounded open lipschitz set and let e ⊂ωbe a set of finite perimeter. then there are sets en ⊂ωof class c2 such that ( ln(e△en) →0, hn−1(∂e△∂en) →0. (2.3) proposition 2.12 (see section 5.10 in [11]). let a ⊂rn be an open set, let e ⊂a be a borel set, let e be an arbitrary unit vector, and e has finite perimeter in a. then ey e has finite perimeter in ay e and ∂ey e ∩ay e = (∂e ∩a)y e, and z ae h0(∂ey e ∩ay e) dy = z a∂e∩a ⟨νe, e⟩dhn−1. conversely, e has finite perimeter in a if there exist n linearly independent unit vectors ei, i = 1, . . . , n such that z aei h0(∂ey ei ∩ay ei) dy < ∞ for all i = 1, . . . , n. 2.5. functions of bounded variation on a manifold we consider several spaces of functions with domains a ⊂rn which are not open. specifically, a will be the boundary of an open and bounded set ωof class c2 and so it will be a compact riemannian manifold (without boundary) of class c2 and dimension n −1 in rn. such a manifold is endowed with a unit normal field ν which is continuous and defined for every x ∈a. in this section we give a brief definition of these spaces. for more details see [3,11,17]. the space of integrable functions on a manifold. let a ⊂rn be a compact riemannian manifold (without boundary) of class c1 and dimension n −1 and define the restriction measure hn−1⌊a(e) := hn−1(e ∩a). a function v is said to be integrable on a, and we write v ∈l1(a; hn−1⌊a), if and only if v is hn−1⌊a-measurable and hn−1⌊a-summable, precisely v−1(j) is hn−1⌊a-measurable for every open set j ⊂r; z a |v(x)| dhn−1(x) < ∞. the space of functions of bounded variation on a manifold. we give a short introduction to the space of functions of bounded variation on a manifold. for more details we refer to [19]. let t ⋆a be the cotangent bundle of a and let γ(t ⋆a) be the space of 1-forms on a. then, given a function v ∈l1(a), define the variation of v by |dv|(a) := sup z a v div w dhn−1 : w ∈γc(t ⋆a), |w| ⩽1  . (2.4) then v ∈l1(a) is said to be a function of bounded variation, i.e., v ∈bv (a) if |dv|(a) < ∞. moreover, if v = χe for some set e ⊂a, then e has finite perimeter if and only if v ∈bv (a), and pera(e) = |dv|(a) = hn−2(e ∩a) < ∞. proposition 2.13. let ω⊂rn be an open bounded set of class c2 and let e ⊂∂ωbe a set of finite perimeter with respect to hn−2. then there are sets en ⊂∂ωof class c2 such that ( hn−1(e△en) →0, hn−2(∂∂ωe△∂∂ωen) →0. higher-order phase transitions with line-tension effect 9 3. characterization of constants lemma 3.1. assume that v : r →[0, ∞) satisfies (hv 1 ) −(hv 3 ). then the constant c defined in (1.4) belongs to (0, ∞). proof. assume by contradiction that c = 0. then there exist two sequences {fn} ⊂h 3 2 loc(r) and {rn} ⊂(0, ∞) satisfying fn(−x) = α, fn(x) = β for all x ⩾rn, (3.1) 1 8 z rn −rn z rn −rn |f ′ n(x) −f ′ n(y)|2 |x −y|2 dx dy + z rn −rn v fn(x)
dx n→∞ − − − − →0. (3.2) let 0 < 2δ < β −α. since fn(−rn) = α, fn(rn) = β, and fn is continuous, there exists an interval (sn, tn) such that fn(sn) = α + δ < β −δ = fn(tn), fn [sn, tn]
= [α + δ, β −δ]. (3.3) by (hv 1 ) and the continuity of v we have that cδ := min z∈[α+δ,β−δ] v (z) > 0. then by (3.2), 0 = lim n→∞ z rn −rn v (fn(x)) dx ⩾lim n→∞ z tn sn v (fn(x)) dx ⩾lim inf n→∞cδ(tn −sn), and so tn −sn →0. for any t ∈[0, 1], define gn(t) := fn tnt + sn(1 −t)
. then gn(0) = α + δ and g(1) = β −δ. changing variables in (3.2) yields z tn sn z tn sn |f ′ n(x) −f ′ n(y)|2 |x −y|2 dx dy = 1 (tn −sn)2 z 1 0 z 1 0 |g′ n(s) −g′ n(t)|2 |s −t|2 ds dt →0. this implies that    g′ n tn−sn    h 1 2 (0,1) →0, and so, up to a subsequence (not relabeled), g′ n tn−sn →constant in l2(0, 1). since tn −sn →0, this implies that g′ n →0 in l2(0, 1). on the other hand, 0 < β −δ −(α + δ) = gn(1) −gn(0) = z 1 0 g′ n(t) dt. letting n →∞, we obtain a contradiction. this shows that c > 0. to prove that c < ∞, take any function f ∈c2 such that f(t) ⩽α for t ⩽−1 and f(t) = β for t ⩾1. it is easy to verify that the energy is finite. remark. from the proof of the previous lemma, it follows that for every 0 < δ < β−α 2 , the constant cδ := inf 1 8 z t s z t s |f ′(x) −f ′(y)|2 |x −y|2 dx dy + z t s v f(x)
dx : f ∈h 3 2 loc(r), f(s) = α + δ, f(t) = β −δ, fn (sn, tn)
= [α + δ, β −δ], for some s, t ∈r  (3.4) also belongs to (0, ∞). lemma 3.2. define the constant c as before by c := inf  7 16 z ∞ −∞ z ∞ −∞ |f ′(x) −f ′(y)|2 |x −y|2 dx dy + z ∞ −∞ v f(x)
dx : f ∈h 3 2 loc(r), f(−t) = α, f(t) = β, for all t ⩾r, r > 0 o , where v satisfies the properties of theorem 1.1. then c ∈(0, ∞). 10 b. galv ̃ ao-sousa proposition 3.3. under the conditions of theorem 1.1, c = c⋆, where c⋆is defined by c⋆:= inf ( 3 2 5 3 z 1 −1 z 1 −1 |g′(x) −g′(y)|2 |x −y|2 dx dy  1 3 z 1 −1 v g(x)
dx  2 3 : g ∈h 3 2 loc(r), g′ ∈h 1 2 (r), g(−t) = α, g(t) = β for all t ⩾1 o . proof. first we prove that c ⩾c⋆. let η > 0, and f ∈h 3 2 loc(r), r > 0 be such that f ′ ∈h 1 2 (r), f(−t) = α, f(t) = β, for all t ⩾r, 1 8 z r −r z r −r |f ′(x) −f ′(y)|2 |x −y|2 dx dy + z r −r v f(x)
dx ⩽c + η. then c + η ⩾ 1 8r2 z 1 −1 z 1 −1 |(f(rx))′ −(f(ry))′|2 |x −y|2 dx dy + r z 1 −1 v f(rx)
dx ⩾ 1 8s2 r z 1 −1 z 1 −1 |g′ r(x) −g′ r(y)|2 |x −y|2 dx dy + sr z 1 −1 v gr(x)
dx ⩾c⋆ where gr(x) = f(rx) which is admissible for c⋆, and sr = arg min s>0  1 8s2 z 1 −1 z 1 −1 |g′ r(x) −g′ r(y)|2 |x −y|2 dx dy + s z 1 −1 v gr(x)
dx  =      z 1 −1 z 1 −1 |g′ r(x) −g′ r(y)|2 |x −y|2 dx dy 4 z 1 −1 v gr(x)
dx      1 3 . let η →0+ to deduce that c ⩾c⋆. the converse inequality follows trivially from following the first part of the proof from the end to the beginning. proposition 3.4. under the conditions of theorem 1.1, c = c⋆, where c⋆is defined by c⋆:= inf ( 3 * 7 1 3 4 z ∞ −∞ z ∞ −∞ |g′(x) −g′(y)|2 |x −y|2 dx dy  1 3 z ∞ −∞ v g(x)
dx  2 3 : g ∈h 3 2 loc(r), g(−t) = α, g(t) = β, for all t ⩾1 o . 4. two auxiliary one-dimensional problems 4.1. compactness for fε theorem 4.1. assume that w : r →[0, ∞) satisfies (hw 1 ) −(hw 2 ). let i ⊂r be an open, bounded interval, let {εn} be a positive sequence converging to 0, and let {un} ⊂h2(i) be such that sup n fεn(un; i) < ∞. (4.1) then there exist a subsequence (not relabeled) of {un} and a function u ∈bv i; {a, b}
such that un →u in l2(i). proof. given a sequence {un} ⊂h2(i) satisfying (4.1), by the compactness result in [13] and (hw 2 ), we obtain a subsequence {un} (not relabeled) and a function u ∈bv i; {a, b}
such that un →u in l2(i). higher-order phase transitions with line-tension effect 11 4.2. lower bound for fε theorem 4.2 (lower bound estimate for fε). let i ⊂r be an open and bounded interval and let w : r → [0, ∞) satisfy (hw 1 ) −(hw 2 ). let u ∈bv i; {a, b}
, let v ∈bv ∂i; {α, β}
, and let {uε} ⊂h2(i) be such that sup ε fε(uε; i) =: c < ∞, uε →u in l2(i) and tuε →v in h0(∂i). then lim inf ε→0+ fε(uε; i) ⩾mh0(s(u)) + z ∂i σ tu(x), v(x)
dh0(x), where m and σ are defined in (1.2) and (1.3), respectively. proof. passing to a subsequence (not relabeled), we can assume that lim inf ε→0+ fε(uε; i) = lim ε→0+ fε(uε; i). since uε →u in l1(i) and ∥w(uε)∥l1(i) ⩽cε, by the growth condition (hw 2 ), we have that, up to a subsequence (not relabeled), uε →u in l2(i), and supε ∥uε∥l2(i) ⩽c. in turn, by proposition 2.6 and the fact that ∥u′′ ε∥l2(i) ⩽cε−3 2 , we deduce that ∥u′ ε∥l2(i) ⩽cε−3 4 , and so lim ε→0+ z i |εu′ ε(x)|2 dx = 0. thus, up to a subsequence (not relabeled), we may assume that εu′ ε(x) →0, and uε(x) →u(x) (4.2) for l1-a.e. x ∈i. since u ∈bv (i; {a, b}), its jump set is a finite set, so we can write s(u) := {s1, . . . , sl}, where s0 := inf i < s1 < * * * < sl< sl+1 := sup i. fix 0 < η < β−α 2 , and 0 < δ0 := 1 2 min {si+1 −si : i = 0, . . . , l}. using (4.2), for every i = 1, . . . , l, we may find x± i ∈(si −δ0, si + δ0) such that |uε(x+ i ) −b| < η, |uε(x− i ) −a| < η, and |εu′ ε(x± i )| < η. (4.3) moreover, since u is a constant in (s0, s1), we assume that u(x) ≡a in this interval (the case u(x) ≡b is analogous), and we have that tu(s0) = a. using (4.2) once more, we may find a point x+ 0 such that |uε(x+ 0 ) −a| < η, and |εu′ ε(x+ 0 )| < η (4.4) for all ε sufficiently small. on the other hand, tuε(s0) →v(s0), and so |tuε(s0) −v(s0)| < η 2 for all ε sufficiently small. since limx→s+ 0 uε(x) = tuε(s0), there is 0 < ρε < x+ 0 < s0 such that |uε(x) −v(s0)| < η for all x ∈(s0, s0 + ρε). there are now two cases. if uε(xε) = v(s0) for some xε ∈(s0, x+ 0 ), then take x− 0,ε := xε. if uε(xε) ̸= v(s0) for all xε ∈(s0, x+ 0 ), then we claim that there exists x− 0,ε ∈(s0, x+ 0 ) such that u′ ε(x− 0,ε) uε(x− 0,ε) −v(s0)
> 0. indeed, if say uε(x) > v(s0) in (s0, x+ 0 ), then for η > 0 such that |v(s0) −a| > 2η, we have that  uε(x+ 0 ) −v(s0)   ⩾|v(s0) −a| −|uε(x+ 0 ) −a| > η, and so there exists a first point xε ∈(s0, x+ 0 ) such that uε(xε) = v(s0) + η. hence, by the mean value theorem, there is x− 0,ε ∈(s0, xε) such that u′ ε(x0,ε) = uε(xε) −tuε(s0) xε −s0 > v(s0) + η −v(s0) −η 2 xε −s0 > 0. thus, we have found x0,ε ∈(s0, x+ 0 ) such that |uε(x− 0,ε) −v(s0)| < η and u′ ε(x− 0,ε) uε(x− 0,ε) −v(s0)
⩾0, (4.5) 12 b. galv ̃ ao-sousa for all ε sufficiently small. for simplicity of notation, we write x− 0 := x− 0,ε and x− l+1 := x− l+1,ε. from the facts that the intervals [x− i , x+ i ] are disjoint for i = 0, . . . , l+ 1, and that w is nonnegative, we have that fε(uε; i) ⩾ l+1 x i=0 z x+ i x− i  ε3 u′′ ε(x)  2 + 1 εw uε(x)
 dx. (4.6) we claim that z x+ i x− i  ε3 u′′ ε(x)  2 + 1 εw uε(x)
 dx ⩾ml−o(η) −o(ε) (4.7) for all i = 1, . . . , l, that z x+ 0 x− 0  ε3 u′′ ε(x)  2 + 1 εw uε(x)
 dx ⩾σ tu(s0), v(s0)
−o(η) −o(ε), (4.8) and that z x+ l+1 x− l+1  ε3 u′′ ε(x)  2 + 1 εw uε(x)
 dx ⩾σ tu(sl+1), v(sl+1)
−o(η) −o(ε). (4.9) if (4.7), (4.8), and (4.9) hold, then from (4.6) we deduce that lim inf ε→0+ fε(uε; i) ⩾ml+ σ tu(s0), v(s0)
+ σ tu(sl+1), v(sl+1)
−o(η). letting η →0+ yields lim inf ε→0+ fε(uε; i) ⩾ml+ z ∂i σ tu(x), v(x)
dh0(x). the remaining of the proof is devoted to the proof of (4.7), (4.8), and (4.9). step 1. proof of (4.7). define the functions g(w, z) := inf z 1 0 w g(x)
+  g′′(x)  2 dt : g ∈c2[0, 1]; r
, g(0) = w, g(1) = b, g′(0) = z, g′(1) = 0  , (4.10) h(w, z) := inf z 1 0 w h(x)
+  h′′(x)  2 dt : h ∈c2[0, 1]; r
, h(0) = a, h(1) = w, h′(0) = 0, h′(1) = z  . (4.11) note that, considering third-order polynomials, one deduces that these functions satisfy lim (w,z)→(b,0) g(w, z) = 0, lim (w,z)→(a,0) h(w, z) = 0. (4.12) from (4.3), for ε sufficiently small, we have g uε(x+ i ), εu′ ε(x+ i )
, h uε(x− i ), εu′ ε(x− i )
⩽η. by (4.10) and (4.11), we can find admissible functions b gi and b hi for g uε(x+ i ), εu′ ε(x+ i )
and h uε(x− i ), εu′ ε(x− i )
, respectively, such that z 1 0  b gi ′′(x)  2 + w b gi(x)
dx ⩽g uε(x+ i ), εu′ ε(x+ i )
+ η ⩽2η, (4.13) z 1 0   b hi ′′(x)  2 + w b hi(x)
dx ⩽h uε(x− i ), εu′ ε(x− i )
+ η ⩽2η. (4.14) we now rescale and translate these functions, precisely, gi(x) := b gi  x −x+ i ε  , hi(x) := b hi  x −x− i ε + 1  . higher-order phase transitions with line-tension effect 13 define wε,i(x) :=                    b if x ⩾x+ i ε + 1, gi(t) if x+ i ε ⩽x ⩽x+ i ε + 1, uε(εx) if x− i ε ⩽x ⩽x+ i ε , hi(t) if x− i ε −1 ⩽x ⩽x− i ε , a if x ⩽x− i ε −1. by construction wε,i ∈h2 x− i ε −1, x+ i ε + 1
and wε,i is admissible for the constant m given in (1.2). hence for all ε sufficiently small, z x+ i x− i  ε3 u′′ ε(x)  2 + 1 εw uε(x)
 dx = z x+ i ε x− i ε  w′′ ε,i(y)  2 + w wε,i(y)
 dy = z x+ i ε +1 x− i ε −1  w′′ ε,i(y)  2 + w wε,i(y)
 dy − z 1 0  b gi ′′(y)  2 + w b gi(y)
 dy − z 1 0   b hi ′′(y)  2 + w b hi(y)
 dy ⩾ml−4η, where we used (4.13) and (4.14). step 2. proof of (4.8). define the functions l(w, z) := inf z 1 0 w f(x)
+  f ′′(x)  2 dt : f ∈c2[0, 1]; r
, f(0) = w, f(1) = a, f ′(0) = z, f ′(1) = 0  , (4.15) j(w, z) := inf z r 0 w j(x)
+  j′′(x)  2 dt : j ∈c2[0, r]; r
, j(0) = v(0), j(r) = w, j′(r) = z, for some r > 0  . (4.16) analogously to (4.10), lim (w,z)→(a,0) l(w, z) = 0, and from (4.4), for all ε sufficiently small we have l uε(x+ 0 ), εu′ ε(x+ 0 )
⩽ η. hence we can find an admissible function b f0 for l uε(x+ 0 ), εu′ ε(x+ 0 )
such that z 1 0 w b f0(x)
+   b f0 ′′(x)  2 dx ⩽l uε(x+ 0 ), εu′ ε(x+ 0 )
+ η ⩽2η. (4.17) we now prove that lim w→v(s0) z(w−v(s0))⩾0 j(w, z) = 0. (4.18) fix η > 0 and let w, z ∈r be such that |w −v(0)| < η and z(w −v(0)) ⩾0. if |z| ⩽√η, then take j(x) := w + z(x −r) + (v(0)−w)+rz r2 (x −r)2, which is admissible for j(w, z), to obtain j(w, z) ⩽c  r + (v(0) −w + rz)2 r3  ⩽c  r + η2 r3 + η r  . choosing r = √η, we deduce that j(w, z) = o(√η). if |z| ⩾√η, then let r := w−v(0) z > 0, which satisfies 0 < r < √η. then, let j(x) := w + z(x −r), which is admissible for j(w, z) because j(0) = w −zr = v(0) and j′′(x) = 0, so j(w, z) = o(√η). this proves (4.18). by (4.18) and (4.5), we may find a function j admissible for j uε(x− 0,ε), εu′ ε(x− 0,ε)
such that z r 0 w j(t)
+  j′′(t)  2 dt ⩽j uε(x− 0,ε), εu′ ε(x− 0,ε)
+ η ⩽2η, (4.19) for some r = r(η) > 0. 14 b. galv ̃ ao-sousa set f0(x) := b f0  x −r − x+ 0 −x− 0,ε ε  , and define wε,0(x) :=              a if x ⩾1 + r + x+ 0 −x− 0,ε ε , f0(x) if r + x+ 0 −x− 0,ε ε ⩽x ⩽1 + r + x+ 0 −x− 0,ε ε , uε ε(x −r) + x− 0,ε
if r ⩽x ⩽r + x+ 0 −x− 0,ε ε , j(x) if 0 ⩽x ⩽r. by construction wε,0 belongs to h2 loc(0, ∞) and is admissible for σ tu(s0), v(s0)
as defined in (1.3). hence for all ε sufficiently small, we have that z x+ 0 x− 0  ε3 u′′ ε(x)  2 + 1 εw uε(x)
 dx = z r+ x+ 0 −x− 0 ε r  w′′ ε,0(y)  2 + w wε,0(y)
 dy = z 1+r+ x+ 0 −x− 0 ε 0  w′′ ε,0(y)  2 + w wε,0(y)
 dy − z 1 0   b f0 ′′(y)  2 + w b f0(y)
 dy − z r 0  j′′(y)  2 + w j(y)
 dy ⩾σ tu(s0), v(s0)
−4η, where we have used (4.17) and (4.19). this proves (4.8). the proof of (4.9) is analogous. 4.3. compactness for gε to prove compactness for the functional gε defined in (1.7), we begin with an auxiliary result. lemma 4.3. let θ ∈l1(j; [0, 1]) and let x := ( x ∈j : − z j∩b(x;δ) θ(s) ds ∈(0, 1) for all 0 < δ < δ0, for some δ0 = δ0(x) > 0 ) be a finite set. then θ ∈bv (j; {0, 1}) and s(θ) ⊂x. theorem 4.4 (compactness for gε). assume that v : r →[0, ∞) satisfies (hv 1 ) −(hv 3 ). let j ⊂r be an open, bounded interval, let {εn} be such that εnλ 2 3 n →l ∈(0, ∞), and let {vn} ⊂h 3 2 (j) be such that sup n gεn(vn; j) < ∞. (4.20) then there exist a subsequence (not relabeled) of {vn} and a function v ∈bv j; {α, β}
such that vn →v in l2(j). proof. since λn →∞, by (4.20) we have that c1 := sup n z j v (vn) dx < ∞. by condition (hv 3 ) and the fact that j is bounded, we have that 1 c l1(j) + c z j |vn|2 dx ⩽ z j v (vn) dx ⩽c1, and so {vn} is bounded in l2(j). thus by the fundamental theorem of young measures (for a comprehensive exposition on young measures, see [26,4,23,12]), there exists a subsequence (not relabeled) generating a young measure {νx}x∈j. letting f(z) := min  v (z), 1 , since λn →∞, we have that 0 = lim n z j f(vn) dx = z j z r f(z) dνx(z) dx. since f(z) = 0 if and only if z ∈{α, β}, we have that for l1-a.e. x ∈j, νx = θ(x)δα + 1 −θ(x)
δβ (4.21) higher-order phase transitions with line-tension effect 15 for some θ ∈l∞j; [0, 1]
. define x := ( x ∈j : − z b(x;δ) θ(s) ds ∈(0, 1) for all 0 < δ < δ0, for some δ0 = δ0(x) > 0 ) . (4.22) we claim that x is finite. to establish this, let s1, . . . , slbe distinct points of x and let 0 < d0 < 1 2 min{|si −sj| : i ̸= j, i, j = 1, . . . , l}. since si ∈x, we may find di > 0 so small that di ⩽d0 and − z b(si;di) θ(s) ds > 0, − z b(si;di) 1 −θ(s)
ds > 0. (4.23) define d := min{d1, . . . , dl}. let 0 < η < β−α 2 , let φη ∈c∞ c r; [0, 1]
be such that supp φη ⊂b(α; η) and φη(α) = 1, and let γη ∈c∞ c r; [0, 1]
be such that supp γη ⊂b(β; η) and γη(α) = 1. using the fundamental theorem of young measures with f(x, z) := χb(si;d)(x)φη(z), we obtain lim n→∞ z b(si;di) φη vn(x)
dx = z r z r f(x, z) dνx(z) dx = z b(si;di) θ(x) dx > 0, . (4.24) similarly, lim n→∞ z b(si;di) γη vn(x)
dx = z b(si;di) 1 −θ(x)
dx > 0. (4.25) in view of (4.24) and (4.25), we may find x± n,i ∈(si −d, si + d) such that |vn(x− n,i) −α| < η, and |vn(x+ n,i) −β| < η. let wn(x) := vn  εnλ −1 3 n x  , which is admissible for the constant cη defined in (3.4). then by (4.20), ∞> c ⩾lim inf n gεn(vn; j) ⩾lim inf n l x i=1 gεn vn; (x− n,i, x+ n,i)
⩾lim inf n l x i=1 εnλ 2 3 ng1  wn;  x− n,i εnλ −1 3 n , x+ n,i εnλ −1 3 n  ⩾cηll. we conclude that h0(x) ⩽ c cηl < ∞. by lemma 4.3, this implies that θ ∈bv j; {0, 1}
. in particular, we may write θ = χe, and so νx = δv(x), where v(x) := ( α if x ∈e, β if x ∈j\e. it follows that {vn} converges in measure to v. by condition (hv 2 ), there are c, t > 0 such that v (z) ⩾c|z|2 for all |z| ⩾t, and so z e∩{|vn|⩾t } |vn(x′)|2 dx′ ⩽1 c z e v (vn(x′)) dx′ ⩽c1 c 1 λn . this implies that {vn} is 2-equi-integrable. apply vitali's convergence theorem to deduce that vn →v in l2(j). 16 b. galv ̃ ao-sousa 4.4. lower bound for gε in this section we prove the following theorem. theorem 4.5 (lower bound for gε). let j ⊂r be an open and bounded interval and let v : r →[0, ∞) satisfy (hv 1 ) −(hv 3 ). assume that ελ 2 3 ε →l ∈(0, ∞). let v ∈bv j; {α, β}
and let {vε} ⊂h 3 2 (j) be such that sup ε>0 gε(vε; j) =: c < ∞ (4.26) and vε →v in l2(j) as ε →0+. then lim inf ε→0+ gε(vε; j) ⩾clh0(s(v)), where c ∈(0, ∞) is the constant defined in (1.4). we begin with some preliminary results. lemma 4.6. let v : r →[0, ∞) satisfy (hv 1 ) −(hv 3 ), and let v ∈h 3 2 (c, d) be such that tv(c) = w and tv′(c) = z, for some c, d, z, w ∈r, with c < d and |z| + |w −α| ⩽1. let f(x) := ( v(x) if c ⩽x ⩽d, p(x) if c −1 ⩽x ⩽c, where p is the polynomial given by p(x) := α + (3w −3α −z)(x −c + 1)2 + (z + 2α −2w)(x −c + 1)3. then f ∈h 3 2 (c −1, d), z d c−1 z d c−1 |f ′(x) −f ′(y)|2 |x −y|2 dx dy − z d c z d c |v′(x) −v′(y)|2 |x −y|2 dx dy = z c c−1 z c c−1 |p′(x) −p′(y)|2 |x −y|2 dx dy + 2 z c c−1 z d c |v′(x) −p′(y)|2 |x −y|2 dx dy ⩽c |z| + |α −w|
2 + 2sv(c), (4.27) and z d c−1 v (f(x)) dx − z d c v (v(x)) dx = z c c−1 v (p(x)) dx ⩽c |z| + |α −w|
2, (4.28) for some constant c = c(v, α) > 0, and where sv(c) := z d c |v′(x) −v′(c)|2 |x −c|2 dx. (4.29) moreover, z c c−1 |p′(x)|2 |x −c + 1|2 dx ⩽c |z| + |α −w|
2, (4.30) z c c−1 |p′(x)|2 |x −d −1|2 dx ⩽c |z| + |α −w|
2. (4.31) proof. since v ∈c2(r), and v (α) = v ′(α) = 0, by taylor's formula, for any t ∈r, there exists t0 between α and t such that v (t) = v ′′(t0) 2 (t −α)2. on the other hand, we have that |p(x) −α| ⩽|3w −3α −z|(x −c + 1)2 + |z + 2α −2w||x −c + 1|3 ⩽5 |z| + |α −w|
for all x ∈[c −1, c], and so z c c−1 v (p(x)) dx ⩽1 2  max ξ∈[α−5,α+5] |v ′′(ξ)|  z c c−1 (p(x) −α)2 dx ⩽c |z| + |α −w|
2. higher-order phase transitions with line-tension effect 17 to estimate the first integral in (4.27), write p′(x) in the following form p′(x) = z + 2(2z + 3α −3w)(x −c) + 3(z + 2α −2w)(x −c)2, for all x ∈[c −1, c]. then, for x ∈[c −1, c], |p′(x) −z| ⩽12 |z| + |α −w|
|x −c|, (4.32) while for x, y ∈[c −1, c], |p′(x) −p′(y)| ⩽18 |z| + |α −w|
|x −y|, (4.33) and so z c c−1 z c c−1 |p′(x) −p′(y)|2 |x −y|2 dx dy ⩽c |z| + |α −w|
2. to estimate the second integral in (4.27), we have that z c c−1 "z d c |v′(x) −p′(y)|2 |x −y|2 dx # dy ⩽2 z c c−1 "z d c |v′(x) −v′(c)|2 |x −y|2 dx # dy + 2 z c c−1 "z d c |p′(y) −z|2 |x −y|2 dx # dy, where we have used the fact that v′(c) = z. by fubini's theorem, z c c−1 "z d c |v′(x) −v′(c)|2 |x −y|2 dx # dy ⩽ z d c |v′(x) −v′(c)|2 |x −c|2 dx = sv(c), while z c c−1 "z d c |p′(y) −z|2 |x −y|2 dx # dy = (d −c) z c c−1 |p′(y) −z|2 (c −y)(d −y) dy ⩽c 2 |z| + |α −w|
2. this concludes the first part of the proof. to estimate (4.30), we write p′(x) = 2(3w −3α −z)(x −c + 1) + 3(z + 2α − 2w)(x −c + 1)2, so for x ∈(c −1, c) we have |p′(x)|2 ⩽c |z| + |α −w|
2(x −c + 1)2. hence z c c−1 |p′(x)|2 |x −c + 1|2 dx ⩽c |z| + |α −w|
2, while z c c−1 |p′(x)|2 |x −d −1|2 dx ⩽c |z| + |α −w|
2 z c c−1     x −(c −1) x −(d + 1)     2 dx ⩽c |z| + |α −w|
2. the estimate for (4.31) is analogous. this completes the proof. corollary 4.7. let v : r →[0, ∞) satisfy (hv 1 )−(hv 3 ) and let v ∈h 3 2 (c, d) be such that tv(c) = w1 and tv′(c) = z1, tv(d) = w2 and tv′(d) = z2, for some c, d, z1, z2, w1, w2 ∈r, with c < d and |z1|+|w1 −α| ⩽1 and |z2|+|w2 −β| ⩽1. let f(x) :=        p2(x) if d ⩽x ⩽d + 1, v(x) if c ⩽x ⩽d, p1(x) if c −1 ⩽x ⩽c, (4.34) where p1 and p2 are the polynomials given by p1(x) := α + (3w1 −3α −z1)(x −c + 1)2 + (z1 + 2α −2w1)(x −c + 1)3, p2(x) := β + (3w2 −3β + z2)(d + 1 −x)2 + (2α −2w2 −z2)(d + 1 −x)3. (4.35) 18 b. galv ̃ ao-sousa then f ∈h 3 2 (c −1, d + 1), z d c z d c |v′(x) −v′(y)|2 |x −y|2 dx dy ⩾ z d+1 c−1 z d+1 c−1 |f ′(x) −f ′(y)|2 |x −y|2 dx dy −c |z1| + |z2| + |α −w1| + |β −w2|
2 −cqv(c, d) −2sv(c) −2sv(d), (4.36) and z d c v (v(x)) dx ⩾ z d+1 c−1 v (f(x)) dx −c |z1| + |z2| + |α −w1| + |β −w2|
2, (4.37) where c = c(v, α, β) > 0, qv(c, d) := |v′(c) −v′(d)|2 |c −d|2 , (4.38) and sv(*) is defined in (4.29). proof. the estimate (4.37) follows by applying twice (4.28) in lemma 4.6. to obtain (4.36), by figure 1, it suffices to estimate the double integrals over the sets s1, s2, and s0. d + 1 c −1 c d s1 c d d + 1 s2 s0 fig. 1. scheme for the estimates. the estimates on s1 and s2 are a direct consequence of lemma 4.6. to estimate the integral over s0, we observe that p′ 1(x) = z1 + 2(2z1 + 3α −3w1)(x −c) + 3(z1 + 2α −2w1)(x −c)2, p′ 2(x) = z2 + 2(−2z2 + 3β −3w2)(x −d) + 3(z2 −2β + 2w2)(x −d)2, so for x ∈(c −1, c) and y ∈(d, d + 1), we deduce that |p′ 1(x) −p′ 2(y)| ⩽|z1 −z2| + c |z1| + |z2| + |α −w1| + |β −w2|
|x −y|. this implies that z d+1 d z c c−1 |p′ 1(x) −p′ 2(y)|2 |x −y|2 dx dy ⩽c |z1 −z2|2 |d −c|2 + c |z1| + |z2| + |α −w1| + |β −w2|
2. corollary 4.8. let v : r →[0, ∞) satisfy (hv 1 )−(hv 3 ) and let v ∈h 3 2 (c, d) be such that tv(c) = w1, d tv′(c) = z1, tv(d) = w2, and tv′(d) = z2, for some c, d, z1, z2, w1, w2 ∈r, with c < d, |z1| + |w1 −α| ⩽1, and |z2| + |w2 −β| ⩽1. let f(x) :=                  β if x ⩾d + 1, p2(x) if d ⩽x ⩽d + 1, v(x) if c ⩽x ⩽d, p1(x) if c −1 ⩽x ⩽c, α if x ⩽c −1, (4.39) higher-order phase transitions with line-tension effect 19 where p1 and p2 are the polynomials defined in (4.35). then f ∈h 3 2 (c −1, d + 1), f ′ ∈h 1 2 (r), and z d c z d c |v′(x) −v′(y)|2 |x −y|2 dx dy ⩾ z ∞ −∞ z ∞ −∞ |f ′(x) −f ′(y)|2 |x −y|2 dx dy −c |z1| + |z2| + |α −w1| + |β −w2|
2 −cqv(c, d) −2(1 + d −c) sv(c) −2sv(d)
−2 log(1 + d −c) |z1|2 + |z2|2
, (4.40) where c = c(v, α, β) > 0, qv(*, *) is defined in (4.38), and sv(*) is defined in (4.29). proof. by corollary 4.7, we know that z c+1 −c−1 z d+1 c−1 |f ′(x) −f ′(y)|2 |x −y|2 dx dy − z d c z d c |v′(x) −v′(y)|2 |x −y|2 dx dy ⩽c |z1| + |z2| + |α −w1| + |β −w2|
2 + cqv(c, d) + 2sv(c) + 2sv(d), so to prove estimate (4.40), it suffices to estimate z ∞ −∞ z ∞ −∞ |f ′(x) −f ′(y)|2 |x −y|2 dx dy − z c+1 −c−1 z d+1 c−1 |f ′(x) −f ′(y)|2 |x −y|2 dx dy = 2(i± 4 + i± 5 + i± 6 ), where the ii's are defined by i− 4 := z c−1 −∞ z c c−1 |p′ 1(x)|2 |x −y|2 dx dy, i+ 4 := z ∞ d+1 z d+1 d |p′ 2(x)|2 |x −y|2 dx dy, i− 5 := z ∞ d+1 z c c−1 |p′ 1(x)|2 |x −y|2 dx dy, i+ 5 := z c−1 −∞ z d+1 d |p′ 2(x)|2 |x −y|2 dx dy, i− 6 := z c−1 −∞ z d c |v′(x)|2 |x −y|2 dx dy, i+ 6 := z ∞ d+1 z d c |v′(x)|2 |x −y|2 dx dy. to estimate i− 4 , we compute i− 4 = z c c−1 |p′ 1(x)|2 x −c + 1 dx ⩽c |z1| + |α −w1|
2 by (4.33) (with p replaced by p1), and analogously, i+ 4 ⩽c |z2| + |β −w2|
2. for i± 5 , we have that i− 5 = z c c−1 |p′ 1(x)|2 d + 1 −x dx ⩽ z c c−1 |p′ 1(x)|2 x −c + 1 dx = i− 4 ⩽c |z1| + |α −w1|
2, and analogously i+ 5 ⩽c |z2| + |β −w2|
2. to estimate i− 6 , we write i− 6 = z d c |v′(x) ± v′(c)|2 x −c + 1 dx ⩽2 z d c |v′(x) −v′(c)|2 x −c + 1 dx + 2 z d c |v′(c)|2 x −c + 1 dx (4.41) ⩽2 z d c |v′(x) −v′(c)|2 (x −c)2 (x −c)2 x −c + 1 dx + 2|z1|2 log(1 + d −c) ⩽2(d −c)sv(c) + 2|z1|2 log(1 + d −c). analogously, i+ 6 ⩽2(d −c)sv(d) + 2|z2|2 log(1 + d −c). this completes the proof. proposition 4.9. let j ⊂r be an open and bounded interval and let v : r →[0, ∞) satisfy (hv 1 ) −(hv 3 ). assume that ελ 2 3 ε →l ∈(0, ∞), and consider a sequence {vε} ⊂h 3 2 (j) such that sup ε>0 gε vε; (x−, x+)
< ∞, for some x± ∈j, with |vε(x−) −α| ⩽η, |vε(x+) −β| ⩽η,  εv′ ε(x±)   ⩽c,  ε3svε(x±)   ⩽c,  ε3qvε(x−, x+)   ⩽c, (4.42) 20 b. galv ̃ ao-sousa where c > 0, η > 0, and sv(*) and qv(*, *) are defined in (4.29) and (4.38), respectively. then lim inf ε→0+ gε vε; (x−, x+)
⩾cl, (4.43) where c ∈(0, ∞) is the constant defined in (1.4). proof. define wε(t) := vε ελ −1 3 ε t
for x ∈j. by the change of variables x = ελ −1 3 ε t, y = ελ −1 3 ε s, we have gε vε; (x−, x+)
= ε 8 z x+ x− z x+ x−  v′ ε(x) −v′ ε(y)  2 |x −y|2 dx dy + λε z x+ x−v vε(x)
dx = ελ 2 3 ε   1 8 z x+ ελ −1 3 ε x− ελ −1 3 ε z x+ ελ −1 3 ε x− ελ −1 3 ε  w′ ε(t) −w′ ε(s)  2 |t −s|2 dt ds + z x+ ελ −1 3 ε x− ελ −1 3 ε v wε(t)
dt    (4.44) let fε be the function given in (4.39) with the choice of parameters v := wε, c := x− ελ −1 3 ε , d := x+ ελ −1 3 ε , w1 := wε x− ελ −1 3 ε ! = vε(x−), w2 := wε x+ ελ −1 3 ε ! = vε(x+), z1 := w′ ε x− ελ −1 3 ε ! = ελ −1 3 ε v′ ε(x−), z2 := w′ ε x+ ελ −1 3 ε ! = ελ −1 3 ε v′ ε(x+). by corollary 4.7, (4.44), and the fact that ελ 2 3 ε →l, we have that gε vε; (x−, x+)
⩾(l + o(1))   1 8 z x+ ελ −1 3 ε +1 x− ελ −1 3 ε −1 z x+ ελ −1 3 ε +1 x− ελ −1 3 ε −1 |f ′ ε(t) −f ′ ε(s)|2 |t −s|2 dt ds + z x+ ελ −1 3 ε +1 x− ελ −1 3 ε −1 v (fε(t)) dt    −c h ελ −1 3 ε |v′ ε(x−)| + |v′ ε(x+)|
+ |α −vε(x−| + |β −vε(x+)| +qwε x− ελ −1 3 ε , x+ ελ −1 3 ε ! + 2swε x− ελ −1 3 ε ! + 2swε x+ ελ −1 3 ε !# . we claim that f ′ ε ∈h 1 2 (r). if the claim holds, since fε is admissible for the constant c defined in (1.4), and by (4.42), we have that gε vε; (x−, x+)
⩾(l+o(1))c−c(2λ −1 3 ε +2η)2−cqwε x− ελ −1 3 ε , x+ ελ −1 3 ε ! −cswε x− ελ −1 3 ε ! −cswε x+ ελ −1 3 ε ! . (4.45) since λε →∞, to conclude that the first part of the proof, it remains to estimate the last three terms on the right-hand side of (4.45). by (4.42), qwε x− ελ −1 3 ε , x+ ελ −1 3 ε ! =         w′ ε  x+ ελ −1 3 ε  −w′ ε  x− ελ −1 3 ε  x+ ελ −1 3 ε − x− ελ −1 3 ε         2 ⩽cελ −4 3 ε , (4.46) while swε x− ελ −1 3 ε ! = z x+ ελ −1 3 ε x− ελ −1 3 ε    w′ ε(t) −w′ ε  x− ελ −1 3 ε     2    t − x− ελ −1 3 ε ,     2 dt ⩽cλ−1 ε , (4.47) higher-order phase transitions with line-tension effect 21 and similarly swε x+ ελ −1 3 ε ! ⩽cλ−1 ε . (4.48) thus, by (4.45)–(4.48), gε vε; (x−, x+)
⩾(l −o(1))c −c(2λ −1 3 ε + 2η)2 −cελ −4 3 ε −cλ−1 ε . letting first ε →0+ and then η →0+ we obtain (4.43). to complete the proof, we show that sup ε |f ′ ε|h 1 2 (r) ⩽c. starting again from (4.44), but using corollary 4.8 in place of corollary 4.7, we obtain c ⩾gε vε; (x−, x+)
⩾(l + o(1))1 8 z ∞ −∞ z ∞ −∞ |f ′ ε(t) −f ′ ε(s)|2 |t −s|2 dt ds −c " ελ −1 3 ε |v′ ε(x−)| + |v′ ε(x+)|
+ |α −vε(x−| + |β −vε(x+)| + qwε x− ελ −1 3 ε , x+ ελ −1 3 ε ! + 1 + x+ −x− ελ −1 3 ε ! swε x− ελ −1 3 ε ! + swε x+ ελ −1 3 ε !! + ε2λ −2 3 ε log 1 + x+ −x− ελ −1 3 ε ! |v′ ε(x−)|2 + |v′ ε(x+)|2
# . (4.49) by (4.42), and (4.45)–(4.48), we have c ⩾gε vε; (x−, x+)
⩾(l + o(1))|f ′ ε|2 h 1 2 (r) −c(2λ −1 3 ε + 2η)2 −cελ −4 3 ε −c λε 1 + x+ −x− ελ −1 3 ε ! −cλ −2 3 ε log 1 + x+ −x− ελ −1 3 ε ! . (4.50) since ελ 2 3 ε →l, it follows that for all ε > 0 sufficiently small, we have (l −o(1))|f ′ ε|2 h 1 2 (r) ⩽c(1 + l) + cη, where c depends also on x+ −x−. this proves that f ′ ε ∈h 1 2 (r), which completes the proof. proof of theorem 4.5. passing to a subsequence (not relabeled), we can assume that lim inf ε→0+ gε(vε; j) = lim ε→0+ gε(vε; j). this will allow us to take further subsequences (not relabeled). by proposition 2.5, (4.26), and the growth condition (hv 2 ), we know that ∥vε∥h1(j) ⩽cε−1. since v ∈bv j; {α, β}
, its jump set s(v) is finite, and we write s(v) = {s1, . . . , sl}, where s1 < * * * < sl. let 0 < d < 1 2 min {si −si−1 : i = 2, . . . , l}, and assume that v = α in (s2j, s2j+1) for j = 0, . . ., where s0, sl+1 are the endpoints of j. then lim k→∞lim inf ε→0+ z s1 s1−d " k|vε(x) −α| + 1 k ε|v′ ε(x)| + ε3 k z j  v′ ε(x) −v′ ε(y)  2 |x −y|2 dy # dx = 0. hence, we may find k0 ∈n such that for all k ⩾k0, lim inf ε→0+ z s1 s1−d " k|vε(x) −α| + 1 k ε|v′ ε(x)| + ε3 k z j  v′ ε(x) −v′ ε(y)  2 |x −y|2 dy # dx ⩽d. 22 b. galv ̃ ao-sousa by fatou's lemma, we have that for k ⩾k0, 1 d z s1 s1−d lim inf ε→0+ " k|vε(x) −α| + 1 k ε|v′ ε(x)| + ε3 k z j  v′ ε(x) −v′ ε(y)  2 |x −y|2 dy # dx ⩽1 fix k1 > max  k0, 1 η . by the mean value theorem, there exists x− 1 ∈(s1 −d, s1) such that lim inf ε→0+ " |vε(x− 1 ) −α| + 1 k2 1 ε|v′ ε(x− 1 )| + ε3 k2 1 z j  v′ ε(x− 1 ) −v′ ε(y)  2 |x− 1 −y|2 dy # < η. so, up to a subsequence (not relabeled), |vε(x− 1 ) −α| < η, ε|v′ ε(x− 1 )| < ηk2 1, and ε3 z j  v′ ε(x− 1 ) −v′ ε(y)  2 |x− 1 −y|2 dy < ηk2 1. (4.51) analogously, considering lim k→∞lim inf ε→0+ z s1+d s1 " k|vε(x) −α| + 1 k ε|v′ ε(x)| + ε3 k z j  v′ ε(x) −v′ ε(y)  2 |x −y|2 dy + ε3 k  v′ ε(x) −v′ ε(x− 1 )  2 |x −x− 1 |2 # dx = 0, we may find x+ 1 ∈(s1, s1 + d) such that (up to a further subsequence) |vε(x+ 1 ) −β| < η, ε|v′ ε(x+ 1 )| < ηk2 2, and ε3 z j  v′ ε(x+ 1 ) −v′ ε(y)  2 |x+ 1 −y|2 dy < ηk2 2, (4.52) and ε3  v′ ε(x+ 1 ) −v′ ε(x− 1 )  2 |x+ 1 −x− 1 |2 < ηk2 2. (4.53) we now repeat the process to find points x± i in (si −d, si + d) with the properties (4.51)–(4.53). by proposition 4.9, we deduce that lim inf ε→0+ gε(vε; j) ⩾ l x i=1 lim inf ε→0+ gε vε; (x− i , x+ i )
⩾llc = clh0(s(v)). 5. the n-dimensional case in this section we prove theorems 1.1 and 1.2. 5.1. compactness in this subsection we prove theorem 1.1. we follow the argument of [13], which we reproduce for the convenience of the reader. theorem 5.1 (compactness in the interior). let ω, w, and v satisfy the hypotheses of theorem 1.1, and let ελ 2 3 ε →l ∈(0, ∞). consider a sequence {uε} ⊂h2(ω) such that c1 := sup ε fε(uε) < ∞, where fε is the functional defined in (1.1). then there exist a subsequence of {uε} (not relabeled) and a function u ∈bv (ω; {a, b}) such that uε →u in l2(ω). proof. for simplicity of notation, we suppose n = 2. the higher dimensional case is treated analogously. higher-order phase transitions with line-tension effect 23 step 1. assume that ω= i × j, where i, j ⊂r are open bounded intervals. for x ∈ω, we write x = (y, z), with y ∈i, z ∈j. for every function u defined on ωand every y ∈i we denote by uy the function on j defined by uy(z) := u(y, z), and for every z ∈j we denote by uz the function on i defined by uz(y) := u(y, z). the functions uy and uz are called one-dimensional slices of u. we recall that by slicing, if u ∈h2(ω), then uy ∈h2(j) for l1-a.e. y ∈i, uz ∈h2(i) for l1-a.e. z ∈j, and ∂2u ∂z2 (y, z) = d2uy dz2 (z), ∂2u ∂y2 (y, z) = d2uz dy2 (y), for l1-a.e. y ∈i and for l1-a.e. z ∈j. since |∇2u|2 ⩾max n   ∂2u ∂z2    ,    ∂2u ∂y2    o , we immediately obtain that c1 ⩾fε(u) ⩾ z i fε(uy; j) dy, c1 ⩾fε(u) ⩾ z j fε(uz; i) dz, (5.1) where fε is the functional defined in (1.6). consider a family {uε} ⊂h2(ω) such that fε(uε) ⩽c1 < ∞. then we have that w(uε) →0 in l1(ω). from condition (hw 2 ), we have the existence of c, t > 0 such that for all |z| ⩾t, w(z) ⩾c|z|2. this implies that {uε} is 2-equi-integrable and, in particular, it is equi-integrable. therefore, fix δ > 0 and let η > 0 be such that for any measurable set e ⊂r, with l2(e) ⩽η, sup ε>0 z e |uε(x)| + |b|
dx ⩽δ. (5.2) for ε > 0 we define vε : ω→r by vε(y, z) := ( uy ε(z) if y ∈i, z ∈j, and fε(uy ε; j) ⩽cl1(j) η , b otherwise. we claim that {vε} and {uε} are δ-close, i.e., ∥uε −vε∥l1(ω) < δ. indeed, let zε := {y ∈i : uy ε ̸= vy ε}. by (5.1), we have c1 ⩾ z i fε(uy; j) dy, and so l1(zε) ⩽l1  y ∈i : fε(uy ε; j) > c1l1(j) η  ⩽ η c1l1(j) z i fε(uy; j) dy ⩽ η l1(j). it follows that l2(zε × j) ⩽η. thus, by (5.2), ∥uε −vε∥l1(ω) ⩽ z zε×j |uε(x) −b| dx ⩽ z zε×j |uε(x)| + |b|
dx ⩽δ. moreover, for every y ∈i we have fε(vy ε; j) ⩽c1l1(j) η , where we have used the face that fε(b; j) = 0, and therefore theorem 4.1, yields l2(j) precompactness of {vy ε}. similarly, we can construct a sequence {wε} δ-close to {uε} so that {wz ε} is precompact in l2(i) for every z ∈j. using proposition 2.2 we conclude that the sequence {uε} is precompact in l2(ω). step 2. general case. this case can be proved by decomposing ωinto a countable union of closed rectangles with disjoint interiors. the fact that the limit u belongs to bv (ω; {a, b})) is a direct consequence of theorem 2.10. theorem 5.2 (compactness at the boundary). let ω, w, and v satisfy the hypotheses of theorem 1.1, and let ελ 2 3 ε →l ∈(0, ∞). consider a sequence {uε} ⊂h2(ω) such that c := sup ε fε(uε) < ∞, where fε is the functional defined in (1.1). then there exist a subsequence of {uε} (not relabeled) and a function v ∈bv (∂ω; {α, β}) such that tuε →v in l2(∂ω). 24 b. galv ̃ ao-sousa to prove this theorem we introduce the localization of the functionals fε: for every open set a ⊂ωwith boundary of class c2, for every borel set e ⊂∂a, and for every u ∈h2(a), we set fε(u; a, e) := z a  ε2  ∇2u  2 + 1 εw(u)  dx + λε z e v (tu) dhn−1. note that for u ∈h2(ω), fε(u) = fε(u; ω, ∂ω). we begin by proving compactness on the boundary in the special case in which a = ω∩b, where b is a ball centered on ∂ωand e = b ∩∂ωis a flat disk. later on we will show that this flatness assumption can be dropped when b is sufficiently small. proposition 5.3. for every r > 0, let dr be the open half-ball dr := {x = (x′, xn) ∈rn : |x| < r, xn > 0} and let er := {x = (x′, 0) ∈rn : |x| < r}. let w and v satisfy the hypotheses of theorem (1.1), and let ελ 2 3 ε →l ∈(0, ∞). consider a sequence {uε} ⊂h2(dr) such that c1 := sup ε>0 fε(uε; dr, er) < ∞. then there exist a subsequence of {uε} (nor relabeled) and a function v ∈bv (er; {α, β}) such that tvε →v in l2(er). proof. to simplify the notation, we write d and e in place of dr and er. the idea of the proof is to reduce to the statement of theorem 4.4 via a suitable slicing argument. fix i = 1, . . . , n −1 and let eei := {y ∈rn−2 : (y, xi, 0) ∈e for some xi ∈r}. for every y ∈eei, define the sets dy := {(xi, xn) ∈r2 : (y, xi, xn) ∈d}, ey := {xi ∈r : (y, xi, 0) ∈e}. for every y ∈eei and every function u : d →r, let uy : dy →r be the function defined by uy(xi, xn) := u(y, xi, xn), (xi, xn) ∈dy, and for every function v : e →r, let vy : ey →r be defined by vy(xi) := v(y, xi), xi ∈ey. if u ∈h2(d), then by the slicing theorem in [27] for ln−2-a.e. y ∈eei, the function uy belongs to h2(dy), for l2-a.e. (xi, xn) ∈d, ∂u ∂xk (y, xi, xn) = ∂uy ∂xk (xi, xn), for k = i, n, and ∂2u ∂xk∂xj (y, xi, xn) = ∂2uy ∂xk∂xj (xi, xn), for k, j = i, n, and the trace of uy on ey agrees l1-a.e. in ey with (tu)y. taking into account these facts and fubini's theorem, for every ε > 0 we get fε(u; d, e) ⩾ε3 z d |d2u(x)|2 dx + λε z e v (tu(x′, 0)) dx′ ⩾ z eei  ε3 z dy |d2 xi,xn uy(xi, xn)|2 dxi dxn + λε z ey v (tuy(xi, 0)) dxi  dy. higher-order phase transitions with line-tension effect 25 we apply the trace inequality (2.1) to each function uy to obtain fε(u; d, e) ⩾ z eei gε(tuy; ey) dy, (5.3) where gε is the functional defined in (1.7) to prove that the sequence {tuε} is precompact in l2(e), it is enough to show that it satisfies the conditions of proposition 2.2. since c1 = sup ε>0 fε(uε; d, e) < ∞, (5.4) we have that sup ε>0 λε z e v tuε(x′, 0)
dx′ ⩽c1. from condition (hv 2 ), we may find c, t > 0 such that for all |z| ⩾t, v (z) ⩾c|z|2, and so z e∩{|t uε|⩾t}  tuε(x′, 0)  2 dx′ ⩽2 c z e v tuε(x′, 0)
dx′ ⩽2c1 c 1 λε . this implies that {tuε} is 2-equi-integrable. in particular, it is equi-integrable. thus to apply proposition 2.2, it remains to show that for every δ > 0 there is a sequence {vε} ⊂l1(e) that is δ-close to tuε, in the sense of definition 2.1, and such that {vy ε} is precompact in l1(ey) for ln−2-a.e. y ∈eei. fix δ > 0, let η > 0 be a constant that will be fixed later, and let vε(y, xi) := ( tuy ε(xi) if y ∈eei, x ∈ey, and gε(tuy ε; ey) ⩽c1 η , α otherwise. (5.5) note that although vε is no longer in h 3 2 (e), for every y ∈eei, either vy ε = tuy ε ∈h 3 2 (ey), or vy ε ≡α, and so vy ε always belongs to h 3 2 (ey). we claim that {vε} is δ-close to {tuε}. indeed, by fubini's theorem, ∥tuε −vε∥l1(e) ⩽ z zei z ey |tuy ε(xi) −α| dxi dy ⩽ z zei z ey (|tuy ε(xi)| + |α|) dxi dy, where zei := {y ∈eei : tuy ε ̸= vy ε} =  y ∈eei : gε(tuy ε; ey) > c1 η . since {tuε} is equi-integrable, to prove that the right-hand side of the previous inequality is less than δ, it suffices to show that the ln−1 measure of the set h := {(y, xi) : y ∈zei, xi ∈ey} can be made arbitrarily small. again by fubini's theorem and the definition of zei, ln−1(h) = z zei l1(ey) dy ⩽2rln−2(zei) ⩽η c1 z zei gε(tuy ε; ey) dy ⩽η, where we have used (5.4) and the fact that l1(ey) ⩽2r ⩽1 for r ⩽1 2. thus if η is chosen sufficiently small, we have that {vε} is δ-close to {tuε}. to prove that {vy ε} if precompact for ln−2-a.e. y ∈eei, it suffices to consider only those y ∈eei such that gε(tuy ε; ey) ⩽c1 η (since otherwise vy ε(xi) ≡α and there is nothing to prove). for these y ∈eei, the precompactness follows from theorem 4.4. hence we are in a position to apply proposition 2.2 to conclude that {tuε} is precompact in l1(e). thus, up to a subsequence (not relabeled), we may assume that there exists a function v ∈l1(e) such that tuε →v in l1(e). note that since {tuε} is 2-equi-integrable, it follows by vitali's convergence theorem that tuε →v in l2(e). it remains to show that v ∈bv (e; {α, β}). indeed, replacing u by uε in (5.3), and passing to the limit as ε →0+, by fatou's lemma we deduce that ∞> lim inf ε→0+ fε(uε; d, e) ⩾ z eei lim inf ε→0+ gε(tuy ε; ey) dy, which implies that lim inf ε→0+ gε(tuy ε; ey) is finite for ln−2-a.e. y ∈eei. since tuε →v in l2(e), up to a subsequence (not relabeled), we have that tuy ε →vy in l2(e) for ln−2-a.e. y ∈eei. then proposition 2.12 yields vy ∈bv (ey; {α, β}) and lim inf ε→0+ fε(uε; d, e) ⩾ z eei clh0(svy) dy. (5.6) 26 b. galv ̃ ao-sousa the right-hand side of (5.6) is finite, so proposition 2.12 implies that v ∈bv (e; {α, β}), and that svy agrees with sv ∩ey for a.e. y ∈eei. to prove compactness in the general case, i.e., where ωis not flat, we introduce the notion of isometry defect following [2]. definition 5.4 (isometry defect). given a1, a2 ⊂rn open sets and a bi-lipschitz homeomorphism ψ : a1 →a2 of class c2(ai; rn), the isometry defect δ(ψ) of ψ is the smallest constant δ such that ess sup x∈a1  dist dψ(x), o(n)
+ dist d2ψ(x), 0
⩽δ, where o(n) :=  a : rn →rn linear mappings, aat = in . proposition 5.5. let ω, w, and v satisfy the hypotheses of theorem 1.1. given a1, a2 ⊂rn open sets and a bi-lipschitz homeomorphism ψ : a1 →a2 of class c2(ai; rn) such that ψ has finite isometry defect and maps a set a′ 1 ⊂∂a1 onto a′ 2 ⊂∂a2. then for every u ∈h2(a2) there holds fε(u; a2, a′ 2) ⩾ 1 −δ(ψ)
n+4fε(u ◦ψ; a1, a′ 1) −δ(ψ) 1 −δ(ψ)
2ε3 z a2  (  d2u   du   + δ(ψ)  du  2 dx. (5.7) proposition 5.6. let ω⊂rn be an open and bounded set of class c2 and let dr := {x ∈rn : |x| < r, xn > 0}. then for every x ∈∂ω, there exists rx > 0 such that for every 0 < r < rx, there exists a bi-lipschitz homeomorphism ψr : dr →ω∩b(x; r) such that (i) ψr maps dr onto ω∩b(x; r) and er := br ∩{xn = 0} onto ∂ω∩b(x; r); (ii) ψr is of class c2 in dr and ∥dψr −in∥∞+ ∥d2ψr∥∞⩽δr, where δr r→0+ − − − − →0+. for a proof of propositions 5.5 and 5.6 we refer to [2]. we now turn to the proof of theorem 5.2. proof of theorem 5.2. in view of proposition 5.6 and a simple compactness argument we can cover ∂ωwith finitely many balls bi centered on ∂ωso that ω∩bi is the image of a half-ball under a map ψi with isometry defect smaller than 1. hence it suffices to show that the sequence {tuε} is precompact in l2(∂ω∩bi) for every i. fix i and let e uε := uε ◦ψi. since the isometry defect of ψi is smaller than 1, proposition 5.5 implies that supε fε(e uε; dr, er) < ∞. hence the precompactness of the traces tuε in l2(∂ω∩bi) is a consequence of the precom- pactness of the traces t e uε in l2(er), which follows from proposition 5.3. this completes the proof. we are finally ready to prove theorem 1.1. proof of theorem 1.1. let {uε} ⊂h2(ω) be a sequence such that c := supε fε(uε) < ∞. then, by theorem 5.1, we may find a subsequence uεn ∈h2(ω) and a function u ∈bv (ω; {a, b}) such that uεn →u in l2(ω). on the other hand, by applying theorem 5.2 to the sequences {εn} and {uεn}, which still satisfy c = supn fεn(uεn) < ∞, we may find a further subsequence {uεnk } of {uεn} and a function v ∈bv (∂ω; {α, β}) such that tuεnk → v in l2(∂ω). note that we still have uεnk →u in l2(ω). this completes the proof. 5.2. lower bound in rn before proving the lower bound estimate in the general n-dimensional case, we state an auxiliary result. lemma 5.7. let μ, μ1, and μ2 be nonnegative finite radon measures on rn, such that μ1 and μ2 are mutually singular, and μ(b) ⩾μi(b) for i = 1, 2, and for any open ball b such that μ(∂b) = 0. then for any borel set e, μ(e) ⩾μ1(e) + μ2(e). proof of theorem 1.2(i). we now have all the necessary auxiliary results to prove the lower bound estimate for the critical regime. consider a sequence {uε} ⊂h2(ω) and two functions u ∈bv (ω; {a, b}) and v ∈bv (∂ω; {α, β}) such that uε →u in l2(ω) and tuε →v in l2(∂ω). higher-order phase transitions with line-tension effect 27 we claim that lim inf ε→0+ fε(uε; ω) ⩾mperω(ea) + x z=a,b x ξ=α,β σ(z, ξ)hn−1{tu = z} ∩{v = ξ}
+ clper∂ω(fα). (5.8) without loss of generality, we may assume that ∞> lim inf ε→0+ fε(uε; ω) = lim ε→0+ fε(uε; ω). (5.9) for every ε > 0 we define a measure με for all borel sets e ⊂rn by με(e) := ε3 z ω∩e |d2uε|2 dx + 1 ε z ω∩e w(uε) dx + λε z ∂ω∩e v (tuε) dhn−1. since με = fε(uε), it follows by (5.9) that by taking a subsequence (not relabeled), we obtain a finite measure μ such that με ⋆ ⇀μ in the sense of measures. for every borel set e ⊂rn define the measures: μ1(e) := mperω∩e(ea); μ2(e) := x z=a,b x ξ=α,β σ(z, ξ)hn−1{tu = z} ∩{v = ξ} ∩e
; μ3(e) := clper∂ω∩e(fα). these three measures are mutually singular and so, by lemma 5.7, (5.8) is a consequence of μ(b) ⩾μi(b) for i = 1, 2, 3 for any ball b with μ(∂b) = 0, which we prove next. take b an open ball such that μ(∂b) = 0. using a slicing argument as in theorem 5.1 (see (5.1) for n = 2) and fatou's lemma, we have μ(b) = lim ε→0+ με(b) ⩾ z ωe∩b lim inf ε→0+ fε(uy ε; by) dhn−1(y) ⩾ z ωe∩b  mh0(suy ε ∩by) + z ∂by σ(tuy(s), vy(s))dh0(s)  dhn−1(y) ⩾mperω∩b(ea) + z ∂ω∩b σ(tu(s), v(s))dhn−1(s) = μ1(b) + μ2(b), where we have used theorem 4.2 and proposition 2.12. by section 2.5, the jump set of v, sv, is (n −2)-rectifiable. hence by the lebesgue decomposition theorem, the radon-nikodym theorem, and the besicovitch derivation theorem, for hn−2-a.e. x ∈sv, dμ dhn−2⌊sv (x) = lim r→0+ μ b(x; r)
hn−2b(x; r) ∩sv
∈r. (5.10) fix a point x ∈sv for which (5.10) holds and that has density 1 for sv with respect to the hn−2 measure. take r > 0 such that μ ∂b(x; r)
= 0. find ψr as in proposition 5.6 and set uε := uε◦ψr and v := v◦ψr. then v ∈bv (er; {α, β}) and tuε →v in l2(er), where er is defined in proposition 5.6. since μ ∂b(x; r)
= 0, we have μ b(x; r)
= lim ε με b(x; r)
= lim ε fε uε; ω∩b(x; r), ∂ω∩b(x; r)
⩾(1 −δ(ψr))n+4 lim inf ε z (er)e gε(tuy ε; ey r ) dhn−2(y) ⩾cl (1 −δ(ψr))n+4 z (er)e h0(sv ∩ey r ) dhn−2(y). hence, dμ dhn−2⌊sv (x) ⩾lim r→0+ μ b(x; r)
αn−2rn−2 ⩾cl lim r→0+ − z (er)e h0(sv ∩ey r ) dhn−2(y) = cl, and so μ(b) ⩾ z sv∩b dμ dhn−2⌊sv (x) dhn−2(x) ⩾clper∂ω∩b(fα) = μ3(b). this concludes the proof of the theorem. 28 b. galv ̃ ao-sousa 5.3. upper bound in this subsection we will obtain an estimate for the upper bound. first we prove the result on a smooth setting, i.e., assuming that both su and sv are of class c2. we define a recovery sequence separately in the different regions of figure 2. in proposition 5.8, we define it on a2, then we construct the recovery sequence on a1 in proposition 5.9 and in corollary 5.10 we glue the last two sequences together to make {u}n. then in proposition 5.11, on the setting of a flat domain where sv has also been flattened, we first construct the recovery sequence on t1 and then glue it to the previously constructed sequence {un} on t2. in proposition 5.12 we adapt the sequence of proposition 5.11 to a general domain, but still under smooth assumptions. finally, using a diagonalization argument, we prove the upper bound result without regularity conditions. a2 a2 u = b u = a v = α v = β a1 b1 t1 t2 fig. 2. partition of ωfor the construction of the recovery sequence. in what follows, given a set e ⊂rn and ρ > 0 we denote by eρ the set eρ := {x ∈rn : dist (x, e) < ρ}. proposition 5.8. let w : r →[0, ∞) satisfy (hw 1 ) −(hw 2 ), let εn →0+, let η > 0, and let u ∈bv (ω; {a, b}) be such that su is an n −1 dimensional manifold of class c2. then there exists a sequence {zn} ⊂h2(ω) such that zn →u in l2(ω), zn = u in ω\(su)cεn, (5.11) ∥zn∥∞⩽c, ∥∇zn∥∞⩽c εn , ∥∇2zn∥∞⩽c ε2 n , (5.12) and fεn(zn; ω, ∅) ⩽(m + η)hn−1(su) + o(1), (5.13) where m is the constant defined in (1.2) and c > 0. proof. by the definition of m, we may find r > 0 and a function f ∈h2 loc(r) such that f(−t) = a and f(t) = b for all t ⩾r, and z r −r |f ′′(t)|2 + w f(t)
dt ⩽m + η. (5.14) since su is a manifold of class c2 in rn, there exists δ0 > 0 such that for all 0 < δ ⩽δ0 the points in the tubular neighborhood uδ := {x ∈rn : dist (x, su) < δ} of the manifold su admit a unique smooth projection onto su. define the function zn : ω→r by zn(x) :=        f  du(x) εn  if x ∈urεn ∩ω, a if x ∈ea\urεn, b if x ∈ω\(ea ∪urεn), where du : rn →r is the signed distance to su, negative in ea and positive outside ea and where we recall that ea := {x ∈ω: u(x) = a}. higher-order phase transitions with line-tension effect 29 we then have fεn(zn; ω, ∅) = z ω " ε3 n     1 ε2 n f ′′ du(x) εn
∇du(x) × ∇du(x) + 1 εn f ′ du(x) εn
hu(x)     2 + 1 εn w  f du(x) εn
# dx, where hu is the hessian matrix of du. change variable via the diffeomorphism x := ψ1(y, t), where ψ1 : su×(−δ0, δ0) → uδ0 is defined by ψ1(y, t) := y + tνu(y), with νu(y) the normal vector to su at y pointing away from ea. let ju(y, t) denote the jacobian of this map. then fεn(zn; ω, ∅) ⩽1 εn z su z rεn −rεn   f ′′ t εn
   2 |∇du(ψ1(y, t))|2 + w  f t εn
 ju(y, t) dt dhn−1(y) + εn z su z rεn −rεn   f ′ t εn
   2 |hu(ψ1(y, t))|2ju(y, t) dt dhn−1(y) + c z su z rεn −rεn   f ′′ t εn
     f ′ t εn
   |∇du(ψ1(y, t))|2|hu(ψ1(y, t))|ju(y, t) dt dhn−1(y), which reduces to fεn(zn; ω, ∅) ⩽1 εn z su z rεn −rεn   f ′′ t εn
   2 + w  f t εn
 ju(y, t) dt dhn−1(y) + c z su z rεn −rεn  εn   f ′ t εn
   2 +   f ′′ t εn
   2   f ′ t εn
   2 dt dhn−1(y) =: i1 + i2, where we took into account the facts that the gradient of the distance is 1, and the jacobian ju and the hessian hu of the distance are uniformly bounded. we have i1 ⩽ sup y∈su, t∈(−rεn,rεn) ju(y, t) ! 1 εn z su z rεn −rεn   f ′′ t εn
   2 + w  f t εn
 dt dhn−1(y) = 1 + o(1)
z su z r −r h |f ′′(s)|2 + w (f(s)) i ds dhn−1(y), ⩽ 1 + o(1)
(m + η)hn−1(su), where we used (5.14) and the fact that since su is a compact manifold, ju(y, t) converges to 1 uniformly as t →0. on the other hand, by (5.14), i2 ⩽cεn z r −r h εn |f ′(s)|2 + |f ′′(s)| |f ′(s)| i ds ⩽cεn. we conclude that fεn(zn; ω, ∅) ⩽(m + η)hn−1(su) + o(1). this completes the proof. proposition 5.9. let w : r →[0, ∞) satisfy (hw 1 )−(hw 2 ), let v : r →[0, ∞) satisfy (hv 1 )−(hv 3 ) let εn →0+ be such that εnλ 2 3 n →l ∈(0, ∞), let η > 0, let ωδ := {x ∈ω: dist (x, ∂ω) < δ} for δ > 0, and let u ∈bv (ω; {a, b}) and v ∈bv (∂ω; {α, β}), with su an n −1 manifold of class c2 such that hn−1(∂ω∩su) = 0 and sv an n −2 manifold of class c2 . then there exist r = r(η) > 0 and a sequence {vn} ⊂h2(ωrεn) such that tvn →v in l2(∂ω), ln n x ∈ωrεn\ωrεn 2 : vn(x) ̸= u(x) o ⩽cε2 n, (5.15) ∥vn∥∞⩽c, ∥∇vn∥∞⩽c εn , ∥∇2vn∥∞⩽c ε2 n , (5.16) and fεn(vn; ωrεn, ∅) ⩽ x z=a,b x ξ=α,β (σ(z, ξ) + η)hn−1({x ∈∂ω: tu(x) = z, v(x) = ξ}) + o(1), (5.17) where σ(z, ξ) is the constant defined in (1.3). 30 b. galv ̃ ao-sousa proof. by the definition of σ(*, *), for every z ∈{a, b} and ξ ∈{α, β} there exist rzξ > 0 and gzξ ∈h2 loc(r) such that gzξ(0) = z, gzξ(x) = ξ for all x ⩾rzξ, and z rzξ 0  |g′′ zξ(x)|2 + w(gzξ(x))  dx ⩽σ(z, ξ) + η. (5.18) define r := max{r, raα, rbα, raβ, rbβ}, where r is the number r given in the previous proposition. since ∂ωis an n −1 manifold of class c2, there exists δ0 > 0 such that every point x ∈ωδ0 admits a unique projection π(x) onto ∂ωand the map x ∈ωδ0 7→π(x) is of class c2. hence we may partition ωδ0 as follows ωδ0 =  [ z=a,b [ ξ=α,β azξ  ∪su ∪π−1(sv), where azξ :=  x ∈ωδ0\ su ∪π−1(sv)
: (tu)(π(x)) = z, v(π(x)) = ξ . let n be so large that rεn ⩽δ0 and define gn : ωrεn →r as follows gn(x) :=    gzξ  d(x) εn  if x ∈azξ ∩ωrεn for some z ∈{a, b} and ξ ∈{α, β}, 0 if x ∈ su ∪π−1(sv)
∩ωrεn, where, as before, d : ω→[0, ∞) is the distance to ∂ω. note that the functions gn are discontinuous across su ∪π−1(sv)
∩ωrεn, and so they are not admissible for fεn. to solve this problem, let φ ∈c∞((0, ∞); [0, 1]) be such that φ ≡0 in 0, 1 3
and φ ≡1 in 1 2, ∞
, and let du : ωδ0 →[0, ∞) and dv : ωδ0 →[0, ∞) denote the distance to su and to π−1(sv), respectively. since su is an n −1 manifold of class c2, it follows that du is of class c2 in a neighborhood p1 := {x ∈ωδ0 : du(x) < δ1} of su. similarly, since sv is an n −2 manifold of class c2 by taking δ0 smaller, if necessary, we may assume that π−1(sv) is an n −1 dimensional manifold of class c2 and thus dv is of class c2 in a neighborhood p2 := {x ∈ωδ0 : dv(x) < δ2} of π−1(sv). let n be so large that rεn < 1 3 min{δ1, δ2} and for x ∈ωrεn define vn(x) := φ du(x) rεn  φ dv(x) rεn  gn(x). since φ ≡0 in 0, 1 3
, it follows that vn(x) = 0 for all x ∈ωrεn such that du(x) < 1 3rεn or dv(x) < 1 3rεn. as gn is regular away from su ∪π−1(sv), it follows that vn ∈h2(ωrεn). we claim that tvn →v in l2(∂ω). indeed, since hn−1(∂ω∩su) = 0, we know that hn−1  x ∈∂ω: du(x) < 1 2rεn  ⩽cεn, (5.19) and similarly, since sv is an n −2 manifold contained in ∂ω, hn−1  x ∈∂ω: dv(x) < 1 2rεn  ⩽cεn. (5.20) on the other hand, if x ∈∂ωis such that du(x) ⩾1 2rεn and dv(x) ⩾1 2rεn, then vn = gn in a neighborhood of x, and so by the definition of the sets azξ and the fact that gzξ(0) = z, it follows that vn(x) = v(x). hence by (5.19) and (5.20), ∥vn −v∥l2(∂ω) →0, which proves the claim. it remains to prove (5.17). let ln :=  x ∈ωrεn : du(x) < 1 2rεn , mn :=  x ∈ωrεn : dv(x) < 1 2rεn . step 1. we begin by estimating fε in the set ωrεn\(ln ∪mn). since in this set vn = gn, we have that fεn(vn; ω\(ln ∪mn), ∅) ⩽ x z=a,b x ξ=α,β fεn (gn; azξ ∩ωrεn, ∅) . thus it suffices to estimate fεn(gn; azξ ∩ωrεn). higher-order phase transitions with line-tension effect 31 let a′ zξ := azξ ∩∂ω, which satisfies a′ zξ = {x ∈∂ω: tu(x) = z, v(x) = ξ}. we have fεn(gn; azξ, ∅) = z azξ " ε3 n     1 ε2 n g′′ zξ d(x) εn
∇d(x) × ∇d(x) + 1 εn g′ zξ d(x) εn
h(x)     2 + 1 εn w  gzξ d(x) εn
# dx, where h is the hessian matrix of d. change variable via the diffeomorphism x := ψ2(y, t), where ψ2 : ∂ω× 0, δ1
→ωδ1, defined by ψ2(y, t) := y + tν(y), with ν(y) the normal vector to ∂ωat y pointing to the inside of ω. we write j(y, t) the jacobian of this map. then fεn(gn; azξ, ∅) ⩽ z a′ zξ z rεn 0  1 εn   g′′ zξ t εn
   2 |∇d(ψ2(y, t))|2 + 1 εn w  g′ zξ t εn
 + εn   g′′ zξ t εn
   2 |h(ψ2(y, t))|2 + c   g′′ zξ t εn
     g′ zξ t εn
   |∇d(ψ2(y, t))|2|h(ψ(y, t))|  j(y, t) dt dhn−1(y), which reduces to fεn(gn; azξ, ∅) ⩽  1 εn z a′ zξ z rεn 0   g′′ zξ t εn
   2 + w  gzξ t εn
 j(y, t) dt dhn−1(y) + c z a′ zξ z rεn 0  εn   g′ zξ t εn
   2 +   g′ zξ t εn
     g′ zξ t εn
    dt dhn−1(y)  =: i1 + i2, where we took into account the facts that the gradient of the distance is 1, and the jacobian j and the hessian h of the distance are uniformly bounded. we have i1 ⩽ sup y∈a′ zξ, t∈(0,rεn) j(y, t) ! 1 εn z a′ zξ z rεn 0   g′′ zξ t εn
   2 + w  gzξ t εn
 dt dhn−1(y) ⩽ 1 + o(1)
(σ(z, ξ) + η)hn−1({tu = z, v = ξ}), where we used the fact that since ∂ωis a compact manifold, j(y, t) converges to 1 uniformly as t →0. on the other hand i2 ⩽cεn z r 0 h εn  g′ zξ(s)  2 +  g′′ zξ(s)  2  g′ zξ(s)  2i ds ⩽cεn. we conclude that fεn(gn; azξ, ∅) ⩽(σ(z, ξ) + η)hn−1({tu = z, v = ξ}) + o(1). step 2. we estimate the energy in ln ∪mn. we have fεn(vn; ln\mn, ∅) = z ln\mn  ε3 n    φ du(x) rεn
g′′ n(x) + 2 rεn g′ n(x)φ′ du(x) rεn
∇du × ∇du + 1 r2ε2 n gn(x)φ′′ du(x) rεn
hu     2 + 1 εn w  φ du(x) rεn
gn(x)  dx, where hu is the hessian matrix of du. then fεn(vn; ln\mn, ∅) ⩽c z ln\mn  ε3 n|g′′ n(x)|2 + εn|g′ n(x)|2 + 1 r4εn |vn(x)|2 + lim sup n 1 εn w  φ du(x) rεn
gn(x)  dx ⩽c 1 εn |ln| ⩽cεn, where we took into account the facts that the hessian hu is uniformly bounded, and that vn is uniformly bounded, g′ n is bounded by c εn , and g′′ n is bounded by c ε2 n . we conclude that fεn(vn; ln\mn, ∅) = o(1). similarly, we may prove that fεn(vn; mn, ∅) = o(1). this concludes the proof. 32 b. galv ̃ ao-sousa corollary 5.10. let w : r →[0, ∞) satisfy (hw 1 ) −(hw 2 ), let v : r →[0, ∞) satisfy (hv 1 ) −(hv 3 ). let εn →0+ be such that εnλ 2 3 n →l ∈(0, ∞), let η > 0, and let u ∈bv (ω; {a, b}) and v ∈bv (∂ω; {α, β}), with su an n −1 manifold of class c2 such that hn−1(∂ω∩su) = 0 and sv an n −2 manifold of class c2 . then there exists a sequence {un} ⊂h2(ω) such that un →u in l2(ω), tun →v in l2(∂ω), ∥un∥∞⩽c, ∥∇un∥∞⩽c εn , ∥∇2un∥∞⩽c ε2 n , (5.21) and fεn(un; ω, ∅) ⩽(m + η)hn−1(su) + x z=a,b x ξ=α,β (σ(z, ξ) + η)hn−1({x ∈∂ω: tu(x) = z, v(x) = ξ}) + o(1) (5.22) where m and σ(z, ξ) are the constant defined, respectively, in (1.2) and (1.3). proof. let φ ∈c∞((0, ∞); [0, 1]) be such that φ ≡0 in 0, 1 2
and φ ≡1 in (1, ∞) and let un(x) := φ d(x) rεn
zn(x) +  1 −φ d(x) rεn
 vn(x), for x ∈ω, where the functions zn and vn are defined, respectively, in propositions 5.8 and 5.9, r is the number given in the previous proposition, and d is the distance to the boundary. α b a x0 zn gaα n gbα n fig. 3. scheme for the gluing of the discontinuity set of u to the boundary ∂ωwhen there is no discontinuity in v. since tun = tvn, it follows that tun →v in l2(∂ω). on the other hand, since ∥vn∥∞⩽c, ln ({x ∈ω: d(x) ⩽rεn}) →0, and zn →u in l2(ω), we have that un →u in l2(ω). moreover, by (5.13) and (5.17), fεn(un; ω; ∅) ⩽fεn(zn; ω\ω2rεn; ∅) + fεn(vn; ωrεn; ∅) + fεn(un; pn; ∅) ⩽(m + η)hn−1(su) + x z=a,b x ξ=α,β (σ(z, ξ) + η)hn−1({x ∈∂ω: tu(x) = z, v(x) = ξ}) + lim sup n fεn(un; pn; ∅) + o(1), where pn :=  x ∈ω: 1 2rεn < d(x) < 2rεn . to estimate the last term, note that by (5.13) and (5.15), ln{x ∈pn : un(x) ̸= u(x)}
⩽cε2 n, and so by the continuity of w, 1 εn z pn w(un) dx = 1 εn  max b(0;l) w  ln{x ∈pn : un(x) ̸= u(x)}
⩽cεn →0, where l := supn ∥un∥∞. on the other hand, we have that ∇un(x) = 0 and ∇2un(x) = 0 for ln-a.e. x ∈en := {x ∈pn : un(x) = u(x)}, while for x ∈pn\en, |∇2un(x)|2 ⩽c  1 ε4 n | |zn(x)|2 + |vn(x)|2
+ 1 ε2 n | |∇zn(x)|2 + |∇vn(x)|2
+ | |∇2zn(x)|2 + |∇2vn(x)|2  ⩽c ε4 n , where we used the bounds on zn and vn given in (5.12) and (5.16). hence ε3 n z pn |∇2un|2 dx = ε3 n z pn\en |∇un|2 dx ⩽c εn ln(pn\en) ⩽cεn, which completes the proof. higher-order phase transitions with line-tension effect 33 proposition 5.11. let w : r →[0, ∞) satisfy (hw 1 ) −(hw 2 ), let v : r →[0, ∞) satisfy (hv 1 ) −(hv 3 ). let εn →0+ be such that εnλ 2 3 n →l ∈(0, ∞), let η > 0, let dr := {x ∈rn : |x| < r, xn > 0}, and let er := {(x′, 0) ∈rn−1 × r : |x| < r}. also let u ∈bv (dr; {a, b}) and v ∈bv (er; {α, β}), with su an n −1 manifold of class c2 such that hn−1(er ∩su) = 0 and sv = {x ∈er : xn−1 = 0}. then there exists {un} ⊂h2(dr) such that un →u in l2(dr), tun →v in l2(er), and lim sup n fεn(un; dr, er) ⩽(m + η)hn−1(su) + x z=a,b x ξ=α,β (σ(z, ξ) + η)hn−1({tu = z, v = ξ}) + (c + η)lhn−2(sv), where m, σ, and c are the constants defined in (1.2), (1.3), and (1.5), respectively. proof. first we prove the result for n = 2 and then treat the n-dimensional case. step 1. assume that n = 2. substep 1a. by the definition of c there exists r > 0 and a function h ∈h 3 2 loc(r) satisfying h(−t) = α and h(t) = β for all t ⩾r and 7 16 zz r2  h′(t) −h′(s)  2 |t −s|2 dt ds + z r v h(t)
dt ⩽c + η. (5.23) define w(t, s) = 1 2s z t+s t−s h(τ) dτ. (5.24) by proposition 2.9, we have that w ∈h2 loc(r × (0, ∞)), tw = h, and zz △r  d2w(t, s)  2 dt ds ⩽7 16 z r −r z r −r  h′(t) −h′(s)  2 |t −s|2 dt ds, where △r := t + 2r −(r, 0) and t + 2r :=  (t, s) ∈r2 : 0 < s < r, s < t < 2r −s . for (x, y) ∈△rρn define wn(x, y) := w  x ρn , y ρn  , where ρn = εnλ −1 3 n . then fε(wn, △rρn, (−rρn, rρn) × {0}) = zz △rρn  ε3 n|∇2 (x,y)wn(x, y)|2 + 1 εn w(wn(x, y))  dx dy + λn z rρn −rρn v (twn(x)) dx = zz △rρn ε3 n ρ4 n   ∇2 (t,s)w x ρn , y ρn
   2 + 1 εn w  w x ρn , y ρn
 dx dy + λn z rρn −rρn v  tw x ρn , y ρn
 dx = zz △r h εnλ 2 3 n|∇2 (t,s)w(t, s)|2 + εnλ −2 3 n w(w(t, s)) i dt ds + εnλ 2 3 n z r −r v (tw(t)) dt ⩽(l + o(1)) " 7 16 z r −r z r −r  h′(t) −h′(s)  2 |t −s|2 dt ds + z r −r v (th(t)) dt # + cε2 n, where we used the fact that w is continuous and ∥w∥∞⩽c. thus fε(wn, △rρn, (−rρn, rρn) × {0}) ⩽l " 7 16 z r −r z r −r  h′ η(t) −h′ η(s)  2 |t −s|2 dt ds + z r −r v (thη(t)) dt # + o(1). (5.25) substep 1b. to complete this step, we need to match the function wn to the function un given in corollary 5.10 (with n = 2 and ω:= dr). consider the function e un(x, y) := ψn(x, y)wn(x, y) + 1 −ψn(x, y)
un(x, y) for (x, y) ∈r2, where ψn ∈c∞(r × (0, ∞); [0, 1]) satisfies ψn ≡1 in t + rρn, ψn ≡0 in rn\t + rεn, and ∥∇ψn∥∞⩽c εn and ∥∇2ψn∥∞⩽c ε2 n . (5.26) 34 b. galv ̃ ao-sousa α b a β zn gaα n gbα n wn ! un r rλ 1 3 ε (a) (b) fig. 4. (a) close-up view of t2rεn and t2rρn.; (b) domain of integration after change of variables and divided in regions. since twn = tun = v in △rεn\△rρn
∩er, we have that t e un = v on △rεn\△rρn
∩er. hence v t e un
= 0 in △rεn\△rρn
∩er. thus, it suffices to estimate fε(e un; ln, ∅) = z ln  ε3 n|∇2e un(x, y)|2 + 1 εn w(e un(x, y))  dx dy, (5.27) where ln := △rεn\△rρn. by young's inequality and (5.26), for (x, y) ∈ln we have |∇2e un(x, y)|2 ⩽(1 + η)|∇2wn(x, y)|2 + cη  |∇2un(x, y)|2 + 1 ε2 n  |∇wn(x, y)|2 + |∇un(x, y)|2 + 1 ε4 n  |wn(x, y)|2 + |un(x, y)|2 , (5.28) and, so ε3 zz ln |∇2e un(x, y)|2 dx dy ⩽(1 + η)ε3 n zz ln |∇2wn(x, y)|2 dx dy + c zz ln  εn|∇wn(x, y)|2 + 1 εn |wn(x, y)|2 + ε3 n|∇2un(x, y)|2 + εn|∇un(x, y)|2 + 1 εn |un(x, y)|2  dx dy =: i1 + i2 = i3. (5.29) to estimate i1, note that ε3 n zz ln |∇2 (x,y)wn(x, y)|2 dx dy = ε3 n ρ4 n zz ln   ∇2 (t,s)wn x ρn , y ρn
   2 dx dy = εnλ 2 3 n zz 1 ρn ln |∇2w(t, s)|2 dt ds (5.30) extend w to tr εn ρn using (5.24). since w(t′, *) is even, by proposition 2.9 and (5.30), we have ε3 n zz ln |∇2 (x,y)wn(x, y)|2 dx dy = εnλ 2 3 n zz 1 ρn ln |∇2 (t,s)w(t, s)|2 dt ds ⩽7 16εnλ 2 3 n zz 1 ρn ln     h′(s + t) −h′(s −t) 2t     2 ds dt ⩽7 32εnλ 2 3 n zz 1 ρn ln     h′(s + t) −h′(s −t) 2t     2 ds dt = 7 64εnλ 2 3 n zz −rλ 1 3 n ,rλ 1 3 n
2 \[−r,r]2     h′(w) −h′(z) w −z     2 dz dw. the integral we are estimating is the integral over the the "square annulus" in figure 4(b). note that on all four corner higher-order phase transitions with line-tension effect 35 squares of figure 4(b), we have h′(z) = h′(w) = 0, so the integral reduces to i1 ⩽7 32εnλ 2 3 n   z r −r z rλ 1 3 n r     h′(w) w −z     2 dz dw + z r −r z −r −rλ 1 3 n     h′(w) w −z     2 dz dw   (5.31) ⩽7 32(l + o(1)) "z r −r z ∞ r     h′(w) w −z     2 dz dw + z r −r z −r −∞     h′(w) w −z     2 dz dw # . to estimate i2, note that since ∥wn∥∞⩽c, we have 1 εn z ln |wn(t, s)|2 dt ds ⩽cεn (5.32) and by h ̆ older inequality, proposition 2.6 and (5.31), we obtain that εn zz ln |∇wn(t, s)|2 dt ds ⩽cεn  ∥wn∥l2(ln)∥∇2wn∥l2(ln) + ∥wn∥2 l2(ln)  ⩽c h ε −1 2 n ∥wn∥l2(ln)   ε 3 2 n∥∇2wn∥l2(ln)  + εn∥wn∥2 l2(ln) i ⩽c(√εn + ε3 n). (5.33) combining (5.32) and (5.33) yields i2 ⩽c√εn. (5.34) we estimate i3 using (5.21). precisely, i3 ⩽c εn l2(ln) ⩽cεn. (5.35) finally, using the fact that e un is bounded in l∞(ln), we have 1 εn zz ln w e un(x, y)
dx dy ⩽c εn l2(ln) ⩽cεn. (5.36) using (5.27), (5.29), (5.31), (5.34), (5.35), and (5.36), we obtain that fε(e un, ln, ∅) ⩽7l 32 "z r −r z ∞ r     h′(w) w −z     2 dz dw + z r −r z −r −∞     h′(w) w −z     2 dz dw # + o(1). (5.37) combine (5.25) and (5.37) to obtain fε(e un, △rεn, (−rεn, rεn) × {0}) ⩽l " 7 16 z ∞ −∞ z ∞ −∞     h′(w) −h′(z) w −z     2 dz dw + z ∞ −∞ v (h(z)) dz # + o(1) ⩽c + η + o(1). (5.38) on the other hand, from corollary 5.10 we know that fε(e un, dr\t + rεn, ∅) = fε(un, dr\t + rεn, ∅) ⩽(m + η)h1(su) + x z=a,b x ξ=α,β (σ(z, ξ) + η)h1({tu = z, v = ξ}) + o(1). (5.39) the result follows by combining (5.38) and (5.39). 36 b. galv ̃ ao-sousa step 2. general n-dimensional problem. in this case, we define un(x) := e un(xn−1, xn) for x = (x′′, xn−1, xn) ∈dr. by fubini's theorem and step 1, we deduce that fε(un, b+ r , er) = z bn−2 r fεn(e un; dx′′ r , ex′′ r ) dx′′ ⩽ z bn−2 r  (m + η)h1(su ∩({x′′} × r2)) + x z=a,b x ξ=α,β (σ(z, ξ) + η)h1({tu = z, v = ξ} ∩({x′′} × r2)) + (c + η)lh0(sv ∩({x′′} × r2))  dx′′ + o(1). using theorem 2.12, we then deduce that lim sup n→∞fε(un, b+ r , er) ⩽(m + η)hn−1(su) + x z=a,b x ξ=α,β (σ(z, ξ) + η)hn−1({tu = z, v = ξ}) + (c + η)lhn−2(sv). this completes the proof. proposition 5.12. let w : r →[0, ∞) satisfy (hw 1 ) −(hw 2 ), let v : r →[0, ∞) satisfy (hv 1 ) −(hv 3 ) let εn →0+ with εnλ 2 3 n →l ∈(0, ∞), let η > 0, and let u ∈bv (ω; {a, b}) and v ∈bv (∂ω; {α, β}), with su an n −1 manifold of class c2 such that hn−1(∂ω∩su) = 0 and sv an n −2 manifold of class c2 . then there exists {un} ⊂h2(ω) such that un →u in l2(ω), tun →v in l2(∂ω), and lim sup n fεn(un) ⩽(m + η)hn−1(su) + x z=a,b x ξ=α,β (σ(z, ξ) + η)hn−1({tu = z, v = ξ}) + (c + η)lhn−2(sv), where m, σ, and c are the constants defined in (1.2), (1.3), and (1.5), respectively. proof. from corollary 5.10, it suffices to prove that lim sup n fεn(un; ω∩b(x0; r), ∂ω∩b(x0; r)) ⩽(m + η)hn−1(su ∩b(x0; r)) + x z=a,b x ξ=α,β (σ(z, ξ) + η)hn−1({tu = z, v = ξ} ∩b(x0; r)) + (c + η)lhn−2(sv ∩b(x0; r)), (5.40) for each point x0 ∈sv and for some neighborhood a of x0. ψ1 ψ2 sv y0 sv z0 sv x0 fig. 5. scheme for the flattening of the boundary and of sv. first we fix a point x0 ∈sv. since the domain is of class c2, we can find r > 0 such that, up to a rotation, ∂ω∩b(x0; r) =  x ∈rn : xn = γ1(x′) , (5.41) for some function γ1 ∈c2(rn−1). so we define ψ1(x) := x′, xn −γ1(x′)
, u(y) := (u ◦ψ−1 1 )(y), and v(y) := (v ◦ψ−1 1 )(y). moreover, sv is also of class c2, so we can find 0 < r < r such that, up to a "horizontal rotation", i.e., r =  r′ 0 0 1  , with r′ ∈so(n −1), we have sv ∩b(x0; r) =  y ∈rn−1 × {0} : yn−1 = γ2(y′′) , higher-order phase transitions with line-tension effect 37 for some function γ2 ∈c2(rn−2). let ψ2(y) := y′′, yn−1 −γ2(y′′), yn
, u(z) := u◦ψ−1 2
(z) = u ◦(ψ2 ◦ψ1)−1
(z), v(z) := v ◦(ψ2 ◦ψ1)−1
(z). let φ := ψ2 ◦ψ1 : rn →rn, which is a bi-lipschitz homeomorphism. moreover, its isometry defect δr vanishes as r →0 due to the regularity of both ∂ωand sv. let z0 := φ(x0) ∈sz0. note that dr := φ ω∩b(x0; r)
is a neighborhood of z0, and set er := φ ∂ω∩b(x0; r)
. let {un} ⊂h2 (dr) be defined as in proposition 5.11 with u and v. then from proposition 5.5, we have that fεn (un ◦φ; ω∩b(x0; r), ∂ω∩b(x0; r)) ⩽(1 −δr)−(n+4)fεn (un; dr, er)) + δr (1 −δr)n+2 ε3 n z dr |∇2un(z)| |∇un(z)| + δr|∇un(z)|2
dz. on the other hand, by h ̈ older and propositions 2.6 and 5.11, we have that ε3 n z b+(z0;r) |∇2un(z)| |∇un(z)| + δr|∇un(z)|2
dz ⩽cεn, and that fεn (un; dr, er) ⩽(m + η)hn−1(su ∩dr) + x z=a,b x ξ=α,β (σ(z, ξ) + η)hn−1({tu = z, v = ξ} ∩er) + (c + η)lhn−2(sv ∩er) + o(1). moreover, hn−1(su ∩dr) = hn−1(s(u ◦φ−1) ∩dr) = hn−1 φ(su) ∩φ ω∩b(x0; r)

⩽hn−1 φ su ∩b(x0; r)

⩽lip(φ)n−1hn−1 su ∩b(x0; r)

= hn−1su ∩b(x0; r)
, because φ is an isomorphism. analogously, we deduce that hn−1({tu = z, v = ξ} ∩dr) ⩽hn−1 ({tu = z, v = ξ} ∩b(x0; r)) , hn−2(sv ∩dr) ⩽hn−2 (sv ∩b(x0; rεn)) . hence lim sup n fεn un ◦φ; ω∩b+(x0; r)
; ∂ω∩b(x0; r)
⩽(1 −δr)−(n+4)  (m + η)hn−1 (su ∩b(x0; r)) + x z=a,b x ξ=α,β (σ(z, ξ) + η)hn−1 ({tu = z, v = ξ} ∩b(x0; r)) + (c + η)lhn−2 (sv ∩b(x0; r))  this proves the result. proof of theorem 1.2(ii). since u ∈bv (ω; {a, b}), we may write u as u(x) = ( a if x ∈ea, b if x ∈ω\ea, where ea is a set of finite perimeter in ω. similarly, since v ∈bv (∂ω; {α, β}), we may write v as v(x) = ( α if x ∈fα, β if x ∈∂ω\fα, where fα is a set of finite perimeter in ∂ω. apply proposition 2.11 to the set ea to obtain a sequence of sets ek of class c2 such that ln(ea△ek) →0 and hn−1(∂ea ∩∂ek) →0. by slightly modifying each ek, we may assume that hn−1(∂ω∩∂ek) = 0. similarly, by proposition 2.13 applied to the set fα, we may find a sequence of sets fk ⊂∂ω of class c2 such that hn−1(fα△fk) →0 and hn−2(∂∂ωfα△∂∂ωfk) →0. define the sequences of functions uk(x) := ( a if x ∈ω∩ek, b if x ∈ω\ek, vk(x) := ( α if x ∈∂ω∩fk, β if x ∈∂ω\fk. 38 b. galv ̃ ao-sousa apply proposition 5.12 to find {uk,n} ⊂h2(ω) such that uk,n n − →uk in l2(ω), tuk,n n − →vk in l2(∂ω), and lim sup n fεn(uk,n) ⩽ m + 1 k
perω(ek) + x z=a,b x ξ=α,β σ(z, ξ) + 1 k
hn−1{tuk = z} ∩{vk = ξ}
+ c + 1 k
lper∂ω(fk). since uk →u in l2(ω) and vk →v in l2(∂ω), we have lim k lim n ∥uk,n −u∥l2(ω) = 0, lim k lim n ∥tuk,n −v∥l2(∂ω) = 0, lim sup k lim sup n fεn(uk,n) ⩽mperω(ea) + x z=a,b x ξ=α,β σ(z, ξ)hn−1{tu = z} ∩{v = ξ}
+ clper∂ω(fα). diagonalize to get a subsequence kn →∞and obtain un := ukn,n →u in l2(ω), tun →v in l2(∂ω), and lim sup n fεn(un) ⩽mperω(ea) + x z=a,b x ξ=α,β σ(z, ξ)hn−1{tu = z} ∩{v = ξ}
+ clper∂ω(fα). this completes the proof. acknowledgements this research was partially funded by funda ̧ c ̃ ao para a ciˆ encia e a tecnologia under grant sfrh/bd/8582/2002, the department of mathematical sciences of carnegie mellon university and its center for nonlinear analysis (nsf grants no. dms-0405343 and dms-0635983), irene fonseca (nsf grant dms-0401763) and giovanni leoni (nsf grants no. dms-0405423 and dms-0708039). the author thanks vincent millot and dejan slepˇ cev for the fruitful conversations, luc tartar for useful conversations on proposition 2.9, and is indebted to irene fonseca and giovanni leoni for uncountable discussions and advice as the work progressed that largely influenced its course. references [1] r. adams, sobolev spaces, academic press, 1975. mr 56:9247 [2] g. alberti, g. bouchitt ́ e, and p. seppecher, phase transition with the line tension effect, arch. rational mech. anal. 144 (1998), 1–46. mr 99j:76104 [3] l. ambrosio, n. fusco, and d. pallara, functions of bounded variation and free discontinuity problems, oxford claredon press, 2000. mr 2003a:49002 [4] j. ball, a version of the fundamental theorem for young measures, pdes and continuum models of phase transitions (nice, 1988), lecture notes in phys., vol. 344, springer, berlin, 1989, pp. 207–215. mr 91b:49021 [5] r. choksi, s. conti, r. kohn, and f. otto, ground state energy scaling laws during the onset and destruction of the intermediate state in a type i superconductor, comm. pure appl. math. 61 (2008), no. 5, 595–626. mr 2009d:82162 [6] r. choksi and r. kohn, bounds on the micromagnetic energy of a uniaxial ferromagnet, comm. pure appl. math. 51 (1998), no. 3, 259–289. mr 2000d:82046 [7] r. choksi, r. kohn, and f. otto, domain branching in uniaxial ferromagnets: a scaling law for the minimum energy, comm. math. phys. 201 (1999), no. 1, 61–79. mr 2000c:49060 [8] , energy minimization and flux domain structure in the intermediate state of a type-i superconductor, j. nonlinear sci. 14 (2004), no. 2, 119–171. mr 2005e:82123 [9] s. conti, i. fonseca, and g. leoni, a γ-convergence result for the two-gradient theory of phase transitions, comm. pure applied math. 55 (2002), 857–936. mr 2003c:49017 [10] e. dibenedetto, real analysis, birkh ̈ auser, 2002. mr 2003d:00001 [11] l. evans and r. gariepy, measure theory and fine properties of functions, crc press, 1992. mr 93f:28001 [12] i. fonseca and g. leoni, modern methods in the calculus of variations: lp spaces, springer monographs in mathematics, springer, 2007. mr mr2341508 [13] i. fonseca and c. mantegazza, second order singular perturbation models for phase transitions, siam j. math. anal. 31 (2000), no. 5, 1121–1143. mr 2001i:49030 [14] e. gagliardo, ulteriori proriet` a di alcune classi di funzioni in pi` u variabili, ricerche di matematica 8 (1959), 24–51. mr 22:181 [15] a. garroni and g. palatucci, a singular perturbation result with a fractional norm, variational problems in materials science, progr. nonlinear differential equations appl., vol. 68, birkh ̈ auser, basel, 2006, pp. 111–126. mr 2223366 [16] e. giusti, minimal surfaces and functions of bounded variation, birkh ̈ auser, 1984. mr 56:9247 [17] e. hebey, nonlinear analysis on manifolds: sobolev spaces and inequalities, ams/cims, 1999. mr 2000e:58011 [18] g. dal maso, an introduction to γ-convergence, birkh ̈ auser, 1993. mr 94a:49001 higher-order phase transitions with line-tension effect 39 [19] m. miranda, d. pallara, f. paronetto, and m. preunkert, heat semigroup and functions of bounded variation on riemannian manifolds, j. reine angew. math. 613 (2007), 99–119. mr 2009b:58053 [20] l. modica, the gradient theory of phase transitions and the minimal interface criterion, arch. rational mech. anal. 98 (1987), no. 2, 123–142. mr 88f:76038 [21] , the gradient theory of phase transitions with boundary contact energy, ann. inst. henri poincar ́ e - analyse non lin ́ eaire 4 (1987), no. 5, 487–512. mr 89c:76108 [22] l. modica and s. mortola, un esempio de γ−-convergenza, boll. un. mat. ital. b(5) 14 (1977), no. 1, 285–299. mr 56:3704 [23] s. m ̈ uller, variational models for microstructure and phase transitions, calculus of variations and geometric evolution problems (cetraro, 1996), lecture notes in math., vol. 1713, springer, 1999, pp. 85–210. mr 2001b:49019 [24] l. nirenberg, an extended interpolation inequality, ann. sc. normale pisa - scienze fisiche e matematiche 20 (1966), 733–737. mr 34:8170 [25] e. stein, singular integrals and differentiability properties of functions, princeton university press, 1970. mr 44:7280 [26] l. tartar, compensated compactness and applications to partial differential equations, nonlinear analysis and mechanics: heriot-watt symposium, vol. iv, res. notes in math., vol. 39, pitman, boston, mass., 1979, pp. 136–212. mr 81m:35014 [27] w. ziemer, weakly differentiable functions, graduate texts in mathematics, vol. 120, springer-verlag, new york, 1989, sobolev spaces and functions of bounded variation. mr 91e:46046

0911.1727

electric field generation by the electron beam filamentation instability: filament size effects

the filamentation instability (fi) of counter-propagating beams of electrons is modelled with a particle-in-cell simulation in one spatial dimension and with a high statistical plasma representation. the simulation direction is orthogonal to the beam velocity vector. both electron beams have initially equal densities, temperatures and moduli of their nonrelativistic mean velocities. the fi is electromagnetic in this case. a previous study of a small filament demonstrated, that the magnetic pressure gradient force (mpgf) results in a nonlinearly driven electrostatic field. the probably small contribution of the thermal pressure gradient to the force balance implied, that the electrostatic field performed undamped oscillations around a background electric field. here we consider larger filaments, which reach a stronger electrostatic potential when they saturate. the electron heating is enhanced and electrostatic electron phase space holes form. the competition of several smaller filaments, which grow simultaneously with the large filament, also perturbs the balance between the electrostatic and magnetic fields. the oscillations are damped but the final electric field amplitude is still determined by the mpgf.

introduction the filamentation instability (fi) driven by counterpropagating electron beams amplifies magnetic fields in astrophysical and solar flare plasmas [1-5] and it is also relevant for inertial confinement fusion (icf) [6] and laser-plasma interactions in general [7, 8]. it has been modelled with particle-in-cell (pic) and vlasov codes [9-17] taking sometimes into account the ion response and a guiding magnetic field. it turns out that the fi is important, when the beam speeds are at least mildly relativistic and if the beams have a similar density [18]. otherwise its linear growth rate decreases below those of the competing two-stream instability or mixed mode instability [19]. the saturation of the fi is attributed to magnetic trapping [9]. more recently, it has been pointed out [13, 14] that the electric fields are also important in this context. an electric field component along the beam velocity vector vb is driven by the fi through the displacement current. this component is typically weak and its relevance to the plasma dynamics is negligible compared to that of the magnetic and the electrostatic fields. the fi is partially electrostatic during its linear growth phase, if the electron beams are asymmetric due to different densities. symmetric electron beams result in purely electromagnetic waves with wavevectors k ⊥vb [19, 20]. a nonlinear growth mechanism is provided in this case by the current of the electrons, which have been accelerated by the magnetic pressure gradient force (mpgf). the electromagnetic and electrostatic components separate in a 1d simulation box, because the gradients along two directions vanish in the maxwell's equations. the electrostatic field is polarized in the simulation direction, while the electromagnetic components are polarized orthogonal to it. if both electron beams have an equal density and temperature, the electrostatic field component along the wavevector k can only be driven nonlinearly. we select here a direction of our 1d pic simulation box that is orthogonal to vb, through which this nonlinear mechanism can be examined in an isolated form. the equally dense and warm counterstreaming beams of electrons have the velocity modulus |vb| = 0.3c. the ions are immobile and compensate the electron charge. the mildly relativistic relative streaming speed ≈0.55c implies, that the growth rate of the fi is significant. at the same time, any relativistic mass changes can be neglected during the growth phase and the saturation of the fi. the initial conditions of the plasma equal those in the refs. [21, 22]. the size distribution of the filaments could be sampled with the help of the long 1d simulation box in ref. [21]. a pair of current filaments, which are small according to this size distribution, has been isolated in ref. [22]. it could be shown that the electrostatic field is indeed driven by the mpgf for this filament pair. the electrostatic field performed undamped oscillations around a background one. the latter excerted the same force on the electrons as the mpgf. here we assess the influence of the filament size. this paper is structured as follows. section 2 discusses briefly the pic code, the initial conditions and the key nonlinear processes. the results are presented in the section 3, which can be summarized as follows. the electrons are heated up along the electric fields of the filamentation instability 3 wavevector k by their interaction with the wave fields. as we increase the filament size the peak amplitudes grow, which are reached by the magnetic and by the electrostatic field when the fi saturates. the electron heating increases with the filament size and large electron phase space holes form, which interact with the electromagnetic fields of the filamentation modes. the large box sizes allow the growth of more than one wave and the filamentation modes compete. the electrostatic field oscillations are damped or inhibited and the amplitude modulus converges to one, which equals that expected from the mpgf. we confirm that the strength of the electrostatic force on an electron is comparable to that of the magnetic force, when the fi saturates. the extraordinary modes are pumped by the fi [14]. the results are discussed in section 4. 2. the pic simulation, the initial conditions and the nonlinear terms the pic simulation method is detailed in ref. [23]. our code is based on the numerical scheme proposed by [24]. the phase space fluid is approximated by an ensemble of computational particles (cps) with a mass mcp and charge qcp that can differ from those of the represented physical particles. the charge-to-mass ratio must be preserved though. the maxwell-lorentz equations are solved. the plasma frequency of each beam with the density ne that we model is ωp = (e2ne/meǫ0)0.5 and ωp = √ 2ωp. the electric and magnetic fields are normalized to en = ee/cmeωp and bn = eb/meωp. the current is normalized to jn = j/2neec and the charge to ρn = ρ/2nee. the physical position, the time and speed are normalized as xn = x/λs with λs = c/ωp, tn = tωp and vn = v/c. the normalized frequency ωn = ω/ωp. we drop the indices n and x, t, ω, e, b, j and ρ are specified in normalized units. the equations are ∇× e = −∂tb , ∇× b = j + ∂te, (1) ∇* e = ρ, ∇* b = 0, (2) dtpcp = qcp (e[xcp] + vcp × b[xcp]) , dxxcp = vcp,x, (3) with pcp = mcpγcpvcp. here vcp,x is the component along x of vcp. the currents jcp ∝qcpvcp of each cp are interpolated to the grid. the summation over all cps gives j, which is defined on the grid. the j updates e and b through (1). our numerical scheme fulfills (2) as constraints. the new fields are interpolated to the position of each cp and advance its position xcp and pcp through (3). all components of p are resolved. two spatially uniform beams of electrons with qcp/mcp = −e/me move along z. beam 1 has the mean speed vb1 = vb and the beam 2 has vb2 = −vb1 with vb = 0.3. both beams have a maxwellian velocity distribution in their respective rest frame with a thermal speed vth = c−1(kbt/me)0.5 of vb/vth = 18. the negative electron charge is compensated by an immobile positive charge background. the initial conditions are ρ, j, e, b = 0. figure 1 displays the k spectrum of the unstable waves. the growth rates of the fi modes are close to the maximum value, while relativistic effects are still negligible. the growth rate spectrum with k∥= 0 relevant for our simulations peaks with δm = 0.29 at kmλs ≈10. a filamentation mode with kmλs = 7 has been electric fields of the filamentation instability 4 figure 1. (colour online) the growth rates in units of ωp as a function of the wavenumber in the full k space, where λsk∥(λsk⊥) points along (orthogonal) to vb. the growth rates of the fi modes with k∥= 0 are comparable to that of the two-stream mode with k⊥= 0 and to those of the oblique modes. the growth rates for k∥= 0 decrease to zero for k⊥→0 and they are stabilized at high k⊥by thermal effects. the growth rate maximum for k∥= 0 is δm = 0.29 and it is reached at kmλs ≈10. considered in detail previously [22], while we investigate here larger filaments. the box length l1 = 2 for the simulation 1 and the filamentation mode with k1 = 2π/l1 grows at the exponential rate 0.92 δm. the box length of the simulation 2 is l2 = 2.8 and the growth rate of the filamentation mode with k2 = 2π/l2 is 0.86 δm. the growth rates decrease rapidly for lower k and these modes are no longer observed in pic simulations [21]. both simulations resolve x by ng = 500 grid cells with the length ∆x and use periodic boundary conditions. the phase space distributions f1(x, v) of beam 1 and f2(x, v) of beam 2 are each sampled by np = 6.05 * 107 cps. the total phase space density is defined as f(x, v) = f1(x, v) + f2(x, v). each electron beam constitutes prior to the saturation of the fi a fluid with the index j, which has the density nj(x) = r v fj(x, v)dv and the mean velocity vj(x) = r v vfj(x, v)dv. the normalized momentum equation for such a fluid is ∂t(njvj) + ∇(njvjvj) = −∇pj −nje + ∇(bb) −∇b2/2 + b × ∂te, (4) where the thermal pressure tensor pj is normalized to 2menec2. the restriction to one spatial dimension implies, that the gradients along y and z vanish. the fi results in this case in the initial growth of by and of a weaker electric ez. the thermal pressure electric fields of the filamentation instability 5 is initially diagonal due to the spatially uniform single-maxwellian velocity distribution. the x-component of the simplified fluid momentum equation is ∂t(njvj,x) + dx(njv2 j,x) = −v2 thdxnj −njex −bydxby + by∂tez. (5) the thermal pressure gradient v2 thdxnj is valid, as long as the electron beams have not been heated up. let us assume that the displacement current and the thermal pressure gradient can be neglected, leaving us with the term njex and the mpgf as the key nonlinear terms. the fluid momentum equations can be summed over both beams and we consider the right hand side of (5). as long as ex is small, the electron density is not spatially modulated and n1 + n2 ≈1. the nonlinear terms cancel out, if ex = −2bydxby. it could be demonstrated for a short filament in ref. [22] that this is the case, even when the fi just saturated. the ex oscillated in time and after the saturation with the amplitude eb = −bydxby around a time-stationary eb. 3. simulation results 3.1. the scaling of by, ex and eb with the box length the beam velocity vb ∥z and the electrons of both beams and their micro-currents are re-distributed by the fi only along x. the initially charge- and current-neutral plasma is transformed into one with jz(x, t) ̸= 0. the gradients along the y, z-direction vanish in our 1d geometry. ampere's law simplifies to dxby = jz + ∂tez, resulting in the growth of by and ez. the mpgf drives ex. the bx = 0 in the 1d geometry and ey, bz remain at noise levels. the right-hand side of (5) depends on ex, ez and by, as well as on their spatial gradients, which should vary with the filament size. we want to gain qualitative insight into the scaling of the field amplitudes with the filament size and determine if ex is driven by the mpgf also for the large filaments. the fields that grow in simulation 1 and 2 are compared to those discussed previously in ref. [22] that used the box size lc = 0.89. figure 2 shows the respective dominant fourier component of by, of ex and of 2eb. the amplitude moduli of the mode with ks = 2π/ls are considered for by and those of the 2ks mode for ex and 2eb. the subscript s is 1, 2 or c and refers to the respective simulation. the amplitudes of by increase with an increasing box size. after the fi has saturated, we find that by(k1, t) ≈2by(kc, t) and by(k2, t) ≈2.5by(kc, t). the increase of the saturation value of by(ks, t) with ls is consistent with magnetic trapping [9]. the magnetic bouncing frequency ωb = (vbksb[ks, t])1/2 in our normalization. the fi should saturate once ωb is comparable to the linear growth rate of the fi, which is approximately constant for the box sizes lc, l1 and l2 (fig. 1). a lower ks supports a larger by(ks, t). the ωb ≈0.2 for simulation 1 is comparable to the linear growth rate ωi ≈0.25. after the saturation, the ex(2k1,2, t) > 2ex(2kc, t) and ex(2k1, t) > ex(2k2, t). the ex(2k1, t) > 3ex(2kc, t) while l1/lc ≈2.2. the electrostatic potential in simulation 1 is thus larger by a factor 6, which should result in a more violent electron acceleration than in the box with the length lc. the thermal pressure gradient force is potentially electric fields of the filamentation instability 6 30 40 50 60 70 80 90 100 0 0.02 0.04 0.06 (a) time by(ks,t) c 1 2 30 40 50 60 70 80 90 100 0 0.01 0.02 (b) time ex(2ks,t) 2 1 c 25 30 35 40 45 50 55 60 10 −6 10 −3 (c) time ex(2k1), 2eb(2k1) 25 30 35 40 45 50 55 60 10 −6 10 −3 (d) time ex(2k2), 2eb(2k2) figure 2. (colour online) panel (a) compares the by(ks, t) and panel (b) the ex(2ks, t) in the boxes with the size l1, l2 and lc (dashed curve). eb(2ks, t) (dashed curve) is compared with ex(2ks, t) (solid curves) for the box size l1 (c) and l2 (d). more important for larger filaments and it may modify the balance between the nonlinearly driven ex and the mpgf. however, an excellent match between ex(2k1, t) and 2eb(2k1, t) is observed for t < 50, due to which the two nonlinear terms on the right hand side of (5) practically cancel for simulation 1. the ex(2k2, t) ≈2eb(2k2, t) in simulation 2 for 30 < t < 42 and for 46 < t < 53. both fields disagree in between these time intervals and a local minimum is observed. the field and electron dynamics is now examined in more detail for the box lengths l1 and l2. 3.2. simulation 1: box length l1 = 2 figure 3 displays the evolution of the relevant field components. the by(x, t) rapidly grows and saturates at t ≈45. it is initially stationary in space but it oscillates in time until t ≈65, which implies that by(x, t) does not immediately go into its stable saturated state. the by(x, t) shows only one spatial oscillation and the filamentation mode with the wavelength k1 = 2π/l1 is thus strongest. however, the interval with the large positive by(x, t ≈45) covers 0 < x < 0.9, while that with the large negative by(x, t ≈45) is limited to 1.2 < x < 1.7. this mode is thus initially not monochromatic. the saturated structure formed by by(x, t) drifts after t ≈65 to lower x at a speed < 0.01 and it remains stationary in its moving rest frame. the ez(x, t) grows initially in unison with by(x, t) and it is shifted in space by 90◦with respect to by(x, t), as expected from ampere's law. oscillations of ez(x, t) are spatially correlated with those of the by(x, t) for 45 < t < 65. the ez(x, t) undergoes a mode conversion at t ≈65 into a time-oscillatory and spatially uniform ez(x, t). figure 3(c) demonstrates that ex(x, t) is following the drift of by(x, t) towards decreasing x, but that its wavenumber is twice that of by(x, t). the by(x, t) is stationary in its moving rest frame, while ex(x, t > 70) is oscillating around an equilibrium electric field with an amplitude and spatial distribution that resembles eb(x, t) in fig. 3(d). the electric and the magnetic electric fields of the filamentation instability 7 figure 3. (colour online) the field amplitudes in the box l1: the panels (a-d) show by, ez, ex and eb, respectively. the amplitude of by reaches a time-stationary distribution, which convects to decreasing x at a speed < 0.01. the ez and ex components are oscillatory in space and in time. the ez is phase-shifted by 90◦ relative to by when the fields saturate at t ≈45. the ex and the eb are co-moving and ex oscillates in time around a mean amplitude comparable to eb for t > 70. 0 2 4 50 100 150 200 250 (a) k / k1 time 0 2 4 50 100 150 200 250 (b) k / k1 time 0 0.005 0.01 0.015 0.02 0 0.005 0.01 0.015 0.02 50 100 150 200 250 0 0.01 0.02 0.03 (c) time ( 2k1 ) amplitude 50 100 150 200 250 0 0.005 0.01 (d) time (4k1) amplitude figure 4. the relevant part of the amplitude spectrum ex(k, t) is displayed for low k in (a) and (b) shows that of eb(k, t). the wavenumbers are expressed in units of k1. the amplitude moduli of the dominant modes are displayed for k = 2k1 in (c) and its first harmonic with k = 4k1 in (d), where the dashed curves correspond to eb. forces are comparable in their strength, but their distribution differs. figure 4 compares in more detail the moduli of the amplitude spectra ex(k, t) and eb(k, t). the amplitudes of the strongest modes fulfill ex(2k1, t) ≈2eb(2k1, t) until t = 50 (see also fig. 2). the ex(2k1, t) thus overshoots eb(2k1, t) and it oscillates around it after t = 50. the oscillation is damped and the amplitudes of ex(2k1, t) and eb(2k1, t) converge. the full spectra ex(k, t) and eb(k, t) reveal that the mode k = 4k1 is also important for t > 100. it probably is a harmonic of the mode with k = 2k1 and electric fields of the filamentation instability 8 figure 5. (colour online) the 10-logarithm of the phase space densities in units of cps at the time t = 50 (a-c) and t = 120 (d-f) in the box l1: panels (a,d) show the total phase space density f(x, pz) with the beam momentum p0 = mevbγ(vb). the phase space density f1(x, px) of beam 1 is shown in (b,e) and the f2(x, px) of beam 2 in (c,f). both beams are spatially separated and (e,f) reveal cool electron clouds immersed in a hot electron background with momenta of up to ≈p0. not an independently growing fi mode. otherwise we would expect that the mode with k ≈3k1 also grows. the amplitude of ex(4k1, t) is close to that of eb(4k1, t). a dissipation mechanism for the interplaying jx and ex is present, which causes the damping and the convergence of ex(x, t) to eb(x, t). the damping persists after t = 65, when by is quasi-stationary in its moving reference frame. the term by∂tez in (5) could, in principle, be one dissipation mechanism. however, even at t ≈50 when ∂tez is largest and by has developed in full, this term is weaker by more than one order of magnitude than the mpgf and the term njex in simulation 1 (not shown). if the term by∂ez would be the damping mechanism, this should have resulted in a noticable field damping also in the short simulation box with length lc. a damping of ex(x, t) has not been observed in ref. [22]. the thermal pressure gradient force may provide this damping and we examine now the electron phase space density distribution. figure 5 displays the phase space distributions f1(x, px) of the beam 1 and the f2(x, px) of the beam 2 at the times t = 50 and t = 120. the total phase space density f(x, pz) is shown for the same times. the beams reveal a high degree of symmetry already at t = 50 and the filament centres are shifted along x by l1/2. the phase space structures in fig. 5(b,c) are, however, different at the filament boundaries x ≈0.5 and x ≈1.5. this difference is responsible for the deviation of the initial by(x, t) from a sine curve in fig. 3(a). the phase space distribution at late times reveals, that the electrons are heated along px but not along pz. the filament drift to lower x is visible from figs. 5(a,d) and agrees with the observed one of by(x, t) in fig. 3. the electrons are accelerated along x to a peak speed ∼vb, which is more than twice that observed in the box with the length lc [22]. the peak electron kinetic energy due to the velocity electric fields of the filamentation instability 9 figure 6. the field amplitudes in the box l2: the panels (a-d) show by, ez, ex and eb = −bydxby, respectively. the amplitude of by reaches a steady state value, which convects to increasing x at a speed < 0.01. the ez and ex components are oscillatory in space and in time. the ez is phase-shifted by 90◦relative to by when the fields saturate at t ≈50. the ex and the eb are co-moving and ex(x) oscillates in time around a mean amplitude comparable to eb for t > 100. component along x thus increases by a factor, which is comparable to the increase in the electrostatic potential as we go from a box with length lc to one with l1. this strong electron heating is likely to result in higher thermal pressure gradient forces. the expression dxn1(x) r vxf(x, vx)dvx has been evaluated (not shown) at t = 75 and its peaks reach values ≈0.1, which are comparable to the mpgf. the width of these peaks is small compared to the electron skin depth. movie 1 animates in time the 10-logarithmic phase space distributions f1(x, px) and f1(x, pz) of the beam 1 in the simulation 1. the formation of the filaments is demonstrated. we observe a dense untrapped electron component immersed in an electron cloud that has been heated along the simulation direction by the saturation of the fi. the spatial width of the plasmon containing the dense bulk of the confined electrons in f1(x, px) oscillates in time. the overlap of the filaments in fig. 5(e,f) is thus time dependent and related through its current jx(x, t) to the oscillating ex(x, t) in fig. 3(b). the phase space distribution f1(x, px) reveals small-scale structures (phase space holes) that gyrate around the centre of the filament. these coherent structures result in jumps in the thermal pressure. 3.3. simulation 2: box length l2 = 2.8 figure 6 displays the fields that grow in the simulation with the box length l2 = 2.8. the growth rate map in fig. 1 demonstrates that the fi can drive simultaneously several modes in the simulation box. the mode with k2 = 2π/l2 ≈2.25 has, for example, a lower growth rate than that with k ≈2k2. we observe consequently oscillations electric fields of the filamentation instability 10 0 2 4 50 100 150 200 250 (a) k / k2 time 0 0.005 0.01 0.015 0.02 0 2 4 50 100 150 200 250 (b) k / k2 time 0 0.005 0.01 0.015 0.02 50 100 150 200 250 0 0.01 0.02 (c) time (2k2) amplitude 50 100 150 200 250 0 0.01 0.02 (d) time (4k2) amplitude figure 7. the relevant part of the amplitude spectrum ex(k, t) is displayed for low k in (a) and (b) shows that of eb(k, t). the wavenumbers are expressed in units of k2. the amplitude moduli of the dominant modes are displayed for k = 2k2 in (c) and its first harmonic with k = 4k2 in (d), where the dashed curves correspond to eb. in by(x, t) along x, which are a superposition of several waves with a k ≥k2 during the initial growth phase 40 < t < 50. these oscillations merge and only one spatial oscillation of by(x, t) and, thus, a single pair of filaments survive after the saturation at t ≈50. the magnetic field structure convects to increasing values of x at a speed < 0.01, but it is stationary in its rest frame after t ≈70. the phase of ez(x, t) is shifted by 90◦with respect to by(x, t) for 40 < t < 60. the oscillations of ez(x, t) undergo a mode conversion during 60 < t < 100 and we observe undamped oscillations with k = 0 for t > 100. the amplitude of these oscillations is modulated on a long timescale. the ex(x, t) and the by(x, t) show no correlation until t ≈70. thereafter the spatial amplitude of ex(x, t) oscillates in time around eb(x, t). the force on an electron imposed by ex(x, t) is comparable to that imposed by vbby(x, t). a more accurate comparison of ex(x, t) and eb(x, t) is again provided by the moduli of their spatial amplitude (fourier) spectra, ex(k, t) and eb(k, t). figure 7 displays ex(k, t) and eb(k, t) and compares in more detail ex(2k2, t) with eb(2k2, t) as well as ex(4k2, t) with eb(4k2, t). the amplitudes ex(2k2, t) ≈2eb(2k2, t) during the exponential growth phase of the fi for 25 < t < 45 (see fig. 2), the amplitude moduli then have a local minimum and continue to grow after this time. we identify the likely reason from eb(k, t) in fig. 7(b). the eb(3k2, t) competes with eb(2k2, t) at t ≈50. a large amplitude modulus of eb(3k2, t) evidences that by(x, t) is not a sine wave at this time. if by ∝sin (k2x), then eb ∝sin (k2x) cos (k2x) and eb(k, t) would be composed of a wave with k = 2k2. the periodic boundary conditions would also allow for a by ∝sin (2k2, t) and here eb would involve a wave with k = 4k2. an eb(3k2, t) can thus not be connected to a single filamentation mode. during the linear growth phase of the fi prior to t ≈40, the jz(x, t) can form structures with a wideband wavenumber spectrum (see fig. 1) and their associated by can grow independently. electric fields of the filamentation instability 11 50 100 150 200 250 0 0.5 1 c) amplitude 50 100 150 200 250 0 0.02 0.04 d) amplitude 50 100 150 200 250 0 0.02 0.04 time e) amplitude figure 8. (colour online) a time-interval of ez(x, t) and the 10-logarithm of its power spectrum pez(k, ω) are displayed in (a) and (b). wavenumbers are given in units of k2. peak 1 is at ω < 0.5 and k = k2. peak 2 is observed at k = k2 and ω ≈1 and peak 3 at k = 0 and ω ≈1. the by(k2, t) is shown in (c), the ez(k2, t) in (d) and ez(0, t) in (e), all normalized to the maximum of by(k2, t). once the mpgf in eq. 5 has reached a significant strength, the fi saturates. the strength of the mpgf increases with k, due to the larger dxby(x, t) of the rapid oscillations. the by ∝sin (k2x) should maximize the magnetic field strength for a given mpgf. this may explain why this mode is the dominant one after t = 70 despite its lower growth rate. the decrease of eb(2k2, t) in fig. 2(d) at t ≈45 is tied to the saturation of eb(3k2, t). the ex(k, t) in fig. 7(a) has a broadband spectrum within 50 < t < 75, which is probably caused by the current jx arising from the rearrangement of the filaments. after this time, ex(2k2, t) ≈eb(2k2, t) and ex(4k2, t) ≈eb(4k2, t). the ex(2k2, t) does not show oscillations around eb(2k2, t) as the simulation 1. the filament rearrangement inhibits an oscillatory equilibrium between jx and ex. figure 8 examines the mode conversion of the electromagnetic ez component observed in fig. 6(b). the pez(k, ω) is the squared modulus of the fourier transform of ez(x, t) over space and over 45 < t < 100. the dispersion relation shows three peaks. peak 1 has a k = k2 and ω < 0.5 and it is tied to the ez(x, t) of the fi mode. this mode grows exponentially and aperiodically. its frequency spectrum is thus spread out along ω. its energy can leak into the peak 2 at k = k2 and ω ≈1. the ez(x, t) is orthogonal to by(x, t) and peak 2 corresponds to an extraordinary mode, similar to the slow extraordinary mode. peak 3 has a k = 0 and ω ≈1 and it corresponds to a spatially uniform oscillation in an extraordinary mode branch. the intermittent behaviour of ez(x, t) in fig. 8(a) results in a broadband spectrum in k and ω. these turbulent wave fields can couple energy directly to the high-frequency electromagnetic modes and excite a discrete spectrum if the boundary conditions are periodic [14]. the interplay of the waves belonging to the three peaks in fig. 8(a) is assessed with the moduli of the amplitude spectra by(k2, t), ez(k2, t) and ez(0, t) in figs. 8(c- electric fields of the filamentation instability 12 figure 9. the 10-logarithmic phase space densities in units of cps at t = 50 (a-c) and t = 120 (d-f) in the box l2: panels (a,d) show the total distribution f(x, pz) with p0 = mevbγ(vb). the beam temperature along pz is unchanged. the distribution f1(x, px) of beam 1 is shown in (b,e) and the f2(x, px) of beam 2 in (c,f). the electrons of both beams spatially separate and (e,f) reveal a dense electron component immersed in a tenuous hot electron background, which reaches a thermal width ≈p0. e). the by(k2, t) and ez(k2, t) grow at the same exponential rate until they saturate at t ≈50, evidencing that they belong to the same fi mode. the by(k2, t) maintains its amplitude after t = 50, while ez(k2, t) decreases until t ≈120 and remains constant thereafter. the ez(0, t) grows in the same time interval to its peak amplitude, which suggests a parametric interaction between these modes. the amplitude modulation in fig. 8(e) must be caused by a beat between two waves, which are similar to the slow- and fast extraordinary modes in the limit k = 0. both modes are undamped on the resolved timescales. one may interpret the parametric interaction as a three-wave coupling between the waves corresponding to the peaks 1-3 in fig. 8(b), resembling the system of ref. [25]. however, here the by(x, t) varies spatially and the parametric interaction may involve more of the waves of the spectrum in fig. 8(b). figure 9 displays the phase space densities f1,2(x, px) and f(x, pz) at the times t = 50 and t = 120. figure 9(a) demonstrates that the electrons of both beams have been rearranged by the fi. the filaments have not yet reached the stable symmetric configuration, because the most pronounced density minima at x ≈1.5 for beam 2 and at x ≈2.5 for beam 1 are not shifted by l2/2. this asymmetry results in the eb(3k2, t) ̸= 0 and in the broadband ex(k, t) at this time in fig. 7. the spatial gradients of by(x, t) and ex(x, t) are high at t ≈50 and the lorentz force changes rapidly with x, explaining the complex phase space structuring in fig. 9(b,c). the phase shift of l2/2 of the density maxima of both beams has been reached at t = 120 in fig. 9(d). the by(x, t) is stationary in its rest frame at this time in fig. 6(a). the electrons are heated up from an initial thermal spread of px/p0 ≈0.05 with p0 = mevbγ(vb) to a peak value of electric fields of the filamentation instability 13 px ≈p0 in fig. 9(e,f). the mean momentum of each beam varies along pz in response to a drift imposed by ex(x, t) and by(x, t) but no heating is observed in this direction. movie 2 shows the 10-logarithmic phase space density projections f1(x, px, t) and f1(x, pz, t) of beam 1 in the simulation 2. it demonstrates that only the core electrons in fig. 9 remain spatially confined. the heated electrons, which have in some cases reached a momentum px that is comparable to the initial beam momentum, are untrapped. the heated electrons move practically freely and they ensure that the beam confinement is not perfect. the trapped electrons maintain the jz(x, t) ̸= 0 and, thus, the by(x, t) ̸= 0. the trapped electrons slowly move to larger values of x. the associated shift of jz(x, t) causes the slow drift of by(x, t) in fig. 6(a). the movie visualizes the formation of the phase space beams and their evolution into phase space holes in f1(x, px). 4. discussion we have examined here the electron beam filamentation instability (fi) in one dimension and in an initially unmagnetized plasma with immobile ions. the fi has been driven by nonrelativistic symmetric electron beams with the same initial conditions as those considered previously [21, 22]. the electric field along the one-dimensional box, which is oriented orthogonally to the beam velocity vector, can only be generated nonlinearly if the beams are symmetric [20]. the fluid equations show that the relevant nonlinear mechanisms can be the magnetic pressure gradient force (mpgf), the thermal pressure gradient force and a term due to the displacement current. the magnetic tension may become important in multi-dimensional simulations, but not for initial conditions similar to ours [26]. the term due to the displacement current is weak in our simulations. it has been observed in ref. [22] that the electrostatic field performs after the saturation of the fi undamped oscillations around a time-stationary background electric field. the amplitude of the oscillatory and of the background electric field are both given by eb(x, t) ≈−bydxby. the phases of both fields are fixed such, that ex(x, t0) = 2eb(x, t0) at the saturation time t0. this amplitude ensures that the nonlinear terms due to the mpgf and due to the electrostatic field cancel each other approximately in the fluid equations when the fi saturates. the thermal pressure gradient force did not visibly contribute in the simulation of the small filament pair [26], possibly because of the only modest heating of the initially cool beams. here we have assessed the importance of the filament size with the help of two 1d pic simulations, which used two different box lengths that were larger than that of the 1d box in ref. [22, 26]. the initial conditions for the plasma were otherwise identical. we summarize our findings as follows. we have demonstrated for both simulations, that ex(x, t ≤t0) ≈2eb(x, t ≤t0) during the full exponential growth phase and not just at the saturation time t0. the fi thus adjusts the electrostatic field during its exponential growth phase such, that the dominant nonlinear terms cancel each other. magnetic trapping states that the fi saturates, when the magnetic bouncing frequency is comparable to the linear growth rate. the exponential growth rates for the two electric fields of the filamentation instability 14 simulations considered here and that in ref. [22] are close. the amplitude reached by the magnetic field prior to its satuation thus increases with the box length. we found that the electrostatic potential driven by the mpgf is 5-6 times stronger for the box sizes used here than for the short box in ref. [22], while the initial mean kinetic energy of the electrons is the same. consequently, the electron heating is stronger and the plasma processes more violent for large filaments. magnetic trapping is, however, not the exclusive saturation mechanism. the electrostatic forces are comparable in strength to the magnetic forces when the fi saturates [13, 14]. the electrostatic field during the intermittent phase has differed in our two simulations from that observed in ref. [22]. the movies demonstrated that this phase involves the formation of large nonlinear structures (phase space holes) in the electron distribution, which can result in steep gradients of the thermal pressure and in the generation of solitary (bipolar) electrostatic wave structures that are independent of the fields produced by the fi. the thermal pressure gradient force is comparable to that of the other nonlinear terms, but only over limited spatial intervals. the electric field component along the beam velocity vector has undergone a mode conversion. its energy leaked into the high-frequency electromagnetic modes [14]. the wavenumber spectrum of the electrostatic field correlated well with that of the mpgf in simulation 1, but the peak electric field overshot the expected one. the electrostatic field performed damped oscillations around eb and both converged eventually to the same value. the wavenumber spectrum of the electrostatic field in simulation 2 deviated from that of the mpgf in the intermittent phase. its wavenumber spectrum was broadband, while that of the mpgf was quasi-monochromatic. the amplitude modulus of the electrostatic field at the wavenumber, which corresponds to the dominant fourier component of the mpgf, jumped to the value expected from the mpgf. it did not overshoot and it was non-oscillatory. both simulations here have evidenced that the magnetic field driven by the fi organized itself such, that we obtained one oscillation in the simulation box after the intermittent phase. this is remarkable, because the exponential growth rate of the fundamental wavenumber is below that of its first harmonic. long waves excert a lower mpgf for a given amplitude and the dominance of the fundamental wavenumber may thus result from the lower nonlinear damping of this mode compared to that of its harmonics. the mode with the fundamental wavenumber considered in ref. [22] has a higher growth rate than its harmonics and the absent mode competition may have facilitated the undamped oscillations around the equilibrium. however, the amplitude of the electrostatic field in the two simulations discussed here eventually converged to that expected from the mpgf and eb is thus a robust estimate for the electrostatic field driven by the mpgf for the considered case. this robustness explains, why a connection between the electrostatic field and the mpgf has been observed in a 2d pic simulation [26], where no equilibrium can be reached due to the filament mergers. this estimate does, however, not apply if positrons are present. their current reduces that of the electrons. if equal amounts of electrons and positrons are present, electric fields of the filamentation instability 15 the electrostatic field driven by the mpgf is suppressed alltogether [27]. mobile protons will react in particular to the stationary electric field [14] and they will modify through their charge modulation the balance between the electrostatic field and the mpgf. highly relativistic beam velocities will probably also modify the balance between the mpgf and the electron currents it drives. we leave relativistic beams to future work. acknowledgements the authors acknowledge the support by vetenskapsr ̊ adet and by the projects ftn 2006-05389 of the spanish ministerio de educacion y ciencia and pai08-0182-3162 of the consejeria de educacion y ciencia de la junta de comunidades de castilla-la mancha. the hpc2n has provided the computer time. references [1] yang t y b, gallant y, arons j and langdon a b 1993 phys. fluids b 5 3369 [2] petri j and kirk j g 2007 plasma phys. controll. fusion 49 297 [3] karlicky m, nickeler d h and barta m 2008 astron. astrophys. 486 325 [4] medvedev mv and loeb a 1999 astrophys. j. 526 697 [5] lazar m, schlickeiser r and shukla p k 2006 phys. plasmas 13 102107 [6] tabak m et al. 1994 phys. plasmas 1 1626 [7] ruhl h, sentoku y, mima k, tanaka k a and kodama r 1999 phys. rev. lett. 82 743 [8] key m h et al. 2008 phys. plasmas 15 022701 [9] davidson r c, wagner c e, hammer d a and haber i 1972 phys. fluids 15 317 [10] lee r and lampe m 1973 phys. rev. lett. 31 1390 [11] molvig k 1975 phys. rev. lett. 35 1504 [12] honda m, meyer-ter-vehn j and pukhov a 2000 phys. rev. lett. 85 2128 [13] honda m, meyer-ter-vehn j and pukhov a 2000 phys. plasmas 7 1302 [14] califano f, cecchi t and chiuderi c 2002 phys. plasmas 9 451 [15] sakai j i, schlickeiser r and shukla p k 2004 phys. lett. a 330 384 [16] medvedev m v, fiore m, fonseca r a, silva l o and mori w b 2005 astrophys. j. 618 l75 [17] stockem a, dieckmann m e and schlickeiser r 2008 astrophys. j. 50 025002 [18] bret a, gremillet l and bellido j c 2007 phys. plasmas 14 032103 [19] bret a, gremillet l, benisti d and lefebvre e 2008 phys. rev. lett. 100 205008 [20] tzoufras m, ren c, tsung f s, tonge j w, mori w b, fiore m, fonseca r a and silva l o 2006 phys. rev. lett. 96 150002 [21] rowlands g, dieckmann m e and shukla p k 2007 new j. phys. 9 247 [22] dieckmann m e, kourakis i, borghesi m and rowlands g 2009 phys. plasmas 16 074502 [23] dawson j m 1983 rev. mod. phys. 55 403 [24] eastwood j w 1991 comput. phys. commun. 64 252 [25] sharma r p, tripathi y k and kumar a 1987 phys. rev. a 35 3567 [26] dieckmann m e 2009 plasma phys. controll. fusion in press (2009) [27] dieckmann m e, shukla p k and stenflo l 2009 plasma phys. controll. fusion 51 065015

0911.1728

mass varying neutrinos, quintessence, and the accelerating expansion of the universe

we analyze the mass varying neutrino (mavan) scenario. we consider a minimal model of massless dirac fermions coupled to a scalar field, mainly in the framework of finite temperature quantum field theory. we demonstrate that the mass equation we find has non-trivial solutions only for special classes of potentials, and only within certain temperature intervals. we give most of our results for the ratra-peebles dark energy (de) potential. the thermal (temporal) evolution of the model is analyzed. following the time arrow, the stable, metastable and unstable phases are predicted. the model predicts that the present universe is below its critical temperature and accelerates. at the critical point the universe undergoes a first-order phase transition from the (meta)stable oscillatory regime to the unstable rolling regime of the de field. this conclusion agrees with the original idea of quintessence as a force making the universe roll towards its true vacuum with zero \lambda-term. the present mavan scenario is free from the coincidence problem, since both the de density and the neutrino mass are determined by the scale m of the potential. choosing m ~ 10^{-3} ev to match the present de density, we can obtain the present neutrino mass in the range m ~ 10^{-2}-1 ev and consistent estimates for other parameters of the universe.

introduction neutrino mass related questions are of great interest for particle physics as well as for cosmology (for reviews see ref. [1] and references therein). current upper limits on the sum of neutrino masses from cosmological observations are of the order of 1 ev [2–4], while neutrino oscillations give a lower bound of roughly 0.01 ev [5, 6], making neutrino mass an established element of particle physics. furthermore, understanding the origin of neutrino mass opens a window into understanding physical processes beyond the standard model of particle physics [7–10]. it is now well established that about seventy four percent of the universe is comprised of dark energy (de) (for reviews see ref. [11] and citation therein). the present stage of evolution of the universe is governed by this dominant de contribution, and the universe experiences an accelerating expansion [12, 13]. the nature of de is still unknown, and it is one of the major questions of modern cosmology. there are, broadly speaking, three major 2 possibilities proposed to explain the de [11]. most straightforwardly, and in good agreement with the current observational data, it can be present just as the cosmological constant [11]. secondly, the de can be accommodated in some framework of the modified non-einsteinian gravity theories (see, e.g., refs. [14, 15]). and lastly, following the original proposals [16, 17] on the de originating from a scalar field action similar to the inflaton field, there has been a lot of activity in constructing and analyzing various trial scalar field lagrangians to model the de [13]. note, that it is even unclear what kind of scalar field potential governs the inflationary expansion of the universe [18], and as the result, the effective quantum field that adequately describes inflation is still under debate [19]. a similar observation can be drawn from analyzing many potentials proposed for the de action [13]. on the other hand, several cosmological and astrophysical observations imply that about twenty two percent of the universe consists of dark matter (dm) [11], if we admit the general relativity theory of gravity. most probably dm is formed through massive weakly interacting particles (wimps), and the nature of these particles is also still unknown. there are several recent observations performed by pamela [20] and glast missions which indicate dm particle annihilations [21]. recently it was proposed that both these observations could be used to test baryogenesis [22] which is one of the important problems of the standard particle physics model. another puzzling question in modern cosmology is the coincidence problem - the density of de is comparable to the present energy density of dm. in turn, the latter is comparable (within the order of magnitude), to the energy density of cosmological neutrinos [1, 2]). is there a mechanism explaining this coincidence? a very convincing answer to this question is given by the mechanism of dm mass generation via various types of dm-de couplings, rang- ing from yukawa to more exotic ones. [23–28] the mass of the dm particle in this approach is naturally time-dependent, and they were coined varying mass particles (vamps). vari- ous de–dm interaction models have been constrained by observations of supernovae type ia [29], the age of the universe [30–32], cosmic microwave background (cmb) anisotropies [33, 34], and large scale structure (lss) formation [35]. fardon, nelson and weiner elaborated on the vamp mechanism in the context of neu- trinos. [36]1. in their model the relic neutrinos, i.e., fermionic field(s), interact with a scalar field via the yukawa coupling. if the decoupled neutrino field is initially massless, then the coupling generates a (varying) mass of neutrinos in this de-neutrinos model. this mass varying neutrino (mavan) scenario is quite compelling, since it connects the origin of neu- trino mass to the de, and solves the additional coincidence problem of why the neutrino mass and de are of comparable scales [38]. (for more on the coincidence, see, e.g. [39]). to consider neutrinos as particles which get their mass through the coupling is attractive for particle physics, as well as for its cosmological consequences. however there are significant issues that have to be resolved for the sake of viability of the mavan scenario. most notably, it has been shown [40] that the model of ref. [36] suffers from a strong instability due to the negative sound speed squared of the de-neutrino fluid (see also [41]). any dm-de coupling induces observable changes in large scale structure formation [42]. the main reason for this is due to the presence of additional dm contributions (perturba- tions) in the equation of motion which determines the dynamics of the scalar field. the changes in the dynamics are drastic when massive neutrinos are coupled to de [40]. in this case the squared sound speed of the de-neutrino fluid defined as c2 s = δp/δρ, (where 1 the de-neutrino coupling and the baryogenesis constraints have been also studied also in ref. [37] 3 δ represents the variation, and p and ρ are pressure and energy density of the de-neutrino fluid) is negative. the negative squared sound speed results in an exponential growth of scalar perturbations. [43–46] after the critique in ref. [40], the issue of stability of the de-neutrinos fluid has been addressed by many authors [41, 46–53]. various physical assumptions were made in those references in order to avoid the exponential clustering of neutrinos. in particular, to achieve stability, proposals were put forward to make the de-dm model more complicated, e.g., by extending it to a multi-component scalar field, or by promoting its supersymmetry. [49, 51] we however are not inclined to pursue this line of thought and will explore the simplest possible "minimal" model. as we will demonstrate, the occurrence of the instability in the coupled de-neutrinos model is meaningful, and we will explore the physical implications of this phenomenon. note that wetterich and co-workers [46] have already analyzed various implications of the instability in the mavan model on the dynamics of neutrino clustering. in this paper we re-address the analysis of the de-neutrinos coupled model. what is really new in our results, to the best of our knowledge, apart from a consistent equation for the equilibrium condition, is the analysis of the thermal (i.e. temporal) evolution of the mavan model and prediction of its stable, metastable and unstable phases. the analysis of the dynamics in the unstable phase results in, for the first time in the framework of the mavan scenario, a picture of the present-time universe totally consistent with observations. our findings are in line with the original proposal [16, 17] of the de potential (quintessence) to model the universe slowly rolling towards its true vacuum (λ = 0). as it turns out, the present universe, seen as a system of the coupled de (quintessence) field and fermions (neutrinos) is below its critical temperature. it is similar to a supercooled liquid which has not crystallized yet: its high temperature (meta)stable phase became unstable, but the new low-temperature stable phase (λ = 0) is still to be reached. the afshordi-zaldarriaga-kohri instability corresponding to c2 s < 0 is just telling us this. the rest of the paper is organized as follows: in section ii we give the outlook of the model and formalism applied and derive the basic equations for the coupled model. in section iii we present the qualitative analysis of the equation which yields the fermionic (neutrino) mass. section iv contains analysis of the coupled model with the ratra-peebles de potential at equilibrium. the dynamics of the model applied to the whole universe is studied in section v. the results are summarized in the concluding section vi. ii. model and formalism. basic equations a. outlook in this paper we focus on the case when the scalar field potential u(φ) does not have a non-trivial minimum, and the generation of the fermion mass is due to the breaking of chiral symmetry in the dirac sector of the lagrangian. a non-trivial solution of the fermionic mass equation is a result of the interplay between the scalar and fermionic contributions. we consider the most natural and intuitively plausible yukawa coupling between the dirac and the scalar fields. the key assumption is that the fermionic mass generation can be obtained from mini- mization of the thermodynamic potential. that is, the coupled system of the scalar bosonic and fermionic fields is at equilibrium, at least at some temperatures. this will be analyzed below more specifically. we assume the cosmological evolution, governed by the scale factor 4 a(t) to be slow enough that the coupled system is at equilibrium at a given temperature t(a). then the methods of thermal quantum field theory [54, 55] can be applied. this problem is rather well studied with quantum field theory and statistical physics in different contexts [54–56]. the major conceptual difficulty in applying quantum field- theoretical methods for the dark-energy scalar field is the lack of "well-behaved" potentials interesting for cosmological applications. for instance, a class of the very popular inverse power law slow-rolling quintessence potentials [13] are singular at the origin. consequently, the field theory should be understood as a sort of effective theory, and we plan to address this issue more deeply in our future work. as far as the fermionic sector of the theory is concerned, one needs to distinguish two different cases pertinent for neutrino applications: (i) an equal number of fermions and antifermions, i.e., zero chemical potential μ = 0; (ii) a surplus of particles over antiparticles, and small non-zero chemical potential. for the bounds on the neutrino chemical potential, see refs. [1, 57]. if experiments confirm neutrinoless double beta decay, i.e., that neutrinos are majorana fermions, then the lepton number is not conserved [8], and one cannot introduce a (non-zero) chemical potential. then case (i) above is applicable, proviso that the majorana fields are utilized instead of the dirac ones. for the case (i) with dirac fermions the ground state corresponds to a complete annihilation of fermion-antifermion pairs, i.e. the fermions completely vanish in the zero-temperature limit. assumption of the fermion-antifermion asymmetry and (conserving) particle surplus, i.e., of a non-zero chemical potential, results in the fermionic contributions which survive the zero- temperature limit. however the smallness of the zero-temperature contribution renders this issue rather academic. indeed, for the neutrinos we are interested in this study, by assuming the maximal particle surplus n◦∼115 cm−3, one gets the fermi momentum kf ∼3*10−4 ev. for m ∼10−2 ev, one obtains μ(t = 0) = εf = p k2 f + m2 = m+ o(10−4 ev). this results in a non-trivial vacuum with the particle surplus frozen within an extremely narrow fermi shell m ≤ε ≤εf. thus, trying to grasp the essential physics in this study from possibly the simplest "minimal model", we assume the fermions to be described by a dirac spinor field with zero chemical potential. in this work we will use the standard methods of general relativity and finite-temperature quantum field theory extended for fields living in a spatially flat universe with the friedmann-lemaˆ ıtre-robertson–walker (flrw) metric where the line element is ds2 = dt2 −a2(t)dx2. here t is the physical time and a(t) is the scale factor, which can be obtained from the friedmann equations [9, 10] h2(t) =  ̇ a a 2 = 8πg 3 ρtot , (1) ̇ h(t) + h2(t) = ̈ a a = −4πg 3 (ρtot + 3ptot) . (2) eqs. (1)-(2) also lead to the continuity equation ̇ ρtot + 3 ̇ a a (ρtot + ptot) = 0 . (3) here the dot represents the physical time derivative and ρtot and ptot are the total energy density and pressure of the universe. in accordance with the (standard) λcdm model, the universe is assumed to consist of (1) de, (2) cold dm (cdm) made of weakly interacting 5 massive particles, presumably mdm > 1 ∼10 gev, (3) photons, and (4) baryons. the dm and baryon density parameters today are ωdm = ρdm(tnow)/ρcr ≈0.22 and ωb = ρb(tnow)/ρcr ≈0.04. here ρcr = 3h2 0/(8πg) = 8.1h2 × 10−47 gev4 is the critical density today, tnow defines the current time, h0 = 2.1h×10−42 gev is the present hubble parameter, g is the newton constant, and h ≈0.72 is the hubble parameter in units of 100 km/sec/mpc. the photon contribution to the energy density today can be neglected. the flatness of the universe leads to the relative energy density of the de-neutrino coupled fluid ωφν ≈0.74. to ensure the accelerated expansion of the universe today, the r.h.s. of eq. (2) must be positive at t = tnow. in this paper we will not assume the existence of the cosmological constant λ, as the λcdm model suggests. instead we accept the hypothesis of the dynamical dark energy described by a scalar field. this is a bold assumption and a highly debatable issue. we vindicate our approach a posteriori by the consistent picture we arrive at the end. for a review and/or alternative approaches, see, e.g., refs. [13, 58, 59]. the massless neutrinos are described by the conventional dirac lagrangian. the resulting model is given by the coupled dirac and scalar fields. the grand thermodynamic potential of the coupled model can be derived from the euclidian functional integral representation of the grand partition function. the dynamics of the coupled model is governed by the friedmann equations. throughout the paper we use natural units where ħ= c = kb = 1. b. bosonic scalar field the bosonic scalar field hamiltonian in the flrw metric reads as [9, 60] hb = z a3d3x h1 2 ̇ φ2 + 1 2a2(∇φ)2 + u(φ) i , (4) where the comoving volume v = r d3x, while the physical volume vphys = a3(t)v . since this field does not carry a conserved charge (number), the chemical potential μ = 0. the grand partition function in the functional integral representation: zb ≡tr e−β ˆ h = z dφ e−se b (5) with the bosonic euclidian action se b = z β 0 dτ z a(t)3d3x h1 2(∂τφ)2 + 1 2a2(∇φ)2 + u(φ) i , (6) where φ = φ(x, τ). it is instructive to find the partition function of the free scalar field u(φ) = 1 2m2 b φ2 following the methods explained by kapusta and gale [54] for the case of the minkowski metric. rescaling of the field ̃ φ = a3/2φ (7) changes the partition function (5) by a thermodynamically irrelevant prefactor. the func- tional integration over ̃ φ of the gaussian action gives log zb = −v z d3k (2π)3 h β q m2 b + k2/a2 + log  1 −e−β√ m2 b +k2/a2i . (8) 6 then the density (with respect to the physical volume) of the thermodynamic potential is given by ωb ≡− 1 βa3v log zb = −pb = z d3k (2π)3  ε + 1 β log 1 −e−βε
 , (9) where ε = p m2 b + k2 and pb is the pressure due to the bosonic field. c. free dirac spinor field the dirac hamiltonian in the flrw metric is [60] hd = z a3d3x ̄ ψ −ı aγ * ∇+ m
ψ . (10) the grand partition function is given by the following grassmann functional integral: zd ≡tr e−β( ˆ h−μ ˆ q) = z d ̄ ψdψ e−se d (11) where the conserved charge (lepton number) operator ˆ q = r a3d3xψ†ψ and the euclidian action se d = z β 0 dτ z a(t)3d3x ̄ ψ(x, τ)  γo ∂ ∂τ −ı aγ * ∇+ m −μγo ψ(x, τ). (12) by rescaling the grassmann fields (7) and using the standard techniques [54], we get the thermodynamic potential density (pressure) as a function of the chemical potential and temperature: ωd ≡− 1 βa3v log zd = −pd = −2 z d3k (2π)3  ε + 1 β log 1 + e−βε−
+ 1 β log 1 + e−βε+
 , (13) where ε(k) = √ m2 + k2 , (14) and ε± = ε(k) ± μ. the first term on the r.h.s. of eq. (13) corresponds to the vacuum contribution to the thermodynamic potential (pressure): −ω0 = p0 = 2 z d3k (2π)3ε(k) (15) introducing the notation for the fermi distribution function nf(x) ≡ 1 eβx + 1 , (16) eq. (13) can be brought to the following form: −ωd = pd = p0 + 1 3π2 z ∞ 0 k4dk ε(k)  nf(ε−) + nf(ε+)  (17) 7 d. coupled model: scalar field and dirac massless fermions let us consider a scalar bosonic field interacting via a yukawa coupling with massless dirac fermions. the euclidian action of the model in the flrw metric reads: s = se b + se d   m=0 + g z β 0 dτ z a3d3x φ ̄ ψψ (18) the path integral for the partition function of the coupled model is: z = z dφd ̄ ψdψ e−s (19) the grassmann fields can be formally integrated out resulting in z = z dφ e−s(φ) = z dφ exp  −se b + log det ˆ d(φ)  , (20) where the dirac operator ˆ d(φ) = γo ∂ ∂τ −ı aγ * ∇+ gφ(x, τ) −μγo (21) the thermodynamic potential ωof the model (18) at tree level can be found by evaluating the path integral (20) in the saddle-point approximation. assuming the existence of a constant (x, τ)-independent field φc which minimizes the action s(φ), the term log det ˆ d can be evaluated exactly, and fermionic contribution to the thermodynamic potential is given by eqs. (13) or (17) with the fermionic mass m = gφc . (22) the bosonic contribution to the partition function in this approximation is simply z ∝ exp[−βa3v u(φc)] . the thermodynamic potential density is given then by ω(φc) = u(φc) + ωd(φc) . (23) self-consistency of the employed saddle-point approximation naturally coincides with the condition of minimum of the thermodynamic potential at equilibrium (at fixed temperature and chemical potential): ∂ω(φ) ∂φ    φ=φc = 0 , (24) and ∂2ω(φ) ∂φ2    φ=φc > 0 , (25) note that a non-trivial solution φc of eq. (24) (if it exists) is called the classical field: it is the average of the bosonic field, i.e., φc = ⟨φ⟩. eqs. (22,23,24) can be brought to the equivalent form: u′(φc) + gρs = 0 , (26) 8 where the scalar fermionic density (a.k.a. the chiral density) ρs is given by the following expression: ρs ≡⟨ˆ n⟩ v = ∂ωd ∂m = ρ0 + m π2 z ∞ 0 k2dk ε(k)  nf(ε−) + nf(ε+)  , (27) and ˆ n = r d3x ̄ ψψ. here ρ0 stands for the vacuum contribution to the chiral condensate: ρ0 ≡∂ω0 ∂m = −m π2 z ∞ 0 k2dk ε(k) . (28) note that even if the time, i.e., a(t), does not enter explicitly in the equations for the thermodynamic quantities of the coupled, fermionic or bosonic models (9,13,23,26,27), and they look like their counterparts in a flat static universe, such parameters as, e.g., the temperature and chemical potential in those equations are time-dependent, i.e., t = t(a) and μ = μ(a). the particular form of the dependencies t(a) and μ(a) must be determined from the friedmann continuity equation (3) which relates the energy density ρ(t) and pressure p(t) to the evolution of a(t)[9, 10]. in addition, the fermionic mass m ∝φc in the coupled model is also time varying, since the time enters into φc (26) via t, μ, and all three functions m(a), t(a) and μ(a) are governed by the friedmann equations (1,2,3). the present theory works consistently for the physical quantities (bosonic or fermionic) measured with respect to their vacuum contributions. so, in the rest of the paper we will employ the thermodynamic quantities with subtracted vacuum contributions, keeping however, the same notations, e.g.: ωd 7→ωd −ω0 , pd 7→pd −p0 , ρs 7→ρs −ρ0 . (29) then, according to volovik [61], the pressure and energy of the pure and equilibrium vacuum is exactly zero. (the renormalization of the vacuum terms is, of course a very subtle issue. there are alternative approaches to this problem known from the literature. see, e.g., [62, 63].) iii. analysis of the mass (gap) equation: general properties in cases interesting for cosmological applications, the scalar field potential u(φ) does not have a non-trivial minimum, and the generation of the fermion mass (i.e. a solution of (24) 0 < φc < ∞) is due to the interplay between the scalar and fermionic contributions to the total thermodynamic potential (23). from now on we adapt our equations for the case of equal number of fermions and antifermions and μ = 0, as discussed in sec. ii a. keeping in mind the neutrinos, we assume an extra flavor index of fermions with the number of flavors s. (for neutrinos s = 3.) we also assume the flavor degeneracy of the fermionic sector. before proceeding further, we need to make some important observations regarding the behavior of the coupled model in two limiting cases. assuming that a non-trivial solution of (24) with finite m exists, the fermionic contribution to the thermodynamic potential (pressure) (17) can be written as: −ωd = pd = 2s 3π2β4ip(βm) , μ = 0 , (30) 9 where the integral defined as ip(κ) ≡ z ∞ κ (z2 −κ2)3/2 ez + 1 dz (31) can be evaluated analytically in two cases: ip(κ) = ( 7π4 120 −π2 8 κ2 + o(κ4) , κ < 1 3κ2k2(κ) + o(e−2κ) , κ ≳1 (32) where kν(x) is the modified bessel function of the second kind. in the (classical) low-temperature regime βm ≡m t ≫1 (33) the above equation results in −ωd = pd = 2sm2 π2β2 k2(βm) + o(e−2βm) . (34) to leading order −ωd = pd ≈ √ 2s π3/2 t(tm)3/2e−m/t . (35) the chiral condensate density (27) ρs = 2sm π2β2 z ∞ βm (z2 −(βm)2) 1 2 ez + 1 dz , μ = 0 (36) can be also evaluated in the low-temperature limit as ρs = 2sm2 π2β k1(βm) + o(e−2βm) , (37) which gives to leading order ρs ≈ √ 2s π3/2 (tm)3/2e−m/t . (38) in this limit the fermions enter the regime of a classical ideal gas. indeed, the fermionic particle (antiparticle) density n+ = n−= s π2β3 z ∞ βm z(z2 −(βm)2) 1 2 ez + 1 dz (39) in the low-temperature limit yields n± = sm2 π2β k2(βm) + o(e−2βm) , (40) and to leading order: n± ≈ s √ 2π3/2(tm)3/2e−m/t . (41) 10 we see from eqs. (34,40) that up to terms o(e−2βm), the fermions satisfy the ideal gas equation of state pd ≈(n+ + n−)t , (42) and the chiral density is equal to the total particle density n. ρs ≈n ≡n+ + n−. (43) in the (ultra-relativistic) high-temperature regime m t ≪1 (44) one obtains −ωd = pd ≈7π2s 180 t 4 −s 12(mt)2 . (45) to leading order the chiral condensate is ρs ≈s 6mt 2 , (46) while the particle density is n± ≈3sζ(3) 2π2 t 3 . (47) now we can make some general observations of the fermionic mass generation in the coupled model: (i) it is obvious from the sign of ρs (cf. 27,36) that non-trivial solutions of (26) are impossible for a monotonically increasing potential u(φ). that rules out some popular potentials, e.g., u ∝log(1 + φ/m) [13, 36] for this yukawa-coupling driven scenario of the mass generation. (ii) the monotonously decreasing slow-rolling de potentials ([16, 17] and for reviews, see [11, 13]), e.g., u ∝φ−α or u ∝exp[−aφγ], do have a window of parameters wherein non-trivial solutions of (26) exist. as we can see from (38), for those decreasing potentials the mass equation (26) always has a trivial solution m = gφc = ∞for the minimum of the thermodynamic potential (23). 2 this solution corresponds to a "doomsday" vacuum state [61], when the universe reached its true ground state with zero dark energy density and completely frozen out fermions. a non-trivial solution of (26), corresponding to another minimum of the potential (23), is totally due to the fermionic contribution. since the latter freezes out in the limit t →0, it is clear qualitatively that such a solution 0 < m < ∞can exist only above a certain temperature. for a more quantitative account of these phenomena we need to assume some specific form of the de potential. this will be done in the following section. (iii) to explain the differences between the present study and earlier related work on mass varying fermions (see [23, 24, 36, 40] and more references there), some clarifications are warranted. it is usually assumed in the literature that the low-temperature regime formulas are applicable, and according to (43) ρs = n. the approximation for (26) then can be written as ∂u/∂m + n = 0. the latter is interpreted as a result of minimization of some 2 recall that the grand thermodynamical potential is equal to the free energy for the case μ = 0. 11 effective potential ueff= u + nm with fixed n, which always has a non-trivial minimum 0 < m < ∞for the class of decreasing potentials u, see, e.g., [23, 24]. it turns out that such an approximation changes the picture qualitatively. in what follows, we explore in detail the predictions of the consistent mass equation (26) on the mass varying scenario for the coupled model with a specific de potential ansatz. iv. coupled model with the ratra-peebles quintessence potential a. mass equation and critical temperature now we analyze in detail our coupled model for a particular choice of u(φ), the so-called ratra-peebles quintessence potential [16] : u(φ) = mα+4 φα , (48) where α > 0. it is convenient to introduce the dimensionless parameters ∆≡m t , κ ≡gφ t , ωr ≡ω m4 . (49) then the mass equation (26) can be written as: απ2 2s gα∆α+4 = iα(κ) , (50) where we introduced iα(κ) ≡κα+2 z ∞ κ √ z2 −κ2 ez + 1 dz . (51) according to the relation eq. (22) between the fermionic mass m and the classical field, we get m = tκc, where κc is the solution of eq. (50) corresponding to the minimum of the thermodynamic potential which reads now as (cf. eq. (31)): ωr = gα∆ κ α − 2 3π2 1 ∆4ip(κ) . (52) the dimensionless yukawa coupling constant g ∼1. to reduce the number of model pa- rameters we can set g = 1. this is equivalent to the simultaneous rescaling gφ 7→ ̃ φ and mg α α+4 7→ ̃ m. 3 for simplicity, we also restrict the number of flavors s = 1. we define the mass of the scalar field as: m2 φ = ∂2u(φ) ∂φ2    φ=φc . (53) 3 one can check this scaling also holds for the dynamics of the model, considered in section v. in particular, the neutrino masses do not depend on the value of g. to avoid cluttering of notations we will drop tildes in the rescaled parameters. 12 in terms of the dimensionless parameters it reads mφ m = p α(α + 1)  ∆ κc  α+2 2 (54) it is important to realize that the integral iα(κ) on the r.h.s. of the mass equation is bounded. the quantitative parameters of the function iα(κ) depend on α, but its shape is always similar to the curve shown in fig. 1 for α = 1. so, there exists a maximal ∆crit (critical temperature tcrit) such that for ∆> ∆crit (t < tcrit) only a trivial solution m = ∞ exists, and the stable vacuum has zero energy and pressure. 0 0.5 1 1.5 2 2.5 3 3.5 0 2 4 6 8 10 12 (d) (c) (b) (a) iα(κ) κcrit κ0 -1 -0.5 0 0.5 0 2 4 6 8 10 12 14 16 18 20 κ (d) (c) (b) (a) ωr κ fig. 1: (color online) left: graphical solutions of the mass equation (50) for different values of ∆≡m/t (α = 1). right: dimensionless density of the thermodynamic potential (52). the thermodynamically stable solutions of eq. (50) indicated by the large dots correspond to the minima of the potential. the arrows indicate the unstable solutions of the mass equation, corresponding to the maxima of the potential. the mass equation eq. (50) is solved numerically for various values of its parameters, and the characteristic results are shown in fig. 1. the numerical results can be complemented by an approximate analytical treatment of the problem. the latter turns out to be quite accurate and greatly helps in gaining intuitive understanding of the results. it is easy to evaluate iα(κ) to leading order: iα(κ) ≈ ( π2 12κα+2 , κ < 1 κα+3k1(κ) , κ ≳1 (55) 13 for the critical point where i ′ α(κcrit) = 0, we obtain: κcrit ≈ν , ν ≡α + 5 2 ; (56) iα(κcrit) ≈ rπ 2 ννe−ν. (57) the most important conclusion we draw from fig. 1 is that there are three phases in the model's phase diagram. we analyze each of them in the following subsections. 1. stable (massive) phase: ∆< ∆◦(t◦< t < ∞) in this range of parameters the equation (50) has two nontrivial solutions. the root κc < κ◦indicated with a large dot in fig. 1 (case a) gives the fermionic mass and corresponds to a global minimum of the potential. so it is a thermodynamically stable state. in this phase ω(κc) < 0, so the pressure is positive p > 0. another non-trivial root of (50) corresponds to a thermodynamically unstable state (maximum of ωindicated with an arrow in fig. 1). there is a trivial third root of the mass equation κ = ∞. at these temperatures it corresponds to the metastable vacuum state ω= 0. in the high-temperature region of this phase where ∆≪1 the fermionic mass is small (see fig. 2): m m ≈ √ 6αm t  2 α+2 ∝t − 2 α+2 (58) the fermionic contribution to the thermodynamic potential is dominant, and it behaves to leading order as the potential of the ultra-relativistic fermion gas (cf. eq. (45)): ω= −p = −7π2 180t 4 + o(t 2α α+2) . (59) one can check that the subleading term in the above expression combines the de potential contribution and the first fermionic mass correction, which are both of the same order. it is important to stress that in this coupled model with the slow-rolling potential eq. (48), the mass generation does not follow a conventional landau thermal phase transition scenario. there is no critical temperature below which the chiral symmetry is spontaneously broken and the mass is generated. instead the mass grows smoothly as κc ∝∆ α+4 α+2, albeit starting from the "point" t = ∞. from physical grounds we expect the applicability of the model to have the upper temperature bound: t ≲trd , (60) where trd is roughly the temperature of the boundary between inflation and the radiation- dominated era. the high-temperature result (59) shows that the stable massive phase of the present model can indeed be extended up to those temperatures. the scalar field and fermionic masses demonstrate opposite temperature dependencies. the scalar field is "heavy" at high temperatures: mφ ≈ r α + 1 6 t , ∆≪1 , (61) however its mass decreases together with the temperature. in contrary, the fermionic mass m monotonously increases with decreasing temperature. the exact numerical results for the two masses are shown in fig. 2 14 0 1 2 3 4 5 6 δ 0 0.2 0.4 0.6 0.8 1 m/m mφ/m fig. 2: (color online) masses of the fermionic and scalar fields (m and mφ resp.) as functions of ∆≡m/t, α = 1. at ∆> ∆crit (t < tcrit) the stable phase corresponds to m = ∞and mφ = 0 2. metastable (massive) phase: ∆◦< ∆< ∆crit (tcrit < t < t◦) upon increasing ∆we reach a certain value ∆◦corresponding to a critical temperature t◦ when the thermodynamic potential has two degenerate minima ω(κ◦) = p(κ◦) = ω(∞) = 0. this is shown in fig. 1 (case b). after this point, when the temperature decreases further in the range ∆◦< ∆< ∆crit (here ∆crit stands for the maximal value of ∆when a non-trivial solution of the gap equation (50) exists, see fig. 1), the two minima of the thermodynamic potential exchange their roles. the root κc now becomes a metastable state with ω(κc) > 0, i.e., with the negative pressure p(κc) < 0, while the stable state of the system corresponds to the true stable vacuum of the universe [61] ω(∞) = p(∞) = 0. see fig. 1 (case c). the system's state in the local minimum ω(κc) is analogous to a metastable supercooled liquid. we disregard the exponentially small probability of tunneling of the fermions from the metastable state ω(κc) into the vacuum state ω(∞) = 0[18]. accordingly, the fermionic mass in this phase is determined by the root κc of (50). in the metastable phase κc ≳1, so by using eqs. (52,32,50) we obtain the potential: ωr ≈ ∆ κc αn 1 −α κc −3α 2κ2 c o . (62) from the above result we can find the metastability point ω(κ◦) = 0 as κ◦≈α 2  1 + r 1 + 6 α  (63) expanding iα(κ) near its maximum and using eqs. (55,56,57) along with the gap equation eq. (50), we obtain the following equation: (κc −κcrit)2 2ν ≈1 −  ∆ ∆crit α+4 . (64) on finds from the above equation, e.g., how the mass approaches its critical value: mcrit −m ∝  t tcrit −1 1/2 , (65) 15 or the ratios of temperatures and masses at the metastable and critical points. these latter parameters are given in table i. table i: masses, critical temperatures and potentials for various values of α. all the parameters used in this table are defined in the text. α tcrit t◦ ∆crit m◦ mcrit mcrit m mcrit φ m tcrit m ωcrit m4 ρcrit m4 w(tcrit) 1 0.90 0.91 0.558 3.86 0.187 1.10 0.15 0.84 -0.18 2 0.95 1.04 0.70 4.35 0.130 0.97 0.02 0.25 -0.09 4 0.98 1.44 0.81 4.52 0.048 0.70 6 * 10−4 0.02 -0.03 10 0.99 3.00 0.91 4.16 2 * 10−3 0.33 7 * 10−8 9 * 10−6 -0.008 3. critical point: ∆= ∆crit (t = tcrit) and phase transition the critical point of the model corresponds to the case when the two roots of the mass equation eq. (50) merge, and the minimum of the potential disappears. one can check that instead of the minimum this is an inflection point of the the potential, i.e., ω′′ r(κcrit) = 0. this situation is shown in fig. 1 (case d). at this point the system is in the unstable state with the fermionic mass mcrit tcrit = κcrit ≈ν . (66) in particular, this implies that the fermions are non-relativistic at the critical temperature. from eqs. (57,50) we find the critical parameter (see table i for its numerical values) ∆crit ≈  √ 2 απ3/2ννe−ν 1 α+4 , (67) which allows us to evaluate the critical temperature tcrit = m ∆crit . (68) we can also find the potential at tcrit: ωcrit ≈5 2ν ∆crit ν α m4 (69) thus, from the viewpoint of equilibrium thermodynamics at t = tcrit the model must undergo a first-order (discontinuous) phase transition and reach its third thermodynamically stable (at t < tcrit) phase corresponding to the vacuum ω(κ = ∞) = p(κ = ∞) = 0. during this transition the fermionic mass given at the critical point by eq. (66) and the scalar field mass mcrit φ ≈ p α(α + 1) ∆crit ν  α+2 2 m (70) both jump to their values in the vacuum state m = ∞and mφ = 0. see fig. 2. 16 however, the above arguments are based on the minimization of the thermodynamic potential (i.e. maximization of entropy) at equilibrium. to address the question of how such a system behaves as the universe evolves towards the new equilibrium vacuum state, we need to analyze the dynamics of this phase transition. more qualitatively, we need to study how the particle at the point κcrit at the critical temperature (see fig. 1) rolls down towards its equilibrium at infinity. this issue will be addressed in section v. b. equation of state we define the equation of state in the standard form: p = wρ , (71) where the total pressure in this model is obtained from eq. (52), while the total energy density (ρ) and its dimensionless counterpart (ρr) are determined by the following equation: ρr ≡ ρ m4 = ∆ κ α + 2 π2 1 ∆4iε(κ) . (72) here we define the integral iε(κ) ≡ z ∞ κ z2√ z2 −κ2 ez + 1 dz , (73) which can be evaluated in two limits of our interest: iε(κ) = ( 7π4 120 −π2 24κ2 + o(κ4) , κ < 1 3κ2k2(κ) + κ3k1(κ) + o(e−2κ) , κ ≳1 (74) in the high-temperature region of the stable massive phase where ∆≪1, the fermionic contribution is dominant, and the energy density to leading order is that of the ultra- relativistic fermion gas (cf. eq. (59)) ρ = 7π2 60 t 4 + o(t 2α α+2) . (75) thus, in this regime the model follows approximately the equation of state of a relativistic gas with w ≈1 3. in the region κc ≳1 which includes the metastable phase and the critical point, we obtain by using eqs. (72,74,50,62): ρ ≈ ∆ κc αn 1 + α + 3α κc + 9α 2κ2 c o , (76) and w ≈− 1 −α κc −3α 2κ2 c 1 + α + 3α κc + 9α 2κ2 c (77) the last equation follows very closely the results of the exact numerical calculations shown in fig. 3. at the critical point we evaluate 17 -1 0 0.2 0.4 δ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 α=1 α=2 α=4 δcr α=1 δcr α=2 δcr α=4 w fig. 3: (color online) w ≡p/ρ for several values of α. at ∆> ∆crit(α), i.e., t < tcrit(α) the equilibrium value w = −1 exactly. ρcrit ≈ ∆crit ν αn 1 + α + 3α ν o m4 , (78) and making a rough estimate, we get a lower bound: w ≈−5 2 1 ν(1 + α + 3α ν ) ≥−1 4, ∀α ≥1 . (79) thus for any power law α ≥1, the parameter w of this model at equilibrium cannot cross the bound w < −1 3, necessary for accelerating expansion of the universe ̈ a > 0. 4 at t < tcrit we obtain the equilibrium value of w in the stable vacuum state from eqs. (52,72): w = lim κ→∞ p(κ) ρ(κ) = −1 . (80) so the true vacuum in this model corresponds to the universe with a cosmological constant in the limit λ →0. c. speed of sound we define the sound velocity as c2 s = dp dt dρ dt = dp d∆ dρ d∆ , (81) 4 the relation (79) w(tcrit) ≳−1 4 holds for the model which contains only the de-neutrino coupled fluid. in a more realistic model for the universe, baryons and dm also contribute to the total energy density, and as a consequence w(tcrit) increases, see sec. v. 18 where to obtain the second expression we used the fact that the time enters our formulas only through the temperature t(a(t)), so d dt = d∆ dt d d∆. (82) let us first consider the temperatures t ≥tcrit, i.e., ∆≤∆crit. then dρ d∆= ∂ρ ∂∆+ ∂ρ ∂κ * dκ d∆    κ=κc , (83) where κ is related to ∆through the gap equation (50): dκ d∆    κ=κc ≡ ̇ κc = α + 4 ∆ iα(κc) i ′ α(κc) = α + 4 ∆ d log iα(κc) dκ −1 . (84) note that for the pressure the following relation dp d∆= ∂p ∂∆ (85) holds, since ∂p ∂κ    κ=κc = 0 (86) is just another form of the gap equation (24). thus c2 s = ∂p ∂∆ ∂ρ ∂∆+ ∂ρ ∂κ ̇ κc      κ=κc . (87) in the high-temperature regime ∆≪1 (κc ≪1), it is even easier to use the explicit asymptotic expansions for p(∆) and ρ(∆) in the definition (81) instead of the above formula (87). a straightforward calculation gives the result c2 s ≈1 3 −b∆ 2(α+4) α+2 , b > 0 , (88) consistent with the earlier observation that for ∆≪1 the model behaves as an ultra- relativistic fermi gas. in the case κc ≳1 we find ̇ κc ≈α + 4 ∆ κc ν −κc , (89) and c2 s ≈ ν −κc α(α + 4)(1 + 4 αν ) . (90) everywhere at t > tcrit, including the stable and metastable massive phases c2 s > 0, so the model is stable with respect to the density fluctuations. the sound velocity vanishes in the limit t →t + crit as cs ∝√ν −κc →0 . (91) qualitatively, the vanishing speed of sound is due to divergent ̇ κc (84,89) at the critical point. 19 -1 0 0.2 0.4 δ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 α=1 α=2 α=4 δcr α=1 δcr α=2 δcr α=4 cs 2 fig. 4: (color online) the square of the sound velocity for several values of α. at ∆> ∆crit(α), i.e., t < tcrit(α) the equilibrium value c2 s = −1 exactly. the above analytical results are in excellent agreement with the numerical calculations of c2 s from the formula (87) shown in fig. 4. at the temperatures t < tcrit there is no gap equation relating κ and ∆, so the sound velocity is easily calculated to yield the value in the equilibrium vacuum state: c2 s = lim κ→∞ ∂p ∂∆ ∂ρ ∂∆ = −1 . (92) that what is expected for a barotropic perfect liquid with a constant w, where c2 s = w. v. dynamics of the coupled model and observable universe a. scales and observable universe in order to make a connection between the above model results and the observable uni- verse, we need to first conclude where we are now with respect to the critical temperatures t◦and tcrit. as one can see from table i for α ∼1, the model has t◦∼tcrit ∼m. we identify the current equilibrium temperature of the universe with the cosmic background radiation temperature t = 2.275 k = 2.4 * 10−4 ev. then we see right away that we cannot be above the critical temperature of the coupled model, since: (i) assumption t > tcrit leads to m ≲10−4 ev, which in turn implies too small densities ρ ∼m4 ∼10−16 ev4, i.e, four orders of magnitude less than the observable density; (ii) at t > tcrit the equation of state has w > −1 4 (see fig. 3), which is not even enough to get a positive acceleration ̈ a > 0, while the observable value w ≈−1. [13] so, the first qualitative conclusion is that we are currently below the critical temperature. the universe has already passed the stable and metastable phases and is now unstable, i.e. it is in the transition toward the stable "doomsday" vacuum m = ∞and ω= 0. since at the temperature of metastability p◦= 0, the transition occurs somewhere be- tween the beginning of the matter-dominated era (tmd ≈16500 k ≈1.42 ev) and now, i.e., 1.4 ev ≳tcrit > tnow ∼2.4 * 10−4 ev. because of eq.(68) this inequality gives us the 20 possible range of the model's single parameter m: 2.4 * 10−4 ev < m ≲1.4 ev. (93) as we will show in the following, other consistency checks of the model bring the upper bound of m much lower. b. universe before the phase transition in order to apply the results of the coupled model for the calculation of the parameters of the observable universe, we need to incorporate the matter (we will just add up the dark and conventional baryonic matter together) and the radiation. assuming a spatially flat universe, the total energy density is critical, so ρtot = ργ,now/a4 + ρm,now/a3 + ρφν(∆) = ρcr = 3h2 8πg , (94) where from now on we denote ρφν the energy density of the coupled model given by eq. (72). to relate our model's parameters to the standard cosmological notations, we assume that the temperature is evolving as that of the blackbody radiation, i.e., t = tnow/a. then ∆≡m t = ma tnow = m tnow(1 + z) . (95) we know that ργ = π2 15t 4 , (96) and we set the current density of the coupled scalar field to the observable value of the dark energy, i.e., 3/4 of the critical density: ρφν,now = 3 4 * 3h2 0 8πg ≈31 * (10−3 ev)4 , (97) and ρm,now ≈1 4 * 3h2 0 8πg . (98) the equations above allow us to plot the relative energy densities ω# ≡ρ#/ρtot (99) as functions of redshift (or temperature) up to the critical point, see fig. 5. 5 in the high-temperature limit, the matter term is sub-leading and ρtot ≈ργ + ρφν ≈π2 15 1 + 7 4
t 4 . (100) 5 we apologize for some abuse of notations, but using the same greek letter for the grand thermodynamic potential and relative densities seems to be standard now. since these quantities are mainly discussed in different sections of the paper, we hope the reader will not be confused. 21 ω 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 z+1 1010 108 106 104 102 100 t (ev) 106 104 102 100 10-2 ωφν ωγ ωm z* zcr fig. 5: (color online) relative energy densities plotted up to the current redshift (temperature, upper axis): ωφν – coupled de and neutrino contribution; ωγ – radiation; ωm – combined baryonic and dark matters. parameter m = 2.39 * 10−3 ev (α = 0.01), chosen to fit the current densities, determines the critical point of the phase transition zcr ≈3.67. the crossover redshift z∗≈0.83 corresponds to the point where the universe starts its accelerating expansion. in this limit, then ωφν = 7 11 ≈0.636 , ωγ = 4 11 ≈0.363 , (101) which agrees well with the numerical results displayed in fig. 5. at the critical point the matter strongly dominates and ρm/ργ,φν ≳102. the equation of state parameter of the entire universe, wtot, is given by ptot = wtotρtot. since the matter contribution pm = 0, then ptot = pγ + pφν, where pγ = 1 3ργ and the pressure of the coupled model pφν is obtained from eq. (52). the numerical results of wtot are given in fig. 6. w -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 z+1 1010 108 106 104 102 100 t (ev) 106 104 102 100 10-2 zcr z* fig. 6: (color online) equation of state parameter wtot = ptot/ρtot for m = 2.39 * 10−3 ev (α = 0.01) plotted up to the current redshift (temperature, upper axis). 22 to analyze the dynamics of the coupled model we need, in principle, to go beyond the saddle-point approximation applied in the previous sections and solve the equation of motion: ̈ φ + 3h ̇ φ + ∂ω ∂φ = 0 . (102) above the transition point (t > tcrit) the dynamics is quite simple. let us analyze pertur- bations to the saddle-point solution of (24): φ(t) ≡φc + ψ(t) . (103) taylor-expanding the thermodynamic potential of the coupled model ∂ω ∂φ = ω2ψ + 1 2ω′′′(φc)ψ2 + ... (104) with ω2 ≡ω′′(φc), we obtain from (102) the equation of a damped harmonic oscillator to the leading order: ̈ ψ + 3h ̇ ψ + ω2ψ = 0 . (105) so, the quintessence field φ(t) oscillates around its saddle-point value φc with ψ(t) ∝ eıωt−3 2ht. the damping is very small, since as one can check ω ≫3 2h . (106) the violation of the above condition and breaking down of the oscillating regime occurs in the vicinity of the critical point, which is the inflection point of the potential (ω = 0). this is the well-known phenomenon of the critical slowing down near phase transition. retaining the first non-vanishing term in (104), the equation of motion in the vicinity of the critical point reads: ̈ ψ + 3h ̇ ψ + 1 2ω′′′(φc)ψ2 = 0 . (107) neglecting the small damping term in this equation, its solution can be found analytically via a hypergeometric function. since the explicit form of this solution is not very interesting at this point, we just emphasize the qualitative conclusion of the analysis: the fluctuation ψ(t) oscillates near the classical field φc in the stable (metastable) phase at t > tcrit, and it enters the run-away (power-law) regime when t →t + crit.[64] c. late-time acceleration of the universe. towards the end of times the equilibrium methods are not applicable below the phase transition, and we study the dynamics of the model from the equation of motion (102) together with the friedmann equations (1,2,3). solution of the dirac equations yields ρs ∝a−3 for the chiral density [66], so the equation of motion (102) at a ≤acrit reads: ̈ φ + 3h ̇ φ = −∂u ∂φ −ρs,crit acrit a 3 . (108) 23 from the results of the previous section we evaluate the chiral density at the critical point: ρs,crit ≈α ∆crit ν α+1 m3 . (109) the system of the integro-differential equations (108,1,2,3) was solved numerically. all the quantities entering those equations are defined in the previous subsection, except that one needs to include the extra term 1 2 ̇ φ2 in the computation of both ρtot and ptot. however the numerical results show that in the regimes of the parameters we are interested, the kinetic term can be safely neglected. since the critical point of the model lies in the matter- dominated regime (cf. fig. 5), we start with the hubble parameter h = 2/3t (a ∝t2/3). at the latest times (z ≲1) the hubble parameter was determined self-consistently from the numerical solution of the friedmann equations. we find numerically that the quintessence field φ(t) from the critical point to the present time oscillates quickly (with the period τ ∼10−27 gyr) around the smooth ("mean value") solution ̄ φ(t), where the "mean" ̄ φ nullifies the r.h.s. of the equation of motion (108). relating the mean values with the physically relevant observable quantities, we can easily obtain the key results analytically. (they are checked against direct numerical calculations and found to be accurate within 5 % at most). thus we get ̄ φ = φcrit * 1 + zcrit 1 + z  3 α+1 , (110) ρ ̄ φ = ρφ,crit *  1 + z 1 + zcrit  3α α+1 , (111) where φcrit ≈ ν ∆critm and ρφ,crit ≈( ∆crit ν )αm4. having a free model parameter m, we'll set it by matching the current density of the scalar field ρφ,now to the observable value of the de density (97), so m = ναρφ,now
α+1 α+4∆−α critt −3α α+4 now . (112) the exponent of the quintessence potential α is now the only parameter which can be varied. we define the time-dependent mass via the solution of the motion equation as m(t) = ̄ φ(t), thus obtaining an estimate for the present-time neutrino mass. results for various α are given in table ii. there we also calculate the critical points parameterized by the redshifts zcrit and the crossover points z∗. the latter is defined as the redshift at which the universe starts its late-time acceleration, i.e., where wtot = −1 3. for the present time we find wnow tot ≈−3 4 . (113) as we infer from the data of table ii, the range of exponents α ≪1 corresponds to more realistic predictions for the neutrino mass [1, 7, 8] and for the crossover redshift z∗[65]. for α = 0.01 we plot the evolution of the relative energy densities, the equation of state parameter, and the neutrino mass in figs. 5,6,7. we consider the quite artificial case of small quintessence exponent α as an ansatz crossing over smoothly from physically plausible potentials with, say, α = 1 or 2 to the logarithmic potential u(φ) = m4 1 + α log m φ  . (114) 24 table ii: model's parameters and observables for various α. all the entries in this table are defined in the text. α m (ev) mnow (ev) zcrit z∗ 2 9.75 * 10−2 167 392 4.9 1 1.69 * 10−2 44.6 76.6 2.3 1/2 6.33 * 10−3 17.0 27.7 1.5 10−1 2.81 * 10−3 2.82 8.73 0.93 10−2 2.39 * 10−3 0.27 3.67 0.83 10−3 2.36 * 10−3 0.027 1.60 0.82 mν -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 z+1 1010 108 106 104 102 100 t (ev) 106 104 102 100 10-2 z* zcr fig. 7: neutrino mass m for m = 2.39 * 10−3 ev (α = 0.01) plotted up to the current redshift (temperature, upper axis). the latter often appears in various contexts [13, 36].6 vi. conclusions in this paper we analyzed the mavan scenario in a framework of a simple "minimal" model with only one species of the (initially) massless dirac fermions coupled to the scalar quintessence field. by using the methods of thermal quantum field theory we derived for the first time (in the context of the mavan or, even more broadly, the vamp models) a 6 the numerical results for small parameter α, as e.g. α = 0.01 taken for the plots, are virtually in- distinguishable for the cases of the ratra-peebles (48) or logarithmic (114) potentials. however the ratra-peebles potential at more "natural" α = 1, 2 allows to probe the coupled fermionic-quintessence models in the search of heavy dm particle candidates. 25 consistent equation for fermionic mass generation in the coupled model. we demonstrated that the mass equation has non-trivial solutions only for special classes of potentials and only within certain temperature intervals. it appears that these results have not been reported in the literature on vamps before now. we gave most of the results for the particular choice of a trial de potential – the ratra- peebles quintessence potential. this potential has all the necessary properties we needed for our task: it is simple, it satisfies the criteria we found for non-trivial solutions of the mass equation to exist, and it has only one dimensionfull parameter- the energy scale m to tune. also, at small values of the exponent α it effectively crosses over to the case of a logarithmic potential. we have checked that other potentials, e.g., exponential, lead to a qualitatively similar picture, but they have at least one more energy scale to handle, which we consider as an unnecessary complication at this point. we analyzed the thermal (i.e. temporal) evolution of the model, following the time arrow. contrary to what one might expect from analogies with other contexts, like, e.g., condensed matter, the model does not generate the mass via a conventional spontaneous symmetry breaking below a certain temperature. instead it has a non-trivial solution for the fermionic mass evolving "smoothly" from zero at the "point" t = ∞. the scalar field is infinitely heavy at the same point. more realistically, we assumed the model is applicable starting at the temperatures somewhere in the beginning of the radiation-dominated era. we found that the de contribution in this regime is subleading, and the model behaves as an ultra-relativistic fermi gas at those temperatures. this regime corresponds to a stable phase of the model given by a global minimum of the thermodynamic potential ω(φ). the temperature/time dependent minimum ⟨φ⟩generates the varying fermionic mass m ∝⟨φ⟩. with increase in time, as the temperature decreases, the model reaches the point of metastability where its pressure (p) vanishes. from our estimates of the model's scales, we showed that this happens during the matter-dominated era of the universe. at this point the system's ground state becomes doubly degenerate, and the potential ω= 0 at the non-trivial (finite) minimum ⟨φ⟩as well as at the trivial vacuum φ = ∞. further on, at lower temperatures the system stays in the metastable (supercooled) state until it reaches the critical point where the local minimum of the thermodynamic potential disappears and it becomes an inflexion point. at this critical temperature the model un- dergoes a first-order (discontinuous) phase transition. at the critical point the equilibrium values of the fermionic and the scalar field masses discontinuously jump to the 'doomsday" vacuum state values m = ∞and mφ = 0, respectively. the square of the sound velocity and equation of state parameter w have the equilibrium values corresponding to the de sitter universe with a cosmological constant, i.e. c2 s = w = −1. it is worth pointing out that c2 s > 0 in both the stable and metastable phases, and the sound velocity vanishes reaching the critical temperature from above. since the equilibrium approach is not applicable below the critical temperature, we find parameters of the model from direct numerical solution of the equation of motion and the friedmann equations. the single scale m of the quintessence potential is chosen to match the present de density, then other parameters of the universe are determined. we obtain a consistent picture: the phase transition has occurred rather recently at zcrit ≲5 during the matter-dominated era, and the universe is now being driven towards the stable vacuum with zero λ-term. the expansion of the universe accelerates starting from z∗≈0.83. setting α = 0.01 for m ≈2.4 * 10−3 ev, we end up with the neutrino mass m ≈0.27 ev. 26 the present results allow us to propose a completely new viewpoint not only on the mavan, but on the quintessence scenario for the universe as well. the common concerns about the slow-rolling mechanism for the de relaxation toward the λ = 0 vacuum are related to the question of what is the mechanism to set the initial value of the scalar field φ where it evolves (rolls down) from. our results demonstrate that up to recent times (i.e. above the critical temperature) the quintessence field was locked around its average (classical) value ⟨φ⟩. its value is determined by the scale m and the temperature. the average ⟨φ⟩gives the fermionic mass at the same time. the scalar field is rigid (i.e. massive), although it softens (i.e., its mass decreases) as the system approaches the critical temperature. above the critical temperature the scalar field can only oscillate around its equilibrium value ⟨φ⟩. at the critical point the minimum of the thermodynamic potential becomes the inflexion point, the scalar field looses its rigidity (mass). then the field can only roll down towards the new stable ground state ω= 0 at φ = ∞. so physically, the critical point corresponds to the transition of the universe from the stable oscillatory to the unstable rolling regime. a more sophisticated numerical study of the kinetics after the critical point is warranted in order to address such issues as the detailed description of the crossover between different regimes, and the clustering of neutrinos. these and some other questions are relegated to our future work. acknowledgments we highly appreciate useful comments and discussions with d. marfatia, b. ratra, and n. weiner. we are grateful to n. arhipova, d. boyanovsky, r. brandenberger, o. chkvorets, h. feldman, a. gruzinov, l. kisslinger, the late l. kofman, s. lukyanov, and u. wi- choski for helpful discussions and communications. we thank the anonymous referee for constructive criticism and comments which stimulated us to undertake deeper analyses of the model, and especially of its dynamics. g.y.c. thanks the center for cosmology and par- ticle physics at new york university for hospitality. we acknowledge financial support from the natural science and engineering research council of canada (nserc), the laurentian university research fund (lurf), scientific co-operation programme between eastern europe and switzerland (scopes), the georgian national science foundation grants # st08/4-422. t.k. acknowledges the support from nasa astrophysics theory program grant nnxloac85g and the ictp associate membership program. a.n. thanks the bruce and astrid mcwilliams center for cosmology for financial support. [1] a. d. dolgov, phys. atom. nucl. 71, 2152 (2008) [arxiv:0803.3887 [hep-ph]]; physics reports 370, 333 (2002). [2] j. lesgourgues and s. pastor, phys. rept. 429, 307 (2006) [arxiv:astro-ph/0603494]. [3] a. strumia and f. vissani, arxiv:hep-ph/0606054. [4] s. hannestad, ann. rev. nucl. part. sci. 56, 137 (2006) [arxiv:hep-ph/0602058]. [5] s. eidelman et al. [particle data group], phys. lett. b 592, 1 (2004). [6] h. abele, prog. part. nucl. phys. 60, 1 (2008). [7] r. n. mohapatra, et al, rep. prog. phys. 70, 1757 (2007). 27 [8] f. t. avignone, s. r. elliott and j. engel, rev. mod. phys. 80, 481 (2008) [arxiv:0708.1033 [nucl-ex]]. [9] s. weinberg, cosmology, oxford university press, new york (2008). [10] s. dodelson, modern cosmology, academic press, san diego (2003). [11] p. j. e. peebles and b. ratra, rev. mod. phys. 75, 559 (2003) [arxiv:astro-ph/0207347]. [12] s. m. carroll, econf c0307282, tth09 (2003) [aip conf. proc. 743, 16 (2005)] [arxiv:astro-ph/0310342]. [13] e. j. copeland, m. sami and s. tsujikawa, int. j. mod. phys. d 15, 1753 (2006) [arxiv:hep-th/0603057]. [14] s. nojiri and s. d. odintsov, phys. rev. d 68, 123512 (2003). [15] s. m. carroll, a. de felice, v. duvvuri, d. a. easson, m. trodden and m. s. turner, phys. rev. d 71, 063513 (2005). [16] b. ratra and p. j. e. peebles, phys. rev. d 37, 3406 (1988) [17] c. wetterich, nucl. phys. b 302, 668 (1988). [18] a. linde, lect. notes phys. 738, 1 (2008) [arxiv:0705.0164 [hep-th]]; particle physics and inflationary cosmology, chur, switzerland: harwood (1990) [arxiv: hep-th/0503203]. [19] s. weinberg, phys. rev. d 77, 123541 (2008) [arxiv:0804.4291 [hep-th]]. [20] e. mocchiutti et al., arxiv:0905.2551 [astro-ph.he]. [21] e. a. baltz et al., jcap 0807, 013 (2008) [arxiv:0806.2911 [astro-ph]]. [22] k. kohri, a. mazumdar, n. sahu and p. stephens, phys. rev. d 80, 061302 (2009) [arxiv:0907.0622 [hep-ph]]. [23] g. w. anderson and s. m. carroll, arxiv:astro-ph/9711288. [24] m. b. hoffman, arxiv:astro-ph/0307350. [25] g. r. farrar and p. j. e. peebles, astrophys. j. 604, 1 (2004) [arxiv:astro-ph/0307316]. [26] m. kawasaki, h. murayama and t. yanagida, mod. phys. lett. a 7, 563 (1992). [27] j. garcia-bellido, int. j. mod. phys. d 2, 85 (1993) [arxiv:hep-ph/9205216]. [28] d. comelli, m. pietroni and a. riotto, phys. lett. b 571, 115 (2003) [arxiv:hep-ph/0302080]. [29] u. alam, v. sahni, t. d. saini and a. a. starobinsky, mon. not. roy. astron. soc. 354, 275 (2004) [arxiv:astro-ph/0311364]; l. amendola, m. gasperini and f. piazza, jcap 0409, 014 (2004) [arxiv:astro-ph/0407573]. m. szydlowski, t. stachowiak and r. wojtak, phys. rev. d 73, 063516 (2006) [arxiv:astro-ph/0511650]; h. li, b. feng, j. q. xia and x. zhang, phys. rev. d 73, 103503 (2006) [arxiv:astro-ph/0509272]. z. k. guo, n. ohta and s. tsujikawa, phys. rev. d 76, 023508 (2007) [arxiv:astro-ph/0702015]. [30] u. franca and r. rosenfeld, phys. rev. d 69, 063517 (2004) [arxiv:astro-ph/0308149]. [31] u. franca, m. lattanzi, j. lesgourgues and s. pastor, phys. rev. d 80, 083506 (2009) arxiv:0908.0534 [astro-ph.co]. [32] g. huey and b. d. wandelt, phys. rev. d 74, 023519 (2006) [arxiv:astro-ph/0407196]. [33] b. wang, j. zang, c. y. lin, e. abdalla and s. micheletti, cosmological parameters," nucl. phys. b 778, 69 (2007) [arxiv:astro-ph/0607126]. [34] r. mainini and s. bonometto, jcap 0709, 017 (2007) [arxiv:0709.0174 [astro-ph]]. [35] t. koivisto, phys. rev. d 72, 043516 (2005) [arxiv:astro-ph/0504571]; r. rosenfeld, phys. rev. d 75, 083509 (2007) [arxiv:astro-ph/0701213]; s. das, p. s. corasaniti and j. khoury, phys. rev. d 73, 083509 (2006) [arxiv:astro-ph/0510628]; x. j. bi, b. feng, h. li and x. m. zhang, neutrinos," phys. rev. d 72, 123523 (2005) [arxiv:hep-ph/0412002]; m. man- era and d. f. mota, mon. not. roy. astron. soc. 371, 1373 (2006) [arxiv:astro-ph/0504519]; l. amendola, g. camargo campos and r. rosenfeld, phys. rev. d 75, 083506 (2007) 28 [arxiv:astro-ph/0610806]; g. j. stephenson, j. t. goldman and b. h. j. mckellar, int. j. mod. phys. a 13, 2765 (1998) [arxiv:hep-ph/9603392]; s. matarrese, m. pietroni and c. schimd, jcap 0308, 005 (2003) [arxiv:astro-ph/0305224]; a. v. maccio, c. quercellini, r. mainini, l. amendola and s. a. bonometto, phys. rev. d 69, 123516 (2004) [arxiv:astro-ph/0309671]; f. vernizzi, phys. rev. d 69, 083526 (2004) [arxiv:astro-ph/0311167]; h. li, m. z. li and x. m. zhang, phys. rev. d 70, 047302 (2004) [arxiv:hep-ph/0403281]; k. cheung and o. seto, phys. rev. d 69, 113009 (2004) [arxiv:hep-ph/0403003]; a. nusser, s. s. gubser and p. j. e. peebles, phys. rev. d 71, 083505 (2005) [arxiv:astro-ph/0412586]. s. s. gubser and p. j. e. peebles, phys. rev. d 70, 123510 (2004) [arxiv:hep-th/0402225]. [36] r. fardon, a. e. nelson and n. weiner, jcap 0410, 005 (2004) [arxiv:astro-ph/0309800]. [37] p. gu, x. wang and x. zhang, phys. rev. d 68, 087301 (2003). [38] r. d. peccei, phys. rev. d 71, 023527 (2005) [arxiv:hep-ph/0411137]. [39] j. grande, j. sola and h. stefancic, jcap 0608, 011 (2006) [arxiv:gr-qc/0604057]; j. grande, a. pelinson and j. sola, phys. rev. d 79, 043006 (2009) [arxiv:0809.3462 [astro-ph]]. [40] n. afshordi, m. zaldarriaga and k. kohri, phys. rev. d 72, 065024 (2005) [arxiv:astro-ph/0506663]. [41] m. kaplinghat and a. rajaraman, phys. rev. d 75, 103504 (2007) [arxiv:astro-ph/0601517]. [42] v. pettorino and c. baccigalupi, phys. rev. d 77, 103003 (2008) [arxiv:0802.1086 [astro-ph]]. [43] o. e. bjaelde, a. w. brookfield, c. van de bruck, s. hannestad, d. f. mota, l. schrempp and d. tocchini-valentini, jcap 0801, 026 (2008) [arxiv:0705.2018 [astro-ph]]; o. e. bjaelde and s. hannestad, phys. rev. d 81, 063001 (2010) [arxiv:0806.2146 [astro-ph]]. [44] r. bean, e. e. flanagan and m. trodden, phys. rev. d 78, 023009 (2008) [arxiv:0709.1128 [astro-ph]]; new j. phys. 10, 033006 (2008) [arxiv:0709.1124 [astro-ph]]. [45] j. valiviita, e. majerotto and r. maartens, jcap 0807, 020 (2008) [arxiv:0804.0232 [astro- ph]]. [46] c. wetterich and v. pettorino, arxiv:0905.0715 [astro-ph.co]; v. pettorino, d. f. mota, g. robbers and c. wetterich, aip conf. proc. 1115, 291 (2009) [arxiv:0901.1239 [astro-ph]]; l. amendola, m. baldi and c. wetterich, phys. rev. d 78, 023015 (2008) [arxiv:0706.3064 [astro-ph]]; c. wetterich, phys. lett. b 655, 201 (2007) [arxiv:0706.4427 [hep-ph]]; d. f. mota, v. pettorino, g. robbers and c. wetterich, phys. lett. b 663, 160 (2008) [arxiv:0802.1515 [astro-ph]]; v. pettorino, d. f. mota, g. robbers and c. wetterich, aip conf. proc. 1115, 291 (2009) [arxiv:0901.1239 [astro-ph]]. [47] a. w. brookfield, c. van de bruck and l. m. h. hall, phys. rev. d 77, 043006 (2008) [arxiv:0709.2297 [astro-ph]]; a. w. brookfield, c. van de bruck, d. f. mota and d. tocchini- valentini, phys. rev. lett. 96, 061301 (2006) [arxiv:astro-ph/0503349]; phys. rev. d 73, 083515 (2006) [erratum-ibid. d 76, 049901 (2007)] [arxiv:astro-ph/0512367]. [48] a. e. bernardini and o. bertolami, phys. rev. d 77, 083506 (2008) [arxiv:0712.1534 [astro- ph]]; phys. lett. b 662, 97 (2008) [arxiv:0802.4449 [hep-ph]]; phys. lett. b 684, 96 (2010) [arxiv:0909.1280 [gr-qc]]; phys. rev. d 80, 123011 (2009) [arxiv:0909.1541 [gr-qc]]. [49] r. takahashi and m. tanimoto, phys. lett. b 633, 675 (2006) [arxiv:hep-ph/0507142]. [50] r. takahashi and m. tanimoto, jhep 0605, 021 (2006) [arxiv:astro-ph/0601119]. [51] r. fardon, a. e. nelson and n. weiner, jhep 0603, 042 (2006) [arxiv:hep-ph/0507235]. [52] c. spitzer, arxiv:astro-ph/0606034. [53] k. ichiki and y. y. keum, jcap 0806, 005 (2008) [arxiv:0705.2134 [astro-ph]]; jhep 0806, 058 (2008) [arxiv:0803.2274 [hep-ph]]. [54] j.i. kapusta and c. gale, finite-temperature field theory: principles and applications, 29 second edition, cambridge university press, cambridge (2006). [55] j. zinn-justin, quantum field theory and critical phenomena, 4th edition, clarendon press, oxford (2002). [56] w. greiner, l. neise, and h. st ̈ ocker, thermodynamics and statistical mechanics, springer- verlag, new york (1995). [57] s. hannestad, jcap 0305, 004 (2003) [arxiv:astro-ph/0303076]; v. barger, j. p. kneller, h. s. lee, d. marfatia and g. steigman, phys. lett. b 566, 8 (2003) [arxiv:hep-ph/0305075]; a. melchiorri and c. j. odman, phys. rev. d 67, 081302 (2003) [arxiv:astro-ph/0302361]; s. d. stirling and r. j. scherrer, phys. rev. d 66, 043531 (2002) [arxiv:astro-ph/0206173]; j. hamann, j. lesgourgues and g. mangano, jcap 0803, 004 (2008) [arxiv:0712.2826 [astro- ph]]. [58] i. l. shapiro and j. sola, phys. lett. b 475, 236 (2000) [arxiv:hep-ph/9910462]; jhep 0202, 006 (2002) [arxiv:hep-th/0012227]; i. l. shapiro, j. sola, c. espana-bonet and p. ruiz- lapuente, phys. lett. b 574, 149 (2003) [arxiv:astro-ph/0303306]; i. l. shapiro, j. sola and h. stefancic, jcap 0501, 012 (2005) [arxiv:hep-ph/0410095]; j. sola, j. phys. a 41, 164066 (2008) [arxiv:0710.4151 [hep-th]]. [59] f. bauer, j. sola and h. stefancic, phys. lett. b 678, 427 (2009) [arxiv:0902.2215 [hep-th]]; phys. lett. b 688, 269 (2010) [arxiv:0912.0677 [hep-th]]; d. a. demir, found. phys. 39, 1407 (2009) [arxiv:0910.2730 [hep-th]]. s. basilakos, m. plionis and j. sola, phys. rev. d 80, 083511 (2009) [arxiv:0907.4555 [astro-ph.co]]. [60] n.d. birrel and p.c.w. davies, quantum fields in curved space, cambridge university press, cambridge (1982). [61] g. e. volovik, phil. trans. roy. soc. lond. a 366, 2935 (2008) [arxiv:0801.0724 [gr-qc]]; f. r. klinkhamer and g. e. volovik, phys. rev. d 77, 085015 (2008) [arxiv:0711.3170 [gr- qc]]; g. e. volovik, the universe in a helium droplet, clarendon press, oxford (2003); g. e. volovik, jetp lett. 80, 531 (2004); jetp lett. 77, 769 (2003). [62] h. j. de vega and n. g. sanchez, arxiv:astro-ph/0701212. [63] e. v. gorbar and i. l. shapiro, jhep 0302, 021 (2003) [arxiv:hep-ph/0210388]; jhep 0306, 004 (2003) [arxiv:hep-ph/0303124]; i. l. shapiro and j. sola, phys. lett. b682, 105 (2009) [arxiv:0910.4935[hep-th]]. [64] a detailed analysis of the vicinity of the critical point shows that it is an unstable degenerate saddle–node of the differential equation (107). see, e.g., n.n. bautin and e.a. leontovich, methods of the qualitative analysis of dynamical systems on a plane, second edition, nauka, moscow (1990). [65] e. e. o. ishida, r. r. r. reis, a. v. toribio and i. waga, astropart. phys. 28, 547 (2008) [arxiv:0706.0546 [astro-ph]]. [66] s. m. r. micheletti, arxiv:1009.6198 [gr-qc].

0911.1730

variational approach for electrolyte solutions: from dielectric interfaces to charged nanopores

a variational theory is developed to study electrolyte solutions, composed of interacting point-like ions in a solvent, in the presence of dielectric discontinuities and charges at the boundaries. three important and non-linear electrostatic effects induced by these interfaces are taken into account: surface charge induced electrostatic field, solvation energies due to the ionic cloud, and image charge repulsion. our variational equations thus go beyond the mean-field theory. the influence of salt concentration, ion valency, dielectric jumps, and surface charge is studied in two geometries. i) a single neutral air-water interface with an asymmetric electrolyte. a charge separation and thus an electrostatic field gets established due to the different image charge repulsions for coions and counterions. both charge distributions and surface tension are computed and compared to previous approximate calculations. for symmetric electrolyte solutions close to a charged surface, two zones are characterized. in the first one, with size proportional to the logarithm of the coupling parameter, strong image forces impose a total ion exclusion, while in the second zone the mean-field approach applies. ii) a symmetric electrolyte confined between two dielectric interfaces as a simple model of ion rejection from nanopores. the competition between image charge repulsion and attraction of counterions by the membrane charge is studied. for small surface charge, the counterion partition coefficient decreases with increasing pore size up to a critical pore size, contrary to neutral membranes. for larger pore sizes, the whole system behaves like a neutral pore. the prediction of the variational method is also compared with mc simulations and a good agreement is observed.

introduction the first experimental evidence for the enhancement of the surface tension of inorganic salt solutions com- pared to that of pure water was obtained more than eight decades ago [1, 2]. wagner proposed the correct physi- cal picture [3] by relating this effect to image forces that originate from the dielectric discontinuity and act on ions close to the water–air interface. he also correctly pointed out the fundamental importance of the ionic screening of image forces and formulated a theoretical description of the problem by establishing a differential equation for the electrostatic potential and solving it numerically to com- pute the surface tension. using series expansions, on- sager and samaras found the celebrated limiting law [4] that relates the surface tension of symmetric electrolytes to the bulk electrolyte density at low salt concentration. ∗email: [email protected] †email: [email protected] ‡email: [email protected] however, it is known that the consideration of charge asymmetry leads to a technical complication. indeed, im- age charge repulsion, whose amplitude is proportional to the square of ion valency, leads to a split of concentration profiles for ions of different charge, which in turn causes a local violation of the electroneutrality and induces an electrostatic field close to a neutral dielectric interface. bravina derived five decades ago a poisson-boltzmann type of equation for this field [5] and used several ap- proximations in order to derive integral expressions for the charge distribution and the surface tension. these image charge forces play also a key role in slit- like nanopores which are model systems for studying ion rejection and nanofiltration by porous membranes (see the review [6] and references therein, and [7] for a re- view of nano-fluidics). several results have been found in this geometry and also for cylindrical nanopores beyond the mean-field approach (using the debye closure and the bbgky hierarchical equations) and averaging all dielec- tric and charge effects over the pore cross section. within these two approximations, the salt reflection coefficient has been studied as a function of the pore size, the bulk arxiv:0911.1730v2 [cond-mat.soft] 31 mar 2010 2 salt concentration and the pore surface charge. more precisely, the strength of electrostatic correla- tions of ions in the presence of charged interfaces without dielectric discontinuity is quantified by one unique cou- pling parameter ξ = 2πq3l2 bσs (1) where q is the ion valency, and σs the fixed surface charge [8–10]. the bjerrum length in water for mono- valent ions, lb = e2/(4πεwkbt) ≈0.7 nm (εw is the dielectric permittivity of water) is defined as the dis- tance at which the electrostatic interaction between two elementary charges is equal to the thermal energy kbt. the second characteristic length is the gouy-chapman length lg = 1/(2πqlbσs) defined as the distance at which the electrostatic interaction between a single ion and a charged interface is equal to kbt. the coupling param- eter can be reexpressed in terms of these two lengths as ξ = q2lb/lg. on the one hand, the limit ξ →0, called the weak coupling (wc) limit, is where the physics of the coulomb system is governed by the mean-field or poisson-boltzmann (pb) theory, and thermal fluctu- ations overcome electrostatic interactions. it describes systems characterized by a high temperature, low ion va- lency or weak surface charge. on the other hand, ξ →∞ is the strong coupling (sc) limit, corresponding to low temperature, high valency of mobile ions or strong sur- face charge. in this limit, ion–charged surface interac- tions control the ion distribution perpendicularly to the interface. for single interface and slab geometries, several perturbative approaches going beyond the wc limit [11] or below the sc limit [8, 12, 13] have been developed. although these calculations were able to capture impor- tant phenomena such as charge renormalization [14], ion specific effects at the water-air interface [15, 16], man- ning condensation [17], effect of monopoles [18] or attrac- tion between similarly charged objects, they also showed slow convergence properties, which indicates the inabil- ity of high-order expansions to explore the intermediate regime, ξ ≃1. this is quite frustrating since the com- mon experimental situation usually corresponds to the range 0.1 < ξ < 10 where neither wc nor sc theory is totally adequate. consequently, a non-perturbative approach valid for the whole range of ξ is needed. a first important attempt in this direction has been made by netz and orland [19] who derived variational equations within the primitive model for point-like ions and solved them at the mean- field level in order to illustrate charge renormalization ef- fect. interestingly, these differential equations are equiv- alent to the closure equations established in the context of electrolytes in nanopores [6]. they are too compli- cated to be solved analytically or even numerically for general ξ. a few years later, curtis and lue [20] and hatlo et al. [21] investigated the partition of symmetric electrolytes at neutral dielectric surfaces using a similar variational approach (see also the review [22]). they have also recently proposed a new variational scheme based on a hybrid low fugacity and mean-field expansion [23], and showed that their approach agrees well with monte-carlo simulation results for the counterions-only case. how- ever, this method is quite difficult to handle, and one has to solve two coupled variational equations, i.e. a sixth order differential equation for the external potential to- gether with a second algebraic equation. within this approach, these authors generalized the study of ion-ion correlations for counterions close to a charged dielectric interface, first done by netz in the wc and sc limits [24], to intermediate values of ξ. they also studied an elec- trolyte between two charged surfaces without dielectric discontinuities at the pore boundary, in two cases: coun- terions only and added salt, handled at the mean-field level [25]. although this simplification allows one to focus exclusively on ion-ion correlations induced by the surface charge, the dielectric discontinuity can not be discarded in synthetic or biological membranes. indeed, it is known that image forces play a crucial in ion filtration mecha- nisms [6]. the main goal of this work is to propose a variational analysis which is simple enough to intuitively illustrate ionic exclusion in slit pores, by focusing on the competition between image charge repulsion and surface charge interaction. moreover, our approach allows us to connect nanofiltration studies [26–28] with field-theoretic approaches of confined electrolyte solutions within a gen- eralized onsager-samaras approximation [4] character- ized by a uniform variational screening length. this vari- ational parameter takes into account the interaction with both image charge and surface charge. we also compare the prediction of the variational theory with monte-carlo simulations [29] and show that the agreement is good. the paper is organized as follows. the variational formalism for coulombic systems in the presence of di- electric discontinuities is introduced in section ii. sec- tion iii deals with a single interface. we show that the introduction of simple variational potentials allows one to fully account for the physics of asymmetric electrolytes at dielectric interfaces (e.g. water–air, liquid–liquid and liquid–solid interfaces, see ref. [30]), first studied by brav- ina [5] using several approximations, as well as the case of charged surfaces. in section iv, the variational approach is applied to a symmetric electrolyte confined between two dielectric surfaces in order to investigate the prob- lem of ion rejection from membrane nanopores. using restricted variational potentials, we show that due to the interplay between image charge repulsion and direct elec- trostatic interaction with the charged surface, the ionic partition coefficient has a non-monotonic behaviour as a function of pore size. ii. variational calculation in this section, the field theoretic variational approach for many body systems composed of point-like ions in the presence of dielectric interfaces is presented. since the field theoretic formalism as well as the first order varia- 3 tional scheme have already been introduced in previous works [19, 20], we only illustrate the general lines. the grand-canonical partition function of p ion species in a liquid of spatially varying dielectric constant ε(r) is z = p y i=1 ∞ x ni=0 eniμi ni!λ3ni t z ni y j=1 drije−(h−es) (2) where λt is the thermal wavelength of an ion, μi denotes the chemical potential and ni the total number of ions of type i. for sake of simplicity, all energies are expressed in units of kbt. the electrostatic interaction is h = 1 2 z dr′drρc(r)vc(r, r′)ρc(r′) (3) where ρc is the charge distribution (in units of e) ρc(r) = p x i=1 ni x j=1 qiδ(r −rij) + ρs(r), (4) and qi denotes the valency of each species, ρs(r) stands for the fixed charge distribution and vc(r, r′) is the coulomb potential whose inverse is defined as v−1 c (r, r′) = −kbt e2 ∇[ε(r)∇δ(r −r′)] (5) where ε(r) is a spatially varying permittivity. the self- energy of mobile ions, which is subtracted from the total electrostatic energy, is es = vb c(0) 2 p x i=1 niq2 i (6) where vb c(r) = lb/r is the bare coulomb potential for ε(r) = εw. after performing a hubbard-stratonovitch transformation and the summation over ni in eq. (2), the grand-canonical partition function takes the form of a functional integral over an imaginary electrostatic aux- iliary field φ(r), z = r dφ e−h[φ]. the hamiltonian is h[φ] = z dr " [∇φ(r)]2 8πlb(r) −iρs(r)φ(r) (7) − x i λie q2 i 2 vb c(0)+iqiφ(r) # where a rescaled fugacity λi = eμi/λ3 t (8) has been introduced. the variational method consists in optimizing the first order cumulant fv = f0 + ⟨h −h0⟩0. (9) where averages ⟨* * * ⟩0 are to be evaluated with respect to the most general gaussian hamiltonian [19], h0[φ] = 1 2 z r,r′ [φ(r) −iφ0(r)] v−1 0 (r, r′) [φ(r′) −iφ0(r′)] (10) and f0 = −1 2tr ln v0. the variational principle consists in looking for the optimal choices of the electrostatic kernel v0(r, r′) and the average electrostatic potential φ0(r) which extremize the variational grand potential eq. (9). the variational equations δfv/δv−1 0 (r, r′) = 0 and δfv/δφ0(r) = 0, for a symmetric electrolyte and ε(r) = εw, yield ∆φ0(r) −8πlbqλe−q2 2 w (r) sinh [qφ0(r)] = −4πlbρs(r) (11) −∆v0(r, r′) + 8πlbq2λe−q2 2 w (r) cosh [qφ0(r)] v0(r, r′) = 4πlbδ(r −r′). (12) where we have defined w(r) ≡lim r→r′  v0(r, r′) −vb c(r −r′)  (13) whose physical signification will be given below. the second terms on the lhs of eq. (11) and of eq. (12) have simple physical interpretations: the former is 4πlb times the local ionic charge density and the latter is 4πlbq2 times the local ionic concentration. the rela- tions eqs. (11)–(12) are respectively similar in form to the non-linear poisson-boltzmann (nlpb) and debye- h ̈ uckel (dh) equations, except that the charge and salt sources due to mobile ions are replaced by their local values according to the boltzmann distribution. on the one hand, eq. (11) is a poisson-boltzmann like equa- tion where appears the local charge density proportional to sinh φ0. this equation handles the asymmetry in- duced by the surface through the electrostatic potential φ0, which ensures electroneutrality. this asymmetry may be due to the effect of the surface charge on anion- and cation-distributions (see section iii b) or due to dielec- tric boundaries and image charges at neutral interfaces, which give rise to interactions proportional to q2, and in- duce a local non-zero φ0 for asymmetric electrolytes (see section iii a). on the other hand, the generalized dh equation eq. (12), where appears the local ionic concen- tration proportional to cosh φ0, fixes the green's function v0(r, r′) evaluated at r with the charge source located at r′ and takes into account dielectric jumps at boundaries. these variational equations were first obtained within the variational method by netz and orland [19]. they were also derived in ref. [31] within the debye closure approach and the bbgky hierarchic chain. yaroshchuk obtained an approximate solution of the closure equa- tions for confined electrolyte systems in order to study ion exclusion from membranes [6]. equations (11)–(12) enclose the limiting cases of wc (ξ →0) and sc (ξ →∞). to see that, it is interesting to rewrite theses equations by renormalizing all lengths and the fixed charge density, ρs(r), by the gouy-chapman length according to ̃ r = r/lg, ̃ ρs( ̃ r) = lgρs(r)/σs (σs is the average surface charge density). by introducing a new electrostatic potential ̃ φ0(r) = qφ0(r), one can 4 express the same set of equations in an adimensional form ̃ ∆ ̃ φ0( ̃ r) −λe−ξ 2 ̃ w ( ̃ r) sinh ̃ φ0( ̃ r) = −2 ̃ ρs( ̃ r) (14) − ̃ ∆ ̃ v0( ̃ r, ̃ r′) + λe−ξ 2 ̃ w ( ̃ r) cosh ̃ φ0( ̃ r) ̃ v0( ̃ r, ̃ r′) = 4πδ( ̃ r − ̃ r′)(15) where ̃ v0 = v0lg/lb, ̃ w = wlg/lb and we have also introduced the rescaled fugacity λ = 8πλl3 gξ [32]. now, one can check that, in both limits ξ →0 and ξ →∞, the coupling between φ0 and v0 in eq. (11) disappears and the theory becomes integrable. finally, it is important to note that this adimensional form of variational equations allows one to focus on the role of v0(r, r) whose strength is controlled by ξ in eqs. (14) and (15). however, even at the numerical level, their explicit coupling does not allow for exact solutions for general ξ. in the present work, we make a restricted choice for v0(r, r′) and replace the local salt concentration in the form of a local debye-h ̈ uckel parameter (or inverse screening length) κ(r) in eq. (12), κ(r)2 = 8πlbq2λe−q2 2 w (r) cosh [qφ0(r)] , (16) by a constant piecewise one κv(r) = κv in the presence of ions and κv(r) = 0 in the salt-free parts of the sys- tem. note that it has been recently shown that many thermodynamic properties of electrolytes are successfully described with a debye-h ̈ uckel kernel [33]. the inverse kernel (or the green's function) v0(r, r′) is then taken to be the solution to a generalized debye- h ̈ uckel (dh) equation  −∇(ε(r)∇) + ε(r)κ2 v(r)  v0(r, r′) = e2 kbt δ(r −r′) (17) with the boundary conditions associated with the dielec- tric discontinuities of the system lim r→σ−v0(r, r′) = lim r→σ+ v0(r, r′), (18) lim r→σ−ε(r)∇v0(r, r′) = lim r→σ+ ε(r)∇v0(r, r′) (19) where σ denotes the dielectric interfaces. we now re- strict ourselves to planar geometries. we split the grand potential (9) into three parts, fv = f1 + f2 + f3, where f1 is the mean electrostatic potential contribution, f1 = s z dz ( −[∇φ0(z)]2 8πlb + ρs(z)φ0(z) − x i λie− q2 i 2 w (z)−qiφ0(z) ) , (20) f2 the kernel part and f3 the unscreened van der waals contribution. the explicit forms of f2 and f3 are re- ported in appendix a. the first variational equation is given by ∂fv/∂κv = ∂(f1 + f2) /∂κv = 0. this equa- tion is the restricted case of eq. (12). as we will see below, its explicit form depends on the confinement ge- ometry of the electrolyte system as well as on the form of ε(r). the variational equation for the electrostatic potential [34] δfv/δφ0(z) = 0 yields regardless of the confinement geometry ∂2φ0 ∂z2 + 4πlbρs(z) + x i 4πlbqiλie− q2 i 2 w (z)−qiφ0(z) = 0. (21) the second-order differential equation (21), which is simply the generalization of eq. (11) for a general elec- trolyte in a planar geometry, does not have closed-form solutions for spatially variable w(z). in what follows, we optimize the variational grand potential fv using re- stricted forms for the electrostatic potential φ0(z) and compare the result to the numerical solution of eq. (21) for single interfaces and slit-like pores. the single ion concentration is given by ρi(z) = λi e− q2 i 2 w (z)−qiφ0(z) (22) and its spatial integral by z dzρi(z) = −λi ∂fv ∂λi . (23) we define the potential of mean force (pmf) of ions of type i, φi(z), as φi(z) ≡−ln ρi(z) ρb (24) by defining w(z) ≡w(z) −wb (25) where wb is the value of w(z) in the bulk and comparing eqs. (22)–(24), we find φi(z) = q2 i 2 w(z) + qiφ0(z) (26) q2 i 2 wb = ln γb i ≡μi −ln(ρbλ3 t) (27) hence, q2 i wb/2 is nothing else but the excess chemical potential of ion i in the bulk and q2 i w(z)/2 = ln γi(z) is its generalization for ion i at distance z from the interface. they are related to the activity coefficients γb i and γi(z). note that the zero of the chemical potential is fixed by the condition that φ0 vanishes in the bulk. the pmf, eq. (26), is thus the mean free energy per ion (or chemical potential) needed to bring an ion from the bulk at infinity to the point at distance z from the interface, taking into account correlations with the surrounding ionic cloud. before applying the variational procedure to single and double interfaces, let us consider the variational approach in the bulk. in this case, the variational potential φ0 is 5 ε 0 z εw ε ε 0 z d εw (a) (b) fig. 1: (color online) geometry for a single dielectric inter- face (e.g. water–air) (a) and double interfaces or slit-like pores (b). equal to 0, and the variational grand potential fv only depends on κv. two minima appear: one metastable minimum κ0 v at low values of κv, and a global minimum at infinity (fv →−∞for κv →∞) which is unphysical since at these large concentration values, finite size effects should be taken into account. it has been shown by intro- ducing a cutoffat small distances [20], that, for physical temperatures, this instability disappears and the global minimum of fv is κ0 v ≈κb given by the debye–h ̈ uckel limiting law:  μi = ln(ρbλ3 t) −q2 i 2 κblb κ2 b = 4πlb p i q2 i ρi,b (28) from eq. (27), we thus find wb ≈−κblb and the poten- tial w(z) reduces to w(z) ≈v0(z, z) −vb c(0) + κblb, (29) which will be adopted in the rest of the paper. further- more, problems due to the formation of ion pairs do not enter at the level of the variational approach we have adopted. let us also report the following conversion re- lations ib ≃0.19(κblb)2 mol.l−1 (30) κb ≃3.29 √ i nm−1, where i = 1/2 p i q2 i ρi,b is the ionic strength expressed in mol.l−1. finally, the single-ion densities are given by ρi(z) = ρi,be− q2 i 2 w(z)−qiφ0(z). (31) iii. single interface the single interfacial system considered in this section consists of a planar interface separating a salt-free left half-space from a right half-space filled up with an elec- trolyte solution of different species (fig. 1). in the gen- eral case, the dielectric permittivity of the two half spaces may be different (we note ε the permittivity in the salt- free part). the green's function, which is chosen to be solution of the dh equation with ε(z) = εθ(−z) + εwθ(z) 0.0 0.2 0.4 0.6 0.0 1.0 2.0 3.0 w(z) z/lg fig. 2: (color online) potential w(z) in units of kbt for ε = 0 (black solid curve), eq. (36), and ε = εw (red dashed curve) and κblb = 4. and κ(z) = κvθ(z) where θ(z) stands for the heaviside distribution, is given for z > 0 by [5] w(z) = lb(κb −κv) (32) +lb z ∞ 0 kdk p k2 + κ2 v ∆(k/κv) e−2√ k2+κ2 vz where ∆(x) = εw √ x2 + 1 −εx εw √ x2 + 1 + εx . (33) and f2 [eq. (a4)] can be analytically computed [20] f2 = v κ3 v 24π + s∆κ2 v 32π , (34) where ∆≡∆(x →∞) = εw −ε εw + ε. (35) the first term on the rhs. of eq. (34) is simply the volu- mic debye free energy associated with a hypothetic bulk with a debye inverse screening length κv and the second term on the rhs. involves interfacial effects, including the dielectric jump ∆, and κv. for the single interface system, as seen in section ii, f3 is independent of κv and φ0(z), which means that it does not contribute to the variational equations. by minimiz- ing eqs. (20) and (34) with respect to κv for fixed φ0(r) and taking v →∞, one exactly find the same variational equation for κv as for the bulk case. hence, as discussed above, we have κv = κb given by eq. (28) and the first term of the rhs. of eq. (32) vanishes. this result was obtained in [20] for the special case φ(r) = 0. it is of course not surprising to end up with the same result for finite φ(r) since we know that the electrostatic potential should vanish in the bulk. this potential combines in an intricate way both the image charge and solvation contri- butions due to the presence of the interface. the image 6 force corresponds to the interaction of a given ion with the polarized charges at the interface and is equivalent to the interaction of the charged ion with its image lo- cated at the other side of the dielectric surface. as it is well known, the image charge interaction is repulsive for ε < εw (e.g. water-air interface) and attractive for ε > εw (the case for an electrolyte-metal interface) [35]. the interfacial reduction in solvation arises because an ion always prefers to be screened by other ions in order to reduce its free energy. hence, it is attracted towards areas where the ion density is maximum (at least at not too high concentrations for which steric repulsion may predominate). this term is non-zero even for ε = εw since for an ion close to the interface, there is a "hole" of screening ions in the salt-free region (where κv = 0). although our choice of homogeneous variational inverse screening length allows us to handle the deformation of ionic atmospheres near interfaces that are impermeable to ions, it does not allow us to treat in detail the local variations in ion solvation free energy arising from ion- ion correlations (except in an average way in confined geometries where κv can differ from the bulk value of the inverse screening length, see section iv below). equation (32) simplifies in three cases : 1) for ε = 0 (∆= 1), where the solvation effect vanishes because the lines of forces are totally excluded from the air region [35], eq. (32) reduces to w0(z) = lb e−2κbz 2z . (36) this is the case where the image charge repulsion is the strongest (see fig. 2). 2)a slightly better approximation for ε ̸= 0 can be ob- tained by artificially allowing salt to be present in the air region. this gives rise to the "undistorted ionic atmo- sphere" approximation [6], for which w(z) in eq. (36) is multiplied by ∆: w(z) = ∆lb e−2κbz 2z . (37) solvation effects are now absent and salt exclusion arises solely from dielectric repulsion. eq. (37) is exact for ar- bitrary κb and ∆= 1, or arbitrary ∆and κb = 0. 3)in the absence of a dielectric discontinuity ε = εw (∆= 0), the potential can be expressed as w(z) = κblbf(κbz) (38) f(x) = (1 + x)2e−2x 2x3 −k2(2x) x where k2(x) is the bessel function of the second kind. one notices that unlike the case ∆> 0, the potential has a finite value at the interface, i.e. w(0) = κblb/3. we note that in this case of one interface, we have limz→∞φ0(z) = 0 and the fugacity λi of each species is fixed by its bulk concentration according to ρi,b = lim z→∞ρi(z) = λie q2 i 2 κblb (39) where we used eq. (22). a. neutral dielectric interface we investigate in this section the physics of an asym- metric electrolyte close to a neutral dielectric interface (e.g. water–air, liquid–liquid or liquid–solid interface) located at z = 0 (σs = 0). for the sake of simplicity, we assume ε = 0, which is a very good approximation for the air-water interface characterized by ε = 1 (see the discussion in ref. [4]). hence we keep the approxima- tion w(z) = w0(z) given by eq. (36). the electrolyte is composed of two species of bulk density ρ+ and ρ−and charge (q+e),−(q−e) with q+ > q−. in order to satisfy the electroneutrality in the bulk, we impose ρ+q+ = ρ−q−. according to eq. (28), the bulk inverse screening length noted κb is given by κ2 b = 4πlbq−ρ−(q−+ q+) (40) and the variational equation (21) for the electrostatic po- tential is a modified poisson-boltzmann equation ∂2φ0 ∂z2 + 4πlbρch(z) = 0 (41) with a local charge concentration ρch(z) = ρ−q−  e− q2 + 2 w(z)−q+φ0(z) −e− q2 − 2 w(z)+q−φ0(z)  . (42) equation (41) can not be solved analytically. its numer- ical solution, obtained using a 4th order runge-kutta method, is plotted in fig. 3(a) for asymmetric elec- trolytes with divalent and quadrivalent cations and the local charge density is plotted in fig. 3(b). fig. 3 clearly shows that, very close to the dielectric interface for z < a, image charge repulsion expulses all ions (since ρch(z) ∼exp(−1/z) has an essential singu- larity) and φ0 is flat. for z > a, but still close to the interface, there is a layer where the electrostatic field is almost constant (φ0 increases linearly), which is created by the charge separation of ions of different valency due to repulsive image interactions. the intensity of image forces increases with the square of ion valency and close to the interface, ρch(z) < 0 since we assumed q+ > q− (the case for mgi2). to ensure electroneutrality, the lo- cal charge then becomes positive when we move away from the surface (fig. 3(b)), and the electrostatic poten- tial goes exponentially to zero with a typical relaxation constant κφ. moreover, in fig. 3(a) one observes that when the charge asymmetry increases, the electrostatic potential also increases. knowing that for symmetric electrolytes, φ0 = 0, our results confirm that the charge asymmetry is the source of the electrostatic potential φ0. fig. 3(b) is qualitatively similar to fig. 1 of bravina who had derived an integral solution of eq. (41) by using an approximation valid for κblb ≪1 [5]. in order to go fur- ther in the description of the interfacial distribution of ions, we look for a restricted variational function φ0(z) which not only contains a small number of variational 7 (a) 0.00 1.00 2.00 3.00 4.00 z/lb -0.16 -0.12 -0.08 -0.04 0.00 electrolyte 1:2 electrolyte 1:4 φ0 (b) 0.00 1.00 2.00 3.00 4.00 z/lb -0.03 -0.01 0.01 0.03 0.05 electrolyte 1:2 electrolyte 1:4 lb 3 ρch fig. 3: (color online) (a) electrostatic potential φ0 (in kbt units) for asymmetric electrolytes: numerical solution of eq. (41) (symbols) and variational choice, eq. (44) (solid lines), for divalent and quadrivalent ions and ρ−= 0.242 m. variational parameters are κφ ≃1.4κb, a/lb = 0.12; 0.21 and φ = −0.10; −0.156. (b) associated local charge density profile (thick lines) and anion (dashed lines) and cation (thin solid lines) concentrations. parameters (such as a and κφ) but also is as close as possible to the numerical solution. as suggested by the description of fig. 3, a continuous piecewise φ0(z) is nec- essary to account for the essential singularity of ρch(z). to show this, let us expand eq. (41) to order φ0: ∂2φ0 ∂z2 ≈−4πlbρ−q−  e− q2 + 2 w(z) −e− q2 − 2 w(z)  (43) +4πlbρ−q−  q+e− q2 + 2 w(z) + q−e− q2 − 2 w(z)  φ0. this linearization is legitimate, as seen in fig. 3: q+|φ0(z)| < 1 is satisfied for physical valencies. the first term in the rhs. of eq. (43) corresponds to an effective local charge source while the second term is responsi- ble for the screening of the potential. if we observe the charge distribution for q2w(z) > qφ0(z) and z > a, i.e. the first term of the rhs. of eq. (43), we notice that it behaves like a distorted peak. the simplest function hav- ing a similar behavior is f(z) = cze−κφz, where c and κφ are constants. hence, we choose a restricted variational piecewise solution φ0(z) φ0(z) =  φ for z ≤a, φ [1 + κφ(z −a)] e−κφ(z−a) for z ≥a. (44) whose derivation is explained in appendix b. the vari- ational parameters are the constant potential φ, the de- pletion distance a and the inverse screening length κφ. the grand potential (b5) derived for this solution was optimized with respect to the variational parameters us- ing the mathematica software. the restricted variational potential (44) is compared to the numerical solution of eq. (41) in fig. 3 for electrolytes 1 : 2 and 1 : 4 and ρ−l3 b = 0.05. the agreement is excellent. one notices that the screening of the effective surface charge created by dielectric exclusion enters into play when z > κ−1 φ . fi- nally, let us note that since κblb = 1.37 and κblb = 1.77 respectively for the monovalent and quadrivalent elec- trolytes in fig. 3, the method adopted by bravina is not valid. to summarize, the charge separation is taken into ac- count by the potential φ (which increases with q+/q−) and the relaxation constant κφ ≃1.4κb is almost inde- pendent of q+/q−. interestingly, the variational parame- ter a/lb ≃0.1 −0.2 is less than 1 nm. indeed, for finite size ions, w(z) differs from eq. (36) very close to the in- terface and reaches a finite value at z = 0. the size of this region exactly corresponds to a which is of the order of an ion radius. this is thus an artifact of our point-like ion model and occurs only for asymmetric electrolytes at neutral surfaces. the surface tension σ is equal to the excess grand po- tential defined as the difference between the grand poten- tial of the interfacial system and that of the bulk system: σ = ∆κ2 b 32π −κφφ2 32πlb (45) −ρ− z ∞ 0 dz  e− q2 − 2 w(z)+q−φ0(z) −1  + q− q+  e− q2 + 2 w(z)−q+φ0(z) −1  . the surface tension for electrolytes characterized by q−= 1 and q+ = 1 to 4 is plotted in fig. 4 as a a function of ρ−, because the anion density is an experimentally accessible parameter. unlike symmetric electrolytes [20], a plot with respect to κ2 b may lead to a different behavior. one notices that the increase in valency asymmetry leads to an important increase of the surface tension. this is of course mainly due to the reduction of the cation density in the bulk by a factor of q−/q+ necessary to satisfy the bulk electroneutrality (see the second term in the integral of eq. (45)). 8 0.01 0.06 0.11 0.16 0.21 0.26 lb 3ρ- 0.02 0.07 0.12 0.17 0.22 electrolyte 1:1 electrolyte 1:2 electrolyte 1:3 electrolyte 1:4 βlb 2σ fig. 4: (color online) surface tension l2 bσ/kbt for asymmet- ric electrolytes vs. the anion bulk concentration, for increas- ing asymmetry q+/q−= 1 to 4 from bottom to top. b. charged surfaces we now consider a symmetric electrolyte in the prox- imity of an interface of constant surface charge σs < 0 located at z = 0. the variational equation (21) simplifies to ∂2 ̃ φ0 ∂ ̃ z2 = 2δ( ̃ z) + ̃ κ2 be−ξ 2 ̃ w( ̃ z) sinh ̃ φ0. (46) the mean-field limit (ξ →0) of this equation corre- sponds to the nlpb equation, whose solution reads ̃ φ0( ̃ z) = 4arctanh γbe− ̃ κb ̃ z
(47) where γb = ̃ κb− p 1 + ̃ κ2 b. in this section, we show that a piecewise solution for the electrostatic potential similar to the one introduced in section iii a agrees very well with the numerical solution of eq. (46). inspired by the existence of a salt-free layer close to the interface and a mean-field regime far from the interface (wc), we propose two types of piecewise variational functions (see appendix c). the first variational choice obeys the poisson equation in the first zone of size h and the non- linear poisson-boltzmann solution in the second zone : ̃ φnl 0 ( ̃ z) = ( 4arctanhγ + 2( ̃ z − ̃ h) for ̃ z ≤ ̃ h, 4arctanh  γe− ̃ κφ( ̃ z− ̃ h) for ̃ z ≥ ̃ h, (48) where γ = ̃ κφ − q 1 + ̃ κ2 φ. variational parameters are h and an effective inverse screening length κφ. the second type of trial potential obeys the laplace equation with a charge renormalization in the first zone and the linearized poisson-boltzmann solution in the second zone : ̃ φl 0( ̃ z) = ( −2η ̃ κφ + 2η( ̃ z − ̃ h) for ̃ z ≤ ̃ h, −2η ̃ κφ e− ̃ κφ( ̃ z− ̃ h) for ̃ z ≥ ̃ h. (49) variational parameters are ̃ h, ̃ κφ, and the charge renor- malization η, which takes into account the non-linear ef- fects at the mean-field level [19]. the explicit form of the φ0 0.0 0.5 1.0 1.5 2.0 z/lg -1.5 -1.0 -0.5 0.0 numerical non-linear pb linear pb 0 2 4 6 8 log ξ 0.0 0.5 1.0 1.5 2.0 2.5 h/lg fig. 5: (color online) electrostatic potential, φ0 (in units of kbt): numerical solution of eq. (46) (symbols) and restricted variational choices eqs. (48) and (49) for ε = 0, κblg = 4, and ξ = 1, 10, 100, and 1000 (from top to bottom). the vari- ational parameters are respectively κφ = 3.83, 3.74, 3.69, 3.66 and η ≃1. markers on the x-axis denote, for each curve, the size, ̃ h, of the sc zone, plotted vs. ln ξ in the inset. associated variational free energies are reported in ap- pendix c. the inset of fig. 5 displays the size of the sc layer h against ξ. our approach predicts a logarithmic dependence ̃ h ∝ln ξ, the factor behind the logarithm being ̃ κ−1 b for ̃ κb ≫1. the restricted choices for φ0 are compared with the full numerical solution of eq. (41) in the same figure for ε = 0. we see that, as in the pre- vious section, the numerical solution and the restricted ones match perfectly. hence salt-exclusion effects are es- sentially carried by the parameter h. furthermore, one notices that ̃ φ0( ̃ z) relaxes to zero between ̃ z = ̃ h and ̃ z = ̃ h + 2 ̃ κ−1 φ . at κblg = 4 we are in the linear regime of the pb equation and therefore one has η ≃1. the charge renormalization idea was introduced by alexan- der et al. [14], who showed that the non-linearity of the pb equation can be effectively taken into account at long distances by renormalizing the fixed charge source and extending the linearized zone where | ̃ φ0| < 1 to the whole domain. a linear solution of the form eq. (49) can be very helpful for complicated geometries or in the presence of a non-uniform charge distribution where the nlpb equation does not present an analytical solution even at the mean-field level. these issues will be discussed in a future work. figure 6 displays the ion concentrations ρi(z)/ρi,b = e−φi, which are related to the ion pmf eq. (24), com- puted with the restricted solution eq. (48) for several val- ues of ξ. as already said in the introduction, in rescaled distance, the coupling parameter ξ measures the strength of the excess chemical potential, w(z). we first see that for coions as well as for counterions, the depletion layer in rescaled units in the proximity of the dielectric inter- face increases with ξ due to the image charge repulsion and/or solvation effect, i.e. the term e−ξ 2 ̃ w( ̃ z) in eq. (46). furthermore, one notices that the counterion density ex- 9 (a) 0.0 0.5 1.0 1.5 2.0 z/lg 0.00 0.25 0.50 0.75 1.00 1.25 1.50 ρ/ρb (b) ρ/ρb 0.0 0.5 1.0 1.5 2.0 z/lg 0.00 0.25 0.50 0.75 1.00 1.25 1.50 fig. 6: (color online) ion densities for κblg = 4, and (a) ε = 0 and (b) ε = εw, for increasing coupling parameter: from left to right, ξ = 1, 10, 100, and 1000. solid lines correspond to counterions, dashed lines to coions and dashed-dotted lines to the poisson-boltzmann result (47). hibits a maximum. this concentration peak is due to the competition between the attractive force towards the charged wall and the repulsive image and solvation in- teractions. it is important to note that in the particular case ε = εw, there is no depletion layer for ξ < 10. iv. double interface in this section, the variational method is applied to a double interface system which consists of a slit-like pore of thickness d, in contact with an external ion reservoir at its extremities (fig. 1). the dielectric constant is εw inside the pore and ε in the outer space. the electrolyte occupies the pore and the external space is salt-free. the solution of the dh equation (a2) in this geometry is [6] w(z) = (κb −κv)lb (50) + lb z ∞ 0 kdk p k2 + κ2 v ∆(k/κv) e2d√ k2+κ2 v −∆2(k/κv) × h 2∆(k/κv) + e2(d−z)√ k2+κ2 v + e2z√ k2+κ2 v i where ∆(x) is given in eq. (33). the variational pa- rameter of the green's function is the variational inverse screening length κv which is taken uniform (generalized onsager-samaras approximation, see [6, 21]). a more complicated approach has been previously developed in ref. [21] where the authors introduced a piecewise form for the variational screening length, i.e. κ(z) = κv over a layer of size h and κv = κb in the middle of the pore. although this choice is more general than ours, the min- imization procedure with respect to κv is significantly longer than in our case and the variational equation is much more complicated. consequently, this piecewise approach is not very practical when one wishes to study a charged membrane where the external field created by the surface charge considerably complicates the technical task (see section iv b). we show that the simple varia- tional choice adopted here captures the essential physics with less computational effort. as in eq. (32), the integral on the rhs. of eq. (50) takes into account both image charge and solvation ef- fects due to the two interfaces, whereas the first term is the debye result for the difference between the bulk and a hypothetic bulk of inverse screening length κv. we should emphasize that, in the present case, the spatial in- tegrations in eqs. (a3)-(a4) run over the confined space, that is from z = 0 to z = d. by substituting the so- lution eq. (50) into eqs. (20)-(a5) and performing the integration over z, one finds [22] f2 + f3 s = dκ3 v 24π + ∆κ2 v 16π (51) +κ2 v 4π z ∞ 1 dxx ln  1 − ̄ ∆2(x)e−2κvdx +κ2 v 8π z ∞ 1 dx ̄ ∆(x) − ̄ ∆3(x)
/x −2κvd ̄ ∆2(x) e2dκvx − ̄ ∆2(x) where we have defined ̄ ∆(x) = ∆ √ x2 −1
. the limiting case ε = 0 allows for closed-form expres- sions. this limit is a good approximation for describing biological and artificial pores characterized by an exter- nal dielectric constant much lower than the internal one. in the following part of the work, we will deal most of the time with the special case ε = 0, unless stated otherwise. in this limit, eq. (51) simplifies to f2 + f3 s = κ3 vd 24π + κ2 v 16π  1 + 2 ln 1 −e−2dκv
 − κv 8πdli2 e−2dκv
−li3 e−2dκv
16πd2 (52) where lin(x) stands for the polylogarithm function and ξ(x) the riemann zeta function (see appendix d). within the same limit (ε = 0), ∆(x) = 1 and we obtain an analytical expression for the green's function eq. (50) w0(z) = (κb −κv)lb −lb d ln 1 −e−2dκv
+ lb 2d h β  e−2dκv; 1 −z d, 0  + d z e−2κvz 2f1  1, z d, 1 + z d, e−2dκv (53) 10 where β(x; y, z) is the incomplete beta function and 2f1(a, b; c; d) the hypergeometric series. the definitions of these special functions are given in appendix d. at this step, the pmf thus depends on three adimensional parameters, namely dκv, dκb, and d/lb. for the system with a single interface, the ion fugacity λi was fixed by the bulk density. in the present case where the confined system is in contact with an external reservoir, λi is fixed by chemical equilibrium: λi = λi,b = ρi,be− q2 i 2 κi,blb, (54) where κb and λi,b are respectively the inverse debye screening length and the fugacity in the bulk reservoir [see eq. (28)]. once this constraint is taken into ac- count, the last term of electrostatic part of the vari- ational grand potential eq. (20) can be written as −p i ρi,b r d 0 dz e− q2 i 2 w(z)−qiφ0(z). eq. (21) then becomes for a symmetric q : q elec- trolyte: ∂2 ̃ φ0 ∂z2 −κ2 be−q2 2 w(z) sinh ̃ φ0 = −4πqlbσs [δ(z) + δ(z −d)] (55) the optimization of fv = f1 +f2 +f3 given by eq. (20) and (52) with respect to the inverse trial screening length κv leads to the following variational equation for κv: (dκv)2 + dκv tanh(dκv) = (dκb)2 z 1 0 dx e−q2 2 w0(xd) × cosh[ ̃ φ0(xd)]  1 + cosh [(2x −1)dκv] cosh(dκv)  . (56) within the particular choice that fixed the functional form of the κv dependent green's function eq. (53), the two coupled equations (55) and (56) are the most general variational equations. in the following, we first consider the case of neutral pores and then the more general case of charged pores. a. neutral pore, symmetric electrolyte in the case of a symmetric q : q electrolyte and a neu- tral membrane, σs = 0, the solution of eq. (55) is natu- rally φ0 = 0. the variational parameter κv is solution of eq. (56) with φ0 = 0 and w(z) = w0(z) given by eq. (53) when ε = 0, which can be written as dκv = f(dκb, lb/d). let us note that eq. (56) can be solved with the mathe- matica software in a fraction of a second. within the debye-h ̈ uckel closure approach, yaroshchuk (see eq. (59) of ref. [6]) obtains a self- consistent approximation for constant κv by replacing the exponential term of eq. (12) with its average value in the pore: κ2 v = κ2 b z 1 0 dx e−q2 2 w(xd), (57) 0.0 0.2 0.4 0.6 0.8 1.0 variational yaroschuk κv/κb 0 1 2 3 d/lb 0.0 0.2 0.4 0.6 0.8 1.0 (b) (a) κblb 0.1 0.6 0 1 2 3 d*/lb κv/κb fig. 7: (color online) inverse screening length inside the neu- tral membrane (monovalent ions) normalized by κb vs. the pore size d/lb for ε = 0 and (a) κblb = 0.1 (ρb = 1.926 mm), (b) κblb = 1 (ρb = 0.1926 m). dashed lines correspond to the mid-point approximation, eq. (58). the inset shows the characteristic pore size corresponding to total ionic exclusion as a function of the inverse bulk screening length. the bot- tom curve corresponds to monovalent ions and the top curve to divalent ions. which should be compared with eq. (56) with φ0 = 0. in order to simplify the numerical task, yaroshchuk intro- duces a further approximation in which he replaces the potential w(z) inside the depletion term of eq. (57) by its value in the middle of the pore, w(d/2). then eq. (57) takes the simpler form κ2 v = κ2 be−q2 2 w(d/2). (58) the self-consistent midpoint approximation is frequently used in nanofiltration theories [6, 28, 36]. for ε = 0, the mid-point potential has the simple form w(d/2) = (κb − κv)lb −2lb ln(1 −e−κvd)/d. this approach is compared with the full variational treatment in fig. 7 where the adimensional inverse screening length in the pore κv/κb is plotted as a function of the pore size d. we first note that as d decreases below a critical value d∗, the pore is empty of salt and κv = 0. the inset of fig. 7 shows d∗ versus the inverse bulk screening length. searching for d such that κv = 0 in eq. (56) leads to the same equation as eq. (57), thus the value of d∗is identical within both approaches. however, fig. 7 shows that the mid-point approximation, eq. (58), overestimates the internal salt concentration as well as the abruptness of the crossover to an ion-free regime for decreasing pore size. indeed, this approximation is equivalent to neglecting the strong ion exclusion close to the pore surfaces (which is larger than in the middle of the pore). a similar behavior was also observed in fig. 6 of ref. [21] for the screening length in the neighborhood of the dielectric interface. the effect of the dielectric discontinuity is illustrated in fig. 8(a) where the inverse internal screening length is 11 (a) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 d/lb 0.0 0.2 0.4 0.6 0.8 1.0 κv/κb (b) 0.0 0.4 0.8 1.2 log10 (d/lb) 0.0 0.2 0.4 0.6 0.8 1.0 κv/κb monovalent ions divalent ions trivalent ions fig. 8: (color online) inverse screening length inside the membrane vs. the pore size d/lb (εw = 78, κblb = 1). (a) from bottom to top: ε = 0 (∆= 1), ε = 3.2 (∆= 0.92), ε = 39 (∆= 1/3), and ε = 78 (∆= 0). (b) log-linear plot for monovalent, divalent and trivalent ions, from left to right (ε = 0). compared for ε between 0 and εw = 78 where the image- charge repulsion is absent and the solvation effect is solely responsible for ion repulsion. first of all, one observes that the total exclusion of ions in small pores is specific to the case ε = 0. moreover, in the solvation only case, the inverse screening length inside the pore only slightly de- viates from the bulk value, 0.8 ≤κv/κb ≤1. this clearly indicates that, within the point-like ion model consid- ered in this work, the image-charge interaction brings the main contribution to salt-rejection from neutral mem- branes. roughly speaking, the image-charge and solva- tion effects come into play when the surface of the ionic cloud of radius κ−1 b around a single ion located at the pore center touches the pore wall, i.e. for κ−1 b > d/2. this simple picture fixes a characteristic length dch ≃2κ−1 b below which the internal ion density significantly devi- ates from the bulk value and ion-rejection takes place. this can be verified for intermediate salt densities in the bottom plot of fig. 7 and the top plot of fig. 8. since image-charge effects are proportional to q2, we illustrate in fig. 8(b) the effect of ion valency q. at pore size d ≃2.5lb ≃1.8 nm, where the inverse inter- nal screening length for monovalent ions is close to 80 % -5.0 -4.0 -3.0 -2.0 -1.0 log10(κb lb) 0.1 0.3 0.5 0.7 0.9 σs d=5, variational d=5, yaroschuk d=2, variational d=2, yaroschuk d=2, variational, plug flow fig. 9: (color online) salt reflection coefficient (dimension- less) against the logarithm of the inverse bulk screening length for ε = 0 and two pore sizes, d/lb = 2 and 5 (red lines corre- spond to the mid-point approximation, eq. (58)). of its saturation value κb, the exclusion of divalent ions from the membrane is total. this effect driven by image interactions is even much more pronounced for trivalent ions. since the typical pore size of nano-filtration mem- branes ranges between 0.5 and 2 nm, we thus explain why ion valency can play a central role in ion selectivity, even inside neutral pores. the salt reflection coefficient, frequently used in mem- brane transport theories to characterize the maximum salt-rejection (obtained at high pressure) is related to the ratio of the net flux of ions across the membrane to that of the solvent volume flux j per unit transverse surface : σs ≡1 − 1 jρb z d 0 v||(z)ρ(z)dz (59) = 1 −12 z 1 1/2 x(1 −x) e−q2 2 w(xd)dx. where we have used, in the second equality, the poiseuille velocity profile, v||(z) = 6j d3 z(d −z) in the pore and the pmf given by eq. (24). it depends only on the parame- ters κblb and d/lb. in certain nanopores with hydropho- bic surfaces, the solvent flux may considerably deviate from the poiseuille profile (see [37]). in this case, the ve- locity profile is flat, v||(z) = j d . we emphasize that since the velocity profile is normalized in both cases, the mid- point approximation is unable to distinguish between a poiseuille and a plug flow velocity profile. fig. 9 displays σs as a function of the inverse bulk screening length for two pore sizes d = 2lb and d = 5lb. as seen by yaroshchuk, decreasing the pore size shifts the curves to higher bulk concentration and thus increases the range of bulk concentration where nearly total salt rejection oc- curs. however, quantitatively, the difference between the variational and mid-point approaches becomes significant at high bulk concentrations and this difference is accen- tuated in the case of plug-flow (for which σs is higher 12 when compared to the poiseuille case because the flow velocity no longer vanishes at the pore wall where the salt exclusion is strongest). this deviation is again due to the midpoint approximation of eq. (58) in which the image interactions are underestimated. however since the velocity profile vanishes at the solid surface for the poiseuille flow, the deficiencies of the mid-point approx- imation are less visible in σs than in κv in this case. finally, we compute the disjoining pressure within our variational approach. we compare in appendix e the result with that of the more involved variational scheme presented in ref. [21] and show that one gets a very similar behaviour, revealing that the simpler variational method is able to capture the essential physics of the slit pore. as stressed above, the main benefit obtained from the simpler approach proposed in this work is that the min- imization procedure is much less time consuming. this point becomes crucial when considering the fixed charge of the membrane, which is thoroughly studied in the next section. b. charged pore, symmetric electrolyte in this section, we apply the variational approach to a slit-like pore of surface charge σs < 0. in the following, we will solve eqs. (55) and (56) numerically in order to test, as in the case of a single charged surface, the validity of restricted trial forms for φ0(z). we define the partition coefficients in the pore for counterions and coions, k+ and k−, as k± ≡ρ± ρb = z d 0 dz d e−φ±(z). (60) where φ±(z) is given by eq. (26). 1. effective donnan potential when one considers a charged nanopore, because of its small size, gradients of the potential φ0 can be neglected as a first approximation. we thus assume a constant po- tential ̄ φ0. the so-called effective donnan potential ̄ φ0 introduced by yaroshchuk [6] will be fixed by the vari- ational principle. by differentiating the grand potential eq. (20) with respect to ̄ φ0 (or equivalently integrating eq. (55) from z = 0 to z = d with ∇ ̄ φ0 = 0), we find 2 |σs| = −2qρb sinh(q ̄ φ0) z d 0 dz e−q2 2 w(z) (61) which is simply the electroneutrality relation in the pore, taken in charge by the electrostatic potential ̄ φ0. by 1.0 1.5 2.0 donnan numerical piecewise cyar 1.0 2.0 3.0 4.0 ξ 0.6 1.2 1.8 κvlg (a) (b) κvlg fig. 10: (color online) inverse internal screening length κv against ξ for κblg = 2, ε = 0 and (a) d = 3lg and (b) d = 10lg. comparison of various approximations: yaroshchuk, eq. (71) (diamonds), variational donnan potential (dashed line), piecewise solutions (solid line), and numerical results (squares). horizontal lines corresponds to the wc limit, eq. (67) (top), and sc limit, eq. (70) (bottom) . defining γ = z 1 0 dx exp[−q2lb ̄ w(xd)/(2d)] (62) = z 1 0 dx exp[−ξ ̄ w(x ̃ d)/(2 ̃ d)], (63) where ̄ w(x) ≡w(x)d/lb, we have k± = γ exp(∓q ̄ φ0) and eq. (61) can be rewritten as k+ −k−= 2 |σs| qρbd = xm qρb = 8 κ2 bdlg = 8 ̃ κ2 b ̃ d (64) where the second equality contains the gouy-chapman length lg and the quantity xm = 2|σs|/d, frequently used in nanofiltration theories, corresponds to the volume charge density of the membrane. hence, the partition coefficient of the charge, k+ −k−, does not depend on ξ, i.e. charge image and solvation forces. by using eq. (61) in order to eliminate the potential ̄ φ0 from eq. (60), one can rewrite the partition coefficients in the form k± = γe∓q ̄ φ0 = v u u tγ2 + 4 ̃ κ2 b ̃ d !2 ± 4 ̃ κ2 b ̃ d (65) by substituting into eq. (56) the analytical expression for ̄ φ0 obtained from eq. (61) (or eq. (65)), one obtains a single variational equation for κv to be solved numeri- cally, ( ̃ d ̃ κv)2 + ̃ d ̃ κv tanh( ̃ d ̃ κv) = ( ̃ d ̃ κb)2 v u u tγ2 + 4 ̃ κ2 b ̃ d !2 ×   1 + z 1 0 dx e−ξ 2 ̃ w(x ̃ d) cosh h (2x −1) ̃ d ̃ κv i γ cosh( ̃ d ̃ κv)   . (66) 13 1.0 2.0 3.0 4.0 ξ 0.0 0.2 0.4 0.6 0.8 1.0 <ρ/ρb> 0.0 0.2 0.4 0.6 0.8 1.0 1.2 counterions coions (a) (b) <ρ/ρb> fig. 11: (color online) ionic partition coefficients, k±, vs. ξ for κblg = 2, ε = 0, and (a) d = 3lg and (b) d = 10lg. the horizontal line corresponds to the sc limit for counterions. as explained in the text,we note that k+ −k−= 8/(κ2 bdlg). the numerical solution of eq. (66) is plotted in fig. 10 as a function of the coupling parameter ξ. we see that as we move from the wc limit to the sc one by increasing ξ, the pore evolves from a high to a low salt regime. this quite rapid crossover, which results from the exclusion of ions from the membrane, is mainly due to repulsive image-charge and solvation forces controlled by γ whose effects increase with increasing ξ. in fig. 11 are plotted the partition coefficients of coun- terions and coions, eq. (60), as a function of ξ. here again, k± decreases with increasing ξ. moreover, we clearly see that the rejection of coions from the mem- brane becomes total for ξ > 4. in other words, even for intermediate coupling parameter values, we are in a counterion-only state. this is obviously related to the electrical repulsion of coions by the charged surface. in the asymptotic wc limit (ξ →0), γ = 1 and we find the classical donnan results in mean-field where k−= k−1 + = eq ̄ φ0 with q ̄ φ0 = arcsinh[4/( ̃ κ2 b ̃ d)]. the variational equations (66) and (65) reduce to κ2 v = κ2 b v u u t1 + 4 ̃ κ2 b ̃ d !2 (67) k± = v u u t1 + 4 ̃ κ2 b ̃ d !2 ± 4 ̃ κ2 b ̃ d (68) quite interestingly, the relation eq. (67) shows that, even in the mean-field limit, due to the ion charge imbalance created by the pore surface charge, the inverse screening length is larger than the debye-h ̈ uckel value κb. in the case of small pores or strongly charged pores or at low values of the bulk ionic strength, i.e. κ2 blgd ≪1 or dρb ≪|σs|/q, we find κv ≃2/√lgd and ρ−= 0 and ρ+ = 2|σs|/(dq). we thus find the classical poisson-boltzmann result for counterions only [24]. the counterion-only case is also called good coion exclusion limit (gce), a notion introduced in the context of nanofiltration theories [6, 38, 39]. hence, in this limit the quantity of counterions in the membrane is independent of the bulk density and depends only on the pore size d and the surface charge density σs. in the case of a membrane of size d ≃1 nm and fixed surface charge σs ≃0.03 nm−2, this limit can be reached with an electrolyte of bulk concentration ρb ≃ 50 mm. in the opposite limit κ2 blgd ≫1, one finds κv ≃κb and ρ± = ρb. in the sc limit ξ →∞, γ = 0 and eq. (66) simplifies to ( ̃ d ̃ κv)2 + ̃ d ̃ κv tanh( ̃ d ̃ κv) = 4 ̃ d[1 + sech( ̃ d ̃ κv)]. (69) for d > lg ( ̃ d > 1), the solution of eq. (69) yields with a high accuracy ̃ κv ≃ p 1 + 16 ̃ d −1 2 ̃ d . (70) the partition coefficients simplify to k−= 0 and k+ = 8/( ̃ d ̃ κ2 b) = 2|σs|/(dqρb) and we find the counterion only case (or gce limit) without image charge forces dis- cussed by netz [24]. partition coefficients in the sc limit and variational inverse screening length in both limits, eqs. (67) and (70), are illustrated in figs. 10 and 11 by dotted reference lines. consequently, one reaches for ξ = 0 the gce limit exclusively for low salt density or small pore size, while the sc limit leads to gce for arbitrary bulk density. it is also important to note that although the pore-averaged densities of ions are the same in the gce limit of wc and sc regimes, the density pro- files are different since when one moves away from the pore center, the counterion densities close to the inter- face increase in the wc limit due to the surface charge attraction and decrease in the sc limit due to the image charge repulsion. it is interesting to compare this variational approach to the approximate mid-point approach of yaroshchuk [6]. for charged membranes, he considers a constant po- tential and replaces the exponential term of eqs. (11) and (12) by its value in the middle of the pore. he ob- tains the following self-consistent equations: κ2 = κ2 be−q2 2 w(d/2) cosh q ̄ φ0
(71) 2 |σs| = −2qdρb sinh q ̄ φ0
e−q2 2 w(d/2). (72) the above set of equations are frequently used in nanofil- tration theories [6, 28, 36]. by combining these equations in order to eliminate ̄ φ0, one obtains an approximate non- linear equation for κv (approximation cyar in fig. 10). in the limit of a high surface charge, the non-linear equa- tions (71)–(72) depend only on the pore size d and the surface charge density σs: κ2 ≃8πlbq|σs| d = 4 lgd. (73) 14 σ lb 2 0.0 1.0 2.0 3.0 κvlb donnan variational cyar asymptotic limit (a) 0.0 0.1 0.2 0.3 0.4 0.0 1.0 2.0 3.0 (b) (a) κvlb fig. 12: (color online) inverse internal screening length κv against the reduced surface charge ̄ σ = l2 bσs for d = lb, ε = 0 and (a) κblb = 1, (b) κblb = 2: constant variational donnan approximation (solid line), asymptotic result eq. (73) (dotted line) and yaroshchuk approximation eq. (71) (dashed line). one can verify that in the regime of strong surface charge, eq. (73) is also obtained from the asymptotic solution eq. (70) since the dependence of the pmf on z is killed when ξ →∞and only the mid-pore value contributes. the numerical solution of eq. (71)-(72) is illustrated as a function of ξ in fig. 10, and as a function of the surface charge in fig. 12, together with the asymptotic formula eq. (73). for the parameter range considered in fig. 10, the solution of eq. (71) strongly deviates from the result of the full variational calculation. for ξ < 2, the mid-point approach follows an incorrect trend with increasing ξ. it is clearly seen that at some values of the coupling parameter, eqs. (71)-(72) do not even present a numerical solution. using the relations d/lb = ̃ d/ξ and lbκb = ξ ̃ κb for monovalent ions, one can verify that the regime where the important deviations take place corresponds to high ion concentrations. this is confirmed in fig. 12: the error incurred by the approximate mid-point solution of yaroshchuk increases with the electrolyte concentration. in section iv a on neutral nanopores, it has been un- derlined that, due to the image charge repulsion, the ionic concentration inside the pore increases with the pore size d (see fig. 8). in the present case of charged nanopores, this result is modified: eqs. (67), (70) and (73) show that for strongly charged nanopores the concentration of ions inside the pore decreases with d. moreover, the very high charge limit is a counterion-only state and eq. (61) shows that, for a fixed surface charge density, electroneutrality alone fixes the number of counterions, n−, in a layer of length d joining both interfaces, and image charge interactions play a little role. this is the reason why κ2 v ∝ρ−= n−/(sd) decreases for increasing d. 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.2 0.4 0.6 0.8 k- d/lb (a) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 d/lb 0.01 0.03 0.05 0.07 σ (nm-2 ) 0 5 10 15 20 25 dcr (nm) k+ (b) fig. 13: (color online) partition coefficient in the pore of coions (a) and counterions (b) vs. the pore size d/lb for increasing surface charge density, σsl2 b = 0, 0.004, 0.02, 0.04, 0.08, 0.12, from left to right, and κblb = 1. inset: critical pore size dcr vs. the surface charge density σs (ε = 0). hence, we expect an intermediate charge regime which interpolate between image force counterion repulsion (case of neutral pores, see section iv a) and counterion attraction by the fixed surface charge. this is illustrated in fig. 13 where the partition coefficients are plotted vs. d for increasing σs. as expected, coions are electrostat- ically pushed away by the surface charge which adds to the repulsive image forces, leading to a stronger coion ex- clusion than for neutral pores. the issue is more subtle for counterions: obviously, increasing the surface charge, σs, at constant pore size, d, increases k+. however, for small fixed σs, a regime where image charge and direct electrostatic forces compete, k+ is non-monotonic with d. below a characteristic pore size, d < dcr, the electro- static attraction dominates over image charge repulsion and due to the mechanism explained above, k+ decreases for increasing d. for d > dcr, the effect of the surface charge weakens and k+ starts increasing with d. in this regime, the pore behaves like a neutral system. the inset of fig. 13 shows that dcr increases when σs increases. for highly charged membranes l2 bσs ≫0.1, there is no min- imum in k+(d), and the average counterion density in- 15 2πξlg3 ρ(z) 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 z/lg counterion, mc coion, mc counterion, variational with four parameters coion, variational with four parameters counterion, variational with a single parameter coion, variational with a single parameter counterion, nlpb coion, nlpb fig. 14: (color online) ion densities in the nanopore for ε = εw , ξ = 1 and h/lg = 2. the continuous lines correspond to the prediction of the variational method with four parameters, the dashed-dotted line the variational solution with a single parameter (see the text), the symbols are mc results (fig. 2 of [29]) and the dashed lines denote the numerical solution of the non-linear pb result. side the membrane monotonically decreases towards the bulk value. experimental values for surface charges are 0 ≤σs ≤0.5 nm−2 (or 0 ≤l2 bσs ≤0.25), which cor- responds to physically attainable values of dcr. the in- terplay between image forces and direct electrostatic at- traction is thus relevant to the experimental situation. the variational donnan potential approximation is thus of great interest since it yields physical insight into the exclusion mechanism and allows a reduction of the computational complexity. however, membranes and nanopores are often highly charged and spatial variations of the electrostatic potential inside the pore may play an important role. in the following we seek a piecewise so- lution for φ0(z). 2. piecewise solution the variational modified pb equation (55) for ̃ φ0 shows that as one goes closer to the dielectric interface, w(z) increases and the screening experienced by the poten- tial ̃ φ0 gradually decreases because of ionic exclusion. this non-perturbative effect which originates from the strong charge-image repulsion inspires our choice for the variational potential ̃ φ0(z). we opt for a piecewise so- lution as in section iii: a salt-free solution in the zone 0 < z < h and the solution of the linearized pb equa- tion for h < z < d/2, with a charge renormalization parameter η taking into account non-linear effects. by inserting the boundary conditions ∂ ̃ φ0/∂z|z=0 = 2η/lg and ∂ ̃ φ0/∂z|z=d/2 = 0 and imposing the continuity of ̃ φ0 and its first derivative at z = h [eq. (b3)], the piecewise -1.0 -0.5 0.0 numerical homogeneous piecewise φ0 (a) 0.0 1.0 2.0 3.0 4.0 z/lg -111.5 -110.0 -108.5 (b) φ0 fig. 15: (color online) variational electrostatic potential (in units of kbt) in the nanopore. comparison of the numeri- cal solution of eq. (55) with the homogeneous (h = 0) and piecewise solution of eq. (c5) for lgκb = 3 and (a) ξ = 1, (b) ξ = 100. the horizontal line is the donnan potential obtained from eq. (61) (ε = 0). potential, solution of eq. (55) with κ2 b exp[−q2w(z)/2] replaced by κ2 φ, takes the form ̃ φ0(z) = ( ̄ φ −2η lg  z −d 2   for 0 < z ≤h, φ − 2η lgκφ cosh[κφ(d/2−z)] sinh[κφ(d/2−h)] for h ≤z ≤d/2 (74) where ̄ φ = φ + 2η lg d 2 −h  − 2η lgκφ coth  κφ d 2 −h  (75) is imposed by continuity, and κφ, φ, h and η are the vari- ational parameters. by injecting the piecewise solution eq. (c5) into eq. (20), we finally obtain f1 s = −2 |σs| q  η(η −2) h lg − η2(d/2 −h) 2lg sinh2[κφ(d/2 −h)] + η(η −4) 2lgκφ coth[κφ(d/2 −h)] + φ  − κ2 b 4πlbq2 z d 0 dz e−q2 2 w(z) cosh ̃ φ0(z). (76) the solution to the variational problem is found by op- timization of the total grand potential f = f1 + f2 with respect to κφ, φ, h, η and κv, where f2 + f3 is given by eq. (52) for a general value of ε and by eq. (52) for ε = 0. this was easily carried out with mathematica software. a posteriori, we checked that two restricted forms for φ0, homogeneous with h = 0 and piecewise with φ = 0, were good variational choices. fig. 14 compares the ion densities obtained from the variational approach (with homogeneous φ0) with the predictions of the mc simu- lations [29] and the nlpb equation for ε = εw , ̃ d = 2 and ξ = 1. two variational choices are displayed in this 16 figure, namely, the homogeneous approach with four pa- rameters κv = 1.68, κφ = 1.36, φ = 0.16, η = 0.97 and a simpler choice with η = 1, κφ = κv and two variational parameters: κv = 1.69, φ = −0.18. in the latter case, one can obtain an analytical solution for φ and injecting this solution into the free energy, one is left with a single parameter κv to be varied in order to find the optimal solution. we notice that with both choices, the agree- ment between the variational method and mc result is good. it is clearly seen that the proposed approach can reproduce with a good quantitative accuracy the reduced solvation induced ionic exclusion, an effect absent at the mean-field level. moreover, we verified that with the sin- gle parameter choice, one can reproduce at the mean-field variational level the ion density profiles obtained from the numerical solution of the nlpb equation (dashed lines in fig. 14) almost exactly. we finally note that the small discrepancy between the predictions of the varia- tional approach and the mc results close to the interface may be due to either numerical errors in the simulation, or our use of the generalized onsager-samaras approx- imation (our homogeneous choice for the inverse effec- tive screening length appearing in the green's function v0 does not account for local enhancement or diminution of ionic screening due to variations in local ionic density). for ε = 0, the piecewise and homogeneous solu- tions are compared with the full numerical solution of eqs. (55)–(56) in fig. 15 for ξ = 1 and 100. first of all, one observes that for ξ = 1, both variational solu- tions match perfectly well with the numerical solutions. for ξ = 100, the piecewise solution matches also per- fectly well with the numerical one, whereas the match- ing of the homogeneous one is poorer. the optimal val- ues of the variational parameters (κv, κφ, η, h) for the piecewise choice are (2.57, 2.6, 0.98, 0.15) for ξ = 1 and (0.83, 0.13, 0.97, 1.37) for ξ = 100. the form of the electrostatic potential φ0(z) is inti- mately related to ionic concentrations. ion densities in- side the pore are plotted in fig. 16 for ξ = 1 and ξ = 100. we first notice that even at ξ = 1, the counterion density is quite different from the mean-field prediction. further- more, due to image charge and electrostatic repulsions from both sides, the coion density has its maximum in the middle of the pore. on the other hand, the coun- terion density exhibits a double peak, symmetric with respect to the middle of the pore, which originates from the attractive force created by the fixed charge and the repulsive image forces. when ξ increases, we see that the counterion density close to the wall shrinks and becomes practically flat in the middle of the pore. hence the po- tential φ0 linearly increases with z until the counterion- peak is reached and then it remains almost constant since the counterion layer screens the electrostatic field created by the surface charge (since in fig. 15, z is renormalized by the gouy-chapman length which decreases with in- creasing σs, one does not see the increase of the slope φ0(z = 0)). in agreement with the variational donnan approximation above, coions are totally excluded from 0.0 0.5 1.0 1.5 coion counterion mean-field ρ/ρbulk z/lg 0.0 1.0 2.0 3.0 4.0 5.0 0.00 0.10 0.20 0.30 0.40 (b) (a) ρ/ρbulk fig. 16: (color online) local ionic partition coefficient in the nanopore (same parameters as in fig. 15 and ε = 0) com- puted with the piecewise solution. (a) ξ = 1 , (b) ξ = 100. the dotted line in the top plot corresponds to the mean-field prediction for counterion density. the pore for large ξ. hence, the piecewise potential al- lows one to go beyond the variational donnan approxi- mation within which the density profile does not exhibit any concentration peak. the inverse screening length κv obtained with the piecewise solution is compared in fig. 10 with the pre- diction of the donnan approximation and that of the numerical solution. the agreement between piecewise and numerical solutions is extremely good. although the donnan approximation slightly underestimates the salt density in the pore, its predictions follow the correct trend. v. conclusion in this study, we applied the variational method to interacting point-like ions in the presence of dielectric discontinuities and charged boundaries. this approach interpolates between the wc limit (ξ ≪1) and the sc one (ξ ≫1), originally defined for charged boundaries without dielectric discontinuity, and takes into account image charge repulsion and solvation effects. the vari- ational greens's function v0 has a debye-h ̈ uckel form with a variational parameter κv and the average vari- ational electrostatic potential φ0(z) is either computed numerically or a restricted form is chosen with varia- tional parameters. the physical content of our restricted variational choices can be ascertained by inspecting the general variational equations (eqs. (11)-(12) for symmet- ric salts). the generalized onsager-samaras approxi- mation that we have adopted for the green's function replaces a local spatially varying screening length by a constant variational one; although near a single interface this screening length is equal to the bulk one, in confined 17 geometries the constant variational screening length can account in an average way for the modified ionic environ- ment (as compared with the external bulk with which the pore is in equilibrium) and can therefore strongly devi- ate from the bulk value. this modified ionic environ- ment arises both from dielectric and reduced solvation effects present even near neutral surfaces (encoded in the green's function) and the surface charge effects encoded in the average electrostatic potential. our restricted vari- ational choice for φ0(z) is based on the usual non-linear poisson-boltzmann type solutions with a renormalized inverse screening length that may differ from the one used for v0 and a renormalized external charge source. the coupling between v0 and φ0 arises because the in- verse screening length for v0 depends on φ0 and vice- versa. the optimal choices are the ones that extremize the variational free energy. in the first part of the work, we considered single in- terface systems. for asymmetric electrolytes at a single neutral interface, the potential φ0(z) created by charge separation was numerically computed. it was satisfac- torily compared to a restricted piecewise variational so- lution and both charge densities and surface tension are calculated in a simpler way than bravina [5] and valid over a larger bulk concentration range. the variational approach was then applied to a single charged surface and it was shown that a piecewise solution, characterized by two zones, can accurately reproduce the correlations and non-linear effects embodied in the more general varia- tional equation. the first zone of size h is governed by a salt-free regime, while the second region corresponds to an effective mean-field limit. the variational calculation predicts a relation between h and the surface charge of the form h ∝(c + ln |σs|)/|σs| where the parameter c depends on the temperature and ion valency. in the second part, we dealt with a symmetric elec- trolyte confined between two dielectric interfaces and in- vestigated the important problem of ion rejection from neutral and charged membranes. we illustrated the ef- fects of ion valency and dielectric discontinuity on the ion rejection mechanism by focusing on ion partition and salt reflection coefficients. we computed within a vari- ational donnan potential approximation, the inverse in- ternal screening length and ion partition coefficients, and showed that for ξ > 4 one reaches the sc limit, where the partition coefficients are independent of the bulk con- centration and depend only on the size and charge of the nanopore. this result has important experimental appli- cations, since it indicates that complete filtration can be done at low bulk salt concentration and/or high surface charge. furthermore, we showed that, due to image inter- actions, the quantity of salt allowed to penetrate inside a neutral nanopore increases with the pore-size. in the case of strongly charged membranes, this behavior is reversed for the whole physical range of pore size. we quanti- fied the interplay between the image charge repulsion and the surface charge attraction for counterions and found that even in the presence of a weak surface charge, the competition between them leads to a characteristic pore size dcr below which the counterion partition coefficient rapidly decreases with increasing pore size. on the other hand, for nanopores of size larger than dcr the system be- haves like a neutral pore. our variational calculation was compared to the debye closure approach and the mid- point approximation used by yaroshchuk [6]. the clo- sure equations have no exact solution even at the numer- ical level. our approach, based on restricted variational choices, shows significant deviations from yaroshchuk's mid-point approach at high ion concentrations and small pore size. finally, the introduction of a simple piecewise trial potential for φ0, which perfectly matches the nu- merical solutions of the variational equations, enabled us to go beyond the variational donnan potential approxi- mation and thus account for the concentration peaks in counterion densities. we computed ion densities in the pore and showed that for ξ > 4, the exclusion of coions from the pore is total. we also compared the ionic den- sity profiles obtained from the variational method with mc simulation results and showed that the agreement is quite good, which illustrates the accuracy of the varia- tional approach in handling the correlation effects absent at the mean-field level. the main goal in this work was first to connect two different fields in the chemical physics of ionic solutions focusing on complex interactions with surfaces: field- theoretic calculations and nanofiltration studies. more- over, on the one hand, this variational method allows one to consider, in a non-perturbative way, correlations and non-linear effects; on the other hand the choice of one constant variational debye-h ̈ uckel parameter is simple enough to reproduce previous results and to illuminate the mechanisms at play. this approach is also able to handle, in a very near future, more complicated geome- tries, such as cylindrical nanopores, or a non-uniform sur- face charge distribution. the present variational scheme also neglects ion-size effects and gives rise to an instability of the free energy at extremely high salt concentration. second order corrections to the variational method may be necessary in order to properly consider ionic correlations leading to pairing [40, 41] and to describe the physics of charged liquids at high valency, high concentrations or low temperatures. introducing ion size will also allow us to introduce an effective dielectric permittivity εp for water confined in a nanopore intermediate between that of the membrane matrix and bulk water, leading naturally to a born-self energy term that varies inversely with ion size and depends on the difference between 1/εw and 1/εp [42, 43]. furthermore, the incorporation of the ion polarizability [44] will yield a more complete physical description of the behavior of large ions [45]. charge inversion phenomena for planar and curved interfaces is another important phenomenon that we would like to consider in the future [46]. note, however, that our study of asymmetric salts near neutral surfaces reveals a closely related phenomenon: the generation 18 of an effective non-zero surface charge due to the unequal ionic response to a neutral dielectric interface for asymmetric salts. a further point that possesses experimental relevance is the role played by surface charge inhomogeneity. strong-coupling calculations show that an inhomogeneous surface charge distribution characterized by a vanishing average value gives rise to an attraction of ions towards the pore walls, but this effect disappears at the mean-field level [47]. for a better understanding of the limitations of the proposed model, more detailed comparison with mc/md simulations are in order[48, 49]. finally, dynamical hindered transport effects [27, 48] such as hydrodynamic forces deserve to be properly included in the theory for practical applications. acknowledgments we would like to thank david s. dean for numer- ous helpful discussions. this work was supported in part by the french anr program nano-2007 (simo- nanomem project, anr-07-nano-055). appendix a: variational free energy for planar geometries (charged planes), the transla- tional invariance parallel to the plane, allows us to sig- nificantly simplify the problem by introducing the partial fourier-transformation of the trial potential in the form v0(z, z′, r|| −r′ ||) = z dk (2π)2 eik*(r||−r′ ||)ˆ v0(z, z′, k). (a1) by injecting the fourier decomposition (a1) into eq. (17), the dh equation becomes  −∂ ∂z ε(z) ∂ ∂z + ε(z)[k2 + κ2 v(z)]  ˆ v0 (z, z′, k; κv(z)) = e2 kbt δ(z −z′). (a2) the translational symmetry of the system enables us to express any thermodynamic quantity in terms of the par- tially fourier-transformed green's function ˆ v0(z, z, k). the average electrostatic potential contribution to fv that follows from the average ⟨h⟩0 reads f1 = s z dz ( −[∇φ0(z)]2 8πlb + ρs(z)φ0(z) − x i λie− q2 i 2 w (z)−qiφ0(z) ) , (a3) the kernel part is f2 = s 16π2 z 1 0 dξ z ∞ 0 dkk z dz κ2 v(z) lb(z) (a4) × h ˆ v0  z, z, k; κv(z) p ξ  −ˆ v0 (z, z, k; κv(z)) i where the first term in the integral follows from f0 and the second term from ⟨h0⟩0. finally, the unscreened van der waals contribution, which comes from the unscreened part of f0 , is given by f3 = s 8π z 1 0 dξ z ∞ 0 dz  1 lb(z) −1 lb  d (∇φ)2e ξ −ln z dφe− r dr 8πlb (∇φ)2 (a5) the technical details of the computation of f3 can be found in ref. [50]. the last term of eq. (a5) simply corresponds to the free energy of a bulk electrolyte with a dielectric constant εw. in the above relations, s stands for the lateral area of the system. the dummy "charg- ing" parameter ξ is usually introduced to compute the debye-h ̈ uckel free energy [51]. it multiplies the debye lengths of ˆ v0 (z, z, k; κv(z)) in eq. (a4) and the dielec- tric permittivities contained in the thermal average of the gradient in eq.(a5). this later is defined as d (∇φ)2e ξ = −(∇φ0)2 (a6) + z dk (2π)2 k2 + ∂z∂z′
ˆ vc [z, z′, k; lξ(z)]|z=z′ where we have introduced l−1 ξ (z) ≡l−1 b +ξ  l−1 b (z) −l−1 b  and ˆ vc [z, z′, k; lξ(z)] stands for the fourier transformed coulomb operator given by eq. (5) with bjerrum length lξ(z). the quantity f3 defined in eq. (a5) does not de- pend on the inverse screening length κv. moreover, in order to satisfy the electroneutrality, φ0(z) must be con- stant in the salt-free parts of the system where lb(z) ̸= lb. hence, f3 does not depend on the potential φ0(z). appendix b: variational choice for the neutral dielectric interface we report in this appendix the restricted variational piecewise φ0(z) for a neutral dielectric interface which is a solution of ∂2φ0 ∂z2 = 0 for z ≤a (b1) ∂2φ0 ∂z2 −κ2 φφ0 = cze−κφz for z ≥a (b2) where φ0(z) in both regions is joined by the continuity conditions φ< 0 (a) = φ> 0 (a), ∂φ< 0 ∂z     z=a = ∂φ> 0 ∂z     z=a . (b3) we also tried to introduce different variational screening lengths in the second term of the lhs. and in the rhs. of eq. (b2) without any significant improvement at the variational level. for this reason, we opted for a single 19 inverse variational screening length, κφ. the solution of eqs. (b1)-(b2) is φ0(z) =  φ for z ≤a, φ [1 + κφ(z −a)] e−κφ(z−a) for z ≥a. (b4) where the coefficient c disappears when we impose the boundary and continuity conditions, eq. (b3). the re- maining variational parameters are the constant poten- tial φ, the distance a and the inverse screening length κφ. by substituting eq. (b4) into eq. (a3), we obtain the variational grand potential fv = v κ3 b 24π + s 32π  ∆κ2 b −κφ lb φ2  −sρ− z ∞ 0 dz ×  e− q2 − 2 w(z)+q−φ0(z) + q− q+ e− q2 + 2 w(z)−q+φ0(z)  (b5) appendix c: variational choice for the charged dielectric interface the two types of piecewise variational functions used for single charged surfaces are reported below. • the first trial potential obeys the salt-free equa- tion in the first zone and the nlpb solution in the second zone, ∂2 ̃ φnl 0 ∂ ̃ z2 = 2δ( ̃ z) for ̃ z ≤ ̃ h, (c1) ∂2 ̃ φnl 0 ∂ ̃ z2 − ̃ κ2 φ sinh φ0 = 0 for ̃ z ≥ ̃ h, whose solution is ̃ φnl 0 ( ̃ z) = ( 4arctanhγ + 2( ̃ z − ̃ h) for ̃ z ≤ ̃ h, 4arctanh  γe− ̃ κφ( ̃ z− ̃ h) for ̃ z ≥ ̃ h, (c2) where γ = ̃ κφ − q 1 + ̃ κ2 φ. variational parameters are h and κφ, and the electrostatic contribution of the variational grand potential eq. (a3) is f1 ̃ s = ̃ h + γ −4arctanhγ 2π ξ − ̃ κ2 b 4πξ z d ̃ ze−ξ 2 ̃ w( ̃ z) cosh ̃ φnl 0 . (c3) • the second type of trial potential obeys the salt- free equation with a charge renormalization in the first zone and the linearized poisson-boltzmann so- lution in the second zone, ∂2 ̃ φl 0 ∂ ̃ z2 = 2ηδ( ̃ z) for ̃ z ≤ ̃ h, ∂2 ̃ φl 0 ∂ ̃ z2 − ̃ κ2 φ ̃ φl 0 = 0 for ̃ z ≥ ̃ h, (c4) whose solution is given by ̃ φl 0( ̃ z) = ( −2η ̃ κφ + 2η( ̃ z − ̃ h) for ̃ z ≤ ̃ h, −2η ̃ κφ e− ̃ κφ( ̃ z− ̃ h) for ̃ z ≥ ̃ h. (c5) variational parameters introduced in this case are ̃ h, ̃ κφ, and the charge renormalization η, which takes into account non-linearities at the mean-field level [19]. the variational grand potential reads f1 ̃ s = 2η(1 + ̃ h ̃ κφ) −η2(1/2 + ̃ h ̃ κφ) 2πξ ̃ κφ − ̃ κ2 φ 4πξ z d ̃ ze−ξ 2 ̄ w( ̃ z) cosh ̃ φl 0( ̃ z). (c6) in both cases, the boundary condition satisfied by φ0 is the gauss law ∂ ̃ φ0 ∂ ̃ z      z=0 = 2η (c7) where η = 1 for the non-linear case. it is important to stress that in the case of a charged interface, eq. (c7) holds even if ε ̸= 0. in fact, since the left half-space is ion-free, ̃ φ0(z) must be constant for z < 0 in order to satisfy the global electroneutrality in the system. appendix d: definition of the special functions the definition of the four special functions used in this work are reported below. lin(x) = x k≥1 xk kn , ξ(n) = lin(1) (d1) β(x; y, z) = z x 0 dt ty−1(1 −t)z−1 (d2) 2f1(a, b; c; x) = x k≥0 (a)k(b)k(c)k xk k! (d3) where (a)k = a!/(a −k)!. appendix e: disjoining pressure for the neutral pore the net pressure between plates is defined as p = −1 s ∂fv ∂d −  2ρb −κ3 b 24π  (e1) where the subtracted term on the rhs. is the pressure of the bulk electrolyte. the total van der waals free energy, which is simply the zeroth order contribution f0 to the variational grand potential eq. (9), is with the 20 0.3 0.5 0.8 1.0 1.3 1.5 1.8 2.0 d/lb 0.1 0.2 0.3 0.4 0.5 -βlb 3(pvar-pvdw) fig. 17: (color online) difference between the pressure and the screened van der waals contribution vs d/lb for κblb = 0.5, 1 and 1.5, from left to right (ε = 0). constraint κv = κb (there is no renormalization of the inverse screening length at this order), fvdw = dκ3 b 24π −κb 8πdli2 e−2dκb
− 1 16πd2 li3 e−2dκb
(e2) and pvdw = −1 s ∂fvdw ∂d + κ3 b 24π . (e3) we illustrate in fig. 17 the difference between the van der waals pressure and the prediction of the variational calculation for κblb = 0.5, 1 and 1.5. we notice that the prediction of our variational calculation yields a very sim- ilar behavior to that illustrated in fig. 8 of ref. [21]. the origin of the extra-attraction that follows from the vari- ational calculation was discussed in detail in the same article. this effect originates from the important ionic exclusion between the plates at small interplate separa- tion, an effect that can be captured within the variational approach. [1] a. heydweiller, ann. d. phys., 4, 33145 (1910) [2] p.k. weissenborn and r.j. pugh, langmuir, 11, 1422 (1995) [3] g. wagner, phys. zs., 25, 474 (1924) [4] l. onsager and n. samaras, j. chem. phys., 2 528 (1934) [5] v.e. bravina, soviet phys. doklady, 120, 381 (1958) [6] a.e. yaroshchuk, adv. colloid interf. sci., 85, 193 (2000) [7] r.b. schoch, j. han and p. renaud, rev. mod. phys. 80, 839 (2008) [8] a.g. moreira and r.r. netz, phys. rev. lett., 87, 078301 (2000) [9] a.g. moreira and r.r. netz, europhys. lett., 52, 705 (2000) [10] h. boroudjerdi et al., physics reports, 416, 129 (2005) [11] r.r. netz and h. orland, eur. phys. j. e, 1, 203 (2000) [12] a.g. moreira and r.r. netz, eur. phys. j. e, 8, 33 (2002) [13] a. naji, s. jungblut, a. moreira and r.r. netz , physica a, 352, 131 (2005) [14] s. alexander, p.m. chaikin, p. grant, g.j. morales, p. pincus and d. hone, j. chem. phys., 80, 5776 (1984) [15] levin y., rep. prog. phys., 65, 1577 (2002) [16] levin y., j. chem. phys., 113, 9722 (2000) [17] g.s. manning, j. chem. phys., 51, 924 (1969) [18] m. kanduc, a naji, y.s. jho, p.a. pincus and r. pod- gornik, j. phys.: condens. matter, 21, 424103 (2009) [19] r.r. netz and h. orland, eur. phys. j. e, 11, 301 (2003) [20] r.a. curtis and l. lue,. j. chem. phys., 123, 174702 (2005) [21] m.m. hatlo, r.a. curtis and l. lue, j. chem. phys., 128, 164717 (2008) [22] m.m. hatlo and l. lue, soft matter, 4, 1582 (2008) [23] m.m. hatlo and l. lue, soft matter, 5, 125 (2009) [24] r.r. netz, eur. phys. j. e, 5, 557 (2001) [25] the case with dielectric discontinuity has been stud- ied without added salt in m.m. hatlo and l. lue, arxiv:0806.3716, (2008) [26] v.m. starov and n.v. churaev, adv. colloid interf. sci. 43, 145 (1993) [27] x. lefebvre, j. palmeri and p. david, j. phys. chem. b 108, 16811 (2004) [28] a. yaroshchuk, sep. purif. technology 22-23, 143 (2001) [29] c. li and h. r. ma, j. chem. phys, 121, 1917 (2004) [30] i. benjamin, annu. rev. phys. chem., 48, 407-51 (1997) [31] s.m. avdeev and g.a. martynov. colloid j. ussr, 48, 632 (1968) [32] note that the factor ξ−1 in front of v0 in eq. (12) found in [19] should be cancelled. [33] j. janeˇ cek and r.r. netz, j. chem. phys., 130, 074502 (2009) [34] this is in fact a maximization with respect to φ0 or to any variational parameter associated to a restricted choice for φ0. indeed, the physical electrostatic potential φ0 corresponds to a pure imaginary auxiliary field φ in the functional integral z. hence the (complex) minimum of ⟨h −h0⟩0 with respect to φ corresponds to a (real) maximum with respect to φ0. [35] j.d. jackson, classical electrodynamics (wiley 2nd ed., new york, 1975) [36] a. szymczyk and p. fievet, j. membrane sci. 252, 77 (2005) [37] f. fornasiero et al., proc. nat. acad. sci. usa, 105, 17250 (2008) [38] x. lefebvre and j. palmeri, j. phys. chem. b 109, 5525 (2005) [39] a. yaroshchuk, adv. colloid interf. sci. 60 1 (1995) [40] m.e. fisher and y. levin, phys. rev. lett., 71, 3826 (1993) [41] j.p. simonin et al., j. phys chem. b, 103, 699 (1999) [42] s. senapati and a. chandra, j. phys. chem. b, 105, 5106 (2001) [43] j. marti el al., j. phys. chem. b, 110, 23987 (2006) 21 [44] s. buyukdagli, d.s. dean, m. manghi and j. palmeri, in preparation. [45] p. jungwirth and b. winter, annu. rev. phys. chem., 59, 343 (2008) [46] c.d. lorenz and a. travesset, phys. rev. e, 75, 061202 (2007) [47] a. naji and r. podgornik, phys. rev. e, 72, 041402 (2005) [48] y. ge et al., phys. rev. e, 80, 021928 (2009) [49] k. leung and s.b. rempe, j. comput. theoret. nanosci., 6, 1948 (2009) [50] r.r. netz, eur. phys. j. e, 5, 189 (2001) [51] d.a. mcquarrie, statistical mechanics chap.15 (univer- sity science book, new york, 2000)

0911.1731

material properties of of caenorhabditis elegans swimming at low reynolds number

undulatory locomotion, as seen in the nematode \emph{caenorhabditis elegans}, is a common swimming gait of organisms in the low reynolds number regime, where viscous forces are dominant. while the nematode's motility is expected to be a strong function of its material properties, measurements remain scarce. here, the swimming behavior of \emph{c.} \emph{elegans} are investigated in experiments and in a simple model. experiments reveal that nematodes swim in a periodic fashion and generate traveling waves which decay from head to tail. the model is able to capture the experiments' main features and is used to estimate the nematode's young's modulus $e$ and tissue viscosity $\eta$. for wild-type \emph{c. elegans}, we find $e\approx 3.77$ kpa and $\eta \approx-860$ pa$\cdot$s; values of $\eta$ for live \emph{c. elegans} are negative because the tissue is generating rather than dissipating energy. results show that material properties are sensitive to changes in muscle functional properties, and are useful quantitative tools with which to more accurately describe new and existing muscle mutants.

introduction motility analysis of model organisms, such as the nematode caenorhabditis elegans (c. elegans), is of great scientific and practical interest. it can provide, for example, a powerful tool for the analysis of genetic diseases in humans such as muscular dystrophy (md) [1, 2, 3] since c. elegans have muscle cells that are highly similar in both anatomy and molecular ∗present address: department of mechanical & aerospace engineering, princeton university, princeton nj 08544, usa †electronic address: [email protected] arxiv:0911.1731v1 [physics.bio-ph] 9 nov 2009 2 makeup to vertebrate skeletal muscles [4, 5]. due to the nematode's small size (l ≈1 mm), the motility of c. elegans swimming in a simple, newtonian fluid is usually investigated in the low reynolds numbers (re) regime, where linear viscous forces dominate over nonlinear inertial forces [6, 7]. at low re, locomotion results from non-reciprocal deformations to break time-reversal symmetry [8]; this is the so-called "scallop theorem" [9]. experimental observations have shown that motility of swimming nematodes including c. elegans results from the propagation of bending waves along the nematode's body length [10, 11, 12]. these waves consist of alternating phases of dorsal and ventral muscle contractions driven by the neuromuscular activity of muscle cells. while it is generally accepted that during locomotion the nematode's tissues obey a viscoelastic reaction [13, 14, 15], quantitative data on c. elegans' material properties such as tissue viscosity and young's modulus remain largely unexplored. motility behavior of c. elegans is a strong function of its body material properties. recent investigations have provided valuable data on c. elegans' motility, such as velocity, bending frequency, and body wavelength [11, 12, 13, 16, 17, 18]. however, only recently have the nematode's material properties been probed using piezoresistive cantilevers [19]. such invasive measurements provided young's modulus values of the c. elegans' cuticle on the order of 400 mpa; this value is closer to stiffrubber than to soft tissues. in this paper, we investigate the motility of c. elegans in both experiments and in a model in order to estimate the nematode's material properties. experiments show that nematodes swim in a highly periodic fashion and generate traveling waves which decay from head to tail. a dynamic model is proposed based on force and moment (torque) balance. a simplified version of the model is able to capture the main features of the experiments such as the traveling waves and their decay. the model is used to estimate both the young's modulus and tissue viscosity of c. elegans. such estimates are used to characterize motility phenotypes of healthy nematodes and mutants carrying muscular dystrophy (md). ii. experimental methods experiments are performed by imaging c. elegans using standard microscopy and a high- speed camera at 125 frames per second. we focus our analysis on forward swimming in shallow channels to minimize three-dimensional motion. channels are machined in acrylic 3 and are 1.5 mm wide and 500 μm deep; they are sealed with a thin (0.13 mm) cover glass. channels are filled with an aqueous solution of m9 buffer [20], which contains 5 to 10 nematodes. the buffer viscosity μ and density ρ are 1.1 cp and 1.0 g/cm3, respectively. under such conditions, the reynolds number, defined as re = ρul/μ, is less than unity, where u and l are the nematode's swimming speed and length, respectively. in fig. 1(a), we display nematode tracking data over multiple bending cycles for a healthy, wild-type nematode. results show that the nematode swims with an average speed < u >= 0.45 mm/s and with a beating pattern of period t = 0.46 s. this periodic behavior is also qualitatively observed in the motion of the nematode tail (fig. 1a); see also video 1 (supplementary material). under such conditions, re ≈0.4. snapshots of the nematode skeletons over one beating cycle (fig. 1b) reveal an envelope of well-confined body postures with a wavelength of approximately 1 mm, which corresponds nearly to the nematode's body length. the displacement amplitudes at the head and tail are similar with 465 μm and 400 μm, respectively. however, the amplitudes of the curvature at the head and tail differ sharply with approximately 6.07 mm−1 and 2.21 mm−1, respectively. the tail/head curvature ratio of 0.36 suggests that the bending motion is initiated at the head. extensive genetic analysis in c. elegans has identified numerous mutations affecting ne- matode motility. one such mutant, dys-1, encodes a homolog of the human dystrophin protein, which is mutated in duchenne's and becker's muscular dystrophy (md). using qual- itative observation, dys-1 mutants have an extremely subtle movement defect [21], which includes slightly exaggerated head bending and time-dependent decay in movement. the quantitative imaging platform presented here is able to robustly differentiate between wild type and dys-1 mutants, as shown in fig. 1(c) and video 2 (supplementary material). results show that the dys-1 mutant swims with an average speed < u >= 0.17 mm/s and re ≈0.15; both values are significantly smaller than for the wild-type nematode. although the dys-1 mutant suffers from severe motility defects [21], it still moves in a highly periodic fashion with t = 0.63 s. snapshots of nematode skeletons over one beating cycle (fig. 1d) also reveal an envelope of well-confined body postures with a wavelength corresponding to the nematode's body length. the dys-1 mutant exhibits a tail/head curvature ratio of 0.23, which is similar to the value found for wild type nematodes. however, the corresponding displacement amplitudes of the mutant are much smaller than those observed for the wild- type (fig. 1b). the displacement amplitudes at the head and tail are 330 μm and 155 μm, 4 respectively. this observation suggests that the dys-1 mutant body, and in particular the tail, are becoming inactive as the bending motion is not able to deliver as much body displacement. to further characterize the motility of c. elegans, we measure the curvature κ(s, t) = dφ/ds along the nematode's body (fig. 2a). here, φ is the angle made by the tangent to the x-axis at each point along the centerline and s is the arc-length coordinate spanning the nematode's head (s = 0) to its tail (s = l). the spatio-temporal evolution of κ for a swimming nematode is shown in fig. 2(a). approximately 6 bending cycles are illustrated and curvature values are color-coded; red and blue represent positive and negative values of κ, respectively. the y-axis in fig. 2(a) corresponds to the non-dimensional body position s/l. the contour plot shows the existence of highly periodic, well-defined diagonally oriented lines. these diagonal lines are characteristic of bending waves, which propagate in time along the body length. note that as the wave travels along the nematode body, the magnitude of κ decays from head to tail. such behavior contrasts sharply with that observed for undulatory swimmers of the inertial regime (e.g. eel, lamprey), where amplitudes of body displacement grow instead from head to tail [22, 23]. the body bending frequency (f) is obtained from the one-dimensional fast fourier trans- form (fft) of the curvature field κ at multiple body positions s/l (fig. 2b). here, the body bending frequency is defined as f = ω/2π. the angular frequency ω is calculated by first extracting multiple lines from the curvature field at distinct body positions s/l, and then computing the one-dimensional fft. the wave speed c is extracted from the slope of the curvature κ propagating along the nematode's body; the wavelength λ is computed from the expression λ = c/f. a single frequency peak f = 2.17 ± 0.18 hz (n = 25) is found in the fourier spectrum, where n is the number of nematodes. this single peak is irrespective of body position and corresponds to a wave speed c = 2.14 ± 0.16 mm/s. iii. mathematical methods the swimming motion of c. elegans is modeled as a slender body in the limit of low reynolds numbers (re) [24, 25, 26]. this model is later used to estimate the material properties of c. elegans. we assume that the nematode is inextensible [26]; the uncertainty in the measured body lengths is less than 3%. the nematode's motion is restricted to the 5 xy-plane and is described in terms of its center-line y(s, t), where s is the arc-length along the filament [27]. the swimming c. elegans experiences no net total force or torque (moments) such that, in the limit of low re, the dynamic equations of motion are ∂⃗ f ∂s = ct⃗ ut + cn⃗ un, (1) ∂m ∂s = −[fy cos(φ) −fx sin(φ)]. (2) in eq. (1), ⃗ f(s, t) is the internal force in the nematode, ci is the drag coefficient experi- enced by the nematode, ⃗ ui is the nematode velocity, and the subscripts t and n correspond to the tangent and normal directions, respectively. the drag coefficients ct and cn are ob- tained from slender body theory [26]. due to the finite confinement of nematodes between parallel walls, corrections for wall effects on the resistive coefficients are estimated for slender cylinders [7, 28]. in eq. (2), m = mp + ma, where mp is a passive moment and ma is an active moment generated by the muscles of the nematode; the active and passive moments are parts of a total internal moment [13, 14]. the passive moment is given by the voigt model [15] such that mp = eiκ + ηpi(∂κ/∂t), where i is the second moment of inertia of the nematode cross section. the voigt model is one of the simplest models for muscle and is extensively used in the literature [29]. qualitatively, the elastic part of the voigt model is represented by a spring of stiffness e while the dissipative part of the voigt model is represented by a dashpot filled with a fluid of viscosity ηp (fig. 3). here, we assume two homogeneous effective material properties, namely (i) a constant young's modulus e and (ii) a constant tissue viscosity ηp. the active moment generated by the muscle is given by ma = −(eiκa + ηai∂κ/∂t), where κa is a space and time dependent preferred curvature produced by the muscles of the nematode and ηa is a positive constant [30]. a simple form for κa can be obtained by assuming that κa is a sinusoidal function of time with an amplitude that decreases from the nematode's head to its tail (see appendix). note that if η = ηp −ηa > 0, there is net dissipation of energy in the tissue; conversely, if η = ηp −ηa < 0, there is net generation of energy in the tissue. experiments have shown that the force generated by active muscle decreases with increasing velocity of shortening [29], so that the force-velocity curve for active muscle has a negative slope (fig. 3). such negative viscosity has been derived by a 6 mathematical analysis of the kinetics of the mechano-chemical reactions in the cross-bridge cycle of active muscles [30]. for live nematodes, we expect η = ηp −ηa < 0 because the net energy produced in the (muscle) tissue is needed to overcome the drag from the surrounding fluid. equations (1) and (2) are simplified by noting that the nematode moves primarily in the x-direction (see video 1, s.m.) and that the deflections of its centerline from the x-axis are small. in such case, s ≈x and cos(φ) ≈1. this results in a linearized set of equations given by ∂fy ∂x −cn ∂y ∂t = 0, (3) ∂m ∂x + fy = 0. (4) differentiating eq. (4) with respect to x and combining with eq. (3), we obtain ∂2m ∂x2 + cn ∂y ∂t = 0. (5) substituting for m(x, t) in terms of κ(x, t) and its time derivative yields a bi-harmonic equation for the displacement y(x, t) of the type ∂4y ∂x4 + ξ∂y ∂t = 0, (6) which can be solved analytically for appropriate boundary conditions, where ξ is a con- stant that depends on the nematode's material properties and the fluid drag coefficient (see appendix). the boundary conditions are such that both the force and moment at the nematode's head and tail are equal to zero. that is, fy(0, t) = fy(l, t) = 0 and m(0, t) = m(l, t) = 0. note that the zero moment boundary conditions at the head and tail imply that eiκ(0, t)+ ηi(∂κ(0, t)/∂t) = eiκa(0, t) and eiκ(l, t) + ηi(∂κ(l, t)/∂t) = eiκa(l, t). experiments show that the curvature κ has non-zero amplitudes (fig. 2a) both at the head (x = 0) and the tail (x = l). in order to capture this observation, we assume that κa(x, t) is a sinusoidal wave with decreasing amplitude of the form κa(x, t) = q0 cos ωt+q1x cos(ωt−b), where q0, q1 and b are inferred from the experiments. note that if the curvature amplitude 7 at the head is larger than that at the tail, then the nematode swims forward. conversely, if the curvature amplitude is smaller at the head than at the tail, the nematode swims backward; if the amplitudes are equal at the head and tail then it remains stationary. we note, however, that other forms of the preferred curvature κa are possible and could replicate the behavior seen in experiments. iv. results & discussion equation (6) is solved for the displacement y(x, t) in order to obtain the curvature κ(x, t) = ∂2y/∂x2. the solution for y(x, t) is a superposition of four traveling waves of the general form ai exp(−βx cos pi) cos(βx sin pi −ωt −φi) where β = (cnω/kb)1/4 and pi is a function of the phase angle ψ. the amplitude ai and phase φi are constants to be determined by enforcing the boundary conditions discussed above (see appendix). the solution reveals both the traveling bending wave and the characteristic decay in κ, as seen in experiments. note that our formulation does not assume a wave-functional form for κ(x, t). rather the wave is obtained as part of the solution. next, the curvature amplitude |κ(x)| predicted by the model is fitted to those obtained from experiment to estimate the bending modulus kb = i p e2 + ω2η2 and the phase angle ψ = tan−1(ηω)/e. the nematode is assumed to be a hollow, cylindrical shell [19, 31] such that i = π((rm + t/2)4 −(rm −t/2)4)/4, where the mean nematode radius rm ≈35 μm and the cuticle thickness t ≈0.5 μm [32]. for the population of wild-type c. elegans tested here (n = 25), the best fit values are kb = 4.19×10−16±0.49×10−16 nm2 and ψ = −45.3o±3.0o. in fig. 4, the experimental values of |κ(x)| along the body of a wild-type c. elegans are displayed together with theoretical values of |κ(x)|, which are obtained by using the best fit value of the bending modulus for this nematode and by changing the phase angle from ψ = 0o to ψ = −90o. figure 4 shows that the model is able to capture the decay in |κ| as a function of body length and the nematode's viscoelastic behavior. the values of young's modulus e and tissue viscosity η can now be estimated based on the values of ψ and kb discussed above. results show that, for the wild-type nematodes, e = 3.77 ± 0.62 kpa and η = −860.2 ± 99.4 pa*s. the estimated value of e lies in the range of values of tissue elasticity measured for isolated brain (0.1 −1 kpa) and muscle cells (8 −17 kpa) [33]. the values of η for live c. elegans are negative because the tissue 8 is generating rather than dissipating energy [30, 34, 35, 36]. we note, however, that the absolute values of tissue viscosity |η| are within the range (102 −104 pa*s) measured for living cells [37, 38]. in order to determine whether the nematode's material properties can be extracted reliably from shape measurements alone, experiments in solutions of different viscosities were conducted [39]. the inferred young's modulus and effective tissue viscosity remain constant for up to a 5-fold increase in the surrounding fluid viscosity (or mechanical load). the nematode's curvature κ(x, t) is now determined from the estimated values of e and η. figure 2(c) shows a typical curvature κ(x, t) contour plot obtained from the solution of the above equations using the estimated values of e and η. while the influence of non- linearities is neglected for the scope of the present paper, the analytical results show that our linearized model, while not perfect, is nevertheless able to capture the main features observed in experiments (fig. 2d). next, the method described above is used to quantify motility phenotypes of three distinct mutant md strains (see table 1 in supplementary material): one with a well-characterized muscle defect (dys-1;hlh-1); one with a qualitatively subtle movement defect (dys-1), and one mutant that has never been characterized with regards to motility phenotypes but is homologous to a human gene that causes a form of muscular dystrophy expressed in nematode muscle (fer-1). note that while both fer-1 and dys-1 genes are expressed in c. elegans muscle, they exhibit little, if any, change in whole nematode motility under standard lab assays [40]. figure 5 displays results of both kinematics (5a) and tissue material properties (5b) for all nematodes investigated here. quantitative results are summarized in table 1 (supplemen- tary material). we find that all three mutants exhibit significant changes in both motility kinematics and tissue properties. for example, fer-1 mutants exhibit defects in motility kinematics which are not found with standard assays [40]. specifically, both the maximum amount of body curvature attained in fer-1 mutants is increased by ∼5%, and the rate of curvature decay along the body is increased by ∼5% (fig. 5a). this data show that fer-1(hc24) mutants exhibit small yet noteworthy defects in whole nematode motility and exhibit an uncoordinated (unc) phenotype. in comparison, kinematics data on dys-1;hlh- 1 show that body curvature at the head of such mutant nematodes increases by ∼70% compared to wild-type nematodes, while the rate of decay along the body is increased by 9 approximately ∼40%. these results are useful to quantify the paralysis seen earlier in the tail motion of such md mutants (fig. 1c). the young's modulus (e) and the absolute values of tissue viscosity (|η|) of wild-type and mutant strains are shown in fig. 5(b). results show that mutants have lower values of e when compared to wild-type nematodes. in other words, dys-1, dys-1;hlh-1, and fer-1 mutants c. elegans are softer than their wild-type counterpart. the values of |η| of fer-1 mutants are similar to wild-type nematodes, within experimental error. however, the values of |η| for dys-1 mutants are lower than wild-type c. elegans. since muscle fibers are known to exhibit visible damage for dys-1;hlh-1 mutants [21], we hypothesize that the deterioration of muscle fibers may be responsible for the lower values of e and |η| found for dys-1 and dys-1;hlh-1 mutants. v. conclusion in summary, we characterize the swimming behavior of c. elegans at low re. results show a distinct periodic swimming behavior with a traveling wave that decays from the nematode's head to tail. by coupling experiments with a linearized model based on force and torque balance, we are able to estimate, non-invasively, the nematode's tissue material properties such as young's modulus (e) and viscosity (η) as well as bending modulus (kb). results show that c. elegans behaves effectively as a viscoelastic material with e ≈3.77 kpa, |η| ≈860.2 pa*s, and kb ≈4.19 × 10−16 nm2. in particular, the estimated values of e are much closer to biological tissues than previously reported values obtained using piezoresistive cantilevers [19]. we demonstrate that the methods presented here may be used, for example, to quantify motility phenotypes and tissue properties associated with muscular dystrophy mutations in c. elegans. overall, by combining kinematic data with a linearized model, we are able to provide a robust and highly quantitative phenotyping tool for analysis of c. elegans motility, kinematics, and tissue mechanical properties. given the rapid non- invasive optical nature of this method, it may provide an ideal platform for genetic and small molecule screening applications aimed at correcting phenotypes of mutant nematodes. our method also sheds new light on our understanding of muscle function, physiology, and animal locomotion in general. 10 appendix: in this appendix, we detail the model for the motion of the nematode c. elegans. the nematode is modeled as a slender filament at low reynolds numbers [26]. in our experiments, the uncertainty in the measured body lengths is less than 3% and we assume inextensibility. the nematode's motion is described in terms of its center-line ⃗ y(s, t), where s is the arc- length along the filament and t is time [27]. we assume that the nematode moves in the xy-plane. the swimming c. elegans experiences no net total force or torque (moments) such that, in the limit of low re, the equations of motion are ∂⃗ f ∂s = ct⃗ ut + cn⃗ un, (a.1) ∂m ∂s ⃗ ez = −ˆ t × ⃗ f, (a.2) where ˆ t(s, t) = ∂⃗ y/∂s is the tangent vector to the center-line, ⃗ f(s, t) is the internal force, and m(s, t)⃗ ez = ⃗ m(s, t) is the internal moment consisting of a passive and active part [13, 14]. tangential and normal velocities are respectively given by ⃗ ut = (∂⃗ y/∂t * ˆ t)ˆ t and ⃗ un = (i−ˆ t⊗ˆ t)∂⃗ y/∂t. the drag coefficients, ct and cn, are obtained from slender body theory [26]. due to the finite confinement of nematodes between the parallel walls, corrections for wall effects on the resistive coefficients are estimated for slender cylinders [7]. the local body position ⃗ y, the velocity at any body position ∂⃗ y/∂t, and the tangent vector ˆ t are all experimentally measured. the constitutive relation for the moment m(s, t) in our inextensible filament is assumed to be given by m = mp + ma, (a.3) where mp(s, t) is a passive moment and ma(s, t) is an active moment generated by the muscles of the nematode. the passive moment is given by a viscoelastic voigt model [15] mp = eiκ + ηpi ∂κ ∂t , (a.4) where κ(s, t) is the curvature along the nematode. here, we assume two homogeneous effective material properties, namely (i) a constant young's modulus e and (ii) a constant 11 tissue viscosity ηp. the active moment generated by the muscle is assumed to be given by: ma = −(eiκa + ηai ∂κ ∂t ), (a.5) where κa = q0 cos ωt + q1s cos(ωt −b) is a preferred curvature and ηa is a positive constant [30]. q0, q1 and b must be obtained by fitting to experiments. q2 1l2 + 2q0q1l cos(b) < 0 means that the curvature amplitudes at the head are larger than those at the tail and we should expect traveling waves going from head to tail causing the nematode to swim forward; similarly, q2 1l2 + 2q0q1l cos(b) > 0 should give traveling waves going from tail to head so that the nematodes swim backward [12]. the total moment m(s, t) can be written as m = mp + ma = ei(κ −κa) + (ηp −ηa)i ∂κ ∂t = ei(κ −κa) + ηi ∂κ ∂t . (a.6) note that if η = ηp −ηa > 0 then there is net dissipation of energy in the tissue; if η = ηp −ηa < 0 then there is net generation of energy in the tissue. for live nematodes that actively swim in the fluid we expect η = ηp −ηa < 0 since the net energy produced in the (muscle) tissue is needed to overcome the drag from the surrounding fluid. in other words, the driving force for the traveling waves seen in the nematode has its origins in the contractions of the muscle. we model the nematode as a hollow cylindrical shell with outer radius ro and inner radius ri [19] such that the principal moment of inertia (second moment of the area of cross-section) i along the entire length of the nematode is given by i = π 4  (rm + t 2)4 −(rm −t 2)4 where rm is the mean radius and t is the cuticle thickness. consistent with experimental observations, we assume that the nematode moves along the x-axis and that the deflections of the centerline of the nematode from the x-axis are small. this allows us to take s = x and write κ(x, t) = ∂φ/∂x = ∂2y/∂x2 where y(x, t) is the deflection of the centerline of the nematode and φ(x, t) is the angle made by the tangent ˆ t to the x-axis. the equations of motion can then be written as ∂fx ∂x = ctvx, ∂fy ∂x = cn ∂y ∂t , (a.7) ∂m ∂x + fy = 0, (a.8) 12 where ⃗ f = fx⃗ ex +fy⃗ ey. vx is the velocity of the nematode along the x-axis and corresponds to the average forward speed u. combining a linearized formulation of eqs. (a.1) and (a.2) along with the viscoelastic model of eq. (a.6) offers a direct route towards (i) a closed-form analytical solution for the curvature κ(x, t) and (ii) an estimate of tissue properties (e and η). note that due to the assumption of small deflections [26], the x-component of the force balance becomes decoupled from the rest of the equations. as a result, we will not be able to predict vx even if y(x, t) is determined. but, we can solve equations (a.6), (a.7), and (a.8) for appropriate boundary conditions on fy and m to see if we get solutions that look like traveling waves whose amplitude is not a constant, but in fact, is decreasing from the head to the tail of the nematode (fig. 2a). to do so, we observe that the nematode's body oscillates at a single frequency irrespective of position x along its centerline (fig. 2b). we therefore assume y(x, t) and m(x, t) to have a form that involves a single frequency ω, so that y(x, t) = f(x) cos ωt + g(x) sin ωt, and m(x, t) = kb(ω)∂2f ∂x2 cos(ωt + ψ(ω)) +kb(ω)∂2g ∂x2 sin(ωt + ψ(ω)), (a.9) where f(x) and g(x) are as yet unknown functions, ψ(ω) is frequency dependent phase angle, and kb(ω) is a frequency dependent bending modulus of the homogeneous viscoelastic material making up the nematode. in particular, tan ψ = ηω e , kb = i p e2 + ω2η2, (a.10) where the parameter kb, e, and η correspond to the effective tissue properties of the nema- tode. to be consistent with the observation of a single frequency ω and non-zero amplitudes of the curvature κ at the head and tail, we apply boundary conditions fy(0, t) = 0, m(0, t) = 0, fy(l, t) = 0, m(l, t) = 0, (a.11) we can make further progress by differentiating the balance of moments once with respect 13 to x and substituting the y-component of the balance of forces into it to get ∂2m ∂x2 + cn ∂y ∂t = 0 (a.12) substituting for m(x, t) from eqn.(a.9) yields a biharmonic equation for the functions f and g which can be solved to give the following solution for y(x, t) y(x, t) = a01 exp(βx cos(π 8 + ψ 4 )) cos(βx sin(π 8 + ψ 4 ) −ωt −φ01) + a23 exp(−βx cos(π 8 + ψ 4 )) cos(−βx sin(π 8 + ψ 4 ) −ωt −φ23) + a45 exp(βx cos(ψ 4 −3π 8 )) cos(βx sin(ψ 4 −3π 8 ) −ωt −φ45) + a67 exp(−βx cos(ψ 4 −3π 8 )) cos(−βx sin(ψ 4 −3π 8 ) −ωt −φ67), (a.13) where β = (cnω/kb)1/4, and a01, a23, a45, a67, φ01, φ23, φ45 and φ67 are eight constants to be determined from the eight equations resulting from the sin ωt and cos ωt coefficients of the boundary conditions. note that these are four waves of the type y(x, t) = b(x) cos(2π(x + vwt)/λ) which is the form originally assumed by gray and hancock based on experiment [26]. in contrast, we have obtained such waves as a solution to the equations of motion. we plot the amplitude and phase of these waves for a particular choice of parameters in fig. 6. but, the exact solution above is cumbersome to use. we need simple expressions that can be easily fit to some observable in the experiment to obtain β and ψ, or equivalently, e and η. one such parameter is the amplitude of the traveling waves as a function of position x along the nematode. we develop a strategy to obtain this amplitude in the following. based on the exact solutions to the equations we approximate the displacement y(x, t) as follows: y(x, t) ≈a1 exp(−βx cos(ψ 4 −3π 8 )) cos(−βx sin(ψ 4 −3π 8 ) −ωt) + a2 exp(βx cos(ψ 4 + π 8 )) cos(βx sin(ψ 4 + π 8 ) −ωt + φ2) + a2 exp(βx cos(ψ 4 −3π 8 )) cos(βx sin(ψ 4 −3π 8 ) −ωt + φ2 −3π 4 ) + a1 exp(−βx cos(ψ 4 + π 8 )) cos(−βx sin(ψ 4 + π 8 ) −ωt −π 2 ). (a.14) 14 for experimental values of κa(l, t)/κa(0, t) = 0.33 we find a1/a2 ≈443 so that a2 is much smaller in comparison to a1. note that y(0, t) = a1 cos ωt + a2 cos(ωt −φ2) + a2 cos(ωt −φ2 + 3π 4 ) + a1 cos(ωt + π 2 ) = √ 2a1 cos(ωt + π 4 ) + 2a2 cos(3π 8 ) cos(ωt −φ2 + 3π 8 ). (a.15) the curvature can be calculated as under: κ(x, t) = ∂2y ∂x2 ≈β2a1 exp(−βx cos(ψ 4 −3π 8 )) cos(−βx sin(ψ 4 −3π 8 ) −ωt −3π 4 + ψ 2 ) + β2a2 exp(βx cos(ψ 4 + π 8 )) cos(βx sin(ψ 4 + π 8 ) −ωt + φ2 + π 4 + ψ 2 ) + β2a2 exp(βx cos(ψ 4 −3π 8 )) cos(βx sin(ψ 4 −3π 8 ) −ωt + φ2 −3π 4 −3π 4 + ψ 2 ) + β2a1 exp(−βx cos(ψ 4 + π 8 )) cos(−βx sin(ψ 4 + π 8 ) −ωt −π 2 + π 4 + ψ 2 ). (a.16) note again that κ(0, t) = β2a1 cos(ωt + 3π 4 −ψ 2 ) + β2a2 cos(ωt −φ2 −π 4 −ψ 2 ) +β2a2 cos(ωt −φ2 + 3π 4 + 3π 4 −ψ 2 ) + β2a1 cos(ωt + π 4 −ψ 2 ) = √ 2β2a1 cos(ωt + π 2 −ψ 2 ) + 2a2 cos(7π 8 ) cos(ωt −φ2 −ψ 2 + 5π 8 ). (a.17) it is possible to determine β and ψ from (a.15) and (a.17) alone. if we recognize that a1 >> a2 at the head then we see that the ratio of the amplitude of the curvature to the amplitude of the displacement at the head is simply β2 and the phase difference between them is (π 4 −ψ 2 ) (the phase difference between ∂y ∂x and ∂2y ∂x2 at the head is 19π 16 −ψ 4 ). we find by comparing with the exact solution that an estimate of β using this method is accurate to within 1% and that of ψ is accurate to within 2 or 3 degrees. the curvature is an oscillatory function and we can determine its amplitude a(x) simply by isolating the coefficients of cos ωt and sin ωt and then squaring and adding them. we get 15 the following expression as a result of this exercise: a2(x) = β4a2 1 exp(−2βx cos(ψ 4 −3π 8 )) + β4a2 2 exp(2βx cos(ψ 4 + π 8 )) +β4a2 2 exp(2βx cos(ψ 4 −3π 8 )) + β4a2 1 exp(−2βx cos(ψ 4 + π 8 )) −2β4a1a2 exp(− √ 2βx sin(ψ 4 −π 8 )) cos(− √ 2βx sin(ψ 4 −π 8 ) −φ2) +2β4a1a2 cos(−2βx sin(ψ 4 −3π 8 ) −φ2 + 3π 4 ) −2β4a2 2 exp( √ 2βx cos(ψ 4 −π 8 )) cos( √ 2βx cos(ψ 4 −π 8 ) + 3π 4 ) +2β4a1a2 cos(2βx sin(ψ 4 + π 8 ) + π 2 + φ2) +2β4a1a2 exp( √ 2βx sin(ψ 4 −π 8 )) cos( √ 2βx sin(ψ 4 −π 8 ) + φ2 −5π 4 ) +2β4a2 1 exp(− √ 2βx cos(ψ 4 −π 8 )) cos( √ 2βx cos(ψ 4 −π 8 ) −π 2 ). (a.18) we can fit the experimental data of curvature as a function of x using the expression above. there are five fit parameters – a1, a2, β, ψ and φ2. the value of φ2 mostly affects the curvature profile near the tail. values of φ2 ≈π/3 seem to give good fits for the curvature data of the nematodes. a1 can be determined from the amplitude of the displacement at the head. this leaves three fit parameters – a2, β and ψ. we can use this fit to check if the parameters β and ψ obtained from analyzing the motion of the head alone are reasonable or not. methods summary c. elegans strain. all strains were maintained using standard culture methods and fed with the e. coli strain op50. the following muscular dystrophic (md) strains were used: fer-1(hc24ts), dys-1(cx18)i and hlh-1(cc561)ii;dys-1(cx18)i double mutant. note that hlh-1 is a myod mutant that qualitatively reveals the motility defects of dys-1 mutants. analysis were performed on hypochlorite synchronized young adult animals. fer-1 mutants were hatched at the restrictive temperature of 25oc and grown until they reach the young adult stage. dys-1(cx18) i ; hlh-1(cc561) ii mutants were grown at the permissive temperature of 16oc. wild-type nematodes grown at the appropriate temperature were used as controls. strains were obtained from the caenorhabditis elegans genetic stock center. 16 acknowledgements the authors would like to thank j. yasha kresh, y. goldman, p. janmey, and t. shinbrot for helpful discussions. we also thank r. sznitman for help with vision algorithms and p. rockett for manufacturing acrylic channels. some nematode strains used in this work were provided by the caenorhabditis genetics center, which is funded by the nih national center for research resources (ncrr). references [1] bargmann, c. i., science 282, 2028 (1998). [2] mendel, j. e., h. korswagen, k. s. liu, y. m. hadju-cronin, m. i. simon, r. h. plasterk, and p. w. sternberg, science 267, 1652 (1995). [3] nelson, l. s., m. l. rosoff, and c. li, science 281, 1686 (1998). [4] white, j. g., e. southgate, j. n. thomson, and s. brenner, phil. trans. r. soc. lond. b biol. sci. 275, 327 (1976). [5] white, j. g., e. southgate, j. n. thomson, and s. brenner, phil. trans. r. soc. lond. b biol. sci. 314, 1 (1986). [6] childress, s., mechanics of swimming and flying (cambridge university press, 1981). [7] brennen, c. and h. winet, annu. rev. fluid mech. 9, 339 (1977). [8] taylor, g. i., proc. r. soc. a 209, 447 (1951). [9] purcell, e. m., am. j. phys. 45, 3 (1977). [10] gray, j. and h. w. lissmann, j. exp. biol. 41, 135 (1964). [11] korta, j., d. a. clark, c. v. gabel, l. mahadevan, and a. d. t. samuel, j. exp. biol. 210, 2383 (2007). [12] pierce-shimomura, j. t., b. l. chen, j. j. mun, r. ho, r. sarkis, and s. l. mcintire, proc. natl. acad. sci. usa 105, 2098220987 (2008). [13] karbowski, j., c. j. cronin, a. seah, j. e. mendel, d. cleary, and p. w. sternberg, j. theor. biol. 242, 652 (2006). [14] guo, z. v. and l. mahadevan, proc. natl. acad. sci. usa 105, 3179 (2008). 17 [15] fung, y. c., a first course in continuum mechanics (prentice hall, 1993). [16] cronin, c. j., j. e. mendel, s. mukhtar, y.-m. kim, r. c. stirb, j. bruck, and p. w. sternberg, bmc genetics 6, 5 (2005). [17] feng, z., c. j. cronin, j. h. wittig, and p. w. s. annd w. r. schafer, bmc bioinformatics 5, 115 (2004). [18] ramot, d., b. e. johnson, t. l. berry, l. carnell, and m. b. goodman, plos one 3, e2208 (2008). [19] park, s.-j., m. b. goodman, and b. l. pruitt, proc. natl. acad. sci. usa 104, 17376 (2007). [20] brenner, s., genetics 77, 71 (1974). [21] gieseler, k., k. grisoni, and l. s ́ egalat, current biology 10, 1092 (2000). [22] tytell, e., and g. v. lauder, j. exp. biol. 207, 1825 (2004). [23] kern, s., and p. koumoutsakos, j. exp. biol. 209, 4841 (2006). [24] qian, b., t. r. powers, and k. s. breuer, phys. rev. let. 100, 078101 (2008). [25] yu, t. s., e. lauga, and a. e. hosoi, phys. fluids 18, 091701 (2006). [26] gray, j. and g. hancock, j. exp. biol. 32, 802 (1955). [27] antman, s. s., nonlinear problems in elasticity (springer-verlag, new york, 1995). [28] katz, d. f., j. r. blake, and s. l. paveri-fontana, j. fluid mech. 72, 529 (1975). [29] linden, r. j., recent advances in physiology (churchill livingstone, edinburgh and london, 1974). [30] thomas, n. and r. thornhill, j. phys. d: appl. phys. 31, 253 (1998). [31] zelenskaya, a., j. b. de monvel, d. pesen, m. radmacher, j. h. hoh, and m. ulfendahl, biophys. j. 88, 2982 (2005). [32] cox, g. n., m. kusch, and r. s. edgar, j. cell biol. 90, 7 (1981). [33] engler, a. j., s. sen, h. l. sweener, and d. e. discher, cell 126, 677 (2006). [34] feit, h., m. kawai, and m. i. schulman, muscle nerve 8, 503 (1985). [35] kawai, m. and p. w. brandt, j. muscle res. cell motil. 1, 279 (1980). [36] tawada, a. and m. kawai, biophys. j. 57, 643 (1990). [37] yamada, s., d. wirtz, and s. c. kuo, biophys. j. 78, 1736 (2000). [38] thoumine, o. and a. ott, j. cell sci. 110, 2109 (1997). [39] purohit, p. k., x. shen, and p. e. arratia, exp. mech. under review, (2009). [40] bessou, c., j.-b. giugia, c. j. franks, l. holden-dye, and l. segalat, neurogenetics 2, 61 18 (1998). 19 figure legends figure 1: motility of wild-type c. elegans and dys-1;hlh-1 muscular dystrophic (md) mutant swimming at low reynolds number. (a) and (c): visualization of c. elegans motion illustrating instantaneous body centerline or skeleton. also shown are the nematode's (i) centroid and (ii) tail-tip trajectories over multiple body bending cycles. (b) and (d): color-coded temporal evolution of c. elegans skeletons over one beating cycle. results reveal a well-defined envelope of elongated body shapes with a wavelength corresponding approximately to the nematode's body length. figure 2: spatio-temporal kinematics of c. elegans forward swimming gait. (a) represen- tative contour plot of the experimentally measured curvature (κ) along the nematode's body centerline for approximately 6 bending cycles. red and blue colors represent positive and negative κ values, respectively. the y-axis corresponds to the dimensionless position s/l along the c. elegans' body length where s = 0 is the head and s = l is the tail. (b) nematode's body bending frequency obtained from fast fourier transform of κ at different s/l. the peak is seen at a single frequency (∼2.4 hz) irrespective of the location s/l. (c) contour plot of curvature κ values obtained from the model. the model captures the longitudinal bending wave with decaying magnitude, which travels from head to tail. (d) comparison between experimental and theoretical curves of κ at s/l = 0.1 and s/l = 0.4; dashed lines correspond to model predictions (root mean square error is ∼10% of peak-to- peak amplitude). figure 3: schematic of the analytical model for the total internal moment m. muscle tissue is described by a visco-elastic model containing both passive and active elements. the passive moment (mp) is described by the voigt model consisting of a passive elastic element (spring) of stiffness e (i.e. young's modulus) and a passive viscous element (dashpot) of tissue viscosity ηp. the active moment (ma) is described by an active muscular element of 20 viscosity ηa and illustrates a negative slope on a force-velocity plot. since there is a net generation of energy in the muscle to overcome drag from the surrounding fluid, we expect η = ηp −ηa < 0. figure 4: typical c. elegans viscoelastic material properties. typical experimental profile of the curvature amplitude |κ| decay as a function of body position s/l. color-coded theoretical profiles of the curvature amplitude decay |κ| at fixed value of the bending modulus kb. curves vary from ψ = 0o (red) to ψ = −90o (blue), which corresponds to η = 0 and e = 0, respectively. figure 5: kinematics and material properties of wild type and three muscle mutants of c. elegans. (a) measured kinematic data and (b) estimated young's modulus e and absolute values of tissue viscosity |η| for wild-type, fer-1(hc24), dys-1(cx18), and dys-1(cx18); hlh- 1(cc561) adult nematodes (n = 7-25 nematodes for each genotype. * - p < 0.01). figure 6: amplitude and phase of traveling waves obtained from enforcing force and mo- ment boundary conditions at x = 0, l. (a) amplitudes of the a23 and a67 waves decrease from x = 0 to x = l while the amplitudes of the a01 and a45 waves increase from x = 0 to x = l. the amplitudes of the former two waves are larger than the latter two. (b) phases φ01, φ23, φ45 and φ67 are all constant from x = 0 to x = l. the phase difference between the a67 wave and a23 wave is approximately 90o. the parameters used to obtain these plots are: ω = 4π radians/sec, l = 1.0mm, kb = 5.0 × 10−16nm2, cn = 0.06ns/m2, ψ = −45o, eiq0 = 4.35 × 10−12nm, q1l = −1.054q0 and b = 198.4o. 21 fig. 1: 22 fig. 2: 23 fig. 3: 24 fig. 4: 25 fig. 5: fig. 6:

0911.1732

a deep dive into ngc 604 with gemini/niri imaging

the giant hii region ngc 604 constitutes a complex and rich population to studying detail many aspects of massive star formation, such as their environments and physical conditions, the evolutionary processes involved, the initial mass function for massive stars and star-formation rates, among many others. here, we present our first results of a near-infrared study of ngc 604 performed with niri images obtained with gemini north. based on deep jhk photometry, 164 sources showing infrared excess were detected, pointing to the places where we should look for star-formation processes currently taking place. in addition, the color-color diagram reveals a great number of objects that could be giant/supergiant stars or unresolved, small, tight clusters. a extinction map obtained based on narrow-band images is also shown.

introduction ngc 604 is a giant hii region (ghr) located in an outer spiral arm of m33, at a distance of 840 kpc. it is the second most luminous hii region in the local group, after 30 doradus in the lmc. both are nearby examples of giant star-forming re- gions whose individual objects can be spatially resolve for further study. ngc 604 has been the target of many studies during the past few decades. a brief summary of known facts about ngc 604 includes the following. this ghr is ionized by a mas- sive, young cluster, with at least 200 o stars (some as early as o3-o4). the cluster does not exhibit a central core distribution. instead, the stars are widely spread over its projected area in a structure called 'scaled ob association' (soba; hunter et al. (1996); ma ́ ız-apell ́ aniz et al. (2004); bruhweiler et al. (2003)). wolf-rayet stars, a con- firmed and many candidate red supergiant stars, a luminous blue variable and a su- pernova remnant are all part of ngc 604's stellar population (conti & massey (1981); d'odorico & rosa (1981); drissen et al. (1993); d ́ ıaz et al. (1996); churchwell & goss (1999); terlevich et al. (1996); barb ́ a et al. (2009)). the age of the central ionizing cluster has been determined by different authors as between 3 and 5 myr (gonz ́ alez delgado r.m.& p ́ erez, e. (2000); bruhweiler et al. (2003); d ́ ıaz et al. (1996); hunter et al. (1996); rela ̃ no & kennicut (2009)). the interestellar medium reveals a complex structure with a high-excitation central re- gion (made up of multiple two-dimensional structures), asymmetrically surrounded by a low-excitation halo. the whole region shows a very complex geometry of cavities, expand- 119 120 c. fari ̃ na et al. table 1. main characteristics of broad-band and narrow-band filters used in our gemini/niri observations. broad bands narrow bands filter central λ (μm) coverage(μm) filter central λ (μm) coverage(μm) j 1.25 0.97-1.07 paβ 1.282 ∼0.1 h 1.65 1.49-1.78 brγ 2.16 ∼0.1 ks 2.15 1.99-2.30 h2(2-1) 2.24 ∼0.1 ing shells and filaments, as well as dense molecular regions. all of these structures show different kinematic behaviour (ma ́ ız-apell ́ aniz et al. (2004); tenorio-tagle et al. (2000); sabalisck et al. (1995); rela ̃ no & kennicut (2009)). aiming to characterize the youngest stellar population and its environment, we performed near-infrared (nir) photometry (j h k) and analysed narrow-band images in paβ, brγ and h2(2-1). taking into account that nir observations are less affected by dust extinc- tion characteristic in star-fomation environments, we can take a deep dive into ngc 604 to study those very young objects which are still immersed in their parental clouds at the sites of current star formation. 2. images and data processing the images were obtained with the near infrared imager and spectrometer (niri) at gemini north. the resulting plate scale is 0.117 arcsecpixel−1, with a field of view of 120 × 120 arcsec2. the filters used and their main characteristics are listed in table 1. the images were taken under excellent seeing conditions, on average ∼0.35" in the j, h, and ks images. a set of approximately 10 individual exposures was taken in each band to combine into the final image. data reduction and processing were performed with specific tasks using the gemini-niri iraf package. images were sky subtracted and flat fielded and short darks exposures were used to identify bad pixels. stellar magnitudes were obtained by point-spread-function (psf) fitting in crowded fields using daophot software (stetson 1987) in iraf. although a standard procedure, psf construction and fitting involves an iterative and careful process in which several tries were made to get the best results. the effective area covered by our photometry is ∼107 × 107 arcsec2 (∼430 × 430 pc2 at the distance of m33). the average photometric errors are 0.09, 0.11 and 0.21 mags in the h, j and ks filters, respectivelly, and the completeness limits are 22 mags in j and 21 mags in h and ks. magnitudes in the individual filters were matched in a unique list containing 5566 objects in the field in which all three j, h and ks magnitudes were measured. astrometry was derived using 35 objects in common in our field of ngc 604 and the gsc-ii catalog, version 2.3.2 (2006) in the icrs, equinox j2000.0. 3. results and discussion the resulting color-magnitude (cm) and color-color (cc) diagrams are shown in fig- ures 1 and 2, respectively. we have included ∼2000 selected objects, located within a radius of 48 arcsec (∼200 pc) and centered on ngc 604, meeting a certain photometric quality level (magnitude error ⩽median(error)+ 1.0 × standard deviation(error)). red symbols in both plots are objects that lie on the right side of the reddening line for a o6-o8 v star.for each of these objects, the error in (h-ks) color is smaller than its distance to the reddening line, so that we can ensure that they undoubtedly show an ir iaus266. ngc 604: near-infrared gemini/niri imaging 121 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 j-h h-ks 06-8v b5v f5v k5v m5v g0iii k2iii m0iii m7iii 06-8v b5v f5v k5v m5v g0iii k2iii m0iii m7iii ngc 604 field objects with ir excess iii v av = 10 mag 15 16 17 18 19 20 21 22 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 ks h-ks 03v 06v 09.5v k2iii m1iii m5iii 03z 09z o7z 09i a5i f8i m4i b3i ngc 604 field objects with ir excess av = 10 mag ms zams i iii figure 1. color-color (left) and color-magnitude (right) diagrams for objects observed in a circular area centered on ngc 604. red squares are objects that show an ir excess. excess. among the objects with intrinsic ir excesses in ghrs we expect to find wolf-rayet stars, early supergiants and of stars, and massive young stellar objects (mysos). many of these types of objects (or candidates) have already been found in ngc 604. based on our deep j h k photometry we found a total of 164 objects that show ir excesses. as can be expected, ∼70% of these lie in a small area near the region's center and a large proportion are tightly grouped in regions coincident with the radio-continuum peaks at 8.4 ghz from churchwell & goss (1999), as shown in figure 3 (left panel). these results are in complete agreement with the analysis of barb ́ a et al. (2009) based on hst/nicmos nir images, where myso candidates also appeared aligned with the radio peak structures. what we found supports the idea mentioned by many authors, that those areas may be embedded star-forming regions, also taking into account that regions coincident with the radio knots show conditions of high temperature and density (tosaki et al. (2007)), and in two of them ma ́ ız-apell ́ aniz et al. (2004) identified compact hii regions. most figure 2. (left) three-color (j, h and ks) composite image of ngc 604 with 8.4 ghz radio– continuun countours overlaid (adapted from churchwell & goss (1999)). red circles are objects that show ir excess in our photometry. (right) extinction map from brγ/hα, regions with higher extinction are in red/white. 122 c. fari ̃ na et al. objects with ir excesses will be the targets for further observing programs (making use of the integral-field spectroscopic facilities at gemini north) to elucidate their nature and derive accurate properties by means of spectroscopic study. also, those objects that are located at the bright, massive end on the cm diagram deserve further observation and study. on the basis of our narrow-band images we generated an extinction map, derived from the observed variations of the brγ to hα ratio (with the hα image taken from bosch et al. (2002)). the color scale used in the map shows regions with higher extinction in red/white. the present and future results of our nir study will be placed in the context of previ- ous studies of ngc 604 to complete the picture of the overall formation and evolution scenario of this ghr. acknowledgements rb acknowledges partial support from the universidad de la serena, project diuls cd08102. references barb ́ a, rodolfo h.; ma ́ ız apell ́ aniz, jes ́ us; perez, enrique; rubio, monica; bolatto, alberto; fari ̃ na, cecilia; bosch, guillermo; walborn, nolan r. 2009, 2009arxiv0907.1419b bosch, guillermo; terlevich, elena; terlevich, roberto 2002 mnras 329, 481 bruhweiler, fred c.; miskey, cherie l.; smith neubig, margaret 2003, aj, 125, 3082 churchwell, e. & goss, w. m. 1999, apj, 514, 188 conti, p. s. & massey, p. 1999, apj, 249, 471 diaz, a. i.; terlevich, e.; terlevich, r.; gonzalez-delgado, r. m.; perez, e.; garcia-vargas, m. l. 1996, asp-cs, 98, 399 d'odorico, s. & rosa, m. 1981, apj, 248, 1015 drissen, laurent; moffat, anthony f. j.; shara, michael m. 1993, aj, 105, 1400 gonz ́ alez delgado, r. m. and p ́ erez, e. 2000, mnras 317, 64 hunter, deidre a.; baum, william a.; o'neil, earl j., jr.; lynds, roger 1996, apj, 456, 174 ma ́ ı-apell ́ aniz, j.; p ́ erez, e.; mas-hesse, j. m. 2004, aj, 128, 1196 rela ̃ no, m ́ onica & kennicutt, robert c. 2009, apj, 699, 1125 sabalisck, nanci s. p.; tenorio-tagle, guillermo; castaneda, hector o.; munoz-tunon, casiana 1995, apj, 444, 200 stetson p. b. 1987, pasp, 99, 191 tenorio-tagle, guillermo; mu ̃ noz-tu ̃ n ́ on, casiana; p ́ erez, enrique; ma ́ ız-apell ́ aniz, jes ́ us; medina-tanco, gustavo 2000, apj, 541, 720 terlevich, e.; daz, a. i.; terlevich, r.; gonzlez-delgado, r. m.; prez, e.; garca vargas, m. l. 1996, mnras 279, 1219 tosaki, t.; miura, r.; sawada, t.; kuno, n.; nakanishi, k.; kohno, k.; okumura, s. k.; kawabe, r. 2007, apj, 664, 27

0911.1733

the interaction of kelvin waves and the non-locality of the energy transfer in superfluids

we argue that the physics of interacting kelvin waves (kws) is highly non-trivial and cannot be understood on the basis of pure dimensional reasoning. a consistent theory of kw turbulence in superfluids should be based upon explicit knowledge of their interactions. to achieve this, we present a detailed calculation and comprehensive analysis of the interaction coefficients for kw turbulence, thereby, resolving previous mistakes stemming from unaccounted contributions. as a first application of this analysis, we derive a new local nonlinear (partial differential) equation. this equation is much simpler for analysis and numerical simulations of kws than the biot-savart equation, and in contrast to the completely integrable local induction approximation (in which the energy exchange between kws is absent), describes the nonlinear dynamics of kws. secondly, we show that the previously suggested kozik-svistunov energy spectrum for kws, which has often been used in the analysis of experimental and numerical data in superfluid turbulence, is irrelevant, because it is based upon an erroneous assumption of the locality of the energy transfer through scales. moreover, we demonstrate the weak non-locality of the inverse cascade spectrum with a constant particle-number flux and find resulting logarithmic corrections to this spectrum.

introduction to the problem to derive an effective kw hamiltonian leading to the lne (8), we first briefly overview the hamiltonian de- scription of kws initiated in [1] and further developed in [4, 13]. the main goal of sec. i b is to start with the so-called "bare" hamiltonian (5) for the biot-savart description of kws (2) and obtain expressions for the frequency eqs. (21), four- and six-kws interaction coeffi- cients eqs. (22) and (23) and their 1/λ-expansions, which will be used in further analysis. eqs. (21), (22) and (23) are starting points for further modification of the kw description, given in sec. i c, in which we explore the consequences of the fact that non-trivial four-wave inter- actions of kws are prohibited by the conservation laws of energy and momentum: ω1 + ω2 = ω3 + ω4 , k1 + k2 = k3 + k4 , (13) where ωj ≡ω(kj) is the frequency of the kj-wave. only the trivial processes with k1 = k3, k2 = k4, or k1 = k4, k2 = k3 are allowed. it is well known (see, e.g. ref. [14]) that in the case when nonlinear wave processes of the same kind (e.g. 1 →2) are forbidden by conservation laws, the terms corresponding to this kind of processes can be eliminated from the interaction hamiltonian by a weakly nonlinear canonical transformation. a famous example [14] of this procedure comes from a system of gravity waves on water surface in which three-wave resonances ω1 = ω2 + ω3 are forbidden. then, by a canonical transformation to a new variable b = a + o(a2), the old hamiltonian h(a, a∗) is transformed to a new one, e h(b, b∗) , where the three-wave (cubic) interaction coefficient v 2,3 1 is eliminated at the expanse of appearance of an additional contribution to the next order term (i.e. four-wave interaction coefficient k k k k 3 2 4 1 k5 v 1 35 v 4 25 ( ) * k k k k k k 3 2 4 5 6 1 k7 t12 47 t37 56 fig. 1: examples of contribution of the triple vertices to the four-wave hamiltonian when three-wave resonances are forbidden (left) and contributions of the quartet vertices to the six-wave hamiltonian in resonances when four-wave reso- nances are forbidden (right). intermediate virtual waves are shown by dash lines. t 3,4 1,2 ) of the type v 3,5 1 v 2,5 4
∗ [ ω5 + ω3 −ω1 ] . (14) one can consider this contribution as a result of the second-order perturbation approach in the three-wave processes k1 →k3 + k5 and k5 + k2 →k4, see fig. 1, left. the virtual wave k5 oscillates with a forced fre- quency ω1 −ω3 which is different from its eigenfrequency ω5. the inequality ω(k1) −ω(k3) ̸= ω(|k1 −k3|) is a consequence of the fact that the three-wave processes ω(k1)−ω(k3) = ω(|k1 −k3|) are forbidden. as the result the denominator in eq. (14) is non-zero and the pertur- bation approach leading to eq. (14) is applicable when the waves' amplitudes are small. strictly speaking our problem is different: as we men- tioned above, not all 2 ↔2 processes (13) are forbidden, but only the non-trivial ones that lead to energy exchange between kws. therefore, the use of a weakly nonlinear canonical transformation (9) (as suggested in [1]) should be done with extra caution. the transformation (9) is supposed to eliminate the fourth order terms from the bse-based interaction hamiltonian by the price of ap- pearance of extra contributions to the "full" six-wave in- teraction amplitude f w 4,5,6 1,2,3, (25), of the following type (see fig. 1, right): t 4,7 1,2 t 5,6 3,7 ω7 + ω4 −ω1 −ω2 , ω7 ≡ω(|k1 + k2 −k4|) . (15) here all wave vectors are taken on the six-wave resonant manifold ω1 + ω2 + ω3 = ω4 + ω5 + ω6 , (16) k1 + k2 + k3 = k4 + k5 + k6 . the danger is seen from a particular example when k1 →k4 , k2 →k5 and k3 →k6, so ω7 →ω2, and the de- nominator of eq. (15) goes to zero, while the numerator remains finite. this means, that the perturbation con- tribution (15) diverges and this approach becomes ques- tionable. however a detailed analysis of all contributions of the type (15) performed a-posteriori and presented in sec. i c 5 demonstrates cancelations of diverging terms with oppo- site signs such that the resulting "full" six-wave interac- tion coefficient remains finite, and the perturbation ap- proach (15) appears to be eligible. the reason for this cancelation is hidden deep within the symmetry of the problem, which will not be discussed here. moreover, finding the "full" hamiltonian is not enough for formulating the effective model. such a model must include all contributions in the same o(1/λ), namely the leading order allowing energy transfer in the k-space. the "full" hamiltonian still contains un-expanded in (1/λ) expressions for the kw frequencies. leaving only the leading (lia) contribution in the kw frequency ω, as it was done in [1, 2], leads to a serious omission of an important leading order contribution. indeed, the first sub-leading contribution to ω shifts the lia resonant manifold, which upsets the integrability. as a result, the lia part of f w 4,5,6 1,2,3 yields a contribution to the effective model in the leading order. this (previously overlooked) contribution will be found and analyzed in sec. i d. b. "bare" hamiltonian dynamics of kws 1. canonical form of the "bare" kw hamiltonian let us postulate that the motion of a tangle of quan- tized vortex lines can be described by the bse model (2), and assume that λ0 ≡ln l/a0
≫1, (17) where a0 is the vortex core radius. the bse can be writ- ten in the hamiltonian form (4) with hamiltonian (5). without the cut-off, the integral in h bse, eq. (5), would be logarithmically divergent with the dominant contribu- tion given by the leading order expansion of the integrand in small z1 −z2, that corresponds to h lia, eq. (6). as we have already mentioned, lia represents a com- pletely integrable system and it can be reduced to the one-dimensional nonlinear schr ̈ odinger (nls) equation by the hasimoto transformation [12]. however, it is the complete integrability of lia that makes it insufficient for describing the energy cascade and which makes it necessary to consider the next order corrections within the bse model. for small-amplitude kws when w′(z) ≪1, we can expand the hamiltonian (5) in powers of w′ 2, see ap- pendix a 1: h = h2 + h4 + h6 + . . . . (18) here we omitted the constant term h0 which does not contribute to the equation of motion eq. (4). assuming that the boundary conditions are periodical on the length l (the limit kl ≫1 to be taken later) we can use the fourier representation w(z, t) = κ−1/2 x k a(k, t) exp(ikz) . (19) the bold face notation of the one-dimensional wave vec- tor is used for convenience only. indeed, such a vector is just a real number, k ∈r, in our case for kws. for further convenience, we reserve the normal face notation for the length of the one-dimensional wave vector, i.e. k = |k| ∈r+. in fourier space, the hamiltonian equa- tion also takes a canonical form: i ∂a(k, t) ∂t = δh{a, a∗} δa∗(k, t) . (20a) the new hamiltonian h is the density of the old one: h{a, a∗} = h{w, w∗}/l = h2 + h4 + h6 + . . . (20b) the hamiltonian h2 = x k ωk ak a∗ k , (20c) describes the free propagation of linear kws with the dis- persion law ωk ≡ω(k), given by eq. (21), and the canon- ical amplitude ak ≡a(k, t). the interaction hamiltoni- ans h4 and h6 describe the four-wave processes of 2 ↔2 scattering and the six-wave processes of 3 ↔3 scattering respectively. using a short-hand notation aj ≡a(kj, t), they can be written as follows: h4 = 1 4 x 1+2=3+4 t 3,4 1,2 a1a2a∗ 3a∗ 4 , (20d) h6 = 1 36 x 1+2+3=4+5+6 w 4,5,6 1,2,3 a1a2a3a∗ 4a∗ 5a∗ 6 . (20e) here t 3,4 1,2 ≡ t (k1, k2|k3, k4) and w 4,5,6 1,2,3 ≡ w(k1, k2, k3|k4, k5, k6) are "bare" four- and six-wave interaction coefficients, respectively. summations over k1 . . . k4 in h4 and over k1 . . . k6 in h6 are constrained by k1 +k2 = k3 +k4 and by k1 +k2 +k3 = k4 +k5 +k6, respectively. 2. λ-expansion of the bare hamiltonian function as will be seen below, the leading terms in the hamil- tonian functions ωk, t 3,4 1,2 and w 4,5,6 1,2,3, are proportional to λ, which correspond to the lia (6). they will be de- noted further by a front superscript " λ ", e.g. λωk, etc. because of the complete integrability, there are no dy- namics in the lia. therefore, the most important terms for us will be the ones of zeroth order in λ, i.e. the ones proportional to λ0 = o(1). these will be denoted by a front superscript " 1 " , e.g. 1ωk, etc. explicit calculations of the hamiltonian coefficients must be done very carefully, because even a minor mis- take in the numerical prefactor can destroy various can- celations of large terms in the hamiltonian coefficients. this could change the order of magnitude of the answers and the character of their dependence on the wave vectors in the asymptotical regimes. details of these calculations 6 are presented in appendix a 1, whereas the results are given below. together with eq. (10), the kelvin wave frequency is: ωk = λωk + 1ωk + o(λ−1) , where (21a) λωk = κ λ k2 4π , (21b) 1ωk = −κ k2 ln kl
 4π . (21c) the "bare" 4-wave interaction coefficient is: t 3,4 1,2 = λt 3,4 1,2 + 1t 3,4 1,2 + o(λ−1) , (22a) λt 3,4 1,2 = −λ k1k2k3k4  4π , (22b) 1t 3,4 1,2 = −  5 k1k2k3k4 + f 3,4 1,2   16π . (22c) the function f 3,4 1,2 is symmetric with respect to k1 ↔k2, k3 ↔k4 and {k1, k2} ↔{k3, k4}; its definition is given in appendix a 2. the "bare" 6-wave interaction coefficient is w 4,5,6 1,2,3 = λw 4,5,6 1,2,3 + 1w 4,5,6 1,2,3 + o(λ−1) , (23a) λw 4,5,6 1,2,3 = 9 λ 8πκ k1k2k3k4k5k6 , (23b) 1w 4,5,6 1,2,3 = 9 32πκ 7 k1k2k3k4k5k6 −g 4,5,6 1,2,3
.(23c) the function g 4,5,6 1,2,3 is symmetric with respect to k1 ↔ k2 ↔k3, k4 ↔k5 ↔k6 and {k1, k2, k3} ↔{k4, k5, k6}; its definition is given in appendix a 3. note that the full expressions for ωk, t 3,4 1,2 and w 4,5,6 1,2,3 do not contain lbut rather ln(1/a0). this is natural be- cause in the respective expansions lwas introduced as an auxillary parameter facilitating the calculations, and it does not necessarily have to coincide with the inter- vortex distance. more precisely, we should have used some effective intermediate length-scale, leff, such that l≪leff≪2π/k. however, since leffis artificial and would have to drop from the full expressions anyway, we chose to simply write lomitting subscript "eff". cance- lation of lis a useful check for verifying the derivations. c. full "six-kw" hamiltonian dynamics 1. full six-wave interaction hamiltonian e h6 importantly, the four-wave dynamics in one- dimensional media with concaved dispersion laws ω(k) are absent because the conservation laws of en- ergy and momentum allow only trivial processes with k1 = k3, k2 = k4, or k1 = k4, k2 = k3. this means that by a proper nonlinear canonical transformation {a, a∗} ⇒{b, b∗}, h4 can be eliminated from the hamil- tonian description. this comes at a price of appearance of additional terms in the full interaction hamiltonian e h6: h{a, a∗} ⇒ e h{b, b∗} = e h2 + e h4 + e h6 + . . . , (24a) e h2 = x k ωk bk b∗ k , e h4 ≡0 , (24b) e h6 = 1 36 x 1+2+3=4+5+6 f w 4,5,6 1,2,3 b1b2b3b∗ 4b∗ 5b∗ 6 , (24c) f w 4,5,6 1,2,3 = w 4,5,6 1,2,3 + q4,5,6 1,2,3 , (24d) q4,5,6 1,2,3 = 1 8 3 x i,j,m= 1 i̸=j̸=m 6 x p,q,r= 4 p̸=q̸=r q p,q,r i,j,m , (24e) q p,q,r i,j,m ≡ t j, m r, j+m−r t q, p i, p+q−i ωr, j+m−r j, m + t q, r m, q+r−m t i, j p, i+j−p ωm, q+r−m q, r , ω3,4 1,2 ≡ω1 + ω2 −ω3 −ω4 (24f) = λω 3,4 1,2 + 1ω 3,4 1,2 + o(λ−1) . the q-terms in the full six-wave interaction coefficient f w 4,5,6 1,2,3 can be understood as contributions of two four- wave scatterings into resulting six-wave process via a vir- tual kw with k = kj + km −kr in the first term in q and via a kw with k = kq +kr −km in the second term; see fig. 1, right. 2. 1/λ-expansion of the full interaction coefficient f w 4,5,6 1,2,3 similarly to eq. (23a), we can present f w 4,5,6 1,2,3 in the 1/λ-expanded form: f w 4,5,6 1,2,3 = λf w 4,5,6 1,2,3 + 1f w 4,5,6 1,2,3 + o(λ−1) , (25) due to the complete integrability of the kw system in the lia, even the six-wave dynamics must be absent in the interaction coefficient λf w 4,5,6 1,2,3. this is true if func- tion λf w 4,5,6 1,2,3 vanishes on the lia resonant manifold: λf w 4,5,6 1,2,3 δ4,5,6 1,2,3 δ λe ω4,5,6 1,2,3
≡0 , (26a) where δ4,5,6 1,2,3 = δ(k1 + k2 + k3 −k4 −k5 −k6) , (26b) λe ω4,5,6 1,2,3 = κλ 4π  k2 1 + k2 2 + k2 3 −k2 4 −k2 5 −k2 6  . (26c) explicit calculation of λf w 4,5,6 1,2,3 in appendix b shows that this is indeed the case: the contributions λw 4,5,6 1,2,3 and λq4,5,6 1,2,3 in eq. (24d) cancel each other (see section 1 in appendix b). (such cancelation of complicated expres- sions was one of the tests of consistency and correctness of our calculations and our mathematica code). the same is true for all the higher interaction coefficients: they must be zero within lia. 7 thus we need to study the first-order correction to the lia, which for the interaction coefficient can be schemat- ically represented as follows: 1f w 4,5,6 1,2,3 = 1w 4,5,6 1,2,3 + 1 1q 4,5,6 1,2,3 + 1 2q 4,5,6 1,2,3 + 1 3q 4,5,6 1,2,3 , (27a) 1 1q ∼ λt ⊗1t λω , 1 2q ∼ 1t ⊗λt λω , (27b) 1 3q ∼−1ω λt ⊗λt [ λω]2 . (27c) here 1w is the λ0-order contribution in the bare vertex w, given by eq. (23c), 1q is the λ0-order contribution in q which consists of 1 1q and 1 2q originating from the part 1t in the four-wave interaction coefficient t , and 1 3q originating from the 1ωcorrections to the frequencies ωin eqs. (24e) and (24f). explicit eqs. (b3) for 1 1q 4,5,6 1,2,3, 1 2q 4,5,6 1,2,3 and 1 3q 4,5,6 1,2,3 are presented in appendix b 2. they are very lengthy and were analyzed using mathematica, see sec. i d 2. d. effective six-kw dynamics 1. effective equation of motion the lia cancelation (26a) on the full manifold (16) is not exact: λf w 3,4,5 k,1,2 δ 3,4,5 k,1,2 δ e ω3,4,5 k,1,2
̸= 0 , e ω3,4,5 k,1,2 ≡ωk + ω1 + ω2 −ω3 −ω4 −ω5 = λe ω 3,4,5 k,1,2 + 1e ω 3,4,5 k,1,2 + o(λ−1) . the residual contribution due to 1e ω3,4,5 k,1,2 has to be ac- counted for – an important fact overlooked in the previ- ous kw literature, including the formulation of the effec- tive kw dynamics recently presented by ks in [2]. now we are prepared to take another crucial step on the way to the effective kw model by replacing the frequency ωk by its leading order (lia) part (21b) and simultane- ously compensating for the respective shift in the reso- nant manifold by correcting the effective vertex e h. this corresponds to the following hamiltonian equation, i ∂bk ∂t = λωk bk (28) + 1 12 x k+1+2=3+4+5 w 3,4,5 k,1,2 b1b2b∗ 3b∗ 4b∗ 5 , where the constraint k + k1 + k2 = k3 + k4 + k5 holds. here w 3,4,5 k,1,2 is a corrected interaction coefficient, which is calculated in appendix b 3: w 3,4,5 k,1,2 = 1f w 3,4,5 k,1,2 + 1 e s 3,4,5 k,1,2 , (29a) 1 e s4,5,6 1,2,3 = 2π 9κ 1e ω4,5,6 1,2,3 x i={1,2,3} j={4,5,6} (∂j + ∂i) λf w 4,5,6 1,2,3 (kj −ki) λ , (29b) where ∂j(*) ≡∂(*)/∂kj. (28) represents a correct effective model and, will serve as a basis for our future analysis of kw dynamics and kinetics. however, to make this equation useful we need to complete the calculation of the effective interaction co- efficient w 3,4,5 k,1,2 and simplify it to a reasonably tractable form. the key for achieving this is in a remarkably simple asymptotical behavior of w 3,4,5 k,1,2, which will be demon- strated in the next section. such asymptotical expres- sions for w 3,4,5 k,1,2 will allow us to establish nonlocality of the ks theory, and thereby establish precisely that these asymptotical ranges with widely separated scales are the most dynamically active, which would lead us to the re- markably simple effective model expressed by the lne (8). 2. analysis of the effective interaction coefficient w 3,4,5 k,1,2 now we will examine the asymptotical properties of the interaction coefficient which will be important for our study of locality of the kw spectra and formula- tion of the lne (8). the effective six-kw interaction coefficient w 3,4,5 k,1,2 consists of five contributions given by eqs. (27) and (29). the explicit form of w 3,4,5 k,1,2 involves about 2×104 terms. however its asymptotic expansion in various regimes, analyzed by mathematica demonstrates very clear and physical transparent behavior, which we will study upon the lia resonance manifold k + k1 + k2 = k3 + k4 + k5 , (30a) k2 + k2 1 + k2 2 = k2 3 + k2 4 + k2 5 . (30b) if the smallest wavevector (say k5) is much smaller than the largest wave vector (say k) we have a remarkably simple expression: w 3,4,5 k,1,2 →−3 4πκkk1k2k3k4k5 , (31) as min{k, k1, k2, k3, k4, k5} max{k, k1, k2, k3, k4, k5} →0 . we emphasize that in the expression (31), it is enough for the minimal wave vector to be much less than the maximum wave number, and not all of the remaining five wave numbers in the sextet. this was established using mathematica and taylor expanding w 3,4,5 k,1,2 with respect to one, two and four wave numbers [20]. all of these expansions give the same leading term as in (31), see apps. b 4 and b 5. 8 the form of expression (31) demonstrates a very sim- ple physical fact: long kws (with small k-vectors) can contribute to the energy of a vortex line only when they produce curvature. the curvature, in turn, is propor- tional to wave amplitude bk and, at a fixed amplitude, is inversely proportional to their wave-length, i.e. ∝k. therefore, in the effective motion equation each bj has to be accompanied by kj, if kj ≪k. this statement is exactly reflected by formula (31). furthermore, a numerical evaluation of w 3,4,5 k,1,2 on a set of 210 randomly chosen wave numbers, different from each other at most by a factor of two, indicate that in the majority of cases its values are close to the asymp- totical expression (31) (within 40%). therefore, for most purposes we can approximate the effective six-kw inter- action coefficient by the simple expression (31). finally, our analysis of locality seen later in this paper, indicates that the most important wave sextets are those which include modes with widely separating wavelengths, i.e. precisely those described by the asymptotic formula (31). this leads us to the conclusion that the effective model for kw turbulence should use the interaction coefficient (31). returning back to the physical space, we thereby obtain the desired local nonlinear equation (lne) for kws given by (8). as we mentioned in the beginning of the present paper, lna is very close (isomorphous for small amplitudes) to the tlia model (7) introduced and simulated in [13]. it was argued in [13] that the tlia model is a good al- ternative to the original biot-savart formulation due to it dramatically greater simplicity. in the present paper we have found a further support for this model, which is strengthened by the fact that now it follows from a detailed asymptotical analysis, rather than being intro- duced ad hoc. 3. partial contributions to the 6-wave effective interaction coefficient it would be instructive to demonstrate the relative im- portance of different partial contributions, 1w, 1 1q, 1 2q, 1 3q and 1 e s [see eqs. (27) and (29)] to the full effective six-wave interaction coefficient. for this, we consider the simplest case, when four wave vectors are small, say k1, k2, k3, k5 →0. we have (see appendix b 5): 1w w →−1 + 3 2 ln (kl) , (32a) 1 1q w →+1 2 −3 2 ln (kl) −1 6 lnk3 k , (32b) 1 2q w →+1 2 −3 2 ln (kl) −1 6 lnk3 k , (32c) 1 3q w →+1 + 3 2 ln (kl) + 1 6 lnk3 k , (32d) 1 e s w → 1 6 lnk3 k . (32e) one sees that eqs. (32) for the partial contributions involve the artificial separation scale l, which cancels out from 1f w = 1w + 1 1q + 1 2q + 1 3q. this is not surprising because the initial expressions eqs. (23) do not contain l but rather ln(1/a0). this cancelation serves as one more independent check of consistency of the entire procedure. notice that in the ks paper [1], contributions (32d) and (32e) were mistakenly not accounted for. therefore the resulting ks expression for the six-wave effective in- teraction coefficient depends on the artificial separation scale l. this fact was missed in their numerical simula- tions [1]. in their recent paper [2], the lack of contribu- tion (32d) in the previous work was acknowledged (also in [13]), but the contribution (32e) was still missing. ii. kinetic description of kw turbulence a. effective kinetic equation for kws the statistical description of weakly interacting waves can be reached [14] in terms of the kinetic equation (ke) shown below for the continuous limit kl ≫1, ∂n(k, t)  ∂t = st(k, t) , (33a) for the spectra n(k, t) which are the simultaneous pair correlation functions, defined by ⟨b(k, t)b∗(k′, t)⟩= 2π l δ(k −k′) n(k, t) , (33b) where ⟨. . .⟩stands for proper (ensemble, etc.) averaging. in the classical limit [21], when the occupation numbers of bose particles n(k, t) ≫1, n(k, t) = ħn(k, t), the collision integral st(k, t) can be found in various ways [1, 4, 14], including the golden rule of quantum mechanics. for the 3 ↔3 process of kw scattering, described by the motion eq. (28): st3↔3(k) = π 12 zzzzz   w 3,4,5 k,1,2    2 δ 3,4,5 k,1,2 δ  λω3,4,5 k,1,2  × n−1 k + n−1 1 + n−1 2 −n−1 3 −n−1 4 −n−1 5
× nkn1n2n3n4n5 dk1 dk2 dk3 dk4 dk5 . (33c) ke (33) conserves the total number of (quasi)-particles n and the total (bare) energy of the system λe, defined respectively as follows: n ≡ z nk dk , λe ≡ z λωknk dk . (34) ke (33) has a rayleigh-jeans solution, nt(k) = t ħλωk + μ , (35) which corresponds to thermodynamic equilibrium of kws with temperature t and chemical potential μ. 9 in various wave systems, including the kws described by ke (33a), there also exist flux-equilibrium solutions, ne(k) and nn(k), with constant k-space fluxes of energy and particles respectively. the corresponding solution for ne(k) was suggested in the ks-paper [1] under an (unver- ified) assumption of locality of the e-flux. in sec. ii c, we will analyze this assumption in the framework of the derived ke (33), and will prove that it is wrong. the n-flux solution nn(k) was discussed in [3]. in sec. ii c, we will show that this spectrum is marginally nonlocal, which means that it can be "fixed" by a logarithmic cor- rection. b. phenomenology of the e- and n-flux equilibrium solutions for kw turbulence conservation laws (34) for e and n allow one to intro- duce the continuity equations for nk and λek ≡ λωknk and their corresponding fluxes in the k-space, μk and εk: ∂nk ∂t + ∂μk ∂k = 0 , μk ≡− z k 0 st3↔3(k) dk , (36a) ∂λek ∂t + ∂εk ∂k = 0 , εk ≡− z k 0 λωk st3↔3(k) dk . (36b) in scale-invariant systems, when the frequency and in- teraction coefficients are homogeneous functions of wave vectors, eqs. (36) allow one to guess the scale-invariant flux equilibrium solutions of ke (33) [14]: ne(k) = aek−xe , nn(k) = ank−xn , (37) here ae and an are some dimensional constants. scaling exponents xn and xe can be found in the case of local- ity of the n- and e-fluxes, i.e. when the integrals over k1, . . . k5 in eqs. (36) and (33c) converge. in this case, the leading contribution to these integrals originate from regions where k1 ∼k2 ∼k3 ∼k4 ∼k5 ∼k and thus, the fluxes (36) can be estimated as follows: μk ≃k5[w(k, k, k|k, k, k)]2n5 n(k) . ωk , (38a) εk ≃k5[w(k, k, k|k, k, k)]2n5 n(k). (38b) stationarity of solutions of eqs. (36) require constancy of their respective fluxes: i.e. μk and εk should be k- independent. together with eqs. (38) this allows one to find the scaling exponents in eq. (37). our formulation (33) of kw kinetics belongs to the scale-invariant class [22]: λωk ∝k 2 , and for ∀η w(ηk, ηk1, ηk2 | ηk3, ηk4, ηk5) = η6 w(k, k1, k2 | k3, k4, k5) . estimating w(k, k, k|k, k, k) ≃k6/κ and λωk ≃κλk2 in eqs. (38), one gets for n-flux spectrum [3]: nn(k) ≃ μκ  λ
1/5 k−3 , xn = 3 , (39a) and for e-flux ks-spectrum [1]: ne(k) ≃ εκ2
1/5 k−17/5 , xe = 17/5 . (39b) c. non-locality of the n- and e-fluxes by 3 ↔3-scattering consider the 3 ↔3 collision term (33c) for kws with the interaction amplitude w 4,5,6 1,2,3. note that in (33c) r dkj are one-dimensional integrals r ∞ −∞dkj. let us ex- amine the "infrared" (ir) region ( k5 ≪k, k1, k2, k3, k4 ) in the integral (33c), taking into account the asymp- totics (31), and observing that the expression δ 3,4,5 k,1,2 δ λe ω3,4,5 k,1,2
n−1 k + n−1 1 + n−1 2 −n−1 3 −n−1 4 −n−1 5
× nk n1 n2 n3 n4 n5 →δ 3,4 k,1,2δ λω3,4 k,1,2
n−1 k + n−1 1 + n−1 2 −n−1 3 −n−1 4
× nk n1 n2 n3 n4 n5 ∼n5 ∼k −x 5 . thus the integral over k5 in the ir region can be factor- ized and written as follows: 2 z 0 k2 5 n(k5) dk5 ∝2 z 1/l k2−x 5 dk5 . (40) the factor 2 here originates from the symmetry of the integration area and evenness of the integrand: r ∞ −∞= 2 r ∞ 0 . the lower limit 0 in this expression should be replaced by the smallest wave number where the assumed scaling behavior (37) holds, and moreover, it depends on the particular way the wave system is forced. for example, this cutoffwave number could be 1/l, where lis the mean inter-vortex separation l, at which one expects a cutoffof the wave spectrum. the crucial assumption of locality, under which both the e-flux (ks) and the n-flux spectra were obtained, implies that the integral (40) is independent of this cutoffin the limit l→0. clearly, integral (40) depends on the ir-cutoffif x ≥3, which is the case for both the e-flux (ks) and the n-flux spectra (39). note that all other integrals over k1, k2, k3 and k4 in (33c) diverge exactly in the same manner as the integral over k5, i.e. each of them leads to expression (40). even stronger ir divergence occurs when two wave numbers on the same side of the sextet (e.g. k1 and k2, or k3 and k4, etc.) are small. in this case, integrations over both of the small wave numbers will lead to the same contribution, namely integral (40), i.e. the result will be the integral (40) squared. this appears to be the strongest ir singularity, and the resulting behavior of the collision integral eq. (33c) is st ir ∼ z 1/l k2−x 5 dk5 !2 . (41) when two wave numbers from the opposite sides of the sextet (e.g. k2 and k5) tend to zero simultaneously, we 10 get an extra small factor in the integrand because in this case n−1 k + n−1 1 −n−1 3 −n−1 4
→0. as a result we get ir convergence in this range. one can also show ir convergence when two wave numbers from one side and one on the other side of the sextet are small (the resulting integral is ir convergent for x < 9/2). divergence of integrals in eq. (33c) means that both spectra (39) with xn = 3 and xe = 17/5 > 3, obtained under opposite assumption of the convergence of these integrals in the limit l→∞are not solutions of the 3 ↔3-ke (33c) and thus cannot be realized in nature. one should find another self-consistent solution of this ke. note, that the proof of divergence at the ir lim- its is sufficient for discarding the spectra under the test, whereas proving convergence would require considering all the singular limits including the ultra-violet (uv) ranges. however, we have examined these limits, too. at the uv end we have obtained convergence for the ks and for the inverse cascade spectra. thus the most dan- gerous singularity appears to be in the ir range, when two wave numbers from the same side of the wave sextet are small simultaneously. d. logarithmic corrections for the n-flux spectrum (39a) note that for the n-flux spectrum (39a) nn(k) ∝k−3, that the integrals (40) and (41) diverge only logarithmi- cally. the same situation happens, e.g. for the direct en- strophy cascade in two-dimensional turbulence: dimen- sional reasoning leads to the kraichnan-1967 [15] turbu- lent energy spectrum e(k) ∝k−3 (42a) for which the integral for the enstrophy flux diverges log- arithmically. using a simple argument of constancy of the enstrophy flux, kraichnan suggested [16] a logarith- mic correction to the spectra e(k) ∝k−3 ln−1/3(kl) , (42b) that permits the enstrophy flux to be k-independent. here l is the enstrophy pumping scale. using the same arguments, we can substitute in eq. (33c), a logarithmically corrected spectrum nn(k) ∝ k−3 ln−x(kl) and find x by the requirement that the re- sulting n-flux, μk, eq. (36a) will be k-independent. hav- ing in mind that according to eq. (38a) μk ∝n5 n, we can guess that x = 1/5. then, the divergent integral (40) will be ∝ln4/5(kl), while the remaining convergent integrals in eq. (33c) will be ∝ln−4/5(kl). therefore, the resulting flux μk will be k-independent as it should be [16]. so, our prediction is that instead of a non-local spectrum (39a) we have a slightly steeper log-corrected spectrum nn(k) ≃ μ κ
1/5 k3 ln1/5(k l) . (43) the difference is not large, but the underlying physics must be correct; as one says on the odessa market: "we can argue the price, but the weight must be correct". conclusions in this paper, we have derived an effective theory of kw turbulence based on asymptotic expansions of the biot-savart model in powers of small 1/λ and small non- linearity, by applying a canonical transformation elim- inating non-resonant low-order (quadric) interactions, and by using the standard wave turbulence approach based on random phases [14]. in doing so, we have fixed errors arising from the previous derivations, particularly the latest one by ks [2], by taking into account pre- viously omitted and important contributions to the ef- fective six-wave interaction coefficient. we have exam- ined the resulting six-wave interaction coefficient in sev- eral asymptotic limits when one or several wave numbers are in the ir range. these limits are summarized in a remarkably simple expression (31). this allowed us to achieve three goals: • to derive a simple effective model for kw turbu- lence expressed in the local nonlinear equation (8). in addition to small 1/λ and the weak nonlin- earity, this model relies on the fact that our findings show, for dynamically relevant wave sextets, the in- teraction coefficient is a simple product of the six wave numbers, eq. (31). for weak nonlinearities, the lne is isomorphic to the previously suggested tlia model [13]. • to examine the locality of the e-flux (ks) and the n-flux spectra. we found that the ks spectrum is non-local and therefore cannot be realized in na- ture. • the n-flux spectrum is found to be marginally non- local and could be "rescued" by a logarithmic cor- rection, which we constructed following a qualita- tive kraichnan approach. however, it remains to be seen if such a spectrum can be realized in quan- tum turbulence, because, as it was shown in [17], the vortex line reconnections can generate only the forward cascade and not the inverse one (i.e. the re- connections produce an effectively large-scale wave forcing). finally we will discuss the numerical studies of kw turbulence. the earliest numerics by ks were reported in [1]. they claimed that they observed the ks spectrum. at the same time they gave a value of the e-flux con- stant ∼10−5 which is unusually small. we have already mentioned that this work failed to take into account sev- eral important contributions to the effective interaction coefficient, and thus these numerical results cannot be trusted. in particular, we showed that their interaction coefficient must have contained a spurious dependence on 11 the scale lwhich makes the numerical results arbitrary and dependent on the choice of such a cutoff. in addi- tion, even if the interaction coefficient was correct, the monte-carlo method used by ks is a rather dangerous tool when one deals with slowly divergent integrals (in this case r 0 x−7/5 dx). on the other hand, recent numerical simulations of the tlia model also reported agreement with the ks scaling (as well as an agreement with the inverse cascade scaling) [13]. how can one explain this now when we showed analytically that the ks spectrum is non-local? it turns out that the correct kw spectrum, which takes into account the non-local interactions with long kw's, has an index which is close (but not equal) to the ks index, and it is also consistent with the data of [13]. we will report these results in a separate publication. acknowledgments we are very grateful to mark vilensky for fruitful dis- cussions and help. we acknowledge the support of the u.s. - israel binational science foundation, the sup- port of the european community – research infrastruc- tures under the fp7 capacities specific programme, mi- crokelvin project number 228464. appendix a: bare interactions 1. actual calculation of the bare interaction coefficients the geometrical constraint of a small amplitude per- turbation can be expressed in terms of a parameter ǫ(z1, z2) = |w(z1) −w(z2)|/|z1 −z2| ≪1. (a1) this allows one to expand hamiltonian (5) in powers of ǫ and to re-write it in terms consisting of the number of wave interactions, according to eq. (18). ks found the exact expressions for h2, h4 and h6 [1]: h2 = κ 8π z dz1dz2 |z1 −z2| h 2re  w ′∗(z1)w ′(z2)  −ǫ2i ,(a2) h4 = κ 32π z dz1dz2 |z1 −z2| h 3ǫ4 −4ǫ2re  w ′∗(z1)w ′(z2) i , h6 = κ 64π z dz1dz2 |z1 −z2| h 6ǫ4re  w ′∗(z1)w ′(z2)  −5ǫ6i . the explicit calculation of these integrals was ana- lytically done in [13], by evaluating the terms in (a2) in fourier space, and then expressing each integral as various cosine expressions [1]. hamiltonian (a2) can be expressed in terms of a wave representation variable ak = a(k, t) by applying a fourier transform (19) in the variables z1 and z2, (for details see [1, 13]). the result is given by eqs. (20), in which the cosine expressions for ωk, t 34 12 and w 456 123 were done in [1]. in our notations they are ωk = κ 2π [a −b] , t 34 12 = 1 4π [6d −e] , w 456 123 = 9 4πκ [3p −5q] , where (a3) a = z ∞ a0 dz− z− k2ck , b = z ∞ a0 dz− z3 −  1 −ck , d = z ∞ a0 dz− z5 −  1 −c1 −c2 −c3 −c4 + c3 2 + c43 + c4 2  , e = z ∞ a0 dz− z3 −  k1k4 c4 + c1 −c43 −c4 2
+ k1k3 c3 + c1 −c43 −c3 2
+ k3k2 c3 + c2 −c43 −c3 1
+ k4k2 c4 + c2 −c43 −c3 2
 , (a4) p = z ∞ a0 dz− z5 − k6k2[c2 −c5 2 −c23 + c5 23 −c4 2 + c45 2 + c4 23 −c6 1 + c6 −c56 −c6 3 + c56 3 −c46 + c456 + c46 3 −c12] , q = z ∞ a0 dz− z7 −  1 −c4 −c1 + c4 1 −c6 + c46 + c6 1 −c46 1 −c5 + c45 + c5 1 −c45 1 + c65 −c456 −c56 1 + c23 −c3 + c4 3 + c13 −c4 13 + c6 3 −c46 3 −c6 13 + c5 2 + c5 3 −c45 3 −c5 13 + c6 2 −c56 3 + c12 + c4 2 −c2  . here the variable, z−= |z1 −z2| and the expressions c, are cosine functions such that c1 = cos(k1z−), c4 1 = 12 cos((k4 −k1)z−), c45 1 = cos((k4 + k5 −k1)z−), c45 12 = cos((k4 + k5 −k1 −k2)z−) and so on. the lower limit of integration a0 is the induced cutoffof the vortex core radius a0 < |z1 −z2|. the trick used for explicit calculation of the analyt- ical form of these integrals was suggested and used in [13]. first one should integrate by parts all the co- sine integrals, so they can be expressed in the form of z ∞ a0 cos(z) z dz. then, one can use a cosine identity for this integral [19], z ∞ a0 cos(z) z dz = −γ −ln(a0) − z a0 0 cos(z) −1 z dz (a5) = −γ −ln(a0) − ∞ x k=1 −a2 0
k 2k (2k)! = −γ −ln(|a0|) + o(a2 0) , where γ = 0.5772 . . . is the euler constant. therefore, in the limit of a small vortex core radius a0, we can ne- glect terms of order ∼a2 0 and higher. for example, let's consider the following general cosine expression that can be found in eqs. (a4): z ∞ a0 z−3 cos(kz)dz, where k is an expression that involves a linear combination of wave numbers, i.e. k = k1 −k4. therefore, integration by parts will yield the following result for this integral: z ∞ a0 cos(kz) z3 dz =  −cos(kz) 2z2 ∞ a0 + k sin(kz) 2z ∞ a0 −k 2 z ∞ a0 cos(kz) z dz = cos(ka0) 2a2 0 −k sin(ka0) 2a0 −k2 2 z ∞ ka0 cos(y) y dy . we then expand cos(ka0) and sin(ka0) in powers of a0, and apply the cosine formula (a5) for the last integral, where in the last step we have also changed integration variables: y = kz. the final expression is then ∞ z a0 cos(kz) z3 dz = 1 2a2 0 −3k2 4 + k2 2 [γ + ln(|ka0|)] + o(a2 0) . by applying a similar procedure to the other cosine in- tegrals, we find that all terms of negative powers of a0, (that will diverge in the limit a0 →0) actually cancel in the final expression for each interaction coefficient. ap- plying this strategy to all interaction cofficients, we get the following analytical evaluation of the hamiltonian functions [13]: λ0 = ln(l/a0) , ωk = κk2 4π h λ0 −γ −3 2 −ln(kl) i , (a6) t 34 12 = 1 16π h k1k2k3k4(1 + 4γ −4λ0) −f 3,4 1,2 i , w 456 123 = 9 32πκ h k1k2k3k4k5k6(1 −4γ + 4λ0) −g 4,5,6 1,2,3 i . explicit equations for f34 12 and g456 123 are given below in appendices a 2 and a 3. in the main text we introduced λ ≡λ0 −γ −3/2. writing λ = ln(l/a), we see that a = a0eγ+3/2 ≃8a0. 2. bare 4-wave interaction function f 3,4 1,2 a rather cumbersome calculation, presented above, re- sults in an explicit equation for the 4-wave interaction function f 3,4 1,2 in eqs. (a6) and (22b). function f 3,4 1,2 is a symmetrical version of f 3,4 1,2: f 3,4 1,2 ≡ n f 3,4 1,2 o s where the operator {. . . }s stands for the symmetrization k1 ↔k2, k3 ↔k4 and {k1, k2} ↔{k3, k4}. in its turn f 3,4 1,2 is defined as following: f 3,4 1,2 ≡ x k∈k1 k4 ln(|k|l) (a7a) +2 x i,j x k∈kij kikj k2 ln(|k|l) . the p i,j denotes sum of four terms with (i, j) =  (4, 1), (3, 1), (3, 2), (4, 2) , k is either a single wave vec- tor or linear combination of wave-vectors that belong to one of the following sets: k1 = −[1],−[2],−[3],−[4],+ [3 2],+ [43],+ [4 2] , k41 = +[4],+ [1],−[43],−[4 2] , (a7b) k31 = +[3],+ [1],−[43],−[3 2] , k32 = +[3],+ [2],−[43],−[3 1] , k42 = +[4],+ [2],−[43],−[4 1] . here we used the following shorthand notations with α , β , γ = 1, 2, 3, 4: [α] ≡kα , [β] ≡−kβ , [α β] ≡ kα −kβ , [αγ] ≡kα + kγ , [βγ] ≡−kβ −kγ , and + or −signs before [. . . ] should be understood as prefac- tors +1 or −1 in the corresponding term in the sum. for example: k4 ln(|k|l) for k ∈{−[1]} is −k4 1 ln(k1l) , k4 ln(|k|l) for k ∈{+[4 2]} is + (k4 −k2)4 ln(|k4 −k2|l) , kikj k2 ln(|k|l) for i = 4, j = 1, k ∈{−[43]} is −k4k1 (k4 + k3)4 ln(|k4 + k3|l) . 3. bare 6-wave interaction function g 4,5,6 1,2,3 function g 4,5,6 1,2,3 ≡ n g 4,5,6 1,2,3 o s. the operator {. . . }s stands for the symmetrization k1 ↔k2 ↔k3, k4 ↔ k5 ↔k6 and {k1, k2, k3} ↔{k4, k5, k6}, and g 4,5,6 1,2,3 is defined as following: 13 g 4,5,6 1,2,3 ≡ x k∈k3 k6k2 k4 ln(|k|l) + 1 18 x k∈k4 k 6 ln(|k|l) , (a8a) where k3 = n +[2], −[5 2], −[23], +[5 23], −[4 2], +[45 2 ], +[4 23], −[6 1], +[6], −[56], −[6 3], +[56 3 ], −[46], +[456], +[46 3 ], −[12] o , (a8b) k4 = n −[4], −[1], +[4 1], −[6], +[46], +[6 1], −[46 1 ], −[5], +[45], +[5 1], −[45 1 ], +[65], −[456], −[56 1 ], +[23], −[3], +[4 3], +[13], −[4 13], +[6 3], −[46 3 ], −[6 13], +[5 2], +[5 3], −[45 3 ], −[5 13], +[6 2], −[65 3 ], +[12], +[4 2], −[2] o . (a8c) appendix b: effective six-kw interaction coefficient 1. absence of 6-wave dynamics in lia according to eqs. (24d) and (24e), the expression for λf w 4,5,6 1,2,3 is given by λf w 4,5,6 1,2,3 = λw 4,5,6 1,2,3 + λq4,5,6 1,2,3 , (b1a) λq4,5,6 1,2,3 = 1 8 3 x i,j,m= 1 i̸=j̸=m 6 x p,q,r= 4 p̸=q̸=r λq p,q,r i,j,m , (b1b) λq p,q,r i,j,m ≡ λt j, m r, j+m−r λt q, p i, p+q−i λωr, j+m−r j, m + λt q, r m, q+r−m λt i, j p, i+j−p λωm, q+r−m q, r , (b1c) where λω 3,4 1,2 ≡λω1 + λω2 −λω3 −λω4. we want to compute this equation on the lia manifold (30). to do this we express two wave vectors in terms of the other four [23] using the lia manifold constraint (30): k1 = (k3 −k) (k2 −k3) k + k2 −k3 −k5 + k5 , (b2a) k4 = (k3 −k) (k2 −k3) k + k2 −k3 −k5 + k + k2 −k3 . (b2b) then λf w 4,5,6 1,2,3 is easily simplified to zero with the help of mathematica. this gives an independent verification of the validity of our initial eqs. (24) for full interaction cof- ficient λf w 4,5,6 1,2,3 which is needed for the calculations of the o(1) contribution 1f w 4,5,6 1,2,3. another way to see the can- celation is to use the zakharov-schulman variables [18] that parameterise the lia manifold (30). 2. exact expression for 1f w we get expressions for 1 1q, 1 2q and 1 3q, introduced by eqs. (27), from eqs. (24d) and (24e). namely: 1 1q4,5,6 1,2,3 = 1 8 3 x i,j,m= 1 i̸=j̸=m 6 x p,q,r= 4 p̸=q̸=r " λt j, m r, j+m−r 1t q, p i, p+q−i λωr, j+m−r j, m + λt q, r m, q+r−m 1t i, j p, i+j−p λωm, q+r−m q, r # , (b3a) 1 2q4,5,6 1,2,3 = 1 8 3 x i,j,m= 1 i̸=j̸=m 6 x p,q,r= 4 p̸=q̸=r " 1t j, k r, j+k−r λt q, p i, p+q−i λωr, j+m−r j, m + 1t q, r m, q+r−m λt i, j p, i+j−p λωm, q+r−m q, r # , (b3b) 1 3q4,5,6 1,2,3 = 1 8 3 x i,j,m= 1 i̸=j̸=m 6 x p,q,r= 4 p̸=q̸=r " λt j, m r, j+m−r λt q, p i, p+q−i λωr, j+m−r j, m
2 * 1ωr, j+m−r j, m + λt q, r m, q+r−m λt i, j p, i+j−p λωm, q+r−m q, r
2 * 1ωm, q+r−m q, r # . (b3c) again, using mathematica we substitute eqs. (b2) into eqs. (b3a) – (b3c). clearly, the resulting equations are too cumbersome to be presented here. but we will ana- 14 lyze them in various limiting cases, see below. 3. derivation of eq. (29b) for 1e s 3,4,5 k,1,2 first of all, let us find a parametrization for the full resonant manifold, by calculation of the correction to the lia parametrization (b2), namely k1 = λk1 + 1k1 , k4 = λk4 + 1k4 , (b4) where λk1 and λk4 are given by the right-hand sides of eqs. (b2) respectively. corrections 1k1 and 1k4 are found so that the resonances in k, eq. (30a), and (full) ω are satisfied. the resonances in k fixes 1k1 = 1k4. then the ω-resonance in the leading order in 1/λ gives e ω4,5,6 1,2,3 = 1k1 ∂λω1 ∂k1 −1k4 ∂λω4 ∂k4 + 1e ω4,5,6 1,2,3 + o(λ−1) = 0 . (b5) thus 1k1 = 1k4 ≈2π λκ 1e ω4,5,6 1,2,3 (k4 −k1) . (b6) this allows us to write down the contribution of λf w from the deviation of the lia resonant surface: 1 e s4,5,6 1,2,3 = 1k1 ∂λf w 4,5,6 1,2,3 ∂k1 + 1k4 ∂λf w 4,5,6 1,2,3 ∂k1 + o(λ−1) ≈ 2π λκ 1e ω4,5,6 1,2,3 (∂4 + ∂1) λf w 4,5,6 1,2,3 (k4 −k1) , (b7) with ∂j(*) = ∂j(*)/∂kj. it is obvious that instead of k1 and k4 we could use parametrizations in terms of other pairs ki and kj with i = 1, 2 or 3 and j = 4, 5 or 6. this enables us to write a fully symmetric expression for 1s: 1 e s4,5,6 1,2,3 = 2π 9λκ 1e ω4,5,6 1,2,3 x i={1,2,3} j={4,5,6} (∂j + ∂i) λf w 4,5,6 1,2,3 (kj −ki) . (b8) this is the required expression eq. (29b). 4. analytical expression for w on the lia manifold when two wave numbers are small let us put together the coefficients to the interaction coefficient w 3,4,5 k,1,2 given in (27a), (27b), (27c), (23c) and (29), and use in these expressions the formulae obtained in the previous appendices and the parametrization of the lia surface (b2). using mathematica, and taylor expanding w 3,4,5 k,1,2 with respect to one wave number, e.g. k5, we obtain a remarkably simple result, - expression (31). now we will consider the asymptotical limit when two of the wave numbers, say k2 and k5 (let them be on the opposite sides of the resonance conditions), are much less than the other wave numbers in the sextet. using mathematica and taylor expanding w 3,4,5 k,1,2 with respect to two wave numbers k2 and k5, we have lim k2 →0 k5 →0 w 3,4,5 k,1,2 = −3 4πκk2k2k2 3k5 , (b9) simultaneously, see eq. (b2): lim k2 →0 k5 →0 k1 →k3 , lim k2 →0 k5 →0 k4 →k . (b10) therefore, (b9) coincides with (31). note that this was not obvious a priori, because formally (31) was obtained when k5 is much less than the rest of the wave numbers, including k2. for reference, we provide expressions for the different contributions to the interaction coefficient w 3,4,5 k,1,2 given in eqs. (27) and (29). for k2, k5 →0: 15 1w →−3 4πκk2k2k2 3k5  + 3 2 ln(kl) −1 24  49 −(1 −x)2(7 + 10x + 7x2) x2 ln |1 −x| + 2x(12 + 7x) ln |x| −7(1 + x)4 x2 ln |1 + x|  , (b11a) 1 1q = 1 2q →−3 4πκk2k2k2 3k5  −3 2 ln(kl) + 1 48  59 −(1 −x)2(9 + 10x + 9x2) x2 ln |1 −x| + 2 9x2 + 14x −6 + 2 1 −x
ln |x| −9(1 + x)4 x2 ln |1 + x|  , (b11b) 1 3q →−3 4πκk2k2k2 3k5  + 3 2 ln(kl) + 1 48  7 + (1 −x)2 1 + x2
x2 ln |1 −x| + 21 −5x + x3 1 −x ln |x| + (1 + x)4 x2 ln |1 + x|  , (b11c) 1s →−3 4πκk2k2k2 3k5  1 6 1 + x 1 −x ln |x|  , (b11d) 1f w →−3 4πκk2k2k2 3k5  1 −1 6 1 + x 1 −x ln |x|  , x ≡k3/k . (b11e) another possibility is for two small wave numbers to be on the same side of the sextet. we have checked that on the resonant manifold, this also leads to (31). 5. analytical expression for w on the lia manifold when four wave numbers are small now let us, using mathematica, calculate the asymp- totic behavior of w when four wave vectors are smaller than the other two; on the lia manifold this automat- ically simplifies to k1, k2, k3, k5 ≪k, k4 (remember that on the lia manifold k1 and k4 are expressed in terms of the other wave numbers using eq. (b2), thus from (b10)) we have lim k 1,2,3,5 →0 w 3,4,5 k,1,2 = −3 4πκ k2k2k2 3k5. (b12) again, we have got an expression which coincides with (31). we emphasize that this was not obvious a priori, because formally (31) was obtained when k5 is much less than the rest of the wave numbers, including k1, k2, k3. therefore we conclude that the expression (31) is valid when k5 is much less than just one other wave number in the sextet, say k, and not only when it is much less than all of the remaining wave numbers. for a reference, we give the term by term results for the limit k1, k2, k3, k5 ≪k, k4: 1w →−3 4πκk2k2k2 3k5  −1 + 3 2 ln(kl) + 0  , 1 1q →−3 4πκk2k2k2 3k5  +1 2 −3 2 ln(kl) −1 6 lnk3 k  , 1 2q →−3 4πκk2k2k2 3k5  +1 2 −3 2 ln(kl) −1 6 lnk3 k  , 1 3q →−3 4πκk2k2k2 3k5  +1 + 3 2 ln(kl) + 1 6 lnk3 k  , 1s →−3 4πκk2k2k2 3k5  0 + 0 + 1 6 lnk3 k  . the sum of this contributions is very simple: 1f w →−3 4πκk2k2k2 3k5 [ +1 + 0 + 0 ] . [1] e. kozik and b. svistunov, phys. rev. lett. 92, 035301 (2004), doi: 10.1103/physrevlett.92.035301. [2] e. kozik and b. svistunov, journal of low temp. phys. 156, 215-267 (2009), doi: 10.1007/s10909-009-9914-y. [3] s. nazarenko, jetp letters 83, 198-200 (2006), doi: 10.1134/s0021364006050031. [4] v. s. l'vov, s. v. nazarenko and o. rudenko, phys. rev. b 76, 024520 (2007), doi: 10.1103/phys- revb.76.024520. [5] v. s. l'vov, s. v. nazarenko and o. rudenko), journal of low temp. phys., 153, 140-161 (2008). [6] e.v. kozik, b.v. svistunov, phys. rev. b 77, 060502(r) 16 (2008) [7] r. j. donnelly, quantized vortices in he ii (cambridge university press, cambridge, 1991) [8] quantized vortex dynamics and superfluid turbulence, ed. by c. f. barenghi et al., lecture notes in physics 571 (springer-verlag, berlin, 2001) [9] k.w. schwarz, phys. rev. b 31, 5782 (1985), doi: 10.1103/physrevb.31.5782, and 38, 2398 (1988), doi: 10.1103/physrevb.38.2398. [10] b.v. svistunov, phys. rev. b 52, 3647 (1995), doi: 10.1103/physrevb.52.3647. [11] r.j. arms and f.r. hama, phys. fluids 8, 553 (1965). [12] h. hasimoto, journal of fluid mech. 51, 477-485 (1972), doi: 10.1017/s0022112072002307. [13] g. boffetta, a. celani, d. dezzani, j. laurie and s. nazarenko, journal of low temp. phys. 156, 193-214 (2009), doi: 10.1007/s10909-009-9895-x. [14] v.e. zakharov, v.s. l'vov and g.e. falkovich. kol- mogorov spectra of turbulence, (springer-verlag, 1992). [15] r. kraichnan, phys. fluids, 10 1417 (1967), doi: 10.1063/1.1762301. [16] r. kraichnan, j. fluid mech, 47 525 (1971), doi: 10.1017/s0022112071001216. [17] s.v.nazarenko, jetp letters 84, 585-587 (2007), doi: 10.1134/s0021364006230032. [18] v.e. zakharov and e.i. schulman, physica d: nonlin- ear phenomena 4, 270-274 (1982), doi: 10.1016/0167- 2789(82)90068-9. [19] i. gradstein and i. ryzhik, table of integrals, series, and products, academic press, new york (1980). [20] the limit of three small wave numbers is not allowed by the resonance conditions. indeed, putting three wave numbers to zero, we get a 1 ↔2 process which is not allowed in 1d for λω ∼k2. [21] here, we evoke a quantum mechanical analogy as an el- egant shortcut, allowing us to introduce ke and the re- spective solutions easily. however, the reader should not get confused with this analogy and understand that our kw system is purely classical. in particular, the plank's constant ħis irrelevant outside of this analogy, and should be simply replaced by 1. [22] it is evident for the approximation eq. (31). for the full expression eq. (29a) it was confirmed by symbolic com- putation with the help of mathematica. [23] it is appropriate to remind the reader, that we use the bold face notation of the one-dimensional wave vector for convenience only. indeed, such a vector is just a real number, k ∈r.